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# How much were the clocks of Apollo craft affected by space tavel?

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I’ve always wondered if the Apollo clocks were slow on Earth clocks at splash down.

Objective: To calculate how much time dilation took place on the longest Apollo voyage (Apollo 17) to the Moon and back. First a timeline is required to work out the leg distances travelled. Blast off at 0533z (GMT/UT +0, zone Zulu) 7th December 1972.

Elapsed Time    Date     Time     Event

000hr   00m     07th      0533z   BLAST OFF (0033r/EST 7th December 1972 Kennedy SC)

000hr   07m     07th      0545z   Insert into 167km (90nm) altitude Earth parking orbit for 2 revolutions.

003hr   12m     07th      0845z   Insert into trans-lunar coast using 5m51s S-IVB engine burn, accelerating Apollo 17 from 25,000ft/s to 35,585ft/s Present altitude 97nm. S-IVB = 225,000lb thrust.

004hr   45m     07th      1018z   S-IVB separation, speed dropped to 22,000ft/s by gravity

073hr   18m     10th      0651z   Apollo 17 enters the Moon’s sphere of influence at velocity 2,354ft/s and 3,355ft/s, distance of 190,647nm & 33,803nm regarding the Earth and Moon respectively.

085hr   45m     10th      1918z   Speed 4,009ft/s and increasing, 7,900nm from the Moon

088hr   54m     10th      2227z   Apollo 17 arrives at the Moon travelling at 7,241ft/s, using 6m38s Service propulsion engine burn to brake (a 2988ft/s          reduction) and establish an initial elliptical lunar orbit of 52-170nm altitude, then nearly circular orbits between 54.5 and 69.7nm altitude. A total of 75 revolutions were made. Service propulsion (SP) engine = 20,500lb thrust.

236hr   42m     17th      0215z   Insert into trans-Earth coast using 2m 25s SP engine burn

Speed is over 8,067ft/s (5500mph).

250hr   40m     17th      1612z   Apollo 17 enters the Earth’s sphere of influence at velocity 3866ft/s and 2,937ft/s, distance of  33,822nm & 171,593nm regarding the Moon and Earth respectively

268hr   25m     18th      0958z   Velocity match of 3839ft/s with both Earth and the Moon

270hr   30m     18th      1203z   Time halfway point 132,654 and 78,504nm fr Earth/Moon

299hr   56m     19th      1729z   speed 10,359ft/s distance 33,435nm

301hr   18m     19th      1851z   Last mid co correction no7 using 9s burn of 2 RCS jets

301hr   38m     19th      1911z   speed 12,611ft/s and will near triple by re entry time

304hr   04m     19th      2137z   CSModule separation, altitude 1,781nm, velocity 20,486ft/s

304hr   08m     19th      2141z   speed 31,253ft/s increasing, 4153miles to go

304hr   19m     19th      2152z   Re-entry blackout at 448,800ft (85miles) speed 36,090ft/s.

Re-entry descent angle – 6.5 degrees, atmospheric braking effect is between 2.0g and 3.1g, splash zone 1044nm ahead

304hr   23m     19th      2156z   Re-entry blackout lifted, voice contact re-established. Speed 4,500ft/sec, braking at 3.1gs speed 4,000ft/s

304hr   26m     19th      2159z   2 drogue parachutes deployed at 23,000ft

304hr   27m     19th      2200z   3 Main parachutes deployed at 10,500ft, (22mph descent)

304hr   31m     19th      2204z   Splash down in the Pacific SE of Somoa, (18S 166W) 1104x (local) 19th December (WST, UT – 11, zone Xray)

The Apollo 17 timeline can now be used to find v in both the Earth and Moon orbits to find the distance travelled using distance = velocity x time (d = vt). So to find v:

(The sq root sign in the equations below should cover all giving v = √GM/r)

v = √GM         v = Velocity in metres per second

r             G = Gravitational constant (6.67384x10-11m/kg/s)

M = Mass kg Centre object (Earth=5.9726x1024kg  Moon=7.342x1022kg)

