A UNITED STATES DEPARTMENT OF COMMERCE

PUBLICATION

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NBS TECHNICAL NOTE 638

*'«ril tX

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U.S. EPARTMENT

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COMMERCE

National au

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A SYNCHRONOUS SATELLITE

TIME DELAY COMPUTER
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards 1 was established by an act of Congress March 3, 1901. The Bureau's overall goal is to strengthen and advance the Nation's science and technology and facilitate their effective application for public benefit. To this end, the Bureau conducts

research and provides: (1) a basis for the Nation's physical measurement system, (2) scientific and technological services for industry and government, (3) a technical basis for equity in trade, and (4) technical services to promote public safety. The Bureau consists of the Institute for Basic Standards, the Institute for Materials Research, the Institute for Applied Technology,

the Institute for Computer Sciences and Technology, and the Office for Information Programs. THE INSTITUTE FOR BASIC STANDARDS provides the central basis within the United

States of a complete and consistent system of physical measurement; coordinates that system

with measurement systems of other nations; and furnishes essential services leading to accurate and uniform physical measurements throughout the Nation's scientific community, industry, and commerce. The Institute consists of a Center for Radiation Research, an Office of Measurement Services and the following divisions: Applied Mathematics — Electricity — Mechanics — Heat — Optical Physics — Nuclear

Sciences 2 — Applied Radiation 2 — Quantum Electronics " — Electromagnetics 3 — Time

and Frequency 3 — Laboratory Astrophysics 3 — Cryogenics 3

. THE INSTITUTE FOR MATERIALS RESEARCH conducts materials research leading to improved methods of measurement, standards, and data on the properties of well-characterized

materials meeded by industry, commerce, educational institutions, and Government; provides

advisory and research services to other Government agencies; and develops, produces, and

distributes standard reference materials. The Institute consists of the Office of Standard Reference Materials and the following divisions: Analytical Chemistry — Polymers — Metallurgy — Inorganic Materials — Reactor

Radiation — Physical Chemistry.

THE INSTITUTE FOR APPLIED TECHNOLOGY provides technical services to promote

the use of available technology and to facilitate technological innovation in industry and Government; cooperates with public and private organizations leading to the development of

technological standards (including mandatory safety standards), codes and methods of test; and provides technical advice and services to Government agencies upon request. The Institute consists of a Center for Building Technology and the following divisions and offices: Engineering and Product Standards — Weights and Measures — Invention and Innovation — Product Evaluation Technology — Electronic Technology — Technical Analysis — Measurement Engineering — Structures, Materials, and Life Safety * — Building Environment * — Technical Evaluation and Application * — Fire Technology.

THE INSTITUTE FOR COMPUTER SCIENCES AND TECHNOLOGY conducts research and provides technical services designed to aid Government agencies in improving cost effec- tiveness in the conduct of their programs through the selection, acquisition, and effective

utilization of automatic data processing equipment; and serves as the principal focus within

the executive branch for the development of Federal standards for automatic data processing equipment, techniques, and computer languages. The Center consists of the following offices and divisions: Information Processing Standards — Computer Information — Computer Services —

Systems Development — Information Processing Technology.

THE OFFICE FOR INFORMATION PROGRAMS promotes optimum dissemination and

accessibility of scientific information generated within NBS and other agencies of the Federal Government; promotes the development of the National Standard Reference Data System and

a system of information analysis centers dealing with the broader aspects of the National Measurement System; provides appropriate services to ensure that the NBS staff has optimum

accessibility to the scientific information of the world. The Office consists of the following

organizational units: Office of Standard Reference Data — Office of Technical Information and Publications —

Library — Office of International Relations.

1 Headquarters and Laboratories at Gaithersburg, Maryland, unless otherwise noted; mailing address Washington, D.C. 20234.

a Part of the Center for Radiation Research.

