Prepared on behalf of the

National Aeronautics and Space Administration
GEOMETRIC INTERPRETATION 

OF LUNAR CRATERS
MONIES ALPES

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MONIES 

SPITZBERGENSIS

ARISTILLUS

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PALUS PUTREDINIS

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Impact craters and other features on the Moon, photographed by Apollo 15 crew while in orbit 106 

km above surface on July 31, 1971. Oblique view is north across eastern Mare Imbrium. The 

horizon lies about 500 km from the lava-flooded 80-km-diameter crater Archimedes. Out-offocus object in lower right corner is part of Command Service Module. Sun is to right. Apollo 

15 mapping-camera photograph 1540.
Geometric Interpretation 

of Lunar Craters

By RICHARD J. PIKE

APOLLO 15-17 ORBITAL INVESTIGATIONS

GEOLOGICAL SURVEY PROFESSIONAL PAPER 1046-C

Prepared on behalf of the

National Aeronautics and Space Administration

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1980
UNITED STATES DEPARTMENT OF THE INTERIOR 

CECIL D. ANDRUS, Secretary

GEOLOGICAL SURVEY 

H. William Menard, Director

Library of Congress Cataloging in Publication Data

Pike, Richard J.

Geometric interpretation of lunar craters.

(Apollo 15-17 orbital investigation) (Geological Survey professional

paper : 1046-C)

Bibliography: p. C60-C65.

Supt.ofDocs.no.: I 19.16:1046-C

1. Lunar craters-Data processing. 2. Lunar craters- Statistical

methods. I. Title. II. Series. III. Series: United States.

Geological Survey. Professional paper ; 1046-C. 

QB591.P73 559.9*1 79-607927

For sale by the Superintendent of Documents, U.S. Government Printing Office

Washington, D.C. 20402
CONTENTS

Abstract ________________________________ Cl

Introduction ________________________________ 1

Acknowledgments ________________________ 3

The new data __________________________________ 3

Crater geometry and crater genesis ____________________ 5

Computer classification of analogs ______________ 5

Experimental methods __________________________ 6

The crater sample _______________________ 7

Interpretation of principal components __________ 9

Classification by principal components __________ 11

Classification by cluster analysis __________ 14

The caldera analogy _______________________ 17

Predicting crater genesis ___________________ 18

The Linne controversy resolved _________________ 18

Secondary-impact craters _______________________ 22

Lunar central volcanoes.-____________________ 23

Cratered domes ________________________ 23

Cratered cones __________________________ 25

Dark-halo craters________________________________ 26

Smooth-rimmed craters ___________________ 26

Summit craters on central peaks ________________ 29

Size dependence in the shape of fresh impact craters ________ 30

Ratio-level variations._________________ 31

Depth ____________________________ 32

Size dependence in the shape of fresh impact craters—Continued 

Ratio-level variations—Continued

Rim height__________________________ C33

Width of rim flank _____________________ 35

Rim-wall slope ______________________ 37

Floor diameter ____ __________________ 38

Rim-crest circularity ______________________ 38

Evenness of rim crest _________________ 40

Nonratio variations-___________________ 41

Index data ___________________________ 43

Frequency data____________________ 44

Discussion ______________________________ 45

Depth versus rim-wall width ___________________ 46

Some alternative explanations_______________ 49

Minor differences in crater shape _________________ 53

Mare-terra contrasts ___________-_________ 54

Far side-near side contrasts ____________________ 54

Azimuth-dependent differences ________________ 56

Summary and conclusions.______________________ 58

References cited ________________________________ 60

Supplemental information-_____________________ 66

Measuring crater dimensions __________________ 66

Tabulation of the new data __________________ 66

ILLUSTRATIONS

FRONTISPIECE Oblique view across Mare Imbrium of craters and other features on the Moon.

Page

PLATE 1. Dendrogram displaying cluster classification of 418 craters __— ———— ______ —————— _____ — ___ In pocket

FIGURE 1. Cross section of crater Proclus showing six topographic dimensions _______— — — —— — -___ ———— --— C4

2. Graph showing correlations of crater types with the first principal component _____ — — —— ————— — — ———— 11

3. Diagram showing scores of crater types on the first three principal components — —— -___ ——— ——— —————— 13

4. Diagram showing cross sections of two crater classes in plate 1 ___________ — ___ —— ______ — _____—————— 15

5. Photograph of the crater Linne __________________________—__ —— ______ —— —— __ —— —— _______ 19

6. Diagrams showing linear topographic profiles and rim-crest outline of Linne —— ——— —— ______ —— _ — ——— __ 20

7-14. Photographs showing:

7. Detail of crater Linne _________________________—__ —— ——— __ —— ___- ——— -- —— --- 21

8. Secondary craters from crater Aristarchus _________-___ ——— ____________-__---_----_ ——— — 22

9. Catena Davy ________________________________________________——————— 23

10. Cratered domes in northern Mare Tranquillitatis _____________ ———— _______ ——— ___ ——— __———— 24

11. Cratered dome near crater Milichius___________________ —— —— _________ ——— __ —— ______ ——— ___ 24

12. Cratered cone in Mare Nubium ___________________-_— — —— — ——— ——— ———— - — - 25

13. Dark-halo craters on floor of crater Alphonsus _________ —— ________ ———— __ ——— ____ —— ___ 26

14. The crater Lassell ____________________________-__-___-——————— ———— --- 27

15. Graphs showing frequency distribution of rim-crest circularity, height/depth ratio, and width/diameter ratio -___ 28

16. Photograph of central peaks of crater Theophilus ________________ ___ ______ _______________ 29

17. Photograph of central peaks of crater Levi-Civita ______________ ———— ________ —— ————— __ —— ——— _——— 29

18. Photograph of central peaks of crater Tsiolkovskiy _________________ —— ———— ——— —— _____ — _____ 30

19-26. Graphs showing:

19. Relation between depth and diameter for 212 lunar craters ____________________________ 33

20. Relation between rim height and diameter for 162 lunar craters ____________________________________ 34
VI CONTENTS

FIGURES 19-26. Graphs showing—Continued

TABLE

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

1.

2.

3.

4.

5.

6.

7.

8.

9. 

10.

	Page

21. Relation between flank width and diameter for 162 lunar craters ________--- —— ————— --——— C36

22. Relation between slope of rim wall and diameter for 107 lunar craters __—____ — _ ———————— -_-— 37

23. Relation between floor diameter and rim-crest diameter for 91 lunar craters ———— ————— ——— --—— 39

24. Relation between rim-crest circularity and diameter for 201 lunar craters _______———————- — — 40

25. Relation between rim-crest relief and diameter for 46 lunar craters _ _______- ———————— - —— - 41

26. Relation between coefficient of variation of rim-crest elevation and diameter for 35 lunar craters———— 42

Photomosaic showing the crater Aristarchus _____________---________ ——— — ———— —————— - —— - 43

Graph showing relation between morphology index and rim diameter for 152 lunar craters _ — ——————— — — 45

Graph showing transition from simple to complex morphologies in lunar craters _______ ——————— __———— 46

Graph showing relation between crater depth and rim-wall width for 107 lunar craters __—__ —— —————— — 47

Photographs of craters Diophantus and Delisle ________________________--————— — ————— --- 49

Graph showing relation between peak height and rim diameter for 29 lunar craters ————— ——— ——————— __ 51

Graph showing relation between depth and diameter for 21 terrestrial impact craters ——————— ————— ——— 52

Photograph of crater Humboldt ___________________________________ ——— _ —————— ____— 53

Graph showing relation between depth and diameter for 950 lunar craters _____________———— ——— ___— 55

Graph showing relation between depth and diameter for 58 far-side and 78 near-side craters __ ——— ——— ___— 56

Graph showing relation between rim height and diameter for 54 far-side and 78 near-side craters ——— —— _—— 57

TABLES

	Page

Measurements for a sampling of fresh lunar craters______________-_________________ —— _————— C5

Statistical summary of untransformed variables for 23 crater types ______— ————————— — ————— ___— 8

Matrix of similarity coefficients for principal-components analysis ____——————————————— —— — _____— 9

Principal-components results for seven variables and 418 craters ____________________--——— 9

Composition of clusters in plate 1 by crater type __________________________ — — _-_____---- 15

Numerical data for geometric model of fresh lunar craters__________-________— — — - —————— — 32

Morphologic index numbers of selected lunar craters _____________-____-_____—-- — ——————— 44

Qualitative deviations from mean rim-crest elevation ____________________________________ 58

Dimensions of 623 lunar craters _________________________________________________ 68

Dimensions of 59 terrestrial craters _______________________________________-—_____ 77
APOLLO 15-17 ORBITAL INVESTIGATIONS

GEOMETRIC INTERPRETATION OF LUNAR CRATERS

By RICHARD J. PIKE

ABSTRACT

Geometric properties of the Moon's craters between about 400 m 

and about 300 km across have been examined at a high and hitherto 

unattainable level of accuracy using new topographic measurements 

made on contour maps compiled from Apollo 15-17 photographs. 

Values of diameter, depth, rim height, rim width, rim-crest circularity, and (where applicable) floor diameter are tabulated here for 623 

lunar craters belonging to 11 morphologic categories. Craters of the 

main sequence—the typical lunar craters customarily interpreted as 

having formed by impact—have remarkably similar morphologies, 

regardless of their location on the Moon. Craters on the far side do 

not differ in shape from near-side craters, and, except for evenness of 

the rim crest, craters on the maria are much like craters on the 

uplands. Nor is there any influence of lunar azimuth on the relative 

height of a crater rim, as once had been thought. Craterlets atop 

central peaks in large craters of the main sequence do not appear to 

be volcanic, and the enigmatic feature Linne is revealed as a quite 

ordinary impact crater.

The origin of the Moon's craters is treated as a formal problem in 

numerical taxonomy, using 11 types of lunar craters and 12 types of 

candidate terrestrial analogs. Lunar craters fall into two main 

morphologic and genetic categories—impact craters and 

volcanoes—according to a principal-components analysis and a cluster analysis of their surface dimensions. Main-sequence craters on 

the Moon most closely resemble terrestrial meteorite-impact craters 

and are much less like terrestrial maars or tuff rings. Terrestrial 

calderas are particularly poor analogs of the typical lunar craters. 

Smooth-rimmed lunar craters also resemble normal impact craters 

rather than calderas. Secondary-impact craters, craters atop lunar 

domes and cones, and dark-halo craters along some rilles all differ 

substantially in shape from main-sequence craters. The lunar domes 

and cones have the shapes of true central volcanoes, but dark-halo 

craters and secondary-impact craters—like terrestrial maars—are 

intermediate between the two main morphologic groups. Although 

single topographic variables such as depth/diameter and circularity 

cannot confidently indicate the genesis of any one crater of the Moon, 

multivariate statistical models constructed from several variables 

can distinguish consistently between volcanoes and large impact 

craters.

Eleven changes in the shape of fresh impact craters on the Moon 

occur within a diameter range of 10 to 30 km (average, 17.5 km) and 

mark the transition from simple small craters to large complex or 

modified craters. The seven ratio-level variations—those of rim-crest 

diameter with depth, rim height, flank width, rim-wall slope, floor 

diameter, circularity, and rim-crest evenness—all are defined for 

fresh-looking craters from the new Apollo data and expressed mathematically where practicable. These relations constitute a shape 

model that constrains interpretations of fresh craters on the Moon. 

The size-dependent changes reflect the occurrence of central peaks, 

rim-wall terraces, and a flat floor within craters over 10 to 20 km 

across. Interpretation of these features remains speculative. The 

threshold diameter probably is the maximum size reached by stable 

crater-landforms having only a simple morphology. The changes in 

crater shape may depend principally upon the manner in which the 

Moon's gravity and rock strength interact with the stresses and pressures generated by impact to control flowage, collapse, or elastic 

recoil of the target materials during crater formation. A similar interpretation probably applies to craters on other planets, but the 

threshold sizes at which craters change shape differ according to 

values of gravitational attraction and other properties.

INTRODUCTION

Rimmed circular depressions of a similar 

morphology dominate the landscape of the Moon at all 

scales. Indeed, the very monotony of crater shapes 

makes them at once both the most interesting and the 

least remarkable of all lunar surface features. Although interpretation of craters and other landforms 

preoccupied most observers throughout the long era of 

telescopic and photographic analysis before the Apollo 

program, establishment of seismic stations on the 

Moon and the availability of lunar rock samples and 

gravity and remotely sensed chemical data now have 

shifted overall emphasis in lunar science from solely 

geomorphology to include geophysics and geochemistry. This broader orientation can be expected to persist 

as a framework for further exploration of the planets, 

because the varied approaches complement one 

another. The development of lunar physics and chemistry has provided information that classical observers of 

the Moon could only speculate on, but this newly found 

emphasis has not displaced the importance of interpreting landforms. Indeed, the discovery of craters on 

other bodies has cast the Moon in the role of a planetary Rosetta Stone. Much of what can be learned from 

analysis of craters on the Moon, for which data are the

Cl
C2 APOLLO 15-17 ORBITAL INVESTIGATIONS

most abundant, applies elsewhere in the solar system 

on bodies for which data are sparse.

The Moon's craters are significant in many respects. 

First, the primary impact craters have been interpreted as recording a late stage or stages in the planet's 

formation (see review by Taylor, 1975, for example). 

From the observed population of lunar craters might be 

estimated the size-frequency distribution of the impacting bodies, from which in turn can be addressed 

the more difficult problems of accounting for their distribution and of locating the Moon's source within—or 

outside—the solar system (see discussion by C. R. 

Chapman, 1976). Second, the largest craters are vast 

basins that constitute the basic stratigraphic and 

structural-tectonic framework for most of the Moon 

(Howard and others, 1974). Material ejected from 

basins is voluminous and widespread across much of 

the planet. Topographic influences imparted by large 

basins, such as the "Imbrium sculpture" (Gilbert, 

1893), have profoundly modified the lunar surface. 

Third, densities of impact craters per unit area record 

relative ages of various surfaces on the Moon 

(Hartmann, 1972a). When correlated with radiometric 

ages of lunar rock samples and with time-stratigraphy 

ascertained from photogeologic mapping, the relative 

chronologies obtained from cratering statistics can 

contribute toward reconstructing an accurate sequence 

of geologic events. Fourth, relative ages of (generally 

later) lunar surfaces also may be interpreted from the 

morphologies of impact craters displaying different degrees of preservation (Offield and Pohn, 1970; Boyce, 

1976). In both of the preceding types of analysis, it is 

essential to exclude secondary-impact craters. Growing 

recognition of the importance of secondary impacts in 

creating landforms over much of the lunar surface is 

another recent development in lunar geology 

(Wilhelms, 1976). Fifth, systematic variations in crater 

shape with respect to crater size provide some clues to 

past lunar events and the operation of internal and 

external geologic processes (Quaide and others, 1965). 

Sixth, the harvest of descriptive data and geologic interpretations from the study of lunar craters provides a 

standard of comparison for craters on Mars and its 

satellites, Mercury, and other planets (Hartmann, 

1972b; Cintala and others, 1976; Gault, Guest, and 

others, 1975; Burt and others, 1976; Schubert and 

others, 1977). Our understanding of the probable ancient craters on Earth also can be improved through 

analogy with the better preserved examples on the 

Moon. The current interest in extraterrestrial craters 

has prompted new interpretations of many important 

but heretofore enigmatic features and deposits in the 

terrestrial geologic record, such as the Sudbury, 

Canada, and Vredefort, South Africa, structures. Finally, a lively debate over the origin of lunar craters 

has continued for some 350 years. Although the issue 

now seems to be resolved to the satisfaction of most 

students of the Moon in favor of impact for craters 

throughout the entire size range, this interpretation is 

by no means accepted universally, especially for craters less than 5 km across (Schultz, 1976a). Regardless 

of whichever view one is inclined to accept, however, a 

few small craters almost certainly are true central 

volcanoes—probably lava shields and cinder cones— 

and some types of large impact craters appear to have 

been modified by post-impact volcanism (Pike, 1968; 

Schultz, 1976b).

Although there are several ways to learn more about 

the significance of lunar craters, geometric analysis is 

the principal approach applied here. The quantitative 

analysis of shape complements visual observation, 

photointerpretation, and size-frequency analysis in examining the Moon's craters. All four of these approaches, which evolved through the long history of 

telescopic work preceding the Apollo program, will 

continue to be essential to the interpretation of craters, 

even if detailed and protracted field study on the Moon 

becomes routine. Although no large craters were examined during any of the six Apollo landings from 1969 to 

1972, the photographs taken from the Command 

Modules in orbit around the Moon on each of these 

missions have proved to be essential in upgrading the 

quality of less direct investigations of the craters. For 

an annotated selection of the best Apollo orbital pictures, see Masursky and others (1978). For quantitative analysis, by far the most useful materials are the 

orbital mapping-camera and panoramic photographs 

from the final three Apollo flights. Contour maps, topographic profiles, and other numerical data that accurately portray for the first time the surface geometry of 

lunar craters larger than about 400 m across have been 

prepared from these pictures. The principal results are 

the 1 : 250,000 Lunar Topographic Orthophotomaps 

(LTO) prepared by the Defense Mapping Agency Topographic Center in cooperation with the National 

Aeronautics and Space Administration (NASA), and 

obtainable through the Lunar and Planetary Programs 

Office, NASA Headquarters, Washington, B.C. 20546. 

Although map products portraying craters smaller 

than 400 m across are not plentiful, interpretations of 

the larger craters no longer need be compromised by 

topographic data of poor precision and unknown accuracy. Throughout the long pre-Apollo era, hypotheses 

dealing with the origin of lunar craters were put forth 

freely and frequently, and almost always without any 

hard descriptive information. Such unconstrained 

speculations will have little credibility unless they are 

consistent with accurate quantitative data, such as
GEOMETRIC INTERPRETATION OF LUNAR CRATERS C3

those now available from Apollo orbital photographs. 

The new topographic maps provide so much more 

accurate data than did older methods, notably measurement of lunar elevations from shadow lengths observed at the telescope, that several issues involving 

the Moon's craters need to be reassessed. First, the 

question of crater genesis can be tested more rigorously 

than in the past by comparing the shapes of different 

classes of lunar craters mathemetically with those of 

various types of terrestrial craters. Second, variations 

of crater shape with lunar geography can be measured. 

It is now possible to determine whether or not lunar 

craters that formed on the mare surfaces are similar to 

those formed on the uplands and to ascertain quantitatively any significant differences between them. Third, 

craters located on the far side of the Moon at last can be 

compared in detail with craters on the near side. 

Fourth, size-dependent differences in crater shape that 

were identified in older work can be defined more accurately, and additional differences of this type that did 

not show up in older less exact data now can be verified. Fifth, the geometric properties of lunar craters 

over a wide size range can be summarized by simple 

statistical models, such as exponential expressions for 

the depth/diameter relations. The resulting equations 

that related paired dimensions of craters furnish a 

rigorous basis for the comparison of lunar craters with 

craters elsewhere. The expressions also may be used, 

with appropriate caution, to model the geometry of 

multiring basins. Most of these points are addressed at 

some length in this paper. Several related areas might 

fruitfully be approached through analysis of the improved data on crater geometry, such as the changes in 

shape of a crater with time (Pike, 1968, 197la), a 

reevaluation of the venerable Schroter's rule (Pike, 

1967), and further development of various models of 

ejecta distribution (Pike, 1974a). I will not touch on 

these topics here.

ACKNOWLEDGMENTS

This report is one of four separately bound chapters 

presenting results of Apollo 15-17 orbital investigations of the following titles: "Stratigraphy of Part of 

the Lunar Nearside" (Chapter A), "Lunar Remote 

Sensing and Measurements" (Chapter B), "Geometric 

Interpretation of Lunar Craters" (Chapter C, this 

paper), and "Experimental Photogrammetry of Lunar 

Images" (Chapter D). It summarizes most of the research I did on behalf of the National Aeronautics and 

Space Administration under contract No. T-1167B 

(Experiment No. S-222) between July 1972 and December 1975. The manuscript was last revised in February 1977, except for tables 9 and 10, which were

supplemented in November 1978 from later work done 

under NASA contract W13,130.

The advice and encouragement of H. J. Moore 

throughout the work are warmly appreciated. Many of 

the results described here were first published in several short papers; criticisms by reviewers and referees 

of those publications are acknowledged here. These 

and other contributions to various aspects of the research were made by D. W. G. Arthur, Joseph 

Ashbrook, Felix Chayes, M. R. Dence, Eric Eliason, R. 

H. Godson, J. W. Head, C. A. Hodges, K. A. Howard, R. 

M. Jordan, D. J. Milton, H. J. Moore, G. M. Nakata, 

Christopher Peebles, F. J. Schafer, P. H. Schultz, Paul 

Switzer, and S. S. C. Wu. I appreciate their help. Special thanks are due certain individuals, especially F. J. 

Doyle of the Geological Survey, who argued successfully for the inclusion of metric quality cameras aboard 

the Apollo 15, 16, and 17 spacecraft. Without their efforts, none of the data upon which this work was based 

would exist. D. C. Kinsler of the Lunar and Planentary 

Institute warrants special thanks for his continuing 

efforts in support of NASA's Lunar Cartographic Program. And finally, J. A. O'Keefe and S. K. Runcorn saw 

to it that I had the chance to present some of these 

results to the International Astronomical Union in 

Grenoble, France, in 1976. The report is dedicated to 

the memory of Thomas Logic MacDonald (1900-1974), 

former head of the Lunar Section, British Astronomical Association, and a pioneer analyst of the geometry 

of lunar craters. His statistical results, based upon 

data gathered by German astronomers in the last century, set the stage for the important breakthroughs 

made by Ralph Baldwin two decades later.

THE NEW DATA

Craters are so numerous on the Moon that no pretense can be made of examining them all, and a modest 

sample must suffice to represent the whole population. 

Previous experience shows that a few hundred craters 

usually are enough to obtain stable results (MacDonald, 1931; Baldwin, 1949; Pike, 1968; Schultz, 

1976b). One of the outstanding contributions of the 

statistical studies by MacDonald (1929) and Baldwin 

(1963) is the list of all measurements used in their 

work; even if none of their analyses and interpretations were proved valid, the raw data would still provide useful information to another investigator. In an 

effort to continue this tradition, all topographic 

measurements—of rim-crest diameter and circularity, 

crater depth, rim height, and flank width (fig. 1)—that 

were made for the 623 lunar craters are tabulated in 

the back of this report, along with a detailed description of how they were made. Only 504 lunar craters
C4 APOLLO 15-17 ORBITAL INVESTIGATIONS

FIGURE 1.—Six crater dimensions. Topographic profile across crater Proclus (rim-crest diameter 28.5 km) from photogrammetry of Apollo 17 

mapping-camera photographs. Corrected for planetary curvature. No vertical exaggeration. Compass directions are approximate.

were measured specifically for this study; 119 additional craters were included from later work. The 

data come from different sources and hence are not 

exactly uniform in quality. The sampled craters range 

generally from about 400 m to about 300 km in diameter and vary in appearance from very fresh to highly 

subdued. Most craters belong to the "main sequence" of 

Wilhelms and McCauley (1971), although a few less 

common varieties—such as rimless craters atop domes 

and cones—also are included. Some unusual types of 

craters, such as concentric-rimmed features, are omitted entirely. Despite efforts to include craters from the 

main sequence within a broad range of relative ages, 

the resulting sample is strongly skewed toward fresher 

craters. This bias was deemed necessary in order to 

establish a reliable geometric model for fresh, uneroded craters. More degraded craters can be added to 

the sample subsequently, as the need arises.

Although most measurements were made on the 

published 1 : 250,000 Lunar Topographic Orthophotomaps (LTO) (Defense Mapping Agency, 

1974-1977), several contour maps and topographic 

profiles generated by stereoplotter at the Photogrammetric Unit of the U. S. Geological Survey Center for 

Astrogeology supplied data that could not be obtained 

from the LTO's. Relatively few fresh craters over 30 

km across are portrayed on LTO's because the LTO 

series primarily covers the sparsely cratered mare regions. To make up the deficiency, topographic profiles 

and rim-crest outlines were prepared from Apollo 

15-17 mapping-camera pictures for 11 large fresh 

craters in the far-side uplands (Pike, 1972a, 1973a). 

Crater measurements from these sources are only 

slightly less accurate than data obtained from the published maps. Conversely, the limitations of map scale 

and the 100-m contour interval preclude accurate 

measurements on the LTO's for craters much less than 

3 km across. Adequate data on craters well under 1 km 

across were obtained from seven large-scale topographic maps that were compiled by the U.S. Geological Survey for small areas on the Moon from Apollo

15-17 panoramic-camera pictures. The contour interval on these maps is as small as 10 m. Topographic 

profiles and rim-crest outlines also were generated for 

various small craters (as small as 2 km across) of special interest, such as mare domes, possible volcanic 

cones, and the impact crater Linne. These large-scale 

data were supplemented by measurements from special 1 : 10,000 and 1 : 50,000 contour maps issued by 

the Defense Mapping Agency along with the LTO's, 

but the latter maps generally show few fresh impact 

craters suitable for statistical analysis.

Dimensions of about 100 additional craters were 

culled from the best pre-Apollo compilations of Lunar 

Orbiter and Earth-based measurements (Pike, 1968; 

Arthur, 1974). These craters were included for comparison with the Apollo data or because they were special types of features, such as summit craters on domes 

or smooth-rimmed craters, that are poorly represented 

by available photogrammetric data. At least two and 

typically about five relative-relief measurements (from 

shadow-length data) went into each estimate of crater 

depth and rim height; rim width and floor diameter 

were measured from Lunar Orbiter images; where possible, circularity was determined directly on rectified 

pictures, Lunar Aeronautical Charts (LAC) being used 

rarely as a last resort.

Some of the results in this paper were obtained partially from measurements other than those provided by 

the Apollo 15-17 pictures, shortly before all of the new 

LTO and other data became available. Overall agreement between older and newer crater measurements, 

with the important exception of depth/diameter for 

craters under 30 km across, is good. This agreement 

assures that the older results, especially those of a 

statistical nature as in the graphical comparison of 

large numbers of craters, are of the same satisfactory 

degree of confidence that is characteristic of the most 

recent findings. Table 1 (from Pike, 1974b) shows a 

sampling of extremes of differences between data from 

different sources. The crater-diameter differences between Lunar Orbiter images (mostly mission IV high-
GEOMETRIC INTERPRETATION OF LUNAR CRATERS 

TABLE 1.—Depth and diameter measurements for a sampling of fresh lunar craters

[From Pike (1974b)]

C5

Diameter, Dr (m)

Crater

BesselH ________ ___

La Hire A

LeMonnier B

Timocharis B

BesselD _ _._ _ _ _.

Linne E

Sulpicius Gallus G

HadleyC ._._._______

Aratus B

Littrow B

CarliniB ___ __

Bessel A

Cauchy D

Jansen F

HadleyB ____________

Vitruvius E

Taruntius E

Isidorus D

Diophantus

Delisle

Madler

Average

LTO1

____ 3,700

____ 5,060

— _ 5,190

_„ 5,250

____ 5,700

____ 5,700

— _ 5,810

____ 5,880

____ 7,000

____ 7,000

— _ 7,120

-___ 7,500

____ 7,570

____ 7,630

____ 8,900

__ __ 8,940

---_ 9,000

_— 9,800

— _ 10,900

--- 11,500

-___ 12,800

— _ 16,500

____ 18,130

____ 25,380

____ 28,000

LO2 -LTO

LTO

-0.05 

0 

-.02 

-.01 

.01 

0 

.02 

-.01 

0 

.06 

.07 

.01 

-.05 

-.01 

.06 

.06 

-.03 

0 

.02 

-.02 

-.03 

-.07 

.02 

-.01 

-.01 4 -.01

LTO

670 

1,210 

925 

1,156 

1,065 

1,125 

1,215 

1,195 

933 

1,430 

1,600 

1,730 

1,494 

1,635 

1,905 

1,810 

2,000 

2,000 

2,190 

2,200 

2,673 

3,300 

3,020 

2,420 

2,830

LO-LTO

LTO

-0.01

-.11 

.04

-.19 

-.07 

-.01 

-.16 

-.04 

-.14 

.01 

-.08 

-.04 

-.03 

-.06 

0 

-.01 

-.05 

.02 

-.05 

-.04 

-.02 

-.09 

-.02 

.06 

-.06 5 -.05

Depth, R, (m)

LO-LAC3

440 

400

550 

630 

620 

710 

620 

620 

300 

730 

640 

1,010 

350 

230 

1,000 

900 

780 

860 

890 

520 

810 

1,000 

385 

140 

950 

644

LO-LAC

LAC

2.00 

.59 

1.34 

2.03 

1.68 

1.78 

1.24 

1.24 

.60 

1.04 

.77 

1.55 

.32 

.18 

1.11 

1.00 

.70 

.74 

.74 

.32 

.45 

.43 

.15 

.06 

.55 

.91

LTO-LAC

450 

530 

515 

846 

695 

725 

715 

695 

433 

730 

770 

1,080 

394 

335 

1,005 

910 

880 

830 

990 

600 

873 

960 

425 

0 

1,110 

701

LTO-LAC

LAC

2.05 

.78 

1.26 

2.73 

1.88 

1.81 

1.43 

1.39 

.87 

1.04 

.93 

1.66 

.36 

.26 

1.12 

1.01 

.79 

.71 

.83 

.38 

.49 

.41 

.16 

0 

.65 

1.00

'LTO—Photogrammetric data, principally from 1:250,000 Lunar Topographic Orthophotomaps.

2LO—Lunar Orbiter IV data (Arthur, 1974).

3LAC—Earth-based telescopic data, mainly from 1:1,000,000 Lunar Aeronautical Charts.

includes 24 additional craters not listed here.

'Includes 20 additional craters not listed here.

resolution pictures) and the new LTO's average a mere 

0.01 of the crater diameter and are not systematic. 

However, contrasts between data obtained from Lunar 

Orbiter images, the LTO's, and the mainly Earthbased Lunar Aeronautical Charts (LAC) are both more 

serious and systematic for the depths of small craters. 

In particular, LAC depths average only half the LTO 

depths. This discrepancy is serious indeed, and Earthbased data on the depths of small craters have not been 

used here. Depth data from Lunar Orbiter images (Arthur, 1974) deviate much less from the Apollo measurements. Arthur's depths average only about 5 percent too low (table 1).

CRATER GEOMETRY AND CRATER GENESIS 

COMPUTER CLASSIFICATION OF ANALOGS

Several aspects concerning the 350-year-old riddle of 

the origin of lunar craters persist despite six manned 

landings on the Moon and general acceptance of the 

impact hypothesis (Taylor, 1975). The problem has 

been compounded by the discovery of craters on Mars 

and its satellites, on Mercury, on Venus and on the 

satellites of Jupiter, and may well be complicated 

further when surfaces of the asteroids and the satellites of Saturn come under closer observation. Until

geologic fieldwork on specific extraterrestrial craters 

again becomes feasible, approaches to solving the crater genesis problem will remain indirect. There are 

many such approaches (Pike, 1968): The oldest, analog 

analysis, compares craters of unknown origin with different types of craters whose origin is known. Given 

certain assumptions concerning magmatism, the 

availability of water, presence of an atmosphere, and 

the differences in gravitational acceleration, extraterrestrial craters that closely resemble terrestrial craters 

in morphology can be inferred to have had the same 

genesis. Most attempts to establish crater similitude 

have been made qualitatively, either at the telescope 

or through photointerpretation (Gilbert, 1893; 

Shoemaker, 1962).

Building upon morphometric work by T. L. MacDonald and earlier German astronomers, R. B. 

Baldwin reduced much of the subjectivity inherent in 

evaluating crater morphology by inspection. He compared lunar and terrestrial craters on logarithmic 

graphs of paired dimensions (for example, depth and 

diameter) that describe geometric characteristics 

common to all craters with raised rims (Baldwin, 1949, 

1963). Subsequently, different groups of lunar craters 

were classified as either volcanoes or as various types 

of impact craters from the results of similarly graphed 

comparisons (Pike, 1967, 1972b; Guest and Murray,
C6 APOLLO 15-17 ORBITAL INVESTIGATIONS

1969; Smith, 1971). However, these and other such 

schemes run a serious risk of misclassifying individual 

craters, because only two of the several measurements 

required for a minimal expression of crater shape are 

examined on one graph simultaneously. The identical 

difficulty was identified in anthropology some time ago 

(Bronowski and Long, 1951). More recently, Siegal 

(1973), Siegal and Griffiths (1974), and myself (Pike, 

1974c) have introduced multivariate techniques, which 

are not limited by the dimensional restrictions of 

graphical methods (Sokal, 1974), for comparing craters 

mutually with respect to several topographic variables. 

Because of this innovation, reservations about the 

geometric approach expressed by Shoemaker (1962) 

and Mutch (1970) with special reference to depth/ 

diameter now are less warranted.

Multivariate classification requires a probabilistic 

comparison of crater morphology rather than the direct 

deterministic evaluation possible with graphical 

analysis. Genetic classifications of craters that are derived graphically from topographic variables all have 

been monothetic in concept (Sokal and Sneath, 1963). 

This means that each of the taxonomic criteria was 

required to uniquely specify the mutually exclusive 

classes, usually either volcanoes or impact craters. Individual parameters do not do this very well (see 

ranges of values in table 2); there is no simple "magic 

number." The statistical approach to crater classification adopted earlier (Pike, 1974c) and probed further 

here is inherently polythetic (Sokal and Sneath, 1963). 

It is an improvement over the monothetic mode because craters are grouped together only if they share in 

several but not necessarily all of the discriminating 

attributes expressed by the variables. In this way no 

one characteristic is vital to classifying a crater, nor 

can one alone place a crater in one group or another. 

The polythetic approach to crater analogs recognizes 

that the problem, like most in geomorphology, is inherently stochastic rather than physical in character 

(Leopold and Langbein, 1963).

The analytical procedures followed here also compare each crater to all other craters in the sample in 

each and all respects. The problem cannot be formulated as a simple choice between impact and explosion 

craters as one class and volcanoes as the other class, 

because there are several candidate terrestrial analogs 

and each differs in shape from the others. Nor has it 

been assumed beforehand that some terrestrial craters 

are better or worse analogs of lunar craters than others 

and thus must be grouped together or somehow accorded special consideration. The first question being 

asked here simply is this: Given the entire spectrum of 

(measurable) cratered landforms on the Earth and the 

Moon, regardless of their genesis, how can individual

craters be grouped according to similarities in their 

gross topography? Other naive but useful queries follow: How many major groups of craters exist within the 

sampled universe of craters, and how are they related 

to each other? Do topographic differences and 

similarities evident in the resulting groups reflect recognized types of craters and particular modes of origin? 

If the last questions can be answered affirmatively, 

then various cratering processes are associated with 

specific crater shapes—as expressed by surface 

geometry—and an attempt can be made to set up numerical criteria for ascertaining the genesis of a given 

lunar crater from its topography.

The issue of crater genesis on the Moon is recast as a 

conventional problem in numerical taxonomy (Sokal 

and Sneath, 1963; Sokal, 1974). This experiment results in three rather similar solutions for 418 representative craters and seven original (four synthetic) descriptive variables. The classifications, which were 

constructed from a principal-components analysis and 

by a cluster analysis, closely resemble the results I 

obtained in a brief study using most of the same craters 

(Pike, 1974c). Contrary to conclusions reached by 

Siegal (1973), also on the basis of multivariate analysis 

although from quite different data, my results show 

that crater morphology is indeed an indicator of origin. 

A genetic scheme for classifying craters by morphology 

alone has been established for craters over 5 km in 

diameter. Interpretations focus on the relation of 

main-sequence lunar craters to three candidate terrestrial analogs: meteorite-impact craters, calderas, 

and maars. Detailed variations among lunar craters 

and among terrestrial volcanoes, which warrant separate study, are not treated in detail here (Pike, 1978a).

EXPERIMENTAL METHODS

Principal-components and cluster analyses are conventional quantitative approaches that have been used 

in many fields to recognize natural groups, more or less 

objectively, within a set of phenomena on which several observations have been made. The two techniques 

have been described sufficiently in the geologic literature so that this account need outline only the essentials of each procedure applied to the craters. 

Background material and details of the programmed 

algorithms are available elsewhere (Sokal and Sneath, 

1963; Harman, 1967; Parks, 1969, 1970). All calculations were made by one computer run in about 10 minutes on an IBM360/65 machine. Compared to the costs 

incurred in measuring the craters and compiling the 

new data, which would have been needed even for a 

conventional graphical analysis, the expense of the 

machine time and data preparation was trivial.
GEOMETRIC INTERPRETATION OF LUNAR CRATERS C7

Principal-components analysis is a statistical 

transformation for simplifying large unwieldy matrices of data with a minimum loss of information (Cattell, 1952; Harman, 1967). In many respects the technique resembles multiple-factor analysis, and results 

of the two procedures often are not very different 

(Harman, 1967). Principal-components analysis is performed on correlation coefficients relating all pairs of 

variables, which reduce to a few independent synthetic 

variables. These linear, uncorrelated combinations of 

the original variable are termed principal components. 

The number of significant components, which are 

extracted sequentially from the correlation matrix to 

explain successively diminished proportions of the 

total variance, is the number of independent aspects 

present in the data. Usually this number is substantially less than the number of original variables. Component scores, similar to the "factor loadings" resulting 

from a multiple-factor analysis (Harman, 1967), are 

calculated to show the correlation between variables 

and principal components as well as the makeup of 

each component in terms of original variables. The 

criteria on which a classification may be devised are a 

second set of scores which express the correlation of 

each crater with each component.

Cluster analysis arranges objects in a twodimensional diagram, resembling a biologic family 

tree in appearance, on which any natural breaks between groups (clusters) of objects are evident by inspection (Sokal and Sneath, 1963). It has numerous applications to geologic problems (Parks, 1966; Harbaugh 

and Merriam, 1968; McCammon, 1969; Adam, 1974). 

The version of cluster analysis used here (Parks, 1969, 

1970) begins with a principal-components transformation to simplify and orthogonalize the variables. The 

scores that were computed between each component 

and each crater are used to determine a statistical 

measure of similarity, known as the simple distance 

function, between each pair among the craters. In this 

experiment, the scores are weighted (multiplied) by the 

percentage of total variance accounted for by each 

component. Although the weighting of original variables or scores remains something of an art (Gower, 

1970), the procedure used here at least assures that the 

clustering is not biased unduly toward the less important components. The distance function measures Euclidean rather than angular distance between any two 

craters in a multidimensional space: one dimension for 

each component. Its formula and properties are given 

by Parks (1966).

From an array of distance-function values, craters 

and subsequently groups of craters are cross-compared 

and then combined by a linkage procedure, one pair at 

a time, into hierarchical clusters. Craters with low

values of the distance-function coefficient group first, 

and craters with similar shapes (similar values of the 

coefficient) group together, apart from differently 

shaped craters. Many linkage techniques, or sorting 

strategies, have been devised for cluster analysis 

(Sokal and Sneath, 1963). The procedure followed here 

(Parks, 1970) is a modification of the unweighted 

pair-group method, a well-known algorithm first described by Sokal and Michener (1958) and since used in 

many clustering applications. The resulting cluster 

diagram, or dendrogram, is printed automatically with 

the computer output in this case. Other types of 

graphic results, commonly involving an X-Y plotter, 

can be obtained as well (McCammon and Wenninger, 

1970).

THE CRATER SAMPLE

In several respects, the data set assembled for this 

experiment is a marked improvement over crater samples that have been used for previous morphometric 

studies (Baldwin, 1949, 1963; Green and Poldervaart, 

1960; Pike, 1967, 1972b; Katterfeld, 1967; Guest and 

Murray, 1969; Siegal, 1973; Smith, 1973; and Siegal 

and Griffiths, 1974). First, the 23 different categories 

better represent the variety of craters relevant to the 

analog problem (table 2). Older modified-looking lunar 

craters and secondary-impact craters are included 

along with the fresh primary craters that customarily 

are preferred for comparison with terrestrial craters: 

the bias toward fresh craters in the overall lunar sample has been eliminated for this analysis. Special attention is accorded terrestrial volcanoes, which have 

been neglected in most morphometric comparison, although the volcano data are not listed here but are 

published elsewhere (Pike, 1978a). Second, the sample 

is larger than previous samples, and so the resulting 

crater classifications are more likely to be statistically 

significant; most types of craters are well represented 

numerically. Third, the data are much more complete 

and systematic in that all seven variables were calculated for each crater. In earlier studies, graphs of different descriptive variables usually did not contain the 

same craters. Fourth, most of the basic measurements, 

including all rim widths and all rim-crest circularities, 

are new and surpass older data in accuracy. Photogrammetry of Apollo pictures furnished data for more 

than half of the lunar craters.

