Smithsonian

Contributions to Astrophysics

VOLUME 1, NUMBER 2

PAPERS ON

REDUCTION METHODS FOR

PHOTOGRAPHIC METEORS

SMITHSONIAN INSTITUTION

Washington, D. C.

1957
Publications of the Astrophysical Observatory

This series, Smithsonian Contribution* to Astrophysics, was inaugurated

in 1956 to provide a proper communication for the results of research conducted at the Astrophysical Observatory of the Smithsonian Institution.

Its purpose is the "increase and diffusion of knowledge" in the field of

astrophysics, with particular emphasis on problems of the sun, the earth,

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other investigators with whom we have common interests.

Another series is Annals of the Astrophysical Observatory. It was started

in 1900 by the Observatory's first director, Samuel P. Langley, and has been

published about every 10 years since that date. These quarto volumes, some

of which are still available, record the history of the Observatory's researches

and activities.

Many technical papers and volumes emanating from the Astrophysical

Observatory have appeared in the Smithsonian Miscellaneous Collections.

Among these are Smithsonian Physical Tables, Smithsonian Meteorological

Tables, and World Weather Records.

Additional information concerning these publications may be secured

from the Editorial and Publications Division, Smithsonian Institution,

Washington, D. C.

FRED L. WHIPPLE, Director,

Astrophysical Observatory,

Smithsonian Institution.

Cambridge, Mass.

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

Washington 25. D. C. - Price 70 cents
Contents

Page

Reduction Methods for Photographic Meteor Trails: Fred L. Whipple

and Luigi G. Jacchia 183

The Method of Reduction of Short- Trail Meteors: Gerald S. Hawkins. 207

A Rapid Graphical Method of Meteor Trail Reduction: Richard E.

McCrosky 215

A Reduction Method for the Motions of Persistent Meteor Trains:

Allan F. Cook and Robert F. Hughes 225

Methods for the Study of Shower Radiants from Photographic Meteor

Trails: Fred L. Whipple and Frances W. Wright 239

m

Reduction Methods for Photographic

Meteor Trails1

By Fred L. Whipple2 and Luigi G. Jacchia*

Methods for determining the space trajectories of meteors observed simultaneously from

two stations have been developed by various

investigators including Bessel (1839), Schaeberle (1895), Turner (1907), Olmsted (1931),

and others. The present photographic techniques of observing meteors simultaneously

from two stations (Whipple, 1938) require a

precise reduction technique suitable for mechanical rather than for tabular methods of calculation. The use of rotating shutters in the

determination of meteoric velocities further

complicates the reduction, as does the necessity,

in many cases, of solving for the instant of the

meteor from the photographic trails. The

solution now includes the determination of

meteoric deceleration as well as the velocity and

usual geometric circumstances of the phenomenon. Hence the reduction process has become

relatively complex.

In the methods of reduction presented here

we borrow freely from established techniques.

Rectangular coordinates and direction cosines

enter extensively, as they are most adaptable

to efficient mechanical computation. The

choice of notation has presented a difficult

problem because of the multiplicity of symbols

required. By the generous use of subscripts

we have avoided duplication as much as

possible.

Measurement of a meteor trail and comparison stars

At each station the meteor trail and the field

stars are photographed simultaneously on an

emulsion. We shall assume that the star images are untrailed, or that some point or points

along the trailed images of the stars can be

identified in time and are measurable. The

optical projection will be assumed approximately gnomonic, as though points on the celestial sphere were projected through its center

onto a plane surface. Graphical corrections in

reduction can make allowance for rather large

deviations from perfect projection. In case the

projection itself deviates markedly from the

gnomonic, as in fast Schmidt camera systems

with curved focal surfaces, the negatives should

be copied on flat glass plates by means of a lens

system approximately to a gnomonic projection

(see pi. 1). Measurements are then made on

the positives.

The gnomonic projection is very closely approximated by current astronomical lens systems, even those of rapid focal ratio and wide

field. This projection has the advantage that

great circles project into straight lines in the

plane; meteor trails lie so nearly on great circles that careful measurement and reduction

usually are required to demonstrate deviations.

Preparation of the photographic material. Stars

1

 Carried out in part under research contracts with the V. 8. Naval Bureau of Ordnance (Re3d), the U. S. Office of Naval Research, Contract No.

N5ori-07647, and the U. S. Army Office of Ordnance Research, Contract'No. DA-19-020-ORD-2556. Reproduction in whole or In part is permitted for

any purposes of the XJ. 8. Government.

' Smithsonian Astrophyslcal Observatory.

* Harvard College Observatory.

183
184 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

to be measured can be identified directly by

marking with ink on the glass side of the photographic plates (see pi. 2). If film is to be measured, it should be placed between two goodquality pieces of glass fastened firmly at the

edges to prevent slippage of the film. Identification marks can then be made on the glass

surface.

Choice of comparison stars. For all goodquality cameras of focal length greater than

about 3 inches, the precision of measurement

and reduction is greater than that of the star

positions in the Bonner Durchmusterung catalog (BD). Astrographic catalog positions are

suitable where right ascension and declination

are tabulated directly, but identification problems are difficult because meteor trails usually

extend over two or more zones. Where only

rectangular coordinates are tabulated the reduction of adjacent zone coordinates to a common system is added to the problem of identification. Hence the astrographic catalog positions are rarely useful in the reduction of meteor trails, with the exception of the Helsingfors

Zone (+39° to +47°) and the Catania Zone

(+46° to +55°) which give right ascension and

declination of all stars.

Normally, star positions should be selected

from the Astronomische Gesellschaft catalogs

(AG) or from the Yale catalog. The meteor

trail should be sketched lightly in pencil on a

BD chart and the AG stars identified in its

neighborhood. Stars for measurement should

be selected at uniform intervals along the trail

(z-direction) and alternately on opposite sides

of the trail (+ and — y) as close to the trail

as is convenient or possible. The number of

stars required increases with trail length and

field distortion. For a focal length of 6 inches

and a meteor trail 10° long, eight or ten stars

are usually sufficient. No trail should be reduced with fewer than six comparison stars.

With film, more comparison stars are usually

necessary. No precise general rule can be

given; the experimenter should determine the

optimum number on the basis of his own experience. To minimize the centering errors in

stars with images distorted by vignetting, it is

recommended that the stars selected be as

nearly of the same photographic magnitude,

and as near the plate limit, as possible.

Measuring engine. The ideal measuring engine for our purpose has two mutually perpendicular precision screws to measure x and y

coordinates simultaneously in a plane. Periodic errors should not exceed one micron, or

should be calibrated and capable of correction.

Precise perpendicularity of the ways is not

important in meteor reductions. The ways

must be straight within approximately one

micron per centimeter. Large deviations from

perpendicularity and some smooth curvature of

the ways, as well as considerable systematic

errors in the screws, can be allowed for precisely

in the reductions.

Single-screw measuring engines in which the

plate can be turned through 180° in its plane

can also be used effectively, as can engines in

which the y-screw is quite short compared to

the x-screw. No accuracy is lost if y must be

measured at a different time from x; in fact,

this technique is recommended.

A relatively low-power microscope is advisable for the measurement of meteor trails.

Mutually perpendicular crosswires, parallel to

the ways of the engine, are required. One of the

excellent measuring engines made by D. W.

Mann and currently used in the Harvard Meteor Program appears in plate 3.

Measurements. The meteor trail should be

carefully oriented parallel to one of the engine

w&ys. The convenience in checking errors and

in combining "direct" and "reverse" measures

well repays a little effort to make the parallelism good to a few microns over the length

of the trail. Measures in y, perpendicular

to the trail, should be made for each comparison star, and at frequent uniform intervals

along the trail, perhaps at every millimeter.

Measures in x, along the trail, should be made

for each star at the beginning and end of the

trail, at the beginning, end, and middle of

each shutter break, and at the beginning, end,

and middle of each flare or noticeable irregularity in the trail. Measures in y along the

trail need not be made at the same points as

measures in x.

It is convenient to identify the shutter

breaks by numbers in time sequence, and to

identify the flares or irregularities in the trail

by consecutive letters. When inspection shows

that it is possible, it is also convenient to use
S & 7

Scale - Inches

& 

A curved film for the Baker Super-Schmidt meteor cameras.
Negative reproduction of a marked glass copy of a Super-Schmidt film. (The original copy is a 
A 2-screw rotatable horizontal measuring engine built for the U. S. Naval Bureau of Ordnance by D. W. Mann, Lin

REDUCTION METHODS FOR PHOTOGRAPHIC METEORS 185

the same letter to identify the flares on the

trails on each of the paired photographs from

the two stations.

The focus of the microscope should be set

most carefully at the beginning, end, and middle

of the trail to establish the best compromise

in case the engine carriage is not precisely flat.

Focus can be tested most accurately by the

minimizing of parallax between the wires and

star images when the eye is moved across

the eyepiece.

On most measuring engines, because of

possible backlash, it is advisable to make all

measures with the screw turning in the direction

of increasing scale readings. Measures on meteor trails are most advantageously made with

the microscope crosswires respectively parallel

and perpendicular to the meteor trail, to the

accuracy within which the wires are straight.

This necessitates measuring not at the intersection of the wires but at some definite position

on each wire near the intersection. It is

convenient and probably advisable to make all

measures in one coordinate separately, and

for all points consecutively in order of increasing x values. A remeasure of the first point at

the end of such a run will indicate immediately

whether the adjustment of the engine has been

disturbed during the run or whether temperature changes have affected the settings systematically. Temperature changes should be

eliminated as much as possible, but in a closed

room their effect tends to stabilize during the

course of an hour or two of measuring. Hence

a repetition of earlier measures in the same

sequence will often show a progressive decrease

in the differences of the readings. Physical

disturbance of the engine will usually show as a

distinct break in the series of differences in

readings. Where systematic temperature variations cannot bo avoided, a few checks at representative points in a run should be made at

the end; if the difference in readings changes

smoothly from beginning to end, the original

set can be accepted without correction.

All measures made "direct," with the plate

in its original position, should be repeated

"reverse" after the plate has been turned 180°

in its own plane.

Each "measure" should be the mean of

three or more fairly rapid settings.

Combination of the measures. The combination of direct and reverse measures not only

eliminates systematic errors between the meteor trail and stars and between various parts

of the meteor trail but also provides an almost

foolproof method of checking for setting or

reading errors. Usually it is most convenient

to reduce the "reverse" set of measures to

values nearly identical with those of the "direct" set by subtracting their values from an

appropriate number. Then differences, direct

measures minus reverse, can be taken easily

and plotted for both x and y on a magnified

(at least 100 times) ordinate scale against x

as abscissa. The resulting points should lie

on a curve that is smooth within the accuracy

of the settings (a very few microns). Conspicuous deviations that are not correlated

with abrupt changes in the brightness of the

trail or in image quality should be rechecked

on the measuring engine. When possible it is

often time-saving to leave the plate in the

measuring engine until the measures have

been combined, so that errors can be checked

quickly after they have been detected.

After gross errors have been eliminated,

the mean of the direct measures and the constant minus the reverse measures should be

obtained. These values constitute the measures to be used in the reductions.

Reductions for the apparent radiant

Corrections for stellar proper motion. The AG

or other position of each star, s, should be corrected for proper motion in right ascension, a,,

and declination, 5,, if values of the proper

motion can be found in the appropriate catalogs. It should be noted that the proper

motions should be applied for the interval from

the epoch of the stellar observations in the AG

or other catalog to the epoch of the meteor,

fractions of years being taken into account.

Also, attention should be given to the fact that

some catalogs give proper motions as Aa and

some as Aa cos 5, per year or per century.

Stellar direction cosines and precession. The

stellar coordinates, after correction for proper

motion, are transformed into equatorial direc-
186 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

tion cosines X,, n,, v,, for the mean equinox of

the catalog, by the formulae:

X,=COS S, COS a,

M«=cos 8, sin a,

rt=sin St

Check:

(1)

(la)

Six significant figures in calculation are

usually appropriate. The check formula is not

adequate for direction cosines smaller than

approximately 0.2, and will not check erroneous

table entries in both functions. A repetition

of this phase of the calculation is advisable to

guard against errors.

Precession can be applied most easily to each

star position by means of nine precessional

direction-cosines, Xx, Xy, etc., applied to the

direction cosines of the star position as follows:

(2)

where the precessional direction cosines apply

to the transformation from Equinox 1 (\lf MI»

PI) to Equinox 2 (X2, m, v2). Tabulations of

these direction cosines from 1850 to 1950

applying to the transformation to 1900 have

been made by E. C. Bower (1932).

The Bower tabulations apply for transformations

from the mean equinox at the beginning of the year to

the mean equinox of 1900.0. The direction cosines to

be applied in equations (2) may be obtained by the

Cracovian calculation (Banachiewicz, 1925, 1929):

(XXYXZX) = (+Xx - Yx-Zx) (XXX,X,)

(XWY.Z,) = (-X,+ Y,+ ZJ (YXY,Y.)

{XtY.Z.) = (- X,+ Y.+ Z.) (ZXZUZ.)

Equinox 1 Equinox 2 to Equinox 1

to 2. 1900.0. to 1900.0.

This notation denotes a calculation convention more

convenient than the usual matrix convention. All

quantities on the left are the sums of three products

of quantities in two vertical columns on the right.

For example,

The tabulations by Bower may also be used less

conveniently in two steps, Equinox 1 to 1900.0 and

1900.0 to Equinox 2.0

Similar quantities for reduction from the

current year to 1950.0 are given in the British

Nautical Almanac.

The direction cosines of the star positions are

rotated from the mean equinox of the star

catalog to the mean equinox at the beginning

of the year nearest in time to the meteor (a

September meteor of 1948 to the mean equinox

of 1949, etc.).

A transfer to the true position at date would

carry advantages too slight to offset the additional computation involved. The meteor positions are affected by aberration in the same

fashion as the star positions, while light-time

is negligible on any basis. The combined

errors in aberration and precession-nutation

over half a year enter the solution chiefly in

terms of absolute time. They amount to a

maximum error of 3* of time in the absolute

position of the meteor, or about 50 feet.

Refraction of light in the atmosphere also

affects the meteor positions and the star

positions in almost the same fashion. Its

minor differential effects can be taken into

account accurately in the general reduction.

Standard coordinates of the star positions.

Standard coordinates, | and 17, are rectangular

coordinates in the plane of celestial-sphere

projection (approximately that of the photographic plate) with origin at the center of

projection. They are expressed in radian

measure. The coordinate jj increases northerly

along the hour circle through the center fo

projection, and | increases easterly along a

tangent to the parallel of declination at the

center.

The center of projection, ae, 5e, may be taken

as the plate center and determined with sufficient precision by means of a ruler on the plate

and a BD chart. Rough precession must be

applied. The direction cosines of the origin

of the new rectangular system and those of the

poles of the two coordinate axes are computed

from the following formulae to the same relative precision as the direction cosines of the

star positions:
REDUCTION METHODS FOR PHOTOGRAPHIC METEORS 187

Plate center{

Xe=cos 8e cos ae

JUV.=COS Se sin ae

pe=sin ae

{

X , = —sin 8e cos ae

M,= — sin 5e sin ac (3)

[X{=—sin ac

I axis < ju{= +cos ae

Uf=O

Check:

(3a)

For each star the cosine of a,, the angle between the star position and the center, and the

standard coordinates of the star position are

computed as follows:

COS O-, =

Check:

(4)

(4a)

Equation (4a) does not provide a satisfactory

end-figure check for values of £ or y less than

0.2. A second or a reverse calculation (equation 10) is generally required.

Standard coordinates along the meteor trail.

SIX-CONSTANT METHOD: The measured positions can be expressed in terms of the standard

coordinates by means of a least-squares solution of the following linear equations for all of

the measured stars:

(5)

Here ax, bx, . . . , cv are plate constants,

the unknowns in the least squares solution.

Residuals are more conveniently expressed in

the reverse sense, computed minus observed:

Az,=:c,(Comp.)— x,(obs.)

Ay,=y,(Comp.) —y,(obs.)

Check:

(6)

(6a)

(In addition, all the three normal equations

must be satisfied.)

At this stage of the calculation the accumulated small errors arising from optical distortion, large-scale emulsion distortion, misplacement of the plate center, differential aberration, differential refraction, systematic screw

error, and systematic deviations of the engine

ways from linearity can all be eliminated

graphically. Deviations of the engine ways

from perpendicularity are already allowed for

precisely in the least-squares solution, as are

all other linear effects along the emulsion.

The residuals, Az, and At/,, for each measured

star position are plotted separately as ordinates

against the measured x, as abscissa. Smooth

curves are drawn through the residual points

on these graphs.

All the measures made on the meteor trail

are corrected from these curves of residuals,

the measured x being used as argument to provide the appropriate correction from either the

Ax or Ay curve. The corrections are applied as

follows:

x (corrected) =x (measured) + Ax (curve) (7)

y (corrected) =y (measured)+Ay (curve).

These corrected measures are henceforth used

in all calculations involving trail coordinate

measures.

It will be seen that this graphical correction

method has the merit of transferring all the

measures from the actual photograph to the

theoretically perfect plane of projection, and

hence eliminates the various errors listed above.

The remaining errors are those arising in the

measurement, from optical or photographic

image distortion, from small-scale errors in the

emulsion or measuring engine, and from errors

in the star positions.

After the measured x's and y's on the meteor

trail have been corrected by the residual curves,

the values of y along the trail should be plotted

as ordinate expanded against x as abscissa. If

the meteor trail is part of a great circle, the

points will lie on a straight line within the

accuracy of measurement. In this case, it is

convenient to express y as a linear function

of x and to use these rectified values of y in

determining the great-circle motion. If the

39U31—57-
188 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

trail shows clear evidence of curvature, the

early straight-line portion only should be used

in determining the great-circle motion for computing the radiant. For flare measures to be

used in calculating the instant of the meteor,

the originally corrected y values are best. For

meteors of exceptionally long duration (>3

seconds), gravity may cause an observable

curvature of the trail. To correct for this

effect, see section on corrections for gravity

(p. 199).

The standard coordinates of any measured

point on the photograph, with corrected coordinates x and y, may be obtained by making

use of inverse plate constants by the following

formulae:

where

Oi = bJCXy

bi——bz/Cxy

Ci=(—byCx

and

a, = — att/czv

ft,= +ax/cxv

) jcxv c, = (+a/i

(8)

(9)

—a^) /cIV

cxy=aj)v—avbx.

Equation (9) may be checked by applying

equation (8) to the measures of a single star

(unconnected). With proper arrangement of

the computation the application of equation (8)

may be checked on the summed values of x, y,

|, and ^.

The direction cosines of points on the meteor

trail are computed from £ and rj as follows:

x=

Check:

= 1 (Recompute X, M, or »<0.2). (10a)

Equations (8), (9), and (10) need be applied

to only a few points on the meteor trail. Usually the following will suffice: beginning point,

b; end point, e; first shutter break, a; last shutter break, j ; middle point, o; and each flare or

irregularity, /<, that can be identified on both

photographs.

FOUR-CONSTANT METHOD: While the sixconstant method may offer some advantages

when the strip within which the comparison

stars are selected is rather wide, it becomes

difficult or even impossible to handle when the

stars selected are very close to the meteor trail.

Six constants will operate a linear transformation from the orthogonal (|TJ) system to an

oblique (xy) system; when, however, the range

in y is very small, it becomes impossible to

determine the angle between the oblique coordinate axes. This difficulty is reflected in

an ill-conditioned system of equations for whose

solution it becomes necessary to carry a very

large number of digits in the computation. As

a safety measure, 10 digits have to be carried

even in ordinary cases, greatly limiting the

possibilities of programming the computation

on ordinary punched-card calculators.

An added disadvantage is that the six-constant method cannot be dissociated from the

lengthy least-squares process, which requires an

overhauling of the whole solution whenever a

mistake is discovered in the input data after the

solution has been computed.

A simple way of avoiding these difficulties

consists in assuming that the (xy) coordinate

system also is orthogonal. If the engine ways

are not exactly perpendicular to each other, we

must then expect that the residuals Ax, and Ay,

will show some dependence on y. This effect,

however, should be quite small if the range in y

for the comparison stars is small. Moreover,

even in a six-constant solution the residuals are

not entirely independent of y, due to the effect

of field distortion.

Instead of equation (5) we shall write, then:

b},— (11)

There is no need to determine the four unknowns a, b, c, d by least squares; a solution

using two stars at the extremes of the meteor

trail is quite satisfactory. Residuals (C-O),

taken according to equation (6) are plotted

against x and the resulting smooth curves are

used to correct the observed s's and y

J

s exactly
REDUCTION METHODS FOR PHOTOGRAPHIC METEORS 189

as before. Since the computed residuals may

show a dependence on y, it is advisable to write

in the diagram the value of yt—2/trau next to

each point, and then draw the curve for

yt—2/tnui—0, as one would draw an isobar on a

pressure map.

Standard coordinates are computed from corrected x's and ?/'s using inverse plate constants:

where

A=-

S,=Ax.+By.+C

vs=Bxs—Ay,+D (12)

ad—bc

'"aTW'

(13)

The observed radiant. If the meteor trail is

straight in the corrected xy plane, the direction

cosines of the beginning, b, and end, e, points

and a middle point, o, should be used in calculating the great-circle motion. If the trail

is curved, three points from the early straightline portion should be used. The direction

cosines of the pole of the great-circle motion,

^•APJ MAP, and vAp for Station A or \Bp, PBP, vBp

for Station B are given by the equations:

\Ap sin lA—

nApsin.lA =

vAp sin lA=

(14)

(lA is the length of the trail as photographed

at Station A).

Checks:

(14a)

and similarly for XB, MB, VB.

The direction cosines of the apparent radiant

X*, HR, VR> are obtained similarly from the

equations:

\R Sin Q^liApVBp—VApV-Bp

HR sin Q=vAp\Bp—\ApvBp (15)

VR Sin Q=\ApfJiBp— V-Ap^Bp

Checks:

Q is the angle of intersection between the two

meteor trails as seen in the sky from Stations

A and B.

The sign of sin Q is chosen to orient the

radiant above the horizon.

For both stations the standard coordinates

of the radiant (£AR, VAR', IBR, VBR) and first

shutter break (j-Aa, VAO', %Ba, VBO) on the photographs are required in later calculations, as are

the theoretical x and y measures of the radiant.