3 Located at Boulder, Colorado 80302.

* Part of the Center for Building Technology.
NOV 7 1973

/GO

V5 ^53

19' A Synchronous Satellite Time Delay Computer

Wallace F. Hamilton

D. Wayne Hanson

Time and Frequency Divison

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

£ fech

NBS Technical notes are designed to supplement the

Bureau's regular publications program. They provide

a means for making available scientific data that are of transient or limited interest. Technical Notes may

be listed or referred to in the open literature.

**-T

,!*

*"»£«„ XOf

U.S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts, Director

Issued July 1973
National Bureau of Standards Technical Note 638

Nat. Bur. Sund.(U.S-), Tech. Note 638. 39 pages (JuK 1973) CODEN: NBTINAE

For sale by the Superintendent of Documents. U.S. Government Printing Office, Washington, D. C. 20402

(Order by SD Catalog No. C13.46:638). Price 45 cents
CONTENTS

Page

1

INTRODUCTION 1

2. SLIDE RULE DESIGN 3

3. COMPARISON WITH THEORY AND EXPERIMENT 13

4. OPERATION OF THE SLIDE RULE 23

5. SUMMARY -9

6. REFERENCES 30
LIST OF FIGURES

Page

Figure 1. Earth satellite geometry for slant range

calculation 4

Figure 2. Geometry used to calculate the central angle, 3 6 Figure 3. Delay error as a function of error in central angle. . 8 Figure 4. Error in central angle as a function of relative

latitude and relative longitude for two values of error in relative longitude 12

Figure 5. The oblateness correction 15

Figure 6. Block diagram of the delay computation procedure . . \j

Figure 7. Slide rule delay error. Theoretical delay minus

slide rule delay 28

Figure 8, Distribution of slide rule delay errors 19

Figure 9. Slide rule delay error with the longitudinal error removed. Theoretical delay minus slide rule delay. 20

Figure 10. Distribution of the slide rule delay errors with the

longitudinal error removed. A, B, C, D 21

Figure 11. Two-way delay between Boulder and ATS-3 at 1700 GMT 22

Figure 12. Sample slide rule top and cursor 14 a Figure 13. Sample slide rule base 14b

LIST OF TABLES

Table 1. Values of 6 for various values of m and \X - X \

1 s s r showing 6 «== \ - X 9

s r

Table 2. Difference between exact and approximate method

for computing a i n degrees exact approximate ... 21 Table 3. Evaluation of the partial derivative of equation 9. Also shown are the delay errors for values of

radius correction equal to 500 and 1000 us caused

by setting this partial derivative equal to 1. 0. . . . 14
A SYNCHRONOUS SATELLITE TIME DELAY COMPUTER

A special purpose slide rule designed to compute the free

space propagation delay between a synchronous satellite and

points on the earth's surface is discussed. The slide rule

was developed to provide users of time information relayed

by geostationary satellites a means of computing the propagation delays without dealing directly with the satellite's orbital

elements. The delays computed with the slide rule are compared with the values obtained from orbital elements using a high precision digital computer. The limitations and accuracy

of the slide rule are discussed. A sample slide rule which

may be cut out and used is included in the report.

Key words; Satellite timing; slant range; synchronous

satellites; time delay.

1. INTRODUCTION

The usefulness of potentially highly accurate timing signals relayed

through geostationary satellites is complicated by the computation of the

propagation delay from the transmitter through the satellite back down to the receiver. The computation of this delay is usually accomplished

through the manipulation of orbital elements. From the orbital elements,

six constants which describe the satellite's position and velocity at a given instant of time, and a complete description of perturbing forces, it

is possible to compute the satellite's position at other times. Once satellite position is known, the free space propagation delay to any reciever

follows directly from simple geometric~considerations. The computation

of satellite position from orbital elements, however, is complicated and

includes the solution of a transcendental equation best accomplished by

iterative techniques using a digital computer. As has been mentioned,

an accurate prediction of the delay using orbital elements must also

NOTE: Preparation of this document was supported in part bythe Air

Force Communications Services under contract EIIIM-6.
account for the forces which perturb the two-body orbit such as the sun's

and the moon's gravitational fields, the non-uniformity of the earth's

gravitational field, and solar radiation pressure, making the computation

all the more laborious.

