NASA TECHNICAL NOTE NASA TN D-6681

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RELATIVISTIC TIME CORRECTIONS

FOR APOLLO 12 AND APOLLO 13

Lavery
Goddard Space Flight Center
Greenbelt, Md. 20771

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION • WASHINGTON, D. C. • AUGUST 1972

1. Report No. , ,_ 2. Government Accession No. 3. Recipient's Catalog No.
NASA TN D-6681
4. Title and Subtitle 5. Report Date
August 1972
Relativistic Time Corrections for
6. Performing Organization Code
Apollo 1 2 and Apollo 1 3
7. Author(s) 8. Performing Organization Report No.
John E. Lavery G-1049 " '
9. Performing Organization Name and Address 10. Work Unit No.
Goddard Space Flight Center
11. Contract or Grant No.
Greenbelt, Maryland 20771
13. Type of Report and Period Covered
12. Sponsoring Agency Nome and Address

National Aeronautics and Space Administration Technical Note

Washington, D. C. 20546 14. Sponsoring Agency Code

15. Supplementary Notes

16. Abstract

Results are presented of computer calculations on the relativistic time corrections

relative to a ground-based clock of on-board clock readings for a lunar mission,
using simple Newtonian gravitational potentials of Earth and Moon and based on
actual trajectory data for Apollo 1 2 and Apollo 1 3. Although the second order
Doppler effect and the gravitational "red shift" give rise to corrections of opposite
sign, the net accumulated time corrections, namely a gain of 560 (± 1.5) microseconds
for Apollo 12 and gain of 326 (±1.3) microseconds for Apollo 13, are still large
enough that with present day atomic frequency standards, such as the rubidium clock,
they can be measured with an accuracy of about ±0.5 percent.

17. Key Words (Selected by Author(s)) 18. Distribution Statement

Relativity
Time Dilation
Spacecraft Clock Unclassified— Unlimited
Red Shift
19. Security Classif. (of this report) 20. Security Classif. (of this page) 21. No. of Pages 22. Price

Unclassified Unclassified 13 $3.00

' For sale by the National Technical Information Service, Springfield, Virginia 22151
 CONTENTS

Page

ABSTRACT i

INTRODUCTION 1

THEORETICAL DEVELOPMENT 1

CALCULATIONS 4

ERROR ANALYSIS 5

ACKNOWLEDGMENTS 9

REFERENCES 10

111
 RELATIVISTS TIME CORRECTIONS
FOR APOLLO 12 AND APOLLO 13
by
John E. Lavery
Goddard Space Flight Center

INTRODUCTION

Interest in a determination of the relativistic time shift as manifested on board a spacecraft has
been sharpened by the recent publication of an estimate by C. O. Alley indicating that a blue shift of
some 300 /xs occurred for the Apollo 8 spacecraft versus a ground-based clock (ref. 1) and of a descrip-
tion of a proposed orbiting clock experiment to determine the gravitational red shift by Kleppner,
Vessot, and Ramsey (ref. 2). However, precise computations using actual tracking data for specific
space missions have yet to be published. To fill this gap, this report presents precise frequency and
time shift computations with tight error bounds for one long-duration lunar mission (Apollo 12) and
one short-duration lunar mission (Apollo 13). Because no precision frequency standards were carried
on these flights, no comparison of these computations with experimental data can be attempted. How-
ever, the error bounds achieved in the computations presented here are to be taken as indicative of the
accuracy achievable in future computations using the methods presented in this report.l

THEORETICAL DEVELOPMENT

Introduced in the theory of relativity is the concept of "proper time" (see, for example, ref. 3),
which is the time indicated by a clock rigidly connected with a moving material frame of reference.
If x (a = 1, 2, 3) are the space coordinates and t the time coordinate of this moving frame with respect
to a given inertial frame of reference, it can be shown that the interval of proper time dr is given by

I .
(1)
where g B are the components of the metric tensor involving the spatial coordinates xa, the subscript
4 labels the time coordinate, and dt is the interval of coordinate time. (Coordinate time is the time