GM=G x M above works, but NASA GM(Earth or Moon) more accurate

GM Earth = 3.986005x1014m3/s2

GM Moon = 4.902794x1012m3/s2

r = Earth radius 6371 km + 167km orbit alt = 6.538x106 metres

r = Moon radius 1737.1 km + 115km orbit alt = 1.8521x106 metres

Earth:    V=√GM          V = √ 3.986005x1014       V= √60966733       V= 7808.12 m/s

r                       6.538x106

Moon: V = √GM          V = √4.902794x1012          V= √2647154        V= 1627 m/s

r                       1.8521x106

Earth orbit time 3hr 5m = 11100s   Dist travelled is 11100s x 7.08012km = 78589km

Moon orbit time 147hr 48m = 532080s   Dist travelled 532080s x 1.627km = 865694km

Moon’s orbital velocity around Earth = 1.022km/s. Dist 532080s x 1.022km=543786km

Earth / Moon distance there & back is 384400km x 2 = 768800km

Adding all four distances to give Apollo 17 com module voyage total d = 2256869km

From the timeline, Apollo 17 command module total time t = 304hr 31m = 1096260s

Now to find voyage average velocity using v = d/t

v = 2256869/1096260    v = 2.0661 km/s   average voyage velocity = 2066m/s (rounded)

Special Relativity

Special relativity means moving clocks run slow. The time dilation equation is below:

(The sq root sign in the equations below should cover all, giving ∆t0 = ∆t√ 1 – v2/c2)

∆t0 = ∆t√1 – v2            ∆t0 = spacecraft time interval     seconds

c2            ∆t = Earth based time interval seconds (1096260s in this case)

v = spacecraft velocity m/s (2.066x103 m/s)   (2.1x103) rounded

c = speed of light (2.99792458x108 m/s)    (3.0x108) rounded

The spacecraft velocity is too small to put directly into the equation, so in this case is multiplied by 10,000. However, because of the nature of the equation, any multiplication is squared and the 104 multiplication becomes 108 in the resultant. Care is needed.

∆t0 = ∆t√1 – (2.1x107)2            ∆t0 = ∆t√1 – (4.41x1014)          ∆ t0 = ∆t√1 – (4.9x10-3)

(3.0x108)2                                   (9.0x1016)

∆ t0 = ∆t√0.9951          ∆ t0 = 1096260x0.997546991 ∆ t0 = 1093570.865sec

∆t - ∆t0 = Time diff     1096260 – 1093570.865 = 2689s   now to undo the 104 therefore multiply 2689 by 10-8   So the clocks on Apollo 17 were slowed a total of 2.7x10-5 s (rounded) by special relativity. The next task is to take into account general relativity.

General Relativity

General relativity means the stronger a gravity field is, the slower clocks run.

The equations for general relativity are much more complicated than for special relativity. Einstein showed by calculation the effect on the orbit of Mercury due to the gravity field of the Sun (0.04m/s at Mercury), but to avoid the difficult calculations data from the NASA Gravity Probe A which was sent to a height of 10,000km in1976 will be used. The probe showed clocks should run faster with respect to Earth by (4.5x10-10) x (time interval) at an altitude of 10,000km above the Earth.

First, it is necessary to calculate the strength of the Earth’s gravity field at 10,000km alt.

F = GME         F= Force of Gravity (field strength)

r2            GME = Gravitational Constant x Earth’s Mass = 3.986x1014 m3/s2

r = Earth’s radius (6.371x106m) plus altitude (1.0x107m)

F = 3.986x1014            F = 3.986x1014

(1.6371x107)2               2.6801x1014

F = 1.487 m/s

Secondly it is necessary to calculate the strength of the Moon’s gravity field at Apollo 17’s orbit altitude of 115km above the Moon.

F = GMM         F = Force of Gravity (field strength)

r2             GMM = Gravitational Constant x Moon’s Mass = 4.9x1012 m3/s2

r = Moon’s radius (1.7371x106m) + altitude (1.15x105m) = 1.8521x106

F =  4.9x1012               F = 4.9x1012

(1.8521x106)2              3.43x1012

F = 1.429 m/s

The gravity field strength while outside of Earth’s orbit was less than the Earth’s gravity field strength at 10,000km above the Earth. The Apollo 17 total time outside Earth orbit is 304hr 15m – 3hr 15m = 301hr = 1083600s.