This type of computational burden would be unmanageable for most

of the expected users of satellite time and frequency signals. Consequently, the National Bureau of Standards has developed a special-purpose

computer in the form of a circular slide rule which, utilizing the satellite's position, allows easy computation of the delay to high accuracy.

An operational procedure for time synchronization using a geostationary satellite and this special purpose slide rule may be envisioned

along the following line. Using satellite tracking data which could be in

the form of orbital elements, the satellite's position in terms of its subsatellite point and radius (the distance from the center of the earth) is

computed at the transmitter in advance of the time broadcast. This position information is incorporated into the time broadcast. The information might be conveyed in digital form or as a voice announcement.

The receiver uses the slide rule to compute his delay from the satellite's

position and his own longitude and latitude. The computed delay, valid

only during a few minutes before and after the announcement because of

satellite motion, is then used in conjunction with his time measurements

to synchronize the receiver's clock.

The slide rule was designed to work with satellites in geosynchronous,

near circular, near equatorial orbits. For such satellites, one may

estimate the orbit dimensional quantities to be: a semi -major axis of

6. 61 earth radii, an eccentricity of 0. 005 or less, and an inclination

angle between the equatorial and the orbit plane of 5 or less.
2. SLIDE RULE DESIGN

The satellite's position at any given time may be specified by its subsatellite point and radius. The sub-satellite point is the point on the

surface of a spherical earth which intersects the line joining the satellite

and the center of the earth. The satellite and any point on the surface of

the earth define the end points of a free space propagation path for which

the slide rule was designed to solve.

The problem of determining the delay from the satellite to an arbitrary site may be approached in several ways. The method used in this

analysis is to solve the triangle formed by straight lines joining the

satellite, the center of the earth and the site (fig. 1). This solution

from plane trigonometry is

R + h

2

- 2Rh cos 8, ( 2 ) where r is the range from the receiver'to the satellite, R is the distance

from the satellite to the center of the earth, h is the distance from the

receiver to the center of the earth, and B is the central angle between

the sub- satellite point and the receiver. The quantity R is a component

of the satellite's position and is assumed to be available via the satellite

broadcast. The quantity h is related to the geodetic latitude, cp. of a site

by the following equation

h = a.

' b

4

z

1 + — . tan cp

, , b 2

1 + — tan cp a

1 (2)

where a = 6378. 2064 km, the earth's semi -major axis; and b = 6356. 5838 km,

the earth's semi -minor axis [l].
h

om

>^

l-i +-> COa

o

CD00

0) +j

•i-t ^| fl

0) +J -rH

ctj -t-J co rti ^ 3u

f-i

.—

rti cd W

CJ

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cu u

^13 C

ao

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CO
For use in the equations below, the geocentric latitude, cp', is computed

from the geodetic latitude, cp, by the following equation.

tan cp' = (b / a ) tan cp. (3)

The sub-satellite latitude is already referenced to the center of the earth

and does not need to undergo this transformation. In the following discussion, X indicates longitude and subscripts s and r denote sub- satellite

point and receiver site respectively.

All that is left then is the computation of cos 3. The direct solution

may be obtained from the triangle consisting of the sub- satellite point,

the site, and the intersection of the z axis with the spherical earth (i.e.,

the North Pole) using spherical trigonometry as follows

cos 3 = sin cp' sin cp + cos cp' cos cp cos | X - X I . (4)

r s r s s r This method, however, does not lend itself to a simple slide rule computation. Instead, the more complicated approach of solving for Ct and 6, the two constructed angles shown in figure 2, leads to the elimination of

terms in the formulation and a simplified slide rule procedure. The angle

6, the perpendicular from the sub-satellite point to the meridian passing

through the receiver, may be calculated by

sin 6 = cos cp sin I X - X I . (5)

s ' s r

'