These quantities may well be computed at this

stage. The appropriate equations are indicated

below for Station A only:

cos <rAR, |AS and AiiB from equation (4),

xAR from equation (5) or (11)

and

cos <rAa= Aa) ^ from equation (4a).

Ranges and heights

For the moment we shall assume that the instant of appearance of the meteor is known. If

the time of the meteor was not recorded from

visual observations, methods for determining it

can be found in the section on page 194.

Fundamental values in the computation of

ranges are the relative rectangular coordinates

in the astronomical equatorial system of Station B with respect to Station A when the sidereal time is zero at Station A. These quantities can be computed once and for all from the

geocentric coordinates of the stations as follows:

ZABO^RB COS 4>'B COS {LB—LA)—RA

VABO=RB COS 4>'B sin (LB—LA)

?ABO=RB sin 4>'B-RA sin +'A.

c o s 4>'A

(16)

Here L stands for the geographic longitude,

positive to the west, <p' for the geocentric latitude, and RA, RB for the distance of the stations from the center of the earth; the subscripts A and B refer to Stations A and B.

respectively. A convenient unit is 100 km.

The distance RAB between the two stations

is obvioushT given by the equation

RAB=¥ABO + VABO + fi

(17)

+ HRHBP + VRVBP = 0

Let 0A be the sidereal time at Station A corresponding to the appearance of the meteor.
190 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

The relative coordinates of Station B with respect to Station A at this instant are

&iB=&ij». cos dA—rjABo sin dA

1AB=VABO COS 6A+1-AB, sin $A

fAB— fABo (18)

Check:

Let us now consider the plane through the

meteor and Station B. The intersection of

this plane with the celestial sphere is the great

circle of motion of the meteor, as seen from

Station B, whose pole is PB=(XB *, HB,, vBp).

The distance SA of Station A from this plane

is given by SA=VLAB-^B, i. e., by the equation

(19)

The range RAi (distance from Station A) of

any point i on the plane is given by RAt=SA

sec c<, where c« is the angle between P« and the

range vector HAf. Points on the meteor

0<AU PAU VA%) are just particular points on the

plane; for them we shall have the equations

and

(20)

(21)

In general, it is convenient to compute ranges

(and heights) for four or five meteor points,

conveniently spaced.

Ranges from Station B can be obtained in

the same fashion as those from Station A,

starting from the relative coordinates (%BAO,

HBAO, $BAO) of Station A with respect to B at

sidereal time 6B=0. This procedure, however,

is unnecessary, since clearly RBA=—RBA- We

can thus write

where

and

(22)

(23)

(24)

In computing heights above sea level for

points on the meteor, it is convenient to

compute first the heights (hAi) hBt) above a

plane through the station, normal to the

zenith direction.

The direction cosines of the zenithal point for

Station A are:

cos 6A

sin 6A (25)

where <t>A is the geographic latitude.

The zenith distance ZAt for points on the

meteor is given by the equation

cos ZAf=\ZA\At-\-nZAiiAt+vZAvAt. (26)

The heights hAt above the plane normal to

the zenith direction are then obtained from the

equation

hAi=Rjn COS (27)

To obtain the heights Ht above sea level, we

have to add the elevation of the station HA

above the level and a correction ShAi:

Ht=hAt+HA+5hAi.

The rigorous form of 8hAi is

(28)

(29)

where pA is the distance of the station from the

mean local center of curvature of the geoid

(i. e., the mean local radius of curvature plus

the elevation of the station).

The computation of 5hAi by equation (29) is

somewhat laborious because it is obtained as

the difference of two large quantities, so that a

large number of digits are required in the square

root. It is therefore much more convenient to

compute 6hAi from the expansion of an approximate formula (in which the zenith of the station is assumed to be the same as the zenith of

the meteor points):

shA

where

m

2

 , m3

m =

(30)

The error resulting from the use of this approximate formula does not exceed 20 meters
REDUCTION METHODS FOR PHOTOGRAPHIC METEORS 191

for meteors photographed within 45° of the

zenith and is less than 150 meters for meteors

70° from the zenith. In ordinary cases it is

quite safe to stop at the first term of the expansion, so that 5hAi becomes

(31)

If this formula is used, instead of (30), the

difference in size of the error is negligible. At

70° from the zenith the errors differ by less

than 4 percent.

It is convenient to plot ShAt against hAi and

to keep the resulting curve for further use in

computing heights above sea level of other

points on the meteor trajectory.

Distances along the meteor trail

When ranges RAt have been computed for

two points (i=l and i=2) on the meteor

trajectory, the spatial distance between them

can easily be determined. We first compute

for each point the components of the vector

R4< in the direction of the coordinate axes of

the astronomical equatorial system:

(32)

Let us now define a system of distances DAi

along the trail, starting from an arbitrary point,

such as the observed beginning of the trail.

The distance between our two points is then

given by the equation

(33)

To compute meteor velocities and decelerations, we must first compute DAi for all the

observed shutter breaks. The use of equation

(33) for this purpose would obviously be prohibitively laborious. Millman and Hoffleit

(1936) have shown that DAi can be computed

by a very simple formula, involving only the

corrected values xAi corresponding to individual breaks, the coordinates of the radiant

(xAB) and of the beginning point (xAa), and a

constant, KA. The formula is

-———^ALSAI,

with (34)

KA=RAa COS aAB

Here for convenience xAa has been identified

with the beginning of the meteor trail. It is

obvious, however, that any other point could

have been taken as the origin of DAt. In

equation (34) it is convenient to compute

first DAi for all shutter breaks, and then to

multiply them by the constant KA. This latter

quantity is a function of the meteor instant,

while the D'Ai'& (which we shall call "relative

distances") are clearly independent of time and

can be computed even before the instant of the

meteor has been determined.

When a long series of xA( has to be converted into

DA{, it is useful to re-write equation (34) in the following manner, which is more suitable for machine computation:

DA{ = 1— , a_ i where 0 = ando=— fixAM.

Equation (34) fails when <rAB is close to 90°,

a rather common occurrence. The difficulty

can be avoided if we re-write equation (34) in

the form

DAt= (35)

FA and QA are constants for each meteor,

since clearly FA=KA/(xA,—xAB) and GA =

FA/KA; they can therefore be determined by

using known values of DAi, as determined from

equation (33), for any two points other than

the beginning points (center and end point

are recommended).

There is a simple relation between hAi and

DA{ which can be used as a check for the fundamental meteor points for which ranges and

heights were computed, as well as to compute

heights for any shutter break:

hA(=hAtt—DAi cos ZAS. (36)

To obtain heights above sea level use equation (28), with a graph of dhAi as function of

hAi.

Equation (34), or (35), with the subscript A

changed to B can be used to compute distances

along the trail photographed from Station B.
192 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

By taking each individual beginning point as

the origin of the distances DAi and DBi, we

obtain two systems of D'a on the spatial trajectory, differing by a constant. To simplify

inter-comparisons between the two trails, it is

convenient to use a single origin for both DAt

and DBt. To avoid negative D's, the earlier

(greater height above sea level) of the two observed beginning points can be taken as the

common origin.

Our task, then, will be to transfer the beginning point on one plate to the (xy) coordinate

system of the other plate. This is only a

particular case of point transfer, so what is

said for the beginning point is valid for any

other point on the trail. To stress the generality of the formulae, which can be useful

for other purposes, we shall use the subscript

"i" for the point to be transferred, instead of

the "a" used for the beginning point.

We shall suppose that the transfer is to be

made from plate A (Station A) to plate B

(Station B). Let us first compute the three

components £At, rfAi, $Ai of ~RAi according to

equation (32). The corresponding components

of the range vector RB< from Station B are:

VBi=VAt VAB

i*B<== f AI f AB »

and

(37)

(38)

The direction cosines of the transferred point

are:

Checks:

HAi=HB

(39)

(39a)

(39b)

Standard coordinates (£B<, "rjat) and trail

coordinates (xBi, 2/s<) are then computed from

equations (4) and (5). In computing KB for a

transferred beginning point it must be remembered

that asa o,nd (TBB must be computed for the transferred, not for the observed, beginning point.

The equations for the transfer from plate

B to plate A are entirely similar when subscripts

are changed. It is perhaps worth mentioning

that in equation (37) we would have to replace

£AB with £BA, which is equal to —£AB, and

similarly for ijxfl and £AB.

If ranges RAf are desired for points (such as

shutter breaks) for which DAt is available, the

following procedure is very useful. Compute

first the value of DAt for the point of minimum

range on the meteor trail and let it be DAt:

Here £Aa, rjAa, fAo are the range components

of the beginning point (origin of DAi).

The minimum range itself, RAt, is computed

from the equation

Check:

RAt=SA/smQ.

(41)

(41a)

Equation (4 la) could be used directly to determine

RA*. We have preferred to use it only as a check, since

both SA and sin Q may become very small in certain

cases.

For any DAt we shall then have the equation

RAt=RA,-\-{DAi—DAt)

2

. (42)

Times corresponding to shutter breaks

If P is the period of revolution of the rotating

shutter and N the number of occupations, all

equally spaced, in one revolution, the instant

tt corresponding to the i-ih break can be

written as

P

 " • (43)

where Att is a function of the position of the

meteor relative to the center of rotation of the

shutter. For frontal shutters ordinarily used

with small cameras, Att can be minimized by

making the distance between the shutter

plane and the first principal point of the

objective lens as small as possible. If it is felt

that Att cannot be neglected altogether, the

correction can be determined empirically by visual or photographic observations in the focal

plane of the camera.

For Super-Schmidt cameras provided with

focal, radial-edge shutters the correction is

large, but can be rigorously determined from

the observed (x, y) trail coordinates on the

glass copy of the original curved film.
REDUCTION METHODS FOR PHOTOGRAPHIC METEORS 193

Let xq, yq be the observed coordinates of the

center of rotation of the shutter, and xc, ye the

coordinates of the projection center—either

observed, or taken from the plate constants

(equation (11); xe=c; ye=d), and let / be the

focal length of the camera. Let us write:

dx=xa—xc # u=djdx t (P

We have then

-p,

where

tan 6i - 2 <x*

t a n (Of—z —.

(44)

(45)

Here u0 designates the value of w corresponding to the origin of t< (usually the first break);

x{ and yt are the observed values of x and y of

the i-th break. The sign of At{ is positive when

the shutter edge chases the meteor, negative

when it runs toward the meteor.

When the shutter center coincides with the

projection center (dx=dtl=0), equation (45) reduces to

tan oj=- (45a)

This formula can be safely used whenever

rf<l cm.

When a meteor is photographed close to the

shutter center, the variation of Att with xt

becomes large and a smoothing of Att as computed from Xi is in order.

Velocities and decelerations

When distances DAi have been computed and

tabularly listed as a function of time tAt, they

must be critically appraised before they can be

used to compute velocities and decelerations.

First of all, a weight must be assigned to

each value of DA{ on the basis of the recorded

quality of the basic xAi observations. Strictly

speaking, the scale of weights should be computed from the relative inner scatter found for

each quality class. If no such analysis is available, the following scale is recommended:

l=very poor, doubtful; 2=poor; 4=fair;

6=good; 8=excellent.

In the second place, the DAi must be in*

spected for inner consistency and to detect

any possible instrumental effect (such as shutter or camera vibrations) that may be amenable to empirical correction. A simple and useful device is the construction of an approximate

"retardation diagram." Take only the first

third of the DAi's and from them compute an

approximate value VAa of the initial velocity

of the meteor. For all shutter breaks compute

AAt=DAi— VAatAi and plot AAi against time tAt.

The range AA{ is only a small fraction of the

range of DAi, so a large scale can be used in

the plot. The scale should be such that the

normal observational scatter is clearly apparent.

Any large deviation from the mean curve should

be checked to detect possible mistakes in the

computation; if the computation is verified,

one should consider whether the observed discrepancy is large enough to warrant rejection

of the point. In a series of good observed

points, there is no reason to retain a small

number of doubtful or bad points, whose inclusion would lower the quality of the results. If

periodic oscillations due to instrumental causes

are present, they should be corrected as far as

possible before any analysis is started. (See

section on shutter and camera vibrations, p. 199.)

A direct numerical differentiation of the observed DAi with respect to tAi is not recommended, even if DA{ were tabulated at equal

time intervals (which is not the case with the

Super-Schmidt meteors), because observational

errors are magnified in the process of numerical

differentiation and also because in this process

it would be impossible to account correctly for

the weight of each single observation.

Experience has shown that a very expedient

and accurate way to compute velocities and

decelerations is to fit to the observed values of

DAi, by least squares, an equation of the type

tAt+cAe (46)

where aA, bA and cA are unknowns, and k is

determined beforehand. A simple way to compute k is to read off four equidistant values of

AA{ (Ai, A2, A3, A4) from a smooth curve drawn

with the help of a French curve through the

points of the retardation diagram. Let the

time interval between two successive values of
194 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

the selected AAt be T. We have then the value:

(47)

When aAi bA, cA have been determined, velocities and accelerations are easily obtained

by differentiation:

(48)

Since equation (46) is at best only an empirical approximation of the true form of DAt

fai), we cannot expect it to hold good for the

whole of the observed trail. When dealing

with a long meteor, we shall find it convenient

to break up the trail into sections (either independent, or overlapping) and to compute separate least-squares solutions for each of them.

An ideal goal is to obtain individual values of

eA which exceed their computed probable errors

by a factor of 20 or 30; and with a little experience one can judge how long each section should

be to yield such a margin of safety. In any

case, only the values of VAi and VA1 computed

for the center of each section should be taken

as having full significance.

From equation (48) it is easy to see that bA

is the extrapolated value of VAl for tAt=—».

Thus the least-squares solution provides us

automatically with an important quantity, the

velocity Vm of the meteor before it was decelerated by the earth's atmosphere. If several values of bA have been obtained for a

meteor, from several sections, the value from

the first section will be the best value of Vm.

Determining the instant of the meteor

Instant from measures of flares. When flares

in the meteor trails can be identified and measured from both stations, the instant of the

meteor can be determined independently from

each flare. This method, when possible, is

more satisfactory than the method of equalizing

the observed velocities from the two stations

(described in the next section).

The flare method depends upon the fact that

the line joining the two observing stations must

lie in a plane determined by the directions of

the flare as observed at the two stations. It

will be apparent that the method fails in certain geometrical configurations. At middle

latitudes the solution is usually precise within

a few seconds of time if the stations are located

along a diagonal to the cardinal directions and

if the flare is sufficiently defined for accurate

measures.

Let us select a flare (or any other clear-cut

feature) which can be recognized on both plates.

From the measured x's and y's we shall compute its celestial coordinates as seen from Station A (\Af, fiAf, vAf) and from Station B (Xfl/,

mr, VBf), respectively.

The quantities

mt— vAf\Bf— \A/i>Bf

Check:

(49)

—0

(49a)

are the direction cosines of the pole of the great

circle through the two projections of the flare

on the celestial sphere multiplied by the sine

of their distance in arc. For these projections

to lie in a plane with the two stations, we must

clearly have the condition

(50)

Replacing SAB, VAB, {AB by the expressions

given in equation (18) and writing

rifSAB.^Ct, (51)

we obtain the equation

A, cos dA +Br sin dA + C,=0, (52)

which, when solved, yields 6A. As the equation

is quadratic in either sin 6A or cos 0A, it has two

solutions, of which only one, obviously, is

significant. When the two solutions fall close

together, the accuracy of the time determination by this method must be considered to be
REDUCTION METHODS FOR PHOTOGRAPHIC METEORS 195

poor. The sidereal time, of course, must lie

within the interval common to the two photographs.

Instant from equalization of velocities. This

method is based on the fact that "relative"

velocities VA—^JJ (D'At) and =-H iP'Bi) can

be determined before the instant of the meteor

is known. For any given point on the meteor

trajectory we must clearly have

(53)

Since KA and KB are functions of the meteor

instant, the latter can be determined when

equation (53) is rewritten to contain 6A in

explicit form. The method is not very sensitive and is recommended only when the flare

method cannot be used for lack of reliable

common points.

Let us suppose for a moment that vA and VB

have been determined. We shall then write

M vA

S

cos <rAa vB cos aBa

SA cos <rAB' SB cos aBB

PA=Z

A = %ABOPBP — VABo^Bp QB —

(54;

(55)

A.=MAPA+MBPB

B,=MAQA+MBQB

C.= (MAvB+MBvA) UBC (56)

We then find 8A by solving the equation

A, cos 6A+B, sin BA + C,=Q. (57)

To determine vA and VB it is best to proceed

by successive approximations. First find the

asymptotic velocities vAm(=bA) and vBm(=bB)

by least squares, fitting to the observed values

D'A{ and D'Bt an equation of the same type as

equation (46). These asymptotic velocities

can be used instead of vA and vB in equation

(57) to obtain a first approximation to 0A.

Using this first approximation, we can transfer

the center of the trajectory on plate A onto

plate B and compute relative velocities for

these two points. Since these are central

points, the relative velocities pertaining to

them will be more reliable than the extrapolated asymptotic velocities and will yield a

better approximation to 8A. The process may

be repeated if the possible results seem to be

justified by the labor.

Instant from an undrwen monitoring plate.

We shall suppose that the monitoring camera

is placed close enough to the meteor camera

that the parallax between the two cameras is

negligible. Let us assume that the cameras

are situated at Station A and that a time break

is impressed on the star trails, centered around

the instant 9^ of local sidereal time.

For simplicity we shall consider only a short,

central portion of the meteor trail, and select

a few (four or five) time breaks on star trails

in the region surrounding this portion. Measure x and y coordinates for the star breaks

and for the meteor portion, and determine the

coordinates (a, 8) of the plate center at the

time of the star breaks. Compute standard

coordinates of star breaks and plate constants

(six-constant method preferable). If the area

covered by the star breaks is not too wide,

the field distortion within the area, as revealed

by the individual residuals Ax and Ay, will be

small; it must be made small because we shall

be forced to neglect it.

Plot y against x for the measured points on

the meteor trail, and derive a smoothed value

yu of V for a given value xM of y near the center.

For this point, compute standard coordinates,

direction cosines, and finally right ascension

and declination (a'M} hM), all in the system of

star breaks. The correct right ascension au

corresponding to hu can be computed from the

equation of the great circle of motion of the

meteor:

cos (au—aP) = — tan &u tan Sr (58)

Here aP and 6P are the right ascension and

declination of the pole of the trail, both known

from the driven meteor plate. The difference

A6=au—a'u is equal to the sidereal time

elapsed between the instant of the star breaks

and the instant of the meteor; i. e., we have

the equation

(59)
196 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

This method fails when the meteor runs

approximately along a parallel of declination.

In that case, the difference Ad can be computed

from aP—a'p, where a'P is the right ascension

of the pole of the meteor trail, referred to the

star breaks. If the meteor runs approximately

along the celestial equator, the method fails

completely unless some use can be made of

identifiable common points.

Space trajectory by direct triangulation

When the angle of intersection Q between

the great circles of motion of the meteor as

seen from the two stations is very small, the

method described in the section on ranges and

heights (p. 189) fails. In such cases direct triangulation can be used.

The instant of the meteor must be known

from visual observations or from a monitoring

plate, and a good common point m (xAn, yAm

on plate A; xBm, VBm on plate B) must be available. A second common point n, as remote as

possible from m, is established by making use

of the shutter breaks. Select a break (xAn,

yAn) on plate A and evaluate the time interval

tnn between this break and the first point m;

on plate B find the point (xBn, 2/sn) for which

the time interval from point m is exactly equal

to tmn. Be sure to eliminate the effect of

observational errors by first smoothing xAi and

xBi (smoothing by differences is not recommended; comparison with a close-fitting polynomial is preferable).

We shall have two separate triangulations

to compute RAm, RBm and RA*, Ran', in the

following formulae i stands for either morn .

RAB is the spatial distance between the two

stations.

The base angles at and fi{ and the vertex

angle yt are first computed from the equations

RAB cos ai = tA

RAB COS /3<=—

(60)

sin 7i=sin a< cos /3<+cos a< sin /9< (60a)

after which the ranges become

At this stage, we must decide whether we

want to use the trail on plate A or the trail

on plate B for the computation of the radiant,

or a compromise between the two. Since the

radiant is no longer defined as the intersection

of the great circles corresponding to the trails,

but rather as the vanishing point of the space

trajectory of the meteor, we must expect to find

a slight discrepancy between the two determinations. Suppose we have selected the trail on

plate A as the most reliable. Then:

Vmn — VAm VAn't ^AR^=

?mn = tAm — fAn

 VAR~£

(62)

sin & -

sina <

Meteors photographed by stationary cameras

General problems. Either one or both meteor

photographs may have been taken with stationary cameras. We shall discuss each case

separately; first, however, we may examine a

few problems which arise with trailed plates in

general.

We shall suppose that the star trails on the

undriven plate show "time breaks" caused by

a mechanical or hand-operated shutter at certain instants which have been carefully timed.

If we refer the meteor and the plate center to

one particular set of time breaks on star trails,

we do not obtain its true position in the sky,

unless the time of appearance of the meteor

happened by chance to coincide exactly with

the instant of the time breaks. All the declinations we determine in such fashion are correct,

but the right ascensions are shifted by an

amount A6=6A—BA, where 0A is the sidereal

time of the meteor, and 0A that of the star

breaks, both taken here as observed at Station

A. We shall call such a hybrid system of coordinates "relative coordinates," and designate

them with a prime (a'At, 8Ai; \'At, n'Ai, vAi, etc.).

The faintest stars recorded on trailed plates

are considerably brighter than those that are

normally selected as comparison stars on diverse plates. Thus fewer stars are available

and it is almost impossible to find time breaks

sufficiently close to the meteor trail to yield

satisfactory field correction (Ax, Ay) curves.

It is therefore convenient to determine the

plate constants from star breaks in a limited
REDUCTION METHODS FOR PHOTOGRAPHIC METEORS 197

area of the plate rich in star breaks and crossed

by the meteor trail—an area sufficiently small

to justify the neglect of field corrections within

its limits. The "relative" position of the pole

(

aA, &A) of the meteor trail must be determined

from (x, y) measures confined to this area alone;

i. e., in the plot of observed y's in function of x,

we must isolate the portion of the diagram that

lies within the extreme values of x measured for

the star breaks, and draw the best possible

straight line through those points only.

