NASA TECHNICAL MEMORANDUM

REPORT NO. 53923

RELATIVISTIC T IME D ILATION ON LUNAR FLIGHTS

By Fred Wills Space Sciences Laboratory

September 16, 1969

NASA

George C. Marshall Space Flight Center Manball Space Flight Center, Alabama

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9 (ACCESSION NUMBER) 3 06

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2 1 2 none LL !PAGES) (CODE)

C --- [NASA CR OR TMX OR AD NUMBER) (CATEGORY)

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Changed to TNI X-53923, September 16, 1969 STANDARD P A G E

2. Government Access ion No. 3. R e c ~ ~ t e n t ' s Catalog No.

4. T t t l s and S u b t ~ t l e

Relativistic Time Dilation on Lunar Flights

I 6. Performing Organizat ion Code I 1 9. Performing Organizat ion Name and Address 110. Work U n i t No. I

7 . Author(a)

Fred Wills 8. Performing Orgonizat ion Report No.

George C. Marshall Space Flight Center Marshall Space Flight Center, Alabama 35812

14. Sponsoring Agency Code

11. Contract or Grant No.

12. Sponsoring Agency Name and Address

15. Supplementary Notes

Prepared by Space Sciences Laboratory, Science and Engineering Directorate

13. Type o f Report and P e r l a d Covered

Internal Note

1 16. Abstract

This report describes briefly the theory of relativistic time dilation. The theory is applied to the Apollo 8 flight to the moon and return. It is found that the astronauts aged some 335 p-sec relative to an earth-being at Cape Kennedy.

17. Key Words 18. D is t r ibu t ion Statement

I 19. Security C l o r s i f . (of t h ~ s report) 1 23. Security C lass i f . (06 t h i s page) 1 21. No. o f Pages 1 22. P r i c e

R E L A T I V I S T I C TIME D I L A T I O N ON LUNAR FLIGHTS

A gravitational field in space is described in a system of coordinates, 2 3 ' xi, x , x , x4, where x4 = ct , and t represents coordinate time ( c is the

speed of light) , by the relationship

i j ds2 = g . . dx dx 9

11

where i and j a r e summed from I to 4 by use of Einstein's summation convention. The relationship holds between each locally measured interval,

i CIS, and the corresponding coordinate difference, dx . The quantities, gij ,

define the metric o r physical properties of space. For events occurring a t a single point, the locally measured interval is given by

where dT denotes the proper time a t that point.

For a spherically symlnetric stationary distribution of mass about the origin, the g.. a r e given by the Schwarzschild solution to the field equations

11 of general relativity:'

and

where p , v = 1 , 2 , 3 and 6 = 1 ( p = v ) ; 6 = 0 ( p Z v ) . PV I-1 v

,. I. These values for the g a r e for a r a ius greater than o r equal to the i j

F radius of the spherical distribution of n ~ a s s .

The coordinates xi' 2 ' a r e Cartesian and

- r =x" + x 2 f +x3 F, r = ( X 1 ) 2 + ( X 2 ) 2 + ( X 3 ) 2

The factor G is the gravitational constant and M is the inass of the spheri- cally symmetric m a s s distribution. The noriiialized velocity of a point moving with respect to the origin is defined as

By substitution of equation ( 2 ) , the g.. , and equation ( 8) into equation ( 1) , 1J

and neglecting ternis higher in order than K and p2 yields

for the relationship between proper tinie and coordinate tinie for a point moving with normalized velocity /3 in a Schwarzschild field.

Within the realm of our forthcoming considerations, the earth is suf- ficiently spherical to be considered to yield aschwarzschild field. A spaceship going to the moon would be governed in its flight trajectory predoniinantly by the earth's Schwarzschild field. However, a s the spaceship gets closer to the moon, the Schwarzschild field is pertrubed by the gravitational field of an as - sumed spherically symnietric moon. Hence, if ? is the position of the

m moon and m is the moon's mass , then define

and the moon's normalized gravitational potential is given by

The effects of earth ohlateness, the sun's potential, solar pressure, etc. , will he taken to he negligible compared to the effects of a zero-order earth and a zero-order moon. In view of the effect in equation ( l l ) , w e will replace t h e factor K in equation ( 9 ) by K i- Q, i . e . ,

The subscript e will be used to designate the relationship between proper time and the coordinate time of a point on the earth's surface corre- sponding to the latitude of Cape Kennedy. The subscript s will be used to designate the relationship between proper time and coordinate t ime of a space traveler moving in the earth-moon field. We have

The proper time difference between the separation and return of a space traveler journeying to the nioon and back can be defined by

where T is the total time of separation. An expansion through the f i r s t or- der of equations ( 13) and ( 14) and substitution into equation ( 15) yields

The following t e r m s a r e defined:

G = 6.67 u 10'" m3/( kg-sec2) , universal gravitational constant

r = 6.37 x l o 6 m , average ear th radius 0

M = 5. 983 x kg, ear th ' s m a s s

c = 3. 00 x 10' m/sec, velocity of light

w = 7. 27 x rad/sec, ear th ' s r a t e of rotation

m = 7.347 x kg, moon's mass .