J
A11 computations presented here were carried out by program E00032 of the Goddard Space Flight Center Computer Program
Library, This program can be used without modification for calculations for any near-Earth mission. With small modifications it can
be used for interplanetary missions.
indicated by a clock at rest with respect to the coordinate system and under no gravitational
potential.)
For a weak static gravitational field having potential 0, the components of the metric tensor can
be taken to be

"afi <xfl

and.
20
^2

where 6 = 1 if a = j3 and 6 = 0 if a =£ 0. Hence

(2)

where 0 is the total gravitational potential at the clock, v is the speed of the clock with respect to the
given inertial frame, and c = 2.997 925 X 108 m/s (the speed of light). Let (f>A and <t>B represent the
gravitational potential at clock A and the gravitational potential at clock B, respectively, and VA and
VB represent the speed of clock A and the speed of clock B, respectively. If we compare clock A to
clock B, both in the same inertial frame, the relative frequency of clock A with respect to clock B is
given by

/ 20 u2
/i + l_d _ — rfr
V c2 c2
t

\/l + — - —2
4 dt
2
V c c

1 14>A V2

V1+ c 2 " c2
(3)
/
1+ ^B 4
V c2 " c2

1 + ^-l
- + (3a)
2c2

Hence, to the first order, the frequency shift of clock A with respect to clock B is

"A V
B v2
+ (4)
2 2
2c 2c
Thus, the total time difference of clock A with respect to clock B accumulated from (coordinate) time
t^ to (coordinate) time t2 is

r
A .c 2c2 2c2_
dt (5)

Now, for points in the neighborhood of the Earth-Moon system, the gravitational potentials <j>Aw
and 4>B may be computed with sufficient precision from the expression
GM, J
3 /RE\
1 [—I (3 sin2 1E- 1) (—] (5 sin3 l£ - 3 sin 1E)

J /A?
4 I E\ I**E\
(35 sin4 1E - 30 sin2 1E + 3) + 3(C22 cos 2\E + S22 sin 2A£.)[—J cos2 l£

GM, KI (R. ,f\

1 - -1(^\ (3 sin2 1M - cos2 1M cos 2\M (6)
rf*
'M M>

where
GMS = (1.327 124 99 ± 0.000 000 15) X 1020 m 3 /s 2 = gravitational parameter of the Sun
rs = distance from the clock to the center of mass of the Sun
GME = (3.986 012 ± 0.000 004) X 1014 m3/s2 = gravitational parameter of the Earth
r£ = distance from the clock to the center of mass of the Earth
/2 = (1.0827 ± 0.0001) X 1(T3
RE = 6.378 166 X 106 m = equatorial radius of the Earth
IE = geocentric latitude (declination) of the clock
/3 =(-2.56 ± 0.1) X ID"6
/4 =(-1.58 ±0.2) X 1(T6
C22 = (1.57 ± 0.01) X 1(T6
S22 = (-0.897 ± 0.01) X 1(T6
\E = geocentric longitude of the clock
GMM = (4.902 78 ± 0.000 06) X 1012 m 3 /s 2
rM = distance from the clock to the center of mass of the Moon
KI =(2.071 08 ± 0.05) X 1(T4
 RM = (1.738 09 ± 0.07) X 106 m = mean lunar radius
IM - selenocentric latitude (declination) of the clock
K22 =(2.0716 ±0.5) X 10-5
\M = selenocentric longitude of the clock (positive eastward with respect to the Moon's prime
meridian)
These constants are taken from pages 6-2 to 7-1 of reference 4. Equation (6) is derived from the
gravitational-potential equations for the Earth and for the Moon on these same pages.