If the instant of the meteor is known, or has

been computed, relative coordinates can be

transformed into true coordinates by the following transformation:

\n=^ < cos A0—p'Ai sin A0

MA<=M4* cos A0+\'At sin A0 (63)

The usual checks must be applied to the

transformed meteor points

equation of the great circle of motion of

the meteor in "relative" coordinates:

(63a)

= co3 A0.

Once all coordinates (of points on the meteor,

of the pole, and of the plate center) have been

thus converted, the reduction can proceed in

normal fashion, as for driven plates.

When the meteor intersects the star trails at

a considerable angle, so that intersection points

can be accurately measured, such points can be

used to determine the field correction at least

in the x coordinate. To avoid interference

with the shutter breaks on the meteor trail, we

should first measure the y coordinate of the

meteor trail at convenient intervals of x (with

one of the cross-wires of the measuring engine

as nearly parallel to the meteor trail as possible),

then plot the observed y's against x and draw a

smooth curve through the points. This curve

can then be used to set the y coordinate of the

measuring engine for each intersection to be

measured; in this fashion one of the cross-wires

replaces the meteor trail, bridging the gaps

caused by the rotating shutter.

At the intersection with a star trail, the

declination 8m of the meteor is equal to the declination of the star. Thus, by feeding the

declination of the intersecting star into the

cos (a'A—am) = — tan SAp tan (64)

we shall obtain the "relative" right ascension

aut of the intersection point. We can do this

for all the measured intersections, from the

actual beginning to the actual end of the

meteor trail; for each intersection we can correct (aUt SM) into standard coordinates and

compute the corresponding (a;, y) coordinates

with the help of the plate constants from the

time breaks in the small area. A comparison

with the measured values of x and y will yield

individual residuals Ax and Ay, from which

field correction curves can be obtained. While

the Ax curve can be considered to be an excellent approximation to the real field distortion relative to the adopted plate constants,

the same is not necessarily true of the Ay curve.

The reasons for this are two:

(1) We determined the (xy) equation of the

trail from a small portion of the trail only, so

that the margin of error at the extremes of the

trail is rather large.

(2) We assumed, in computing intersections,

that the meteor followed a great circle in the

sky, and this may not have been rigorously

true.

A final word should be said about time

breaks. If several time breaks are impressed

on star trails during the course of the exposure,

breaks belonging to different sets may be used

to determine plate constants provided they are

all reduced to a common set. On glass plates,

the indiscriminate choice of time breaks of

different sets should cause no trouble when the

timing is accurate. On the other hand, in film

photographs taken with conventional cameras

some trouble may be caused by progressive

film distortion under stress.

Case of one driven and one stationary camera.

Suppose the stationary camera is located at

Station A. If the instant of the meteor is

known, there is no problem. First compute

"relative" coordinates of points on the meteor

and of the pole of the trail, and then convert

them to true coordinates, as described in the

preceding section.

If the instant of the meteor is not known, it

must first be determined by using flares or
198 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

other common points on the two photographic

meteor trails. Only when good, sharp common

points can be found is a solution worth trying.

Compute first:

Z'AB=Z'AB, COS e'A—

IIAB=VABO COS

sin B'A

s in O'A

Check:

(65)

(65a)

For each flare (relative coordinates \'Af,

/A/, »Af) compute

/

/ / yj I

t—MA/SAB—^A

TTlf= (66)

and the coefficients

(67)

Then solve the equation,

.4', cos Afl+B'/sin A0+ C,=0. (68)

Since Ad is always relatively small, solve for

sin Ad. If the computation was done for several

flares, take a weighted mean of all the resulting

values of A0 and proceed to convert all relative

coordinates to true coordinates, as described

in the preceding section.

Case of two stationary cameras. If the meteor

is photographed with stationary cameras at

both stations, the solution is independent of

time. By this we mean that correct heights,

distances, and velocities will be obtained no

matter what instant is assumed for the meteor.

Unless, however, we know the instant from

observation, the position of the radiant remains

indeterminate; i. e., its declination can be

computed, but not its right ascension.

If the instant of the meteor is not known,

assume for simplicity that the meteor appeared

at the time of a given set of time breaks on star

trails on plate A. Then the plate constants

on plate A can be computed in a straightforward manner by directly correlating the

observed (xy) coordinates of star breaks with

($, 17) coordinates computed from catalog

positions for the stars. On plate B, however,

the right ascensions of the stars (and of the

plate center, as referred to star breaks) must

be augmented by 0%—O'B, where 0% is the

sidereal time at Station B corresponding to

the assumed meteor instant, and B'B is the

sidereal time, at the same station, of the time

breaks.

From here on the solution proceeds as for

meteors on driven plates with observed time.

One should remember that while the positions

of the radiant and of the zenith, as computed,

are purely fictitious, their distance ZR is correct.

Since the radiant position is indeterminate,

no orbit can be computed.

Plate center from three stars

Estimate the approximate position of the

plate center and select three stars Su S2, S3, as

nearly equidistant as possible from the center

and at 120° (counted from the center) from

each other. We recommend taking a plate of

clear glass or a transparent plastic sheet and

drawing on it three lines issuing at 120° from a

common point, with distances from this point

marked on each line. This plate or sheet can

then be superimposed on the meteor plate to

help select the stars. The three stars should

not be too close to the center, but not so far that

differential atmospheric refraction might affect

the results; also, they should be of the same

brightness and rather faint, to avoid excessive

coma distortion.

On a measuring engine measure the distance

between pairs:

di=distance between S2 and S3

<^2=distance between $1 and Sa

d\=distance between Si and S2

Compute the corresponding angular distances in the sky:

COS 7l =

COS 72 =

COS 73 = (69)
REDUCTION METHODS FOB PHOTOGRAPHIC METEORS 199

By successive approximations solve the following equations for Pi(p<=/2

+d?, where/ is the

focal length of the camera):

2-2 COSY,

2-2 COST,

IJ,

2-2 COSTS

. /IS

In first approximation it can be assumed that

P2—P3=Pi—P3=Pi—p,!=0; proceed to further

approximations until all three p's are stabilized

to the nearest micron (0.001 mm). The coordinates of the plate center (Xe, ne, ve) are obtained from the equation

=//P« (*=1, 2, 3). (71)

Suggested procedure: Solve

(72)

for \e/f, pe/f, vjj; compute / from the obvious

relation

Corrections for gravity

The spatial deflection of a meteor from its

purely inertial trajectory, due to gravity, at a

given point i is

/?,=! a t? (14)

where g is the acceleration of gravity (=0.0096

km/sec2 at average meteor heights) and t{ is the

time elapsed from the time origin (i. e., from

the instant of the first break, or from any other

convenient point on the meteor trail).

The corresponding deflection Sy on the photographic plate, in the y direction (perpendicular

to the direction of motion of the meteor), is

Sy= Gt

 cos ^ (75)

where R{ is the distance of the meteor from the

station, Sv is the scale of the photographic plate

in the y direction (see section on photometry,

p. 203), and ^ is the zenith distance of the pole

of the meteor trail, for Station A (and similarly for Station B), computed from the

equation

COS TI>A= (76)

As a correction to the observed y'a on the

trail, Sy should be applied with such a sign as

to bring the meteor positions closer to the

zenith. Ordinarily Rt is computed only at a

later stage, but since a very limited accuracy

in R{ is needed to obtain 5t, a rough solution,

requiring only a few minutes, is in order.

To correct distances on the trail:

De0Tr=Dt~Gt cos ZB. (77)

To correct velocities:

VC0TT=V0-g (t,-t0) cos ZR. (78)

Here Vo is the velocity determined for a

given instant t0 and Zs stands for either ZAR or

ZBB- A correction to the observed velocities

is justified only when it becomes necessary to

inter-compare several velocity determinations

in the course of a trajectory. For orbital

purposes, the correction for zenith attraction

(see p. 201) takes into account the effect of

gravity from infinity to the point of observation.

To correct accelerations:

VCOrr=V— g cos Zfi. (79)

This correction is not always negligible in

early parts of meteor trajectories.

Corrections for vibrations

A common trouble in the photography of

meteors with small cameras provided with

frontal rotating shutters arises from vibrations

imparted by the shutter to the camera. In

Super-Schmidt cameras the shutter is too light

to transmit any vibrations to the camera;

frequently, however, a modulated flutter is

observed in the rotational motion of the

shutter. Both of these effects are reflected in

a semiperiodic oscillation of the observed position of the meteor in its trajectory. In the first
200 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

case, however, the oscillation affects directly

the measured (xy) coordinates, while in the

second case it affects the computed time of the

shutter breaks. We must, therefore, treat

the two cases separately.

Camera vibration. In dealing with camera

vibrations, we shall assume that the rotating

shutter showed no error arising from eccentricity or from asymmetry, or that any such

errors have been determined experimentally

and corrected for. A true camera oscillation

is revealed by its occasional effect on the

transverse (y) coordinate of the meteor trail,

resulting in a sinuous trajectory. We can

easily eliminate this latter effect either by

trying to measure the mean rather than the

actual y coordinate or by drawing a straight

line through the observed oscillations in the

(xy) diagram.

In general, the oscillation of a given point

on the plate will be along an ellipse, whose

shape may be subject to semiperiodic changes.

If the oscillation is imparted by the shutter,

its period will be either the period of rotation of

the shutter, or a submultiple of it. To study

the longitudinal component of the oscillations,

one should analyze the observed x's; this task,

however, is made difficult by the fact that x

on the meteor trail is a rather complicated

function of time. It is much more convenient

to do the analysis when the x's have been

reduced to distances D, and then to convert

residuals AD to Ax by means of simple formulae.

When D's have been computed, a retardation

diagram must be plotted; if the range of the

retardation is large, it is convenient to compute

approximate values of the coefficients aA, bA, cA

in equation (46) from three points on the

retardation curve and compute residuals A2D

from this equation. These residuals can then

be converted to the corresponding residuals

in x (A&) by multiplying them by dxfdD

(=Vmm/V; see equation (106)), which can

easily be computed to two or three figures

even at this stage.

Once the period of oscillation has been established, divide the shutter breaks into as many

groups as there are breaks in one cycle of the

oscillation, and for each group draw a separate

curve of A& against i, the order number of the

break. The differences between the curves will

supply the systematic corrections to be applied

to the observed x of each individual break.

Instead of applying this correction to x, one

may, of course, multiply it by dD/dx and apply

it to Dt.

Shutter flutter. The flutter in the shutter of

the Super-Schmidt cameras causes the computed instants of individual shutter breaks to

be in error by a visible amount At; conversely,

if we assume that the computed instants are

correct, we shall observe corresponding oscillations AD=vAt in the distances. Since we cannot detect At except by analyzing AD, it is convenient, once AD has been isolated, to apply

corrections to the observed distances Dt rather

than to the instants U.

To isolate the fluctuations AD, follow the

same procedure as for camera vibrations: construct a retardation diagram and plot residuals

A2D from equation (46), with coefficients computed from three points, against i. For the

Super-Schmidt cameras A2D occasionally shows

slow, semiperiodic oscillations (most frequent

period=0*23), superimposed on a slower variation, due to the imperfect fit of equation (46).

Draw first a smooth curve through the observed

points, following all the oscillations; then draw

a smooth curve to bisect all the waves of the

oscillations. The difference between the two

curves can be taken as the correction to be

applied to D{. If the trail covers only a little

more than one cycle of the oscillation, it is convenient to fit a sine term to the observed values

of A2D.

Computation of orbital elements

When the position of the apparent radiant (X«,

MB, VR) and the apparent velocity of the meteor

before atmospheric deceleration (Va>) have been

determined, it is possible to compute the orbital

elements of the meteor.

If Va is determined separately from two

plates, take a weighted mean of the two

quantities.

Correction for earth's rotation (aberration).

Compute the (negative) velocity components

of the velocity:

(80)
REDUCTION METHODS FOR PHOTOGRAPHIC METEORS 201

The corrected velocity and radiant are computed from the equations

Vc\Rc=$'+~RA cos <f>'A sin 6A

-*«

VCHRC=T)'—~RA cos <}>A cos 0A (81)

•*»

Here T, is the sidereal period of rotation of

the earth in seconds=86164. Station A is

chosen as the observers' point, although no

substantial difference is found if Station B is

used.

Corrections jor zenith attraction. Let us take

a central point on the meteor trajectory as

photographed from Station A (X^o, HAO, VAO) and

compute its distance Ro from the center of the

earth and the zenith direction through it (XZo,

too, vZe):

RO^ZO^RA COS <f>'A cos 6A+RAo\Ao

RoliZo=RA cos 4>'A sin dA+RAonAo (82)

AOVAOCalculate the direction cosines XJV, pN, vN and

X«, HK, VK of the normals to the zenith, and the

corrected zenith distance ZRe by the equations

\N sin ZRe=nRevZo— vRenzo

nN sin ZRc— vRe\Zo— \RevZo (83)

vN sin ZRC=\RCHZO—^RC^ZO

Checks:

X.vXZo+UNPZO + vNvZo=0

(84)

(83a)

(84a)

The velocity Vee corrected for zenith attraction is given by the equation

COS ZRe= XR<.XZo4" HRCHZO + VRCVZO

sin ZJee=X/rXZ(,+/iKMzo+»'Jt»'zowhere go is the acceleration of gravity corresponding to a distance Ro from the center of

the earth. A good approximation for 2g0R0 is

given by the equation

20O#O=O.79464/#O (in units of 100 km/sec2). (86)

The correction AZ to the zenith distance is

given by the equation

AAZZ Ve-Vee sinZRc

t a n

 2 -

ZRee=ZRe-{-AZ,

(87)

in which

sin AZ=

2 tan —

. , UkZ

1 + tan2 —

; cos AZ= (88)

Compute the direction cosines of the final

corrected radiant:

XKW= XRC COS AZ+ XK sin AZ

HRCC=VRC COS AZ+MJC sin AZ

VRCC^VRC COS AZ+VK sin AZ (89)

Checks:

,=cos ZBet

Be = CO8 AZ

= 0

=sin AZ. (89a)

Heliocentric velocity. The geocentric velocity

of the meteor in a.u./1/fc mean solar days is

TV-=3.3595Ffc (90)

when Vet is expressed in units of 100 km/sec.

In the conversion constant the solar parallax

has been assumed to be 8T80. Compute the

velocity components,

Ym= — \ mURct

Z'm=-VnvRct. (91)

Vle=V*-2goR.t

(85)

These components are referred to the equinox

of the year of the meteor and should now be

converted to the standard equinox of 1950.0

by using the same equations and tables as for

the precessional rotation of direction cosines.

From an astronomical ephemeris interpolate,

for the instant of the meteor, the equatorial

rectangular coordinates of the sun for the
202 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

standard equinox (1950.0) (Xo, Fo , Zo ), and

their first derivatives (X'Q, YQ, Z'Q) with respect

to time, expressed in mean solar days. It is

convenient to compute the derivatives (using

all significant differences) for midpoints between tabulated values for a few days before

and after the date of the meteor, and then to

interpolate for the exact instant of the meteor.

The interpolated positions and derivatives

can be checked against the ephemeris of the

radius vector rT of the earth in its orbit, given

elsewhere in the almanac:

XQ -\- YQ + ZQ = TT

XoX'Q+YQYQ+ZQZ'Q=-rTr'T. (92)

To obtain the heliocentric position and

velocity components of the earth in the same

system and units, we have simply to change

sign in the above quantities. The velocities

are expressed in a. u. per mean solar day; to

add them to the meteor velocity components,

we must first multiply them by 1/k (=58.132).

Neglecting the distance of the meteor from the

center of the earth, we can thus compute the

heliocentric equatorial coordinates and velocity

components of the meteor:

X=-XQ X'=X'mY=-Yo; Y' = 7^-58.

Z = -ZQ Z' =Z'm-58.132Z'Q (94)

r

i=X2+Yi+Z2

rr'=XXf+YY'+ZZ'

V*=X'2+Y'2+Z'a. (95)

The heliocentric velocity of the meteor in

km/sec is 29.767 V.

Orbital elements. The customary symbols

used in orbit calculations are given in the following section. Some slight duplication of the

previous symbols should not cause confusion.

o=semimajor axis \

<7=perihelion distance I.

3 \. i- J- i. >m a. u.

q — aphelion distance

p=parameter J

e=eccentricity

i=inclination

«=obliquity of the ecliptic

»=true anomaly

«=argument of perihelion

Qt = longitude of ascendingequinox of 1950.0

node

o=longitude of perihelion

P= period of revolution (sidereal years)

X=elongation of the corrected radiant from

the apex of the earth's orbital motion.

The elements are derived as follows:

Check:

e sin v=r'plf2

e cos v—-—1

r

1—e2

(96)

(96a)

Direction cosines of the major axis:

(97)

Direction cosines of the minor axis:

p—r Y'

r ~ . p—rZ' (98)

Checks:

Q

2X+QY+QI=1

PXQX+PYQY+PZQZ=O. (98a)
so. 2 REDUCTION METHODS FOR

Ecliptic elements:

sin i cos u=Qz cos t—QY sin «

sin i sin u=Pz cos «—Pr sin « (99)

cos ft = Px cos w—Qx sin w

sin ft = (Pr cos o>—Qr sin w)-4-cos e. (100)

Check:

—cos i sin ft = @x cos «-f-Px

 8 m

 «• (100a)

If Y' sin €—Z' cos e>0, «=180°—© and ft

=Longitude of Sun.

If Y' sin «—Z' cos e<0, «=360°—1> and ft =

Longitude of Sun+180°.

Additional elements:

PHOTOGRAPHIC METEORS 203

ra=ft+«

q'=a(l+e)

P=a3/2

cos X= j~i

(101)

Photometry

In this section we shall briefly describe the

techniques used in the Harvard Meteor Project

to measure the apparent brightness of meteors

photographed on glass plates (small cameras)

and on curved films (Super-Schmidt cameras).

We shall start with a few general remarks and

useful formulae.

Scale on photographic plates. Apart from

relatively small distortions, the scale S (in

radians/mm) can be considered to be constant

on Super-Schmidt films, and equal to /, the

effective focal length of the camera (in mm).

We can compute / from the plate constants on

the glass copy:

f

2=a?-\-bi

 (4-constant method)

or J'=ai+bl (6-constant method). (102)

The same formulae can be used to derive

the focal length of an ordinary camera using

glass plates.

On plates, the scale varies at each point with

direction. In the absence of distortions, at a

distance d from the projection center (where

the optical axis of the camera, or of the copying

system, encounters the plate) and in a direction

which makes an angle 4> with the radial direction, the scale of the plate is

f

2+d> (103)

In the case of a meteor, in which the x coordinate is measured on the trail, and y is

constant, the scale factor Ss in the direction of

motion of the meteor, and the scale factor S9

in the perpendicular direction are, respectively:

g _yp—\*—*e). g _\p—yy—ye) Q ^

P

2

 p1

Here xc, ye are the coordinates of the projection center; p

t=ft

-\-d2; d can be computed from

d*=(x-xey+(y-ye)>.

Trailing velocity of stars and meteors. On a

plate taken with a stationary camera, the trailing velocity of stars (in mm/sec) is

2x

(105)

where T, is the sidereal period of rotation of

the earth (86164 seconds of mean time).

The trailing velocity of the meteor on the

plate, vmM=dx/dt, can be determined either by

numerical differentiation, from the measured

values of x, or from the following formula:

Vi(xAtKA(xA.-

(106)

Here the subscript A indicates the plate taken

at Station A; with changed subscript, of course,

it can be used for Station B. Vt is the instantaneous velocity of the meteor and xAl, xAm, xAm

and KA have the same meaning as in equation

(34). &tA is read off the field-correction curve

in function of xA; the last factor, which was

introduced to account for field distortions, is

always very close to 1 and may be disregarded

in ordinary cases.

Reciprocity-law failure. If a given star region

is successively photographed by a driven camera

with different exposure times tx and U, and the

photographic images on the two plates are

intercompared for brightness, we find that there

is a constant difference between the two sets of
204 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

star images, corresponding to a magnitude

difference,

m.-m^-2.5 log £+/(«„*,), (107)

where/ (tiy t2) is a term arising from reciprocitylaw failure (Mees, 1942). If the reciprocity-law

curve, plotted with log / (intensity) as abscissa

and log (It) as ordinate, is nearly a straight line

in the interval between U and t2, equation (107)

can be written in the form

m1 -m1=-(2.5+F,)log;-f. (108)

In fast emulsions FB is usually quite large

(+0.3 to +0.9) for exposures ranging from 1

minute to a few hours; for short exposures

(<1 socond) FR may range from 0.0 to —0.2,

according to the typo of emulsion. For the

Kodak X-Ray emulsion, FR is nearly zero in

the short-exposure branch.

Absolute magnitudes. It is convenient to reduce the apparent (observed) meteor magnitudes to a standard distance of 100 km. If the

ranges Rt (i. e., RAt or RBt) are expressed in

units of 100 km, the reduced, "absolute" magnitudes are, clearly:

Mt—mt— 5 log Rt (109)

Photometry on small-camera plates. The

brightness of meteors is measured, emulsion

against emulsion, with free-trailed star images

on comparison plates taken with the same

instrument as the driven meteor plate (Jacchia,

1949).

The comparison plates are all taken without

rotating shutter and are centered on the

equator, in a region rich in bright stars (first

choice: Orion; second choice: Aquila); the most

suitable exposure seems to be one that gives

star trails half millimeter in length. It is useful

to take plates with different focal settings, and

to have out-of-focus images to compare with

fuzzy-looking meteors; also, it is convenient to

take several plates of each type, so that plates

whose emulsions have been damaged by use

can be replaced when necessary. Two standard, driven plates must be taken with rotating

shutter, one before and one after the series of

comparison plates; the center must be the same

as that of all the comparison plates and the

exposure equal to that of standard meteor

plates. To avoid complications arising from

extinction, the whole series of plates must be

so timed that its center roughly coincides with

the culmination of the selected region. The

series can be considered to be acceptable if no

difference in brightness can be noticed between

the two extreme plates.

Photographic magnitudes in the standard

region can be taken from the Henry Draper

Catalogue when the following rules are observed:

(1) Take only the stars whose magnitudes are

given to two decimal places (the others are

simply reduced B. D. magnitudes, too unreliable for our purpose).

(2) Select only stars of spectral types from

Bo to F5, for which the color index is reliable,

because the photographic magnitudes in the

Draper catalog are really visual magnitudes

reduced to photographic magnitudes with standard color indices.