The angle a, the "corrected satellite latitude" or merely the arc length

from the intersection of the perpendicular and the meridian to the equator,

is given by

cos a = cos cp C os \X - X \ I cos 6. (6)

s ' s r
<u A-u

CD

+-> rd 1 .—

1 3

LU

u

Q .—

13 cti h- a

i—t o o u +-> z << o T3

CD _l 1 CD3

LU CO > << >^ —I Jh 1— CD _l s

LU

O

oc CDc CD. o

CD » O

a)M

fl CM rt

<D i—

U u 3 •u

CO fl • H CD h O
Equation (4) then becomes

cos 3 - cos (cp' - a) cos 6 . (7) r Using eqs (1) through (7), the "down-link" free space propagation delay

from the satellite to the receiver is easily determined by dividing the

range by the velocity of free space propagation (0. 2997925 km/|~ts). The

procedure must be repeated substituting the transmitter for the receiver

location to determine the "up-link" delay. The total free space propagation delay, then, is the sum of the delays computed using the transmitter

and receiver locations. The change in signal velocity through the troposphere and ionosphere and the accompanying ray bending can be shown to introduce only a few microseconds difference in the round-trip free space

propagation time when operating above 100 MHz L2j.

In order to incorporate these computations on a slide rule, several

approximations were made. Most of the approximations ultimately affected the accuracy of the calculation of 3. Consequently, it was necessary

to know the relationship between errors in 3 and errors in delay. Figure

3 shows this delay error for several error values of 3 as a function of 3 . Here it may be seen that in all cases an error of 0. 05 in 3 will cause

less than 20 |_is of error, which is acceptable for most timing applications.

Two approximations dealing with the computation of 6 and a were

possible because we were dealing with geostationary satellites. For these

satellites, the inclination of the orbit is small, usually less than 3 . The

resultant sub- satellite latitude is a function that is periodic in time about

the equator and has an amplitude equal to the inclination. The sub -satellite

latitude, CO , therefore, is normally less than 3 . Referring to eq (5), it s may be seen that since cos cp is very nearly unity, sin 6 «= sin \\ - \ | or

6 «* I A. _ X I . The slide rule design sets 6 = |\ - X I . Table 1 gives the 1 s r

5 ' s r

b
20 30 40 50 60 70 CENTRAL ANGLE (3) IN DEGREES

Figure 3. Delay error as a function of error in central angle.

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o o o o o o o o o o o o o o o

o o o o o o o o o o o o o o o

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i—I —> CM CM 0O in o in o in o inoo -st1 Tf m m vO no o in

tN- fN9
actual values of 6 for various values of cp and | A. - X | . As may be seen from eq (7) the error in central angle introduced by this approximation

depends also upon the value of |cp - a | or the relative latitude. Figure 4

shows this error in central angle for the various parameters involved.

As will be shown in example 4 of section 4 of this report, it may be desirable to compute the error introduced by this assumption in order to reduce

the overall delay computation error for some sites.

Because cos cp is very nearly linear over the small range of cp , one

s s may make an approximation in the calculation of a. From this linearity

and the near equality of 6 and | A. - X | , it may be seen in eq (6) that cos a must also be nearly linear. Therefore, if one knows a for some

non-zero value of cp , he may approximate any other a by extrapolating

(cp =0 implies a = ). Taking an assumed value for cp of 1. 25 then,

the slide rule calculates a as

(cos 1.25 cos | A. _A. |/ cos (sin (cos 1.25 sin | X -X | ))). (J -1 1.25 ~~~ x

' s "r

Naturally, this approximation introduces some error. Table 2 shows the

difference between this method of caluclation of cc and that of eq (6). By

interchanging longitude and latitude in figure 4, one may determine the error in central angle caused by approximation 8. This error amounts

to less than 0. 01 of error in central angle or less than 5 \jls of range

error for better than 90% of the table.