Terrestrial and extraterrestrial craters are equally 

represented (table 2). The 206 lunar craters range from 

400 m to 275 km in diameter and include mainsequence Copernican and Eratosthenian and preImbrian craters on the near-side uplands (Wilhelms 

and McCauley, 1971), craters on the maria, smoothrimmed craters interpreted as possibly volcanic
C8 APOLLO 15-17 ORBITAL INVESTIGATIONS

TABLE 2.—Statistical summary of untransformed variables 1-4 for 23 crater types

[Entries under each variable are, from left to right, arithmetic mean, standard deviation, and range. __, not calculated. These three statistics may be approximated for variables 5-7 from 

values given here for variables 2-4. Some statistics are for values of n (see Pike, 1974c) that differ slightly from those here]

Crater type, name Type 

symbol

n Variable 1— 

rim-crest circularity

Variable 2— 

rim height/crater depth

Terrestrial

Oceanic island (mainly

Tuff ring _ _ _ __ ___ __ __ _T

13 

18 

26 

6 

5 

39 

20 

20 

10 

9 

20 

22

0.54 

.61 

.59 

.58 

.36 

.59 

.62 

.70 

.48 

.58 

.81 

.85

0.16 

.13 

.14 

.14 

.18 

.14 

.15 

.10 

.24 

.23 

.06 

.08

0.24-0.84 

.35- .75 

.25- .78 

.32- .68 

.14- .55 

.23- .85 

.32- .85 

.52- .86 

.19- .87 

.25- .85 

.68- .92 

.66- .99

14.33 

4.50 

4.71 

5.80 

1.60 

.39 

1.58 

3.55 

4.22 

3.94 

.30 

.18

Variable 3— 

flank width/rim diameter

Variable 4 — 

rim height/diameter

craters

12.54 

2.92 

7.50 

9.03 

.41 

.23 

1.04 

1.80 

5.22 

2.44 

.11 

.06

3.28-48.49 

1.22-12.89 

.86-40.00 

.67-24.16 

1.05- 2.07 

.08- 1.00 

.73- 4.95 

1.28- 6.74 

1.00-19.20 

1.13- 9.00 

.10- .59 

.10- 35

4.03 2.04 

2.61 2.01 

1.96 1.48 

1.25 .54 

.82 .28 

.37 .25 

.51 .25 

1.15 .45 

5.70 3.88 

1.91 1.22 

.20 .04 

.26 .08

1.30- 6.82 

.60- 8.00 

.61- 6.67 

.62- 2.13 

.59- 1.30 

.11- 1.52 

.18- 1.18 

.50- 2.13 

1.75-12.59 

.81- 4.20 

.14- .30 

.12- .48

0.623 

.415 

.198 

.138 

.048 

.042 

.110 

.483 

.539 

.915 

.034 

.040

0.335 

.424 

.150 

.077 

.012 

.030 

.066 

.253 

.586 

.580 

.016 

.025

0.156-1.167 

.054-1.473 

.013- .545 

.038- .237 

.033- .063 

.005- .171 

.018- .253 

.103-1.107 

.043-1.655 

.459-2.333 

.006- .067 

.005- .083

Lunar craters

Mare-filled basin _ _ _ _ _ _ MB

Dark-halo crater within.

Mare-flooded crater __ . __ __ MF

Isostatically compensated crater ___ 1C 

Cratered dome on mare surface - - MD

3 

29 

40 

23 

40

3 

17 

19 

7 

15 

14

0.81 

.57 

.83 

.82 

.76

.56

.81 

.78 

.79 

.58 

.55

.05 

.07 

.08

.06 

.06 

.05 

.20 

.16

0.80-0.83 

.34- .84

.70- .92 

.66- .91 

.52- .88

.67- .88 

.66- .91 

.74- .87 

.15- .88 

.28- .80

.0.41 2b

.24 

.25 

.60

.12

.75 

.49 

.69 

2.25 

2.52

.07 

.09 

.20

.19 

.17 

.20 

1.39 

1.62

0.32- 0.48 

.08- .51

.12- .40 

.13- .52 

.24- 1.29

.07- .15

.36- 1.00 

.25- .72 

.36- 1.00 

.73- 6.07 

.73- 6.67

0.25 — —

.22 .05 

.22 .05 

.11 .04

.53 ______

.14 .04 

.12 .05 

.10 .07 

2.25 1.31 

.85 .40

0.18- 0.31 

.13- .35

.13- .33 

.13- .35 

.05- .21

.39- .68

.07- .22 

.06- .23 

.05- .21 

.43- 6.35 

.35- 1.55

0.003 

.025

.026 

.027 

.016

.017

.018 

.022 

.011 

.285 

.255

.009 

.011 

.008

.007 

.008 

.006 

.224 

.156

0.002-0.003 

.007- .041 

.012- .051 

.012- .052 

.007- .038

.009- .022 

.008- .029 

.010- .044 

.005- .023 

.051- .851 

.077- .521

"Statistics omit Schiller and Wargentin.

(Wilhelms and McCauley, 1971), craters flooded with 

mare material or evidently affected by isostatic compensation, craters located atop domes on the mare surfaces, small raised cones located on mare surfaces or on 

the floors of fresh craters such as Copernicus and King, 

small dark-halo craters on the floor of the crater Alphonsus, and secondary-impact craters of Copernican 

age. Examples of most of these types of craters are 

illustrated in photographs throughout the report. The 

six martian craters and calderas included in the source 

multivariate classification have been removed (for 

later results, see Pike, 1978a), as have the unclassified 

craters on the iunar far side (Pike, 1974c). The latter 

omission is not important because near-side and farside craters are similar in shape. Among the 212 terrestrial craters, which range in diameter from 1.3 m to 

65 km, are calderas representing all the major petrographic associations, maars, cinder cones, tuff rings, 

small lava shields and domes, meteorite-impact craters, and experimental craters excavated by chemical 

and nuclear explosives. All these craters have been 

mentioned at one time or another as suitable analogs of 

lunar craters.

Seven dimensionless variables express the overall 

shape of each crater in plan and in profile. Rim-crest 

circularity, height/depth, width/diameter, height/ 

diameter, depth/diameter, height/width, and depth/ 

width provide the minimum geometric signature 

needed to identify and cross-compare the craters. 

Statistical properties of the first four variables have 

been summarized for all 23 types of craters (table 2).

The circularity measure (C) is described in the back of 

this report (see also fig. 24). The last six variables are 

simply ratios of the four linear dimensions used by 

Baldwin (1963) to describe the basic profile shape of 

any rimmed crater: rim-crest diameter (Dr), depth (Rt \ 

rim height (Re ), and rim width (We ) (fig. 1).

The four profile dimensions are not used individually 

as variables, because they measure only crater size, an 

aspect that often is related to neither crater 

morphology nor crater genesis. Particularly in the case 

of lunar craters, which occupy at least six orders of 

magnitude, it is shape not size that reflects the fundamental mode of origin. Moreover, it has become customary in lunar analog work to compare any terrestrial crater with any lunar crater, regardless of the size 

disparity. Although the premise makes geologic sense 

only for those few types of terrestrial craters that could 

conceivably attain the great size of the larger lunar 

craters, it is interpreted liberally here—almost to the 

point of absurdity—to insure that the widest possible 

variety of potential analogs is fairly represented. For 

these reasons, size is eliminated as a morphologic discriminant. Failure to observe this practice in multivariate analysis can lead to incorrect or inconclusive 

results and interpretations (Siegal, 1973; Siegal and 

Griffiths, 1974). Although statistical problems can 

arise in applying correlation analysis to ratio data 

(Chayes, 1971; Atchley and others, 1976), forming 

ratios of linear dimensions is judged the most effective 

way to remove crater size from the analysis. Subsequent evaluation of the final results suggests that
GEOMETRIC INTERPRETATION OF LUNAR CRATERS C9

ratio correlation has not produced a spurious outcome. 

The multivariate techniques applied here can yield 

misleading results unless the input data conform to the 

normal or Gaussian statistical model. The frequency 

distributions of all seven variables are highly skewed, 

a condition brought about in part by combining 23 constituent distributions of very different craters that 

vary widely in mean, variance, and numbers of craters 

(table 2; see also older data in fig. 15). This is not the 

usual case, where mixing frequency distributions 

usually removes rather than generates skewness (F. 

Chayes, written commun., 1974). The circularity data 

are skewed negatively, toward slightly lower values, 

whereas the remaining six variables all are skewed 

positively, toward higher values (see older data in figs. 

15, 16, and 17). Each variable was transformed to 

make its distribution as nearly symmetric as possible 

(table 3), but not to overcompensate (R. P. Chapman, 

1976). Different transformations were required because the seven frequency distributions vary in their 

degree of skewness, especially in the distance of extreme values from the median. Although these simple 

transformations do not yield the Gaussian ideal, they 

eliminate the most disruptive effects of extreme values 

from the analysis (Sokal and Sneath, 1963; Lindqvist, 

1976).

INTERPRETATION OF PRINCIPAL COMPONENTS

Correlation (both positive and negative) between the 

seven variables with respect to all 418 craters reveals 

that the variables are not equally interdependent. According to the matrix of similarity coefficients (table 3), 

height/depth, width/diameter, height/diameter, and 

depth/width are the most closely associated variables 

and describe a similar dominant pattern of covariance 

in the data.

The seven original variables reduce to four 

significant principal components (table 4), all of which 

can be interpreted in terms of crater morphology. The 

highest component scores correspond roughly to the 

highest absolute values of the similarity coefficients 

listed in table 3. The analysis is so nearly complete by

TABLE 4.—Principal-components results for seven variables and 418

craters

[Format after Harman (1967). Component scores are listed for each variable on the first four 

principal components (P|-P.,) only. Including the last three components (not shown), total 

variance = 7.0 (the number of variables); four components yield an exceptionally complete solution]

Variable Symbol

Height/depth

Height/width

Depth/width

Percentage of total variance

C 

RJRt 

Wf/Dr 

Re/Dr 

R,/Dr 

Re /We

R,We

P,

-0.573 

.924 

.900 

.863 

-.003 

.170 

-.805

- 3.412

- 48.75

PI

0.223 

-.053 

.072 

.498 

.851 

.772 

.554

1.932 

27.6

P3

0.358 

.373 

-.334 

.019 

-.516 

.552 

-.061

.945 

13.6

P.

0.703 

.026 

.262 

.077 

.082 

-.258 

-.149

.665 

9.5

Variance

1.000 

.996 

.995 

.999 

.996 

.997 

.980

6.954 

99.5

the time four of the seven possible components are 

extracted that virtually no information is lost. This 

highly favorable outcome is evident in two respects. 

First, the variance (summed squares of the component 

scores) accounted for by the four components for each 

variable approximates unity. Second, and similarly, 

total variance accounted for by the seven variables for 

the first four components is 99.5 percent.

The first and dominant (49 percent of variance) principal component expresses the most important descriptive aspect present among the variables: size, position, and shape of the crater rim relative to that of 

the crater depression. The three highest scores in table 

4 indicate its main constituents: height/depth, width/ 

diameter, and height/diameter. Depth/width is somewhat less important and varies inversely; circularity 

also varies inversely and is even less significant. Scores 

of the various types of craters on the first principal 

component suggest that the component is oriented 

within the data to account for most of the variance 

resulting from the difference between volcanoes and 

craters usually attributed to impact (fig. 2). Terrestrial 

impact and experimental-explosion craters and fresh 

lunar craters typically have the lowest scores on the 

first component, whereas terrestrial shield volcanoes 

that contain either calderas or simple summit craters 

have the highest scores. These two extremes are only 

indicative of the dichotomous pattern within this component, which is interpreted accordingly as a good 

criterion of primary crater genesis.

TABLE 3.—Matrix of similarity coefficients for principal-components analysis

[Although technically these values were derived from the formula for the product-moment coefficient of correlation (r) between pairs of transformed variables, the numbers should be 

regarded as only indicators of similarity rather than as values of a testable statistic. Statistical tests of significance do not apply to these values because some correlation is inherent 

among ratios showing similar crater dimensions (Chayes, 1971)]

Variable

Circularity

Height/depth . _

Depth/diameter

Height/width

Deoth/width _ _ _ _

Transformation

loe NXjrT

logVjc~

loe "^xT

log*

..__.____.____logx

VX

Symbol

c

RjRi

w in

Re lDr

R,/Dr

Re IWe

RS IW»

c

1.00

-.47

-.44

-.38

.07

-.01

.46

RtlR,

1.00

.73

.77

— 1Q

.29

-.79

We/Dr

1.00

.84

.29

-.01

-.68

Rf/Dr

1.00

.48

.52

-.40

Ri/Dr

1.00

.41

.46

RtIDT

1.00

.31

R,/W,

1.00
CIO APOLLO 15-17 ORBITAL INVESTIGATIONS

The three remaining components are not clear-cut 

indicators of crater origin but represent other important descriptive attributes. The second principal component accounts for variance (28 percent) that is attributed to the next most important aspect of crater 

shape present in the data: slope of the crater depression 

(depth/diameter) and slope of the rim flank (height/ 

width). Depth/width and height/diameter also contribute to this component, but not as much. Neither 

depth/diameter nor height/width is at all diagnostic of 

crater genesis (Green, 1959; Steinberg, 1969; Pike, 

1972b), although both variables do reflect sizedependent differences in the shape of impact craters, as 

well as degradation of the crater rim with time and 

commensurate filling of the bowl. Depth/diameter correlates roughly with the relative age of many lunar 

craters (Baldwin, 1963; Pike, 197 la). Small steep features such as terrestrial volcanic domes and cones, as 

well as experimental-explosion craters, characteristically have the highest scores on the second component. 

Large gently sloping landforms such as mare-filled 

lunar basins and terrestrial ash-flow calderas have the 

lowest scores.

The third principal component, which explains 14 

percent of the total variance within the data, is less 

amenable to a physical interpretation because none of 

the scores (table 4) is very high. This component reflects primarily the coincidence of a steep rim flank 

(height/width) with a shallow crater (depth/diameter). 

On Earth, this configuration is not very common 

among cratered landforms except for cinder cones (fig. 

3). On the Moon, a relatively steep rim flank is more 

often associated with a shallow crater interior, especially where fresher, younger craters are filled or 

modified catastrophically and do not age gradually (for 

example, Archimedes, Taruntius). Conversely, low 

scores on the third component indicate exceptionally 

deep craters with low exterior rim flanks. These most 

often include dark-halo and secondary-impact craters 

on the Moon, as well as small lava shields (such as 

Burfell, in southwest Iceland) on Earth.

Although the fourth principal component contains 

only 9 percent of the total variance, it warrants interpretation by virtue of the relatively high score of 

circularity on it (table 4). To a limited extent, scores on 

this component by individual craters increase proportionately with circularity of the crater rim, but 

primarily the fourth component seems to account for 

variance arising from atypically high or low circularities. Highest scores on this component are made 

by craters that are nearly circular but have an overall 

profile geometry that is more characteristic of a volcano. On Earth, this configuration is found in many 

cinder cones (Wizard Island, in Crater Lake, Oregon),

some maars (Ndubot in Tanzania), and in a few calderas (Volcan Darwin in the Galapagos Islands), but 

very high scores on this component are not typical of 

craters on the Moon. The lowest scores on the fourth 

principal component indicate highly acircular craters 

that have a profile geometry characteristic more of impact craters than volcanoes. Terrestrial craters of this 

sort tend to be irregular maars (Lojotipullur in Iceland). On the Moon, low scores on the fourth component often are found among secondary-impact craters 

and irregular or highly elongate main-sequence craters such as Zollner, Daniell, Messier, and Schiller.

The 418 craters have been arranged and sorted into 

groups according to similarities and differences in 

shape, using the principal-components outcome in 

three of several possible ways (figs. 2 and 3 and pi. 1). 

The resulting classifications are similar overall— 

notably in the separation of most impact craters from 

most volcanoes—but differ in detail. Each grouping 

emphasizes some variations in crater shape that are 

not well shown in the others. The need for more than 

one graphic display for the principal-components outcome underscores the inherent difficulty of presenting 

results of a multivariate analysis in only two or three 

dimensions. The end product is always a compromise.

The three crater classifications are discussed in order 

of increasing complexity. The first is simply a plot of 

the scores of each crater (here grouped by types) on the 

first principal component (fig. 2). It emphasizes primary crater genesis without statistical "noise" from 

the last three components, which reflect postformational modification or less important variations in crater shape. An advantage of this kind of classification is 

the ease and directness with which individual craters 

can be compared (although labeling each crater is very 

awkward). However, figure 2 is oversimplified because 

the first component includes only half of the variance 

in the analysis and is incapable of breaking down the 

418 craters into more than two large clusters. The information content of such a classification can be increased considerably by plotting the scores of individual craters on more than one principal component. The 

resulting diagrams are limited to only two or three 

components and can be virtually impossible to interpret if a great many craters are graphed. A triaxial plot 

(fig. 3) is a compromise, whereby 90 percent of the total 

variance in the analysis is represented and the craters 

are not graphed individually but are grouped by type. 

Although single craters cannot be compared in this 

diagram, relations among the different types are more 

fully and faithfully represented than they can be in 

figure 2. More than two major clusters of crater types 

can be distinguished. Finally, the 418 craters were 

grouped by a formal cluster analysis of all four (weigh-
GEOMETRIC INTERPRETATION OF LUNAR CRATERS Cll

ted) principal components (pi. 1). Largely prefigured by 

the dichotomous grouping evident in figure 2, the cluster sorting yields a more detailed and highly structured grouping of the 418 craters than do the first two 

classifications. Many subclusters are evident. All 100 

percent of the variance is included, and all craters are 

labeled. A disadvantage of the cluster technique is that 

the dendrogram is confined to only two dimensions and 

some of the relations among crater types evident 

within the three-dimensional plot in figure 3 do not 

show up very well in plate 1.

CLASSIFICATION BY PRINCIPAL COMPONENTS

Scores on the first principal component classify the 

23 crater types into two mutually exclusive and widely 

separated groups (fig. 2), with the conspicuous exception of terrestrial maars (E) and tuff rings (T) and 

lunar raised cones (RC). High scores indicate craters 

with high wide rims, floors lying well above the exterior datum, and (usually) highly acircular rim crests. 

The mare domes (MD) and most terrestrial volcanoes 

score over 0.250, whereas meteorite-impact (M) and 

experimental-explosion craters (X) and all other lunar 

craters except the raised cones usually score under

0.150. The two contrasting classes of crater shape reflect processes corresponding to one of two general 

modes of origin, excavation and accumulation. In excavational (mostly impact) craters, the rim is composed of 

material derived primarily from the crater bowl. In 

craters of net accumulation, or cratered edifices (most 

volcanoes), the rim material either comes from depth 

or is unrelated to formation of the present crater bowl. 

The three lunar dark-halo craters (DH) also are exceptions to the genetic dichotomy in figure 2: Their scores 

on the first principal component place them within the 

impact group—if only marginally—but their albedo 

and geologic setting on the floor of the larger crater 

Alphonsus (Carr, 1969) indicate an internal origin.

Terrestrial tuff rings (T) and maars (E) and lunar 

raised cones (RC) scatter widely and overlap both genetic clusters in figure 2. Considered as crater classes, 

maars and tuff rings resemble neither impact craters 

nor the more conventional volcanoes in shape, but 

rather bridge the two main groups. The transitional 

position of maars and tuff rings in figure 2 probably 

reflects a tendency for both excavational and accumulational processes to combine in forming specific craters. The difference between the two transitional 

classes of terrestrial craters in figure 2 suggests a sysCrater type

Meteorite-impact craters 

Experimental-explosion craters 

Young mare craters 

Young upland craters 

Pre-Imbrian craters 

Isostatically compensated craters 

Mare-flooded craters 

Mare-filled basins 

Smooth-rimmed craters 

Secondary -impact craters 

Dark-halo craters 

Maars 

Tuff rings 

Raised cones 

Mare domes

"3 

xi

>,

W

M 

X 

PM 

TY 

PI 

1C 

MF 

MB 

IS 

SC 

DH 

E

T 

RC 

MD

Chiefly

craters

1 —————— • —— 1 

1 ———— • ———— 1

H

Chiefly

volcanoes

1 —— • ———————— 1

1 ——————— • —————————————— 1

1 —————— • ——————————— 1

1 ——————— • ———— 1

1 —————— • ——————— 1 

1 ———— • —————— 1

1 ——————— • ———————— 1 

1 ———————— • ———————— 1

1 ———————————— o ——————— ,

i i 1 1

"3

XJ

x

C/D

KO 

KA 

KC 

KP 

AP 

D 

S 

C

E 

T 

RC

MD

Crater type

Tholeiite shields 

Alkalic calderas 

Calc-alkalic calderas 

Potassic calderas 

Ash-flow plains 

Small lava domes 

Small lava shields 

Cinder cones

Maars 

Tuff rings 

Raised cones 

Mare domes

0.10 0.20 0.30 0.40 0.50

NORMALIZED COMPONENT SCORES (WEIGHTED)

FIGURE 2.—Correlations of crater types with first principal component. Mean values and ranges of scores for 23 crater types listed in table 2. 

Closed circles, terrestrial craters; open circles, lunar craters. Normalization (transforming scores of each component to 0.0 to 1.0 scale) 

and weighting (see text) do not affect relative order of distribution. Mean scores omitted for mare-filled basins and dark-halo craters (too 

few) and for small lava shields (Icelandic shields group far to right of other small shields). Strongly dichotomous distribution of most 

crater types reflects fundamental contrasts between impact craters and most volcanoes and prefigures two-part cluster classification 

(pl. 1).
C12 APOLLO 15-17 ORBITAL INVESTIGATIONS

tematic mix of both mechanisms, maars being mostly 

excavational in origin and tuff rings being mostly accumulational. However, maars and tuff rings belong to 

a morphologic and genetic continuum, shared by cinder 

cones as well (Pike, 1978a), in which case this difference is only an artifact of the crater nomenclature. 

Most lunar raised cones (RC), which according to scores 

on the first component are shaped more like terrestrial 

cinder cones than tuff rings, are implied to be predominantly constructional landforms.

Classification of craters on the first principal component excludes terrestrial calderas of any type as acceptable topographic analogs of typical lunar craters. 

The most caldera-like of the main-sequence craters on 

the Moon are the badly battered pre-Imbrian crater 

Schroter (score 0.160) and the odd, boat-shaped crater 

Schiller (0.166). The evidently inundated crater 

Wargentin (0.137) is much less caldera-like in shape. 

Three of the secondary craters attributed to ejecta from 

Aristarchus also score over 0.160. The most lunar-like 

of the terrestrial calderas in this data set are Ilopango 

in El Salvador (0.240) and Suswa in Kenya (0.236). 

Meteorite-impact craters are the only natural features 

on Earth that consistently resemble all types of mainsequence lunar craters. Accordingly, these lunar craters probably formed by impact. Although many terrestrial maars (MacDougal Crater in the Pinacate volcanic field in Mexico; score 0.085) and a few tuff rings 

(Hverfjall in Iceland; 0.162) individually resemble 

some lunar craters—notably the more degraded 

types—as a class neither type of crater is consistently 

similar in shape to the lunar craters. Moreover, terrestrial maars and tuff rings rarely reach 1,500 m 

across and do not exceed 5 km in diameter, whereas 

impact craters appear to have no such upper limit in 

size (Gilbert, 1893; Shoemaker, 1962).

Ten types of craters make up the tightly clustered 

and low-scoring group attributed to impact or explosion (fig. 2). These types of craters vary systematically 

within the group, as reflected by average scores on the 

first principal component, according to degree of complexity of morphology and geologic history. The freshest and least modified craters have the lowest scores, in 

order of increasing mean scores: experimentalexplosion craters (X), meteorite-impact craters (M), 

young mare craters (PM), and young upland craters 

(TY). Next come younger craters that either have 

slightly anomalous original shapes—smooth-rimmed 

craters (IS)—or else have been modified somewhat by 

isostatic rebound (1C). Clearly older main-sequence 

craters that have undergone much degradation (PI) are 

next, followed closely by mare-filled basins (MB). 

Secondary-impact craters (SC) and deeply flooded 

main-sequence craters (MF) both differ considerably in

shape from fresh, young impact craters and are last. 

Dark-halo craters (DH) lie even farther to the right in 

figure 2 and thus do not really belong to the impact 

cluster, even on the basis of morphology. These craters 

occupy a mutually exclusive field with terrestrial 

meteorite-impact and experimental-explosion craters 

on figure 2 and are all but mutually exclusive with 

fresh craters on the lunar maria as well.

The scores of volcanoes scatter much more widely 

than do the scores of impact craters in figure 2. The 

disparity indicates that volcanoes intrinsically are 

more variable in overall shape than are impact craters. 

Certainly a more regular landform would be expected 

from the instantaneous and comparatively uniform 

impact mechanism than from volcanic central eruptions, which can vary considerably in duration, intensity, vent location, and in the chemistry and physical 

properties of their ejecta.

Some of the variance within the volcanic group (fig. 

2) may have arisen from a connection between gross 

morphology of calderas and silica content of the 

erupted products and the percentage of pyroclastic 

material. First, most caldera-bearing tholeiite shield 

volcanoes (KO) (Hawaii) and those of the small shield 

volcanoes (S) of Iceland lie well to the right of other 

types of volcanoes. Average component scores also decrease systematically from tholeiite calderas (KO) 

through alkalic calderas (KA) to calc-alkalic calderas 

(KG), calderas of the potassic (Mediterranean) association (KP), and cratered ash-flow plains (AP). However, 

so much overlap occurs among the different caldera 

types that the dominant rocks of extraterrestrial caldera volcanoes, such as those on Mars, cannot be predicted with much certainty from this relation. Certainly the correlation is less substantial than that between petrography and geographic location of Cenozoic 

volcanoes (Chayes and Velde, 1965; Chayes, 1964). 

Moreover, although none of the mare domes (MD) occupy the same range of principal-component scores as 

do the large terrestrial tholeiite shields (KO) (fig. 2), 

the lunar mare basalts are more like tholeiite than any 

other type or terrestrial basalt. The lunar domes do 

coincide with about half of the smaller tholeiite shields 

(S), however, and in fact the lunar domes may more 

closely resemble this type of volcano than the others. I 

have pursued these problems in later work (Pike, 

1978a).

The marked contrast between craters of excavation 

and cratered landforms of accumulation is again evident in a graph of crater scores on the first three principal components (fig. 3). Adding the second and third 

components has, if anything, enhanced and clarified 

the dichotomy. Weighted scores on each component 

were averaged for each crater type from scores of single
GEOMETRIC INTERPRETATION OF LUNAR CRATERS C13

EXPLANATION 

O Lunar crater type 

0 Terrestrial crater type

1C

0.14

o

FIGURE 3.—Normalized scores of 21 crater types (table 2; pi. 1) on first three principal components (Pu P2, PS), averaged from 

individual scores on 418 craters. For simplicity and clarity five categories of calderas in table 2 are reduced to three: calderas on 

shields (V), calderas on stratovolcanoes (K), and calderas surrounded by ash-flow sheets (A). Impact and explosively formed craters 

cluster in a tight linear distribution in upper left. They are arrayed from freshest to most degraded (declining scores on P2 axis): 

experimental-explosion craters (X), meteorite-impact craters (M), fresh lunar impact craters (PM, TY), smooth-rimmed craters 

(IS), pre-Imbrian craters (PI), mare-flooded craters (MF), isostatically compensated craters (1C), and mare-filled basins (MB). 

Volcanoes cluster somewhat more loosely, on right. In addition to three types of calderas mentioned above, this group comprises 

tuff rings (T), mare domes (MD), raised cones (RC), small lava shields (S), cinder cones (C), and lava domes (D). A third loose 

grouping, comprising terrestrial maars (E), lunar dark-halo craters (DH), and secondary-impact craters (SO, is situated midway 

between the two primary clusters.

craters and plotted for 21 crater types: the fourfold 

chemical distinction among calderas is dispensed with. 

In figure 3 calderas are recognized on a somewhat simpler structural basis as being either on a shield volcano 

(V), atop a stratovolcano (K), or surrounded by an

ash-flow sheet (A). Although the contrast between impact craters and volcanoes dominates the threedimensional array of crater types in figure 3, some important relations concealed by the simpler method of 

data display in figure 2 also are evident.
C14 APOLLO 15-17 ORBITAL INVESTIGATIONS

Several observations may be made on both of the 

principal clusters of crater types in figure 3. The impact craters (M, PM, TY, PI, IS, 1C, MF) cluster tightly 

together with experimental-explosion craters (X) as in 

figure 2, but the mare-filled basins (MB), secondaryimpact craters (SC), and dark-halo craters (DH) here 

are detached from this group on the basis of significant 

differences expressed by the second and third principal 

components. Dark-halo craters (DH) are unique, by 

virtue of their negligible score on the third component, 

and form a loose cluster with secondary-impact craters 

(SC) and terrestrial maars (E). This cluster falls between the two principal groups but may be interpreted 

as a third major subdivision of the 21 crater types 

rather than as a satellite of one of the two larger 

groups. Judging from graphical distances between the 

grouped craters in figure 3, maars are not very good 

analogs of main-sequence craters on the Moon but easily could be confused with impact secondaries. Indeed, 

just this question arose concerning interpretation of 

the craters of Catena Davy (Oberbeck and Morrison, 

1973). Differences among the seven types of impact and 

explosion craters with respect to components 2 and 3 

seem to reflect systematic differences in relative age 

and state of degradation. This sequence starts with the 

experimental-explosion craters (X) and ends with the 

mare-flooded (MF) and isostatically compensated (or 

volcanically modified) craters (1C). According to distances between plotted points in figure 3, contrasts 

among these seven types are less than the contrasts 

between them as a group and any other type of crater; 

the impact interpretation of the lunar craters in this 

group remains compelling.

The volcanic craters scatter appreciably in figure 3, 

as they do in figure 2, but with some significant differences. The shallow calderas within ash-flow sheets (A) 

stand quite apart from the other volcanoes, a reasonable outcome for a shape classification in view of their 

unique morphologic characteristics and mode of origin. 

All three types of calderas are rather distinct from one 

another on the graph, presumably reflecting structural 

contrasts of the different kinds of volcanic edifices. 

Conversely, the two types of terrestrial shield volcanoes are fairly similar in figure 3. The mare domes 

(MD) seem to resemble more closely the calderas on 

large shields (K) than craters on small lava shields (S). 

However, the analysis presented here is restricted only 

to crater shape and does not take into account the disparity in size of the volcanic piles. Domes on the lunar 

mare surfaces still may resemble the smaller terrestrial shields. Raised cones (RC) on the Moon lie quite 

apart from lunar impact craters in figure 3, a distinction that is not made in figure 2. An interpretation of 

the mode of origin of the cones really cannot be extracted from either diagram, for these features have no 

clear-cut Earth analog. Terrestrial tuff rings (T), like 

maars (E), as a group still are transitional between 

most volcanoes and most impact craters with regard to 

morphology, but they do appear to lie closer to other 

types of volcanoes in figure 3 than they do in figure 2. 

Terrestrial cinder cones, tuff rings, and maars lie along 

a linear systematic trend in figure 3 that suggests a 

genetic, as well as morphologic, continuum of small 

pyroclastic volcanoes. This progression, which could 

reflect the relative abundance of ground or surface 

water in the crater-forming environment (Lorenz, 

1973)—maars being the most dependent upon 

water—almost certainly is irrelevant to lunar volcanism. The great steepness of the terrestrial lava 

domes (D) and cinder cones (C) relative to other, mostly 

larger, types of volcanoes is clearly evident by their 

position in figure 3. This contrast sets the two types of 

craters apart from all other kinds of volcanoes, with 

respect to overall morphology.

CLASSIFICATION BY CLUSTER ANALYSIS

The overall dichotomous classification of craters that 

results from cluster analysis (pi. 1) differs only in 

minor detail from a cluster classification obtained in an 

earlier study of this kind (Pike, 1974c, fig. 2). Because 

most of the data are the same, it was judged unnecessary to test the new results. The stability of the earlier 

outcome, however, was tested several times because so 

few descriptive variables were used, because all clustering procedures do not necessarily yield similar or 

equally valid classifications of the same data (Anderson, 1971; Howarth, 1973), and because the analysis is 

based upon correlation of ratio data—a practice that is 

best avoided (Chayes, 1971). The craters first were divided evenly, and each subset was clustered separately 

by Parks' (1970) procedure. Next, the data were clustered by two quite different techniques that did not use 

the result of a principal-components analysis (Chernoff, 1973; Wishart, 1969). Finally, the 402 craters 

were increased to 420 and reclustered by Parks' program. In the latter two classifications (see also pi. 1), 

class I craters (Pike, 1974c, fig. 2, upper cluster) tended 

to separate into two groups each equal in rank to class 

II (Pike, 1974c, fig. 2, lower cluster). In all cases, some 

maars (E) and tuff rings (T) tended to shift back and 

forth between class I and class II depending upon the 

size and makeup of the crater sample, a clear indication that these craters fall between the two principal 

clusters and fit comfortably into neither. However, all 

five test outcomes agreed with the fundamental 

twofold division in plate 1 in that most class I craters 

differ markedly in shape from class II craters.
GEOMETRIC INTERPRETATION OF LUNAR CRATERS CIS

Although the cluster classification of 418 craters (redone for this report, pi. 1) generally preserves the same 

two major classes of crater shape evident in figures 2 

and 3, many subclasses are distinguishable within the 

hierarchical structure of the dendrogram. Compositions of resulting subclasses by crater type may be 

determined from the letter code; these are summarized 

in table 5. The threefold division of caldera types has 

been carried over to plate 1 from figure 3, and in table 5 

fresh lunar craters (PM, TY) have been combined and 

then divided into large craters (over 15 km across) and 

small craters. Average topographic differences between craters in the two principal classes, which also

can be expressed by citing individual craters that have 

the same respective geometry, are very great indeed 

(fig. 4). The upper profile in figure 4 looks most like the 

terrestrial meteorite-impact crater Henbury No. 3, and 

the lunar craters Flamsteed and Aristillus. The lower 

profile in figure 4 most closely resembles the calderas 

Sete Cidades (Azore Is.), Black Peak (Aleutian Is.), and 

Nemrut Dagi (eastern Turkey). Although clusters I 

and II are very different on the average, certain craters 

in each group are quite similar; these tend to be tuff 

rings (T) and maars (E). The two most nearly alike 

craters in classes I and II, Potrillo maar (New Mexico, 

U.S.A.) and Laja maar (Tanzania), respectively, are

TABLE 5—Composition of clusters in plate 1 by crater type

Crater type (name)

All ________________________________________

Crater 

type 

(symbol) N

________________ 418

Craters of excavation

IA ffl

120 29

1C ID

67 44

Clusters

Craters of accumulation

HA IIB HC IID

53 30 26 24

Others

25

Lunar craters

Fresh mare and terra crater, small

Fresh mare and terra crater, large

Mare-flooded lunar crater

Mare-filled basin

Secondary-impact crater

Dark-halo crater in Alphonsus

Raised cone

PM TY1 27

________PM, TY2 36

______ ________PI 40

T/^1 rj

__ _ _MF 17

MB 3

_______ __ _IS 19

Qp OQ

T*|TT O

____________ RC 14

_____ _ __ _MD 15

1

26 

34

7 

17

16 II 

2 4

II ~3

24 2 

9 1 

2 2

~2 "I 

5 18 

__ 2

__

"2 ~4 ~5 II

625-

~2

"I

~2

Terrestrial craters

Stratovolcano with caldera

Maar

Tuff ring, tuff cone

Cratered lava dome

'Under 15-km diameter. 

2Over 15-km diameter. 

Type KO and some KA

_______ ________X 22

M 20

V3 21

__________ _ _K4 38

__——— __.___A5 9

.__________„__£ 39

T 20

_______________C 20

__________ — S 10

D 9

1 

9

7 12 

10

__

14 6 

11 -

__ __

— 12

~3 II "I 13 

26 1 6 4

7

~4 ~5 II II 

__ 16 3 1

5 - - 5 

2 6 1

1

~4

1

2

8 

1

Type KA, KC, KP. 

Type AP and some KC.

\_ ^s

FIGURE 4.—Contrasting models illustrating overall geometric properties of two largest crater classes in plate 1. Dimensionless profiles, scaled to same rim-crest diameter, from averages of six of seven transformed variables for all 272 craters in 

class I (upper profile) and all 146 craters in class II (lower profile). Respective average values of circularity index are 0.78 

and 0.64. Dashed line is exterior topographic datum, vertical exaggeration is 2x. Absence of transitional maars and tuff 

rings (fig. 2) would enhance contrast.
C16 APOLLO 15-17 ORBITAL INVESTIGATIONS

clearly transitional in topographic character and are 

not accommodated by a strict twofold division of the 

sample. In fact, both craters fall in small clusters that 

do not fit in very well with the other craters of the two 

major classes.

Further breakdown of the two resulting classes of 

craters is expressed by their subclasses, eight of which 

have been labeled in plate 1. Several minor subgroups 

have not been so designated. The selection of subgroups within a dendrogram is always somewhat 

subjective and largely depends upon the level of 

generalization appropriate to the interpretation. The 

subgroups shown here constitute a fairly general 

breakdown of classes I and II, commensurate with the 

nature of the 23 original crater types, most of which 

contain a wide variety of morphologies. A more detailed subdivision of plate 1 probably is not warranted 

by the data. Class I includes all terrestrial meteoriteimpact and experimental-explosion craters, all but two 

maars, half of the tuff rings, the three mare-filled 

basins, and all lunar craters except those situated on 

the 15 domes and on 11 of the 14 raised cones. Like the 

craters with low scores in figure 2, most of these 272 

craters have low narrow rims, a floor lying well below 

the exterior datum, and relatively high rim-crest circularity (fig. 4, upper profile). Most terrestrial craters in 

class I formed primarily by processes of exc'avation, in 

the case of natural craters either by meteorite impact 

or by maar volcanism. The lunar craters are implied to 

have formed similarly.

Subclustering within class I supports the interpretation evident from figures 2 and 3 that fresh lunar craters resemble terrestrial meteorite-impact craters more 

closely than maars. Most meteorite-impact craters 

group with the experimental-explosion craters and 

with small (under 15 km diameter) fresh lunar craters 

(Cauchy, Linne) throughout subclass 1C and with 

larger and fresher lunar craters (Copernicus, Clavius) 

in subclass I A. On the other hand, most maars, (along 

with lunar dark-halo craters and secondary-impact 

craters, which they resemble) group separately, in subclasses IB and ID. Tuff rings in class I occur only in 

subclass IB, which contains not one main-sequence 

lunar crater. There is not one maar in subclass 1C. 

Only a few maars associate with main-sequence lunar 

craters, which tend to be the larger, older, or severely 

modified craters (McClure D, Capuanus, Briggs, 

Yerkes) in subclass IA, not the smaller fresh craters. A 

few terrestrial maars (Badillo, Bishoftu 4B, Kino, Lake 

Pawlo, and MacDougal; see Pike, 1978a) do indeed associate with meteorite-impact craters (Wolf Creek and 

Flynn Creek) and with some fresh main-sequence 

lunar craters in subclass IA (Theophilus, Aristillus, 

Plinius). However, postcrater sedimentation may have

brought about even this minor agreement: both Lake 

Pawlo and Wolf Creek Crater are now known to be 

deeply filled with sediment (Fudali, 1979) and at least 

three of the remaining four maars have unknown 

thicknesses of postcrater fill. In plate 1 the differences 

between maars and impact craters substantially outweigh the similarities.