When the meteor trail is compared with the

comparison star trails, we obtain fictitious magnitudes m[, which must be corrected for the

different trailing velocities of the stars and the

meteor on the plate. The correction, according

to equation (108) is

(110)

The effective exposure time for star trails

is of the order of Is and that for meteor trails

of the order of 10~8 to 10~4 seconds.

Due to the progressive dimming of images

with increasing distance r from the center of

the plate, a distance correction /(r) must be

applied to the observed meteor magnitudes to

reduce them to the plate center, where the

standard comparison stars are located. This

correction must be determined separately for

each camera and for each focal setting. A

simple way to determine/(r) is to compare the

trails on two plates of the same series, on the

north-south line through the center, in the

northern half of the plate. It is easy to see

from equation (105) that on the north-south

line vt is the same for all stars if the plate center

is at 5=0; by taking the northern half, which

is close to the zenith, extinction problems are
NO. 2 REDUCTION METHODS FOR PHOTOGRAPHIC METEORS 205

minimized. It appears that /(r) is larger for

fainter stars than for brighter stars. The

limited number of bright stars available on the

plate makes it impossible, however, to determine separate curves of j(r) for different classes

of star brightness. A good approximation can

be obtained by determining/(r) for stars 1 to 2

magnitudes brighter than the plate limit, and

assuming for brighter trails the correction can

be expressed as j(I)f(r), where /(/) is a factor

dependent on brightness alone, which can be

determined by using a few brighter stars.

If we compare stars at the center of the

meteor plate with stars of the standard region

at the center of the driven comparison plates,

we shall find a constant difference between the

estimated magnitudes m« and the catalog

magnitudes me. When allowance is made for

the difference in the exposure times of the

meteor plate (tm) and of the comparison plate

{t0), we obtain a corrected difference,

T=me—mg—2.5 log t?, (111)

which may be due to any combination of the

following factors:

1. Difference in sky transparency.

2. Difference in differential extinction.

3. Difference in the sensitivity of the emulsion.

4. Difference in the development.

5. Clouds on the meteor plate (it is assumed that the

comparison plates were taken on a clear night).

The first four factors affect the meteor and

the background stars equally; but the fifth

does not. Therefore, it is safe to apply the

correction to the meteor magnitudes only

when there is absolute certainty that no clouds

were on the central region during the whole

exposure. Otherwise, it is better to ignore the

correction and take the risk of introducing an

error of a few tenths of a magnitude in the

reduction.

To summarize: absolute magnitudes Mt are

obtained from "observed" magnitudes m\ by

the equation

Mt=m't-{2.&+FB) log - 5 log R.

(112)

Photometry on Super-Schmidt films. The

brightness of meteors photographed with SuperSchmidt cameras is estimated on the original

films with the help of free-trailed star images

on comparison films. Since the rotating

shutter has openings totalling only a quarter of

the circumference, the meteor trail is reduced

to a succession of widely separated dashes.

In most cases these dashes appear diffuse due

to the difference in focus between stars and

meteor. To avoid systematic errors, it is

necessary to compare these dashes with star

trails of equal length and equal degree of

diffuseness. To this end, comparison films are

taken (with stationary cameras without shutter) at various focal settings and with different

exposures. Specifically, four focal settings

were used in the Harvard Meteor Project and

at each focal setting four films taken, with

exposures of 3*, 6*, 12*, and 24*, respectively—

a total of 16 films for each camera (actually,

the whole series was taken in triplicate, to

supply a reserve for replacements).

To enable a direct comparison, emulsion

against emulsion, with the meteor films, the

comparison films are turned inside out, so that

the emulsion is placed on the concave side; the

operation can be done with a sharp rap of the

hand without injuring the film. As a support

for the films, we find that the hemispherical

surface of an opalescent glass globe for overhead

lighting, with a strong light source inside,

provides an almost ideal device. The globes

which are available have a radius of 7 inches,

one inch less than the radius of curvature of the

films, but the difference is well within the

tolerable limit. It is actually better to use

supporting globes a little more curved than the

films: if the curvature is only a small fraction

of an inch less than that of the films, these

cannot be brought in contact with the supporting surface and the gadget becomes useless.

Stars in Harvard Standard Regions at declination + 15° are used for comparison with

meteor trails. All comparison films are centered on the celestial equator, so that the

center of the standard region is always a little

more than halfway from the film center to the

edge. No appreciable magnitude correction in

function of distance from the film center has

been detected.

The trailing velocity of stars on the films, V,,

can be computed from equation (105) with
206 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

S+=llf; the trailing velocity »£,„ of the meteor

on the film can bejobtained from vmn as computed for the glass plate:

Vmrn==jS^Omm- (113)

Since compact, small cumulus clouds'are frequently present in the New Mexico sky, we

deem it safer to do away entirely with the

intercomparison of stars on meteor films and

driven standard films to determine the correction T. Since the standard exposure for SuperSchmidt films is only 12 minutes, the dimming

effect of a passing cloud on field stars can be

quite large.

Notation symbols

Sky and plate coordinates.

a, &—The spherieal'equatorial coordinates right ascension and declination.

X, ft, *—Direction cosines referred to the equatorial

astronomical system.

i, n—Standard rectangularlcoordinates with the origin

at the optical plate center.

x, y—Measured rectilinear coordinates on photographic

plate.

9—Angular distance from optical plate center.

Single-letter subscripts:

s—Coordinates of stars.

c—Projection center of plate.

q—Center of rotating shutter.

Double-letter subscripts are used for points

on the meteor trail, or related to the trail.

The first letter is always A (for Station A) or

B (for Station B). A list of second subscript

letters follows:

i—A generic point on the trail, in particular any shutter

break or segment.

b—Beginning of photographic trail,

e—End of photographic trail.

a—First measured break or segment.

o—Central break or segment.

j—Last measured break or segment.

/—A specific flare.

«—Point of minimum range,

p—Pole of trail, considered as a great circle.

R—Meteor radiant.

Points not on meteor:

ZA, ZB—Zenith at Station A or Station B.

Primes (a'; X', y.' /) are used for "relative"

celestial coordinates on undriven plates.

Processional direction cosines:

•A«J Ay, "B

Y,, Y,, Y.

Zx, Zt, Z,

Rectangular coordinates in space.

R—Range or distance vector, with origin at the

observing station.

R—Modulus of R.

£, ij, f—Components of R in the astronomical equatorial

system.

Double-letter subscripts are affixed to these

symbols, and they have the same meaning as

for sky and plate coordinates.

Additional symbols:

RAB {£AB, VAB, TAB)—Vector from Station A to Station

B.

RIM (ZBA, VBAJ $BA)—Vector from Station B to Station

A.(~*-RAB).

RABO ({is., VABO, £AB»)—Vector from Station A to

Station B at sidereal time 0^=0.

RBAO (bioi VBAO, £BAO)—Vector from Station B to

Station A at sidereal time 0a=0.

Rx, R-a—The single subscripts stand for the distance

of the station (A or B) from the center of the earth.

XQ, YQ, ZQ—Geocentric coordinates of the sun in the

astronomical equatorial system (in a. u.).

Xt Y, Z—Heliocentric coordinates of the meteor in the

astronomical equatorial system (in a. u.).

rT—Radius vector of the earth in its orbit,

r—Radius vector of the meteor in its orbit.

References

BANACHIEWICZ, T.

1925. Cracow Obs. Circ. No. 17.

1929. Acta Astron., ser. C, vol. 1, p. 63.

BESSEL, W.

1839. Astron. Nachr., vol. 16, p. 323.

BOWER, E. C.

1932. Lick Obs. Bull., vol. 16, p. 34.

JACCHIA, L. G.

1949. Harvard Coll. Obs. and M. I. T. Tech. Rep.

No. 3. (Harvard Obs. Repr., ser. 2, No. 31.)

MEES, C. E. K.

1942. The theory of photographic process. Macmillan & Co. (Especially pp. 236-250.)

MILLMAN, P. M., and HOFFLEIT, D.

1936. Ann. Harvard Obs., vol. 105, p. 601.

OLHSTED, M.

1931. Ann. Harvard Obs., vol. 87, p. 221.

SCHAEBERLE, J. M.

1895. Contr. Lick Obs., No. 5, p. 31.

TURNER, H. H.

1907. Monthly Notices Roy. Astron. Soc. London,

vol. 67, p. 562.

WHIPPLE, F. L.

1938. Proc. Amer. Philos. Soc, vol. 79, p. 499.
The Method of Reduction of

Short-Trail Meteors1

By Gerald S. Hawkins2

The preceding paper gives a comprehensive

account of the reduction of photographic meteor

trails and indicates that a great deal of time is

spent in evaluating the distortions of the plate,

the deceleration of the meteoroid, and the vibrations of the camera and shutter. These

three stages of the analysis require a critical

assessment by the computer and cannot be

handled by existing machines. In the case of

short trails, however (angular length <C5° on

Super-Schmidt plates), distortion, deceleration,

and vibration are usually below the limits of

measurement, and we may use a simplified

method of reduction. This method embodies

the minimum requirements for the precise

determination of the radiant point, height, and

velocity of a meteor when photographed at two

separate stations. It is assumed that the

cameras are driven to follow the diurnal motion

of the sky and that one at least is equipped with

an occulting shutter.

Methods of measurements

Plate marking and selection of comparison stars.

This method presumes that we are dealing with

flat plates which approximate to a gnomonic

projection of the sky (loc. cit. Whipple and

Jacchia, p. 183) .

3

 Four stars are selected for

comparison purposes and marked on the glass

side of a plate. They are labeled 1, 2, 3, and 4

in the direction of motion of the meteor, which

is usually given by a visual observation. The

direction may also be found by placing the star

field of one plate on top of the other and noting

the apparent intersection point or radiant of the

meteor trails. Two of the stars are used subsequently for determining the plate constants

and two are used as checks. The stars are

identified in the B. D. catalog and marked

together with the meteor trail on B. D. charts

(loc. cit., p. 184). Accurate star positions are

found from the Yale and A. G. catalogs. In

selecting stars for short trail reductions, we

should observe certain criteria to obtain the

utmost accuracy. These criteria are based on

experience with Super-Schmidt plates and a

measuring engine which can be read to ±0.0001

cm.

1. Star positions must be available which are accurate to 0T5 of arc.

2. The diameter of the star image should not be

greater than 120", or 0.012 cm on Super-Schmidt

plates.

The above standards severely limit the choice

of star images; the Yale and A. G. catalogs do

not, in general, give positions for stars fainter

than a visual magnitude of 9.2, and stars

brighter than magnitude 7.5 produce large

1

 Carried out In part under research contract with the Massachusetts Institute of Technology, Lincoln Laboratory, Contract No. AF19(122)-458,

Subcontract No. 57. Reproduction In whole or in part Is permitted for any purposes of the TJ. S. Government.

1

 Ilarvard College Observatory, Cambridge, Mass.

J

 All references given thus (loc. cit.) refer to the previous paper by F. L. Whipple and L. Q. Jacchia on "Reduction Methods For Photographic

Meteor Trails."

207
208 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

images. In the ideal case the stars must be

close to 9th magnitude visual.

3. The stars should be equally spaced along the trail

with not more than 5° between the end stars. They

should be not more than %° from the great circle path

of the meteor.

4. Any two of the stars may be chosen for the determination of plate constants, but it is advisable to use

a pair with good images which are from 1° to 3° apart

and on the same side of the trail.

5. A single star image that is recorded as a double

star in the catalogs should be avoided if the separation

of the components is greater than 1T0.

Measurement of the projection center of the

plate. In the process of copying SuperSchmidt negatives onto flat plates, the projection center is marked automatically by a white

circle. Normally there is sufficient overlap between a pair of plates so that the star fields can

be superimposed in the region of the projection

centers, and the position of the center can be

read from a star chart. An accuracy of ±0°.l

is sufficient and readily obtained.

Measurement of comparison stars and trail.

The plate is set, emulsion down, on the machine with the trail parallel to the x axis to

within ±20 microns and with x values increasing in the direction of motion of the meteor.

Standard precautions, as given in the previous

paper (loc. cit., p. 185), are taken in using the

machine. In the short trail method it is found

expedient to take only one series of measurements, because there are checks at the end of

the computing that will detect gross errors such

as misreading a vernier or misidentifying a star.

Measurements in rectangular coordinates x and

y on the plate should be recorded to the nearest

micron.

Marked stars: Three readings of x and three

readings of y are taken from the center of the

star images.

Trail: To obtain a linear equation relating

x and y, 10 points on the trail are selected at

approximately equal intervals of x. The value

of x is measured to the nearest tenth of a mm

and three values of y are observed at each point

and recorded to within ±0.0001 cm.

Segments of the trail: With an occulting or

rotating shutter the meteor trail is divided into

bright sigments. With a double-bladed shutter, rotating at 1800 rpm, the segments are

short and the vertical crosswire may be set

accurately at the midpoint of a segment. Three

values of x are obtained for b and e, the first

and last segments visible on the plate. These

give the beginning and end heights of the trail.

(The center and not the end of the dash is

measured in these cases, to give additional

values of velocity if they should be required.

The consequent errors in heights of the beginning and end of the trail are negligible.) The

segments are numbered consecutively along

the trail with dash b designated zero. Three

dashes, a, o, and j , are now selected along the

trail, and three values of x are observed for

each, with the crosswire set at the center of

the dash. It is essential to take great care in

selecting a, o, and j as the accuracy of the

method depends to a great extent on the quality

of the image. Asymmetrical dashes whose

image is distorted, for example by a background star, should be avoided. The good

quality dashes are selected to make the separation between a-o, and o-j as large as possible.

One x value is taken for point /, the point of

maximum light. If more than one maximum

is present, / refers to the brightest maximum.

To check that the plate has not been disturbed during the series of readings, the x and

y values of the first star are measured again.

Finally, one reading is taken of the x and y

values of the projection center. The accuracy

required in this reading is ±0.01 cm.

Measurement of the magnitude of the meteor

and the limiting magnitude of the plate. Photometric estimates are made following the general procedure of the Harvard Meteor Project

(loc.cit., p.205). The brightest segment of the

meteor trail is compared with trailed star

images which are approximately equal in

length to the segment. For this comparison

the original Super-Schmidt film is used with

the emulsions in contact. The maximum magnitude of the meteor, MmiX, and the faintest

magnitude visible on the comparison plate,

•fl^iim > are estimated to ±0.1 mag.

Equations used in the computations

The computing program for short trails is a

simplified and abbreviated version of the full

treatment described by Whipple and Jacchia

(loc. cit.); their notation is used in the equations given here, where possible; new symbols
REDUCTION METHODS FOR PHOTOGRAPHIC METEORS 209

are listed at the end of this paper. In addition,

methods are given for estimating the errors of

the observations. Formulae for checking the

computing are given in the previous paper.

Reductions are carried out in equatorial rectangular coordinates with the X axis pointing

towards the vernal equinox (R. A. 0b, Dec. 0°),M

axis pointing towards R.A. 6b, Dec. 0°, and

v axis pointing towards the north celestial pole.

All measurements are made in "trail coordinates" (x, y), with the x axis along the trail and

x increasing in the direction of motion of the

meteor. The first stage of the computing is to

determine the plate constants which enable one

to convert the measured values (r, y) into equatorial direction cosines (X, fi, v).

Equatorial coordinates. The direction cosines

of a point with R. A.=a, Dec.=5 are given by

the formulae,

Xro=cos 5 cos a"

Mro=cos 5 sin a (1)

o=sin 5.

In the case of stars a correction has to be

made to a and S for proper motion (loc. cit.,

p. 185). The direction cosines are computed for

the four comparison stars and for the projection centers of each plate. These values apply

to the equinox, To, of the catalog or chart from

which a and S were obtained.

To convert equatorial direction cosines to

R. A. and Dec,

dn 5= v"\

an «=£ J (2)

where sin a has the same sign as ft, cos a has

the same sign as X.

Precession. To convert direction cosines from

equinox To to the equinox of the meteor T,

(3)

(4)

(5)

Mr=i7z^ro+ YvnTo-\-YzvTo.

vT=Zx\To-\-ZvixTo-{-ZzvTo.

The values of Xx, Xy, X, may be found by

the Cracovian calculation (loc. cit., p. 186).

Values of (Xr, V-T, VT) are computed for the projection center and comparison stars of each

plate. The equinox T is taken as the beginning

of the year which is nearest in time to the date

of the meteor.

Standard plate coordinates. The standard

plate has the same projection center as the

measured plate, but is tangential to a unit

sphere. These coordinates are a rectangular,

linear system (|, ij) with origin at the projection

center and the tj axis directed northward along

a meridian of R. A. The direction cosines of

the £ axis are

=Pc= — sin oie ( = — ^

v=re=O

and the direction cosines of the rj axis are

/ X p \ " \=lc=—sin 8e cosac I =— — 1

• * • / Vcfic

fi=mc=—aux Se sm ae ( =—^ ~

\ fie

(6)

(7)

where the subscript c refers to the projection

center. These components are computed for

both plates. To convert equatorial direction

cosines (X, n, v) to standard plate coordinates

(i> v), we may use the equations,

£ cos <r= i

^ COS ff=l = (Ue+fime+vnc) J

where

COS ff—

(8)

(9)

To convert standard plate coordinates (£, y) to

equatorial direction cosines (X, n, v), we may

use the equations

j : ) COS <J

M— G*«+weij-f-g<£) cos av^ivc+Tlcy) COS <r

where

sec <r=

(10)

(11)
210 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

Equations (6)-(ll) may be applied to both

plates.

Relations between plate and trail coordinates.

To convert standard plate coordinates (|, rj) to

trail coordinates (x, y) we assume that

(12)

The coefficients a, b, c, and d are called the

plate constants.

To convert trail coordinates (x, y) to plate

coordinates Q, rj) we use the inverse equations,

l=Ax+By+C 1

i=Bx-Ay+D.J (13)

The coefficients A, B, C, and D are called the

inverse plate constants.

Determination of plate constants. If the plate

constant stars are designated 1 and 2 we have

measured values (xu yt), fe, y2), and computed

plate coordinates (fr, rji), (&, ija).

Hence

=(*?2—50 fa— —ft) (yt—

(14)

(15)

c=x1—a£l—btji, (16)

. . (17)

As a rough check, c and d should agree with the

measured x and y values of the projection

center. The inverse plate constants are given

by the expressions

A

 a

A==tf+b2

'

b

V+ft2*

B

ac+bd

~~ a

2+b2

ad—be

~'a2+b2'

(18)

Eight constants are computed for each plate.

To check the distortions of the plate, (x, y) is

computed for stars 3 and 4 from equation (12)

and the residuals (Ax, Ay) in the sense "computed minus observed" are found. These

should be of the order of the error of measurement.

Determination of the pole of the trail. The

equation

y=a'+b'x (19)

represents the best straight line which may be

drawn through the 10 measured points on the

meteor trail. Values of y for the beginning,

b, and end, e, of the trail are obtained by inserting the observed x values in equation (19).

Equatorial direction cosines are then computed

from equations (12), (11), and (10). For the

pole of the trail we may write

\A, sin

nAp sin

vAv sin

(20)

where

(21)

and similarly for trail B.

Determination of the radiant point. The d irection cosines of the radiant point are given by

the formulae

\B sin <2=2

M« sin Q=

vR sin Q=

(22)

where

sin2Q=2j+22+22.-

(23)

The sign of Q is chosen to orient the radiant

above the horizon, and XB fiR, and vR are computed for the equinox T and for the equinox

of 1950. For later computation we require

the quantities cos aB, xB, and yR for both plates.

These may be found from equations (9) and

(12). We also require EA and EB, the elonga-
NO. 2 REDUCTION METHODS FOR PHOTOGRAPHIC METEORS 211

tions of the trails from the radiant. The elongation is given by the expression

COS EA= (24)

where the direction cosines of the approximate

center of the trail, o, are found in the manner

described for points b and e in section 6.

The third stage is to compute the space trajectory of the meteor.

Hour angle, or sidereal time at the apperance

of the meteor. The sidereal time for the instant

of the meteor is computed as follows:

OA=0O+OJ.T.)-LA + A$

(25)

where 60 is the sidereal time at Greenwich for

0

h

 U. T. on the night of occurrence of the

meteor. U. T., the instant of appearance in

universal time, is given for New Mexico by the

expression

U. T.=Mountain Standard Time +7h. (26)

The correction Ad is given by the expression

A6*= 236.555 Q^£r\ (27)

The longitudes LA, LB of the Harvard stations

are

Dona Ana: LA = '

(28)

na: LA = 7b7

m

ll\9>\

y

: ZB=7h6

m

26e.8. J Soledad: Z B =

Space vector. The equatorial direction cosines

of the vector from Station A to Station B at the

instant of the meteor are given by the formulae

sin 6A

sin BA

(29)

=ZAB=$ABO COS 6A —

=VAB=VABO COS

The quantities with subscripts ABo, BAo refer

to the components at sidereal time 0h at Stations

391131—57 3

"ABRAB ZAB£ABO- J

The components of the vector from B to A are

>>BARAB = SBA = ZBAO COS QB — VBAO sin 9B*

HBARAB=VBA = VBAO COS 6B+$BA, sin 6B • (30)

A and B, respectively. They are given by

equation (16) (loc. cit., p. 189) and for Soledad

and Dona Ana are, in units of 100 kms, as

follows,

£4*0=0.12128228 ZBAO= -0.12186172

VABO=0. 17695427 VBAB= -0.17655574

UBO=- 0.188639 fB^0=0.188639.

(31)

Components of the space vector are computed

from equations (29) and (30). As a check it

may be noted that ZAB=—£BA and VAB=— VBADirection oj zenith. The direction of the

zenith in equatorial direction cosines is given

by the formulae

Xr4=cos 4>A cos 0A

Hz A = cos <f>A si n BA

p Zjl=si n 4>A.

(32)

Similar formulae give \XB, UZB, *ZB, and the

directions of both zeniths are calculated.

Dona Ana:

Soledad:

cos ^=0.8433343

sin ^ = 0.5373893

cos 0«=0.8452266

sin 4>B=0.5344081 J

(33)

Range. The range RAt, in units of 100 km,

from Station A to a point i on the trail is given

by the expression

Tf __ (34)

Range is calculated for points b, a, o, j,j, and e

on trail A. Ranges between Station B and

points on trail B are also computed.