Since the orbits of geostationary satellites are somewhat elliptical,

the value of the radius, R, varies as a function of time. Referring to figure 1, it can be envisioned that these small changes in radius will cause nearly the same change in slant range (r). For typical eccentricities, the peak-to-peak variation in R amounts to about 300 km or roughly

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o o o o o o o o o o o o o o r—l CO o o o o o o o o o o o o o o o o

o o o o o o o o o o o o o o o o

o o o o o o o o o o o o o o o o

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o o o o o o o o o o o o o o CO o o o o o o o o o o o o o o o o

o o o o o o o o o o o o o o o o

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ro CM co CO * -f in m o ^3 r- rI X

X

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X\ spn^iSuoT; sAp-exs'H

11
RELATIVE LONGITUDE

( .2° ERROR)

RELATIVE LONGITUDE

( .1° ERROR)

20 30 40 50 60

RELATIVE LATITUDE (DEGREES)

70

Figure 4. Error in central angle as a function of relative latitude and relative

longitude for two values of error in

relative longitude.

12 I 3-.I.-MMI "1
1 ms. In the extreme case, the variation should amount to little more

than twice this amount or 2 ms. The slide rule computes the delay for a value of R equal to 42, 143.4517 km. The difference between the actual

radius and this value has been termed the "radius correction. " The

value of the radius correction is one of the three parameters defining the

satellite's position and is given in microseconds. This value is added to the slant range delay computed by the slide rule; i. e. , the computation

procedure used by the slide rule sets Ar = AR. The error introduced by

this approximation may be calculated from

|1 = (R - hcos P) /r. (9)

Evaluating this shows a zero error when p = 0, increasing to 1. 2% of the

value of the radius correction when p = 75 . Table 3 gives the value of

gr/oR and the accompanying delay error for the normal (±500 M- s and the

extreme (±1000 |j.s) values of the radius correction. This table indicates

a maximum error of 1 Z M- s introduced by this method of dealing with variations in R.

Finally, the variations of h introduced by eq (2) are not accounted

for in the initial calculation of the delay. Instead, the value of h in eq (1)

is set to the semi-major axis, a. The error introduced by this procedure

is then removed. Figure 5 gives this error as a function of relative longitude and relative latitude. This graph, called the "oblateness correction, "

is included on the face of the slide rule.

3. COMPARISON WITH THEORY AND EXPERIMENT

In order for the slide rule to be useful for the computation of the

delay from satellite to ground, two factors must be valid. First,

the slide rule delay must agree with the theoretical delay as compute;:

13
Table 3. Evaluation of the partial derivative of equation 9. Also shown

are the delay errors for values of radius correction equal to 500 and

1000M-S caused by setting this partial derivative equal to 1.0.

Sr/jR

Error with 500 Error with 1000

[is radius ijs radius

correction correction

. 059366 . 118732

. 237167 .474334

. 524044 1. 04809

. 906289 1. 81258

1. 36608 2. 73216

1. 88273 3. 76546

2.43396 4. 86791

2. 99656 5. 99313

3. 54844 7. 09689

4. 06903 8. 13806

4. 54038 9. 08077

4. 9473 9. 89461

5. 27841 10. 5568

5. 52535 11. 0507

5. 68348 11. 367

1.000000

5 .999881

10 .999526

15 .998952

20 .998187

25 .997268

30 .996235

35 .995132

40 .994007

45 .992903

50 .991862

55 .990919

60 .990105

65 .989443

70 .988949

75 .988633

14

TOP PIECE

CURSOR

Assembly Instructions (figs. 12 and 13).

1. Cut out top piece, cursor, and base of the slide rule,

2. Cut out the two windows on the top piece.

3. Using a straight pin or other suitable fastner, assemble the

cursor, top piece, and base at the central point, indicated with

a dot in the middle of a small circle.

Figure 12. Sample slide rule top and cursor.

14a
Figure 13. Sample slide rule base.