A possible tendency of cluster algorithms to misclassify transitional objects is evident in the arbitrary 

attachment of most maars, tuff rings, and lunar darkhalo craters to the impact craters of class I (pi. 1) rather 

than their separation into a third major class (as in fig. 

3) or the inclusion of more of them in class II with the 

other volcanoes. The misclassification, if any, probably 

arises from three causes. First, the dendrogram is constrained to only two dimensions, and thus any tendency of subclasses IB or ID to separate from subclasses 

IA and 1C cannot be shown effectively. Second, the 

"misclassified" craters were produced by explosiveexcavational processes and belong in class I. Third, 

class II has much greater variance than does class I. 

This disparity, which shows up well in figure 2, evidently results in part from residual skewness not eliminated from the seven frequency distributions by the 

transformations. The difference in variance has placed 

the division between clusters I and II of the dendrogram at a higher value of the first principal-component 

score (0.225) than normally would be selected (0.200) 

from an inspection of figure 2 (Chayes and Velde, 

1965). Shortcomings of this sort can be recognized by 

examining simple plots of principal-component scores 

(figs. 2 and 3) before attempting to interpret the more 

complex dendrograms.

However, the clustering procedure has correctly separated nearly all small (<15-km diameter) lunar craters (subclass 1C) from both older and larger craters 

(subclass IA). The distinction probably was made on 

the basis of the second principal component. Smaller 

(simple or unmodified) lunar craters have long been 

known to differ from the larger (complex or modified) 

craters in being relatively deeper, in having relatively 

higher rims, and in lacking well-developed flat floors, 

rim terraces and central peaks (Gilbert, 1893; Quaide 

and others, 1965; Pike, 1967). These size-dependent 

contrasts appear to characterize impact craters on 

Earth and the other planets as well (Dence, 1968; Losej 

and Beales, 1975; Pike, 1971a; Hartmann, 1972b; 

Gault, Guest, and others, 1975).

Time-dependent, postformational effects also are reflected in the affiliation of craters with different subgroups within class I. The most evident of these effects 

is the shallowness of many older lunar craters in subclass IA, which contains most pre-Imbrian and 

smooth-rimmed craters and every one of the mare-
GEOMETRIC INTERPRETATION OF LUNAR CRATERS C17

flooded and isostatically compensated craters. Postformational events also may have caused some of the 

unsystematic mixing of the types of lunar craters that 

make up class I; morphologies within the various crater types are by no means homogeneous. The most important processes evidently are isostatic adjustment, 

volcanic flooding of mare basins, ballistic erosion and 

sedimentation, and mass-wasting (Baldwin, 1963; 

Pike, 1971a, b; Ross, 1968; Howard, 1973; Schultz, 

1976a).

Class II comprises the cratered mare domes, most of 

the raised lunar cones, and all terrestrial volcanoes 

except the maars and the remaining half of the tuff 

rings. It corresponds to the group of craters with high 

scores in figure 2. In contrast to the shapes of the craters in class I, these 146 craters typically have high 

wide rims, a floor situated well above the exterior 

datum, and relatively low rim-crest circularity (fig. 4, 

lower profile). All terrestrial craters in class II formed 

primarily by accumulation of erupted material rather 

than by processes of excavation. By implication, the 

cratered mare domes and raised cones also are volcanoes, a conclusion that is consistent with most other 

interpretations of the two types of features. Conspicuously absent from class II are such purportedly "volcanic" lunar craters as Schiller, Wargentin, Sabine, 

Ritter, and Kopff (Wilhelms and McCauley, 1971). The 

anomalous appearances of these craters, all of which 

associate with other main-sequence lunar craters in 

class I, have yet to be explained satisfactorily, although post-impact volcanism may have been at least 

partially responsible (Schultz, 1976b).

Although the four subgroups that are distinguished 

within class II (pi. 1) differ substantially from one 

another in crater geometry, their geologic and geomorphic interpretations are not as clear-cut as they are for 

the class I subgroups. The probable lunar volcanoes 

(RC, MD) neither cluster apart from terrestrial volcanoes nor predominate in any particular subclasses 

within class II (table 5). Some patterns are evident, 

however. The larger craters are in subclasses IIA and 

IID, whereas the smaller craters tend to occur in subclasses IIB and IIC. Mare domes (MD) and raised cones 

(RC) are all but mutually exclusive with large terrestrial tholeiite shield volcanoes (KO) and the smaller 

basalt shields (S). The highest concentrations of lunar 

domes are in subgroups IIA and IIC, whereas the lunar 

raised cones tend to occur in subgroups IIB and IIC. 

Most of the terrestrial lava domes are in subgroup IIC, 

but this may not be significant in terms of the origin of 

the lunar features. Terrestrial cinder cones (C) tend to 

occur only in subclass IIB, probably on the basis of 

their greater relative rim-steepness and crater depth. 

Too few lunar domes or cones are in this group to warrant any special interpretation. Subclasses IIA and 

IID, which comprise mostly terrestrial calderas, appear 

to be distinguished from each other by size of the crater 

relative to size of the host volcano. This difference 

seems to reflect both rock composition and the nature 

of caldera-forming eruptions: shield volcanoes dominate subclass IID, whereas most ash-flow calderas are 

located in subcluster IIA.

Given the scattered distribution of lunar domes and 

cones among the first three subgroups in plate 1, a good 

analogy cannot be drawn between probable lunar volcanoes and specific types of terrestrial volcanoes. 

Inasmuch as the topographic data on lunar domes and 

cones probably are the least accurate of all the measurements used here, detailed comparisons of this kind 

may be premature. The lunar domes probably are volcanoes and they appear to be some type of shield, but 

an exact interpretation remains problematical. The 

lunar raised cones are somewhat similar in shape to 

terrestrial cinder cones and lava domes, but their origin is not uniquely specified by analogy with terrestrial features, especially in view of possible effects of 

differing gravitational acceleration and energy of 

eruption (Wright and other, 1963; McGetchin and 

Head, 1973).

THE CALDERA ANALOGY

The inclusion of all terrestrial calderas exclusively 

within class II (pi. 1) and within the high-scoring group 

in figure 2—as well as their separation from all impact 

craters in figure 3—has clear-cut implications for genetic interpretation of most of the Moon's craters. Although calderas have been proposed repeatedly as terrestrial analogs of large main-sequence lunar craters 

since the time of Dana (1846), all three classifications 

based on the principal-components results verify the 

contention of Gilbert (1893) that caldera shapes differ 

dramatically from those of the lunar craters.

Large lunar craters usually are interpreted as volcanic calderas by inspection, on the basis of supposed 

morphologic similarities (such as flat floors, scalloped 

inner rims, and structural terraces and lineaments) 

that are presented in paired sketches and photographs 

of single lunar and terrestrial craters (Fielder, 1965; 

McCall, 1966; Moore and Cattermole, 1967; Peyve, 

1969; Green, 1971; Firsoff, 1961; Erlich and others, 

1974; Leonardi, 1976). Although minute details of the 

crater or caldera proper are discussed at great length 

in such comparisons, the form and extent of the overall 

structure are almost always ignored. The latter aspects 

are of critical importance. Two landforms so clearly 

different in gross surface geometry as terrestrial calderas and main-sequence lunar craters hardly can be
CIS APOLLO 15-17 ORBITAL INVESTIGATIONS

proved to be analogous on the basis of minor 

morphologic characteristics. Auxiliary geomorphic 

traits that previously were interpreted in terms of primary volcanic cratering on the Moon probably are 

either of exogenic origin or are secondary endogenic 

features superposed on initial impact landforms (Pike, 

197la). Many odd-shaped craters on the Moon are 

large basin-impact secondaries (Wilhelms, 1976; Oberbeck and Morrison, 1976). If any of the larger mainsequence craters on the Moon are of internal origin, 

then they have developed through volcanic processes 

and events distinctively unlike those that form terrestrial calderas. The close morphologic and geometric resemblance of the quite normal-looking calderas on 

Mars to their terrestrial counterparts (Carr, 1973; 

Pike, 1974c) suggests that such exceptional volcanic 

processes and events, whatever one might imagine 

them to be (Schultz, 1976a), have not formed any conspicuous craters on the Moon.

PREDICTING CRATER GENESIS

Whereas single topographic parameters can distinguish volcanoes as a class from impact craters as a 

class (Guest and Murray, 1969; Murray and Guest, 

1970; Pike, 1972a, b; Oberbeck and others 1972), they 

cannot predict the mode of origin of individual craters 

with any certainty. Even if the lunar raised cones and 

the terrestrial maars and tuff rings are omitted, values 

of the first four morphologic variables (table 2)—taken 

one at a time—simply do not divide craters completely 

into two genetic groups (the last three variables are 

even poorer discriminants). Width/diameter data are 

only just mutually exclusive for the two main groups of 

craters (at an untransformed value of 0.49). Because 

the width of the rim flank often is difficult to measure 

accurately and can be in serious error when obtained 

from photointerpretation alone, especially of spacecraft 

images, a wider separation is needed. For all other variables, the two categories overlap to some extent. Although a height/depth value of about 0.93 would correctly separate most craters into either volcanoes or 

impact craters, rim height possibly is the least reliable 

of all crater measurements and should not be relied 

upon as the sole indicator of origin. Height/diameter 

(at a value of 0.053) would be an even less effective 

predictor of crater origin. Finally, the circularity data 

overlap so badly (centered at a value of 0.71) that this 

often-used parameter must not be regarded as a reliable genetic discriminant for individual craters. The 

considerable overlap explains the only moderate score 

for circularity on the first principal component (table 4) 

and the existence of the fourth component.

The chances of misclassifying an object decrease substantially as the number of taxonomic criteria increases and description of the object becomes more 

complete (Sokal and Sneath, 1963). The strength of the 

multivariate approach, when properly implemented, 

for classifying craters lies in its ability to combine several variables in a parsimonious statistical model that 

automatically emphasizes the most important properties present in the descriptive data. The adoption of 

shared attributes of crater geometry as a topographic 

signature eliminates the reliance on individual 

criteria, which are especially vulnerable to additional 

uncertainties resulting from unavoidable errors in 

measurement. Such errors can arise easily when the 

only sources of data are degraded spacecraft pictures. 

The results of this experiment suggest a method for 

determining or at least narrowing the alternatives for 

the genesis of any large (over 5 km across), rimmed 

extraterrestrial crater on the basis of gross 

morphology. Five kilometers is the size of the largest 

terrestrial maar and a quite conservative cutoff diameter. As the original classification was based on the 

principle of indeterminacy, the genetic criterion too, 

must be stochastic. A suitable set of topographic measurements, preferably no fewer than the number used 

here, can be subjected to a multivariate analysis together with data for a representative sample of terrestrial craters whose origin is known. The origin of a 

given ("unknown") crater reflects that of the other 

("known") craters in its cluster. Although this method 

appears to lack the exactness of the simple graphical 

criteria proposed previously (Pike, 1972b; Oberbeck 

and others, 1972; Smith, 1973), the chances of misclassifying craters are considerably less. The present 

investigation has developed three simple statistical 

models, illustrated in figures 2 and 3 and in plate 1, by 

which large craters may be tested for geometric 

similitude. Discriminant analysis—a multivariate 

technique by which "unknown" objects are assigned to 

one of several predetermined classes of "known" objects 

(Chayes and Velde, 1965; Griffiths, 1966)—offers yet a 

fourth approach for comparing a crater with craters 

belonging to the major types recognized here. Numerical criteria for determining the genesis of craters less 

than 5 km across using landform data continue to remain elusive, however, and may even prove to lie beyond the information content of the five basic topographic characteristics and their ratios.

THE LINNE CONTROVERSY RESOLVED

Apollo 15 pictures have both ended a century of controversy over the genesis of the lunar crater Linne 

(Pike, 1973b) and given us a new morphologic standard 

for simple impact craters on the Moon (fig. 5). Conflict-
GEOMETRIC INTERPRETATION OF LUNAR CRATERS C19

FIGURE 5.—Oblique view of Linne, unusually fresh example of simple impact crater of lunar main sequence, located in northern Mare 

Serenitatis. Diameter of rim crest is 2,450 m. Note steep interior walls, large blocks of ejecta on rim, dunelike textures on flank, and 

secondary-impact craters about 1.3 crater radii from rim crest. Compare with much larger fresh crater Aristarchus (fig. 27), which has a 

complex morphology. View is to north; sun is to right. Apollo 15 panoramic camera photograph 9353.

ing visual observations of Linne, a small crater in 

western Mare Serenitatis, were responsible for rekindling interest in lunar research during the late 

nineteenth century and for encouraging belief in recent volcanic eruption on the Moon. Since its initial 

recognition as a crater in 1823, Linne had been described as a well-formed deep crater between 6 and 11 

km in diameter. In 1866, the astronomer Julius 

Schmidt announced that the crater had vanished and 

had been replaced by a bright patch surrounding a 

craterlet only about 500 m across. Reports of still other 

changes followed, and Linne became one of the most 

famous "problem" craters on the Moon. Further details 

of the ensuing controversy and references to the vast 

literature on this enigma, which accumulated as the 

misunderstandings were clarified, are available 

elsewhere (Ashbrook, 1960,1963b; Moore, 1965; Moore 

and Cattermole, 1967).

Apollo 15 photographs reveal Linne as an extremely 

fresh but otherwide quite ordinary impact crater (fig. 

5). Two topographic profiles across Linne and a detailed elevation profile on the rim crest from Apollo 15 

mapping-camera frames 0407 and 0408 (fig. 6) furnish 

the following measurements: rim-crest diameter, Dr, 

2,450 m; width of the rim flank, We , 650 m; depth, Rt , 

600 m; height of rim crest above outer surface datum, 

Re , 125 m; index of rim-crest circularity, C, 0.89. Although Ashbrook (1963a) obtained good values for 

depth and height visually, at the telescope, his estimate of diameter was low by a factor of 2. According to 

the new measurements, the first four of which are averages, the crater interior lies far below the exterior 

datum; the outer rim flank is both low and narrow, and 

the rim crest is highly circular. The interior profile is 

approximately parabolic; the inner wall just below the 

rim crest is as steep as 40°, and the slope decreases to
C20 APOLLO 15-17 ORBITAL INVESTIGATIONS

1440 1413 

1436 ^-~~ 

1418/' 

1392/ 

1435' 1 440.' 

1458-' 

1435 I

1436- 

1 448 \

Linne 1451 ^._ 

1453

1000 2000 3000 METERS

1443

1409 

"s1445

M485

'1458 

/1462

.^-'1424 

1414 

1392

FIGURE 6.—Topographic profiles and rim-crest outline of crater Linne. Profiles dashed where shadowed. No vertical exaggeration; north 

approximate only. Numbers on rim-crest profile are elevations in meters above arbitrary lunar datum. Data from photogrammetry of 

Apollo 15 mapping-camera frames 0407 and 0408 by Raymond Jordan, U.S. Geological Survey.

between 25° and 30° about 250 m below the rim. From 

comparison with shapes of several different types of 

lunar and terrestrial impact-explosion and volcanic 

craters (Pike, 1972b), the geometric signature of Linne 

clearly is that of a very young impact crater, intermediate in size between Meteor Crater (Arizona) and 

the New Quebec (Canada) crater. It falls into subcluster 1C in plate 1 along with both terrestrial meteorite 

craters.

Linne is perhaps the freshest lunar crater of its size 

yet photographed in such detail (fig. 7); superposed 

craters are both scarce and small. Linne has all the 

features diagnostic of a primary impact crater. Below a 

crisp and almost circular rim crest lies a hummocky 

outer rim surface, thickly strewn with boulders up to 

35 m across. Boulders distributed singly and in roughly 

radial rays persist well beyond the topographic break 

in slope at the base of the rim flank. The surrounding 

bright patch, seen as coalescing rays on smaller scale 

high-sun photographs (Pike, 1973b), resolves into an 

annulus of superbly preserved dunelike structures, 

formed as ejecta swept out from the impact focus. This 

deposit may be best interpreted as "ground-surge" 

(Sparks and Walker, 1973). Radially outward, the 

morphologic character of Linne's ejecta changes from a 

coarse pattern of braided hummocks to finer, individually resolvable transverse dunes, and finally to clusters and chains of secondary-impact craters. Relative 

spectral reflectivity data also show that Linne is very 

young (McCord and Adams, 1973). The spectral 

signature of Linne resembles that of (crushed) young

mare rock (Apollo 12 site) better than it does the 

signatures of upland or older mare rock.

Linne is a simple—as opposed to a complex—lunar 

crater (Quaide and others, 1965; Pike, 1967), in which 

terraces, central peaks, and a flat floor are characteristically absent. Moreover, little modification appears to 

have taken place within the crater since the initial 

impact (fig. 7). Linne's inner walls vary in detailed 

morphology with distance below the rim crest. About 

50 m from the top a discontinuous narrow band of 

shadow around the visible part of the inner rim suggests an outcrop of relatively resistant material, which 

might be interpreted as the top of structurally uplifted 

(bedrock) material underlying the ejecta. Below this 

possible outcrop, and extending about a third of the 

way down the inner bowl, the surface is scarred by 

numerous subradial grooves. Their crudely dendritic 

pattern suggests shallow debris channels. The many 

blocks and the deposits of finer loose material present 

below the grooved zone may indeed indicate mass wasting of the steeper slope above. Radial streaks arid 

bands, which are most conspicuous in the loose material lying below the eroded, fluted zone, may reflect 

mass-wasted deposits of different ages and textures.

The detailed geometry and morphology of Linne constitute as compelling an argument for primary impact 

genesis as occurs anywhere on the Moon. There is no 

evidence that this classic impact crater has ever 

changed substantially in shape. Totally absent is the 

rim asymmetry or concentricity which would be expected if, by some bizarre coincidence, a second impact
GEOMETRIC INTERPRETATION OF LUNAR CRATERS C21

FIGURE 7.—Detail of crater Linne; enlargement of central portion of figure 5. Note large blocks on rim and within 

crater, suggestion of stratification on upper rim wall, scarcity of superposed impact craters, radial furrows on 

rim wall, and slump deposit below furrows. Apollo 15 panoramic camera frame 9353.

had been scored directly on the alleged larger crater 

observed before 1866. Nor is there a trace of endogenic 

modification to any part of the crater or its ejecta

facies; the small craters superposed on Linne look like 

impact pits, not volcanic vents. Even more unlikely is 

the possibility that the crater seen before 1866 was a
C22 APOLLO 15-17 ORBITAL INVESTIGATIONS

different small crater in western Mare Serenitatis. 

Evidently, optical effects were responsible for the 

enigmatic changes in Linne (Robinson, 1970), an apt 

testament to the perils of visual lunar observation near 

the resolution limit of Earth-based telescopes and to 

the need for more complete coverage of the Moon by 

data of Apollo quality.

SECONDARY-IMPACT CRATERS

The significance of secondary-impact craters on the 

Moon to the interpretation of mechanics and ballistics 

of large primary craters has been recognized for some 

time (Shoemaker, 1960; Roberts, 1964; Oberbeck and 

Morrison, 1973), but detailed shape properties of the 

satellitic craters themselves have been accorded little 

attention. Apollo photogrammetry now has made it 

possible to draw some quantitative distinctions between primary and secondary craters on the Moon. Although the morphologic characteristics of the two types 

of craters have been contrasted (Shoemaker, 1962), the 

small size of most secondaries and the resulting absence of topographic data have precluded a more exact 

description of their shape. The 29 secondaries that 

were measured for this report provide a fair first sampling of satellitic craters in the 450 to 4,000-mdiameter range. Larger older secondaries, particularly 

those surrounding the Imbrium and Orientale basins 

(Wilhelms, 1976; Oberbeck and Morrison, 1976), are 

also common but were not included in this preliminary 

study. Since the work was completed, I have examined 

larger secondary craters (Pike and Wilhelms, 1978). 

The 29 craters used here all are of Copernican age, as 

nearly as can be determined, and are mostly from Aristarchus (fig. 8) and Catena Davy (fig. 9); a few are from 

Copernicus and elsewhere (table 9). Many of the craters are clustered and share portions of their rims with 

adjoining secondaries, although others are single (but 

not isolated) craters.

The contrast in shape between most impact primaries and most secondaries is clearly evident in averaged values of five ratios of crater dimensions, as 

shown below for the 29 secondaries and for 44 primary 

craters in the same diameter size range.

Descriptive variable

Height-depth, ReIRt _ _ _ Width/diameter, WeIDr _

Depth/diameter, Rt/Dr

Primaries Secondaries 

(n=44) (n=29)

_ _ 0.81 0.57

_ - _ .19 .22

_ _ .26 .26

._ ___ .04 .02

.19 .13

Mean circularity value for the primaries is 0.81, 

whereas it is only 0.57 for the secondaries. The latter

FIGURE 8.—Nearly vertical view of inverted V-shaped cluster of 

secondary-impact craters in Oceanus Procellarum; formed by 

material ejected from crater Aristarchus to the south. Legs of V are 

4 km apart; north is up. Sun is to right. Compare with craters in 

figure 9. Apollo 15 panoramic camera photograph 0320. Eleven 

largest craters were measured on a 1 : 10,000 topographic map 

compiled from this picture and from frame 0325.

figure is too high, moreover, because the most complex 

and irregular secondary craters are difficult to measure and hence are underrepresented in the sample. 

Rim height is another rough indicator of crater genesis 

(Pike, 1972b). According to the new data shown above, 

the raised rims of primary craters are about twice as 

high above the surrounding terrain as those of the 

satellitic craters. Aside from about a 50 percent greater 

depth, however, primary craters are rather like secondary craters with respect to the remaining characteristics tabulated above: both the height/depth ratio and 

the relative width of the rim flank are about the same. 

Further examination of the data on secondary-impact 

craters in table 9 at the back of this report reveals that 

the 11 Aristarchus secondaries exhibit systematically 

shallower depths and greater rim heights relative to 

the other larger secondaries in the sample (relation not 

illustrated here). The explanation for this curious difference, which does not appear to reflect either map 

accuracy or relative crater age, may lie in the fact that 

the Aristarchus secondaries (fig. 8) all are clustered in 

a V-shaped pattern, whereas the larger craters are 

single or arrayed in a chain (fig. 9). If the Catena Davy 

craters are Orientale secondaries, they would also have 

formed in higher velocity impacts. This possible 

morphologic difference among secondaries may depend 

upon the angle of impact, the distance from the primary crater, the shape of the projectile, and the nature 

of the ejected material; it should be examined subsequently in greater detail. 

The ability to distinguish primary craters from sec-
GEOMETRIC INTERPRETATION OF LUNAR CRATERS C23

FIGURE 9.—Vertical view of Catena Davy, a chain of about two dozen small craters (mostly on floor of large flooded crater Davy Y) that probably 

formed as a result of secondary impact from large and distant primary crater to the west (unknown but possibly Orientale). These craters 

also have been interpreted as volcanic vents (Schultz, 1976a). Chain is about 48 km long. Sun is to right. Compare with figure 8. Apollo 16 

mapping-camera photograph 2198. Nine of the larger craters were measured for this study on 1 : 50,000 topographic maps compiled from 

Apollo 16 panoramic camera photographs.

ondaries is useful in assessing the extent and volume of 

ejecta from lunar impact basins and perhaps the original lengths of ray systems from the basins. Where the 

secondary craters are alined or clustered, telling them 

apart from primaries is not difficult, especially when 

the craters are all fresh. However, the best topographic 

criteria in attempting this distinction for individual 

craters, circularity and rim height, become less effective as both types of craters are degraded with time. 

Because rim height is difficult to measure for a small 

fresh crater under even the most favorable circumstances, rim-crest circularity may remain the best clue 

to the origin of an older small isolated crater on the 

Moon. With appropriate care and allowing for filling of 

a crater with time, the depth-to-diameter ratio also 

might be helpful in deciding whether a particular crater is of primary or of secondary origin.

LUNAR CENTRAL VOLCANOES

The frequency of genuine volcanic landforms on the 

Moon still remains somewhat indefinite (Chapman and 

others, 1970; Smith, 1973; Fielder and Wilson, 1975; 

Guest and Murray, 1976; Head, 1976a; Schultz, 1976a). 

Because lunar volcanic craters are small in size (those 

measured here do not exceed 3 km in rim diameter) 

and evidently few in number, almost no data on their 

shape have been made available. For the first time 

enough topographic information has been generated 

for a preliminary quantitative assessment of the 

morphology of several types of likely volcanoes. The 

lunar volcanoes for which measurements could be

made from Apollo 15-17 data occur on mare surfaces 

and on the flat floors of a few large craters, the two 

main loci of endogenic activity on the Moon. Cratered 

volcanic edifices on the Moon fall into at least three 

distinct categories: domes, cones, and dark-halo rille 

craters (figs. 10-13). The dimensions of each feature 

are listed in the back of this report, and the averaged 

geometric properties of the three types are summarized 

in table 2. As the data are few and sometimes of doubtful reliability, I have made little attempt to distinguish 

morphologic subtypes within any of the three major 

classes of lunar volcanoes. Other kinds of volcanic 

edifices familiar from terrestrial experience, such as 

stratocones with a summit crater or caldera, have not 

been identified with certainty on the Moon. Although 

the peculiar crater Hyginus and the chain of smaller 

pits along the adjacent Rima Hyginus probably are endogenic, little effusive activity appears to have occurred (raised rims are either absent or exceedingly 

low), and the craters all may be largely collapse pits 

and not true volcanoes (Pike, 1976a). Other smaller 

types of volcanoes or pseudovolcanic features may exist 

on the mare surfaces (Greeley and Schultz, 1975: 

Schultz and others, 1976), but measurements are not 

yet available.

CRATERED DOMES

The 15 cratered domes are a small but probably representative sampling of the many similar features that 

have been recognized and mapped on the Moon, 

usually on mare surfaces (Jamieson and Rae, 1965;
C24 APOLLO 15-17 ORBITAL INVESTIGATIONS

Smith, 1973). Coverage of domes by Apollo data is poor; 

only four of the fifteen domes listed in table 9 are 

measureable on the new photographs and maps. 

Named domes include Herodotus Omega, Hortensius 

Sigma, Hortensius Phi, Hortensius Omega, Kies 

Alpha, and Cauchy Omega. Two unnamed domes are 

located near the crater Maraldi B (fig. 10), one north of 

Rima Aristarchus VIII, one west of the crater Milichius 

(fig. 11), another in the Marius Hills, and three west of 

the crater Fra Mauro. An enigmatic and very shallow 

D-shaped depression situated in a patch of mare material between the Montes Haemus and Apenninus 

(Whitaker, 1972) has been interpreted as a caldera 

(El-Baz, 1973); its overall form is that of a very low 

dome or shield. The domes west of Fra Mauro crater

FIGURE 10.—Vertical view of two small cratered domes in northern 

Mare Tranquillitatis, just west of crater Maraldi B. Summit 

craterlets are about 7 km apart. North is up. Apollo 17 mappingcamera photograph 0306. Sun is to right.

FIGURE 11.—Oblique view to northwest of cratered dome on mare 

surface between craters Milichius and Milichius A (near Kepler). 

Dome is nearly 12 km across at base but only 250 m high. Apollo 

12 photographs 51-7552 (Hasselblad camera). Sun is to right.

conceivably could be only upland hills that have been 

cratered by chance impacts (Smith, 1973). All domes 

appear at very low sun angles as low flattish hills, 

roughly circular to oval in outline with convex margins, and display a more or less centrally disposed, rimless depression at the summit (Figs. 10, 11). A rough 

model of surface form, comprising values of the arithmetic mean and standard deviation for five topographic 

variables, is shown below for the 15 lunar domes (see 

also table 2):

Descriptive variable Mean

Standard 

deviation

Rim-crest circularity, C 

Crater diameter, Dr (m) 

Crater depth, Ri (m)__ 

Dome height, Re (m) __ 

Flank width, We (m)__

_ 0.58 0.20

.1525 640

_ 160 76

_ 350 175

_3525 1750

The more complex, perhaps two-stage lunar domes that 

are found in the Marius Hills region on the Moon 

(McCauley, 1965, 1968) are not represented in the 

sample. The simpler domes treated here usually are 

arranged in groups, although isolated domes (Kies 

Alpha) are known, and may exhibit some tendency toward alinement along trends of the lunar grid system. 

Two types of simple dome may exist on the Moon. The 

larger flat cratered domes, which are restricted in location to the mare surfaces exclusively, are the kind 

sampled here. A smaller relatively steeper dome that 

may occur on the floors of some craters (Copernicus) 

can be confused with cratered cones.

Although it is generally agreed that the lunar domes 

formed as the result of some type of endogenic activity, 

several alternative hypotheses have been advanced 

and the exact mechanism remains in doubt (Baldwin, 

1963). The well-established mafic character of the 

lunar mare rocks suggests that the flat cratered domes 

considered here could be shield volcanoes of the 

Hawaiian type (Pickering, 1906; Baldwin, 1949), and 

their overall geometry lends some support to this interpretation (table 2). However, lunar domes differ 

from terrestrial shield volcanoes in several respects. 

The lunar domes are a good deal smaller than most of 

their alleged terrestrial counterparts, the depth of the 

summit crater is a larger fraction of the height of the 

edifice, and the lunar summit craters are not the intersecting and stepped, shallow, pan-shaped depressions 

commonly found on terrestrial shields. In some ways, 

the lunar domes are more like the smaller terrestrial 

lava shields found in the Hawaiian Islands, the western United States especially in southern Idaho and 

northern California, and Iceland. Although lunar 

domes usually are quite flat and do not seem to have 

the conical profile of the classic Icelandic-type lava 

shields, to which they are often similar in size, the
GEOMETRIC INTERPRETATION OF LUNAR CRATERS C25

overall geometry of these two types of features is 

somewhat more comparable (table 2). The shapes of the 

summit craters in particular are more alike.

The data presented here tentatively support the 

analogy first drawn by Rae (1966) and later adopted by 

Head (1976a) that the lunar mare domes are small flat 

shield volcanoes, more akin to small terrestrial lava 

shields with simple craters than to Hawaiian-type volcanic piles with complex summit craters (see also Pike, 

1974c). The exceedingly low relief of the lunar domes 

relative to their areal extent, together with their flat 

configuration, suggests that the constituent rocks were 

once highly fluid lava, low in volatile content, emitted 

in quiet central eruptions with virtually no intercalated pyroclastic material. The lunar mare basalts, 

which approximately resemble terrestrial tholeiite 

basalt that is highly enriched in titanium, would have 

been emplaced in this fashion—almost certainly they 

are the materials of which the simpler lunar domes 

were built. The similarity of the domes to the surrounding mare surfaces with respect to albedo and colorimetric properties supports this speculation.

CRATERED. CONES

The 14 cratered raised cones are a good sample of a 

rare lunar feature that has received less attention than 

the mare domes (Moore, 1971; Mattingly and others, 

1972; El-Baz, 1972; Scott, 1973; Scott and Eggleton, 

1973). Coverage of cones by Apollo is better than that 

for domes, half of the sampled ones being measureable 

on the new photographs and maps. The cones include 

two in southern Mare Serenitatis, three on the floor of 

King crater and four on the floor of Copernicus, a single 

cone west of the crater Lassell (fig. 12), two north of the 

Aristarchus Plateau, and one located in the Marius 

Hills. The examples within Copernicus conveivably 

may be only hillocks cratered by fortuitous impacts 

(Howard and Wilshire, 1975). No measurements were 

available for any of the alleged "cinder cones" near the 

Apollo 17 site (El-Baz and Worden, 1972). A preliminary model of shape, comprising values of the arithmetic mean and standard deviation for five topographic 

variables, is given below for the 14 raised cones (see 

also table 2):

Descriptive variable

Rim-crest circularity, C

Crater diameter, Dr (m)

Flank width, We (m)_ __ __

Mean

0.55

___ _ 715

75

. _ _120

440

Standard 

deviation

0.16

540

65

45

150

Lunar cones resemble mare domes in that the bottoms 

of their summit craters tend to lie at or above the surFIGURE 12.—Oblique view to south of cratered cone in Mare Nubium 

between craters Lassell H and Lassell J. Cone, which is darker 

than surrounding mare surface, measures about 2,500 m across at 

its base. Apollo 16 Hasselblad photograph 19237. Sun is to left.

rounding datum, their crater rim crests are quite acircular, and their height/diameter and depth/diameter 

aspects are similar (table 2); none of these characteristics is typical of impact craters. The summit craters of 

domes are about three times the diameter of craters 

atop cones. Moreover, cratered cones have much 

steeper and narrower rim flanks than lunar domes, and 

unlike domes, the lunar cones are found on the flat 

floors of a few large fresh impact craters (Copernicus, 

King) as well as on mare surfaces. Cones also tend to be 

disposed in clusters or alinements; isolated cones (Lassell H) are exceedingly rare. Cratered cones tend to have 

somewhat more subdued morphologies than domes; although cones may be complete and well defined, more 

often they are breached and highly irregular.

Like mare domes, the cratered lunar cones are directly comparable in both size and shape with specific 

types of terrestrial volcanoes. Scott (1973) draws an 

analogy between cones in southern Mare Serenitatis 

and terrestrial cinder cones. His interpretation is supported by the topographic data presented here (table 9), 

except perhaps that terrestrial cinder cones tend to be 

somewhat higher, deeper, and more circular than their 

possible counterparts on the Moon (table 2). These differences might be ascribed to erosion of the lunar 

cones. The rim flanks of lunar cones also tend to be 

slightly narrower than the flanks of the terrestrial cinder cones (table 2), in direct contradiction of calculations predicting the widespread distribution of ejecta 

around exceedingly low lunar cinder cones (Wright and 

others, 1963; McGetchin and Head, 1973). However, 

the observations of "cinder cones" in the TaurusLittrow area that precipitated these calculations (ElBaz and Worden, 1972) have not yet been verified by 

detailed measurements of the alleged volcanoes. Two of
C26 APOLLO 15-17 ORBITAL INVESTIGATIONS

these craters, Shorty and Van Serg at the Apollo 17 

site, were later interpreted as impact craters 

(Muehlberger and others, 1973). If in fact the lunar 

cones described here are pyroclastic volcanoes, then 

the high rim flanks may have to be explained by taking 

into account still other differences between lunar and 

terrestrial eruptive environments. Again, as in the 

case of the mare domes, if the lunar magmas were exceedingly low in volatiles and the resulting eruptive 

activity was correspondingly less explosive, then 

perhaps the ejected material would not travel as far 

from the vent as it would from a comparable eruption 

on Earth. All in all, minor pyroclastic eruptions on the 

Earth and Moon may have yielded fairly comparable 

landforms despite certain and likely differences in the 

two physical environments.

DARK-HALO CRATERS

Topographic measurements could be obtained for 

only three dark-halo craters, Alphonsus KG, Alphonsus MD, and a nearby unnamed crater—all from 

Apollo data on a 1:50,000 map (fig. 13). The 

morphology of these craters, which are situated along 

Rima Alphonsus I on the northeastern part of the flat 

floor in Alphonsus crater, has been described in detail 

by Kuiper, Strom, and LePoole (1966) and by Carr 

(1969). Although this small sample is an inadequate 

basis for drawing any firm genetic conclusion (no 

geometric model is give here), the gross geometric 

characteristics of the three craters do not vary much 

(table 9). The dark-halo craters are not raised cones 

(Schultz, 1976a) but rather share some geometric 

similarities with both lunar secondary-impact craters

I.

FIGURE 13.—Oblique view of three dark-halo craters, Alphonsus KG 

(left arrow), Alphonsus MD (right arrow), and unnamed depression 

(middle arrow) situated along rilles on floor of large impact crater 

Alphonsus. View is to southwest. Two named craters are 15 km 

apart. Apollo 16 mapping-camera photograph 2477. Sun is to lower 

left.

and with terrestrial maars: low rim-crest circularity, 

low rims, and low depth/diameter ratio. Their occurrences only along rilles and their wide, dark rim flanks 

distinguish them readily from lunar secondaries. 

Hartmann (1967) and Peterfreund (1976) estimated 

rim and bowl volumes and found that the rim deposits 

consistently occupy greater volumes than the craters, 

but preliminary calculations from the new map data 

suggest rather the reverse. Bowl volumes average 

about one and a half times the rim volumes. Clearly, 

good data are needed on the geometry of other craters 

within Alphonsus.

Carr (1969) regarded the dark-halo craters in Alphonsus as volcanoes but did not make a more specific 

interpretation. Kuiper, Strom, and LePoole (1966) interpreted them as maars. Although these craters are 

quite similar in surface geometry to terrestrial maars, 

there are difficulties with the prevalent maar interpretation because of the absence of abundant water on the 

Moon (Taylor, 1975; Mason and Melson, 1970; Waters 

and Fisher, 1970; Lorenz, 1973). Conceivably, a nonphreatic maar mechanism could apply (Shoemaker, 

1962; Oilier, 1974). The dark-halo craters along the 

rilles in Alphonsus also are larger than typical terrestrial maars, although this difference could have resulted from the difference in gravity or from their 

being enlarged by widening of the rilles. Thus, the 

exact mode of origin—although almost certainly 

endogenic—remains in doubt. It is possible that a 

pyroclastic eruption producing volumetrically less, but 

more widely dispersed, ejecta than lunar cratered 

cones may occur along a few rilles on the Moon, particularly in the case of features—like the Alphonsus 

dark-halo craters—situated near the center of the near 

side (Pike, 1976a). These dark-halo craters could be 

lunar equivalents of terrestrial cinder cones (Peterfreund, 1976). The morphologic differences between 

raised cones and dark-halo craters might have arisen 

from the presence of some fluid lava in the cone eruptions but none in the formation of the dark-halo craters. Alternatively, different magmas may have given 

rise to the two different types of lunar volcanoes.

SMOOTH-RIMMED CRATERS

Craters such as Ritter, Sabine, Herodotus, and Kopff 

have been classified on the 1:5,000,000 geologic map of 

the Moon as a type (Ics) that "may differ sufficiently 

from main-sequence (impact) craters to suggest a different origin" (Wilhelms and McCauley, 1971). These 

craters span an usually narrow diameter range, about 

20 to 45 km. On Lunar Orbiter images, smooth-rimmed 

craters lack the well-developed hummocky texture and 

radial ridges that are characteristic of the rims of other
GEOMETRIC INTERPRETATION OF LUNAR CRATERS C27

craters, and conspicuous ejecta blankets are absent. 

The crater Kopff in Mare Orientale (Lunar Orbiter IV 

frame H 187) is a particularly good example; unfortunately it was not photographed by Apollo. In addition, 

they are comparatively shallow; "floors are commonly 

about level with the surrounding terrain" (Wilhelms 

and McCauley, 1971). Following Morris and Wilhelms 

(1967) and especially McCauley (1968), Wilhelms and 

McCauley suggested a caldera genesis for these craters, an interpretation that seems to have gained some 

support (Mutch, 1970; DeHon, 1971; Smith and Sanchez, 1973; Short, 1975: Fielder and Wilson, 1975). 

This idea badly needs to be tested. Nineteen smoothrimmed craters are included in the sample compiled 

here, of which five (Langrenus F and B, Yerkes, Lick, 

and Jansen) have been measured from Apollo photogrammetric data, all on LTO's. The remainder include 

Vitello, Kopff, Damoiseau, Herodotus, Sabine, Ritter, 

Grove, Daniell, Gambert, Lassell (fig. 14), McClure D, 

McClure M, Crozier, and Sosigenes. A crude model of 

surface form, comprising values of the arithmetic mean 

and standard deviation for six topographic variables, is 

shown below for these 19 craters (see also table 2).

Descriptive variable

Rim-crest circularity, C ...........

Rim diameter, Dr (km) ...........

Depth, Rf (km) ...................

Rim height, Re (km) ..............

Flank width, We (km) ............

Floor diameter, Df (km) ..........

Mean

....... 0.78

...... .31

....... 1.4

....... .6

....... 4

...... .23

Standard 

deviation

0.06

8

.6

.2

2

7

i

FIGURE 14.—Oblique view of Lassell, a smooth-rimmed crater in 

Mare Nubium. Note shallow floor, absence of radial textures on 

rim flank. Crater is 23 km across. View is to south. Apollo 16 

Hasselblad photograph 19066. Sun is to lower left.