Heights. The heights of points Au Bt, above

the sea level surface of the earth are given by

equation (28) of Whipple and Jacchia (loc. cit.).

For Dona Ana and Soledad

(35)

where

. (36)
212 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

A similar equation holds for hBi. HAi and HBt

are computed for segments b, J, and e. As a

check, heights other than hAb may be computed

from the equation

hAl=hAb—100 HA

COS <Tx» (X XAb)

-——-f-—-—r

coaaAB (x—xAR)

. (37)

A similar equation holds for hBi- The value of

cos aAb may be found from equations (11) and

(13). Values of hAt and hBi are computed for

points 6, a, o, j , e, and / from equations (36)

and (37). Agreement between the values of h

found from the two methods shows that there

are no errors in the computation of heights and

ranges.

Time at segment No. nt. In the case of SuperSchmidt cameras, a double-bladed shutter rotates in the focal plane about an axis which is

aligned with the center of projection of the

plate, and the time lapse between dash b and

dash nt is given by the expression

tAt=,

S0nAt r+

30

(R.P.M>

i~ <aAb) (38)

where the number of revolutions of the shutter

per minute are given first as a modulus and

secondly with a sign + or —. If the shutter

appears to rotate in a clockwise direction over

the sky, the + sign is taken and vice versa.

The angle u is measured clockwise in the sky

and is given by the equation,

(39)

(aA—dA)

Similar relations hold for the times on plate

B. Values of tAt and tBi are computed for a,

o, and j .

Velocity of the meteor. The velocity of the

meteor determined from segments dA and

oA is VA\\ and the velocity from oA and jA is

VA2- Equation (35) of Whipple and Jacchia

(loc. cit.) gives

VA1=

<rAb

,— *„) cos aAR

r(^i4o %Ab)

A similar equation holds for VA2, VB\,

all four velocities are computed.

(40)

and

Correction jor deceleration. In the case of a

short trail it is usually not possible to measure

the deceleration directly (loc. cit. p. 193). It is

possible, however, to apply a correction for

the drag of the atmosphere based on the

measured decelerations from longer trails. We

may write

1=^^-31.69 log. (41)

From the exact reduction of long trails we

know that the ratio of the mass at any instant

to the mass of the meteoroid outside the atmosphere is given to a good approximation by the

assumption that the light curve is formed from

two half parabolae joining at nf. Then before

the position of maximum light, when n'<nff

ne

2- 0.99814+

(42)

and after the position of maximum light, when

0.09374

n, [»•

99814 +

(43)

where n' is the difference between the position

of maximum light and the mean dash number,

hence

_ i _

(44)

(45)

and

Similar equations hold for VmAi, VmBi, V.saAbsolute magnitude. The maximum photographic magnitude, reduced to a range of 100

km, is given for Super-Schmidt plates by the

expression

M^M^-9.063-2. 5 log,.

(46)

Equation (104) of Whipple and Jacchia (loc.

cit.) gives jsx. The last term is a small correction which can be added to Mmtx at the time
REDUCTION METHODS FOE PHOTOGRAPHIC METEORS 213

the photometric comparison is made. It is a

function of the position of the trail on the film,

and for Super-Schmidt films with the center of

projection at the shutter center, the correction

has been tabulated below.

TABLE 1.—Photometry correction for distance of trail in

cms from center of film

Correction

—2.5 logn/f

+0.0

+0.1

+0.2

+0.3

Trail parallel

to radius

0.0

4.4

7. 6

9.9

Trail at 45° Trail perpendlcto radius ular to radius

0.0

5. 1

8. 8

0. 0

6. 2

10. 9

In critical cases descend. The absolute

visual magnitude is given by the expression

M=.i,M,m+1.8. (47)

Mass of meteoroid. By assuming, as in the

section on correction for deceleration (p. 212),

that the light curve is a double parabola it can

be shown that

t

1

-

where w is visible energy in ergs emitted by a

meteor of zero magnitude and r0 is the luminous

efficiency. In all previous work it has been assumed that the value of the term in brackets is

47.71. If T0 is subsequently revised the value

of maA must be scaled accordingly. A similar

equation holds for mmB.

The final stage of the computation for radiant

and trajectory is to estimate the errors of the

computed quantities. This is essential in short

trail reductions since only a few measures are

available and the accuracy depends on many

variable factors.

Probable error in the radiant position. The

error in position in radians is given by the

expression

SE= cosec QV(sin EA 8b'A)

2+(sin EB bb'Bf (49)

where bb'A and 8b'B are the estimated errors in

the angle of the trail. Usually the largest component of 6b' is due to plate distortions, and if

391131—57 4

1 and 2 are the plate constant stars and 3 and 4

are the check stars, then 56' is given by the

approximate equation

(50)

where Ayt, Ayt are the residuals of the check

stars as defined in the section on determination

of plate constants (p. 210).

This equation applies to both trails with the

appropriate change of values.

Probable error in velocity. The error in individual determinations of velocity is given by

the expression

VAl

T Vv2

(xo—xa)

(cot EA« (51)

where At is the estimated error due to vibration

of the shutter. For the Harvard SuperSchmidt cameras, with the original motors and

shafts,

and

ST

and BT is the estimated error in the time of

appearance of the meteor in units of 6 minutes.

In the measurements at Harvard the error in

x measurements has been found to be 6(x0—z.)

= 5X 10~* cm. Similar equations hold for

and

Time of appearance by equalization of velocities. If the time of appearance is not known,

but assumed to be at the midtime of the exposure, then the time of appearance is found

by adding a correction term A TV where

minutes. (54)

irA

The correction should not be applied if

VmA—VmB is less than the error in velocity
214 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

estimated from the first three terms of equation

(51). It holds for exposure times that are less

than 20 minutes from Dona Ana and Soledad

stations.

Time of appearance by equalization oj heights.

The time of appearance may also be calculated

by adding a correction A TH to the assumed

time. The correction is given by the expression

minutes (55)

6T ST

where

(56)

and the differential 8RA/RA&T is found from

equation (53) by putting ST=1. The correction term ATS should also be evaluated for

points b and e on the trail and a mean value

taken. However if it is suspected that one

camera has not recorded the beginning or end

portion of the trail, due to clouds and other

causes, then the equalization of heights at that

end of the trail should be ignored.

From a weighted mean value of A TV and

ATB we may computed the correction to

velocity and height. The correction to height

is obtained from equation (56); the correction

to velocity is obtained from the differential

equation

V &T~R 8T (57)

Notation symbols additional to those of

Whipple and Jacchia (loc. cit., p. 206)

To—Equinox of star position or R. A. and Dec. of

projection center.

T—Equinox of meteor. This is taken as the beginning

of the year nearest in time to the date of the

appearance of the meteor.

VAi—Velocity determined from points a, o on trail A.

VA2—Velocity determined from points o, j on trail A.

U. T.—Universal or Greenwich Civil Time.

LA—Geographical longitude of Station A.

tAi—Time in seconds for meteoroid to move from middle

of segment b to middle of segment i on plate A

R. P. M.—Rotational speed of shutter in revolutions

per minute.

rtimA—Mass of meteoroid outside atmosphere as

measured on plate A.

mAi

:—Mass of meteoroid at any instant, i.

Mmax—Photographic magnitude of a star which produces a trailed image on a stationary plate equal

to the maximum intensity of the meteor trail.

Mvm—Photographic magnitude of a star which produces the same intensity as the meteor's maximum

photographic intensity when the star is trailed

at the angular velocity of the meteor. This

magnitude is also corrected for the inverse square

law to correspond to a standard range of 100 km.

MW{,—Visual magnitude corresponding to Mvm.

Mum—The photographic magnitude of the faintest star

visible on the comparison plate.

EA—Angle in the sky between the radiant and point o

on trail A.

SE—Probable error in position of radiant in units of

circular measure.

Sb'A—Probable error in the gradient of the trail equation

of plate A.

/—The point of maximum light on the trail.

n—The number of the trail segment with segment b

numbered 0.

a'A, b'A—Constants in the linear equation chosen to represent the meteor trail on plate A.
A Rapid Graphical Method of Meteor Trail

Reduction1

By Richard E. McCrosky

Since February of 1952 when the Harvard

Meteor Project initiated double-station operations with the Baker-Super-Schmidt cameras,

more than 4,000 meteor pairs have been photographed. Both the magnitude and the quality

of this recent photographic material have made

it comparable with visual and radar observations as a basis for statistical analysis. However, a previous method of analysis described

by Whipple and Jacchia (in the first paper of

this series) is far too time-consuming to be considered practical as a routine reduction method

for statistical data. A more rapid method was

needed, and is presented here.

When the approximate reduction program

was first considered, the criteria for a "satisfactory" method were rather vague. However,

from the experience gained in the accurate reduction program one could be certain that data

with errors of as little as 1 percent could not be

obtained with a small expenditure of time. At

the other extreme, one could question the value

of statistics derived from data with mean errors

as large as 10 to 15 percent. With these limitations on the accuracy, a search for a sufficiently

rapid method was made. The general result

of this attempt is a method, about to be described, that yields mean errors of about 5

percent and requires 30 minutes for the complete reduction of a meteor pair. The method

can be best explained if we first derive the

necessary equations, next discuss separately

the methods of measurements, and finally, return to the equations and estimate the magnitude of errors due to various simplifying assumptions and to errors of measurement.

Meteor heights

The geometry of the problem can be seen in

figure 1, where RAB is the distance between

stations A and B; RA is the distance from Station A to some point on the meteor path C;

and RB is the distance from Station B to the

same point in space. This point, on either

meteor trail, will be referred to as a common

point (C). The orientation in space of the

triangle (A, B, C) is described by the declination (5) and the hour angle (t) coordinates.

(See figure 1). RAB, &AB and tAB are constants

of the stations. The hour angle, tAB, and declination, SAB, represent the direction of Station B

as seen from Station A.

From the spherical triangle, (Pole, PAB, Pt),

we solve for yif the angle between the direction

to the meteor and the direction of the line joining the stations. The subscripts i and j may

refer to either Station A or Station B.

cos7,=sin St sin (tAB—

(1)

> This paper is based upon research supported by the XT. 8. Office of Naval Research, Contract N5cri-O7M7. Reproduction in whole or In part is

permitted for any purposes of the U. S. Government.

> Harvard College Observatory.

215
216 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

Station B Station A

FIGURE 1.—Relationship of geometrical quantities used in determining meteor heights.

With y determined, the values of the ranges

RA and RB are found from the equation:

Rt=RAB

sin 7,

sin (7B—

(2)

The height of the common point above ground,

if we assume a flat earth, is then:

i=Rt cos Zt, (3)

where Z is the zenith distance of the common

point and cos Zt is given by the equation,

cos Z,=sin 4> sin 5,+cos <f> cos St cos tf, (4)

where, for the latitude <f>, we have used the

average value of the two stations.

Clearly the same point in space can be described by only one height, and thus hA=hB.

This comparison gives the first check on the

computations. The height above sea level,

H, is found by assuming h to be measured

from an elevation equal to the mean elevation

of the two stations.

Solutions for equations (1) and (2) were prepared in graphical form in order to eliminate

the tedious and time-consuming task of directly

forming such solutions many thousands of times.

Since both yt and cos Z, are functions of 8t

and tt, a solution for coz Z was also prepared

and superimposed on the graph for y. These

charts will not be reproduced here. They are

applicable only to the data acquired at the

Soledad Canyon and Dona Ana Stations of the

Harvard Meteor Project, and therefore are not

of general interest. Table 1 summarizes the

information obtained from these graphs and

from another to be discussed later.

TABLE 1.—Graphs utilized in the approximate reductions

Graph

la

Ib

II

III

IV

Enter with—

f-io°<a,<9O°l

t-50°</,<50°J

{ Same }

l60o<7B<135°J

{ Same }

f.50<#<1.00l

\o0<tAB<200j

Read

COS Zi

«'

RB

sin a

Accuracy

0°. 1

0.01

1 km

1 km

.001

Velocities and radiants

The meteor trail is interrupted every %o sec by

one of the occulting sections of the rotating

shutter. These breaks in the trail allow one to
REDUCTION METHODS FOR PHOTOGRAPHIC METEORS 217

measure the apparent angular velocity of the

meteor at some point along its trajectory. This

measure is referred to as the apparent angular

velocity, since only the component normal to

the line of sight is measured. In general, such

measures are made on both photographs of the

meteor at the position of the common points.

Figure 2 shows the relationships between the

apparent angular velocity, «'; the angular velocity, «; and the radiant. We see that:

(5)

or

sin

where tt is the angular distance from the radiant

to the common point. The space velocity, V,

is then given by the equation:

(6)

sin e<

where Rt is the range found in the solution for

heights. As in the case of the heights, the velocities obtained from the two photographs should

agree.

Co'

FIGURE 2.—Relationships of geometrical quantities used in

determining meteor velocities.

For equation (6) to be applicable, we must

have determined the position of the radiant.

When the two meteor trails intersect at a small

angle, Q, it is difficult to find an accurate radiant

point. In such cases another approach to determine the velocity is desirable. From equation (6) we have:

y y WARA U>'BRB /- \

A

~

 fl~~sin «,!~sin tB

sin u'ARA

sin

:E, by definition. (8)

If the apparent angular velocity is measurable

on both trails of a meteor pair, E may be computed. This gives one relationship between eA

and eB. For meteors with a small Q, it is possible to measure €A—eB (or eA+tB) with good

accuracy. That this is true can be most easily

seen by considering the extreme case of a meteor

pair intersecting at Q=0° (or #=180°); i. e.,

both trails lie on the same great circle. Then

the distance between the two common points

will be exactly:

«A—«B=«XB (or tA+tB=eAB), by definition. (9)

The simultaneous equations (8) and (9) may be

solved for sin eA and sin eB. This solution was

prepared in graphical form. (See table 1, graph

IV.) With these values of et, Vf is found from

(6). We have called velocities determined in

such a fashion "indirect velocities," as distinct

from the velocities obtained from meteors where

a direct measure of the radiant is possible

(direct velocities). It will be noted from equation (7) that we have forced the indirect velocities determined from each trail to agree.

The comparison of these velocities offers a

check on the computations.

The position of the radiant may be obtained

by finding the intersection of two small circles,

with radii of tA and tB with centers at PA and

PB, respectively. In general such circles will

intersect twice, but a quick inspection of the

trails is sufficient to distinguish the spurious

radiant point.

Reduction procedure

In this section we shall carry through the complete reduction of an imaginary meteor pair.

Our purpose is not primarily to instruct in

technique but to present, in logical sequence,

the various difficulties and approximations that

are present so that the reader may acquire a

realization of the limitations of the method.

A separate computing form is used for each

meteor. Certain data necessary for each meteor
218 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

are copied from the card catalog maintained for

all Harvard meteors. These include:

(a) Meteor serial number for each trail.

(b) Camera designation (SS, ST, SK, SL) and plate

number for each film.

(c) Region of the plate center, given by S and a.

(d) Astronomical date of the exposure.

(e) Time of meteor occurrence, to 0.01 minute if the

meteor was observed visually; or the mean time

of the exposure if no visual observation was

available.

The local sidereal time, in degrees, of the

nearest midnight is obtained for the date. To

this is added a correction to obtain the local

sidereal time of the meteor instant.

The preceding section makes it clear that the

entire method depends heavily on our ability

to locate on each film a point on the meteor

trails that represents the same point in space,

the common points. If the meteor shows a

burst or some other discontinuity in the light

curve, a common point is obvious. However,

the faint meteors, with which we are primarily

concerned here, usually have smooth light

curves. In these cases a more subtle technique

is necessary. The number of dashes visible on

each trail is counted and, if it is the same, we

assume that dashes with the same ordinal number are common points.

Such an assumption fails in three minor respects. Firstly, the shutters are not synchronized; i. e., we do not know the orientation of

one shutter with respect to the other at any

given instant. Consequently the common point

can be in error by as much as half the distance

between dashes. Secondly, the focal plane

shutter interrupts the same position of the film

at constant time intervals. The rate at which

the meteor is interrupted depends on its direction of motion and its apparent velocity on the

film. In the extreme case of a meteor trail

passing through the film center, an error of one

dash can be made. But since the two trails of

a meteor pair are usually similar in their direction of travel on the film, and since their apparent velocities on the film are small compared

to the shutter velocity, the error will generally

be only a small fraction of a dash. Thirdly, in

assuming that breaks with the same ordinal

number are common points, we assume that

both cameras have photographed the meteor to

the same limiting absolute magnitude or, alternatively, that the differential distance correction

from the beginning to the end of the meteor is

small or the same for both trails. Since most

of our trails are short, the differential distance

correction is small.

We can conclude that the common points

chosen on trails showing the same number of

breaks will be accurate to within one dash.

However, the two cameras often do not record

the same number of breaks for the following

reasons:

1. The sensitivities of the cameras are not equal.

With increasing experience in figuring the correcting

plate, the manufacturer has been able to improve the

quality of each succeeding camera. The effect is most

noticeable when we compare trails obtained on the

first (SS) and second (ST) Super-Schmidts.

2. The apparent magnitude of the meteor as seen

from the two stations may differ by several tenths of

a magnitude because of distance corrections. Effects

1 and 2 often compensate one another in part, since

the earlier camera, at Soledad Station, is directed more

nearly toward the zenith than is the mate camera at

the Dofia Ana Station.

3. The effective exposure time per dash is proportional to o>i. Thus, the trail nearer the radiant is

photographed as a brighter image and the plate limit

is reached at a fainter absolute magnitude.

Effects 1 and 2 combined are usually minor

compared to effect 3. By noting the general

shape of the meteor light curve of the brighter

trail one can obtain an acceptable common

point by estimating the number of dashes

that failed to be recorded at the beginning and

at the end of the fainter meteor trail. In only

a very few cases does the combination of the

factors of effective sensitivity of the camera,

and the apparent velocity of the meteor, differ

so much that no acceptable common point can

be found. For each common point, we record

under "Quality" an estimate of the number of

dashes by which the common point may be

in error. This never exceeds three for an

acceptable meteor and rarely exceeds two.

To mark the common points, a small piece

of Scotch tape is placed on the reverse side

of the film at the location of the trail. Ink

dots placed on this tape indicate the common

point. For longer meteors (20 dashes or more)

two common points are chosen in order to

make a check on the entire reduction. The

common point is recorded as the ordinal number

of the dash, the dash nearest the radiant being
MCCROSKY PLATE 1

The transparent plotting globe. The declination scale pivots about the horizontal rod which passes

through the globe at the celestial poles. The computer is positioning the curved rules to be coincident with the meteor image on each film. The juncture of the rulers (under the right hand)

represents the meteor's radiant.

REDUCTION METHODS FOR PHOTOGRAPHIC METEORS 219

called "1. " The duration of the meteor is

measured in terms of the total number of

visible dashes, N.

To measure the celestial coordinates of the

common points and radiant, we utilize a transparent Plexiglass hemisphere of 8-inch radius,

calibrated in hour angle and declination. The

scale of these calibrations and the radius of

the globe correspond to the scale and radius

of the Super-Schmidt films. To read the

coordinates of the common point, we need only

to position the film properly on the globe.

We accomplish this by choosing three or four

bright stars which appear on one of the films,

determining the declination and hour angle

of these stars at the time of the meteor, and

plotting these star positions on the globe.

The film is then placed on the globe so that

the star images and their plotted positions

coincide. Since the regions of the two mate

films overlap in an area of approximately

one-quarter of the film, we can position the

second film by superimposing stars in this

region. The hour angle and declination of the

common points are then read from the globe

scales.

We determine the radiant and radiant

distances, eA and eB, with the aid of a pair of

curved rulers of 8-inch radius and 90° length,

attached to one another by a hinge. They are

calibrated in degrees, the pivot point of the

hinge being zero degrees. When each rule

lies parallel to one of the meteor trails, the zero

point represents the radiant. The radiant

distances are read from the rules and the

coordinates of the radiant point are read from

the hemisphere (see pi. 1). The cosine of the

zenith distance of the radiant (cos ZB) is

determined from graph Ib of table 1.

The quantity Q is the angle of intersection

of the two trails. An estimate of this quantity,

accurate to about 10 per cent, is made when

the radiant is found. The value of this angle

is not used in the reductions but it serves as a

measure of quality of the directly determined

radiant.

The apparent angular velocity, or distance

per break, is measured on the Harvard Coast

and Geodetic measuring machine. This has

been equipped with a section of an 8-inch radius

Plexiglass sphere for the support of the film.

The spherical section is large enough to permit

all measurements to be made with the optical

axis perpendicular to the image when the trail

occurs at the edge of the film. This reduces

the focusing problem and also allows us to

neglect any correction for the projection effect

which would be necessary if the curved film

rested on the flat carriage of the measuring

engine.

The number of breaks measured depends on

the apparent angular velocity of the trail. In

general, we attempt to measure a distance of

from 1 to 3 mm, which may represent 2 to 8

breaks. The measurements are usually made

from the end of one dash to the end of another;

that is, an integral number of dashes is measured. Corrections, by eye, are made for the

photographic spreading of the image if the two

terminal dashes of the measured trail section

do not appear to be of about the same intensity.

The dashes to be measured are chosen in such

a way that the common point lies in the center

of the measured section.

The trails are inspected for any wake, terminal blending, or marked abnormalities in

the distribution of light over the trail. The

position and apparent magnitude of the brightest dash may be measured.

This completes the measurements made on

a meteor pair. We then determine, from the

equations given earlier, cos Zt, yt, Rit ht H,

E, and Vit in that order.

To complete this outline of the method, we

will add a description of an earlier approach to

the problem of finding common points. The

argument, due to C. P. Olivier,3 proceeds as

follows: Consider the plane defined by the two

stations and a point on the meteor. The lines

RA, RB, and RAB lie in this plane which intersects the celestial sphere on a great circle.

Then the position of the meteor point as seen

from each station must lie on a great circle

which also includes the celestial position of one

station as seen from the other. Conversely, if

the two meteors are located properly on the

globe, we can draw a great circle through the

point defining the direction between the stations. Then, by the argument above, the intersections of this great circle with the meteor

* Meteors, 276 pp., 1925. Williams and Wilkins Co., Baltimore. (Esp. pp. 156-157.)
220 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

trails must represent common points. This

technique was tried and discarded as being too

insensitive for most cases. In practice, we

employed a movable great circle which was

attached to the measuring globe and pivoted

at the points (8AB, tAB) and (— SAB, <UB—180°).