14b I B- 1 03344~l

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(U -t-j <u u

[S

ctj +j 0)

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(1)

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en

CD

cti o >-H

CD

u COo CD

?H +-" o ni i—i 1a CD -•H ^ G

+-> o &0 U CO <U d

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t—t

t—1

«1 CD

CO +-> <D ni

fl CO

CCJ

(ti +j O CDbO

CD

ccj h> h

O

<x; o -—- C

1/1

O

LU »rH LlJ CJ a; CD CD Li UJ U o o o

*d- UJ

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O CO

CD 3) c 1— CD •—1

+-> 1— ccj <c .—1

X>O o C_> CD CO i—, X

Cd

H

h-z

LU in

(_) O CD LU ^1 CD 3h

15
by a high precision digital computer using orbital elements. Second, it must be shown that the prediction of satellite position (and the resultant

delay) from orbital elements agrees with actual delay measurements.

Remember that the orbital elements resulted from high resolution

tracking of the satellite. From the orbital elements and a complete

description of the orbit perturbations, the satellite's position at any other

time is predicted. Satellite position in terms of sub-satellite point and

radius is given to the user who in turn uses the slide rule to calculate

the total free space delay. The process is illustrated in figure 6.

A comparison of slide rule delay and the delay derived from the

orbital elements was accomplished using over 500 different sites and

three different satellite positions. The sites were chosen in 1 increments

of relative longitude from to 70 for a fixed relative latitude. Relative

latitude was varied from to 70 in 1 increments. No computations

are shown for central angles greater than 75 which is considered the

limit of usefulness of the slide rule. Figure 7 shows the error of the

delay computed using the slide rule. Figure 8 gives the distribution of

this error and shows that the slide rule is weighted towards giving a high delay prediction. Figures 9 and 10 give the corresponding results

after the errors in relative longitude listed in table 1 have been removed.

(The method of correction for this longitudinal error is illustrated in

example 4 below. ) The distribution now is nearly normal and the rms

error has been reduced from 9. 88 us to 6. 79 [is. This longitudinal error

is large with respect to this rms error for only about 15% of the satellite

coverage, but for this region (the shaded area of table 1), it is suggested

that the error be removed.

The question of how well the orbital elements describe or can be

used to predict satellite position is answered in figure 11. Over a period of several months, it is seen that measured and predicted values

16
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1—

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Figure 8. Distribution of slide rule delay errors.

A. Satellite at -1. 928°N, 69. 027°W

B. Satellite at 0.518°N, 68. 601°W

C. Satellite at 2. 177°N, 68.496°W

D. Total of A, B, and C.

19
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Figure 10. Distribution of the slide rule

delay errors with the longitudinal error removed. A, B, C, D.

21

IB- 102055
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22
fall to within an rms value of 10 |is which was approximately the measurement resolution. The particulars of the measurements are reported

upon in a later publication L 3 J

.

4. OPERATION OF THE SLIDE RULE

For computation of the up -link and down-link delay by this slide

rule, several pieces of information are needed. They are:

1. The transmitter and receiver longitude and latitude;

2. The longitude and latitude of the sub -satellite point; and

3. The difference between the actual and nominal value of the

radius to the satellite (i. e. , the radius correction).

The location of the observing site may be determined from a good

quality map to about 0. 01 . Referring to figure 3, it may be seen that

this position error will not introduce a significant amount of error in the

delay computation.

The position of the satellite is constantly changing. If the subsatellite point is supplied to the user to the nearest 0. 01 , updates of

positon would have to be supplied once every few minutes for a typical

geostationary satellite. An obvious way of disseminating this information

is to broadcast, either by voice or code, the satellite position along with

the timing signals.

In the explanations and examples of slide rule operation it may be

useful to cut out and assemble the sample slide rule (figures 12 and 13).

The steps to calculate delay are the following:

1. Determine the longitude of the receiver and transmitter sites

relative to the satellite's longitude (i. e. , determine relative longitude).

This may be computed as the longitudinal difference between the site and

the sub-satellite point, | A. - X |.