The available data show that smooth-rimmed craters 

do not differ in shape from ordinary main-sequence 

craters or from otherwise normal main-sequence craters that have been flooded with mare material to the 

level of the exterior terrain. This conclusion is evident 

from the multivariate analysis (figs. 2 and 3; pi. 1) as 

well as from percentage-frequency distributions for the 

individual variables circularity, rim height/crater 

depth, and rim width/diameter (fig. 15). The three 

graphs contain representative samples of terrestrial 

volcanic craters, terrestrial meteorite-impact craters 

and explosive craters, and rimmed main-sequence 

lunar craters. The five sample smooth-rimmed craters 

(LTO data) are compared with four extraterrestrial 

cratered edifices (on Mars) that without doubt are true 

caldera volcanoes. According to the three criteria, all 

the smooth-rimmed craters exhibit an impact-crater 

morphology rather than a volcanic-crater shape and 

simply do not have the correct shape to be calderas. 

The somewhat anomalous Jansen is discussed below in 

greater detail. In all three graphs, the overlap between 

volcanic and lunar craters mainly reflects the inclusion 

of 43 maars, many of which tend to resemble lunar 

craters in their geometry. However, the possibility that 

smooth-rimmed craters are colossal lunar maars is remote; these craters are roughly 20 to 30 times larger 

than typical terrestrial maars. On the Moon, where 

unambiguously volcanic craters are of uniformly small 

size (none over 3 km across), the maar interpretation of 

large craters is as tenuous as the caldera hypothesis.

Explanations other than primary volcanic genesis 

will have to be found for the unusual morphologies of 

smooth-rimmed craters. Indeed, perhaps these craters 

should not even be treated uniformly as a single class. 

A modification of the initial impact process by some 

unknown perturbation or else some form of secondary 

postimpact volcanic activity seem to be the most likely 

prospects for interpreting many of these craters (Pike, 

1968; Young, 1972; Brennan, 1975; Schultz, 1976b). 

Possibly the craters all formed by impact, but some of 

the target material, probably mare lavas, had not 

wholly solidified and the characteristic hummocky 

rims and ejecta aprons never developed. Alternatively, 

they may have formed as main-sequence impact craters, but some of the later mare lavas partially inun^ 

dated them, obliterating the ejecta and much of the 

hummocky rim material. Just such flooding is suggested by the typically low narrow rim flanks and the 

wide flat interior floors of most smooth-rimmed craters.

The likely involvement of extrusive volcanism in 

producing the anomalous characteristics of many 

smooth-rimmed craters is strongly evident in Jansen, 

perhaps the most unusual of all craters of this type. 

Contours on LTO sheet 60B-2 (Defense Mapping 

Agency, 1974-1977) show that the floor of the flooded
C28 APOLLO 15-17 ORBITAL INVESTIGATIONS

0.2 0.4 0.6 0.8 

RIM-CREST CIRCULARITY

1.0

70

60

m 50 

O 

CC 111

^ 40

O 30

HI

D

a

m 20

CC

10

B

0.01 0.1 1.0 10 

RIM HEIGHT/DEPTH

100

70

60

LU 50

O 

CC 

LU

^ 40

30

20

10

C

0.01 0.1 1.0 10 100

RIM WIDTH/DIAMETER

FIGURE 15.—Frequency distributions of three descriptive variables 

for 325 craters. Apollo measurements used for smooth-rimmed 

lunar craters Langrenus B, Langrenus F, Lick, Yerkes, and Jansen. Topographic information for martian calderas Olympus Mons, 

Pavonis Mons, Arsia Mons, and Ascraeus Mons from contoured 

editions of Mars charts MC-4 and MC-17. Data from Pike (1972a); 

data used in later analysis (table 2; Pike, 1974c) modified somewhat from those shown here. In each case, curve for volcanic craters differs significantly in shape and location from other two 

curves. Smooth-rimmed lunar craters resemble impact and exploEXPLANATION

Lunar craters of the main sequence, all

ages, n=167 

Terrestrial meteorite-impact and explosive

test craters, n=41 

Terrestrial calderas, tuff cones, maars,

and cinder cones, «=117 

Lunar smooth-rimmed crater

Martian caldera

sion craters, not volcanoes, and probably are impact craters that 

have had an unusual formational or postformational history. A, 

Frequency distribution for index of rim-crest circularity (defined in 

text). B, Frequency distribution for rim height/depth ratio. Displacement of lunar data toward higher ratio values reflects large 

number of older and hence shallow craters included in sample 

(Pike, 197la). C, Frequency distribution of rim width/rim-crest 

diameter ratio. Displacement of lunar data toward lower ratio values reflects diminution of rim width of older lunar craters, or possibly underestimation of rim widths for these objects.
GEOMETRIC INTERPRETATION OF LUNAR CRATERS C29

23-km-diameter crater is elevated about 260 m above 

the main exterior datum. Craters with this odd configuration, the best known of which is Wargentin, are 

exceedingly rare on the Moon. The raised floor within 

Jansen, which otherwise has all the characteristics 

typical of other deeply flooded impact craters, suggests 

that unusual hydrostatic conditions prevailed locally 

during the emplacement of basin-filling lavas in this 

area of Mare Tranquillitatis. Possibly related features 

in the vicinity of Jansen that seem to reflect true volcanic activity include rilles, a rimless boat-shaped depression 450 m deep, and several 5-km-diameter craters that appear to be buried nearly to the rim crest by 

mare materials. Although with respect to depth/ 

height, Jansen and Wargentin superficially resemble 

the lava-filled summit crater of the North Tongariro 

stratocone in New Zealand, the relatively much wider 

flank and higher rim of the terrestrial volcano rule out 

the analogy completely. The data interpreted here 

suggest that neither lunar crater is of primary volcanic 

origin, and neither are any of the other smooth-rimmed 

craters on the Moon. Later contributions to the 

1: 1,000,000 geologic atlas of the Moon (such as 

Hodges, 1973) have abandoned this interpretation entirely.

SUMMIT CRATERS ON CENTRAL PEAKS

The depressions 1 to 5 km across that have long been 

observed at or near the tops of some of the central 

peaks in large lunar craters have been interpreted as 

the summit craters of stratovolcanoes (Moore, 1953), as 

chance impact craters on piles of nonvolcanic material 

(Baldwin, 1963), or as the collapsed summit of a rebound structure (Baldwin, 1963; Schultz, 1976a). Of 

the 40 or more near-side craters on the Moon once believed to contain central peaks with summit craterlets 

(Warner, 1962), six are portrayed on the LTO's: 

Timocharis, Theophilus (fig. 16), Kant, Taruntius, 

Capella, and Langrenus. Several craters on the far 

side, including Tsiolkovskiy and Levi-Civita, are also 

shown on the Apollo pictures (figs. 17 and 18). Although measurements could not be made of these small 

craters, the new large-scale maps enable at least two of 

the three contending interpretations to be examined 

more closely. The collapse hypothesis could not be 

evaluated from the new data.

Inspection of both contoured and uncontoured versions of the LTO's as well as supplemental maps and 

photographs from the Apollo and Lunar Orbiter 

missions, shows not one convincing volcanic crater on 

the summit of any of the central peaks (figs. 16-18). 

The photographs and maps do show numerous irregular clefts and protuberances that easily could have

FIGURE 16.—Oblique view of multiple central peaks within young 

100-km diameter impact crater Theophilus. Note paucity of wellformed small impact craters on sloping mountain flanks. Craters 

are common on level terrain of the floor of Theophilus, around 

peaks. Base of peak complex is about 25 km across. View is to 

southwest. Apollo 16 mapping-camera photograph 0694. Sun is to 

left.

FIGURE 17.—Vertical view of central peak within old upland crater 

Levi-Civita, at 23° S. and 146° W. on far side. Note double craterlet 

at summit. Well-formed superimposed impact craters are rare on 

sloping flanks of peak but common on level surfaces. Base of peak 

is about 20 km across. Apollo 15 mapping-camera photograph 

1019. Sun is to left.
C30 APOLLO 15-17 ORBITAL INVESTIGATIONS

FIGURE 18.—Vertical view of central peaks within very young crater 

Tsiolkovskiy, at 20° S. and 129° E. on far side. Note scarcity of 

well-formed impact craters on sloping mountain flanks. Craters 

are common on level terrain at foot of peaks. Base of peak complex 

is about 40 by 25 km across. Apollo 15 mapping-camera photograph 0889. Sun is to left.

been mistaken for craters at the telescope or on the 

small-scale photographs available when such "summit 

craters" first were recognized. These illustrations also 

show a great many small, often rimmed, pits that appear to be impact craters. Most of the central peaks in 

question have been cratered liberally by small impacts, 

judging from the densities of craters on adjacent level 

surfaces, but preservation of the craters is highly selective. Impacts on the sloping flanks of central peaks 

result in misshapen or barely recognizable shallow 

craters that soon are obliterated by mass wasting. Impact craters formed on the flat floor near the base of the 

peaks and directly on gently sloping summit areas, on 

the other hand, are less distorted and tend to persist for 

a longer time. Impact craters with both degrees of preservation may be seen on central peaks in the craters 

Timocharis, Capella, Kant, and Theophilus. They also 

occur on the peaks of Gassendi, Regiomontanus, Tsiolkovskiy, and other large craters (figs. 16-18).

Regiomontanus A, which is located centrally within 

the old 118-km-diameter crater Regiomontanus, is one 

of the largest and most striking examples of a cratered 

central peak on the Moon. The 5.65-km-diameter crater has been interpreted as a caldera, mainly on the 

strength of Earth-based shadow measurements that 

put the crater depth at something less than 500 m but

the height of the rim above the floor of Regiomontanus 

at about 680 m (Kopal and others, 1965); width of the 

rim flank is about 6 km. A more recent determination 

of the crater's depth that was made from shadow 

lengths on Lunar Orbiter images (Arthur, 1974), however, reveals that Regiomontanus A is 1,200 m deep. 

Unfortunately, this crater was not photographed by 

Apollo (see Lunar Orbiter IV high-resolution frame 

107, framelets 398-400), but for most small craters Arthur's measurements are comparable in accuracy to 

those made later from Apollo photogrammetry (Pike, 

1974b); there is no reason to question the 1,200-m 

figure, which now places the bottom of the crater some 

700 m below the floor of Regiomontanus, not above it. 

The gross morphology of the crater and central peak no 

longer suggests a caldera origin, but rather indicates 

fortuitous superposition of a fresh, comparatively large 

impact crater on the summit of a central peak within a 

large, old lunar crater. This interpretation may hold 

for most other such features on the Moon, although 

other alternatives may better explain the observations 

for some craters (Alien, 1975).

SIZE-DEPENDENCE IN THE SHAPE OF FRESH 

IMPACT CRATERS

Fresh craters on the Moon form a continuous 

morphologic sequence within which they display 

greater complexity with increasing size. A general size 

continuum of lunar crater shapes has been known to 

exist for some time (Gilbert, 1893). Results of the 

Ranger, Orbiter, Surveyor, and Apollo missions have 

since confirmed that the sequence extends over at least 

11 orders of magnitude (Horz and Ronca, 1971;. Most of 

the craters throughout this remarkably broad size 

range, from microscopic pits to the largest basins, 

share some overall characteristics (such as radial 

symmetry, a depressed cavity, and ejecta) and probably 

originated by impact. Variations in 33 morphologic 

properties of craters between 1 and 1,000 km across 

have been reviewed and summarized in considerable 

detail by Howard (1974).

The size-morphology sequence is not perfectly continuous, however, in that craters of adjacent size 

ranges differ significantly with respect to characteristics such as the proportion of depth to diameter and the 

presence or absence of central peaks. Some of the more 

apparent changes of crater shape with size were described and illustrated briefly by Gilbert (1893) for 

craters that could be resolved telescopically from 

Earth. Seventy-five years later, Mackin (1969, p. 743) 

wrote of the same continuum of larger lunar craters, 

which he described aptly as "a single genetic series, 

with simple bowl-shaped impact-explosion craters at 

one end and the largest maria [that is, multiringed
GEOMETRIC INTERPRETATION OF LUNAR CRATERS C31

basins] at the other. There are two breaks in the series, 

both transitional, but both well-defined and meaningful: (a) at a diameter of 20±5 km between craters that 

show no slumping and those bounded by a wreath of 

slumps ***, and (b) between the latter and *** the 

maria [that is, basins]."

The first of these two discontinuities is elucidated 

here; other morphologic thresholds, such as the crucial 

change from large craters to small basins (StuartAlexander and Howard, 1970), interrupt the size continuum of lunar craters but will not be treated further 

in this report.

Fresh impact craters on the Moon exhibit changes in 

no fewer than 11 attributes of shape in the diameter 

range between about 10 and 30 km. At this size there is 

a transition from smaller, simple craters to larger, 

more complex craters. The changes usually are described as being either morphologic (qualitative) or 

geometric or morphometric (quantitative) in character, 

but the distinction is artificial; some are just measured 

more exactly than others (Griffiths, 1960). Four differences in crater shape are documented here at the nominal and interval levels of measurement: the restriction of swirl-textured floors mostly to craters bridging 

the simple-to-complex transition, and presence and 

elaborateness of central peaks, flat interior floors, and 

terraces on the rim wall. Seven size-dependent contrasts in crater geometry are defined at the ratio level 

of measurement (Griffiths, 1960): graphs of rim-crest 

diameter (D,.) against crater depth (Rj ), rim height (Re \ 

flank width (We ), floor diameter (D/), rim-wall slope, 

circularity of the rim crest (C), and (profile) evenness of 

the rim crest (Vc ) all inflect at values of rim diameter 

ranging from about 10 to about 30 km. The first four 

ratio relations are logarithmic and the last three 

semilogarithmic. The last five relations are evident 

only from the new topographic data available from the 

Apollo missions. Three quantities for which an inflection with increasing crater size has not yet been found 

are width of continuous ejecta, length of rays, and 

perhaps relative height of the central peak.

The value of average crater diameter at which the 11 

changes in shape occur is 17.5 km, in excellent agreement with earlier estimates of the threshold size 

(Baldwin, 1968b; Pike, 1972b), although it is more 

realistic to think in terms of a diameter range (about 

10-20 km) because the scatter of data points in the 

depth/diameter and other diagrams can be quite severe. Moreover, flat floors and other features do not 

appear all at once in craters exactly 17.5 km across. 

The seven ratio-level changes occur over an interval of 

about 10 km, and the four other changes are transitional over a diameter interval of about 40 km. The 

seven inflections in plotted distributions differ in

crispness: those of diameter with depth, rim height, 

and slope are quite obvious, but the remaining four are 

more subtle. Analogous size-morphology thresholds 

that separate simple from complex craters have been 

identified in terrestrial impact structures (Dence, 

1964, 1968; Lozej and Beales, 1975) and are evident in 

craters on Mercury (Gault, Guest, and others, 1975) 

and Mars as well (Pike, 197la; Hartmann, 1972b).

RATIO-LEVEL VARIATIONS

The relation of depth and other geometric attributes 

of lunar craters to their overall size, as expressed by 

the rim-crest diameter or radius, can best be ascertained on bivariate graphs with both quantities plotted 

as logarithms (Baldwin, 1949; 1963; Pike, 1968, 

1972b). The high quality of the Apollo photogrammetric data enable depth/diameter and other plots to be 

determined with more accuracy for craters between 

about 400 m and 300 km across than they could from 

the old shadow-length data. Fresh craters, Copernican 

and Eratosthenian in age as mapped by the U.S. 

Geological Survey, predominate in all graphs so as to 

avoid contaminating resulting trends with the topographic effects of crater degradation. The largest craters in the sample, such as Gagarin and Mendeleev, are 

substantially older, but they also are so large that their 

gross form may not have changed very much since 

their formation; no harm is done by including them. 

According to the new data, seven geometric properties 

of fresh lunar craters exhibit some sort of change or 

inflection at a crater diameter of about 10 to 30 km: 

depth, rim height, width of rim flank, slope angle of the 

rim wall, diameter of the flat floor, circularity of the 

rim crest, and evenness of the rim crest.

Seven of the 10 ratio-level relations of crater rim 

diameter to other dimensions are linear regressions of 

the form log y = log b + a logx, with four of them— 

depth/diameter, height/diameter, width/diameter, and 

rim diameter/floor diameter—requiring two such expressions. This form does not apply to the 

semilogarithmic plots of rim slope, circularity, and 

evenness against diameter, although the constituents 

of rim slope (crater depth and rim-wall width) do plot 

exponentially as two linear fits. Calculated leastsquares fits to data in the graphs are presented where 

applicable. All equations written for the new data are 

summarized in table 6, which gives values of a (slope) 

and b (intercept on the y axis at 1-km rim diameter) for 

11 exponential equations (variables in kilometers), 

along with the sample size and the standard error of 

estimate of y on x (x always being the rim diameter). 

Because these fits are statistical relations rather than 

physical laws, the y value calculated for a particular x
C32 APOLLO 15-17 ORBITAL INVESTIGATIONS

TABLE 6.—Numerical data for geometric model of fresh lunar craters

Relation

Width/diameter .

Floor diameter/

Width continuous

Relief of central

Length (radius) of

Intercept 

Size at 1-km Standard 

range diameter error 

Symbol (km) N Slope (km) (km) Source

_ _ __R,/Dr <15 171

>15 33 

JteIDT <15 124

>15 38 

„. _ ,WfIDT <15 117

>15 46

>20 53 

———W^IDr All 84

-Rn/Dr >27 22

________Rr/Dr All 50

1.010 

.301 

1.014 

.399 

1.011 

.836

1.765 

1.249

1.001 

.900 

1.25

0.196 

1.044 

.036 

.236 

.257 

.467

.031

.187

.674 

.032 

4.41

+0.038 (') 

-.027 

+.067 (') 

-.063 

+ .0075 

-.0062 

+ .036 

-.031 

+ .055 

-.046 

+ .081 

-.069

+.011 

-.009 

+.012 

-.011

-.162 O 

-.157

+ .011 

-.008

+2.94 (3 ) 

-1.76

'Pike (1974b).

2Moore and others (1974).

"Moore and others (1974); data from Baldwin (1963), table 32.

from any of the equations is not unique but only lies 

within a range of probable values. The dependability of 

each estimate of y varies according to the severity of 

scatter in the original data. To provide some measure 

of this variability to users of the equations, the standard error (Croxton and others, 1967, p. 395) was calculated for each fit.

The new equations (table 6) constitute fairly complete models, or average representations, of the 

geometry of both simple and complex fresh craters on 

the Moon that may be helpful in estimating the initial 

dimensions of multi-ring basins and large craters as 

well as the volume and distribution of their ejecta. 

They also serve as standards of comparison for the 

shapes of craters on Earth, Mars, Mercury, and other 

planets and their satellites. For some purposes, it will 

be found more convenient to convert crater diameter to 

radius (Quaide and others, 1965; McGetchin and 

others, 1973; Pike, 1974a). The equations supplant and 

revise older models of crater shape (Baldwin, 1963; 

Pike, 1972b) and should not change very much with 

additional data. The seven size-dependent changes in 

crater geometry indicate a qualitative increase in the 

complexity of fresh lunar craters that correlates with 

other types of size-dependent changes at the threshold 

diameter of about 10 to 20 km. An interpretation of the 

geometric changes in terms of dimensional analysis 

and allometry has appeared elsewhere (Pike, 1967; 

1972b).

DEPTH

The relation of depth to diameter for craters on the 

Moon has a long history, reviewed in detail elsewhere

(Baldwin, 1949; Pike, 1968). Previous work, commencing recently with that of Baldwin (1963, p. 141-142) 

and Quaide and others (1965), but actually going back 

to Ebert (1889), has shown statistically that lunar 

craters less than 20 to 30 km across are significantly 

deeper per unit diameter than are larger craters. Using 

more abundant and refined measurements than had 

been available before, I have located the inflection at a 

crater diameter near 15 km and fit separate linear 

equations to the data for craters both above and below 

the size threshold (Pike, 1967). The depth/diameter 

plot recently was revised (Pike, 1974b) from Apollo 

photogrammetric data for 204 craters between 60 m 

and 275 km across. A similar graph—with eight additional craters (fig. 19)—shows a sharp diminution in 

the rate of increase of depth as crater diameter increases beyond 20 km.

The distribution of 171 craters smaller than about 15 

km across (excluding the eight new craters) is described by the expression

Rt = 0.196 D, 1.010 (1)

where Ri is crater depth (km) andDr is rim-crest diameter (km). Equation (1), a regression line that was calculated by the method of least squares, applies equally 

to fresh mare and to fresh upland craters. The eight 

added craters follow this trend. The depth/diameter 

distribution of the 33 craters over about 15 km in diameter follows the least-squares expression

R. = 1.044 D 0.301 (2)

Large mare craters in figure 19 do not appear to differ 

systematically from large upland craters. The two 

linear fits intersect at a crater diameter of about 10.6 

km. Equation (1), which resembles depth/diameter expressions for experimental-explosion and meteoriteimpact craters on Earth, reveals that small lunar craters are about 50 percent deeper than older telescopic 

data showed them to be. The difference between old 

and new depth data is much less for the larger craters. 

The new depth/diameter equations have been compared with earlier expressions and discussed in greater 

detail elsewhere (Pike, 1974b).

The two subgroups of fresh craters in figure 19 overlap somewhat between 12 km and about 25 km in diameter. First noticed independently in 1965 by both R. 

B. Baldwin and myself, the trend established for 

smaller craters—here, equation (1)—extends well beyond the inflection diameter of about 11 km (see also 

David, 1975). The 13 largest craters that are described 

by equation (1) are Proclus, Alfraganus, Macrobius A
GEOMETRIC INTERPRETATION OF LUNAR CRATERS C33

10

t/3

£

LU

I

t 

UJ 

Q

DC 

UJ

<

o

0.01

00 -Po

•o b

S:o*

• • O O

O. .CD 

Of ;* o

o 

o o

EXPLANATION

O Upland crater, n=74 

• Mare crater, «= 138

0.05 0.1 1 10 100 500

RIM-CREST DIAMETER.IN KILOMETERS

FIGURE 19.—Relation between depth, Rt , and diameter, Dr, for 204 fresh lunar craters from Pike (1974b) and eight additional craters. All 

data from maps and topographic profiles compiled photogrammetrically from Apollo 15, 16, and 17 photographs. Distribution inflects 

sharply at a diameter of about 15 km, marking transition from simple to complex, or modified, craters on the Moon. This 15-km inflection 

appears in several of the following figures. Simple craters extend beyond transition diameter (see text).

and B, Kastner F, Diophantus, Dario, Vitruvius A, 

Dubyago S, Isidorus D, Li Po, Benedict, and Langrenus 

C. All of these anomalous craters are exceptionally 

deep for their size (see table 9). Only one, Diophantus, 

has a central mound, and only Proclus has rim-wall 

terraces; in neither crater are these features well developed. The flat interior floors are quite small, especially in those craters under 18 km across. Clearly 

these are simple, rather than complex craters, despite 

their comparatively large size. It is not clear why these 

13 craters and others like them have turned out differently from most lunar craters in the 13- to 28-km size 

range. About as many of them have formed on mare or 

upland-plains surfaces as have formed on uneven 

upland terrain, and so geographical location—and any 

material strength or other contrasts related to mareupland differences—evidently was not a factor. Probably the anomalously deep craters reflect chance variations in the transition from simple to complex 

morphologies on the Moon, rather than any systematic 

physical influence on crater shape or internal structure.

RIM HEIGHT

Before the Apollo missions, the relation of rim height 

to crater size on the Moon was less certain than the 

depth/diameter relation, because rim height could not 

be measured with as much accuracy as crater depth. 

Baldwin (1963, p. 143) first recognized that rim height 

also is significantly greater per unit diameter for lunar 

craters below about 30-km diameter than it is for 

larger craters. Although I placed the inflection nearer
C34 APOLLO 15-17 ORBITAL INVESTIGATIONS

to 15-km diameter and derived linear equations for rim 

height/diameter that had much the same form as the 

depth/diameter fits (Pike, 1967, 1972b), the greater 

scatter in the height/diameter data and the vagueness 

of the inflection at about 10-20-km diameter left the 

true form of the relation tenuous. Any doubts now are 

dispelled. The height/diameter relation has been revised from the latest Apollo photogrammetric data, including most of the 212 craters plotted in figure 19. The 

resulting graph for 162 fresh craters between 400 m 

and 320 km across (fig. 20) indisputably establishes a 

well-defined inflection of rim height at a crater diameter between 10 and 20 km.

The 162 craters divide into two groups at a diameter 

of about 17 km, and each distribution was fitted with a 

regression line. The two linear fits intersect at a diameter of about 21.3 km. The 124 smaller craters follow 

the expression

RP = 0.036 Dr (3)

where Re is crater rim height and all units are in 

kilometers. Fresh upland and mare craters do not appear to differ. Unlike the case with depth/diameter (fig. 

19), smaller craters do not extend beyond the inflection

10

1.0

LU

I

OC 0.1

0.01

EXPLANATION

o Upland crater, «=65 

• Mare crater, «=97

o o

o o q

o o o 

o o ° 

a. o 00° •f f °

• •o

1 o ••

••%

0«

o

«0

0.1 1.0 10 100 500

RIM-CREST DIAMETER, IN KILOMETERS

FIGURE 20.—Relation between rim height, Re , and diameter, Dr, for 162 of the 212 fresh lunar craters plotted in figure 19. Data from 

photogrammetry of Apollo 15-17 pictures. Distribution inflects at diameter of about 15 km. Equations fitted to two linear segments of 

distribution supplant previous expressions for height/diameter (see text).
GEOMETRIC INTERPRETATION OF LUNAR CRATERS C35

diameter. The height/diameter distribution of the 38 

larger craters follows the expression

Re = 0.236 Dr (4)

Neither of these expressions differs radically from the 

visually fitted regression lines given previously for 

fresh craters on the Moon (Pike, 1967, 1972b), although the slope of equation (3) is now essentially 1.0, 

the value expected from dimensional theory (Pike, 

1967, 1968). Equations (3) and (4) provide a new 

standard of accuracy that defines the rim height/ 

diameter of fresh lunar craters better than do the equations currently in use (Baldwin, 1963; Pike, 1972b). 

Equation (3) now should be used instead of earlier expressions in applications where the height/diameter relation of craters under 15 km across is deemed appropriate for modeling the transient (pre-slump) profile 

geometry, interval volumes, and ejecta volumes of 

fresh impact basins and craters over 15 km across. 

Although further improvement in the quantity and 

quality of topographic measurements on the Moon may 

result in equations (3) and (4) being refined, neither 

expression is likely to change appreciably from that 

given here.

The rim height/diameter relation described for fresh 

lunar craters in figure 20 by equation (3) is quite consistent with the height/diameter characteristics of 

fresh experimental-explosion and meteorite-impact 

craters on Earth. Equation (3) is very much like an 

expression derived previously for a sample of 28 freshly 

formed (or reexcavated) terrestrial meteorite craters 

(Pike, 1972b),

Re = 0.033 (5)

and also resembles an expression derived for a sample 

of 100 experimental-explosion craters (Pike, 1972b),

Re = 0.042 Dr (6)

variables for both equations are in kilometers. Equation (3) differs only somewhat more from the height/ 

diameter expression describing 35 laboratory craters 

formed in loose particulate material by hypervelocity 

impact of small projectiles (Gault and others, 1966),

RP = 0.022 Dr (7)

variables converted to kilometers. The new lunar 

height/diameter data do not appear to require any 

qualitative revisions of hypotheses or conclusions that 

had been based on older rim-height measurements, 

now obsolete.

WIDTH OF RIM FLANK

According to earlier studies over a wide size range of 

craters on the Moon, the relation of flank width to diameter was linear throughout (Baldwin, 1963; Pike, 

1968, 1972b; Guest and Murray, 1969). The apparent 

absence of any inflection in the width/diameter relation posed a problem, because diminution of crater 

depth at a diameter of about 15 km customarily has 

been attributed to slumping of the rim and formation of 

terraces (Quaide and others, 1965; Mackin, 1969). The 

substantial increase in crater diameter believed to result from slumping would require a commensurate retreat of the rim flank. Although earlier I appealed to 

isostatic compensation to lower the rim height of craters over 15 km in diameter without substantially increasing crater diameter and suggested that horizontal 

dimensions such as rim width were less susceptible to 

postimpact modifications than were vertical dimensions such as rim height (Pike, 1967), this interpretation probably is incorrect. Gravity data (Phillips and 

others, 1975; Moore and others, 1979) have shown that 

isostatic movements are insignificant for all but the 

largest craters.

The width-diameter relation has been reexamined 

using the same 162 craters as in figure 20. The resulting graph (fig. 21) shows a perceptible diminution of 

rim-flank width as crater diameter increases beyond 20 

km, although the inflection is not as strong as those 

evident for depth/diameter and rim height/diameter. 

Evidently earlier studies did not show this subtle 

change, because flank width is especially susceptible to 

errors in visual measurements on photographs. The 

inflection in the width/diameter data at the same crater diameter as the height/diameter inflection suggests 

strongly that diminution in rim height at the threshold 

diameter of 10-20 km occurs through retreat of the 

rim, probably by slumping, rather than through isostatic subsidence.

The 162 craters in figure 21 fall into two linear segments at a crater diameter of about 17 km; each segment has been fitted with a least-squares regression. 

Again, the distribution of smaller craters does not extend beyond the inflection diameter, as it does in the 

depth/diameter plot (fig. 19). The 116 smaller craters 

follow the trend

(8)

and the 46 larger craters are described by the expression

(9)

where Wf is rim-flank width and both variables are in
C36 APOLLO 15-17 ORBITAL INVESTIGATIONS

100

10

cc

LU

ILU

IQ

1.0

0.1

I

EXPLANATION

o Upland crater, n=65 

• Mare crater, n=97

o

• o o 

ooo

,0

v«*

•

•o

0.1 1.0 10 100 500

RIM-CREST DIAMETER, IN KILOMETERS

FIGURE 21.—Relation between width of rim flank, We , and rim-crest diameter, Dr, for same 162 fresh lunar craters as in figure 20. 

Distribution inflects at diameter of about 15 km rather than remaining linear for craters of all sizes. Equations fitted to two linear 

segments of distribution supplant previous expressions for width-diameter, all of which were linear throughout (see text).

kilometers. The two linear fits intersect at a crater diameter of about 30.4 km, although visually the inflection seems closer to 20 km. The discrepancy probably 

reflects some nonlinearity in the data, particularly 

those describing smaller craters. As in the case of 

depth/diameter and rim height/diameter, fresh upland 

craters do not appear different from mare craters.

Equation (8) shows that the rim flanks of fresh lunar 

craters under about 20 km diameter are between 25 

and 30 percent wider than had been ascertained from 

pre-Apollo observations by Baldwin (1963, fig. 22) for 

about 100 craters:

We = 0.206 Dr 

and by Pike (1972b) for 400 craters:

(10)

We = (ID

The latter measurements all were visual estimates 

from photographs, which tend to result in narrower 

rim flanks because terrain of the outer rim is not sufficiently steep or rugged to cast perceptible shadows on 

photographs. The new values of We for smaller craters 

on the Moon also are systematically higher by 30 percent or so than rim-flank widths of 15 terrestrial 

meteorite-impact craters:

We = 0.200 A-097

and 20 experimental-explosion craters: 

Wf = 0.190ZX0 - 98

(12)

(13)
GEOMETRIC INTERPRETATION OF LUNAR CRATERS C37

(both from Pike, 1972b). The explanation for these latter discrepancies may involve effects of differences in 

gravity scaling that favor slightly broader blankets of 

ejecta on the topographic rim and subjacent annular 

zones of uplifted bedrock on the Moon than on Earth.

RIM-WALL SLOPE

Fauth (1894) discovered that inner walls (between 

rim crest and floor) of lunar craters under about 30 km 

across did not vary significantly in "effective" slope 

with changing diameter, whereas the slope angle decreased appreciably with increasing diameter for craters above this size. On the supposition that rim-wall 

slopes should reflect the several changes in rim and 

floor geometry that had been shown to occur in lunar 

craters at a diameter of 10 to 20 km, I examined slopes 

for a small sample of far-side craters from the first 

Apollo 15 and 17 photogrammetric data and identified 

a possible discontinuity in the relation between rimwall slope and crater size at a diameter of about 17 km 

(Pike, 1973a). The slope/diameter relation has been 

studied more thoroughly for this report for 107 craters,

mostly fresh, from the recently published Apollo 

photogrammeteric maps.

The latest graph of rim-wall slope against diameter 

(fig. 22) shows the tangent of slope increasing gradually for small craters from about 0.34 (19°) in craters 

0.5 km across to a maximum of about 0.55 (29°) at a 

diameter of about 10-20 km, then dropping sharply for 

larger craters to a value of about 0.25 (14°) at 50-60 

km diameter, and then declining somewhat less 

rapidly to a value of 0.12 (7°) for the largest crater. The 

revised relation confirms the thrust of Fauth's original 

findings but places the change in slope angle at a crater 

diameter of about 15 km rather than 30 km. Craters 

that have flat floors and central peaks have been so 

indicated in figure 22. In this sample, flat floors develop 

at a much smaller crater diameter, 7 km, than do central peaks, 18 km. On the slope/diameter diagram, 

craters with flat floors and craters without flat floors 

occupy all but mutually exclusive fields. The steepest 

interior walls all occur in craters that have small flat 

floors. The relation between slope of the rim flank and 

crater size, which led to no conclusive results in the 

preliminary test (Pike, 1973a), has not been redone 

here.

0.70

0.60

S 0.50 

o

CO

u- 0.40 

O

.

< 0.30 h0.20

0.10

X

x x

x xx

x x

X •>

X

X

EXPLANATION

• Crater without floor or peak, «=44 

X Crater with flat floor, no peak, «=27 

o Crater with flat floor and peak, n=36

oXX

X

X

X

X

0.3 1.0 10 100 400

RIM-CREST DIAMETER, IN KILOMETERS

FIGURE 22.—Relation between tangent of slope of rim wall and rim-crest diameter, Dr, for 107 fresh lunar craters. Data from photogrammetry of Apollo 15-17 pictures. Slope values increase with crater diameter to maximum of about 29° (0.55) for craters about 15 km 

across, and then decrease sharply (see text).
C38 APOLLO 15-17 ORBITAL INVESTIGATIONS

FLOOR DIAMETER

Although the size of the flat floor within larger craters on the Moon has been studied in little detail, the 

ratio of floor diameter to rim-crest diameter correlates 

inversely with depth/diameter and thus roughly indicates the relative age of a crater (Warner, 1961; Pike, 

1968). Warner's raw data, which he fit with a single 

linear equation, also suggested that the floor 

diameter/rim-crest diameter relation for fresh craters 

inflected at a rim diameter of about 10 km; my data 

(Pike, 1968), which were fit with two linear equations, 

seemed to suggest a very slight inflection at 80 km. The 

new photogrammetric data show that the floor 

diameter/rim-crest diameter plot for fresh craters (fig. 

23) inflects perceptibly at a rim diameter of about 20 

km and a floor diameter of about 8 km. Craters smaller 

than this threshold value have increasingly narrow 

floors, down to about a kilometer across in craters 7 km 

or so in diameter. There is considerable scatter in the 

latter data, possibly because floors are more difficult to 

measure in the smaller craters; floors occur quite infrequently in craters much under 5 km across. Flat 

floors make up increasingly larger fractions of the rim 

diameter in fresh craters over 20 km across, the proportion reaching nearly 75 percent for Tsiolkovskiy, 

with very little scatter in the data.

The 91 craters in figure 23 were divided by inspection into two linear segments at a diameter of roughly 

20 km; each segment has been fitted with a leastsquares equation. The 51 smaller craters are described 

by the trend

Df = 0.031 ZV- 765 - K, (14)

and the 40 larger craters follow the expression

Df = 0.187ZV-249 , (15)

where Df is diameter of the flat floor, K is some constant between 2 and 5 km, and the two variables are in 

kilometers. The equations intersect at a crater diameter of 33 km, again a higher figure than visual inspection would lead one to expect. The difference is ascribed 

to curvature in the field of smaller craters (fig. 23). 

Both relations now are more clearly defined than they 

had been from the older data. Equation (14), which 

does not extend to craters less than 2 to 5 km across 

(approximately where Df goes to zero), differs entirely 

from my previous expression for smaller craters (Pike, 

1968). Equation (15) is much less different from the 

equations fit to larger craters by me (Pike, 1968) and 

by Warner (1961) and also seems to fit data for the 

flooded mare basins up to a diameter of 900 km. 

Upland craters have been distinguished from mare 

craters in figure 23, but again, no difference in the two 

subsamples is evident in the plot.

Unlike the other logarithmic relations between 

linear variables that describe lunar impact craters, the 

newDf/Dr equation describing smaller craters does not 

follow the slope of 1.0 that normally would be expected 

from dimensional theory for an isometric phenomenon 

(Pike, 1968). However, if a flat floor is not present in 

the transient phase of an impact or explosion cratering 

event (Pike, 1967), floor size should not be expected to 

relate similarly to rim-crest diameter and other variables that describe primary features of craters in both 

transient and fresh postimpact configurations. A slope 

of 1.0 is found in all logarithmic plots of linear variables describing primary morphologic aspects: those 

that always are present in craters or that reflect no 

changes in crater shape. Measurements for other types 

of morphologic aspects, such as flat floors and central 

peaks or the depths of modified craters (see table 6), 

characteristically do not plot with a slope of 1.0. Thus, 

Siegal and Wickman (1973) are mistaken when they 

contend that allometry—actually Gibrat's law of proportionate effect (see Aitchison and Brown, 1957)— 

does not apply to empirical relations among crater dimensions. Linear measurements of crater geometry 

graph logarithmically with a slope of 1.0 only if the 

craters all have a similar shape. Deviations from a 

slope of unity are important indications of differences 

in shape and hence suggest differences or complications in mode of origin (Pike, 1967).

RIM-CREST CIRCULARITY

The planimetric symmetry of craters on the Moon— 

expressed by circularity or polygonality of the rim 

crest—has been examined quantitatively by several 

investigators, but the findings disagree. Ronca and 

Salisbury (1966), Adler and Salisbury (1969), and 

Murray and Guest (1970) found no functional relation 

between crater size and the value of a circularity index. 

Similar negative results are evident for Mars from 

Mariner 6 and 7 pictures (Oberbeck and others, 1972). 

However, Fielder (1961) and Quaide and others (1965) 

both observed that craters more than 20 km across are 

more polygonal than are smaller craters, and Schultz 

(1976a) judged the rims of most craters over 15 km 

across to be either scalloped or polygonal. I found increases, statistically, in the frequency of rim-crest 

polygonality and in strength of polygonality with increasing crater size (Pike, 1968). Pohn and Offield 

(1970) divided the size continuum of fresh lunar craters 

over 8 km across into three morphologic classes "on the 

basis of the planimetric shape of their rim crests" as 

determined by inspection from Lunar Orbiter IV images. The diagnostic criterion was the presence or absence of polygonality. Fresh craters between about 16
GEOMETRIC INTERPRETATION OF LUNAR CRATERS C39

300

100

CO 

QC 

LU

ILU

QC

LU

H 10

LU

Q 

DC 

O 

O

1.0

0.5

EXPLANATION

O Upland crater, n=57 

• Mare crater, n=34

o»o o

o o o

°0

o o o

A °» o

1.0 10 100 1000

RIM-CREST DIAMETERJN KILOMETERS

FIGURE 23.—Relation between diameter of the inner flat floor, Df, and rim-crest diameter, Dr, for 91 fresh lunar craters. All but a half-dozen 

points from photogrammetry of Apollo 15-17 pictures. Distribution inflects at diameter of about 20 km. Equations fitted to two linear 

segments of distributions supplant earlier expressions.

and 48 km across were observed by Pohn and Offield 

(1970) to have markedly polygonal rim crests, whereas 

the rim crests of fresh craters both larger and smaller 

than this size range were observed to be more nearly 

circular. The qualitative tripartite classification can be 

tested to some extent quantitatively, using an index of 

crater circularity. Although circularity, the twodimensional equivalent of sphericity, is not the same 

as polygonality, the two-dimensional equivalent of 

roundness, a polygonal crater nonetheless must be

somewhat less circular than a similar-sized crater that 

has perfect radial symmetry.