When the meteor trails formed a moderately

small angle with this circle, the common points

could be varied by several dashes with only a

slight shift of the films. If the time of the

meteor occurrence is lacking, the precise position of the film on the celestial sphere is unknown.

Single-station shower meteors and the

approximate method

The program of meteor astronomy at the Harvard College Observatory includes the investigation of the origin and histories of the shower

meteors by study of the distribution of the

radiants over the period of the shower. Singlestation meteors are used in a least-squares solution to determine the radiant, if they appear

to belong to the shower—that is, if an extension

of the trail passes through (or near) the assumed

radiant point for the time of the meteor, and

if visual inspection determines that the apparent

angular velocity of the meteor is reasonable for

the shower velocity and the radiant distance.

A small amount of work with the globe used in

the approximate method removes the guesswork

from this visual determination. By assuming

that the meteor belongs to the shower, we can

estimate a height that corresponds to that of

meteors of the shower velocity. The apparent

angular velocity (wj), cos Zt, and the distance

from the assumed radiant (e{), are measured

for some point on the trail. From equations

(3) and (6), we have:

V—

cos Z{ sin tt

(10)

If the meteor belongs to the shower, the measured values should yield the shower velocity.

We may be deceived occasionally by meteors

whose true radiants and velocities are not

those of the shower but combine, by chance, in

such a way that

( F Sin €i)8hower=(F Sm €,)nOMhow«- (11)

Such cases must be far more rare than the 10

percent of single-station meteors which we

have been able to eliminate from those meteors

thought to belong to the shower.

It is obvious that the usual approximate

method may also be used to eliminate those

double-station meteors that have the proper

radiant but a nonshower velocity.

Height errors

The problem of errors in our data reduction

includes two major questions, neither of which

has yet been finally answered. First, the

velocities and radiants determined by the

direct and indirect methods are, in general, at

variance with one another. We need some

quantitative criteria for making a choice

between the two results. Second, we desire a

more exact knowledge of the mean errors as a

function of the various parameters of the

solution. The final answers to these questions

will be found only after the completion of the

reduction project. Eventually we intend to

acquire approximate data on 2000 meteors,

including about 300 faint meteors that have

been reduced by the method described by

Hawkins in the preceding paper. An intercomparison of results should supply the information we want. Comparison can be made

with several hundred brighter meteors already

reduced by Jacchia. However, the accuracy

we can obtain in determining the radiant of

these long meteors will seldom be approached

for fainter meteors. On the other hand,

common points are often more difficult to

obtain on long meteors. All in all, we do not

consider bright meteors to be comparable to

faint ones with regard to our system of measuring. A comparison of velocities obtained by

accurate methods and by the present method

has been made for some 25 bright meteors.

Our average error in velocity was about 3

percent.

With respect to the order-of-magnitude

estimate we can study the results of measuring

errors of probable amounts. We may also

compare our velocity results for shower meteors

with their known values.

Let us begin with the errors in height introduced by our assumption that the earth is a

plane surface. The correction for this was
REDUCTION METHODS FOB PHOTOGRAPHIC METEORS 221

ignored as being small compared with the

intrinsic errors of measurement. This is true

if the meteors are in the vicinity of the zenith.

It is easily shown that:

Ah=h-h'-- R

2

-h'2

'' D (12)

where h is the computed height above the

station level, h' is the true height above the

station level, R is the range from meteor to

station, and D is earth's diameter. With the

assumption that

.R2-h2

we obtain

Employing equation (3), we may write

D+2h

(13)

(14)

(15)

For meteors of 100 km altitude, the correction

factor reaches 0.5 km at a zenith distance of

37°.7. Essentially none of the meteors reduced

exceed this value. The average zenith distance

is of the order of 20°.

If the common point is improperly chosen or

measured, or if the two films are not correctly

positioned on the globe, the two range lines, as

defined by (8A, tA) and (5s, tB) will either not

intersect in space or will not intersect at a point

on the meteor trail. Whether the intersection

occurs for any given set of measures or not, our

computations still lead to a complete description of some triangle which represents, to some

degree of approximation, the true triangle defined by the two stations and a point on the

meteor. We are interested in knowing how

good an approximation our measures probably

give. We will discuss the errors involved in one

particular case. Computations for other cases

show that the total errors will be similar in

other parts of the sky where meteors have been

photographed.

It is a bit difficult to estimate a reasonable

amount for the errors in the common point,

positioning, or measuring which combine to

give the total errors, AS and ht. The error

involved in reading the globe scale should not

391131—57 5

exceed 0°.2. For fast meteors with a high apparent angular velocity and with a common

point of only moderate quality, say two, the

common point error would be about 1°. We

believe this to be considerably greater, perhaps

by a factor of two, than the common point

error for the average meteor.

Positioning errors can occur in two ways.

First, the two films may not be properly superimposed. Because the globe is not perfectly

spherical and because its mean radius is not

exactly that of the films, we can not always

superimpose the entire star field common to

both films. The attempt is always made to

carry out the superposition in the vicinity of

the meteors and an error of about 0°.5 would

probably be large. The second positioning

error, resulting from an unknown time of the

meteor occurrence, is a special case and will be

treated separately.

From the preceding extreme figures, we may

estimate that the average error, in 5, or tt, will

almost certainly not exceed 1°. It is unfortunate that this figure cannot be verified by

more rigorous methods than those used. However, another check on our errors, to follow

later, will supply additional information tending to confirm this as being an extreme value.

Let us assume that the common point at

Station B has been properly located, positioned

and measured, and that, for any of the aforementioned reasons, a 1°.O error exists in the

measures of the common point at Station A.

The percentage errors for this case have been

computed from the differentials of equations

1, 2, 3, and 4 and are listed in table 2. The

quantity t\ is defined by the equation

At'i=Mt cos 5<.

TABLE 2.—Errors resulting from a 1°.O error in the

Percentage

•nor in:

RA*=

hA=

common point

3.7

3.7

3.5

3.8

3.6

3.7

3.7

3.7

The similarity of values within either column

should be expected. The similarity between

the two columns is the result of chance and

indicates that the maximum error for the
222 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

meteor will occur for departures in a direction

roughly half way between the directions of the

5 and the t axes.

When the instant of the meteor is unknown,

we choose the instant of the middle of the

exposure for reduction purposes. Thus, with

12-minute exposures our maximum error is

Mt=6 minutes=l°.5. However, in this case

the common points do not suffer a shift relative

to one another and the resulting errors in the

ranges will be the algebraic sums of the errors

caused by the displacement of both common

points. Since these are of opposite sign and

of about the same magnitude, the final errors

will be small. Table 3 shows the percentage

errors resulting from a 1°.5 shift in tt of both

films. We can see that the timing error will

be negligible even in those cases where the

maximum possible error results from a displacement along the f-axis.

TABLE 3.—Errors resulting from a displacement of Af<

1°.B of both films

Quantity RA RB hA hB

Error (km) +0.34 -0.14 +0.20 -0.17

Percentage error. 0.4 0.2 0. 2 0.2

Velocity errors

Errors in velocity may result from errors in any

of the measures. Errors in the apparent angular velocity, w<, are probably small compared

to any other type and we will neglect these.

From differentials of equation (6) we see that

the percentage errors due to range and radiant

errors are, respectively:

\VL.M~ cot eAc. (17)

Thus the error in velocity cannot be less than

the range error, which we found to be about

4 percent with the assumption of a common

point error of l°.0. The function A* is itself a

function of e as one can understand by visualizing the extreme cases when the meteor appears

at the radiant (e=0°, Ae=0°), and when the

meteor appears at a great distance from the

radiant (say, c=;90°, Ae=?^0°). We might

estimate this unknown Ac to be of the order of

5° or 10° in the worst cases. We may say that

certainly Ac varies less rapidly than tan c.

This leads to the apparently contradictory result

that, in general, meteors must have a badly

determined radiant to allow us to produce an

accurate velocity. However, another independent error in e, that imposed by our scale

reading accuracy, 0°.2, weighs more heavily

against meteors of small e. These two errors

may combine in such a way that meteors at

some intermediate e give the best velocities.

We have no method of determining a satisfactory relationship between e and Ac and we

must therefore approach this problem from

another side. Among the meteors reduced,

there are 36 Orionid and 45 Geminid shower

meteors for which we know velocities. These

groups will determine our velocities errors.

Furthermore, since most meteors in these

showers were reduced by both the direct and

indirect methods, we can find some estimate

of a criterion for choosing between the results

of the two methods.

In general, the direct velocities from the two

films of a given meteor do not agree with one

another to within several percent. The direct

velocity chosen is usually the average of the

two. However, in some examples, the radiant

distance of one common point greatly exceeds

that of the mate photograph and, in these instances, the velocity derived from the more

distant trail is used or weighted more heavily

in the average. Average velocities, mean errors,

and percentage errors were found for both sets

of shower meteors for the following cases:

(a) Direct velocity used for all meteors.

(b) Indirect velocity used for all meteors.

(c) Indirect velocities used when |«4BI^10°.0 , direct

when |«XB|<10 0. 0 .

(d) Same as (c), with division made at |«.*fl| = 9o.0.

(e) Same as (c), with division made at |«XB| = 8°.O.

Table 4 gives the average values, mean deviations, and percentage errors for each of these

velocity criteria and for both groups of shower

meteors. The letters refer to the categories

outlined above.

We should qualify the character of these data

before discussing the results. First, the Geminids were somewhat brighter meteors than
REDUCTION METHODS FOR PHOTOGRAPHIC METEORS 223

normal and, hence, easier to measure. Furthermore, the radiant of this shower is not far distant from a Geminii, a star that appears on

most of the Geminid meteor photographs.

Since these meteors were reduced specifically

to check for errors, it was necessary for the

measurer to "forget" the position of the radiant

for each measure, a difficult task with a radiant

so clearly marked. Still, we believe this was

accomplished.

TABLE 4.—Errors in velocity for two sets of shower meteors

Geminids:

(a)

(b)

(c)

(d)

(e)

Orionids:

(a)

(b)

(c)

(d)

(e)

Number of

Number indirect deof

meteon

44

38

45

45

45

34

34

36

36

36

termina- i

i tions used

0

38

16

21

26

0

34

18

24

28

^/km\

\sec/

35.7

35.7

36.3

36. 1

36.0

68.3

67.2

68.2

67.7

67.6

/km\

\sec/

1.42

1.74

0.92

1.05

1. 13

3.01

2.52

2.59

2. 17

2.05

Percentage error

4.0

4.9

2.5

2.9

3. 1

4.4

3.8

3.8

3.2

3.0

In the case of the Orionids, the situation is

quite different. The radiant was not present

on the region being photographed and, more

important, we did not realize we were reducing

shower meteors until after the measures had

been made. Also, these meteors did not produce such bright trails as the Geminids. Perhaps, then, the Orionids should be regarded as

giving the best test, although, as can be seen

in table 4, the errors for both showers are

comparable.

In regard to the indirectly determined velocities, our measured quantities are E and tAB as

defined by equations (8) and (9). It may easily

be shown that if the radiant is properly chosen

and there are no measuring errors,

sin t\

sin eAB

(18)

where et, in this case, is the angle from the

radiant to the common point of the trail most

distant from the radiant. A poorly determined

common point has little effect on E, since this

quantity is determined from a ratio of the

ranges, R. A reasonable error in the common

point results in comparable changes, of the

same sign, in the two ranges. We may write

Ecc

Then:

. J?_

R%

(19)

(20)

Thus, the errors in the range tend to compensate one another in the determination of E.

However, since we must still apply the velocity

equation (6) in its original form after determining et) any error in range will affect the velocity

in the usual fashion.

One can see that an error in the common

point will directly affect eAB, which is essentially

the distance between common points. It is also

clear from the — factor in equation (18)

sin eAB

that the indirect method will be more powerful

when tAB is sufficiently large. From the shower

meteors we can obtain an idea of how large.

Table 4 indicates that the deviation from the

mean velocities of the showers is of the order of

3 to 4 percent, but we have yet to show that

these mean velocities actually correspond to

those expected for these showers. Our velocities are those at some point in the atmosphere,

uncorrected for deceleration and therefore somewhat lower than the velocity outside the atmosphere which is usually quoted for a shower. In

general, we choose common points as near the

beginning of the trail as possible to minimize

the deceleration correction. From Orionid and

Geminid meteors reduced by Jaccbia, we have

obtained the velocities at the beginning point

of the meteor (F«). These results, as well as

Jacchia's values for the no-atmosphere velocity

(F.) are given in table 5. They are compared

with the average velocity we obtained for the

method which yielded the smallest a in table 4.

The agreement is excellent and there is no

evidence of a systematic error with velocity.

We may conclude that our velocity errors, for

these cases, are about 3 percent and that the

TABLE 5.—Average velocities obtained by accurate reductions of shower meteors (by Jacchia)

Geminids

Orionids

Number of

meteors

17

7

Va.

36.6

67.5

V.

36.3

67.4

Vmmu

36.3

67.6
224 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

E

31.7 33.7 35.7 37.7

Velocity (km/sec)

FIGURE 3.—Velocity distribution of Geminid shower meteors, approximate reductions.

optimum minimum value of «AB for the indirect

reductions is about 8° to 10°. It seems unlikely that measures on somewhat shorter and

fainter nonshower meteors would yield errors

that exceed 5 percent. When this result is

compared with the errors expected for ranges

and heights, we see that we must have overestimated the probable error in the common

point, for, as was pointed out earlier, the

velocity error cannot be less than the range

error, which was of the order of 4 percent on

the assumption of a 1°.O error in common

point.

There still remains the possibility that in

selecting the shower meteors, we passed over

some cases which were so badly determined

that they were unrecognizable as members of

the shower. A study of the frequency diagram

of the velocities of the individual meteors makes

such a hypothesis unlikely if, on the basis of

such a diagram, one is willing to grant that the

scatter may be represented to a fair degree of

approximation by the usual error curve. We

have reproduced in figure 3 the frequency

diagram for the directly determined velocities

of our Geminid meteors.
A Reduction Method for the Motions of

Persistent Meteor Trains1

By Allan F. Cook2 and Robert F. Hughes2

Two Baker Super-Schmidt cameras, located

35 kilometers apart, at Mayhill and at Sacramento Peak, New Mexico, operate continuously

to obtain multiple exposures of persistent

meteor trains. The method of operation of the

cameras and the general process of obtaining

wind velocities normal to such meteor trains

have been described by F. L. Whipple (1953).

Further results from the program have been

presented by Liller and Whipple (1954). No

details of the reduction methods have been

published, however, nor have the methods been

previously extended to include determination

of the air motion parallel to the direction of

the meteor. Hence we present, in considerable

detail, the procedures of measurement and

reduction that we have found to be useful in

determining atmospheric motions from multiplyphotographed persistent meteor trains.

For measurement the spherical SuperSchmidt films undergo an approximate gnomonic

projection to a flat glass-plate emulsion by

means of a copying camera. With the Mann

measuring engine, train positions are measured

with respect to the infinitely distant reference

system provided by the stars. The loci of the

multiple trains are readily converted to linear

plate displacements from the trail in the reduction system of the concurrent meteor photographs, which provide complete geometric

and time coordinates of the meteor's flight.

From the length of the exposures, as determined

by photometric measures and clock records,

the velocity components of the train motion on

the plate are calculated and transformed to two

components in space. One component of the

train motion is normal to both the meteor trail

and the line of sight, and the second is parallel

to the projection of the meteor trail on the

plane normal to the line of sight. If the former

component is observed at both stations, the

air motion is determined as a composite of a

mean wind field plus a superimposed turbulent

field. Given the train motion component

parallel to the projected trail at even one

station, one may investigate the existence of

any systematic motion of the train along the

actual meteor trail.

Selection of train films

The Super-Schmidt cameras used by the

Harvard Meteor Expedition for photographing

meteor trains are triggered photoelectrically or

manually. The exposure durations of roughly

1/2", 28, 2s, etc., are separated by o!3 shifts

of the camera in declination, and are programmed automatically with a special long

exposure added manually at the end. On the

train film, the various star exposures appear in a

line at angular separations of about 1/3°, or

1 mm linearly, where the fainter end exposure

is the first exposure. Plate 1 is an example of

1

 This paper is based upon research supported by the U. S. Air Force under Contract NTo. A F19(122M82. Reproduction in whole or in part is permitted

for any purposes of the U. 8. Government.

1

 Harvard College Observatory, Cambridge, Mass.

225
226 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

such a photograph. The observers generally

use a greased crayon to indicate the trains,

which may be continuous, diffuse, or varying

in intensity and amount of distortion from a

straight line. The possible appearance of the

meteor trail on the train film, or alignment

with the concurrent meteor film, confirms

the identification. Trail films marked SS

(Sacramento Peak) or ST (Mayhill) have

concurrent meteor films SL and SK, respectively. The initial "processing" of the train

films may be outlined as follows:

/. Standards for selection:

(a) Concurrent meteor films are successful.

(b) Train film is in good focus.

(c) At least four train exposures or 3 trains

and a trail are observable

2. Copying:

(a) The train and concurrent meteor films

are both copied.

(b) If the meteor or the train occurs very

near the film edge, an offset position of the

desired projection center is indicated so that

the containing rim of the camera will not

eclipse the image.

(c) The plates of the films are inspected in

order to see that all detail observed on the

train film has been reproduced. Several copying attempts are sometimes necessary to insure

optimum results.

S. Marking:

(a) Reference stars are marked on the train

and meteor plates during the reduction for the

meteor's trajectory, velocity, and deceleration.

At Harvard this is done by the Meteor Analysis

Laboratory.

(b) If the number of train exposures is greater

than four, the first and last images of the fainter

stars are indicated.

(c) Stars, lettered A to Z in the direction of

meteor flight, are placed every 5-10 mm, as

near the train as possible, and extending slightly

beyond the extremities of the trains.

4- Photometric reduction:

(a) After the films are successfully copied,

they are subjected to photometric analysis.

Magnitude differences between successive images are determined by eye intercomparison of

the images of a selected standard star sequence

(Payne-Gaposchkin, 1937).

(b) The film center is indicated from record

sheets to aid in the identification of the star

field.

Measuring procedures

Careful application of the parallax test is necessary to adjust the focus of the measuring: engine

properly. To measure a diffuse line with poorly

defined edges which are susceptible to distortions from grain structure in the emulsion requires an eye integration along nearly 1 mm

of the cross-hair. Any one position is best

defined with the aid of neighboring points,

which tend to average out the small irregularities of the train shapes. Train images are

measured to the limit of visibility. The horizontal cross-hair, which is parallel to the engine

carriage, is the X-axis; the vertical cross-hair

is the F-axis. The procedure for measurement

is as follows:

1. Plate alignment:

(a) The plate is secured in the frame with

the emulsion down.

(b) The lighting is adjusted for maximum

train visibility.

(c) The train is placed (similar to trail)

parallel to the X-axis and the frame locked

securely in position.

(d) The four verniers are read as accurately

as possible (to 10" on the Mann engine).

2. Stars and trains:

(a) Stars. For each exposure one reading is

made in X and Y to the nearest micron (0.001

mm) in the region of the respective train image.

The reading is repeated after the train measurements to obtain averaged star readings and to

check for the possibility that a plate movement

has occurred during the train measurement.

Two fiducial marks (sharp images of faint

stars, emulsion defects, etc.) are chosen and

their positions read before and after the measurements as a further protection against disturbance of the plate during the measures.

(b) Trains. For convenience in making the

reduction, the Y coordinate of the train is

measured at equal intervals of X along the

train. X intervals need only be small enough

to define and confirm the varying shape of the

train. Generally 0.5 to 1.0 mm is adequate

for earlier images. This is sometimes diminished to 0.1 to 0.25 mm for later images where
REDUCTION METHODS FOR PHOTOGRAPHIC METEORS 227

distortions occur and difficulty in measurement

increases. Regions about singular points in

the trains are specially defined in each image.

(c) Elimination of personal equation. The

systematic error, due to personal equation, in

centering the cross-hairs on the train images

is averaged out by taking two sets of readings

measured from opposite directions. After the

plate has been rotated 180°, the measurements

described in section 2-b are repeated. Comparison of the vernier readings with the first

position allows a very close approximation to

180° rotation. A shift in the Y scale is often

necessary to make the train region observable.

Reduction to a single train system

The transformation of the two sets of train

readings into a common train system consists

of a rotation and translation, where the angle

of rotation may be expressed as 180°+e.

Then,

where the constants of rotation (e) and translation (a, b) are found from the star measurements

as follows:

1. Averaging. The star readings of each of

the two sets of measurements are averaged.

2. Conversion from one system to the other.

Three stars in good focus are selected, two of

which have a long base line in X. The third is

taken near the midpoint as a check. If

{Xx-\-X2) and (Y1 + Y2) are constant for these

stars, £=0, and the average sums are taken as

a and b, respectively. A nonzero « will introduce in {Yi-\-Y^) a marked linear change in

proportion to X. In this case, a larger number

of points (Yi+Y2, X{) is plotted to determine

the slope «. The X and Y sums are corrected

for c, and then a, b determined from the average

values of (X1-\-X2-{-€Y1) and (Y1 + Ya-€Xl),

respectively.

S. Choice of adopted system. The single train

system is taken as that in which the X coordinate of stars A, B, to Z increases positively.

Finally all star and train readings are transformed by equation (1).

Reduction to the concurrent meteor system

The precise reduction of the concurrent meteor

plates (see Whipple and Jacchia in the first

paper of this group) provides accurate geometric

and time coordinates of the meteor's flight in

space and as projected on the tangent plane of

the sky at the plate center. In order to relate

the train positions to the original trail position

and to the time of train formation, measurements in the common train system are transformed to the concurrent computed meteor

system, the coordinates of which are equal to

the original plate measure plus field corrections.

The coordinates of identical stars in the computed meteor system and hi the train system

define the transformation of X and Y coordinates for each train image. The difference

in the coordinate values for the image (t),

(X—Xt, Y—Yt) will be a nonlinear function in

Xt, with a possible additional linear term in the

Y transformation, proportional to the distance

of the star from the meteor trail. Although a

least-squares solution is possible, a graphical

representation is sufficient and much easier,

since (X—Xt) can readily be found graphically

to 0.01 mm and (F— Yt) to 0.003 mm. The

detailed procedure of this transformation is as

follows.