CALCULATIONS

Calculations of the frequency shift and resulting time difference of a spacecraft clock (clock A)
versus an Earth-based clock (clock 5) were made by equations (4) and (5), respectively, where the
inertial frame of reference is a nonrotating Cartesian coordinate system with origin at the center of
mass of the Earth. The spacecraft clock was assumed to be on the command module (CM) of the
Apollo 12 and Apollo 13 spacecraft and the Earth-based clock was assumed to be at NASA's Network
Test and Training Facility (NTTF) (38°59'56.7" N, 76°50'22.7" W, 52 m above the Earth model
ellipsoid). To an accuracy of 10~ 14 , the frequency of a clock at NTTF is the same as the frequency of
an identical clock anywhere on the Earth model ellipsoid. Hence the results presented in the tables
and figures in this report are typical of the results expected for a comparison of a spacecraft clock to

Table 1.-Frequency and Time Corrections for a Clock on Apollo 12 for Selected Values of Ground
Elapsed Time (GET)
Frequency shift of CM standard Time
Distance Distance versus Earth standard, 10~10 correction
fCT of CM from of CM from Speed (advance) of
(jt.1,
Event center of center of ofCMu, CM standard
s Due to gravity of- Due to Total versus Earth
Earthy, MoonrM, m/s
m m velocity shift standard,
Sun Earth Moon
MS

Begin TLC 10 380.0 6 724 000 370 883 000 10805 -0.0061 0.3633 -0.0000 -6.4873 -6.1301 0.0
TLC 247 080.0 347 003 000 63 883 000 674 .0822 6.8248 -.0071 -.0180 6.8819 153.797
In orbit
around Moon 425 580.0 378 498 000 1 846 000 2660 .1570 6.8354 -.2941 -.3865 6.3120 271.328
In orbit
around Moon 514 380.0 385 457 000 1 851 000 613 .2012 6.8376 -.2934 -.0136 6.7318 329.228
TEC 671 280.0 346213000 64 183 000 905 .2099 6.8245 -.0071 -.0383 6.9890 434.334
Reentry 879 769.6 6 464 000 401 185 000 11033 -.0056 .0975 .0000 -6.7650 -6.6731 570.430
a
Splashdown 880557.6 6 377 000 402 727 000 442 -.0063 .0032 .0000 -.0036 -.0067 570.345

a
This speed is due mainly to the rotational speed of the Earth, not the slight downward speed of the CM.
 Table 2.-Frequency and Time Corrections for a Clock on Apollo 13 for Selected Values of GET

Frequency shift of CM standard Time

Distance Distance
versus Earth standard, 10~10 correction
of CM from of CM from Speed
GET, (advance) of
Event center of center of ofCMu,
s Due to gravity of- CM standard
Earth r£, Moon rM, m/s Due to Total
m m velocity shift
Sun Earth Moon standard, MS

Begin TLC 11 197.0 13 361 000 387 728 000 7631 -0.0065 3.6375 -0.0000 -3.2327 0.3983 0.0
Oxygen tank
explosion 201 420.0 336 373 000 91 536 000 993 .0599 6.8208 -.0046 -.0475 6.8286 122.841
TLC 226 620.0 360 077 000 63 750 000 930 .0637 6.8295 -.0072 -.0408 6.8452 140.074
Closest approach
to Moon 278 470.0 406 425 000 2 002 000 1481 .0702 6.8435 -.2715 -.1148 6.5274 175.499
TEC 325 020.0 355 207 000 63 771 000 1 170 .0573 6.8278 -.0072 -0689 6.8090 207.174
Begin reentry 513 645.7 6 495 000 404 547 000 11037 .0045 .1325 .0000 -6.7696 -6 6326 327.559

any clock near sea level on the Earth. The Earth model ellipsoid referred to has equatorial radius
RE and polar radius (1 - 1/298.3)RE. Tables 1 and 2 give frequency and time differences at selected
times during translunar coast (TLC), near the Moon, during trans-Earth coast (TEC), and during
reentry for the flights of Apollo 12 and Apollo 13, respectively. Figures 1 and 2 are the plots for
Apollo 12 of the frequency shift of the CM standard versus the Earth standard and the time advance
of the CM standard relative to the Earth standard. Figures 3 and 4 are the same plots for Apollo 13.
From the columns in tables 1 and 2 giving the frequency shift of the spacecraft standard versus
the Earth standard, it is clear that the main factor in the blue shift of the spacecraft clock with respect
to the Earth clock is that the spacecraft standard is far away from the Earth's relatively strong gravita-
tional field. From table 1, we see that the total time advance of a hypothetical standard on the CM
of Apollo 12 versus an Earth standard accumulated during TLC, lunar orbit, TEC, and reentry (up to
splashdown) is 570.3 MS. From table 2, the time advance of a clock on Apollo 13 accumulated from
TLC up to reentry is 327.6 us.