Using the Gravity Probe A result: 4.5x10-10 x 1083600 = 4.876x10-4s spacecraft clock speed gain due to general relativity.

Therefore the Apollo 17 Command Module experienced a 2.7x10-5s clock time loss due to special relativity, but a 4.876x10-4s clock gain due to general relativity.

This means the Apollo 17 clocks would have been ahead of Earth clocks by 4.606x 10-4s on splash down.

No time dilation occurred on Apollo 17 with respect to Earth therefore, in fact, quite the opposite. As a footnote, I have just found a more accurate, official distance for the Apollo 17 voyage as 2389769 km, which is 132,900km more than my figure. This 5.5% difference does not make a significant change to the findings though.

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Not my field, but in special relativity doesn't the true form of the 'twins paradox' apply? The problem being that there is no way in SR of deciding that one twin moved and one stayed still, since movement is relative. So if, in SR, 'moving clocks run slow' which one is defined as moving? We can only state that the two clocks will differ but we cannot say that either of them is definitive.

Of course in GR we can distinguish between the twins because the 'Apollo Twin' was accelerated while the Houston Twin was not.

Olly

Edited by ollypenrice

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Not my field, but in special relativity doesn't the true form of the 'twins paradox' apply? The problem being that there is no way in SR of deciding that one twin moved and one stayed still, since movement is relative. So if, in SR, 'moving clocks run slow' which one is defined as moving? We can only state that the two clocks will differ but we cannot say that either of them is definitive.

Of course in GR we can distinguish between the twins because the 'Apollo Twin' was accelerated while the Houston Twin was not.

Olly

Good point. In special relativity the twins paradox applies. The problem of deciding which twin moved and which twin stayed still is impossible. However we can say the lengthening of the time interval, due to special relativity, between the launch and splash down measured by the Apollo clocks with respect to the clocks on Earth implied that the moving Apollo clocks slowed down.

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would time not of speeded up for them as they moved away from earths gravity, stabailzed in zero g then slowed at the moon and speeded up  on the way back once more until back at earth when they would of been normal on landing?

what effect does acceleration and deceleration have during these gravitational differnces?

Would they, if there was a measurable effect actually be a little foward in time?

Edited by Earl

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I would expect the dilatory differences due to the differing gravity Wells to be so low as to be immeasurable.

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That's a very detailed analysis Spartan45.

For what it is worth, I've done a bit of Googling and I found, according to Wikipedia (!), that the record for time dilation during manned space flight is held by the Russian Cosmonauts Sergei Avdeyev and Sergei Krikalev. Their cumulative missions mean that they have each aged about 20mS less than those on earth!

Ian

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I would expect the dilatory differences due to the differing gravity Wells to be so low as to be immeasurable.

I don't know. Atomic clocks are known to differ once one is taken down a shaft and closer to the earth's C of G.

Olly

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would time not of speeded up for them as they moved away from earths gravity, stabailzed in zero g then slowed at the moon and speeded up  on the way back once more until back at earth when they would of been normal on landing?

what effect does acceleration and deceleration have during these gravitational differnces?

Would they, if there was a measurable effect actually be a little foward in time?

This interesting question requires an answer covering the two effects influencing the Apollo craft’s clocks. First, general relativity: As you rightly say, as the distance from the Earth increases, the gravity field weakens, having the effect of speeding up the Apollo craft’s clocks. Once at the point in the timeline where the gravity influence between Earth and Moon is crossed the gravity field strengthens due to the Moon ahead. This has the effect of slowing the Apollo clocks. The process is reversed on the return trip.