/. Transformation plots. Averages for the

two sets of star positions in the common system

are formed. For each image the differences

(X—Xi) and (Y— Yt) are formed and plotted

against Xt. It is preferable to plot these on

the same graph by using one abscissa scale and

introducing various ordinate scales, because the

X and Y transformations for all images have

the same general variation; this aids in the

curve drawing, especially when one or two star

positions are found to be inconsistent. The X

transformation function is well defined (to 0.01

mm by the above difference plots). In other

words, the scatter of points about a freehand

curve smoothed by a French curve is small. If

the scatter in (F—F<) is too large to define a

function to 0.003 mm, a linear correspondence

of the residuals from a first approximation function to the star's displacement from the trail

may be noted. Introduction of a linear term

in displacement from the trail should reduce

the scatter enough to allow a determination of

the transformation to within 0.003 mm.

2. The X, Y coordinates. The X, Y coordinates for both sets of tram measurements are

next transformed individually according to
228

SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS VOL. 1

the above difference functions. The transformed X is expressed to the nearest 0.01 mm

and Y to the nearest 0.001 mm.

Train displacements

The systematic error in the train measurements

due to personal equation depending on intensity

of the train image is now averaged out by

freehand curves drawn through the plots of

the train images in the meteor system. This

elimination is possible since points from both

the direct and reversed measurements will

have been plotted. Since the equation of the

trail is known, displacements of train images

from the trail at any position along the trail (X)

are readily computed. A more detailed description follows.

1. Plotting. Abscissa scales {X) for all

images should allow estimation of X coordinate

to 0.01 mm. Ordinate scales (F) for earlier

images (1-3) should allow estimation to 0.001

mm, but in later images the Y scale must be

compressed to facilitate curve drawing, because

the magnitude of the scatter in train measures

increases in the later exposures. However, the

percentage (scatter) generally remains rather

constant (10 percent).

2. Averaging. The two sets of train measures

will differ systematically from each other with

an increased separation in the later images, if

this separation has not already been eliminated

by the large scatter that results from the

extreme difficulty of measurement. A freehand

curve averaging the two sets of readings is

drawn to represent the most likely train

position. All curve variations consistent with

the two independent sets of measurements are

drawn regardless of the scales adopted in the

plotting. At this stage no attempt should be

made to correct for distortion introduced by

the emulsion and varying scales.

S. Displacement computation. The displacement of train image (i) from the meteor trail,

at a point X along the trail, may be defined

as AiY=Yi— YT, where the subscript Tdenotes

the trail. Since YT is a very slowly varying

linear function of X, AtY may be tabulated

directly by reading the smoothed plots of the

train images at appropriate intervals of X and

subtracting YT. Again, the X interval need

only be small enough to define the freehand

curve.

Epochs of exposure

Photometric measures of star exposures give the

relative exposure times for different images.

The total time of shutter opening during the

automatic program is indicated by the clock

record, and thus determines the actual duration

of the exposure. Clock readings and the

observations provide consistency checks. If

reasonable assumptions are made for the decay

function of train luminosity, the effective

epoch of exposure for a train image will not

deviate significantly from a simple average of

the times at the beginning and end of the

exposure. Since the time interval required to

shift the camera from one exposure to another

is negligible compared with the duration of

all exposures after the first, the effective time of

exposure of image (i) with respect to the

beginning of the first exposure is the sum of the

durations of the preceding exposures plus

one-half the duration of image (i). In order

to determine the time since formation of the

train, the time interval (T0) between the

beginning point (t=0) on the meteor's trail

and the beginning of the first exposure must be

known. The reduction of the photometric

data, consistency checks, and the determination

of r0 are carried out as follows:

1. Photometric determination of exposure duration:

(a) Given the magnitude differences between

successive images, the magnitude difference

(Am,) between image (i) and the first image is

obtained. Negative differences are taken conventionally to indicate greater brightness of

image (i) with respect to the first image.

(b) Relative exposure durations. If the exposure duration (d() is proportional to /<, then

the exposure time of image (i) relative to the

total exposure (program plus special exposure)

is given by the expressions (/J//i)/(S</1//1),

where the summation is over all exposures.

(c) As a rough check, the ratio of program

time to special exposure time from photometric

analysis and clock record should agree to better

than 40 percent.

(d) The relative time of exposure of image (i)

to the program time (P) is given by the expres-
Multiple photograph of meteor train of June 29-30, 1954, on film No. ST 5488 from Dona Ana, N. Mex., by 

REDUCTION METHODS FOR PHOTOGRAPHIC METEORS 229

sion: (/i/ZO/CZ^/Ji), where the summation is

only over the automatically programmed exposures.

(e) The program time (P) is read directly

from the clock record; the exposure duration is

the product of P and the expression for the

relative times given above. The exposure duration for the last exposure (special) is taken

directly from the clock record. The duration

times (dt) should be expressed to three significant figures, which generally means to 09

01 for

the early images.

2. Clock check for consistency. Time readings

as given on the clock tape (0.01 min.) are about

one order of magnitude less precise than

desired, but nevertheless they provide limits

to the possible range in the exposure durations

determined photometrically. If ±0*004

maximum error is allowed in each clock tapereading for beginning and ending of exposures,

the allowable values of dt may be determined,

and checked with the photometry for consistency. If disagreement occurs, the photometric values must be adjusted to fit those of

the clock in the following manner:

(a) A di and a beginning or ending time of

an exposure which are well substantiated by

the clock and by photometry should be sought.

As an example, if the clock records the time

exactly midway between successive hundredths

of a minute, both digits will be observed on the

tape straddling the proper position; i. e., the

dial for 0.01 minute intervals moves continuously. This increases the accuracy of the

clock so that it becomes comparable to that

of the photometry. If agreement for these

durations is good, the exposure time is well

substantiated.

(b) Small magnitude differences given photometrically are more accurate than large differences OO^) . Successive exposures with small

magnitude differences (^O^S) are used as

much as possible; dt+i then may be computed

from Sd—— 0.92 dt 8m where 8d and hm denote

small increments in duration and magnitude

respectively. This procedure allows simulation

of the clock tape record of beginning and

ending times. If di was well chosen, successive

exposure times computed from the small

difference equation should be consistent with

the clock readings.

(c) For a large magnitude difference, di+i is

computed from the expression,

If a computed end or beginning time of an exposure is not consistent with that of the clock,

the allowable clock time nearest to the photometric value is the best to assume.

(d) All program exposures can be adjusted

except possibly the first exposure. The magnitude differences between the first and second

image may be as large as 2m and consequently

subject to large errors. In this case, the clock

will generally be of no aid in fixing a lower limit

on dx, so one usually has no choice but to

assume the photometric value. If photometry

is also done on the train images the decay law

for its radiation may be found and the epochs

of exposure corrected accordingly.

(e) The validity of the adjusted durations

can be checked by comparison of the final

magnitude difference for the adopted times

with the original photometric values. Errors

of about O1^ are possible in large differences,

especially when star images are extremely

distorted, or when one star is excessively faint.

(f) The epoch of exposure with respect to the

beginning of the first exposure (T\) is given by

the expression («?!+<?!-}- . . . +</<_,+ 1/2*^).

S. Observational test for consistency of adopted

durations. Assume that the train displacements from the trail a t a given value of X are

directly proportional to the time elapsed since

the train was formed for all time intervals.

In the meteor system, f=0 is taken as the time

at the first measurable dash on the trail; the

time of meteor passage at all other points is

given by t(X). If T0 is the time between t=0

and the beginning of the first exposure, then

for an image (i) at point X, the time elapsed

since the train was formed (time of meteor

passage) is given by T O +TI—t(X), and consequently

&iY_ _

TO+T< t(X)

'X^V—A Y— ' ' '__' * v*5/

The ratio AtY/(AjY—AfY) is a linear function

of t(X). The determination of the slope provides an observational check on the exposure

epochs which, if successful, leads immediately
230 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

to an evaluation of TO from the intercept. Details of the observational test follow:

(a) Generally, only the first and second train

images are long enough to provide a sufficient

time base for a slope determination. A plot

of AiF/(A2F— AXF) against t(X) is required.

(b) The instability of the displacement ratio

due to the small values of AtY necessitates an

estimate of the ordinate error originating from

the scatter in the train measurements. If the

root mean-square scatter in Yt from the

smoothed train position for image (i) at X is

given by €it then the corresponding root meansquare scatter in the ratio A,F/(A2F— AiF) is:

(3)

When the distribution of the scatter magnitude

has been noted, the most reasonable slope for

the above linear function may be drawn. The

slope — (TJ—TJ)"1 gives di+d?. If the observational estimate is consistent (20 percent), the

clock-photometric durations are adopted as the

final values.

4- Epoch oj exposure with respect to first break

on trail. In order to decrease the percentage

errors in T\— T\ and AtY/(AjY—AiY), two widely

separated images are chosen to determine a

point on the linear function as follows:

(a) For i and j , the first and last images

respectively are usually used.

(b) A small range of t(X) in a region of welldefined train images giving A«F/(AyF—A<F) is

selected and its scatter computed. An averaged point AiF/(A,F— A,F), t(X0) may be

estimated graphically.

(c) If the adopted epochs are taken, then T0

is given by the equation

TO=(T>—U) I +t(X0). (4)

(d) For a rough check, T0 may be computed

from the (AXF/A2F—AXF, t) plot by use of the

slope from the adopted times. If many images

are observed, several pairings of first and later

images may be employed to find a weighted

average of T0.

(e) The epoch of exposure with respect to

t=0 for image (i) is given by T<=TO+T<.

(f) If the trail is present on the train film,

then TO is equal to the time of meteor passage

at the point along the trail where the train

ends and the trail begins.

Linear plate velocity and correction for

rotation

The displacement of the train normal to the

meteor trail AtF and the times since formation [TI—t(X)] determine the velocity on the

plate normal to the projected trail. Because

the stars are used as a reference system, a

spurious train motion is introduced corresponding to the westward diurnal motion of the

stars. The correction to the normal velocity

is generally large enough to be considered.

Corrections to the X coordinate of the measurement of A,F may be neglected if three or fewer

images exist, but they are otherwise required.

Details of plotting the corrected normal velocities are as follows:

1. Normal velocities. The normal train velocities (Vfn) for each image should be tabulated:

(5)

2. Correction for diurnal rotation. The detailed equations may be found in Appendix III.

Note that the time rates of change of the plate

coordinates X and F are proportional to the

time rate of change of the right ascension of the

corresponding spatial point of the train. A

westward motion of the reference system

increases the right ascension of the train point

spuriously, and hence its X and F points also.

The time during which a train point has been

tracking is simply the time elapsed since its

formation, namely r<—t(X). Let the rate of

change of X, Y due to rotation of the reference

system be Xr(X) and YT{X) respectively in

mm sec"1. Then the correction procedure is

as follows (the formulae are derived and given

in Appendix III):

(a) The correction to X,is —XT(X)[Ti—t(X)].

(b) The correction to F, is — YT(X)[r{—t(X)].

Since VNi is proportional to [T<—t(X)]~x, the

correction to the normal velocity is simply

— YT(X), a constant for all images.

(c) The magnitude of YT is about 0.007 mm

and varies linearly but slowly with X. The

magnitude of XT is about 0.01 mm and varies

quadratically in X.
REDUCTION METHODS FOR PHOTOGRAPHIC METEORS 231

(d) The tabulated correction of the normal

velocity is applied and the abscissa X corrected

and plotted in a single operation. Velocity

points for each image are plotted on one graph

with an indication of the scatter in VNi given

by ««/[r,-

Train motion components

Inspection of the plots of velocity on the plate

determines semiquantitatively whether one or

two components are defined. If the Vst agree

within the scatter of the measurements, a

single component of train motion is defined;

namely, that which is normal to the projected

meteor trail. In this case a freehand curve

may immediately be drawn through the points,

weighted inversely proportional to the scatter

in velocity for each train image. If the plots

of the train velocities have similar shapes and

show a systematic shift parallel to the trail,

then the train motion component that is

parallel to the projected meteor trail may be

found. Measurement of a normal displacement

of a train over a small range in X adequately

defines the displacement in Fa t the given X}

but a given displacement in Y defines X poorly.

Consequently, detection of a motion parallel

to the trail requires a large relative shift in

X (0.5 to 3 mm) over several images that form

distinctive shapes. Large relative shifts become increasingly apparent, the greater the

time since formation of the train; hence, a large

number of images (more than five) is required

for a solution for velocity parallel to the trail.

No parallel motion will be apparent from the

scatter of the normal velocity component for

two-image to four-image trains. The procedure

for defining the parallel component VP{X) is

as follows:

1. Displacements parallel to trail. Displacements in X between images widely separated

in time are measured from the normal velocity

plots at those velocities for which the normal

curve has a non-zero slope.

2. Determination of parallel velocity. The

parallel plate velocity component (VP) at a

given normal velocity (VN) is computed from

the X displacement between images j and k

by the equation:

(6)

The determinations of VP from various image

pairings are plotted against VN, and a freehand

smooth curve is drawn through the points which

are weighted roughly in proportion to the time

base (TJ.—T)). The scatter will be considerably

larger than for the normal velocity component,

but a general agreement on the shape of the

function must exist in order to establish VP.

No systematic deviations should be observed

from any highly weighted image pairings.

8. Correction of normal velocity for parallel

displacement. A normal velocity, VKi(X), is

now plotted on a separate graph at the abscissae

[X— VP (VN)[(Tt—t (X)]}. Enough points in V*

should be used to prevent introduction of any

systematic error in the definition of the transposed curves other than the abscissa correction.

4- Final normal velocity component. After

the parallel velocity correction, all train images

occur at the point in X along the trail where the

train was originally formed. A smooth curve

is drawn as in the case of a single component

solution to define V#(X).

5. Plot of parallel velocity component. The

normal velocity as a function of X, V^iX), is

defined by the smoothed curve in section 4,

which in turn defines VP from section 2 above.

The tabulation of (X, VP) is plotted with X as

the independent variable.

6. The accuracy of the results. Steps 1-5 may

be considered as the first step in successive approximations for VN(X) and VP(X), since the

times used to compute V^< were taken at points

along the trail where the train image was observed rather than at the points where the train

was actually formed. But t(X) enters as

Tfc—t(X), where X here is the point at which

the train was observed. Since

\t(X)-t(X-VP(X)[rk-t(X)})\<^rt, (7)

the first approximation to the normal and

parallel velocity components is entirely satisfactory.

Coordination of velocity components from

two stations

Double-station observation of the train motion

requires the matching of the X coordinates in

the two meteor systems according to a common
232 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS VOL. 1

height above sea level (H) in the following

manner:

Let subscripts A and B refer to the Mayhill

and Sacramento Peak Stations respectively.

The distance (D) along the trail from the

first measurable dash at one station, say Mayhill, must be computed from the equation,

DA = 100KA(X-XAa)/(X-XAR), (8)

where KA, XAa, XAR are constants provided in

the meteor reduction. At Harvard these appear on the reduction sheets of the Meteor

Analysis Project. See the notation list in Appendix V for more detailed definitions.

The height of the point XA above the plane

tangent to the earth at one station, e. g., Mayhill, is given by the equation,

hA=h(DA=0)-DA. cosZA> (9)

where h(DA=Q) and the zenith angle ZA are

given quantities.

From the AhA versus hA graph, the height

above sea level (HA) is given by hA+AhA

The mated meteor systems are arranged so

that points with the same D have the same

height; therefore the XB corresponding to XA

is given by the equation

D(XA)XBR-100KBXB

D(XA)-l00KB

(10)

where the subscript B denotes the second station, e. g., Sacramento Peak.

Intervals in height should be small enough to

define the projected velocity curves. A table

of D, XA, XB, and the respective normal and

parallel velocity components versus H as independent variable may then be prepared.

Spatial components of train motion

The plate velocities normal and parallel to the

trail are readily transformed to two spatial

components normal and parallel to the meteor

trail as projected on a plane perpendicular to

the line of sight. These velocities in turn may

be expressed in terms of the three components

of velocity of the train. These components

are taken parallel to the east, north, and zenith

directions from the midpoint of the observed

train region. Since the X axis is parallel to the

projected meteor trail, a systematic motion of

the train along the trail will not influence the

component normal to the meteor trail in space.

Thus this normal component is indicative only

of air motion flowing transversely to the meteor

trail. When the component normal to the projection of the meteor trail is plotted against

distance along the trail, large variations in amplitude occur over ranges in distance along the

trail of the order of 10 kilometers. Smaller

variations in amplitude over a considerably

smaller distance scale are superimposed on

these. A smooth curve is drawn through the

large-scale fluctuations and the smaller fluctuations are smoothed out. This curve defines

the component of the mean wind field normal

to the projected trail. The difference between

the two curves represents the component of

turbulent motion. Because the parallel velocity component is much less accurately measured than the normal component, only the

latter can be used to define the small-scale

velocity variations.

Observation of the two normal and parallel

velocity components at both stations defines

three linear equations in terms of the unknown

train motion components (TB, TN, Tz). The

solution of the simultaneous equations for T

is not singularly significant for several reasons.

First, the combining of the less accurate parallel

function with the more accurate normal function decreases the accuracy of all the train

components. Second, the relatively small base

of triangulation makes the determinant of the

coefficients of (Tg, TN, Tz) nearly zero. This

results in a solution in which the errors inherent

in the train measurements are noticeable fractions of the derived train-velocity components.

Third, in the first double-station train reduction attempted, an observed systematic motion

of the train along the trail had a vertical component six times the maximum value (~.r

)ins~I)

allowable for vertical air transport in the thermally stable region above about 80 kilometers.

The theory of the persistence of meteoric luminescence and the momentum exchanges from

meteor to air particles can explain such a

coasting train-velocity, but has not been developed to a degree of accuracy such that the

parallel component functions can be corrected

to represent only the mean wind projection.
REDUCTION METHODS FOR PHOTOGRAPHIC METEORS 233

The analysis is therefore aimed at investigating

the existence of a coasting train-velocity from

the solutions for T after the mean wind field

has been defined by the normal components

alone.

The reasonable assumption is made that the

vertical component of the mean wind (W) is

very small compared to the horizontal component, or Wz=0. The horizontal mean-wind

components, WB and WN, are computed from

the smoothed spatial component normal to the

projected trail as observed at both stations. It

is to be noted that, under certain geometrical

conditions, some small variations on the plate

will become greatly enlarged when projected on

the horizontal plane. The reduction of T by

the mean-wind solution leaves the train components (TB—WB, TN—WN, TZ) independent of

any mean air motion. The only remaining

wind contribution is a fluctuating component

due to turbulence. Since the random component fluctuates about a zero point, each component is smoothed as in the procedure of

separating the mean wind component from the

normal wind component. If the resulting

smooth curves actually can be regarded as

representing a coasting velocity along the

meteor trail, then the direction cosines of the

velocity vector will coincide with the direction

of meteor flight at all heights.

Deriration of normal and parallel spatial

velocity components

1. Basic linear equations. For a general point

at a distance R from the observing station and

with direction cosines (I, m, n), equation (11)

gives the components of the point with respect

to axes parallel to those of the meteor system

(see Appendix I):

r\—Rm (11)

Because the train velocity components

(ms"1), are very small (10~3) compared to R,

the time rates of change in the coordinates of a

train point may be represented as the projection

of T(H) on the respective axes of the meteor

system:

, + TNlN+ Tzlz (12a)

ii=Rm+mR=TBmB+TNmN+Tzmz (12b)

(12c)

where (I, m, n) B, #, z are direction cosines of

the train coordinate axes in the meteor system

(see Appendix II).

2. Linear expression jor rate of change of

distance. The distance by equations (11) is,

and by equations (11), R=l£-{-

or,

m (TBmB-\- TNmN-{- Tzmz) +

n(TBnB+TNnN+Tznz). (13)

3. Distance as a function of position on the

plate. If Rt is the minimum distance to the

trail, RJR is the cosine of the included angle,

or the scalar product of the direction cosines of

R« and R expressed in any system. In the local

system (see Appendix I) Rt/R is given by equation (14) where the subscript (e) denotes the

point of minimum distance:

(14)

and

from equation A(2)3 (Appendix I). From equation A(9), xxt+yy, = jf,+ljfljf

 where x, y

are dimensionless as defined in Appendix I.

Y may be taken as the same value used in the

rotation correction formulae; Xt is defined by

formulae (8) and (10) for D=D, (see Appendix

IV). Also,

By equations A(2) (Appendix I),

and from equation A(9) (Appendix I),

(15)

Substitution of these expressions in equation

(14) gives:

l+xx.+yy.

4. Rates of change of direction cosines as functions of position on the plate. From Appendix

• Equation numbers preceded by the letter "A" indicate formulae appearing in the appendices.
234 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

I, x=l(n, y=m/n, and n=f'=

by equation (15). Straightforward differentiation with time gives:

I =n[(l—n

2

x*)x—n2xyy],

m=n[—n2xyx+ (l—n2y

2

)y],

n=—n

3

[xx+yy].

(17)

5. Equations jor velocities on the plate as junctions oj components of velocities oj the train.

Direct substitution into equation (12) of A

from equation (13), R from equation (16), and

(I, m, n) from equation (17) gives three nonindependent equations for T(H). Multiplication of expression (12c) by x and subtraction

from expression (12a), and multiplication of

expression (12c) by y and subtraction from

expression (12b) reduces the three expressions

to equations (18a) and (18b), respectively:

(ls~nBx) TB+(lN-nNx) 7>+

(mB—nBy) TB+{mN—nNy) TN

Here x= VP/f and y = V^//have units of inverse

seconds; if R, is in units of meters, the train

VlB— nEx

VmB~nBy mN—nNy

VmE--nEy

La+2)*

mN-nNy~\

J

7. Formulae jor mean horizontal wind. If

Wz—0 is assumed, then the simultaneous solution of expressions for the normal train component defines the mean wind, where the

smoothed normal components (A0 of the mean

wind motion are used:

L

+ L d+y 2

) ^ J f l ) J

(21)

8. Formulae jor investigation oj coasting velocity. As functions of height, the train meteor

velocities are expressed in meters per second.