23
2. The corrected satellite latitude, a, is determined by setting the

appropriate index of scale #1 to the sub -satellite latitude on scale #2 and

reading the corrected satellite latitude on scale #2 opposite the relative

longitude. Possible decimal point and sign confusion will be clarified in

examples 2 and 3. The relative latitude is then computed as the difference

between the site latitude and the corrected satellite latitude, cp - a I . r

3. An uncorrected delay is determined by setting the index of scale

#3 at the relative longitude on scale #4, positioning the cursor at the

relative latitude on scale #3 and reading the delay on scale #5. The

central angle, 3, is also read on scale #4 under the cursor. Two corrections are then applied to this delay. The oblateness correction, found

from the graph using the site's latitude and the relative longitude, is

added. The radius correction is also added.

This procedure must be followed for both the transmitter and receiver

sites in order to determine the round-trip propagation delay.

For the purposes of the following examples, the transmitter is asI to be loca

West Longitude.

sumed to be located at NBS, Boulder, 40. 00° North Latitude, 105. 26°

24
Example #1: In this example the satellite, transmitter, and receiver are north of the equator. The transmitter is west and the receiver east

of the sub -satellite point.

Sub-satellite point 2.25 North Latitude

70.37° West Longitude

Receiver site 47. 85 North Latitude

56.11 West Longitude

Radius correction 135 Microseconds

Relative longitude

Up -link Delay

105.26°

-70.37

° 34.89°

Down-link Delay

70.37°

-56.11°

^4^6°

Corrected satellite

latitude 2.74 2.32

Relative latitude

Uncorrected delay

Oblateness correction

Radius correction

One-way delay

Round-trip delay

40. 00

-2.73°

37727°

127670 M s 16 u

s 135 m

127821 ,js

255008 u.8

47.85

-2.32°

45. 53°

127030 u s 22 |js

135 ^

127187 [js

25
Example #2: The same receiving site is used, but the satellite is

south of the equator. Note that the sub-satellite latitude is considered

to be a negative number.

Sub-satellite point 2.25 Sough -Latitude

70.37° West Latitude

Receiver site 47. 85 North Latitude

56. 11 West Longitude

Radius correction 135 Microseconds

Relative longitude

Up -link Delay

34.89°

Down -link Delay

14.26°

Corrected satellite

latitude • 2.74 2.32

Relative latitude

40. 00

(-2.73°)

42.73°

47.85

-(-2. 32°)

50. 17°

Uncorrected delay

Oblateness correction

Radius correction

One-way delay

Round-trip delay

128830MS

16MS

135 Us

128981 U-s 257528 [is

128390 |-is

22 |is

135 [is 128547 [is

26
Example #3: Here the receiving site is south of the equator yielding a negative cp . The relative latitude, however, is still a positive number.

Also, note the procedure for handling sub-satellite latitudes of less than

1°.

Sub- satellite point 0.98 North Latitude

69.84° West Longitude

Receiver site 34.47 South Latitude

58.40° West Longitude

Radius correction 348 Microseconds

Relative longitude

Up -link Delay

105.26

-69.84

35.42 c

Down -link Delay

69

-58.

84°

40°

11 44°

Corrected satellite

latitude 1.20 1. 00

Relative latitude

40. 00

-1.19'

38. 81'

34.47

1. oo

c 33.47c

Uncorrected delay

Oblateness correction

Radius correction

One-way delay

Round-trip delay

128085|J.s

l6|J.s

348 [as 128449 Ms

252560p.s

123745 Ms

18 Ms

348 Ms

124111 Ms

27
Example #4: This example delineates the procedure for correcting

the longitudinal error introduced by setting the relative longitude | A. - \

equal to 6 in eq (7). The receiver is assumed to be located on Tristan da Cunha in the south Atlantic.