Second, special relativity: As the departure velocity relative to the Earth decreases due to gravity pull, the special relativity effect of slowing the Apollo clocks decreases, but the special relativity effect increases again (slowing the clocks) as Apollo starts to gain speed from the Moon’s gravity pull and orbital velocity. On leaving the Moon, Apollo is slowed by the Moon’s gravity pull, the special relativity effect of slowing the Apollo clocks decreases, but the special relativity effect increases again (slowing the clocks relative to Earth) as Apollo starts to gain approach velocity from the Earth’s gravity pull. What is important here is the Apollo craft’s velocity relative to the Earth, meaning the higher the relative velocity with respect to Earth, the bigger the special relativity effect.

Understanding direction of travel changing the relative velocity is challenging. The best example I can think of is the experiment carried out in the 70’s using an aircraft heading W and then repeating the procedure heading E. Atomic clocks were carried on the aircraft having been synchronised with a master atomic clock at an observatory in the United States. The clock having travelled W all around the world showed a reading of about 96 nano seconds more and the E travelling about 184 nano seconds less than the observatory master due to special relativity. This is because the W bound plane was moving against the speed of the Earth’s rotation while the E bound plane was adding to it. So the W bound plane had a relative velocity less than the observatory master, speeding the plane Atomic clocks up, while the W bound plane had a relative velocity greater than the observatory master, slowing it down. I hope this helps.

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That's a very detailed analysis Spartan45.

For what it is worth, I've done a bit of Googling and I found, according to Wikipedia (!), that the record for time dilation during manned space flight is held by the Russian Cosmonauts Sergei Avdeyev and Sergei Krikalev. Their cumulative missions mean that they have each aged about 20mS less than those on earth!

Ian

Thanks for your post, it made me think that perhaps the command pilot of Apollo 17 holds the record for negative time dilation meaning he aged 0.00046 seconds more than those on Earth.

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For what it is worth, I've done a bit of Googling and I found, according to Wikipedia (!), that the record for time dilation during manned space flight is held by the Russian Cosmonauts Sergei Avdeyev and Sergei Krikalev. Their cumulative missions mean that they have each aged about 20mS less than those on earth!

Ian

Only a matter of time before the L'oreal 'scientists' recommend it as an anti-ageing treatment then.

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Yes, the astronauts were in lower gravity, which made their clocks seem to run faster, but at higher speed, which made their clocks seem to run slower. Had they travelled at a constant speed, and their gravity been constant, a calculation of their time dilation might be straightforward.

As it is, the calculation is difficult.

However, I once read that the trip to the Moon and back lasted about one second less for the astronauts than for their colleagues who stayed behind in Houston.

This may interest you: Sky and telescope once published Rocket.bas, a BASIC program that calculates travel times for a spaceship travelling at a constant acceleration of 1 g. I turned it into an Excel worksheet which is included below. The units used are light-years for distance and years for time.

This is the worksheet: Einstein_Rocket.xls

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Hang on, acceleration and gravity are equivalent and the crew were accelerated (both positively and negatively, though it doesn't matter which) during their flight.

Olly

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This interesting question requires an answer covering the two effects influencing the Apollo craft’s clocks. First, general relativity: As you rightly say, as the distance from the Earth increases, the gravity field weakens, having the effect of speeding up the Apollo craft’s clocks. Once at the point in the timeline where the gravity influence between Earth and Moon is crossed the gravity field strengthens due to the Moon ahead. This has the effect of slowing the Apollo clocks. The process is reversed on the return trip.

Second, special relativity: As the departure velocity relative to the Earth decreases due to gravity pull, the special relativity effect of slowing the Apollo clocks decreases, but the special relativity effect increases again (slowing the clocks) as Apollo starts to gain speed from the Moon’s gravity pull and orbital velocity. On leaving the Moon, Apollo is slowed by the Moon’s gravity pull, the special relativity effect of slowing the Apollo clocks decreases, but the special relativity effect increases again (slowing the clocks relative to Earth) as Apollo starts to gain approach velocity from the Earth’s gravity pull. What is important here is the Apollo craft’s velocity relative to the Earth, meaning the higher the relative velocity with respect to Earth, the bigger the special relativity effect.