ERROR ANALYSIS

The errors in the calculation of these numbers are (1) error due to the use of the approximation
(3a) for drA /drB ; (2) error in equation (6); (3) errors in the raw data, particularly the ranging and
velocity data; (4) integration errors in equation (5) due to integrating over GET rather than coordinate
time t and calculating the integral by the trapezoidal rule; and (5) roundoff errors in the values 570.3
and 327.6 jus. The error bounds which will now be calculated are not optimum bounds. They are,
however, sufficiently sharp for our calculations.
The error due to the use of expression (3a) to approximate drA /drB can be estimated by means
of the Taylor-series remainder terms. It is bounded by 7.0 X 10~19 (resulting in a time-difference
 TLC ORB TS AROUND MOON TEC
' • —
wimim WWWWWM mimm Hi
/ \

REENTRY

2 3 4 5

GET, 10s s

Figuie 1.-Frequency shift of clock on Apollo 12 CM versus Earth clock.

ORBITS AROUND MOON

4 5 8 9
GET, 10s s

Figure 2.—Time shift of clock on Apollo 12 CM versus Earth clock (curve A, left-hand
time shift scale); the results of subtracting out the average offset frequency
(6.5544 X 10~10) are given in curve B (using the right-hand time shift scale).
 TLC TEC

CLOSEST APPROACH TO MOON

Figure 3.—Frequency shift of clock on Apollo 13 2 3 4

CM versus Earth clock. GET, 105 s

600

CLOSEST APPROACH TO MOON

S O N /

-2

Figure 4.—Time shift of clock on Apollo 13

CM versus Earth clock (curve A, left-hand time
shift scale); the results of subtracting out the
average offset frequency (6.5193 X 1(T10) are
given in curve B (using the right-hand time
1 2 3
GET, 10s s
shift scale).
error of less than 1 ps for both Apollo 12 and Apollo 13). The error resulting from errors in equation
(6) can be bounded by using the error bounds on the constants given after equation (6).
Although error estimates for the internal consistency of the tracking and ephemeris data used in
the calculations are available (ref. 5), no absolute error bounds for the range and velocity data are
available. However, the following error bounds, based on the error estimates for the internal con-
sistency of the tracking and ephemeris data will be assumed. The ranging errors of both the CM stand-
ard and the Earth standard to the center of the Sun are bounded by 200 km. The ranging error of the
CM standard to the center of the Earth is bounded by 2 km when the range is less than 100 000 km
and,by 20 km when the range is greater than 100 000 km. The increased error bound for ranges
greater than 100 000 km is due to large ranging errors near the Earth-Moon interface. The ranging
error of the Earth standard to the center of the Earth is bounded by 10m. The ranging error of the
CM standard to the center of the Moon is bounded by 2 km when the range is less than 10 000 km
and by 20 km when the range is greater than 10 000 km. Again, the increased error bound for larger
ranges is to account for large ranging errors near the Earth-Moon interface. The ranging error of the
Earth standard to the center of the Moon is bounded by 2 km. The velocity error of the CM standard
is bounded by 2 m/s and the velocity error of the Earth standard is bounded by 10~3 m/s.
Let/(0 denote the integrand in equation (5). The error in integrating/(O over GET (r fl ) rather
than over coordinate time t is

f(t)dt (7)

and is bounded by
max /(O (8)

which is less than 1 ps for both Apollo 12 and Apollo 13. For our data/(0 is piecewise twice con-
tinuously differentiable. Then, if n + 1 represents the number of equispaced points on a twice con-
tinuously differentiable segment of the curve/(/) from ?3 to /4, which are used to calculate

r /(O dt

by the trapezoidal rule, the error in the numerically calculated value of the integral is bounded by

max (9)
12«2
Su.mming these error bounds over all segments of each flight gives an error bound for the final time
corrections.
Table 3 gives the magnitudes of the error bounds for the final time corrections listed in tables 1
and 2. All error bounds in table 3 have been rounded upward to the nearest nanosecond except those
less than 0.1 ns, which are entered as "negligible." As a result, the relativistic blue shift from