Sub- satellite point 2. 50 North Latitude

74.67 West Longitude

Receiver site 37.15 South Latitude

o 12.30 West Longitude

Radius correction 176 Microseconds

Relative longitude

Corrected satellite

latitude

Relative latitude

Uncorrected delay

Oblateness correction

Radius correction

Up -link Delay

105.26°

-74.67°

30. 59°

2.90~

40. 00°

-2,90°

37. 10u

126880 Ms

17 Ms

-176 Ms

126721 \is

Down-link Delay

74.67°

-12. 30°

62. 37°

5.40

37. 15

-5.40

42. 55 u

134765 [is 6 Ms

-176 [is 134595 Ms

Before determining the round-trip delay, the down-link should be cor- rected for the longitudinal error since the relative longitude and satellite

latitude combinations fall into the shaded area of table 1. Interpolating

from this table, we find that 6 should be 62. 26 . Using this value there

instead of 62. 37 to compute the down-link delay yields an uncorrected

28
delay of 134740 |_is, a one-way delay of 134570 |-is, and a round-trip delay

of 261291 us. An alternate procedure for correcting this down-link delay requires

figures 3 and 4 in addition to table 1. The longitudinal error determined

from table 1 is found on figure 4. This shows an error in central angle

of 0. 075 . The delay error resulting from this error in (3 is found on

figure 3 to be about 27 |_is using a central angle of 70 . This value does

not differ significantly from the 25 |Js error found by recomputing the

delay with the slide rule.

5. SUMMARY

The special purpose slide rule developed in this report offers a simple, fast, convenient and accurate method of computing the free space

propagation delay to a geostationary satellite. A satellite time synchronization system utilizing this slide rule would require that the operating

agency compute the satellite's position coordinates and relay them to the user along with the rest of the timing format. Following the five-step

procedure of section 4, the user enters his own longitude and latitude

along with the satellite coordinates on the slide rule to determine the

theoretical free space propagation delay for comparison with his delay

measurement. The slide rule delay has been shown to agree with the

delay as computed using a high speed digital computer to within 10 [is rms. A satellite time and frequency system which requires only an

easily operated slide rule for the computation of the propagation delay

should find expanded applications for its use.

29
6. REFERENCES

[lj. Lewis, E. A., "Parameteric Formulas for Geodesic Curves and

Distances on a Slightly Oblate Earth, " Air Force Cambridge

Research Laboratory Report-63-485 (April 1963).

L2J. Weisbrod, S. , and Anderson, L. J. , "Simple Methods for Computing

Tropospheric and Ionospheric Refractive Effects on Radio Waves, " Proceedings of the IRE, 47, No. 10, pp. 1770-1777 (October 1959).

L3]. Hanson, D. W. , and Hamilton, W. F. , "Experimental Time and

Frequency Broadcasts from the ATS-3 Satellite", (to be published).

30
raRM HBS-H4A (1-71)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBS-TN-638

2. Gov't Accession No.3. Recipient's Accession Nc

4. TITLE AND SUBTITLE

A SYNCHRONOUS SATELLITE TIME DELAY COMPUTER

5. Publication Date July 1973

6. Performing Organization Code

7. AUTHOR(S)

Wallace F. Hamilton and D. Wayne Hanson

8. Performing Organization

10. Project/Task/ Work Unit No. 2730114

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS, Boulder Labs.

DEPARTMENT OF COMMERCE

Boulder, Colorado 80302

11. Contract/Grant No.

12. Sponsoring Organization Name and Address Same as No. 9

13. Type of Report & Period CoveredFinal

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant bibliography or literature survey, mention it here.) A special purpose slide rule designed to compute the free space propagation delay

between a synchronous satellite and points on the earth's surface is discussed. The

slide rule was developed to provide users of time information relayed by geostationar-jr

satellites a means of computing the propagation delays without dealing directly with th

satellite's orbital elements. The delays computed with the slide rule are compared

with the values obtained from orbital elements using a high precision digital computer

The limitations and accuracy of the slide rule are discussed. A sample slide rule

which may be cut out and used is included in the report.

17. KEY WORDS (Alphabetical order, separated by semicolons)

satellites; time delay.

Satellite timing; slant range; synchronous

18. AVAILABILITY STATEMENT

1^1 UNLIMITED.

I I FOR OFFICIAL DISTRIBUTION. DO NOT RELEASE

TO NTIS.

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(THIS REPORT)

UNCLASSIFIED

20. SECURITY CLASS

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21. NO. OF PAGES

39

22. Price

. 50

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