Understanding direction of travel changing the relative velocity is challenging. The best example I can think of is the experiment carried out in the 70’s using an aircraft heading W and then repeating the procedure heading E. Atomic clocks were carried on the aircraft having been synchronised with a master atomic clock at an observatory in the United States. The clock having travelled W all around the world showed a reading of about 96 nano seconds more and the E travelling about 184 nano seconds less than the observatory master due to special relativity. This is because the W bound plane was moving against the speed of the Earth’s rotation while the E bound plane was adding to it. So the W bound plane had a relative velocity less than the observatory master, speeding the plane Atomic clocks up, while the W bound plane had a relative velocity greater than the observatory master, slowing it down. I hope this helps.

This is a retype to correct the W to read E on the last line :

This interesting question requires an answer covering the two effects influencing the Apollo craft’s clocks. First, general relativity: As you rightly say, as the distance from the Earth increases, the gravity field weakens, having the effect of speeding up the Apollo craft’s clocks. Once at the point in the timeline where the gravity influence between Earth and Moon is crossed the gravity field strengthens due to the Moon ahead. This has the effect of slowing the Apollo clocks. The process is reversed on the return trip.

Second, special relativity: As the departure velocity relative to the Earth decreases due to gravity pull, the special relativity effect of slowing the Apollo clocks decreases, but the special relativity effect increases again (slowing the clocks) as Apollo starts to gain speed from the Moon’s gravity pull and orbital velocity. On leaving the Moon, Apollo is slowed by the Moon’s gravity pull, the special relativity effect of slowing the Apollo clocks decreases, but the special relativity effect increases again (slowing the clocks relative to Earth) as Apollo starts to gain approach velocity from the Earth’s gravity pull. What is important here is the Apollo craft’s velocity relative to the Earth, meaning the higher the relative velocity with respect to Earth, the bigger the special relativity effect.

Understanding direction of travel changing the relative velocity is challenging. The best example I can think of is the experiment carried out in the 70’s using an aircraft heading W and then repeating the procedure heading E. Atomic clocks were carried on the aircraft having been synchronised with a master atomic clock at an observatory in the United States. The clock having travelled W all around the world showed a reading of about 96 nano seconds more and the E travelling about 184 nano seconds less than the observatory master due to special relativity. This is because the W bound plane was moving against the speed of the Earth’s rotation while the E bound plane was adding to it. So the W bound plane had a relative velocity less than the observatory master, speeding the plane Atomic clocks up, while the E bound plane had a relative velocity greater than the observatory master, slowing it down. I hope this helps.

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It's worth having a look at the link that JamesF gave in post #6. Even though the maths is a torture, Figures 2 & 4 on pp 6-7 nicely show how the time shift varies according to the phase of flight, and Tables 1 & 2 list the time deviations due to both the gravity effect and the velocity effect. In fact, it's a case of a picture speaking a thousand (indecipherable) words!

Ian

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It's worth having a look at the link that JamesF gave in post #6. Even though the maths is a torture, Figures 2 & 4 on pp 6-7 nicely show how the time shift varies according to the phase of flight, and Tables 1 & 2 list the time deviations due to both the gravity effect and the velocity effect. In fact, it's a case of a picture speaking a thousand (indecipherable) words!

Ian

This is an excellent post which gives focus to JamesF information link. Using table 1 (page 4) it shows the clock time correction (advance) of Apollo 12 command module versus Earth standard was 5.70345 x 10-4 seconds on splash down.

Using table 2 (page 5) it shows the clock time correction (advance) of Apollo 13 command module versus Earth standard was 3.27559 x 10-4 seconds on splash down.

The Gravity Probe A figure of (4.5x10-10) x time interval to give the Apollo 17 clock advance appears too small, so calculating from table 1 a figure of (6.85x10-10) x time interval to give the Apollo 17 clock advance (using the Apollo 17 timeline interval): (6.85x10-10) x 1083600seconds = 7.42266x10-4 seconds Apollo 17 clock advance due to general relativity. Subtracting the 2.7x10-5 sec Apollo 17 clock time loss due to special relativity means the Apollo 17 clocks would have been in advance of Earth clocks by 7.15 x 10-4 seconds on splash down, a significant increase due to the revised general relativity figures.

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