The latitude of Cape Kennedy is about 27" above the equator and the speed of Cape Kennedy in a space-fixed f r a m e of reference is approximately

( w r ) c o s (27"). So we calculate 0

fo r all t imes th rougho~~t separation.

The speed of the spaceship in a 100 nautical-mile orbit is about 7. 8 x lo3 m/sec. Hence, a calculation yields

The spaceship remains in the L O O nautical-mile orbi t for about 2 1/2 hours before insertion into an earth-moon trajectory. Therefore, fo r the f i r s t 2 1/2 hours we have

After insertion into an earth-moon trajectory and lunar orbit , the curves of Figure 1 show the actual distance-velocity profile fo r the Apollo 8 spaceship. These curves , along with the t ime, distance, and velocity data of Table 1, w e r e furnished by Mr . Williani D. McFadden of Mr. Robert Benson's group of Aero-Astrodynamics Laboratory at MSFC , Huntsville, Alabama. From these curves and data, d (Ar ) /dt was ultimately computed for each value of universal time2 in Table 1. Figure 2 is a plot of d ( Ar ) /d t a s a function of universal t ime from launch to insertion into lunar orbit. Also, Figure 2 shows d ( A r ) / d t a s a function of universal t ime for a typical lunar orbit. We compute the shaded a rea under the curves to obtain:

A r = 3.30 p-sec, from launch to about 2 3/4 hours af ter launch

= -158.00 p-sec, f rom 2 3/4 hours after launch to about 71 hours after launch

A r 3 e-2.30 p-sec, fo r a typical lunar orbit of about 2.10 hours.

We note that insertion into lunar orbit occurred about 71 hours after launch f rom Cape Kennedy. We assume that the return t r ip f rom the moon t o the ear th is nearly symmetr ical and that there were approximately 10 lunar orbi ts . Hence, we compute the integral, equation ( 1 6 ) , to be

2, Astronomical observations a r e usually re fer red to universal t ime, or Us T, , w i ~ i c h is the -mean so l a r t ime at the meridian of Greenwich.

'This means that tl~e astronauts aged 335 ,u-see more than an earth man at Cape Kennedy during their trip to the moon and back. It is interesting to note that if an astronaut made a similar trip to the moon and back every week for a period of time between 50 and 60 years, he would only age about one second more than if he had remained on earth. I do not thinli he would have to worry about old age creeping up on him for making s o many tr ips to the moon!

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6 T H I S GRAPH SHOWS A PLOT OF THE T I M E RATE OF T H I S PORTION OF CHANGE OF THE PROPER T I M E DIFFERENCE BETWEEN GRAPH S H W S A A POINT AT CAPE KEWNEDY AND THE APOLLO 8 TYB l CAL LUWAR

6 ASTRONAUTS AS A FUNCTION OF T IME OW THEIR ORBIT PROPER T R I P TO THE HOOW.

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0 10 20 40 5 0 60 70 0 I 2 T l M E IN HOURS AFTER APOLLO 8 LAUNCH TlME IN HOURS

Figure 2. Proper time difference as a function of universal time.

TABLE i. RELATIVISTIC TIME DILATION DATA FOR APOLLO 8

I I I I I DATA FOR A TYPICAL LUNAR ORBIT I I I I I

+U.T. = DEC. 21, 1969: 15.785 hrs. (EARTH-LUNAR TRAJECTORY INSERTION TIME **kl.T. = DEC. 24, 1969: 12.320 hrs. (LUNAR ORBIT IMSERTlON TIME 1 .

APPROVAL

RELATIV IST IC T I M E B ILATION ON LUNAR FLIGHTS

Fred Wills

The information in this report has been reviewed for security classifi- cation. Review of any information concerning Department of Defense o r Atomic Energy Commission programs has been made by the NiSFC Security Classifi- cation Officer. This report, in its entirety, has been determined to be unclas- sified.

This document has also been reviewed and approved for technical accuracy.

HENRY E. STERN Act. Cliief, Nuclear and Plasma Physics Division

DISTRIBUTION

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S&E-SSL-C R e s e r v e ( 15)

S&E-SSL-N Mr. Henry E. S t e r n D r . Daniel P. Hale

S&E-SSL-NR Mr . Mar t in 0. B u r r e l l

S&E-SSL-NP Dr. I l m a r s Dalins Mr. Eugene W. Urban

S&E-SSL-NA Dr . Nat Edmonson, Jr. Mr. L e s t e r Katz Mr. F r e d D. Wil ls ( 5) Dr . P e t e r Eby Dr . A. C . deLoach

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S&E-SSL-TT D r . Mona J. Hagyard

S&E-SSL-S Dr . W e r n e r S ieber Dr . 0. K. Hudson

S&E-AERO-FT Mr. Will iam D. McFadden Mr. Rober t He Benson Dr . Helmut G. L. K r a u s e

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National Aeronaut ics and Space Adniini s t ra t ion

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