8
 Table 3.-Error Bounds for Final Time Corrections

Error bound, us
Error source
Apollo 12 Apollo 13

Approximation (3a) Negligible Negligible

Equation (6) 0.015 0.007
CM standard .
Ranging to Sun 0.012 0.007
Ranging to Earth 0.009 0.006
Ranging to Moon 0.017 0.006
Velocity 0.065 0.039
Earth standard:
Ranging to Sun 0.012 0.007
Ranging to Earth 0.001 0.001
Ranging to Moon Negligible Negligible
Velocity Negligible Negligible
Integration over GET instead of coordinate time Negligible Negligible
Trapezoidal integration error 0.084 0.038
Roundoff in final value 0.045 0.041
Total 0.260 0.152

10 380.0 s GET to 880 557.6 s GET of a standard on Apollo 12 versus an Earth standard is
570.3 ± 0.3 jus and the relativistic blue shift from 11 197.0 s GET to 513 645.7 s GET of a standard
on Apollo 13 versus an Earth standard is 327.6 ± 0.2 MS.
It remains now only to indicate the precision to which an actual experiment could measure these
relativistic time corrections. The only atomic frequency standards of sufficiently small size and weight
to be carried on Apollo missions are cesium and rubidium standards. For the intervals of time used
to make the theoretical calculations for both flights, the standard deviation of the frequency 2 of a
cesium standard is bounded by 5.0 X 10~13, and that of a rubidium standard is bounded by 2.0 X 10~ 12 .
Multiplying these standard deviations by the duration of the flights and rounding them off to the
nearest 0.1 jus yields the experimental error estimates of 0.5 MS (cesium) and 1.8 MS (rubidium) for
Apollo 12 and 0.3 MS (cesium) and 1.1 MS (rubidium) for Apollo 13. Hence, if an experiment were
performed, one could expect a fractional statistical error of 0.1 percent with a cesium standard and
0.33 percent with a rubidium standard.

ACKNOWLEDGMENTS

Dr. Fouad G. Major of GSFC originally suggested to the author the topic of the present report
and provided the author with much help in understanding the theoretical background of the material
presented here as well as in writing the report. The author is also grateful to Raymond V. Capo of •

2
Operational conditions such as those on a space flight are considered (See J. E. Lavery Operational Frequency Stability of
Rubidium and Cesium Frequency Standards. NASA Technical Note, to be published.) It is assumed, however, that the spacecraft
standard is sufficiently rugged to be operationally unaffected by acceleration and shock.

V
Goddard Space Flight Center and Donald J. Incerto of Manned Spacecraft Center for their help in
obtaining the trajectory tapes for Apollo 12 and Apollo 13.

Goddard Space Flight Center

National Aeronautics and Space Administration
Greenbelt, Maryland, September 15, 1971
312-02-08-20-51

REFERENCES

1. ' "A Matter of Overtime." Time 93(10): 42, 1969.

2. Kleppner, D.; Vessot, R. F. C.; and Ramsey, N. F.: "An Orbiting Clock Experiment To Determine
the Gravitational Red Shift. "As trophys. Space Sci. 6:13-32, 1970.
3. Bergmann, P.: Introduction to the Theory of Relativity. Prentice-Hall, Inc., 1942.
4. Natural Environment and Physical Standards for the Apollo Program and the Apollo Applications
Program. NASA Rept. M-DE 8020.008C, July 10, 1969.
5. Devine, C. J.: JPL Development Ephemens Number 19. JPL Tech. Rept. 32-1181, Nov. 15, 1967.