The left-hand sides of expressions (18a) and

(18b) are similar to the scalar products of T(fl)

and the vector parallel and normal to the projected trail, respectively. Hence, normalization of the vectors in equation (18a) by the

factor (1-f-a;2)"^ and of the vector in equation

(18b) by the factor (1+y2)"** represents the

spatial components P(H) and N(H) corresponding to the plate components VP and VN, respectively, or

and

where

(19)

N(H) and P(H) are directly proportional to

Vu and VP, Rt, and a slowly varying function

6. Formulae jor components oj velocity oj train.

The normal component expressions (18b) from

both stations are combined with one parallel

component of the form of expression (18a) to

give three simultaneous equations in three unknowns. Two solutions for T(H) are obtained

by taking the parallel component from each of

the two stations individually. As an example,

for Station A,

pz-nzxl T =p ff

(20)

components corrected for the mean air motion

are: (Tg-Wg, 7V-W^, Tz). The magnitude and

direction cosines of the motion are readily

determined. If (XB, nR, vR) are the direction

cosines of the meteor radiant, then the velocity

component is truly a coasting train motion if

the condition (22) is satisfied at all heights,

where Vc is the magnitude of the above vector:

Vmz—nzy\ T _

Ui+y^L2

\mz—nzy\ T _

— WE= — VC(XflXjy "f HRHE + VRVE),

Tz= —

(22)

+ HRUZ + vRvz).
REDUCTION METHODS FOR PHOTOGRAPHIC METEORS 235

The bar denotes smoothed expressions; X*, etc.,

are the equatorial direction cosines of the east,

north, and zenith points at the midpoint of the

observed train region.

Outline of computational steps with

formulae

Following is a convenient outline of the steps in

reducing a meteor train with the numbers of

the appropriate formulae.

1. Transformation constants: A(l).

2. Direction cosines of train system in meteor system:

3. Plate coordinates, x, y, xt, y, in dimensionless form:

A(20), A(6), (8), (10).

4. Minimum distance between trail and station: A(20),

5. Tabulation of velocity components relative to trail

atff: (19).

6. Tabulation of coefficients in solution for T: (20).

7. Computation of total train motion components: (20).

8. Computation of mean wind components: (21).

9. Analysis of coasting velocity: (22).

Appendix I: Coordinate transformation

formulae

1. Equatorial system. Direction cosines are

given by (\, A,) „).

2. Local system. The equatorial system is

transformed so that the vertical axis is parallel

to the direction of the plate center and the

other axes are directed towards the east and

north points. Direction cosines (X, n, v) are

transformed to (£', rj', f') by equation A(l),

where the subscript (e) denotes the plate center.

where 72+i»J=l.

S. Meteor system. The spherical SuperSchmidt film is projected on the tangent plane

of the sky at the point of the plate center.

The £', V axes, toward the east and north

points respectively, become the standard coordinate axes £, rj as given by equations A (2).

A(2 )

A(2 )

The X, Y axes in the tangent plane are taken

so that the X axis is parallel to the projected

meteor trail; they represent a rotation and

translation from the £, lj axes except for additional plate distortion effects. The plate

constants at, av, etc., are given at Harvard on

the reduction sheets for the meteor.

or sufficiently,

where,

X aJ+brH-c,, 1

==a/

^-\-b'rj-\-cx, I

. _ ;_ y

==b £— a ri~\~Cyt J

a'=2 (a,— by),

b'=\{bx+av).

A(3)

A(4)

A(5)

Let the nondimensional quantities x, y be

defined as follows:

x=

X—cx "

/ A(6)

Here / denotes the focal length of the camera

and is given by

/=(a"+&")K. A(7)

The corresponding nondimensional plate constants are

a=a'ljy 6=67/- A(8)

Substitution into equation A (4) simpUfies the

form as shown in equation A(9),

W \b-a)\v),

A(9)

where a 8+6*=l. If (/, m, n) are the direction

cosines of a point in the meteor system, then

the dimensionless x, y, are equal to l/n and

m/n respectively. Substitution of the expressions for x, y, and J, fl into equation A(9) gives

the direction cosines in the meteor system:

W \b-a)\tj A(10)
236 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

The vertical component (n) is invariant under a

rotation about the vertical axis.

Appendix II: Direction cosines of train

coordinate axes in the meteor system

The train coordinate system has its origin near

the midpoint of the train region with axes

directed toward the east, north, and zenith

points at the midpoint. Let (I, m, n) B, N, z,

respectively represent the direction cosines of

these vectors in the meteor system. Computational formulae are outlined below.

/. Equatorial direction cosines of the zenith.

(a) Given the equatorial direction (XOT, nm, vm)

of the midpoint of the train, and its distance

Rm from a station, the rectangular coordinates

from the station are:

Sm—

(b) Referred to a geocentric system, equations A(ll) become:

7=(p f cos <t>s) cos ds+Sm,*

5=(p, cos 4>,) sin 6,+rim,

€=p, sin 0,—{-|*m,

A(12)

where p, is the geocentric distance of the

station, <f>t is its geocentric latitude and 6, is tbe

local sidereal time. Equations A(13) represent

the equatorial direction cosines corresponding

to (y, 8, e).

A(13)

where pm=(ys-\-82+t2)X is the geocentric distance of the midpoint.

2. Equatorial direction cosines of east and

north points.

(a) Given the equatorial coordinates of the

zenith at the midpoint of the train as (azm,

5«m)> the east and north points are in the

directions toward (a^+x/2, 0) and (azm±ir,

ir/2—Stm), respectively.

(b) Let 7 m =cos 8,m, whence y\m+vin=\.

Then, for the east point:

A(14)

and for the north point,

A(15)

(c) As a check, X1X>+/i<^+vJ^=5j, where 5}=0

for i^j and 5j=l for i—j.

3. Direction cosines on the meteor system. The

equatorial coordinates of the east, north, and

zenith points of the train system are transformed by equations A(l) and A(10) to

(/, m, n)K,ifz. As a check:

A(16)

Appendix III: Correction for rotation

For a given point on the train with plate coordinates (X, Y) and equatorial coordinates

(a, 5), (XT, YT), the change in (X, Y) from the

increase in the right ascension, due to the westward motion of the reference system, is desired.

The X, Y coordinates are dimensional [mm] as

measured. From equation A(3),

A(17)

i, rj are functions of i', rj' and f' by equations

A(2). Differentiation of equation A(l) with

respect to time for the point X=cos 5 cos a,

/i=cos 8 sin a, v=sin 5 and straightforward substitution into the expressions for XT, YT define

the spurious plate velocities by equations A (18),

Xr=a[axyc+ vc(—axrj-\- b

a=7.272X10-6[s-1]

e\{X—

— Cy)],

A(18)
REDUCTION METHODS FOR PHOTOGRAPHIC METEORS 237

where £, ij, derived from the inverse of equation

A (3), are given by

A(19)

The inverse plate constants A%, B$, etc., are

given at Harvard on the reduction sheets. Y

may be taken as a constant equal to a mean

position of the observed train images. The

differential expression of YT for a change in Y

indicates that XT and YT are insensitive to a

change in Y of about one half millimeter. One

may even choose Y— YT, the trail ordinate.

Appendix IV: Minimum distance to meteor

trail from station

On the reduction sheets of the meteor as observed from Station A, the direction cosines of

the point t=0 are given by \Aa, nAa. vAa. Let

the direction of the radiant be \R, nR, vR; then

the distance along the trail from £=0 to the

point of minimum distance is given by the

equation,

DAt=100 A(20)

where RAa is the distance in units of 100 km

from Station A to the point for t~0. Then

A(21)

where RAt is in units of 100 km. Similar formulae hold for Station B.

Appendix V: Notation

The following symbols and units have been

arranged in an order roughly corresponding to

their use in train analysis. It is understood

that each symbol is applicable at either station

and may carry a subscript (for the Harvard

Meteor Expedition, vl=Mayhill, i?=Sacramento Peak) when distinction is necessary.

A and B are carefully indicated on the meteor

reduction sheets.

X, Y—Meteor plate system coordinates [mm].

YT—Ordinate of meteor trail in meteor plate system

[mm].

AiY—Train displacement of image i from trail [mm].

di—Duration of exposure i [s].

T{—Epoch of exposure (i) with respect to beginning of

the first train exposure [s].

TO—Time interval between (=0 and the beginning of

the first train exposure [s].

T,—Epoch of exposure (i) with respect to t=0 [s].

t(X)—Time of meteor passage at a general point on

the trail [s].

Ox, bx, cx—Plate constants for transformation of standard coordinates to X, Y system [mm].

|, ij—Standard coordinates.

I', ij', ?'—Direction cosines in local system.

X, p, p—Direction cosines in equatorial system.

Xe, Me> **—Direction cosines of plate center.

Xs, M«I "R—Direction cosines of radiant.

it—Time rate of change of hour angle due to the earth's

rotation [s-1].

V,—Normal train velocity component from image (i)

[mms-'J.

Vp—Plate velocity component of train parallel to the

trail [mms-1].

VN—Plate velocity component of train normal to the

trail and corrected for Vp [mm-1].

Xa—Position of <=0 in meteor system [mm].

Xs—Position of the radiant in the meteor system [mm].

ZR—Zenith distance of radiant.

D—Distance along the trail from <=0 [km].

h—Height above tangent plate [km].

H—Height above sea level [km].

R—Distance to trail from station [km].

a—Subscript denotes first measurable dash.

t—Subscript denotes point of minimum distance from

station.

m—Subscript denotes midpoint of train region.

R—Subscript denotes meteor radiant.

I, m, n—Direction cosines in the meteor system for a

general point.

(I, m, n)g, if, z—Direction consines in the meteor system

for east, north and zenith axes of the train system

respectively.

x, y—Dimensionless plate coordinates.

TE, TN, TZ—Train velocity components in the train

system [ms-1].

WK, WN—Horizontal components of the mean wind

field [ms-'].

N(H)—Spatial component of T(H) normal to the projected meteor trail [ms-1

].

P(H)—Spatial component of T parallel to the projected meteor trail [ms-1

].

N—Smoothed functions are denoted by a bar.

v—Coasting velocity [ms-1].

References

LILLER, WILLIAM, AND WHIPPLE, F. L.

1954. In Boyd and Seaton, eds. Rocket exploration of the upper atmosphere, pp.

112-130. Pergamon Press.

PAYNE-GAPOSCHKIN, C.

1937. Ann. Harvard Obs., vol. 89, No. 8.

WHIPPLE, F. L.

1953. Journ. Meteor., vol. 10, p. 390.

Methods for the Study of Shower Radiants

from Photographic Meteor Trails1

Fred L. Whipple2 and Frances W. Wright3

The techniques described here were developed

for the rapid analysis of single-station photographic meteor trails used in determining the

nature of shower radiants. The measurements

are made graphically on Durchmusterung star

charts; the reductions have been simplified

as far as possible without sacrifice of significant

accuracy. Continued application of the methods to many of the trails photographed by the

Baker-Super-Schmidt meteor cameras has led

to improvement and elaboration of the techniques described earlier (Whipple, Proc. Amer.

Philos. Soc, vol. 91, No. 2, p. 189, 1947). We

present the method in some detail.

Measurement

To measure single-station meteors, we plot each

photographic meteor trail carefully, with the

aid of a magnifying glass, on a large-scale Bonn,

Cordoba, or Cape Photographic Durchmusterung chart, according to the region of the

meteor's occurrence. Through the observed

points, with as much care as possible, we pass

a smooth curve which represents closely a great

circle; the chart scale is approximately 1° to

2 cm. For convenience and greater accuracy

we select five points that are nearly equally

spaced on the smooth trail, at intersections of

the trail with hour or declination circles. After

making visual measurements, under a magnifier, of distances on the chart, we determine

the equatorial coordinates of these five points.

Then, using formulae of analytic geometry, we

derive the direction cosines, given below, of the

pole of the great circle through four of the

points selected above (the mean for points 1-4

and 2-5); the fifth central point is used for a

check. The probable error of the independently measured point from the great circle

through the plotted points is approximately 30

seconds of arc. The precision of plotting and

measurement exceeds the intrinsic accuracy

with which shower meteors usually emanate

from the mean radiant at a given instant of

time. Hence no greater precision is required

in practice.

Reduction of measurement of meteor trail

The direction cosines, X, n, v, for each of the

above points are given by the formulae,

X=cos a cos S

M=sin a cos 5

y=sin S.

(1)

1

 Carried out in part under research contracts with the U. S. Naval Bureau of Ordnance (Task I of NOrd 10449) and the U. S. Office of Naval Research

(Contract No. N5orl-07647). Reproduction in whole or in part is permitted for any purposes of the U. S. Government.

• Smithsonian Astrophysical Observatory.

' Harvard College Observatory.

239
240 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

The direction cosines X,, nP, v, of the pole of

the great circle through any two points a and b

follow from the expressions:

X, sin Cab=iiavb—vanb

Hj, sin Cttb=va\b— \avb

vp sin Cab=Knb—

(2)

where Cab is the great circle distance between

the points a and b. The square of sin Cab is

obtained from the sum of the squares of the

right-hand members of equations (2).

Analysis of a meteor shower

When we begin the identification and analysis

of shower meteors we use, when possible, a

"skeleton" path of the radiant from the photographic double-station meteors with accurate

radiants, velocities, and orbital elements. These

positions, elements, and velocities—all of which

are similar for any one shower—then furnish

criteria for membership in any particular

shower.

First, we eliminate as possible members of

the stream all extended meteor trails more than

10 degrees from the appropriate point along the

skeleton path of the moving radiant. This

elimination is easily made by quick plotting on

a large globe, or by calculation in case the trail

is short and distant from the assumed radiant.

Second, the apparent velocity of these accurate

double-station meteors fixes the mean velocity

in any one shower, a second criterion of membership. To find velocities of other doublestation meteors, we use the extremely rapid

approximation developed by McCrosky (in the

third paper of this group)—the so-called "whirlwind" reduction technique. This same technique may also be used to determine approximate velocities for single-station trails with

shutter breaks. We derive the velocity from

quick measurement of breaks in the trail on the

basis of an assumed radiant. If the approximate velocity of a possible member of a shower

differs by more than 15 percent from the mean

velocity determined by the precise techniques,

we then eliminate the suspected member.

When trails of double-station meteors have not

been measured and reduced precisely, we measure and reduce them by the rapid techniques

used for a single-station meteor. Even though

such reductions may lead to fairly accurate

radiants, we usually prefer to treat the two trails

of these doubles as two single trails in the final

least-squares solution.

Assumed radiant and pole oj radiant motion.

The direction cosines of all radiants or poles of

trial trails must be reduced to a common

equinox. We recommend the method presented by Bower (Lick Obs. Bull., vol. 16,

No. 445, p. 34, 1932). We then adopt an

assumed preliminary radiant, usually the corrected radiant, aR, SB, of the accurate doublestation meteor nearest to the mean position of

the radiant as shown by these accurate radiants.

The direction cosines \PM, nPU, vPM of the preliminary pole of radiant motion can be obtained by choosing for a second point along the

great circle of radiant motion one of the earlier

(or later) accurate radiants. When the circle

of radiant motion has a small inclination to the

celestial equator we often take, for convenience,

the second point at the same declination as SB.

Next, we establish a rectangular coordinate

system with origin at aR, SB and z-axis coincident with the great circle of motion of the

radiant, x increasing with a and y with 6.

Formulae jor distance. The position of a

single trail j can be specified conveniently by

the intersection point Xj of the extended trail

with the x-axis, and by the intersection angle

yf/} which is taken from 0° to 180° as indicated

in figure 1. For each single trail j, where X^,

MP/, vPj denote the direction cosines of the pole

of the trail, we calculate these positional quantities from the equations

Xj sin ^=XP ,XB+HPJUR+VPJVR, (3)

COS ^j — \pj\pM+ flpjflpM + VpjVpM, (4)

sin $, (from trigonometric tables), (5)

X) sin tyt (in minutes of arc) follows from

(3) and X'} (in minutes of arc) and ^ from

(5) and (4) respectively.

Signs and the quadrant of ^ are determined

from the geometry of the situation for each case.

Zenith corrections. Xj sin ^ gives the nearest point of each trail j to the assumed radiant

of the shower, and corrections for both diurnal
REDUCTION METHODS FOR PHOTOGRAPHIC METEORS 241

Assumed Great Circle

of Radiant Motion

Preliminary

Assumed Radiant

FIGURE 1.—Geometry of single trail and radiant.

motion and zenith attractions must be applied

to correct the position of this point. When we

have approximately 30 or more members of a

meteor shower, we make up tables and graphical curves of corrections which we can then

enter to find the proper corrections for an individual meteor. If the shower has fewer members, we usually apply each correction separately, without making tables. The basic

formulae used for zenith correction are

and

where

(6)

(7)

Vm = Velocity of meteor outside the atmosphere.

Vc=Vm corrected for rotational motion.

VQ== VC corrected for zenith attraction.

9=Attraction of gravity (the term introduced by the

rotation of the earth is subtracted) at the station.

p=Radius of the earth.

ZR — Zenith distance of an assumed radiant that approximates the mean value of the apparent radiant.

If there is no correlation between time and

velocity, we adopt the mean Vm and the mean

Vo of the accurate double-station meteors. We

find the mean Vc from equation (6).

The zenith distance, ZR, may be obtained

quickly from the navigation tables of the U. S.

Hydrographic Office, H. O. 214. In these

tables declination arguments in whole and half

degrees head the main columns of each page,

while hour-angle arguments in whole degrees

appear at the side; we enter the tables, therefore, with different values of hour angle and

with a declination as close to the radiant as

possible. Then with the interpolation tables of

H. O. 214, we find the altitude of the radiant

and hence the zenith distance ZB for any hour

angle of the radiant at any specific station.

The interpolation tables are convenient when

the latitude of the station is not a whole number of degrees.

A third formula determines, by means of the

sine law, the sine of the angle v at the "radiant

vertex" in the pole-zenith-radiant triangle:

sin i>=cos iXsin (H. A.)/sin ZB. (8)

From formulas (7) and (8), the relations

Aa cos S=AZB sin v (9)

and

AS=AZR COS v (10)

give the zenith corrections in Ax and Ay for any

one trail. The signs of these corrections appear

from the geometry of the situation in each case.

Diurnal-motion corrections. The diurnal-motion corrections require a knowledge of the

station's rotational velocity Vs, given by the

equation

Vs=

2 T

86,164

p COS <f>'. (11)
242 SMITHSONIAN CONTRIBUTIONS TO ASTROPHYSICS

where p and 4>' are, respectively, the radius of

the earth and the geocentric latitude of the

station; time is measured in seconds of mean

solar time. We have used the following specific

values of Vs for various Harvard Meteor Stations :

Station

Cambridge, Mass. 1

Agassiz Station, Mass.]

Arequipa, South America

Bloemfontein, South Africa

Dofia Ana, N. Mex.l

Soledad, N. Mex. J

V, (km/sec)

0.344

0.447

0.406

0.392

The following equations in Ax and Ay suffice

for the diurnal-motion correction:

Aa cos S=Aa sin 0+Ba cos 6, (12)

Ah=At sin 6+Bt cos 0, (13)

where 0 is the local sidereal time and the coefficients are functions of the velocity Vm and

of the position of the radiant aR and 5B (which

can be the assumed mean radiant for singlestation meteors). The above coefficients are

given by the formulae

Aa= — (VslVJ) sin aB,

Ba=-(Vs/V-)co8aB,

At= — (Vs/V*,) cos aR sin 5B,

B,= + (Vs/Vm) sin aB sin SB. (14)

Finally, the sums of (9) and (12) and of (10)

and (13), respectively, provide the total corrections for zenith correction and diurnal motion.

The corrections may range from only a minute

of arc to several degrees, depending largely upon

the velocities of the meteors and the altitudes

of the radiants for the respective showers.

Corrected (X} sin ^j). The corrected quantity, X) sin tyj, designated by A}

, is given by the

formula

AJ=(X) sin sin

c o s

If the path of radiant motion roughly parallels the celestial equator, Ax}— total Aa}

 cos it\

and Ayj=total Adj with sufficient accuracy.

General formulae for Ax} and Ay}. When, on

the other hand, the inclination (smallest angle)

fi of the path of radiant motion to the celestial

equator is large over a long time interval, more

accurate formulae are

Axj=\Aaj\ cos Sj cos sin 0, (16)

and

COS 8J sin /S+|A5j COS /3. (17)

Formulae for double-station meteors. The date

of the meteor is taken as tt for a double-station,

or as tj for a single-station meteor, and is measured in degrees of true longitude of the sun at

the time of observation and referred to a fixed

equinox. The daily motion of the radiant can

now be determined for double-station meteors

from the observational equations

Az+Bzti=xt-\-Axi and A,+BtU=yt+6yti (18)

where the coefficients Ax, Bx, A9) and Bv are to

be found by least squares.

For any radiant Rt, xt and yt can be computed

from the formulae

and

(19)

sin xi=\pxXBi-\-npInBi+vPxvBh (20)

where Px is the pole of a great circle though the

assumed radiant orthogonal to the great circle

of radiant motion. When the path of radiant

motion is nearly parallel to the celestial equator,

we take, for convenience, aj.^as+90 0 and

SPx=0} with sufficient accuracy.

For single-station meteors, the distance D of

the calculated radiant xc), yti from the great

circle of the meteor trail is to be minimized by

a least-squares solution; i. e., D} is to be considered as a final residual. As seen from figure 1,

Dj is given by the equation,

sin cos

(21)

A substitution of the corrected xt and y< from

equations (18) into equation (21) leads to the

following equations of condition for a leastsquares solution to determine the six constants

and to minimize Dt:

sin fj+A, cos

=XJ sin

sin Vj+Jtfytj cos Yj

sin yj/j+Ay, cos ^ . (22)
NO. 2 REDUCTION METHODS FOR PHOTOGRAPHIC METEORS 24 3

Least-squares equations of condition. The co- the radiant motions over many years can easily

efficients Ax and Ay, after the least-squares be introduced into the least-squares solution,

solution, provide corrections to be applied to No such effect has yet become detectable bethe assumed radiant, while Bx and By provide cause of the limited span of precise photothe daily motion of the best radiant path graphic observations. When such secular efthrough the calculated radiant, with minimized fects become measurable, most investigators

distances of extended meteor trails to the mean would probably prefer to divide the material

radiant of the corresponding instant. into groups of about two or more epochs and

Terms to include possible secular changes in investigate each group separately.