The relation between rim-crest circularity and crater 

size was reexamined for this report using data from 

Apollo photogrammetric maps and orthophotomaps. 

The outcome for 201 fresh lunar craters between 400 m 

and 370 km across is a graph (fig. 24) that shows a 

systematic dependence of the circularity index— 

discussed earlier in this paper—on diameter: circularity is not constant with crater size. Despite the consid-
C40 APOLLO 15-17 ORBITAL INVESTIGATIONS

1.00

0.90

h; 

OC

0.80

O 

QC

O

CO 

UJ 

DC 

O

0.70

0.60

0.50

• •• • Jo o • •

• • o* * • • * «6*V#~ rf "•" * o

• o • o. • % oo • •

• • • «o o.% 8 • o o • o

_ • o«o/oo *o*o .

o • • • o • • ••

• •• o* •• ••oo • o 	• • • * • o o>

• • ••• • o o

• . . °° • • o ° ° o°

EXPLANATION

o Upland crater, n=65 

• Mare crater, «=136

0.3 1.0 10 100 400

RIM-CREST DIAMETER, IN KILOMETERS

FIGURE 24.—Relation between index of crater circularity, C—defined as ratio of area of inscribed circle to area of circumscribed circle fitted to 

outline of rim crest—and diameter, Dr, for 201 fresh lunar craters. Data from photogrammetry of Apollo 15-17 pictures. Average 

circularity increases with rim diameter to maximum of 0.85-0.90 in craters about 10 km across, and then decreases (see text).

erable scatter in the diagram, it is evident that fresh 

lunar craters have maximum planimetric symmetry at 

a rim diameter of 10 km or so, in much the same way as 

rim-wall slope (see fig. 22). Circularity declines in both 

larger and smaller craters, but the decrease is more 

perceptible among the larger craters. Fresh craters on 

the upland do not appear to differ significantly from 

fresh mare craters in circularity, save where rims of 

upland craters have been distorted by uneven terrain. 

Although these results do not expressly contradict the 

observations of Pohn and Offield (1970), their contention that fresh craters in the 16- to 48-km size range 

are markedly more polygonal than are either larger or 

smaller craters is not supported by circularity data. Of 

the sample of craters plotted in figure 24, those in the 

class I size interval of Pohn and Offield (over 45 km) 

have a median circularity of 0.78; class II craters 

(16-48 km) have a median circularity of 0.83; and class 

III craters (under 20 km, but restricted to craters between 8 and 20 km to assure a similar sample to that of 

Pohn and Offield) have a median circularity of 0.87. 

Hence the three size classes of craters adopted by Pohn 

and Offield on the basis of rim-crest polygonality may 

be artificial. As in the case of the new rim width/ 

diameter and floor diameter/rim-crest diameter results, a functional relation between circularity and 

crater size has become evident now only because the 

Apollo data portray lunar landforms with the requisite

accuracy. Interpretation of the results in figure 24 remains speculative. Fulmer and Roberts (1963) earlier 

had suggested an explanation of noncircular crater 

rims in terms of regolith thickness, but the observations here seem to require a more complex mechanism 

or combination of mechanisms.

EVENNESS OF RIM CREST

High-quality data resulting from the last three 

Apollo missions have made it possible to examine 

quantitatively a neglected aspect of a crater shape— 

configuration of the rim-crest profile. Millman (1956) 

made a detailed numerical analysis of the raised rim of 

the New Quebec meteorite crater, but excepting the 

work of MacDonald (1929), earlier evaluations of rimcrest evenness for lunar craters have been qualitative. 

I previously studied two different aspects of rim-crest 

configuration from a small sample of 15 lunar craters 

(Pike, 1973a); here, the sample is increased to 30, and 

the two problems are reexamined in greater detail. The 

relation of rim-crest roughness to crater size is taken 

up here, and azimuth-dependent tendencies are treated 

later in the paper. A common telescopic observation 

has been that rim crests of lunar craters below a certain diameter appeared to be relatively even and 

smooth, whereas rim crests of larger craters were 

significantly rougher (Baldwin, 1949). The critical di-
GEOMETRIC INTERPRETATION OF LUNAR CRATERS C41

ameter has been placed at 15 km by Firsoff (1959) and 

at 50 to 60 km by Shoemaker (1965). Dependence of 

rim-crest evenness upon crater size was treated using 

rim-crest relief and standard deviation of rim-crest 

elevation.

Relative relief, the vertical distance between highest 

and lowest elevation, was measured along the rim crest 

for 46 fresh-looking lunar craters from 1:250,000 LTO 

and larger scale published Apollo maps as well as from 

unpublished rim-crest profiles (Pike, 1973a). The sample is evenly divided between mare and upland craters 

1.6 to 275 km across. Rim-crest relief increases linearly 

with crater diameter (fig. 25) and does not change 

significantly at any well-marked threshold diameter. 

The upland craters have systematically rougher rim 

crests than mare craters, as might be expected from 

their having formed on a rougher surface. The slope of 

each logarithmic distribution is about 1.0 throughout, 

and no inflection or discontinuity can be detected at 

any discrete range of crater diameters. Relative relief 

has shortcomings as a roughness parameter, however, 

because of its undue sensitivity to extreme altitudes 

along the rim crest. Statistically it is the equivalent of 

the range, the crudest of all measures of dispersion 

(Croxton and others, 1967, p. 191).

The standard deviation of elevation along the rimcrest, a far more comprehensive measure of dispersion,

10

CO

cc

UJ 

LLJ

i.o

UJ

_l

UJ

cc

(- 

co

UJ

cc 

o

0.1

0.01

EXPLANATION

Mare or upland-plains

crater, n=22 

Upland crater, «=24

1.0 10 100 1000

RIM-CREST DIAMETER, IN KILOMETERS

FIGURE 25.—Relation between relief along rim crest and crater diameter, Dr, for 46 fresh lunar craters. Data from photogrammetry 

of Apollo 15-17 photographs. Distribution is linear throughout, 

and much of the scatter can be accounted for by less even surfaces 

upon which upland craters formed. Upland craters have about 

three times the rim-crest relief of mare craters.

was calculated for 35 of the 46 craters in an attempt at 

a more sensitive test of the supposed dependency of 

rim-crest roughness upon crater size. Thirty-six elevations were determined along the rim crest of each crater at intervals of 10° from due north. Interpolation of 

elevation values is easy on the circular topographic 

profiles obtained specifically for the purpose (Pike, 

1973a, 1973b) and for LTO's and other published maps 

where the contour interval was relatively small compared to the crater size. Interpolation is less certain on 

LTO's for craters between 6.5 and 11 km across, although the problem was minimized by using craters 

with the maximum number of observed contours and 

spot heights along the rim crest. Some bias toward 

greater roughness thus may have been introduced for 

the latter craters. Because the 35 sampled craters lie at 

different lunar elevations, each resulting standard deviation was divided by its mean elevation to measure 

relative rather than absolute dispersion (Croxton and 

others, 1967, p. 198). The resulting statistic, the coefficient of variation, provides a fairly sensitive index of 

rim-crest evenness for the 35 selected craters.

A plot of the coefficient of variation for rim-crest elevation (Vc ) against the logarithm of crater diameter 

(fig. 26) indicates a significant increase in rim-crest 

roughness for craters over 10-20 km across. For 13 of 

the 21 mare craters (15 km in diameter and less), Vc 

remains relatively constant, at a value of about 0.006, 

whereas the statistic increases systematically to a 

maximum of about 0.05 for the eight mare craters over 

15 km across. The 14 upland craters have less even rim 

crests than the mare craters. Vc values for the upland 

craters also increase with the crater size, but the relation is not very systematic and any size-dependent inflection is much less evident than that shown by the 

mare craters. The sample of small upland craters may 

not be large enough to show an inflection, if indeed one 

exists. Most of the scatter among the larger upland 

craters in figure 26 is attributed to the inherently less 

even surface upon which they formed. Apollo data appear to show that evenness of the crater rim crest 

starts to increase at a diameter of 10 to 20 km for 

craters that formed on surfaces with a similar (flat and 

even) configuration, thus supporting the older qualitative observations of yet another size-dependent change 

in crater shape.

NONRATIO VARIATIONS

At least four discontinuities or transitions in more 

qualitatively expressed aspects of shape also interrupt 

the morphologic continuum of lunar craters in the 

same size range indicated by the seven changes in crater geometry described above. Gilbert (1893) was 

perhaps the first to recognize that well-developed flat
C42 APOLLO 15-17 ORBITAL INVESTIGATIONS

0.24

O 0.2

HI 

COin 

cc O

cc 

cc 

O

O

IE

LL 

O

\-

z

UJ 

O

LL 

LL 

LU 

O 

O

0.1

EXPLANATION

° Upland crater, H=14 

• Mare crater, «=21

cr

1.0 10 100 300

RIM-CREST DIAMETER, IN KILOMETERS

FIGURE 26.—Relation between evenness of rim crest, expressed by coefficient of variation for 36 measured rim-crest elevations per crater, Vc , 

and diameter, Dr, for 35 fresh lunar craters. Data from photogrammetry of Apollo 15-17 pictures. Upland craters have much more 

irregular rims than mare craters. Values of Vc for mare craters remain constant at small diameters but begin to increase with crater size 

at diameter of about 10-20 km.

floors, rim-wall terraces, and central peaks (not related 

to regolith thickness) all are relatively rare in fresh 

craters on the Moon under about 10-20 km in diameter, such as Linne (fig. 5), but are common in larger 

craters, such as Aristarchus (fig. 27). His observations 

have been confirmed repeatedly by independent telescopic, photointerpretive, and statistical work 

(Baldwin, 1949; Firsoff, 1959; Levin and RiTskol, 1962; 

Quaide and others, 1965; Pike, 1968). Morphologies of 

the larger and smaller craters have been described in 

detail by Howard (1975) and Pike (1973b), respectively, and the contrasts were nicely summarized 

by Mutch (1970). One recent study (Smith and Sanchez, 1973) at first appears to contradict the sizedependent onset of occurrence of crater floors, terraces, 

and peaks but in fact strongly supports the 

morphologic discontinuity at a crater diameter of

10-20 km. The same paper also documented the existence of a fourth size-dependent feature: the presence 

on crater floors of a distinctive morphologic characteristic that has been termed "swirl texture."

From an analysis of four semiquantitative index 

numbers that reflect the degree of complexity of crater 

floors, terraces, and central peaks, Smith and Sanchez 

(1973) concluded that fresh lunar craters between 4 km 

and about 150 km across make up a continuous 

morphologic sequence. Although their data show convincingly that the morphologies of lunar craters between about 10-20 km and about 100 km in diameter 

gradually become more complex with increasing size, 

smaller craters do not belong to this unbroken sequence. Upon further analysis, the data of Smith and 

Sanchez documented a size threshold that divides their 

sample of 115 fresh craters into two contrasting shape
GEOMETRIC INTERPRETATION OF LUNAR CRATERS C43

FIGURE 27.—Oblique view of Aristarchus, an especially fresh example of a well-defined complex or modified impact crater of lunar 

main sequence, in Oceanus Procellarum. Rim crest is about 40 km across. Compare with small simple crater Linne (fig. 5) and with 

craters that fall between Aristarchus and Linne in both size and morphology (fig. 31). View is to south. Mosaicked from Apollo 15

panoramic camera photographs 0331 and 0333. Sun is to left.

categories at a diameter of 10-20 km (Pike, 1975). The 

two types of data, index numbers and frequencies of 

occurrence (figs. 28 and 29), gathered by Smith and 

Sanchez from Lunar Orbiter and Apollo photographs 

are examined at some length here because they provide 

an especially systematic and comprehensive body of 

observations showing changes in crater morphology at 

this size range.

INDEX DATA

Smith and Sanchez (1973) obtained a semiquantitative description of crater terraces, floors, and peaks by 

assigning numerical values to specific morphologic 

characteristics in a way such that more complex craters bear proportionately higher values. The index 

numbers derived from this information reveal a 

significant morphologic change in lunar craters within 

the same narrow size range over which seven geometric changes also occur (fig. 28). This result differs from 

that obtained initially by Smith and Sanchez, but the 

use of arithmetic scales on the diagrams and the inclusion of too few small craters prevented the inflection in 

the size-morphology continuum from being recognized 

in the original investigation. First, the arithmetic 

scale is unsuited to most graphical analyses of the dimensions of natural and experimental craters; it unduly compresses smaller values and thus conceals possible variations in the data. Use of the logarithmic 

scale, which accommodates the great size range over 

which lunar craters occur, has been conventional practice for some time (see MacDonald, 1931; Baldwin, 

1949), although it does tend to obscure some constants. 

Second, Smith and Sanchez did not, examine enough 

small craters. Only 3 of their 115 sample craters lie 

below the 5-10-km diameter range where the pronounced differences in morphology start to become evident. Similarly, because the sample was limited to only 

three craters larger than 100 km in diameter, their 

conclusions do not necessarily apply to craters larger 

than about 100 km across.

The Smith-Sanchez conclusions were tested by additional data. A representative sample of fresh primary 

craters in the 0.1- to 10-km diameter range (table 7) 

was obtained to determine whether or not extension of 

the sample to smaller craters would continue to show a 

gradual sequence of shape characteristics. Expansion 

of the sample to larger craters was not attempted. The 

37 small craters were selected primarily from published maps and reports prepared for, or resulting 

from, the Apollo missions. According to the 

morphologic-index (MI) criteria listed in table 2 of
C44 APOLLO 15-17 ORBITAL INVESTIGATIONS

TABLE 7.—Morphologic index numbers of selected lunar craters

Crater identification

Diameter 

(km) TSI' CPI1 MI2

Sunrise (Apollo 14) _____________0.18 5 5 10 20

L.O. Ill H 133 _______________,.20 5 5 5 15

Cone (Apollo 14) ________—____.34 5 5 5 15

AS16, PAN-5400______-______.43 5 10 5 20

Horatio (Apollo 17)____________.45 5 5 5 15

Sherlock (Apollo 17) ___________.50 5 5 5 15

L.O. Ill H 189 _______________.51 10 5 5 20

Near Rima Prinz I _____________.60 5 5 5 15

Emory (Apollo 17) ____________.65 5 5 5 15

South Ray (Apollo 16) ,_________.65 5 10 5 20

Cochise (Apollo 17)_____________.70 5 5 5 15

Camelot (Apollo 17) ___________.73 5 5 5 15

Kiva (Apollo 16) _.____________.85 5 10 5 20

North Ray (Apollo 16) __________.93 5 10 5 20

L.O. III H 143 _____________-1.05 15 5 5 25

Near Apollo 17 site ___________1.30 5 5 5 15

L.O. III H 143 ______________1.50 15 5 5 25

In Sklodowska3 ______________1.60 5 5 5 15

NWofKriegerC _____________1.96 5 5 5 15

Linne ___________________2.45 5 5 5 15

Near Gagarin3 ______________2.75 5 5 5 15

In Sklodowska3 ______________2.75 5 5 5 15

InHirayama3 _________________3.45 5 5 5 15

Censorinus_________________3.82 5 5 5 15

LittrowBD_________________4.05 5 5 5 15

In Curie3 __________________4.25 5 5 5 15

KriegerC _________________4.30 5 5 5 15

Secchi X __________________4.80 5 5 5 15

In Tsiolkovskiy3 ____ _________4.90 5 5 5 15

HyginusB _________________5.20 5 5 5 15

HadleyC__________________5.82 5 5 5 15

In Hilbert3 _________________6.00 5 5 5 15

Censorinus A_______________6.52 5 5 5 15

Moltke______ ____________..6.80 5 5 5 15

LittrowB _________________7.00 5 5 5 15

AS10-29-42073 ______________8.00 10 10 5 25

Messier B _________________8.00 5 5 10 20

'Index numbers that reflect, from left to right, the morphology of crater-wall terraces, 

central peaks, and floors, according to criteria in table II of Smith and Sanchez (1973) 

2Sum of these index numbers. 

3Far-side craters; all but AS10-29-4207 from Pike (1973a).

Smith and Sanchez (1973), most of the additional craters (MI values were taken from photographs) are quite 

similar in shape and have the minimum possible value, 

15, of the cumulative morphology index. For craters 

under 2 km in diameter, cumulative MI values over 15 

may indicate complexities in shape imparted by a thin 

regolith on the Moon (Oberbeck and Quaide, 1967). 

Figure 28 is the resulting revision of Smith and Sanchez's figure 1, with crater diameter plotted on a 

logarithmic scale. The morphologic contrast provided 

by the added smaller craters in table 7 reveals a sharp 

inflection, rather than a gradual change, in the trend of 

the cumulative morphology index at about 10-km diameter. Most lunar craters larger than this size clearly 

are much more complex than are the smaller craters. 

Although each of the constituent index numbers—TSI, 

CPI, and FI of the Smith and Sanchez (1973) cumulative morphology index, MI, shows a similar inflection 

at about 10-km diameter, the sudden change in shape 

is adequately illustrated by the Ml/diameter data 

alone.

FREQUENCY DATA

Data on the prevalence of certain features in craters 

on the Moon also support the existence of a marked 

morphologic change in craters at the restricted diameter range in which the seven geometric changes occur. 

Smith and Sanchez (1973) show that crater-wall terraces, textured (flat) floors, and central peaks increase 

gradually in frequency of occurrence from nearly zero 

in craters under 5 km in diameter until they are present universally in fresh lunar craters that exceed 55 

km in diameter. Replotting their data for all three features (fig. 29) reveals a parallel transition from simple 

to complex crater morphologies within a diameter 

range of about 50 km. This interval is remarkably narrow in view of the 11 orders of magnitude spanned by 

lunar impact features. As craters increase in size, 

statistically a flat floor is the first complication in the 

simple bowl shape of the smaller craters. Development 

of a floor is followed by the appearance of central peaks 

and finally terraced walls.

The frequency data of Smith and Sanchez (1973) 

show that the transition between 100 percent absence 

and 100 percent presence of peaks, terraces, and flat 

floors in all craters falls within a restricted size range, 

about 9 to 22 km in diameter (fig. 29), and averages 15 

km for the three features. Corroboration appears in 

figures 7, 9, and 11 of Smith and Sanchez (1973), 

wherein percentages of craters possessing terraces, 

floors, or peaks within each of the 10 discrete diameter 

intervals are plotted against crater diameter. Presence 

of these features in craters is transitional and does not 

happen at one diameter value. Thus the threshold or 

transitional diameter is statistical in nature and requires a measure of central tendency. The center of the 

transition from a preponderance of craters not having 

these features to one of craters having them is located 

correctly at the crater diameter at which the frequency 

of occurrence of each feature is 0.5 (fig. 29). This median diameter marks the focal point of each transition 

statistically. It lends greater precision to the qualitative statements of earlier workers such as Gilbert 

(1893) and Quaide and others (1965) that various features tend to appear or disappear within fresh lunar 

craters above or below some particular size. The crater 

diameter marking the 0.5 frequency of occurrence is 

preferable to the diameters at which features begin to 

occur (the "onset diameter"; see Cintala and others, 

1976) or stop occurring, because the latter two values 

are based wholly on single occurrences of peaks, floors, 

or terraces and hence are susceptible to large errors.

According to figure 7 of Smith and Sanchez (1973), 

the transition from unterraced to terraced craters is 

centered at a crater diameter of 22 km. Next, their 

figure 9 shows that at a diameter range centered at 

about 9 km, 50 percent of the sampled fresh craters 

exhibit no floor texturing. This curve also describes 

data that mark the presence or absence of flat interior 

floors within 749 lunar craters (Pike, 1968, table 14).
GEOMETRIC INTERPRETATION OF LUNAR CRATERS C45

100 T———I——T I I I——————I———I—

EXPLANATION

• Crater from table 1 of Smith and Sanchez (1973), n=115 

0 Crater from table 7 of this report, «=37

—i—i—r~r

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• • • • ••

• • • •

20

o o

O O O OO

o o 

o o o ooX o o o o o

o • •

o :: :

0.1 10 100 300

CRATER DIAMETER, IN KILOMETERS

FIGURE 28.—Inflection in complexity with size of large lunar impact craters. Relation between cumulative morphology index, MI, of Smith 

and Sanchez (1973) and crater diameter for 152 fresh lunar craters ranging in diameter from 180 m to 195 km. Abrupt elbow in 

distribution at about 10-km diameter marks transition from simple to complex craters on the Moon.

Finally, figure 11 of Smith and Sanchez (1973) shows 

that the transition from craters without central peaks 

to craters with peaks is centered at a crater diameter of

14 km. These three values can be read from figure 29. 

The average of the three transition-center diameters,

15 km, indicates a characteristic size-that divides most 

craters with a relatively uncomplicated shape from 

most craters with relatively complex shapes, however 

continuous the change in morphology (from simple to 

complex or modified) may appear to be. The value of 15 

km, although based on only three morphologic criteria 

and hence approximate, agrees well with the 10-20- 

km diameter range containing the seven geometric inflections.

Moreover, a crater diameter of 15 km exactly bisects 

the frequency distribution derived by Smith and Sanchez (1973) for the occurrence of "swirl texture" on the 

floors of certain lunar craters (their fig. 9). The texture, 

which is particularly well developed in the craters 

Dawes and Bessel, is significant because it may arise 

from the transition of relatively smooth crater walls to 

terraced walls. "Swirl texture" may simply describe 

the surface configuration of material that has slumped 

from the crater walls: not cohesive terraces that have

separated from the upper wall, but rather unconsolidated material that has slid down the walls rapidly in 

rock slides and spread across the flat floor in festooned 

piles of debris. Unlike flat floors, central peaks, and 

terraced walls, "swirl texture" does not increase in 

prevalence from zero to 100 percent over the craterdiameter range of about 4 to 55 km (fig. 29). Rather, it 

is most common in craters about 20 km across and is 

not observed at all in craters under 5 km or over 40 km 

in diameter. Location of the median frequency of occurrence of "swirl texture" at 15-km diameter, the average of the three transition-center diameters (fig. 29), 

suggests strongly that this feature is yet another manifestation of the fundamental change from simple craters to complex craters on the Moon.

DISCUSSION

Inflections in the plot of morphology index and diameter (fig. 28) and in the four morphologic transitions 

evident from the frequency data of Smith and Sanchez 

(fig. 29) coincide with seven size-dependent differences 

in the geometry of lunar craters (figs. 19-26). It is 

likely that all 11 shape changes—which occur at an 

average crater diameter of 17.5 km—have a common
C46 APOLLO 15-17 ORBITAL INVESTIGATIONS

i——r

1.0

0.8

O

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HI

O 0.6

HI 

CC 

CC 

D 

O 

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O 

u- 0.4

O

z

HI

D

o

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DC 0.2

EXPLANATION 

O Flat inner floor (floor texturing) 

D Central peak 

A Terraces on crater wall

•'.•.•.'•'• Swirl texture on crater floor

i i i i i i i i i i i

0.5 1 10 100 500

CRATER DIAMETER, IN KILOMETERS

FIGURE 29.—Transition from simple to complex morphologies in fresh lunar craters at approximately 10- to 20-km diameter. Frequencies of 

occurrence, at various diameter intervals, for four salient features in craters between 0.5 and 500 km in diameter. Solid curves are data 

from Smith and Sanchez (1973). Dashed curves used where data are sparse or obtained elsewhere (Pike, 1968). Barred circle shows range 

and mean diameter for first three features at 50 percent frequency (see text).

origin related to formation of central peaks, a flat floor, 

and rim-wall terraces in fresh craters of a critical size 

(Gilbert, 1893; Shoemaker, 1959; Quaide and others, 

1965; Pike, 1967: Mackin, 1969; Schultz, 1976a). The 

changes in crater geometry reflect two principal effects. First, diminution of rim height and flank width 

indicate lowering of the rim crest and enlargement of 

the crater rim diameter, almost certainly as the result 

of terracing of the rim wall. This wholesale slumping 

probably has decreased slope, depth, circularity, and 

evenness as well. Second, changes in slope, floor diameter, and depth reflect emplacement and widening of the 

flat floor in larger craters—in part probably from uplift 

accompanying formation of the central peak. The net 

result of these changes is to reduce both rim and crater 

volumes for craters exceeding some threshold size, that 

is, to diminish the topographic disturbance created at

the Moon's surface by the original impact. The changes 

in impact-crater shape mark the transition from one 

type of equilibrium landform to another (Pike, 1967): 

simple craters seem to be stable only up to 15-20 km 

across. The larger complex craters evidently cannot 

reach equilibrium with the postimpact environment 

without departing significantly from the geometry of 

the smaller simple craters.

DEPTH VERSUS RIM-WALL WIDTH

The inflected relation between depth and rim-wall 

width (fig. 30) provides further insight into the transition from simple to complex craters. The two variables 

are components of rim-wall slope (fig. 22) for a sample 

of 107 craters ranging in size from 400 m to 370 km 

across. The presence of flat floors and central peaks, 

which generally are accompanied by rim terraces, is
GEOMETRIC INTERPRETATION OF LUNAR CRATERS C47

10

cc

UJ 

UJ

i 

t

UJ 

Q

DC 

UJ

hQC 

O

0.1

0.05

EXPLANATION

• Group I - No flat floor, no peaks, n=44

x Group II - Flat floor, no peaks, n=27

o Group III - Central peaks, flat floor , n=36 oo c

00 O

xT X »X

X •

•

0.1 1.0 10 100

WIDTH OF RIM WALL, IN KILOMETERS

FIGURE 30.—Relation between crater depth,fl,, and width, W{ , of rim wall for 107 fresh lunar craters. Data and crater types same as in figure 

22. The sample of craters divides into three fields (see text). Terraces usually present in craters with peaks. Transition from group II to 

group III craters marks change from simple to complex or modified craters.

indicated for each crater. The depth/wall-width data lie 

in three relatively discrete fields on the graph, each of 

which is dominated by morphologic characteristics 

that mark the transition from simple to complex impact craters. For convenience in discussing the data in 

figure 30, the three fields are referred to as group I (no 

special features), group II (small flat floors only), and 

group III (conspicuous flat floors and central peaks). 

Group I meets group III at a rim width of about 5 km 

and a depth of about 2.5 km, but group II (with craters 

as deep as 3.9 km) spans this inflection without any 

change in slope on the graph. Although flat floors are 

the first complication of the simple shape as craters 

increase in size, depth does not diminish with increasing width of the wall until central peaks begin to appear (group III).

Group I contains 44 craters and exhibits a positive 

slope of about 1.0 (fig. 30). The simple impact craters in 

the group I sample, which range from 400 m to 12.8 km 

in diameter, have no terraces, peaks, or flat floors. Wall 

slopes typically are between 18° and 24°, averaged from 

rim crest to crater center, but reach 30° and more on 

upper slopes of the wall. Cauchy, at 2,675 m, is the 

deepest crater in the group; the slope of its wall (6.4 km 

long) averages about 23°. The depth of Cauchy approaches the 3.0 km upper limit proposed on the basis 

of older data by Quaide and others (1965,) for the depth 

of a stable crater landform on the Moon. A more typical
C48 APOLLO 15-17 ORBITAL INVESTIGATIONS

example of a group I crater is Linne (fig. 7). The symmetrical simple shapes of group I craters and their 

similarity to much smaller experimental impact and 

explosion craters suggest that they differ little in form 

from the transient craters that developed at the climax 

of the original impact event, except for the addition of 

some fallback ejecta and minor rockslides on the walls. 

Group II and III craters are increasingly complex and 

evidently have not retained the simple pre-slump 

morphology.

Group II in figure 30 contains 27 craters and lies 

above and parallel to group I. The craters range from 

about 5 to 29 km in diameter and have no prominent 

central peaks or rim terraces. Group II craters have 

very small flat floors (the same craters less than 20 km 

across in fig. 23) and are comparatively deep for their 

size; their rim walls are narrow and steep. Evidently 

moderate infilling, not accompanied by terrace slumping, has formed a small floor that reduces the rim-wall 

width substantially without much reducing crater 

depth. Four group III craters, which plot with the 

group II craters, contain central peaks, Proclus, Peirce, 

Diophantus, and an unnamed far-side crater 14.6 km 

across, but their peaks are abnormally small for the 

size of the crater (fig. 32). These four anomalous craters 

also are among the smallest craters that have measurable peaks (excepting, of course, the much smaller 

craters excavated in thin regolith), and their rim walls 

are among the steepest in the sample. Some craters 

within the group II size range—such as Messier B 

(Pike, 197Ib) and Taruntius E—have small flat floors, 

but only low hummocks instead of well-developed central peaks. These observations and floor-frequency 

data in figure 29 suggest that emplacement of small 

flat floors is the first manifestation (with respect to 

increasing crater size) of the change in shape at the 

10-20-km diameter threshold. The small peaks within 

larger craters in the group may be large blocks of 

fallback ejecta or rocks that fell from the upper crater 

wall onto the crater bottom but were not buried in the 

flat floor. Alternatively, these protuberances may be 

blocks of material that have been thrust up from depth 

by recoil from the impact. The flat floor in these 

smaller craters is interpreted as a more or less leveled 

deposit of impact fallback and loose rock debris that 

moved rapidly in sheetlike slides down the crater 

walls, probably right after impact, until the slope 

stabilized (Pike, 1971b). The deposit probably is not 

impact melt (Howard and Wilshire, 1975). Typically, 

the floor forms a gently sloping conical surface that is 

not quite flat and level, but inclined 4° to 5°, as can be 

seen in the 18-km crater Gilbert D on LTO sheet no. 

81A2. The floor material covers the originally concave 

crater bottom, which can still be seen in group I craters, and in most cases any small central mounds or 

peaks; only a few peaks protrude above the surface of 

the floor deposit. The maximum depth for group II 

craters is about 4 km, corresponding to a crater diameter of about 20 km (equation 1)—deeper craters on the 

Moon all are much larger and more complex. The limiting slope angle of the crater wall is about 31°. The 

largest craters in the group II sample appear to contain 

the longest steep slopes on the Moon, 6 km to nearly 8 

km. By way of comparison using measurements from 

the 1:50,000 Apollo contour maps, the steepest slopes 

measured on Mons Hadley—which is considerably 

older than the fresh group II craters—are about 7 km 

long and lie at about 29° or less. Thus, the limiting 

conditions for stability of partly constructional and 

partly excavational slopes on the Moon may be about 7 

km and 30°, presumably in badly fractured basalt or 

similar material.

Group III in figure 30 contains 36 craters and slopes 

positively away from the other two fields at about 0.40. 

The craters in group III, which range in diameter from 

18 to 370 km, are full-fledged complex or modified craters, with well-developed terraces, flat floors, and central peaks. The large floors in group III craters are 

considerably more complicated topographically than 

the small floors in group II craters and may consist of 

uplifted material veneered with impact melt (Pike, 

1971a; Howard and Wilshire, 1975). Compared with 

group II craters, the floors are wider, the peaks (if present at all in group II) are larger (fig. 32), and the depths 

are significantly less per unit diameter. Walls slope at 

only 17° to 22° for the smaller group III craters, and 

slopes are much less in larger craters (fig. 22). Small 

craters in group III are only about 2.7 km deep; the 

larger group III craters all are over 4.0 km deep (the 

maximum for group II) once rim-wall slopes are wider 

than 15 km. The limiting depth of the largest crater 

exceeds 6.0 km, but its sloping walls are stable owing 

to their great width. Three group II craters, which lack 

well-developed central peaks—Conon, Vitruvius A, 

and an unnamed crater on the far side—and appear to 

mark the transition from group II to group III, plot at 

the lower end of the group III field. These somewhat 

anomalous craters have low hummocks on their small 

flat floors, tend to be deeper than other group III craters of similar size, and are the three smallest craters 

in the group III sample; their walls, which are only 6.5 

km to 8.0 km wide, slope at 20° to 22°.

With respect to surface geometry, the change from 

simple to complex, or modified, craters on the Moon at 

the 10-20-km size may be defined as the change from 

group II to group III craters in figure 30. Two good 

illustrations of craters that are intermediate in the 

transition are Diophantus and Delisle (fig. 31). Both
GEOMETRIC INTERPRETATION OF LUNAR CRATERS C49

FIGURE 31.—Vertical views of fresh impact craters in western Mare 

Imbrium. Larger craters contain features that illustrate transition 

from simple to complex craters on the Moon (see text, fig. 30). A, 

Diophantus C (5 km across), a simple impact crater in group I (fig. 

30), is much like the crater Linne (fig. 5). Diophantus (18 km 

across), in group II, is between Diophantus C and Delisle in 

morphology. B, Delisle (25 km across), in group III (fig. 30), is a 

complex impact crater but lacks the well-formed slump terraces 

that characterize full-fledged complex craters such as Aristarchus 

(fig. 27). Apollo 15 mapping-camera photograph 2738. Sun is to 

right.

craters have small peaks and floors with these features 

better developed in the larger crater but differ 

significantly in that Delisle has several massive terraces on the rim wall, whereas Diophantus has only 

one or two slump blocks.

Figure 30 suggests that for fresh lunar craters less 

than 10 km or so across and 3 km deep (group I), the 

observed and presumably stable form might not differ 

much from that of the transient cavity reached at the 

height of the impact. The profile of this cavity may be 

parabolic (Dence, 1973), but it could be hemispherical 

or even hemielliptical (Pike, 1968). Although the observed configurations of the larger craters (groups II 

and III) probably are not like those of the transient 

craters, the degree of difference between the two forms 

remains unknown. Figure 30 might provide one way to 

estimate the size and shape of transient cavities for 

these larger craters. Group II craters differ only 

slightly from those in group I. Hence the group II field 

probably could be shifted up to about 0.5 km and to the 

right on figure 30 until it is alined with the upper part 

of group I, in order to obtain a fair approximation of the 

shape of the transient cavities for group II craters. An 

analogous restoration of group III craters would involve rotating the field counterclockwise about an origin marked by a rim width of 6 km and a depth of 2.5 

km. Such a restoration for the larger group III craters 

would involve very great differences between observed 

and hypothesized depths (Pike, 1968). Whether or not 

this practice is justified remains one of the more intriguing questions concerning craters on the Moon, because of possible extrapolation to initial cavities of excavation of large basins and the radial distribution of 

their ejecta (Pike, 1967, 1974a; McGetchin and others, 

1973).

SOME ALTERNATIVE EXPLANATIONS

The problem of explaining the break at 10-20 km 

within the crater continuum now lies less in relating 

changes in crater geometry to the appearance of various morphologic features than in determining the underlying physical mechanisms and how and why they 

operate. The latter are difficult if not impossible to 

judge from a scrutiny of lunar topography alone, so 

some recourse to experimental work, computer modeling and simulation, and especially analogies with terrestrial impact structures is necessary. However, the 

best known terrestrial impact craters are small features compared to the larger craters on the Moon, and 

the fundamental cause of the related features in lunar 

craters 10-20 km across remains uncertain and controversial. Currently the issue seems to hinge upon 

origin of central peaks and rim-wall terraces, for which 

there are two main interpretations in contention: centripetal collapse and elastic recoil.
C50 APOLLO 15-17 ORBITAL INVESTIGATIONS

Various hypotheses for the discontinuity in crater 

shape—including the cometary-impact model developed by Roddy and others (1969), isostasy (Baldwin, 

1963), and variable depths of impact focus (Baldwin, 

1963)—have been summarized by Quaide and others 

(1965) and reviewed more recently by Alien (1975). 

The layered-substrate idea of Wegener (1921) has been 

revived (Head, 1976b). Quaide and his colleagues concurred with Shoemaker's (1959, 1962) view that deepseated slumping of the crater rim to form terraces also 

converged at the crater center to thrust up a peak (see 

also Gilbert, 1893). Subsequently this model was 

adopted by Dence (1968), Mackin (1969), Gault, 

Quaide, and Oberbeck (1968), and Gault, Guest, and 

others (1975). Centripetal collapse of the rim and uplift 

of the central peak, likened to formation of a central jet 

in a liquid splash (Harlow and Shannon, 1967), are 

presumed to result from gravitational instability of 

lunar rock materials making up long slopes that exceed some critical value, shown here to be about 30°, in 

transient craters larger than 10-20 km across and 

deeper than 4 km (fig. 30). According to the collapse 

hypothesis, peak-forming energy is potential energy 

that is stored momentarily in the mass of the rim of the 

transient crater and then released in a late stage of the 

cratering event. This mechanism would produce most 

of the size-dependent changes in crater shape, with a 

veneer of impact melt perhaps accounting for the level 

floors in larger craters such as Tycho and Aristarchus 

(Howard and Wilshire, 1975). Explanations for the increase of rim-crest circularity and rim-wall slope in 

craters up to 10-20 km across are less evident and may 

not even be related to the main shape discontinuity. 

Quantitative evaluation of the gravity-collapse 

hypothesis currently is underway (Melosh, 1977).

Geologic evidence from impact structures on Earth, 

however, suggests that central peaks within lunar 

craters did not result from deep-seated centripetal collapse of the crater rim but are rebound phenomena as 

Baldwin (1963) suggested (see also Gilbert, 1893). 

Elastic recoil depends upon crater-forming energy—as 

manifested by compression of the ground surface by the 

shock wave—rather than upon potential energy stored 

in the rim of the transient cavity. Data of Wood (1973) 

imply that the amount of uplift estimated to have occurred in the centers of terrestrial impact craters is 

directly proportional to the size of the crater and hence 

to impact energy (fig. 32). Theoretical calculations by 

Dent (1973) are consistent with this view.

Study of the Gosses Bluff, Australia, and Sierra 

Madera, Texas, structures on Earth shows that rocks of 

the central uplifts came from stratigraphic horizons 

well below those that could have been affected by rim 

slumping (Milton and Brett, 1968; Milton and others,

1972; Wilshire and others, 1972). Additionally, the 

central uplift appears to develop in early, not late, 

stages of a cratering event (Milton and Roddy, 1972). 

The centripetal movement that seems to accompany 

formation of a central uplift may have caused the rims 

of these structures to collapse into slump terraces with 

attendant loss of crater depth, not vice versa. If the 

terrestrial analogy can be extended to large craters on 

the Moon, then rim terraces and flat floors with "swirltexture" material (Smith and Sanchez, 1973) probably 

develop consequent to central peaks rather than the 

other way around. Much of the shallowness of large 

craters may result from accompanying (and instantaneous) uplift of the floor area between the peaks and 

the rim (Offield and Pohn, 1977), rather than exclusively from terracing of the rim. This structural shallowing also would enable a comparatively small volume of impact melt and fallback material to form a 

broad flat floor in large craters. Great volumes of fill 

might not be required except where it can be shown 

that central peaks have been completely buried, 

usually in subsequent events unrelated to formation of 

the crater.

Although the mechanisms responsible for central 

uplifts in lunar craters according to the recoil, or rebound, hypothesis remain conjectural, the geology of 

terrestrial impact craters and analyses of experimental-explosion craters imply that peaks result from reflection of the impact-rarefaction waves by target 

material at the excavated ground surface (Milton and 

Roddy, 1972, Roddy, 1976). Laboratory experiments 

with small impact craters had suggested earlier that a 

stratified target—comprising a less cohesive surface 

layer and a more cohesive substrate—is required for central peaks to form. From the results of 

experiments, first Wegener (1921, 1975) and later 

Sabaneyev (1962) applied this interpretation to large 

lunar craters. Similar results were obtained in more 

elaborate series of tests by Oberbeck and Quaide (1967) 

and Quaide and Oberbeck (1968), who explained only 

the central mounds in small (less than 250 m across) 

craters on the lunar mare surfaces in terms of the 

strength-discontinuity mechanism. I subsequently 

suggested (Pike, 197 la) that Oberbeck and Quaide's 

interpretation might be scaled up to account for the 

transition from simple to complex morphologies in 

lunar craters over 1 km across. Head (1976b) proposed 

more specifically that the "megaregolith" believed to 

veneer the lunar highlands (Hartmann, 1973) is the 

low-cohesion surface layer by which the strength-discontinuity mechanism can effect the shape transition 

in craters in the 10-20-km diameter range. However, 

the highland "megaregolith," about 2.5 km deep (Head, 

1976b), is two orders of magnitude thicker than typical
GEOMETRIC INTERPRETATION OF LUNAR CRATERS C51

10

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°- 1.0

CCho

LL

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(D 

Ul I

0.1

1.0 10 100 500

RIM-CREST DIAMETER, IN KILOMETERS

FIGURE 32.—Relation between relative height of central peak and rim diameter for 29 fresh lunar craters. Peaks in seven craters under 28 

km across are systematically lower than peaks in the 22 larger craters, for which an equation has been fit (table 6), supplanting the fit to 

shadow data done by Wood (1973). Solid curve is Wood's (1973) fit to central uplifts estimated within terrestrial impact structures. All 

data points from photogrammetry of Apollo pictures.

mare regolith, about 10 m (Oberbeck and Quaide, 

1967; Quaide and Oberbeck, 1968), a difference that 

should be reflected by a dramatic contrast between 

maria and uplands in the crater size ranges at which 

the shape transition occurs. The absence of any such 

difference in the geometric data presented in this report suggests that the strength-discontinuity 

hypothesis for the formation of peaks in large lunar 

craters requires further evaluation.