10 NASA-Langley, 1972 23
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 National Aeronautics and Space Administration

ERRATA

NASA Technical Note D-6681

August 1972

RELATIVISTIC TIME CORRECTIONS

FOR APOLLO 12 AND APOLLO 13

by John E. Lavery

To correct this document, replace pages i and 1 with the enclosed pages.

1972 Issued November 1972

 1. Report No. 2. Government Accession No. 3. Recipient's Catalog No.

4. Title and Subtitle 5. Report Date

Relativistic Time Corrections for Apollo 12 and
Apollo 13 6. Performing Organization Code

7. Author(s) 8. Performing Organization Report No.

John E. Lavery G-1049
9. Performing Organization Name and Address 10. Work Unit No.

Goddard Space Flight Center 11. Contract or Grant No.

Greenbelt, Maryland 20771
13. Type of Report and Period Covered
12. Sponsoring Agency Name and Address
Technical Note
National Aeronautics and Space Administration
Washington, B.C. 20546 14. Sponsoring Agency Code

15. Supplementary Notes

16. Abstract

Results are presented of computer calculations of the relativistic time corrections relative to a
ground-based clock of onboard clock readings for a lunar mission, using simple Newtonian
gravitational potentials of Earth, Moon, and Sun, and based on actual trajectory data for Apollo 12
and Apollo 13. Although the second-order Doppler effect and the gravitational "red shift" give
rise to corrections of opposite sign, the net accumulated time corrections, namely a gain of
570.3 ± 0.3 MS for Apollo 12 and gain of 327.6 ± 0.2 MS for Apollo 13, are still large enough that
with present-day atomic frequency standards, such as the rubidium clock, they can be measured
with an accuracy of about ±0.33 percent.

17. Key Words (Selected by Author(s)) 18. Distribution Statement

Relativity
Time Dilation
Spacecraft Clock Unclassified— Unlimited
Red Shift
19. Security Clossif. (of this report) 20. Security Classif. (of this page) 21. No. of Pages 22. Price*
Unclassified Unclassified 10 $3.00

•For sale by the National Technical Information Service, Springfield, Virginia 22151.
 RELATIVISTS TIME CORRECTIONS
FOR APOLLO 12 AND APOLLO 13
by
John E. Lavery
Goddard Space Flight Center

INTRODUCTION

Interest in a determination of the relativistic time shift as manifested on board a spacecraft has
been sharpened by the recent publication of an estimate by C. O. Alley indicating that a blue shift of
some 300 jus occurred for the Apollo 8 spacecraft versus a ground-based clock (ref. 1) and of a descrip-
tion of a proposed orbiting clock experiment to determine the gravitational red shift by Kleppner,
Vessot, and Ramsey (ref. 2). However, precise computations using actual tracking data for specific
space missions have yet to be published. To fill this gap, this report presents precise frequency and
time shift computations with tight error bounds for one long-duration lunar mission (Apollo 12) and
one short-duration lunar mission (Apollo 13).1 Because no precision frequency standards were carried
on these flights, no comparison of these computations with experimental data can be attempted. How-
ever, the error bounds achieved in the computations presented here are to be taken as indicative of the
accuracy achievable in future computations using the methods presented in this report.

THEORETICAL DEVELOPMENT

Introduced in the theory of relativity is the concept of "proper time" (see, for example, ref. 3),
which is the time indicated by a clock rigidly connected with a moving material frame of reference.
If xa(a = 1, 2, 3) are the space coordinates and t the time coordinate of this moving frame with respect
to a given inertial frame of reference, it can be shown that the interval of proper time dr is given by

dr ,=

where g are the components of the metric tensor involving the spatial coordinates xa, the subscript
4 labels the time coordinate, and dt is the interval of coordinate time. (Coordinate time is the time

J
A11 computations presented in this document, except those for the error bounds, were carried out by program E00032 of the
Goddard Space Flight Center Computer Program Library. This program can be used without modification for calculations for any
mission in the vicinity of the Earth-Moon system. With small modifications it can be used for interplanetary missions.
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
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