Critical to explaining the existance of central peaks 

or uplifts according to any process is their obvious dependence upon crater size. Why peaks form only in 

craters above a particular diameter is not clear, but 

two influences, an energy threshold and gravitational 

acceleration, may be involved. The role of increased 

energy in changing crater form was addressed implicitly by Gilbert (1893), discussed briefly by Baldwin 

(1963, p. 184), and still is speculative. A threshold in 

the relation between impact energy and strength of the 

target materials could provide physical circumstances 

that favor development of central peaks and other 

structural displacements. It is suggested that the mechanics of impact cratering change qualitatively once 

energy exceeds some critical value. Perhaps beyond 

this level, a significant fraction of the target 

material—some of which becomes impact melt— 

behaves as a fluid rather than as fractured and comminuted rock (Baldwin, 1963). The energy level at which 

the postulated change in shock-hydrodynamic regime 

occurs on the Moon is unknown, but it may not be far 

from the 1024 to 1025 ergs (Innes, 1961; Baldwin, 1963) 

thought to be required for development of a central
C52 APOLLO 15-17 ORBITAL INVESTIGATIONS

uplift in terrestrial meteorite craters. Chao's (1976) 

suggestion that the basic mechanism producing shallow (complex) impact craters such as the Ries may differ substantially from that responsible for bowl-shaped 

(simple) craters such as the Arizona Meteor Crater is 

consistent with the existence of a critical energy 

threshold.

Gravity almost certainly affects the way energy 

couples with the target materials to form peaks, because central uplifts (Dence, 1968) and diminished 

depth/diameter ratios (fig. 33, see also David, 1975; 

Losej and Beales, 1975) first appear in terrestrial impact craters 2 to 4 km across, or about one-sixth the 

crater size at which similar changes take place on the 

Moon. Some influence of gravity also is indicated for 

the appearance of central peaks in craters on Mars 

(Hartmann, 1972b) and Mercury (Gault, Wedekind, 

and others, 1975; Gault, Guest, and others, 1975) and 

possibly for the relative height of peaks and uplifts in 

terrestrial and lunar craters (fig. 32; Wood, 1973; 

Gault, Wedekind, and others, 1975; Gault, Guest, and 

others, 1975). Recently I have summarized observa10

cc

ULJ 

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EXPLANATION

Terrestrial impact crater 

Lunar curve

0.1 1.0 10 

DIAMETER IN KILOMETERS

100

FIGURE 33.—Depth/diameter plot for 21 terrestrial impact craters 

(some unpublished data, but most from Pike, 1972b). Craters tend 

to be relatively fresh, otherwise depths were determined from results of drilling or gravity work. Solid curves are equations (1) and 

(2) for fresh lunar craters, simple and complex, respectively. Array 

of terrestrial craters inflects at about one-sixth of the diameter at 

which curve inflects for lunar craters, a consequence of the sixfold 

difference in gravitational acceleration at the planetary surface.

tions showing that the simple-to-complex transition is 

an inverse function of gravitational attraction on 

Earth, the Moon, Mars, and Mercury (Pike, 1978b). 

Although the influence of gravity has been interpreted 

in the context of the centripetal-collapse hypothesis 

(Quaide and others, 1965; Gault, Wedekind, and 

others, 1975; Gault, Guest, and others, 1975), it has not 

yet been evaluated in terms of elastic recoil. Currently 

the role of elastic recoil in forming peaks is being reexamined by computer-modeling large experimental 

craters from more elaborate hydrodynamic and 

material-strength assumptions than have been used in 

past simulations of this type (Ullrich, 1976; Ullrich 

and others, 1977).

A third explanation for the size- or energy-dependent 

occurrence of central peaks might be systematic decrease of scaled depth of burial with increasing crater 

size, although the role of gravity scaling in such a 

scheme is difficult to follow at this time. There are at 

least two ways this might occur. Baldwin (1963) contended that larger meteoroids flatten more upon impact and disintegrate at correspondingly shallower 

depths of burial. He also suggested that if central 

peaks are rebound phenomena, then they must result 

from relatively shallow releases of energy or else be 

buried by fallback. The work of Roddy (1968), Roddy 

and others (1969), and Ullrich (1976) now suggests 

that only shallow impacts are capable of producing 

central peaks in terrestrial craters. If the cometary impact hypothesis for peak formation can be dismissed, 

then the occurrence of central uplifts may instead be 

governed by depth of penetration of the projectile. This 

depth in turn could relate directly to the bulk of the 

impacting body and its flattening, as discussed previously by Baldwin and several others (see Baldwin, 

1963, p. 167). However, cometary impact may not be so 

easily ignored, despite the seeming illogic that only 

comets (low density) can make large craters anfl only 

meteoroids (high density) can make small craters. At 

this time there is no evidence on which to test the idea. 

Until more is known about the composition of impactproducing projectiles in the solar system (Wetherill, 

1977), the possibility of size-dependency in the density 

distribution of impacting bodies remains open.

Other gravity-driven mechanisms for modifying 

craters operate at a more leisurely pace than those 

above. Much of the diminution of crater depth and rim 

height at the 10-20-km size range has been attributed 

to isostatic compensation (Danes, 1962; Baldwin, 

1963; 1968c, 1971; Scott, 1967) and to isostasy plus 

consequent volcanism (Pike, 1967). There seems little 

doubt that viscous flow of the substrate has raised the 

floors of the multi-ringed basins and is intimately involved with mare volcanism (Wise and Yates, 1970).
GEOMETRIC INTERPRETATION OF LUNAR CRATERS C53

Isostasy also has deformed some large craters such as 

Taruntius (Masursky, 1964) and Humboldt (Baldwin, 

1968c), where the central peak appears to have been 

carried up along with the uplifted, cracked crater floor. 

Many of these craters are located along the margins of 

basins (Pike, 1968, 197 la). However, inflection of the 

statistical relation between flank width and rim diameter suggests that crater rims generally lose height by 

retreat of the rim crest through slumping. Thus, isostasy probably does not affect most craters as small as 

10 to 30 km across, and indeed, with few exceptions 

(such as Taruntius) only craters over about 150 km 

across have the substantial positive Bouguer gravity 

anomalies that indicate deformation of the lunar mantle, as in response to isostasy (Phillips and others, 

1975). The minor negative gravity anomalies that typically accompany the smaller complex craters, like 

Copernicus, can be accounted for entirely by their topography (Sjogren and others, 1973; Bowin, 1975); isostasy does not seem to have been involved. The 200-km 

crater Humboldt (fig. 34) has a bulged and cracked 

floor that does suggest some isostatic uplift (Baldwin, 

1968c), along with annular deposits of presumably volcanic material that may have accompanied the deformation (Pike, 1968), but the central peak complex does 

not appear large enough to have been uplifted rather 

than partially buried. Moreover, Schultz (1976a, b) has 

identified over 200 craters that possess similar or

nearly similar modified floors. These "floor-fractured 

craters" range in size from about 10 km to over 200 km 

across and may reflect post-impact modification in 

which isostasy has not played a major role (see also 

Brennan, 1975). Under normal circumstances isostasy 

does not appear to explain enough of the sizedependent morphologic observations summarized here, 

especially emplacement of the central peak.

MINOR DIFFERENCES IN CRATER SHAPE

The most conspicuous contrasts in morphology 

among craters on the Moon reflect mode of origin, size, 

and relative age. Genetic and size-dependent variations among craters have been treated here, whereas 

age-related differences have been analyzed elsewhere 

(Baldwin, 1949; Pike, 1971; Pohn and Offield, 1970; 

Schubert and others, 1977) and some of them recently 

reviewed by Head (1975). Other kinds of contrasts 

among lunar craters have been recognized by observers 

at different times, but they are of less importance, 

where real. Three of these variations are briefly examined here from Apollo 15-17 data: results from the new 

information show that craters on mare surfaces are 

much like craters on uplands, except for the expected 

differences in evenness of the rim imparted by irregular upland terrain; there are no differences between 

far-side and near-side craters on the Moon; and the

FIGURE 34.—Oblique view of large crater Humboldt on far eastern limb of the Moon. Cracks and dark patches on floor of this 200-kmdiameter crater indicate isostatic compensation (Pike, 1968; Baldwin, 1968c). Note unusual double-rimmed crater on floor of Humboldt. 

View is to south. Apollo 15 mapping-camera photograph 2513. Sun is to right.
C54 APOLLO 15-17 ORBITAL INVESTIGATIONS

height of crater rims does not appear to vary systematically with azimuth.

MARE-TERRA CONTRASTS

Some older Earth-based measurements of lunar craters over 20 km across have suggested that craters 

formed on mare surfaces are systematically shallower 

than craters formed on upland surfaces (MacDonald, 

1929, 1931; Marcus, 1967; Pike, 1972c). These data appear to have been inaccurate or unrepresentative. 

There is no such contrast in the new photogrammetric 

depth/diameter data (fig. 19), and a difference so small 

as to be negligible and readily explained is evident in a 

second set of new depth/diameter data, which resulted 

from updating of a standard catalog, the System of 

Lunar Craters (Arthur and others, 1963, 1964, 1965, 

1966). The latter depths, which were generated from 

measurements of shadow lengths on Lunar Orbiter IV 

high-resolution images of the near side, already have 

been described in detail (Wood, 1972; Arthur, 1974). 

Nearly half of the depths from the first section of the 

revised catalog (Arthur, 1974) are plotted against their 

diameters in figure 35 (see also Pike, 1974b). The sample of 950 craters was stratified according to conventions adopted in the System of Lunar Craters (Arthur and others, 1963). Only the freshest looking 

(Class 1) craters are included, and craters on mare surfaces are plotted separately from craters on upland surfaces.

Depth/diameter distributions of mare and upland 

craters less than about 15 km across barely differ from 

each other (fig. 35). Visually this minor contrast is evident only upon overlaying the two plots. The equations 

(fitted by inspection only) for mare and upland craters 

are, respectively,

and

0.18ZV-03 

0.15ZX. 1 - 09

(16)

(17)

(variables in kilometers). The two distributions are 

only slightly lower and steeper than the distribution in 

figure 19, a small difference that also can be seen only 

by overlaying the three graphs. The significance, if 

any, of differences between mare and upland craters 

suggested by the new shadow-length data may be obtainable subsequently from more exact regression and 

an analysis of variance. The contrast could well reflect 

the slightly greater scatter in the upland data. Some 

such scatter is to be expected from inadvertent 

classification of Class 2 upland craters (somewhat older 

looking; rims less crisp than Class 1) as Class 1 objects, 

and from a greater variance in the depths of the upland

craters relative to that of mare-crater depths. Unevenness of the lunar upland terrain almost certainly introduces some minor scatter from both these sources.

Unlike all other aspects of crater geometry, none of 

which show any significant differences between fresh 

upland and mare craters, evenness of the rim crest is 

markedly less for upland than for mare craters. The 

contrast is evident both in the plot of rim-crest relief 

against crater diameter (fig. 25) and in the graph showing coefficient of variation for rim-crest elevation (Vc ) 

as a function of crater size (fig. 26). The influence of 

pre-crater topography upon this difference is obvious 

from perusal of the LTO's. For example, relief along 

the rim of the 27-km-diameter crater Menelaus is 

2,000 m, more than twice the average rim height and 

nearly equal to the depth, because the crater formed on 

the slopes of Montes Haemus rather than on a mare 

surface or on less irregular upland terrain. In contrast, 

relief along the rim crest of Madler, a mare crater similar to Menelaus in size and relative age but located on 

much smoother and more level terrain, is only 700 m. 

These two craters lie on opposite margins of the rimcrest relief/diameter distribution (fig. 25); there are 

numerous other examples.

The uneven upland terrain also is responsible for the 

marked contrast between mare and upland craters in 

figure 26. The inflection in the Fc/diameter distribution would not be as recognizable without the mare/ 

upland distinction. Within the larger sample of figure 

26, the mare craters serve as a control group by which 

rim-crest roughness that is the result of irregular 

background terrain can be separated from roughness 

that is related only to crater size. Among the larger 

upland craters Proclus, followed by King and Tsiolkovskiy, is the most seriously affected by an irregular 

background, whereas Sklodowska has not been distorted at all. Particular care must be taken when 

measuring the dimensions of the more deformed 

among such craters in the lunar uplands in order to 

secure only representative data for subsequent 

analysis. It is conceivable that the mare/terra contrast 

in rim evenness may accompany analogous differences 

in the crater sizes at which central peaks and rim-wall 

terraces begin to occur in mare and highland craters 

(Cintala and others, 1977). If real, these differences 

may reflect only the difference in pre-crater topography rather than any significant contrasts in properties of the target materials.

FAR SIDE-NEAR SIDE CONTRASTS

Through photometric processing of Zond 3 images, 

Mironova (1970a, b) generated the first topographic 

measurements ever made of craters on the Moon's far
GEOMETRIC INTERPRETATION OF LUNAR CRATERS C55

EXPLANATION

Mare crater, «=500

EXPLANATION

Upland crater, «=450

RIM-CREST DIAMETER. IN KILOMETERS

FIGURE 35.—Relation between depth and diameter for 950 fresh (Class 1) lunar craters from Pike (1974b). All data from Arthur (1974). 

Crater depths measured from shadow lengths on Lunar Orbiter IV high-resolution images. Distributions differ only slightly from one 

another and from that for small craters in figure 19 (see text).

side. The 29 craters that were profiled and measured 

range in diameter from 16 to 100 km and vary in relative age. From a brief analysis of surface geometry, 

Mironova concluded that the far-side craters were 

similar in shape to near-side craters and were volcanic 

in origin. I examined both conclusions using preliminary photogrammetric measurements of 25 different 

far-side craters (Pike, 1972a, 1974c). Although the 

Apollo craters and the Zond craters are not the same, 

the statistics of each group are judged to adequately 

represent the larger population of far-side craters. My 

two studies revealed neither any distinction between 

far-side and near-side craters nor evidence for a volcanic origin of the craters. The more abundant Apollo 

data on fresh far-side craters that were collected to 

update the preliminary work (figs. 36 and 37) are consistent with these conclusions.

Compared to the new photogrammetric data on farside craters, the photoclinometric measurements of 

Mironova are not very accurate. Depth measurements 

from the Zond 3 data generally represent craters as 

being excessively shallow, even when taking relative

age into consideration (fig. 36). The rim-height data 

from Zond 3, which also are inconsistent with photogrammetric measurements, are too low for smaller 

craters and too high for larger craters (fig. 37). Despite 

the absolute error, photometric analysis of the Zond 3 

imagery at least preserved the relative proportion between crater depth and rim height for many of the 

craters. The height/depth ratio for fresh lunar craters 

over 15 km across varies between 0.32 and 0.38 (figs. 

19 and 20, equations 2 and 4); for the 18 far-side craters 

in the fresh and intermediate-age categories of 

Mironova, the ratio averages 0.34; however, the Zond 

data do range widely, from 0.09 to 1.20. Under favorable circumstances, useful information can be extracted from the photoclinometric technique (Lambiotte and 

Taylor, 1968; Leighton, 1966), but the low-quality 

measurements obtained by Mironova reflect problems 

that need to be overcome before photoclinometry can be 

applied confidently to large topographic features of 

other planets.

Using the depth/diameter proportion, Mironova 

(1970a) deduced a volcanic genesis for both near-side
C56 APOLLO 15-17 ORBITAL INVESTIGATIONS

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EXPLANATION

• Far-side crater 

O Near-side crater

1.0 10 100 1000

DIAMETERJN KILOMETERS

FIGURE 36.—Depth/diameter plot for 58 far-side and 78 near-side craters on the Moon. Most craters are fresh-looking and were measured 

from photogrammetric data. Polygon contains Zond 3 photoclinometric data for 29 far-side craters, which do not coincide with newer data

(see text).

and far-side craters from the same type of analog argument by which Baldwin (1949) originally inferred an 

impact origin for near-side craters. Although the 

depth/diameter relation has become the most oftenused morphometric criterion of crater genesis, the generous overlap among depth/diameter fields describing 

all types of lunar and terrestrial craters demonstrates 

that it is also the least reliable of all available geometric discriminants (Green, 1959; Steinberg, 1968; 

Baldwin, 1968a: Pike, 1972b) and cannot support any 

genetic hypothesis. Crater-depth measurements are 

better applied to the analysis of relative age or postimpact modification (Baldwin, 1949: Pike, 1968). Rim 

height is a more reliable—although still imperfect— 

indicator of crater genesis, but the Zond 3 rim height/ 

diameter data in figure 37 are characteristic of impact 

craters rather than volcanoes (Pike, 1973a, 1974c).

According to photogrammetric measurements of 

lunar craters, (1) differences between near-side and 

far-side craters are negligible and (2) far-side craters 

more probably formed by the impact of large cosmic

bodies than by volcanism. Mironova (1970a) also 

reached the first conclusion, but as shown here, not for 

the same reasons. Evidently the photoclinometric 

method, the only way to obtain quantitative data from 

Zond images, did not yield topographic information as 

accurate as that of even the older shadow-length crater 

measurements. Mironova's conclusions that the lunar 

craters formed from volcanic eruptions is not supported 

by the available data. Instead, the results obtained 

from Apollo metric cameras, and indeed the Zond 3 

measurements themselves, are more consistent with 

an impact genesis for craters on the Moon's far side.

AZIMUTH-DEPENDENT DIFFERENCES

According to early morphometric work by MacDonald (1929) on data collected and published by 

Schmidt (1878), the rim crests of large craters on the 

Moon generally are much higher along the eastern and 

western parts of their walls than along northern and 

southern segments. Although photointerpretation and
GEOMETRIC INTERPRETATION OF LUNAR CRATERS C57

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EXPLANATION

• Far-side crater 

O Near-side crater

0.01

1.0 10 100 1000

DIAMETERJN KILOMETERS

FIGURE 37.—Rim-height/diameter plot for 54 far-side and 78 near-side craters on the Moon. Most craters are fresh-looking and were 

measured from photogrammetric data. Polygon contains Zond 3 photoclinometric data for 29 far-side craters, which do not coincide with 

newer data (see text).

visual observation of the lunar surface do not show 

such a strong azimuth-related tendency, certain preApollo contour maps of the Moon indeed portray many 

craters with a systematic excess in height of the east 

and west walls (Pike, 1968). The 1:5,000,000 U.S. 

Army Map Service Topographic Lunar Map and various 1:1,000,000 U.S. Air Force Lunar Aeronautical

Charts both show this tendency. A contour map of the 

crater Aristarchus from stereophotogrammetry of 

Lunar Orbiter V images clearly carried this systematic 

difference in rim height (unpublished U.S. Geological 

Survey data, 1968), although the difference was much 

reduced in another map of the same area from the same 

data by the U.S. Air Force Aeronautical Chart and
C58 APOLLO 15-17 ORBITAL INVESTIGATIONS

Information Center (1972). If real, the systematic 

rim-height relation could have interesting structural 

implications—possibly involving influence of broadscale fracture networks—and would further complicate mensuration of crater average depths and rim 

heights. The considerable importance ascribed by some 

investigators to lineament patterns and possible 

planetwide grid systems of tectonic features (Fielder, 

1965) prompted a test for systematic dependence of crater rim-crest profile on lunar azimuth.

The existence of the height excess in the east-west 

direction was tested with an updated sample of 36 

rim-crest elevations from each of 34 craters used previously in this report for examining the comparative 

roughness of crater rim crests (table 8). Following procedures used in a preliminary test (Pike, 1973a), six 

quantities were calculated for each crater: (1) total relief along the rim crest, (2) mean elevation of the rim 

crest, and (3-6) deviation from mean elevation of the 

heights of eastern, western, northern, and southern 

walls, expressed as a plus or minus percentage of total 

rim-crest relief. Deviations were averaged from three 

elevations centered on the four cardinal directions and 

tabulated as in Pike (1973a) for each of the craters, 

which range in diameter from 1.6 to 197 km.

Except for a slight tendency for crater rims to be 

higher in the south and lower in the north, there is no 

systematic relation among the rim-crest results. When 

averaged for all 34 craters, the four percentage deviations from mean elevation are +0.02 (east), -0.01 

(west), -0.04 (north), and -0.05 (south). The absolute 

values average even lower than the deviations resulting from the preliminary study (Pike, 1973a). The essential randomness of rim-crest height-azimuth also is 

evident in table 8. The systematic arrangement sugTABLE 8.—Qualitative deviations from mean rim-crest elevation

90° Azimuth 

270° 360° 180° (34 craters)

19+ 15- 18+ 16- 12+ 22- 22+ 12- 71+ 65-

gested by the data of MacDonald and the early contour 

maps, + + - - in table 8, occurs only in Linne, Euler 

K, and Littrow BD, three of the 12 small mare craters 

examined in this analysis. Summed plus and minus 

deviations confirm that southern segments of crater 

rims tend to be somewhat higher than average and 

northern rims tend to be lower, whereas east and west 

rims are roughly comparable in elevation. The 

significance, if any, of this contrast is not evident. The 

results, which do not change if only mare craters are 

included in the tabulation, imply that variation in 

height along the rim crest of a lunar crater occurs more 

or less randomly with respect to azimuth. Croft and 

Kaula (1976) reached similar conclusions from their 

analysis of rim-crest ellipticity and azimuth for nearly 

300 craters portrayed on the LTO's. The systematic 

variations in height found by MacDonald and observed 

by compilers of the earlier contour maps and charts 

probably are not real. They may have resulted from the 

high-contrast, east-west illumination of the lunar surface and from inadequate photogrammetric separation 

in the available Earth-based photographs.

SUMMARY AND CONCLUSIONS

This report samples the variety of results obtained 

by statistical analysis of the topography of the Moon's 

craters, principally using new data reduced photogrammetrically from Apollo orbital pictures. Many of 

these results have been published elsewhere as separate contributions. The systematic measurements of 

rim and floor diameter, rim height and flank width, 

and crater depth and circularity for some 500 lunar 

craters between about 400 m and about 300 km across 

provide the planetary science community with a new 

data set. The new information supplants earlier compilations of telescopic measurements, and is more complete and precise than data sets compiled from Lunar 

Orbiter results. The new data emphasize freshly 

formed craters and are more accurate than the older 

measurement. The principal aim of this series of investigations was to reevaluate several old problems that 

might benefit from the application of improved data. 

The work addresses four overall areas of possible influences on the variation in shape of lunar craters: mode 

of origin, size, geography, and azimuth.

The new data confirm quite conclusively the thesis of 

Gilbert (1893) that different modes of origin produce 

distinctively different crater shapes. The most fundamental contrast is that between volcanic craters and 

impact craters. According to multivariate statistical 

models of dimensionless variables, the two processes 

yield crater shapes that are all but mutually exclusive. 

With respect to the distribution of mass, most vol-
GEOMETRIC INTERPRETATION OF LUNAR CRATERS C59

canoes are positive landforms, whereas impact craters 

are always negative features in the terrain. To some 

extent, results of the multivariate analysis can be used 

to discriminate impact craters from volcanoes on the 

basis of their gross geometry. Although maar volcanoes on Earth share some geometric characteristics 

with impact craters, they differ sharply in others. 

Secondary-impact craters on the Moon differ similarly 

in geometry from primary craters of the lunar main 

sequence and somewhat resemble terrestrial maars. 

The closest terrestrial volcanic analog of lunar impact 

craters is the maar, not the caldera. Of the several 

categories of calderas examined here, only the large 

ash-flow plains even remotely approach lunar impact 

craters in shape. Calderas are particularly poor 

analogs of lunar craters of any type. Some special 

classes of lunar landforms evidently are true central 

volcanoes, but they are small and rare. The low domes 

on mare surfaces probably are a type of shield volcano; 

the small raised cones on mare and crater-floor surfaces could be some sort of spatter-cinder edifice. The 

dark-halo craters along rilles on the floor of Alphonsus 

are negative landforms, but probably did not form in 

phreatic eruptions like most terrestrial maars. One 

class of lunar craters roughly 20 to 40 km across, often 

described as smooth-rimmed craters, may have had an 

atypical impact origin or may have been modified by 

mare lavas after their formation. These craters almost 

certainly are not calderas. The depressions atop summits of central peaks within many large craters on the 

Moon likewise do not seem to be volcanic vents, but 

rather superposed and distorted impact craters or collapse depressions in the irregular peak topography. 

The small, mysterious crater Linne turns out to be a 

quite ordinary—if exceedingly young—impact crater; 

it appears to have undergone none of the several "volcanic changes" that were ascribed to it by classical 

selenologists. Most of the volcanism associated with 

craters on the Moon is post-impact modification, especially on the flat floors of large craters, and does not 

arise from primary crater genesis.

Aside from the fundamental contrast between impact craters and multi-ring basins, the outstanding 

size-dependent variation among fresh lunar craters between 400 m and about 300 km across is the transition 

from small simple craters to large complex craters at a 

threshold diameter of 10 to 20 km. Eleven of the constituent changes in shape are documented here in considerable detail: the appearance of central peaks, flat 

floors, rim-wall terraces, and swirl texture, and the 

variation with rim diameter of crater depth, rim 

height, flank width, rim-wall slope, floor diameter, 

rim-crest circularity, and rim evenness. Observations 

of some of these changes in craters on Earth, Mars, and

Mercury suggest that the transition is a widespread 

and fundamental phenomenon of planetary impact 

cratering. There are several contending explanations 

for the onset of these features, especially peaks, two of 

the more attractive being centripetal collapse and elastic recoil of target rocks. Size-dependence in the density distribution of impacting bodies is a third alternative. Gravitational acceleration, an impact-energy/ 

strength-of-materials threshold, and layering of the 

target material may modulate the initiating 

process—which remains uncertain. Future research in 

this area should concentrate on developing criteria for 

testing all of the alternative hypotheses. A solution to 

the problem may eventually emerge from some combination of theoretical modeling of the impact process, 

interplanetary comparison of crater shapes, and the 

geologic investigation of terrestrial impact structures 

and experimental-explosion craters.

The variation of several of the crater dimensions 

with crater size, as represented by rim-crest diameter, 

has been expressed in linear power functions and standard errors of the estimate of the dependent variable 

on rim diameter. These new relations from Apollo 

measurements of depth/diameter, rim height/diameter, 

floor diameter/rim diameter, and flank width/diameter 

constitute an average geometric model for fresh craters 

on the Moon. Separate calculations have been made for 

simple and for complex craters. These models serve as 

standards of comparison for older degraded lunar craters, for both fresh and eroded impact craters on the 

other planets, and for experimental-impact and explosion craters excavated on Earth in both the field and 

the laboratory. If simplifying assumptions are allowed 

concerning the shape of the crater interior and the extent of ejecta beyond the edge of the topographic rim 

flank, then these models also can be used to estimate 

volumes for lunar impact craters.

The generalized statistical analysis—mainly curve 

fitting—carried out in this report suggests that fresh 

impact craters on the lunar uplands do not differ 

significantly in shape from craters on the mare surfaces and further, that craters on the Moon's far side 

are similar to near-side craters. The rim crests of 

upland craters are systematically less even than those 

of craters formed on mare and upland plains surfaces, 

it is true, but this difference is ascribable wholly to the 

more uneven surfaces on which the upland craters 

formed rather than to any contrast in substrate composition, strength properties, or structure. It is possible 

that differences in the shapes of mare and upland craters may emerge from more detailed types of analysis 

than those carried out here, but the evidence thus far 

indicates no important differences. In light of current 

knowledge of impact cratering, it would have been sur-
C60 APOLLO 15-17 ORBITAL INVESTIGATIONS

prising indeed to have found any significant disparity 

between craters located on the lunar near side and on 

the far side. There are no such differences according to 

the analysis done here, but it was still advisable to 

address the issue in case an unexpected variation 

might have turned up.

The test for systematic dependence of crater rimcrest profile on lunar azimuth shows that the relative 

height of rim crests is distributed randomly with respect to compass direction. Rims are not systematically 

higher or lower along preferred orientations. It is also 

well known, however, that segments of crater rim 

crests in plan—as opposed to profile—often appear to 

follow local and regional lineament patterns. These 

two results do not necessarily conflict and in any case 

demonstrate that structural patterns are pervasive 

enough to influence the planimetric outline of a crater, 

but insufficiently strong to effect a preferential distribution of mass in the rim.

The work reported here suggests that impact craters 

are among the most orderly of all landforms. Impact 

craters have an even more uniform surface geometry 

than sand dunes, drumlins, alluvial fans, fluvial 

watersheds, and other landscape features. This orderliness arises from the brief duration and high energy of 

the formational process. Unlike most other planetary 

landforms, craters form instantanously, by a shockwave mechanism that within limits is fairly uniform 

over a very wide range of formational energies. An 

equilibrium geometry—that of a "landform of 

explosion"—is imparted to impact craters at the outset, whereas most landforms reach an equilibrium 

configuration only after prolonged operation of less 

energetic geomorphic processes. The geomorphic history of an impact crater is one of degradation of a pristine and geometrically a very well defined shape, 

rather than one of gradual development of a characteristic morphology as the end product.

The analysis of degradational histories of lunar craters is only one of many different problems that might 

well be examined in further analysis of the new quantitative data now available from the orbital cameras of 

the last three Apollo flights to the Moon. Other problem areas in crater geometry include a reevaluation of 

volumetric models, more detailed study of flat floors, 

relations between apparent crater dimension, better 

information on the size and location of central peaks, 

testing various models of rim-wall slumping, more detailed evaluation of any relation between rim-crest 

planimetry and planetary lineaments, regional variations in degree of crater infilling, improved geometric 

models of the shape of secondary-impact craters from 

multi-ring basins, and more precise models of ejecta 

distribution.

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GEOMETRIC INTERPRETATION OF LUNAR CRATERS C63

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C64 APOLLO 15-17 ORBITAL INVESTIGATIONS

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GEOMETRIC INTERPRETATION OF LUNAR CRATERS C65

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C66 APOLLO 15-17 ORBITAL INVESTIGATIONS

SUPPLEMENTAL INFORMATION

MEASURING CRATER DIMENSIONS

Previous work has established at least six quantities that 

adequately specify the overall surface geometry of a lunar crater 

(MacDonald, 1931; Ronca and Salisbury, 1966): (1) rim-crest diameter (or radius); (2) width of exterior rim flank; (3) diameter (radius) of 

flat inner floor, if present; (4) depth; (5) height of rim crest above 

surrounding terrain; and (6) degree of circularity of the rim crest (fig. 

1). Average values of the first five quantities (only four if there is no 

flat floor) were calculated for the 504 lunar craters studied here and 

all but 17 of the 119 craters added later. Other variables can be 

derived from the first five: average slope of the rim flank, width and 

average slope of the wall between the rim crest and the crater bottom 

or flat floor, and the depth of the crater below the pre-crater datum. 

Volumes of a crater and its raised rim can be estimated with fair 

accuracy from average values of these variables, although contour 

planimetry always yields more reliable results. Two additional 

quantities that are useful in the analysis of lunar craters, the radius 

of the bright rays surrounding the crater (Baldwin, 1963, table 32) 

and the diameter of the continuous deposit of ejecta material (Moore 

and others, 1974), can be obtained from photointerpretation rather 

than from measurement of contours but were not gathered 

specifically for this report. The six quantities and combinations of 

them provide a comprehensive geometric signature by which lunar 

craters can be described, compared with each other, and then compared with craters on the Earth and elsewhere.

Similar techniques of mensuration were applied to craters of all 

sizes, although they became exceedingly time-consuming for the 

largest features. The procedure is given in detail here so that other 

workers can more easily supplement the present sample with uniform data of their own. First, the planimetric outline of the rim crest 

was carefully traced on an overlay (a step sometimes dispensed with 

for small sharp craters). The rim crest is defined as the marked break 

in slope where the (typically gentler) rim-flank or rim-summit 

topography—the latter is the "wreath" described by Gilbert 

(1893)—meets the (typically steeper) rim wall. Especially for larger 

complex craters (over 30 km across), the rim crest does not necessarily coincide with maximum elevations along the rim topography, 

which may occur on overlapping segments of concentric ridges located a short distance outward from the rim crest as defined here.

After the circumference of the rim crest was traced, it was fitted 

with circumscribed and inscribed circles of known diameter, using a 

set of transparent templates. The two circle diameters were averaged 

and multiplied by a factor appropriate to the map scale to yield the 

crater rim-crest diameter, Dr, in kilometers. The index of circularity, 

which is defined as the ratio of area of the inscribed circle to area of 

the circumscribed circle, is obtained by dividing the squared diameter of the smaller circle by the squared diameter of the larger. First 

used by the author on lunar craters in unpublished experiments in 

1967, this method yields rapid and reasonably sensitive results, especially for very small craters that would be less amenable to the 

more elaborate techniques. More sophisticated measures of circularity were judged not to warrant the considerable effort required for 

their calculation (Murray and Guest, 1970; Oberbeck and others, 

1972).

Average width of the rim flank, We , was obtained by subtracting 

rim-crest diameter from the diameter at the base of the rim flank and 

halving the result, or it was measured directly with a compass at 

intervals along the circumference. The latter technique is more practicable for larger craters. The base of the rim flank commonly can be 

located without difficulty in terrain of low relief at a topographic 

break in slope (Baldwin, 1963), which may in fact represent the outer 

limit of structurally upthrust bedrock beneath the ejecta blanket 

(Pike, 1968). The topographic inflection is difficult to locate accurately on LTO's for the larger craters without close scrutiny of the 

terrain, especially in the uplands, and in some few cases it was necessary to draw several (radial) topographic profiles. Although this 

break in slope defines a convenient outer limit to the main structure 

of an impact crater, it omits some of the crater's ejecta and thus 

should not be used in volumetric calculations (Pike, 1967) without 

provision for including ejecta beyond the rim flank of the crater.

If a flat floor was present within a crater, its diameter (D/) was 

measured by fitting a circle to the topographic break at the foot of the 

inner wall slope. In cases where slump blocks from the walls above 

produced an irregular floor outline, then an average diameter was 

calculated from readings at several azimuths. In some craters within 

the 10-20-km diameter range, the inflection at the juncture of floor 

and inner wall is poorly defined, and the floor diameter must be 

estimated from changes in contour spacing near the bottom of the 

crater.

Average crater depths (R,) were obtained in one of two ways. For 

craters with no flat floors, the spot height shown on the map at the 

crater bottom (unless that elevation did not represent the lowest 

within the crater) was subtracted from the average elevation of the 

rim crest, or for very small craters, from the spot height given for the 

rim crest. The number of elevation readings along the rim crest was 

increased considerably for large craters or for craters of any size 

where the rim crest appeared irregular. For craters with flat floors, 

the mean elevation of the floor was determined either from several 

spot elevations or by inspection of contours and then subtracted from 

the mean rim-crest elevation. Craters that evidently formed on sloping surfaces, such as Menelaus, posed a special problem because 

average elevations of the highly variable levels of rim crest and floor 

are misrepresentative. Wherever possible in such cases, the relief 

measurements were referenced to a level pre-crater datum. Where 

such adjusted values could not be obtained, the crater was omitted 

from further analysis to avoid contaminating the sample.

Average height of the crater rim above the surrounding terrain 

(Re ) was calculated by subtracting the mean elevation at the foot of 

the rim flank from the rim-crest elevation. The former figure is obtained easily once the width of the rim flank has been determined. 

Many elevation readings are essential if the crater is large or 

situated in irregular topography. In calculating rim height for many 

overlapping craters in the uplands, it is necessary to distinguish 

floors of large adjacent craters from the actual pre-crater datum, as 

the two figures may differ considerably in elevation. In such cases, 

indiscriminately measuring elevations at the foot of the rim flank all 

around a large crater can lead to gross overestimates of its rim 

height. Also, the rim flanks of some large craters appear to have 

little or no relief at all along parts of their circumferences, because 

the crater formed on a highly irregular surface. Upland craters in 

which the rim height could not be determined accurately or objectively were omitted from the sample.

TABULATION OF THE NEW DATA

Descriptive data on the 504 lunar craters and the 59 terrestrial 

impact and experimental craters that have been included for comparison are listed in tables 9 and 10, respectively. Data on 119 lunar 

craters that were measured after the work had been completed also 

are given in table 9. Following the precedent of MacDonald (1931) 

and Baldwin (1963), craters in each of the various categories are 

tabulated in order of decreasing size (rim diameter). All in all, this 

arrangement is thought to be more convenient than an alphabetical 

array (Arthur, 1974) or a listing by position on the Moon (Arthur and 

others, 1963), although each has its advantages for specific purposes. 

Information on primary impact craters of the lunar main sequence is 

arranged in twelve columns; there are only six or seven columns for 

other types of craters.
GEOMETRIC INTERPRETATION OF LUNAR CRATERS C67

The first column (Moon only) gives the crater's identification 

number, which is an exact function of geographic location, from the 

University of Arizona catalog prepared under the direction of Arthur 

(Arthur and others, 1963, 1964, 1965, and 1966). To date only the 

near-side lists, quadrants A-D, have been published, but the far-side 

listing is in an advanced state of preparation (E. A. Whitaker, personal communication). No numbers have been assigned the very 

smallest craters, however. Names are given, where possible, in the 

second column; (IS) denotes a smooth-rimmed crater and (1C) denotes 

a crater that probably has undergone considerable isostatic compensation. Diacritical marks are absent from the names because a 

computer-printout format was used. The third column lists sources of 

the measurements. Besides the l:250,000-scale topographic orthophotomaps (designated LTO plus the sheet number, or only the 

sheet number for larger scale maps), these are primarily Earth-based 

shadow-length measurements from the LAC program (SHADOW 

DATA), topographic profiles compiled stereoscopically by the U.S. 

Geological Survey from Apollo pictures (USGS PHOTOGRM), and 

large-scale contour maps made from Apollo pictures (for example, 

USGSMAP 1/100). A "P" in the fourth column denotes presence of a 

well-defined central peak, and "R" a ridge. The numbers 1-4 in the 

fifth column denote the relative-age classification of the Arthur 

catalog (Arthur and others, 1963), 1 being youngest in appearance 

and 4 oldest; "F" denotes flooding of the crater by dark mare material. The sixth column gives the "background," or type of surface on 

which the crater formed, usually mare (M) or uplands (C), as indicated in the Arthur catalog.

The next five columns list, respectively, the crater rim diameter, 

depth, rim height, width of the rim flank, diameter of the flat floor 

(where present), and circularity of the rim-crest as defined above. 

Secondary impact craters, cratered cones, dark-halo craters, and cratered domes are distinguished from fresh and modified impact craters of the main sequence. Measurements generally are given to a 

precision that accords with accuracy of the raw data, but some 

figures for the smallest craters have been rounded off. Although the 

bulk of the data are from Apollo photographic sources, many of the 

better known near-side craters that were not included in the new 

photogrammetric coverage are listed here where the old shadowlength relief data are much better than average. Shadow data for the 

domes and cones are less reliable than for the normal craters.

Users of the data in table 9 are cautioned that grouping craters for 

analysis according to the classification of morphology by Arthur and 

others (1963) (or any similar type of classification, for that matter) 

does not necessarily guarantee uniform subsamples. Considerable 

variance in the different dimensions, especially depth and rim

height, has been found in the course of analyzing this information, 

and it is clear that the original relative-age ranking on the basis of 

morphology will require some systematic updating. Misclassified 

craters tend to be more common in the lunar uplands (Pike, 1974b).

Measurements of terrestrial meteorite-impact and experimentalexplosion craters were compiled from more than three dozen scattered sources (not all listed in the references), and some unevenness 

in the quality of the data is to be expected. Virtually all of the impact 

craters have been degraded to some extent. Wherever possible, depth 

values were determined from subsurface data in order to approximate the original topography. The depth given for Wolf Creek Crater 

is not that used in the analysis (pi. 1) but rather an older figure that 

included a considerable thickness of sedimentary fill (Fudali, 1979). 

Some caution is advised in comparing the dimensions of impact craters on either Earth or the Moon with those of the experimentalexplosion craters. The scaled depths of burst for these test craters 

(see discussion by Baldwin, 1963) vary considerably, resulting in 

systematically different structures within the raised rim (Pike, 

1974a, 1976b), and only the most shallow experiments may have 

produced suitable analogs for meteorite craters (Roddy, 1968, 1976).

Finally, mistakes invariably find their way into a numerical listing of the magnitude and detail undertaken here even when, as in 

this case, all measurements were made or compiled by one individual 

and photographed directly from a computer listing that has been 

carefully checked. Although every effort has been made to assure 

that the data are correct, some errors almost certainly remain and 

are the sole responsibility of the author. Doubtful entries or outright 

mistakes that are found by users of this listing should be called to the 

author's attention so that corrected figures can be promptly published. Since the initial release of these data (Pike, 1976c), entries for 

the following 21 craters in table 9 have been revised: Gagarin, Tsiolkovskiy, Neper, Sklodowska, Abul Wafa, Brunner, Schubert B, 

Ctesibius, Proclus, Pytheas, Vitruvius A, Dawes, Richards, Macrobius B, Jansen B, Langrenus N, D'Arrest, Lebesgue, Le Monnier B, 

one unnamed lunar crater, and Wolf Creek Crater. Additionally, 

entries for 12 named primary craters have been inserted in table 

9—Aitken, Chaplygin, La Perouse, Langrenus A, Kapteyn E, Euler, 

Langrenus M, La Perouse D, Pytheas A, Jansen K, Picard Y, and 

Cauchy CA; there are nine unnamed craters as well. Data for the 

Siberian crater El 'Gygtgyn have been added to table 10. Five lunar 

craters that I listed as primary (Pike, 1976c) have since been identified as large secondaries: Zollner, Schroter, D'Arrest, Secchi, and 

Capella M. Finally, table 9 includes data for the 118 additional secondary craters that I analyzed after this work had been completed 

(Pike and Wilhelms, 1978).
C68 APOLLO 15-17 ORBITAL INVESTIGATIONS

TABLE 9.—Dimensions of 623 lunar craters

[See text for table explanation]

1.0. CRATER DATA p CL

PRIMARY IMPACT

C1835

00089

A9185

08848

08165

00179

C3798

C6310

04139

A4522

C2500

B0798

C2180

B3136

A1796

C1648A

C5766

09158

B0469

06540

06154

00457

C0615

C5487

A6376

A3576

D0432A

D6197A

A8182

C2230

C2194

A7029

B1295

A4552

01072

A0515

A1899

02427

A9084

00014

A8077

09023

A9037

B0029

02260

C4416

05163A

C3488

09213

M. SERENITATIS

M. CRISIUM

M. SMYTHH

MENDELEEV

GAGARIN

CLAVIUS

TSIOLKOVSKIY

HIL8ERT

CURIE

HIPPARCHUS

NEPER

VENOELINUS

AITKEN

LANGRENUS

ALBATEGNIUS

SKLODOWSKA

CHAPLYGIN

SCHILLER

GASSENOI (1C)

LANGEMAK

BABCOCK

THEOPHILUS

POSIDONlUS (1C)

PITATUS (1C)

PLATO

FRA MAURO

COPERNICUS

ARISTOTELES

TYCHO

WARGENTIN

LA PEROUSE

ARCHIMEDES

KING

KIESS (1C)

REICHENBACH

GUTENBERG

HERNER

ERRO

MUGGINS

OOPPELMEYER (1C)

MACROBIUS

CAPUANUS

BLANCHINUS

GOCLENIUS

FIRMICUS

RITZ

GUERICKE

BONPLANO

TARUNTIUS do

ERATOSTHENES

HIPPALUS (1C)

LADE

ABUL WAFA

ARISTILLUS

KANE

PONT ANUS

SCHUBERT

SCHORR

MAUNDER

REAMUR

APOLLONIUS

MACLAURIN

BRUNNER

OUBIAGO

SO OF SAHA

PALLAS

DESCARTES

TOLSTOY

CAMPANUS

CAPELLA

MERCATOR

KAPTEYN C

LASER PROFILE

LASER PROFILE

LASER PROFILE

LASER PROFILE

LT0102B3

SHADOW DATA

LT0101B2t3

USGS PHOTOGRM

USGS PHOTOGRM

SHADOW DATA

LT063B3»4Cli2

SHADOW DATA

LT086D4/104A1

LT080C1

SHADOW DATA

LT0100A1-2

LT084B3/LOC-4

SHADOW DATA

ORBITER 5 MAP

LT083D2/C1

LT064D1

LT078C2

SHADOW DATA

SHADOW DATA

SHADOW DATA

SHADOW DATA

SHADOW DATA

SHADOW DATA

SHADOW DATA

SHADOW DATA

LT081D2

SHADOW DATA

LT065C1

LT081B4/3

SHADOW DATA

SHADOW DATA

SHADOW DATA

LT064D2

SHADOW DATA

SHADOW DATA

USGS PHOTOGRM

SHADOW DATA

SHADOW DATA

SHADOW DATA

LT062C1

USGS PHOTOGRM

SHADOW DATA

SHADOW DATA

LT061C2

SHADOW DATA

SHADOW DATA

SHADOW DATA

LT06503

SHADOW DATA

SHADOW DATA

SHADOW DATA

LT063C4

USGS PHOTOGRM

ORBITER DATA

SHADOW DATA

LT062C1

LT080B2

LT082D1

LT062C2

USGS PHOTOGRM

SHADOW DATA

SHADOW DATA

LT082A4

SHADOW DATA

LT079A3/4

SHADOW DATA

USGS PHOTOGRM

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

p

P

p

p

p

p

p

p

p

p

3

3

1

2

3

4

2

3F

2

3

3

3F

1

3

3F

2F

4

2

2

1

3F

2

2F

1

4F

3

3

1

3

4F

2

3F

3

3

3F

3

4

4

2F

2

4F

4F

2

1

3F

3

1

2

1

4F

3F

3

2F

2

3

4

2

2F

3

2F

3F

BGD DIAM. DEPTH HT. WIDTH FLOUR CIHC

CRATERS

C

C

C

C

C

C

C

C

C

C

PMC

AMC

C

PMC

C

C

C

C

AMC

C

C

PMC

AMC

AMC

C

AMC

PM

PM

C

C

C

AMC

C

AMC

C

AMC

C

C

C

AMC

C

AMC

C

AM

C

C

AM

AMC

PMC

PM

AMC

C

C

PM

AMC

C

C

C

PMC

AMC

C

C

C

AMC

C

C

C

C

PM

C

AMC

C

680.

480.

450.

370.

275.

235.

197.

178.

158.

150.

148.5

147.

137.1

136.

135.

128.

127.9

125.

110.

106.

102.5

102.

101.

100.

100.

95.

93.

87.3

85.

85.

81.

80.

78.2

72.1

71.4

71.

71.

70.

68.

65.

64.5

64.

62.

60.

59.8

59.

59.

59.

58.5

58.3

57.

56.

55.9

55.3

55,

54.0

53.9

53.5

53.5

53.

52.6

52.5

51.6

50.6

50.5

50.

49.

48.

48.

47.0

47.

46.

2.5

3.75

3.5

6.5

5.3

4.90

5.0

4.1

3.85

1.10

3.97

2.2

4.90

4.50

3.2

4.28

5.10

2.5

1.42

4.50

2.3

4.10

1.37

.68

2.0

.83

3.80

3.50

4.6

.3

4.18

1.6

4.15

.80

3.2

1.75

4.2

1.98

1.85

1.1

4.05

1.6

1.16

2.2

2.14

3.75

.74

.73

1.10

3.43

.9

.82

3.7

3.30

.57

1.80

3.35

3.7

3.2

.82

2.75

3.38

3.63

2.7

3.64

1.5

.85

3.00

2.

3.5

1.1

.5

1.2

1.2

1.5

2,7

2.1

1.6

.9

.5

.5

.99

.15

.5

.90

.SO

.98

1.40

1.75

1.5

1.04

1.5

1.1

1.3

.82

.53

1.6

.80

1.10

1.25

2.4

.6

1.44

1.6

1.28

.55

1.7

1.0

.83

1.03

.99

.4

1.2

1.5

.66

.85

.90

l.b

.66

.69

.80

1.35

.9

.57

1.05

1.00

.74

1.06

1.2

.73

1.1

.74

.93

.85

1.35

.70

1.38

.77

.66

.90

1.2

.9

.7

.35

120.

150.

120.

75.

43.

40.0

40.

35.

29.

15.

27.0

12.

30.0

35.

?8.

23.0

26.0

15.

5.

23.

13.

25.

10.

9.

19.

10.

21.

17.

20.

12.

20.0

13.

22.0

7.0

9.5

15.5

12.

10.0

6.

3.

8.2

9.

5.5

5.

11.5

15.5

3.

4.

14.

19.

2.

4.

12.5

14.

4.

7.5

13.0

13.

10.4

3.5

10.0

8.0

13.

8.0

13.2

5.2

5.

9.0

7.

8.

6.

6.8

600.

400.

400.

260.

200.

190.0

145.

125.

100.

139.0

91.0

132.

85.7

80.

93.

77.0

8b.3

94.

103.

60.

75.

65.

90.

87.

87.

90.

54.

52.

46.

82.

42.5

66.

45.0

70.0

38.

59.

40.

48.0

50.

58.

32.5

51.

52.0

37.

43.8

24.

55.

52.

42.

31.5

51.

50.

22.0

30.

47.

39.0

28.

26.

32.8

46.

32.5

25.

26.

30.0

24.0

35.

43.

23.

28.

20.

35.

39.

.80

.83

.80

.75

.79

.76

.67

.80

.73

.78

.76

.77

.77

.76

.87

.76

.74

.18

.84

.72

.72

.80

.75

.76

.83

.76

.82

.76

.84

.85

.*5

.86

.80

.75

.86

.81

.86

.65

.86

.80

.86

.78

.70

.41

.80

.71

.76

.75

.79

.83

.84

.64

.79

.80

.85

.76

.78

.77

.81

.79

.79

.77

.85

.72

.83

.71

.79

.81

.73

.60

.80

.75
GEOMETRIC INTERPRETATION OF LUNAR CRATERS C69

TABLE 9.—Continued

1.0.

A0080

C2163

A1087

08120

C5520

C2448

D1057

A3657

B3085

09108

A5333

A3286

05143

C0039

B7250

0

B6470

C8078

C4040

A5432

A3730

09125

B8434

D2133

D2089

A1836

A9082

A7255

B7309

D1104

A6396

A1073

0

D9073

D2118

A7271

D8048

B2404

D9072

C6216

02122

B6104

A8187

D3148

A3023

A3042

D

05047

A4390

06217

B3423

D

08140

A4614

A4527

A7207

A2268

04189

04078

A3726

CRATER

RHAETICUS

PARRY

AGRIPPA

LANGRENUS F

VITELLO

HESIODUS

SAUNDER

PLANA

REINHOLO

OELPORTE

GANSKIY

LANGRENUS A

MARALDI

KOPFF

PLINIUS

ISIOORUS

W. HERSCHEL

MARIUS

LITKE

RUNGE

ARISTARCHUS

HALDANE

DAMOISEAU

LANSBERG

ROME**

BURG

VIVIANI

KAPTEYN E

BRIGGS

BALZAC

OOLLOND 8

TAYLOR

C. MAYER B

SCHUBERT B

MILTON

YERKES

HERODOTUS

HALLEY

TISSERAND

GOO IN

WARNER

GILBERT M

KATCHALSKY

ANOEL

NECHO

LICK

LANGRENUS B

TIMOCHARIS

CTESIBIUS

IZSAK

GILBERT N

GASSENDI A

DOLLOND C

KEPLER

AUZOUT

KANT

RITTER

SABINE

DOYLE

ISIDORUS B

FISCHER

VITRUVIUS

BOHNENBERGER

LAMBERT

CELLINI

LANGRENUS K

GROVE

OANIELL

PROCLUS

MONTESQUIEU

AUSTIN

MENELAUS

MADLER

TORRICELLI

BAILY

(IS)

(IS)

(IS)

(1C)

(IS)

(1C)

(IS)

(IS)

(1C)

(IS)

(IS)

(IS)

(IS)

(IS

(IS)

(IS)

DATA

SHADOW DATA

SHADOW DATA

SHADOW DATA

LT080A3

SHADOW DATA

SHADOW DATA

SHADOW DATA

SHADOW DATA

SHADOW DATA

LT083C4

LT08202

LT080C2

LT043D4

ORBITER DATA

LT060B1

LT079A4/01

USGS AS12 PGM

SHADOW DATA

LT0101B1

LT081B2

ORBITER 5 MAP

LT081B1

SHADOW DATA

USGS PHOTOGRM

SHADOW DATA

SHADOW DATA

LT065D2

LT080C2

SHADOW DATA

LT082D1

SHADOW DATA

LT078A3

SHADOW DATA

LT063C4

LT082A]

LT062A1

SHADOW DATA

SHADOW DATA

USGS PHOTOGRM

SHADOW DATA

LT081B2

LT081A2

LT065D2

LT078D1

LT083B4

LT062A1

LT080A3

LT040B3

LT065D3

USGS PHOTOGRM

LT081A2

ORBITER 5 MAP

SHADOW DATA

SHADOW DATA

LT062B4

LT078C1

SHADOW DATA

SHADOW DATA

LT063C4/3

LT079A4

LT066C1

LT04304

SHADOW DATA

LT040A3

LT081B4/C1

LT080A3

SHADOW DATA

SHADOW DATA

LT043C3

LT082A4

LT082D1

LT04203

LT078C2

LT078B3

SHADOW DATA

P CL

P 4

3F

P 2

3

2

3F

4F

P 3

P 1

P 2

P

P 2

3F

F

P 1

2

P 1

2F

P 2F

F

P 1

P F

P 3

P 1

P 1

P 1

P 3

P 2

P 2

4F

P 3

P 3

P 1

F

P 4F

2F

P 3F

3

P 2

P F

P 1

P

P 3

1

P 4F

3

P 1

P

P 2

2

P 2

4F

P 2

P 2

P 1

2F

2

F

P 3

P 2F

P 2

P 2

R

P 3

P 1

1

P 1

2

2

P 1

P 1

3

4F

PGD

C

AMC

C

CM

C

AMC

C

AMC

PM

C

C

C

C

PMC

.PM

C

C

AM

C

C

PMC

AMC

C

PM

C

PM

C

C

PM

C

C

C

C

C

AM

AMC

AMC

C

C

C

C

C

C

C

C

AMC

C

PM

C

C

C

C

C

PMC

C

C

AM

AM

C

C

C

AMC

PMC

PM

C

C

PM

PM

C

C

C

PM

PM

PM

AMC

DIAM.

46.

46.

46.

45.12

45.

45.

45.

45.

44.7

43.38

43.38

43.25

43.1

43.0

42.1

41.8

41.6

41.

40.5

40.25

40.2

40.12

40.

40.0

39.6

39.6

39.50

39.13

39.

38.88

37.

36.25

36.

35.8

35.38

35.25

35.

35.

35.

34.8

34.75

34.5

34.12

34.0

33.75

33.7

33.6

33.50

33.25

33.2

33.0

33.

32.

32.

31.75

31.63

31.

31.

30.88

30.63

30.38

30.10

30.

29.9

29.63

29.25

29.

28.5

28.5

28.38

27.75

27.40

27.13

27.0

27.

DEPTH

1.2

.87

3.0

1.75

1.7

.45

.64

1.8

2.70

3.19 1

3.37 1

2.91 1

1.5

2.0 0

3.07

2.5

3.33

1.5

2.70 1

.23

3.15

.325

1.25

2.75

3.5 1

3.07

2.51

1.75

1.2

2.57

.88

3.1

1.12

2.9

.43

.62

1.3

2.47 1

2.93 1

3.2

.25

3.03 1

3.06

1.35

3.3

.30

1.87

3.00

3.06

3.4 1

2.58

2.9

.62

2.7

3.85

3.72

1.3

1.4

.42

1.6

3.1

1.88

2.4

2.60

2.99 1

2.50

2.37

1.9

4.04 1

3.06

3.05

2.6

2.80 1

2.5

.52

HT.

.87

.86

.80

.30

.61

.38

.55

.73

.78

.0

.0

.11

.55

.9

.95

.9

.90

.6

.0

.13

.80

.300

.6

.65

.1

.85

.45

.b2

.9

.60

.66

.85

.87

.75

.30

.62

.94

.11

.05

.73

.20

.00

.88

.79

.50

.30

.47

.85

.83

.03

.60

.9

.57

.85

.95

.8

.57

.61

.32

.77

.65

.46

.68

.70

.14

.63

.61

.6

.19

.71

.70

.80

.00

.6

.45

WIDTH

3.

4.

5.5

R.O

6.4

3.

2.5

3.

13.

9.

10.

10.5

4.5

6.0

13.

8.

10.0

8.

8.

6.0

15.

7.0

3.

12.5

10.

9.

5.0

8.0

6.5

9.5

4.

6.0

4.

8.0

3.5

4.0

4.5

4.

6.

5.

5.0

9.0

8.0

6.0

11.0

3.0

5.

8.0

7.0

7.7

5.0

8.

3.

9.

6.5

9.

3.

4.

3.0

6.5

9.0

4.0

7.7

10.

8.0

8.0

4.7

2.

9.0

6.5

7.0

7.00

7.5

4.5

2.5

FLOOW

32.

40.

27.

?9.

37.

34.

40.

32.

27.

13.

19.

19.5

31.

27.0

20.

24.

19.0

33.

19.

36.0

17.5

36.0

33.

20.

20.

?0.

10.5

22.0

29.

22.

31.

17.

25.

18.

31.

31.5

28.

22.

16.

10.

31.5

17.0

13.0

24.0

15.0

29.

19.

15.

14.0

13.0

15.0

20.

28.

19.

12.0

13.

23.

24.

26.5

12.

11.5

19.0

24.

14.

12.0

19.

17.

20.5

13.0

9.

12.5

14.0

13.5

10.

?0.

CIRC

.72

.82

.80

.76

.80

.67

.80

.76

.83

.77

.77

.72

.67

.80

.80

.79

.99

.88

.78

.84

.84

.83

.73

.83

.70

.80

.83

.73

.74

.78

.78

.68

.71

.82

.74

.84

.83

.80

.77

.72

.84

.79

.63

.74

.62

.72

.80

.84

.76

.86

.78

.81

.66

.86

.75

.74

.79

.83

.86

.71

.86

.74

.84

.86

.88

.81

.80

.64

.75

.74

.84

.8?

.76

.44

.60
C70 APOLLO 15-17 ORBITAL INVESTIGATIONS

TABLE 9.—Continued

1.0.

B4349

C1001

A3160

B2061

B4499

B3067A

D

C4544

A5003

A7076

A7285

C1236

A4263

86428A

08061

C6225

07255

07253

07254

A0336

C6097

03029

A9176

A6313

09176

63325

C<»082

A7361

09106

B4496

66370

A5330

A2195

09074

0

A4229

C1338

A6326

A9054

A7232

41025

A6315

A4148

A2387

05057

B4029

B4161

A6253

B5365

07099

A7272

01198

CRATER

N. OF KATCHALSKY

EULER

HOSTING

ARAGO

NEAR GAGARIN

GAMBART <is>

DELISLE

REINHOLD B

SWASEY

RAMSOEN

MASKELYNE

w or NEUJMIN

TARUNTIUS M

PICAKO

LASSELL (IS)

JANSEN (IS)

KRIEGER

WEBB

BERGMAN

GASSENDI B

MCCLURE o us>

UNNAMED

CROZIER (IS)

MOISSAN

MCCLURE M us)

CONON

IN GAGARIN (OLD)

FLAMSTEED

ALFRAGANUS

NEPER G

HACROBIUS A

KASTNER F

PYTMEAS

LANSBERG c

PEIRCE

LANGRENUS M

DIOPHANTUS

ARISTARCHUS F

VITRUVIUS A

OARIO

SOSIGENES (IS)

GILBERT D

ON Nfc RIM ERRO

ELMER

OAWES

RICHARDS

UNNAMED

IN HILBERT

BIRT

MACROBIUS L

DUB I A GO S

GLAISHER

SE OF DANJON

* RIM FLANK ERRO

E. PICKERING

MACROBIUS B

JANSEN B

IN SAHA

IN GAGARIN (OLD)

BESSEL

NO. OF ABUL HAFA

ISIOORUS 0

HARDEN

KOSBERG

UNNAMED

MORTENSIUS E

MORTENSIUS

PROCLUS A

BAUDELAIRE

BRAYLEY

LI PO

BENEDICT

MESSIER G

LICK D

ANDEL A

DATA

LT065D2

LT040D1/LO-4

SHADOW DATA

SHADOW DATA

USGS PHOTOGRM

SHADOW DATA

LT039B2

SHADOW DATA

LTD81B3

SHADOW DATA

SHADOW DATA

LT0101C1

LT061C1

LT062A1

SHADOW DATA

LT060B2

LT039A1

LT080A2/B1

LT06602

OR8ITER 5 MAP

SHADOW DATA

LT083C3 ODD

SHADOW DATA

LT066P2

SHADOW DATA

LT041C1

LT0102B3

SHADOW DATA

LT078A3

LT063B4

LT043C4

LT081C1

LT040D2/L04

SHADOW DATA

LT04404

LT080C2

LT039B3

ORBITER 5 MAP

LT043D4

LT082D1

SHADOW DATA

LT081A2

LT064D2

LT081C1

42C3S4(50)

LT066C1

LT082A4

USGS PHOTOGRM

SHADOW DATA

LT043C1

LT06JD4

LT061B2

LT083C4

LT064C2

SHADOW DATA

LT043C1

LT060B3

USGS PHOTOGRM

LT0102B3

LT042D2

LT065D3

LT079A4

LT066C1

LT0102B3

LT0100A2

SHADOW DATA

SHADOW DATA

LT06181

LT0101B4

LT039C1/L04

LT082A1

LT066C1

LT080A4

LT062A1

LT078D1

P

P

P

P

P

P

P

P

P

P

P

P

P

R

P

P

P

CL

1

2

2

2

2F

2

3F

F

2F

2

3

3

1

2F

2F

3F

2F

2

3F

2F

4F

1

2

1

2

1

1

1

2F

2

1

1

3F

1

1

IF

1

1

2

1

3F

1

1

1

1

2

3

1

1

3F

1

1

2

1

1

1

1

2

BGD

C

PM

PMC

PM

C

AMC

PM

AM

C

AM

PM

C

AMC

PMC

AM

AM

AM

PMC

C

c

c

MC

c

c

c

c

PM

c

PMC

PM

c

PM

M

PM

C

PM

AM

C

C

AMC

C

C

C

PM

C

C

C

PM

AMC

C

C

C

C

C

PM

PM

C

c

PM

C

c

c

c

c

AM

PM

C

C

PM

PM

C

PM

PMC

C

DIAM.

26.88

26.63

26.5

26.0

26.

25.6

25.38

25.

25.0

24.

23.8

23.5

23.50

23.00

23.

22.88

22.88

22.56

22.3

22.

22.

21.88

21.5

21.3

21.

20.75

20.66

20.6

20.38

20.0

19.88

19.6

19.56

19.

18.5

18. 2b

18.13

17.9

17.88

17.88

17.8

17.75

17.75

17.5

17.40

17.25

17.13

17.0

17.

17.0

16.5

16.37

16.25

16.19

16.

15.9

15.88

15.7

15.63

15.50

15.37

15.25

15.13

15.06

15.05

15.

14.7

14.63

14.63

14.47

14.38

13.99

13.81

13.88

13.75

DFPTH

2.71

2.24

2.80

2.68

2.6

1.1

2.42

.8

.45

1.9

2.50

2.65

.85

2.32

.9

.34

.95

1.85

2.7

1.36

1.1

2.77

1.3

2.9

.74

2.93

2.0

2.16

3.8

1.5

3.80

3.17

2.54

.78

2.16

3.685

3.02

.29

2.93

3.33

1.5

2.97

2.17

2.35

2.31

3.94

2.45

1.8

3.47

.59

2.91

3.29

2.51

2.33

2.74

3.83

2.19

.88

1.45

1.77

2.73

3.22

2.75

1.25

2.635

.20

2.86

3.01

2.69

2.84

3.59

3.14

1.89

2.41

.80

HT.

.71

.74

.55

.73

.73

.57

.72

.51

.20

.7

.80

.4

.50

.54

.5

.60

.70

.60

.70

.40

.72

.32

.94

.60

.52

.90

.35

.66

.78

.60

.90

.57

.765

.57

.64

.785

.82

.29

.66

.55

.4

.60

.42

.8

.585

.64

.35

.40

.66

.31

.61

.65

.54

.40

.70

.60

.510

.28

.30

.61

.35

.60

.55

.60

.515

.20

.43

.42

.4

.48

.74

.60

.31

.56

.30

WIDTH

6.0

11.5

4.3

5.5

6.0

1.5

7.5

3.

2.5

3.

4.

4.25

5.5

6.0

4.3

4.75

3.5

4.0

5.0

3.5

3.4

4.50

2.

4.8

1.5

5.3

4.3

3.5

5.5

7.0

6.00

4.5

7.0

2.

8.

5.

6.0

2.5

5.50

3.5

2.5

4.0

4.5

4.5

6.5

4.0

4.0

3.0

3.

2.3

4.8

2.8

3.5

4.0

4.

5.

4.5

3.6

3.0

5.75

3.5

5.0

4.0

3.5

4.2

1.2

2.6

4.0

3.5

3.25

4.3

3.5

3.2

4.06

1.8

FLOOR CIRC

12.

12.0

10.

15.

13.0

21.

9.5

21.

22.5

15.

15.

9.

19.

7.5

17.

17.5

19.

14. P

8.

12.

16.

7.0

16.

4.

16.

5.0

8.5

12.5

6.5

10.5

6.

6.5

(5.0)

14.5

10.

5.5

7.

15.3

4.0

4.

11.

0.

5.6

7.

(8.0)

3.5

4.5

5.0

4.

14.

5.

4.5

6.

4.0

6.

3.

0.

9.0

6.0

3.5

2.5

0.

4.0

9.0

5.0

12.5

6.

3.9

4.0

0.

5.

2.7

0.

5.0

9.«5

.70

.84

.78

.76

.86

.79

.84

.86

.82

.72

.69

.84

.84

.88

.80

.82

.75

.76

.74

.83

.73

.51

.74

.71

.67

.82

.81

.80

.88

.74

.Rfl

.88

.76

.82

.81

.90

.87

.84

.87

.92

.88

.80

.84

.80

.80

.84

.82

.79

.80

.61

.83

.81

.83

.84

.88

.90

.75

.80

.85

.fifl

.80

.88

.91

.86

.86

.86

.85

.84

.90

.88

.84

.88

.79

.80

.69
GEOMETRIC INTERPRETATION OF LUNAR CRATERS C71

TABLE 9.—Continued

1.0.

A6189 ,

A0016

C2176

A9381

09105

07036

D8069

C0095

C1161

A

06005

A6116

A7152

D

A3234

A6229

0

0

053^0

04097

A1393

A5338

07033

A6049

D2148

A7110

A8019

A7053

A4361

D8063

C1290

A6046

03100

A7353

C4290

A0470

A5329

A8171

B6520

A5479

A8035

B1445A

B6428

B1445

A6137

01194

A6201

A0476

A5201

A7060

A5368

A3281

A4267

com

B5396A

02114

A7168

A0279

A6300

B5426

CRATER

ISIDORUS E

CHLADNI

PARRY A

MACROBIUS D

LANGRENUS N

MESSIER A

NO. OF FOX

LANGRENUS c

UNNAMED

MOSTING A

LALANOE A

PEEK

CENSORINUS F

CAUCMY

TARUNTIUS A

TALBOT

E OF PURKYNE

NW OF TSIOLKOVSK

TACQUET A

PROCLUS D

SLOCUM

LEBESGUE

ROSSE

TORR1CELLI A

SULPICIUS CALLUS

ROMER K

MESSIER

TARUNTIUS E

DOLLOND

TARUNTIUS C

APOLLONIUS K

TARUNTIUS G

VITRUVIUS E

GILBERT U

WEBB H

GUERICKE C

TARUNTIUS F

ZASYAOKO

ALFRAGANUS C

PEIRCE B

EUCLIOES B

ARATUS

UNNAMED

ROMER L

UNNAMED

FIRMICUS F

IN PASTEUR

KOLLASTON

ANGSTROM

UNNAMED

APOLLONIUS C

UNNAMED

FEUILLET

UNNAMED

KRIEGER B

BEER

CAUCHY o

ANDEL F

CAUCHY A

HADLEY B

JANSEN F

UNNAMED

UNNAMED

TARUNTIUS H

ROMER J

ROSS 0

UNNAMED

JANSEN C

PTOLEMAEUS A

BRAYLEY C

OOLLOND D

LICK E

MANILIUS F

MARALDI M

DIOPMANTUS A

DATA P

LT079A4

SHADOW DATA

LT076C1

LT043C3

LT080C2 P

LT079B2

LT064D3

LT080B4

LT084B3

SHADOW DATA

USGS PHOTOGRM

LT063C3

LT079A2

LT061A3

LT061C2

LT081B2

LT082A2

LT0101B2

LT060B1

LT043C4

LT081B2

LT081B3

SHADOW DATA

LT078B3

LT042D4

LT043D3

LT079B2

LT061CI

LT078D1

LT061C2

LT062D1

LT061C3

LT042C3

LT081B1

LT080A2

LT076C2

LT061C4

LT064D4

LT078A3

LT044D4

LT075C2

LT041CI

LT0100A1

LT043D1

LT082D1

LT062C1

USGS PHOTOGRM

LT039A1

LT039A2

LT082A1

LT062D3

LT064D1

LT041A4

LT082A4

39A1S01 (50)

LT041A4

LT061B4

LT078D1

L T 061A2

LT041B4

LT061A1

LT064DI

LT066D2

LT061C3

LT043D2

LT060B1

LT066C1

LT042C3

LT077D?

LT039C1/L04

LT078D1

LT062A4

LT041C4

LT043D3

LT039B4

CL

1

1

2F

2

2

1

1

F

1

IF

1

F

F

1

1

IF

1

1

1

1

1

1

1

1

1

3

1

1

F

1

1

1

1

1

1

2F

1

1

1

1

1

1

1

1

1

1

2F

1

1

BGD

C

PM

PM

C

C

PM

C

C

C

C

c

PM

C

PM

PMC

C

C

c

AMC

PMC

C

C

PM

PM

PM

PM

PM

PM

C

PMC

C

PM

PMC

PM

PM

PM

PM

PM

C

PM

PM

C

C

c

c

c

c

PM

PM

AM

C

C

PM

C

PM

PM

PM

C

PM

C

PM

C

C

PM

PMC

PM

C

PM

C

PM

C

C

PMC

PM

PM

DIAM.

13.75

13.6

13.5

13.5

13.38

13.25

13.19

13.13

13.13

13.05

13.

13.0

12.85

12.80

12.75

12.6

12.5

12.5

12.4

12.0

12.3

12.25

12.1

11.94

11.88

11.75

11.63

11.50

11.0

10.88

10.78

10.75

10.75

10.75

10.75

10.56

10.56

10.54

10.44

10.4

10.28

10.19

10.13

10.04

10.

10.0

9.9

9.75

9.75

9.75

9.69

9.69

9.56

9.5

9.48

9.44

9.42

9.0

9.00

9.0

8.94

8.94

8.80

8.75

8.75

8.75

8.63

8.62

8.5

8.50

8.5

8.3

B.25

8.25

8.18

DEPTH

3

2

2

1

2

3

2

2

2

2

2

1

2

2

2

2

1

2

2

2

2

2

1

2

2

2

2

2

2

2

2

2

2

2

2

2

2

1

1

2

1

1

1

2

2

1

1

1

1

1

1

1

1

2

1

1

1

1

1

1

1

1

1

1

1

1

1

2

.05

.63

.98

.56

.65

.24

.01

.90

.3

.70

.7

.34

.80

.67

.55

.30

.59

.27

.04

.40

.48

.35

.42

.16

.17

.60

.90

.20

.04

.19

.07

.22

.15

.08

.02

.41

.2

.4

.30

.04

.15

.88

.985

.34

.81

.93

.1

.23

.00

.30

.87

.97

.89

.1

.61

.67

.90

.8

.40

.0

.81

.76

.75

.90

.83

.8

.63

.07

.82

.66

.56

.73

.60

.69

.01

H7. WIDTH FLOOR CISC

.50

.3*

.45

.54

.23

.53

.38

.71

.40

.43

.6

.5?

.4R

.51

.40

.20

.29

.3

.36

.26

.18

.25

.34

.54

.32

.30

.40

.40

.30

.35

.35

.38

.45

.38

.OR

.38

.40

.4

.32

.46

.45

.35

.285

.40

.31

.ao

.11

.38

.40

.20

.34

.25

.18

.23

.40

.2R

.20

.45

.05

.20

.35

.18

.42

.30

.as

.24

.30

.20

.32

.35

.20

.15

.18

.28

.30

2.5

2.7

2.75

4.0

3.1

2.50

3.45

4.0

2.7

3.

3.6

1.5

3.5

3.0

4.0

1.6

3.5

2.75

2.6

2.3

1.6

1.6

2.

3.2

3.80

3.62

2.0

3.0

2.5

3.0

2.5

2.4

2.5

3.75

.68

3.2

3.1

3.6

2.2

4.3

3.10

2.3

2.935

2.5

2.5

2.0

1.75

2.7

2.3

2.0

2.4

2.95

2.0

1.75

3.0

2.5

2.0

2.5

.0

.88

.88

.8

.50

.9

3.0

2.13

1.95

2.5

3.0

2.75

2.2

1.75

1.13

3.4

1.56

2.0

5.

7.60

3.0

2.5

3.5

1.5

4.

3.0

0.

2.

3.75

3.5

0.

2.0

9.4

3.2

0.

7.0

2.2

9.0

9.5

0.

0.

0.

2.0

0.

1.75

2.0

2.5

2.0

1.5

3.0

2.5

0.

0.

0.

0.

2.2

2.5

0.

3.3

2.8

0.

3.

1.6

2.75

0.

0.

7.R

0.

1.6

1.5

4.

0.

2.5

1.4

1.7

6.0

0.

1.0

0.

1.0

2.0

0.

0.

0.

3.5

1.8

0.

1.0

0.

5.0

0.

0.

.75

.92

.89

.86

.71

.43

.91

.R3

.83

.90

.RB

.86

.90

.RR

.85

.86

.85

.79

.92

.85

.84

.85

.92

.86

.81

.84

.36

.84

.83

.87

.90

.91

.91

.91

.87

.65

.85

.78

.85

.87

.90

.88

.86

.85

.82

.89

.79

.90

.90

.90

.88

.79

.88

.81

.84

.83

.83

.82

.95

.76

.37

.78

.R4

.89

.RO

,R8

.84

.R4

.89

.R9

.79

.Rl

.R9

.80

.86
C72 APOLLO 15-17 ORBITAL INVESTIGATIONS

TABLE 9.—Continued

I.D.

D9020A

07026

A8004

A5354

D

A3203

A4285

A4257

A4254

A6380

A3421

A7025

A6279

B4468

A7080

D

D8141

B3500

B2441

D8101

A5284

A8013

A3218

08044

07041

A0480

D

B6456

D9158

D4001

A4366

A8252A

C3176

A2229

D7010

A3326

A2353

B3344

A4189

A

C4126

A6370

A7081

A6078

A0472

A1303

B5561

D2167

A1402

A6182

A6066

D0167A

A 3405

A2454

A5254

B1486

A3483

A6065

83447

A7167

B5405

D7015

A

A4318

A6059

B4395

86418

CRATER

NO. MACLAURIN

MESSIER D

TARUNTIUS N

UNNAMED

MARALDI A

TUCKER

AUWERS A

SW OF HOUTERMANS

JANSEN L

JANSEN D

JANSEN E

PROCLUS W

BESSEL A

TARUNTIUS B

PROCLUS Y

DIOPHANTUS B

TARUNTIUS P

UNNAMED

LANGRENUS KA

CARLINI B

TIMOCHAHIS A

LANGRENUS FF

MARALDI B

TARUNTIUS o

TACOUET

WEBB 0

MESSIER B

ARATUS B

UNNAMED

ARISTARCMUS C

LA PEROUSE D

UNNAMED

MOLTKE

LITTROW B

PICARO X

BONPLAND E

MENELAUS A

SECCHI K

DESEILLIGNY

BESSEL E

PYTHEAS A

JANSEN K

PICARD Y

EUCLIDES D

UNNAMED

N. OF KATCHALSKY

PROCLUS X

TARUNTIUS K

IN HILBERT

TARUNTIUS EA

HADLEY C

SULPIC. GALLUS G

ANGSTROM A

REGIOMONTANUS A

DOLLOND E

HADLEY A

TARUNTIUS MA

SECCHI B

ALBATEGNIUS C

BESSEL 0

LINNE E

VITRUVIUS G

TIMOCHARIS B

FLOOR TSIOLKOVSK

LE MONNIER B

SECCHI A

LA HIRE A

LICK G

OIOPHANTUS C

MESSIER E

UNNAMED

LE MONNIER C

TARUNTIUS EB

EULER K

KRIEGER D

DATA P CL

LT080B2

LT079B2

LT062D4

LT066D2

LT043D2/D3

LT081B3

LT060A2

LT081C2

LT061A1

LT060B2

LT060B2

LT043C3

LT042B4

LT061C3

LT043C4

LT039B2

LT062D4

LT081D2

LT080A3

LT040A2

LT040B4

LT080A4

LT061A2

LT062D4

LT042D3

LT080A2

LT079B2

LT041B3

LT081B3

LT039A4tl

LT081D2

LT010201

USGS PHOTOGRM

LT042C2

LT062B1

LT076D2

LT042D4

LT079B2

LT042C1

LT042D3

LT040D2

LT060B3

LT062B1

LT076D1

LT064D1

LT065D2

LT043C3

LTOL2D4

USGS PHOTOGRM

LTOMC1

41B4S02(50>

LT041C3

LT039A2

SHADOW DATA

LT078D2

LT041B3

LT061C1

LT061C4

LT077C1

LT042A3

LT042A3

LT061A1

LT040B3

LT0101B2

LT042B3

LT061C4

LT040A2

LT062A4

LT039B3

LT079B2

LT063B3

LT042C2

LT061C1

39C2 51(50)

39AISI (50)

1

1

1

2

1

1

1

1

1

1

1

1

2

1

1

2

1

1

1

1

IF

1

1

1

2

1

2

2

1

1

1

1

1

1

1

1

1

1

1

1

1

2

1

1

1

1

1

1

2

1

1

1

1

1

1

2

1

1

2

2

1

1

1

1

1

1

1

BGD

PMC

PM

PM

C

PM

PM

C

C

PM

PM

PM

C

PM

PM

C

PM

PM

C

PM

PM

PM

PM

PM

PM

PM

PM

PM

C

C

PM

C

C

PM

PM

PM

PM

PMC

PM

PM

PM

PM

PM

PM

PM

C

C

C

PM

C

PM

PM

PM

PM

C

C

PM

PM

C

PM

PM

PM

PM

PMC

PM

PM

PM

C

PM

PM

PM

PM

PM

PM

PM

DIAM.

8.0

8.00

7.88

7.84

7.81

7.75

7.63

7.63

7.57

7.56

7.50

7.50

7.5

7.38

7.38

7.38

7.38

7.35

7.12

7.12

7.1

7.0

7.00

7.0

7.00

7.0

7.06

7.00

7.0

7.0

6.85

6.81

6.8

6.75

6.62

6.59

6.50

6.5

6.38

6.25

6.20

6.19

6.19

6.13

6.12

6.1

6.0

6.0

6.0

5.90

5.80

5.81

5.75

5.65

5.56

5.5

5.50

5.5

5.5

5.44

5.44

5.31

5.25

5.25

5.19

5.10

5.06

5.02

5.0

4.94

4.9

4.88

4.81

4.72

4.63

DEPTH

2

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

2

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

.32

.46

.95

.84

.50

.6

.64

.4

.49

.34

.50

.57

.73

.5

.00

.10

.46

.62

.37

.60

.46

.14

.54

.35

.10

.33

.37

.93

.40

.30

.615

.35

.30

.44

.5

.55

.11

.33

.25

.26

.54

.425

.90

.26

.25

.28

.24

.0

.95

.83

.16

.22

.46

.2

.11

.20

.90

.15

.35

.07

.13

.01

.16

.88

.93

.02

.21

.1

.14

.85

.04

.09

.65

.73

.92

HT. WIDTH FLOOK CIRC

.40

.35

.15

.24

.19

.23

.20

.10

.20

L.5

1.3

1.6

1.83

1.85

1.6

1.59

1.2

1.9

.27 2.0

.25

.20

1.75

1.25

.35 2.4

.22

.09

.19

.25

1.5

1.0

1.0

1.80

.395 3.075

.15

.30

1.6

1.5

.23 2.54

.25

.25

.08

.10

.18

.15

.10

.23

.20

.265

.3

1.25

1.8

.8

1.0

1.2

1.25

1.75

1.0

1.25

1.95

1.85

.30 2.0

.25 2.10

.23

.17

.21

.13

.23

.11

.280

.345

.145

.13

.15

.21 ]

.11

.25

.15 ]

.10

.14

.20

.13

.68 «

.2 1

.15

.15

.15

1.7

1.7

1.50

.75

1.81

1.0

.65

.91

.56

1.19

1.4

.2

.2

.6

.0

t.l

.70

1.15

1.36

5.80

.62

1.3

1.0

1.25

.2 2.1

.13 ]

.25

.13

.18

.20 ]

.20

.15

.20

.10

.22

.20

,1?5

.20 1

.0* ]

.IS

.18 ]

L.25

1.50

.85

1.4

.25

.15

1.2

.90

1.2

1.25

.03

1.8

.56

.0

.55

.60

0.9

1.0

0.

0.

4.

1.5

0.

0.

0.

0.

0.

0.

0.

1.2

0.

0.

0.

1.4

1.6

1.5

1.6

0.

.75

0.

0.

0.

1.7

0.

0.

0.

0.

0.

0.

0.

0.

0.

1.75

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

1.3

0.

0.

.0

l.b

0.

0.

0.

0.

0.

0.

0.

0.

1.5

1.5

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

.83

.98

.83

.81

.R5

.P6

.82

.82

.90

.85

.88

.PR

.86

.82

.54

.82

.82

.«0

.70

.81

.87

.87

.87

.86

.75

.91

.90

.84

.80

.77

.85

.83

.85

.86

.86

.60

.86

.8<S

.91

.*5

.73

.82

.82

.92

.78

.85

.85

.85

.79

.86

.84

.88

,8B

.74

.87

.82

.85

.83

.88

.79

.87

.87

.83

.74

.79

.85

.76

.82

.86

.86

.83

.81

.77

.90

.90
GEOMETRIC INTERPRETATION OF LUNAR CRATERS C73

TABLE 9.—Continued

I.D.

A7060A

B6426

A1431

A1450

B2401

A4364

B0374

C6300

D5030

82421

A3403

B6591

B

A

A

A

C

A

A1486

B

B

B

B

B

A

B

B

B

B

A

B

A

B

B

B

B

B

B

B

B

B

B

B

B

A53

B

B

B

A

A

A

CRATER

SE OF TARUNT. H

KRIEGER C

IN CURIE.

ARATUS 0

ARATUS C

TIMOCHAHIS E

LITTROW BD

WALLACE B

GASSENDI N

CENSORINUS

TIMOCHARIS C

BESSEL H

IN HlRAYAMA

WOLLASTON V

UNNAMED

WOLLASTON U

PICAHD z

LEMONNIER LB

CAUCHY CA

ESE DESEILLIGNY

IN SKLODOWSKA

NEAR GAGARIN

12D20»S/358D04»E

LITTROW BB

LINNE

39.6KM SSW WOLLV

4.9 KM ESE WOLLU

28D30«N/325D42»E

N. OF DIOPHANTUS

NW OF KRIEGER C

17D32»N/25D55»E

57.8KM SSE WOLLV

22.5KM SW WOLLV

IN SKLODOWSKA

51.7KM S WOLLV

NE OF KRIEGER

29D25»N/334D20»E

30D40tN/325D37»E

APOLLO 17 SITE

SW AS16 SITE-E*

12.9 KM SW WOLLV

NE AS16 SITE

23.2 KM E WOLL.V

KIVA (APOLLO 16)

SW AS16 SITE-W

PALMETTO ASI&

19.5KM SSE WOLLU

60 KM SW WOLL U

NO. RAY APOLLO 16

GATOR APOLLO 16

31.3KMSSE WOLL.V

24D33'N/329D15»E

24.2KM SE WOLL V

28D28«N/325D47»E

N. OF DIOPHANTUS

SO. RAY APOLLO 16

NEAR RIMA PRINZ

32. IKM SSE WOLLV

25.7KM SSE WOLLU

NE AS16 SITE

50.5KM SSE WOLLU

N. OF DIOPHANTUS

34.0KM SSW WOLLU

24.2KM SSW WOLLU

3D29»N/36D46»E

3026»N/36D45'E

3D28»N/36D46»E

DATA P

LT062D4

LT039A3

USGS PHOTOGRM

LT041B3

LT041B3

LT040B4

42C3S3(50>

LT041D2

ORBITER 5 MAP

USGS PGM AS10

LT040B3

LT042A3

USGS PHOTOGRM

LT038B2

39A1SK50)

LT038B2 IRREG

LT062A2

LT042C?

61D2SK50)

LT042C1

USGS PHOTOGRM P

USGS PHOTOGRM

770350(50)

42C2SK50)

USGS PHOTOGRM

USGSMAP 1/100

USGSMAP 1/100

39B2SK25) P

USGSMAP 1/50M

USGSMAP 1/250

42C3SM50)

USGSMAP 1/100

USGSMAP 1/100

USGS PHOTOGRM

USGSMAP 1/100

39A1S1 (50)

40A1SK50)

39B2S2(25)

USGS MAP1/50M

APOLLO PGM

USGSMAP 1/100

APOLLO PGM

USGSMAP 1/100

APOLLO 14 MAP

APOLLO PGM

APOLLO PGM

USGSMAP 1/50M

USGSMAP 1/50M

APOLLO 14 MAP P

APOLLO 14 MAP

USGSMAP 1/50M

39B3S/U(50)

USGS 1/50MMAP

39B2SK25)

USGSMAP 1/50M

78DISK50)

USGSMAP 1/50M

USGSMAP 1/50M

USGS 1/50MMAP

APOLLO PGM

USGS 1/50MMAP

USGSMAP 1/50M

USGS 1/50MMAP

USGS 1/50MMAP

TM-ORBIIS2<2)

TM-ORBIIS2<2)

TM-ORBIIS2(2)

CL

2

1

1

1

1

1

1

1

1

1

1

2

2

F

1

1

1

1

1

1

1

1

2

2

1

2

1

2

2

2

SECONDARY-IMPACT

D3123

A7250

B1024

D3087

ZOELLNEK

GLAISHER C

SCHROETER

HYPATIA

LT078A3/D2 P

LT062A1

SHADOW DATA

LT078B4

4

4

4F

3

BGD

PM

PM

C

PM

PM

PM

PM

PM

C

C

PM

PM

C

PM

PM

PM

PM

PM

PM

PM

C

C

C

PM

PM

PM

PM

PM

PM

PM

PM

PM

PM

C

C

PM

PM

PM

PM

C

PM

C

PM

C

C

C

PM

PM

C

C

PM

PM

PM

PM

PM

C

PM

PM

PM

C

PM

PM

PM

PM

PM

PM

PM

DIAM. DEPTH

4

4

4

4

4

4

4

4

3

3

3

3

3

3

3

3

3

2

2

2

2

2

2

2

2

2

2

2

2

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

.3

.3

.25

.13

.13

.10

.05

.00

.9

.79 1

.75

.7

.45

.2

.18

.1

.00

.90

.85

.75

.75

.75

.600

.60

.45

.25

.03

.025

.00

.96

.95

.90

.75

.6

.50

.48

.375

.363

.30

.200

.15

.050

.03

.03

.95

.950

.95

.95

.95

.95

.92

.910

.78

.766

.73

.70

.60

.60

.60

.550

.53

.50

.47

.40

.064

.048

.042

.71

.85

.60

.96

.88

.84

.76

.77

.6

.039

.86

.67

.50

.58

.58

.50

.62

.60

.578

.57

.55

.63

.54

.50

.60

.53

.40

.373

.39

.35

.370

.45

.33

.3

.30

.29

.260

.235

.20

.160

.35

.094

.23

.15

.14

.120

.15

.14

.22

.14

.14

.190

.13

.118

.13

.13

.06

.10

.10

.064

.10

.08

.09

.07

.007

.005

.005

HT. WIDTH FLOOR CIRC

.06

.10

.18

.18

.10

.13

.10

.15

.05

.15

.13

.08

.18

.09

.10

.02

.13

.OR

.072

.05

.13

.18

.09

.08

.13

.10

.08

.065

.08

.05

.040

.OR

.08

.1

.04

.05

.030

.019

.03

.011

.05

.009

.04

.03

.007

.006

.02

.03

.04

.04

.03

.019

.01

.018

.03

.03

.01

.02

.02

.006

.02

.02

.02

.02

.002

.001

.001

.70

1.01

.88

.88

.75

1.33

1.00

1.10

0.45

1.2

1.0

1.0

.96

.80

.90

.10

.8

.95

1.025

.50

.50

.85

.75

.85

.65

.53

.64

.70

.50

.44

.77b

.55

.38

.4

.25

.45

.39

.344

.30

.250

.23

.250

.30

0.2R

.160

.225

.23

.25

.21

.28

.27

.270

.21

.207

.14

.19

.10

.15

.IP

.100

.16

.15

.14

.15

.008

.008

.008

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

.0

0.

0.

.6

0.

0.

0.3

.0

0.

0.

0.

0.

.65

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

.0

0.

.0

0.

0.

.0

.0

0.

0.

0.

0.

0.

.0

0.

0.

0.

0.

0.

0.

0.

.0

0.

0.

0.

0.

.0

.0

.0

.80

.82

.64

.89

.88

.83

,R6

.80

.88

.82

.77

.85

.84

.86

.85

,f>8

.80

.90

.87

.7R

.88

.83

.80

,8W

.89

.83

.86

.74

.74

.83

.ftO

.80

.86

.85

.80

.82

.86

.77

.85

.81

.73

.70

.77

.72

.71

.80

.81

.Rl

.71

.77

.84

.R5

.71

.84

.76

.Rl

.78

.76

.73

.73

.77

.80

.78

.82

.76

.75

.66

CRATERS

C

C

AMC

C

42

35

35

34

.75 2

.88 4

,

.50 1

.50

.13 1

.83

.50 1

.620

.180

.81

.10

8.60

8.000

4.

7.50

26.0

0.

29.

?1.0

.53

.35

.57

.31
C74 APOLLO 15-17 ORBITAL INVESTIGATIONS 

TABLE 9.—Continued

1.0.

00296

A2054

A7169

A7231

Dm9

C0440

C0483

00308A

A6084

A2033A

A2286

03243

00133

A2167

01262

00167

05078A

00320

05047

D1509

C0476

06103

D0465

05182

D5195

05175

00136

D0153

05161

A6099

05097

05089

05191

C1144

D0132

B7435

00143

05099

05049

05181

05196

DO 134

D8037

08028

08059

06101 A

05170

D8028A

D8037A

08035

08027

D8028C

08113

D8045A

08028B

D

08038F

CRATER DATA P CL BGD

VOGEL SHADOW DATA 2 C

D'ARREST LAC 60 (NEW) 3 C

LICK B , LT062A4 4F C

GLAISHER E LT061B2 3 C

RITCHEY LT077C2 P 3 c

PURBACH G SHADOW DATA 2 C

PURBACH H SHADOW DATA 3 C

LA CAILLE AB SHADOW DATA 3F C

SECCHI LT061C4 R 3 C

D'ARREST M LAC 60 (NEW) 4 C

AUWERS LT062A2 3F C

TACITUS D LAC 78 (NEW) 3 C

MUELLER LT077B4 3 C

JULIUS CAESAR G LAC 78 (NEW) 4F C

ABULFEDA 0 SHADOW DATA 3F C

ALBATEGNIUS B LT077C1 2 C

IMBRIUM 2NDARY LT079A3 3 C

PARROT H SHADOW DATA 3F c

ISIDORUS B (SO.) LT079A4 3 C

KAISER A SHADOW DATA 2 C

PURBACH C SHADOW DATA 3 C

CAPELLA E LT079A3 2 C

PURBACH B SHADOW DATA i c

CAPELLA GA/N LT079A3 2 c

IMBRIUM 2NDARY LT079D2 2 C

CAPELLA F LT079D2 2 C

ALBATEGNIUS G LT077C1 2 C

HIPPARCHUS J LT077B4 1 C

CAPELLA TA LT079A4 3 c

TARUNTIUS L LTO&ICI 2 c

CAPELLA M LT079A3 2F c

CAPELLA RA LT079A3 3 C

CAPELLA G LT079A3 2 c

PALISSA T LT077D1 2 c

HIPPARCHUS K LT077B4 1 C

HERODOTUS 0 LT038B4 3 C

MUELLER o LT077B4 2 c

CAPELLA C LT079A3 3 C

POSS.IMBR.2NDARY LT079A4 3 C

TSIDLKOV. 2ND LT066D2 04050 'N/138D30 'E

AITKEN 2NOARY LT085C1 08055' S/162DOO 'E

TSIOLKOV 2NDY LT066C1 05D35'N/140D1 0 'E

CAPELLA CA LT079A3 2 C

CAPELLA B LT079D2 1 C

TSIOLKOV 2NDY LT066A3 08025 'N/138D55'E

MUELLER A LT077C1 2 C

LANGRENUS 2NDARY LT080A3 2 PM

LANGRENUS FB LT080A3 1 PM

TSIOLKOV 2NDY LT066C1 06D35'N/143D15 'E

LANGRENUS BB LTOSOAS 2 PM

TSIOLKOV. 2ND LT066D2 06DOO«N/138D55'E

TSIOLKOV. 2ND LT066A3 08D30»N/139D1 0 'E

TSIOLKOV. 2ND LT066D2 07040 'N/139035'E

TSIOLKOV 2NDY LT066C1 06D45'N/143D25'E

TSIOLKOV 2NOY LT066C1 06D45'N/141D25 'E

CAPELLA DA LT079A3 2 C

CAPELLA R LT079A3 2 C

TSIOLKOV. 2ND LT066A3 08D35'N/138D55'E

LANGRENUS 2NOARY LT080A3 1 PM

LANGRENUS 2NDARY LTOSOAS 2 PM

AITKEN 2NDARY LT085CI 08045*5/161050 'E

TSIOLKOV 2NOY LT066C1 05DOO 'N/140D38»E

TSIOLKOV. 2ND LT066D2 05D25'N/138D45'E

LANGRENUS 2NOARY LT080A2 2 PM

TSIOLKOV 2NOY LT066C1 06D55'N/1*1D40 'E

LANGRENUS FC LT080A3 2 PM

AITKEN 2NDARY LT085C1 08037*5/161020 'E

TSIOLKOV. 2ND LT066D2 05D10'N/138038 'E

LANGRENUS 2NDARY LT080A3 2 PMLANGRENUS ?NDARY LTOSOAS 2 PM

LANGRENUS 2NDARY LT080A2 2 PM

TSIOLKOV, 2ND LT06602 06D32'N/139D07'E

LANGRENUS 2NDARY LT080A3 2 PM

LANGRENUS 2ND LT080A2 03D20'S/ 58055'E

LANGRENUS 2NDARY LT080A3 2 PM

DIAM.

34

31

30

30

29

26

26

24

24

23

23

22

21

21

21

19

18

18

18

17

17

16

16

16

16

15

15

14

14

13

12

12

11

11

11

10

10

10

10

10

10

10

9

9

8

9

9

9

8

8

7

7

7

7

7

7

7

7

6

6

6

6

6

5

5

5

5

5

5

5

4

4

4

4

4

.00

.5

.625

.25

.00

.70

.10

.45

.0

.50

.40

.60

.63

.50

.50

.00

.69

.00

.00

.35

.30

.44

.20

.20

.00

.70

.63

.38

.00

.250

.75

.000

.875

.50

.40

.625

.50

.500

.500

.50

.38

.16

.94

.90

.940

.69

.625

.50

.375

.315

.88

.875

.75

.750

.685

.625

.57

.000

.375

.280

.190

.125

.00

.960

.940

.690

.560

.310

.190

.160

.940

.910

.875

.690

.625

OFPTH

2

1

2

1

1

1

1

1

1

1

2

1

1

1

1

1

2

1

1

2

1

1

1

1

2

1

1

2

1

1

1

1

1

1

1

1

1

1

1

.40

.90

.965

.560

.400

.67

.35

.33

.35

.900

.07

.275

.030

.05

.63

.435

.825

.36

.670

.330

.31

.960

.250

.750

.135

.595

.750

.45

.465

.750

.80

.838

.990

.819

.535

.660

.690

.853

.714

.844

.745

.007

.168

.475

.735

.676

.688

.240

.910

.836

.005

.620

.900

.726

.470

.332

.040

.419

.725

.846

.990

.600

.112

.650

.605

.765

.780

.400

.66b

.507

.631

.400

.542

.531

.670

NT. WIDTH FLOOH CI&C

.

!99

.640

.650

.400

.

.

.

.73

.750

.46

.375

.44b

.35

.665

.400

.300

.

.500

.500

.

.385

.600

.306

.24b

.657

.470

.68

,4?3

.35U

.08

.075

.225

.440

.247

.160

.200

.100

.225

.120

.200

.207

.143

.380

.075

.250

.235

.200

,3?0

.130

.305

.075

.100

.184

.120

.100

.363

.069

.100

.150

.075

.150

.075

.125

.085

.102

.185

.075

.100

.100

.133

.150

.125

.056

.119

7

6

4

8

6

5

.00

.0

.69

.000

.00

.

.

.50

4.0

6

6

5

6

5

6

3

5

6

2

4

3

4

4

3

4

?

1

1

1

3

1

2

2

2

3

1

2

2

2

2

2

3

2

2

2

2

2

1

1

2

1

1

3

1

1

1

1

1

1

1

1

1

1

1

.0

.2

.20

.20

.75

.75

.50

.50

.

.500

.

.

.675

.

.100

.250

.600

.700

.3

.500

.625

.9

.875

.250

.250

.810

.500

.750

.250

.500

.750

.060

.400

.030

.560

.000

.150

.475

.500

.300

.590

.810

.800

.625

.125

.660

.935

.775

.750

.560

.735

.400

.560

.750

.270

.150

.700

.470

.345

.650

.920

.030

.300

.065

.780

.600

15

20

22

0

20

19

15

13

15

15

11

8

8

8

13

9

7

7

0

0

6

5

0

0

8

0

5

3

0

8

6

0

6

fl

0

5

0

0

0

0

0

3

2

0

2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

.5

.0

.50

.

.0

.0

.0

.0

.

.0

.0

.0

.0

.5

.5

.0

.0

.0

.

.

.0

.0

.

.

.0

.

.0

.9

.

.0

.5

.

.25

.

.

.80

.

.

.

.

.

.60

.50

.

.50

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.43

.58

.40

.33

.43

.61

.63

.58

.58

.39

.51

.55

.59

.43

.63

.53

.35

.62

.51

.43

.72

.55

.80

.38

.3R

.63

.66

.73

.70

.50

.6?

.36

.68

.54

.80

.36

.68

.46

.31

.46

.59

.57

.43

.63

.49

.61

.37

.42

.46

.39

.43

.60

.46

.52

.38

.63

.53

.36

.42

.63

.64

.40

.72

.52

.58

.43

.51

.30

.59

.43

.46

.36

.54

.44

.58
GEOMETRIC INTERPRETATION OF LUNAR CRATERS C75

TABLE 9.—Continued

I.D.

D8028D

D8057

D

D

D

D

0

D

D

D

D

D

B

B

B

B

B

B

6

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

CRATER DATA P CL BGD

LANGKENUS 2NDARY LT080A3 2 PM

TSIOLKOV. 2ND LT066D2 06D40 'N/139D10 »E

TSIOLKOV. 2ND LT066D2 07D15»N/1 39D25 »E

LANGRENUS 2NDARY LTOSOAS 3 PM

UNNAMED LT062D4 POSS. SECONDARY

LANGRENUS 2ND LT080A3 04D32»S/ 55D3PE

AITKEN 2NDARY LT085C1 08D33»S/160D25 »E

CATENA DAVY 1 SECONDARY USGS AS16 PGM

THEOPHIL. 2ND LT078B3 07D03»S/ 29D16»E

TSIOLKOV. 2ND LT066A3 08D20 »N/139D23 »E

CATENA DAVY 2 SECONDARY USGS ASI& PGM

CATENA DAVY 3 SECONDARY USGS AS16 PGM

THEOPHIL. 2ND LT078B3 07D23'S/ 29D34»E

TSIOLKOV. 2ND LT066A3 08D12»N/139D?7 »E

SECONDARY 100C1SK50) 25D56»S/1 02D59»E

TSIOLKOV. 2ND LT066D2 07D05'N/139D21 »E

THEOPHIL. 2ND LT078B3 07D15»S/ 29D56»E

THEOPHIL. 2ND LT078B3 06D59'S/ 29D08'E

THEOPHIL. 2ND LT078B3 07D13»S/ 29D50'E

THEOPHIL. 2ND LT078B3 06D50»S/ 29D10'E

THEOPHIL. 2ND LT078B3 07D38»S/ 29D26'E

THEOPHIL. 2ND LT078B3 07D04«S/ 2BD29»E

THEOPHIL. 2ND LT078B3 06D55»S/ 28D49»E

COPERNICUS SECONDARY 40A4SOK10)!

SECONDARY 39C2SK50) 20D26»N/328DOO »E

CATENA DAVY 5 SECONDARY USGS ASI& PGM

COPERN 2ND. 39C2SK50) POD27 'N/328DOO »E

CATENA DAVY 7 SECONDARY 77Disoiuo)

CATENA DAVY 8 SECONDARY 77D1SOK10)

CATENA DAVY 6 SECONDARY USGS AS16 PGM

DIOPHAN 2ND 39B2S0125 28D02»N/326Dl 1 «E

EULEH 2NDY. 39B3SK50) 25D18»N/329D42 »E

CATENA DAVY 4 SECONDARY USGS ASI& PGM

SECONDARY 39C2SK50) 20D24»N/328D1 1 »E

SECONDARY 41A3SK50) 25D24»N/359D52»E

COPERNICUS SECONDARY 40A4S01(10)3

CATENA DAVY 9 SECONDARY 77D1SOK10)

DIOPHAN 2ND 39B2S0125 28D10 »N/325D5«5»E

COPERNICUS SECONDARY 40A4SOK10I2

COPERN 2ND. 39C2SK50) 20D20 'N/328D12 »E

SECONDARY 39B3SK50) 25D08'N/329D1 1 «E

COPERN 2ND. 40A1SK50) 29D45»N/333D38»E

ARISTARCHUS SECONDARY-7 USGS 1/lOM MAP

KEPLER 2ND, 39B3SK50) 25D03»N/328D51 »E

COPERN 2ND. 39C2SK50) 20D55»N/328D08«E

ARISTARCHUS SECONDARY-4 USGS 1/lOM MAP

EULER 2NDY. 39B3SK50) 24D25»N/329D06»E

KEPLER 2ND. 39B3S1(50) 25D13»N/329002 I E

COPERN 2ND. 39C2SK50) 20D57»N/328D20'E

ARISTARCHUS SECONDARY-1 1USGS 1/lOM MAP

EULER 2NDY. 39B3S1 (50) 25D19»N/329D07 'E

EULER 2NDY. 39B3SK50) 25D12»N/329D1 3'E

ARISTARCHUS SECONDARY-2 USGS 1/lOM MAP

ARISTARCHUS SECONDARY-1 USGS 1/lOM MAP

KEPLER 2ND. 39B3S1 (50) 25D12'N/328D34»E

EULER 2NDY. 39B3SK50) 24D21 »N/329D01 »E

KEPLER 2ND. 39B3SK50) 25D06'N/328D53 »E

ARISTARCHUS SECONDARY-6 USGS 1/1 OM MAP

ARISTARCHUS SECONDARY-8 USGS 1/lOM MAP

KEPLER 2ND. 39B3SK50) 25D15»N/329D04 »E

ARISTARCHUS SECONDARY-3 USGS 1/lOM MAP

ARISTARCHUS SECONDARY-5 USGS 1/lOM MAP

ARISTARCHUS SECONDARY-9 USGS 1/lOM MAP

ARISTARCHUS SECONDARY-10USGS 1/lOM MAP

ARISTARCHUS SECONDARY-15 USGS 1/lOM HAP

ARISTARCHUS SECONDARY-20 USGS 1/lOM HAP

ARISTARCHUS SECONDARY-16 USGS 1/1QH HAP

ARISTARCHUS SECONDARY-18 USGS 1/lOM HAP

ARISTARCHUS SECONDARY-13 USGS 1/lOM HAP

ARISTARCHUS SECONDARY-14 USGS 1/lOM HAP

ARISTARCHUS SECONDARY-17 USGS 1/lOM HAP

ARISTARCHUS SECONDARY-12 USGS 1/lOM HAP

ARISTARCHUS SECONDARY-19 USGS 1/lOM HAP

DIAM. DEPTH

4,

4,

4,

4,

4,

3.

3,

3.

3,

3,

3,

3,

3,

3,

3,

3,

2,

2,

2,

2,

2,

2.

2,

2,

2,

2,

2,

1,

1,

1,

1,

1,

1.

1,

1,

1,

1,

1,

1,

1,

1,

1,

1,

1,

1,

1.

1,

1

1

1

<

1

1

1

1

1

1

1

I

I

4

1

1

1

4

4

4

I

4

4

1

,500

,440

,375

,125

,0

,940

,750

,5

,440

,440

,4

,2

,19

,150

,125

,060

,750

,750

,750

,66

,475

,375

,325

,265

,160

,1

,100

,97

,85

,85

,850

,740

,7

,675

,650

,580

,50

,490

,410

,325

,290

,175

,120

,100

,100

,045

,025

,975

,940

,930

,890

,875

,740

,735

,725

,700

,675

,665

,660

,640

,605

,550

,470

,445

,390

.365

,355

,320

,285

,265

,220

.220

.210

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

415

300

500

525

530

475

626

607

300

525

497

541

400

275

220

500

225

200

250

300

200

225

200

367

200

316

193

275

300

206

119

270

195

265

145

343

180

151

218

182

200

149

109

131

119

077

097

174

118

076

102

155

060

072

113

095

131

073

078

063

063

066

053

060

047

039

030

029

028

027

028

021

027

HT. WIDTH FLOO«

.065

.075

.050

.075

,07b

.075

.100

.083

.

.075

.074

.072

.

.050

.040

.050

.

.075

.050

.075

.

.

.

.045

.015

.045

.015

.040

.040

.040

.061

.015

.050

.030

,0?5

.040

.030

.057

.040

.015

.020

.022

.023

.020

.020

.039

.010

.015

.014

,0?6

.010

.015

.025

.030

.020

.006

.020

.024

.020

.010

,0?0

.022

.010

.015

.013

.009

.010

.005

.005

.005

.010

.005

.006

1.190

.780

.900

1.060

.500

1.090

1.000

.900

.

1.400

.900

.900

.

.675

.700

.850

.

.

.

.

.

.

.

.500

.440

.600

.750

.550

.650

.550

.760

.500

.500

.430

.360

.500

.500

.480

.350

.290

.280

.300

.240

.360

.425

.225

.225

.290

.280

.235

.230

.260

.180

.135

.210

.275

.235

.165

.170

.140

.210

.175

.090

.113

.130

.140

.150

.110

.080

.070

.060

.080

.065

0,

0,

0,

0,

1

0,

0,

0,

0,

I

1

0,

0,

I

0,

0,

0,

0,

0,

0,

0,

0,

4

I

1

0,

1

I

0,

0,

1

1

1

1

I

0,

1

0,

1

0,

<

0,

0,

1

0.

0.

0.

1 0,

0,

1

4

0.

0,

0,

1

0,

4

1

4

o!

0,

0,

0,

0,

0,

0,

0,

0,

I

I

I

»

,0

1

I,0

I!o

,0

1

»,p

>

1

»

»

»

»

»

1

,0

,0

,0

,,0

,0

,0

»

>

,0

,0

,0

,0

,0

,,0

I,0

1

,0

»

I,0

»

1

1,0

»

»

,0

,0

»

»

1

,0

,0

I

,0

,0

,0

,0

1

1

I

I

1

1

»

I

»

CIRC

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

*

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

45

42

35

69

38

56

58

77

52

80

48

78

42

28

62

66

69

44

69

73

69

53

62

62

59

68

56

60

46

68

39

73

79

74

42

84

34

64

58

63

76

65

36

27

76

49

27

73

62

44

75

80

17

56

66

74

74

76

54

67

46

60

46

61

68

68

54

43

46

59

42

48

78
C76

l.o,

APOLLO 15-17 ORBITAL INVESTIGATIONS 

TABLE 9.—Continued

CRATER DATA P CL BGO DlAM. DEPTH HT. WIDTH FLOOR CI&C

CRATERED CONES

B ARISTARCH.CONE 1 ORBITER 4 SHADOW DATA 

B ARISTARCH.CONE 2 OHBITER 4 SHADOW DATA

LASSELL HJ CONE isooo'S/iooso'w

SERENITAT.CONE 1 18D38'N/27038'E«42C3S2

B MARIUS HILLS CONE L.O.4-157 SHADOW DATA

SERENITAT.CONE 2 18D57«N/27028«E»42C3S1

KING FLOOR CONE NE 5D35«N/i20D5i«s 

KING FLOOR CONE C. 5D32«N/120D48«E 

B COPERNICUS FLOOR CONE LOB-IB* SHADOWS 

KING FLOOR CONE SO 5D27«N/120D«B«E 

KING FLOOR CONE W. 5D31«N/120D43'E 

B COPERNICUS FLOOR CONE LOB-152 SHADOWS 

B COPERNICUS FLOOR CONF Los-153 SHADOWS

B COPERNICUS FLOOR CONE L05-152 SHADOWS

1.865

1.505

1.5

.925

.750

0.635

.500

.488

.475

.440

.330

.260

.240

.120

.194

.136

.240

.060

.030

.030

.056

.050

.060

.062

.038

.030

.040

.015

.145

.116

.175

.072

.200

.060

.152

.147

.160

.068

.145

.090

.12b

.055

.655

.523

.525

.700

.273

.550

.450

.430

.440

.380

.510

.270

.285

.180

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.0

.28

.41

.80

.53

.5?

.55

.35

.48

.67

.79

.54

.69

.41

.65

DARK-HALO CRATERS

C ALPHONSUS KC 77D3S0150 12D52'S/358D23'E

C ALPhONSUS MD 77D350150 12D31»S/358D03'E

C IN ALPHONSUS 77D3S0150 12D33«S/358D17«E

	CRATERED DOMES

B HERODOTUS OMEGA DOME L04-150 SHADOWS

	"D-CRATER" DOME 18D40'N/05D20»E 

C DOME WEST OF KIES L04-125 SHADOWS 

B DOME WEST OF MILICHIUS L04-133 SHADOWS 

A MARALDI 8 DOME 2 NW. 61A2S01 

B HORTENSIUS PHI DOME L04-133 SHADOWS 

B CAUCHY OMEGA DOME L04-225 SHADOW DATA 

A MARALDI B DOME 1 SE. 61A2S01 

B HORTENSIUS OMEGA DOME L04-133 SHADOWS 

B HORTENSIUS SIGMA DOME L04-133 SHADOWS 

B RIMA ARIS.8 DOME 30D30«N/49D10»W USGS 

C DOME W.OF FRA MAURO-S. L04-120 SHADOWS 

B MARIUS HILLS DOME L04-157 SHADOW DATA 

C DOME W.OF FRA MAURO-W. L04-120 SHADOWS 

C DOME W.OF FRA MAURO-E. L04-120 SHADOWS

2

2

1

2

2

1

1

1

1

1

1

1

1

1

1

.550

.250

.725

.90

.650

.830

.730

.70

.670

.650

.60

.530

.485

.275

.200

.950

.670

.470

.376

.336

.212

.300

.050

.265

.165

.185

.190

.180

.070

.190

.180

.025

.210

.120

.070

.095

.050

.050

.015

.460

.200

.690

.260

.090

.370

.300

.130

.590

.350

.065

.540

.270

.4?b

.400

1.

5.

6.

5.

4.

3.

2.

4.

3.

3.

2.

2.

1.

6.

1.

1.

725

875

900

460

175

590

700

4

260

150

4

240

770

235

500

030

070

100

.0

.0

.0

.0

2.25

.0

.0

0.

.0

.0

0.

.0

.0

.0

.0

.0

.0

.0

.58

.64

.47

.15

.54

.71

.75

.49

.66

.59

.45

.71

.73

.30

.50

.88

.65

.80
GEOMETRIC INTERPRETATION OF LUNAR CRATERS 

TABLE 10.—Dimensions of 59 terrestrial craters

C77

CRATER DATA DIAM. DEPTH HT. WIDTH FLOOR CIRC

METEORITE-IMPACT CRATERS

RIESKESSEL BAVARIA»GER.

EL'GYGTGYN E. SIBERIA* USSR

LAKE BOSUMPTWI GHANA

FLYNN CREEK TENNESSEE *USA

NEW QUEBEC UNGAVA* CAN.

ROTER KAMM NAMIBIA

TENOUMER MAURETANIA

LONAR LAKE INDIA, OECCAN

METEOR CRATER ARIZONA* USA

WOLF CREEK W. AUSTRALIA

TEMIMICHAT G. MAURETANIA

AOUELLOUL MAURETANIA

BOXHOLE NT* AUSTRALIA

ODESSA-1 TEXAS* USA

KAALIJARV ESTONIA

ILUMETSA-l ESTONIA

CAMPO 0. CIELO-2 ARGENTINA

HENBURY-8 NT, AUSTRALIA

HENBURY-3 NT. AUSTRALIA

ILUMETSA-2 ESTONIA

OALGARANGA W. AUSTRALIA

ODESSA-2 TEXAS* USA

SIKHOTE-ALIN-60 SIBERIA. USSR

EXPERIMENTAL-EXPLOSION

SEDAN THERMONUCL NEVADA 1962

CABRIOLET NUCL 2.3 KT 1968

LACROSSE THERM. ENIWETOK 1956

CACTUS THERMONUC ENIWETOK 1958

TEAPOT-ESS NUCL. NEVADA 1955

JANGLE-U NUCL. NEVADA 1951

OANNY BOY NUCL. NEVADA 1962

JOHNIE BOY NUC 0.5 KT 1963

FAULD (UK) CHEM. 1944

SCOOTER TNT NEVADA I960

PRE-SCHOONER-II NITROMET 1965

STAGECOACH-3 20-TON 1960

CINDER LAKE-3 ARIZONA 1967

BUCKBOARD-12 TNT NEVADA 1960

PRE-SCHOONER-D NITROMET 1965

STAGECOACH-2 20-TON I960

DISTANT PLAIN CANADA 1966

CINDER LAKE-2 ARIZONA 1967

CINDER LAKE-5 ARIZONA 1967

BUCKBOARD-11 TNT NEVADA 1960

CINDER LAKE-1 ARIZONA 1967

PRE-MINE THROW-* ioo-TON/6 »74

CINDER LAKE-4 ARIZONA 1967

CHARIOT TNT ALASKA 1960

SANDIA 256-LB-8 TNT 1959

SAND I A 256-LB-9 TNT 1959

PRE-MINE THROW-* 7,i-TON/3 »73

SANDIA 256-LB-2 TNT 1959

CUNGARD CRATER PENNA. 1970

TONOPAH 211-56 0.5 TON 1968

TONOPAH 211-52 0.5 TON 1967

TONOPAH 211-40 64 LBTNT 1966

TONOPAH 211-51 64 LBTNT 1967

TONOPAH 211-53 64 LBTNT 1968

TULALIP W2.5B TNT 1964

TULALIP D2A-E TNT 1964

22.5 .500 .125 5.0 0.0 .77

18.0 .620 .200 5.0 0.0 .77

10.42 .400 .135 2.0 0.0 .79

3.600 .225 .075 .500 0.0 .72

3.4* .415 .153 .610 0.0 .88

2.438 .425 .110 .670 0.0 .85

1.920 .329 .100 .300 0.0 .90

1.830 .275 .025 .125 0.0 .81

1.21 .207 .055 .248 0.0 .74

.875 .145 .043 .145 0.0 .75

.686 .069 .023 .100 0.0 .80

.389 .051 .021 .074 0.0 .85

.175 .015 .004 .034 0.0 .77

.168 .029 .003 .030 0.0 .68

.097 .016 .006 .020 0.0 .80

.080 .012 .004 .015 0.0 .78

.070 .015 .002 .015 0.0 .86

.070 .004 .002 .012 0.0 .85

.061 .005 .002 .012 0.0 .80

.050 .005 .002 .015 0.0 .*5

.025 .005 .001 .006 0.0 .78

.021 .OOb .001 .006 0.0 .83

.010 .002 .0004 .001 0.0 .83

CRATERS

.408 .110 .015 .062 0.0 .93

.135 .047 .012 .027 0.0 .85

.123 .015 .002 .015 0.0 .85

.113 .017 .006 .030 0.0 .89

.107 .033 .006 .031 0.0 .81

.091 .019 .0025 .008 0.0 .80

.085 .026 .007 .028 0.0 .84

.045 .0126.0035 .010 0.0 .86

.243 .033 .006 .061 0.0 .66

.104 .027 .004 .022 0.0 .75

.070 .024 .005 .018 0.0 .83

.042 .0109.0020 .0095 0.0 .82

.041 .003 .0003 .016 0.0 .95

.041 .013 .0027 .010 0.0 .75

.037 .010 .003 .007 0.0 .79

.036 .0089.0017 .0075 0.0 .84

.035 .006 .001 .008 0.0 .84

.035 .003 .0003 .011 0.0 .99

.034 .003 .0004 .009 0.0 .94

.034 .009 .0015 .007 0.0 .76

.033 .002 .0003 .Oil 0.0 .98

.026 .0055.0011 .007 0.0 .83

.020 .001 .0001 .006 0.0 .90

.011 .003 .0008 .002 0.0 .79

.0095.0028.0006 .0024 0.0 .85

.0088.0023.0004 .0024 0.0 .86

.0087.0017.0002 .002 0.0 .82

.0082.0027.0004 .004 0.0 .75

.0076.0036.00078.0024 0.0 .74

.0075.0017.00027.0021 0.0 .80

.0069.0015.0002 .0018 0.0 .81

,0051. 0015. 00023. 0021 0.0 .83

.0026.0006.00010.00058 .0 .78

.0024.0006.00011.00055 .0 .85

.0014.0002.00005.0002 0.0 .82

.0013.0001.00002.0003 0.0 .83