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This answer to the question starts with:

This video may help to answer your question. Starting at about 00:24, you can see an astronaut running around the "exercise wheel" of Skylab (an early NASA space station program, which followed the Apollo moon landings). Basically after some time, NASA told the astronauts to stop running around like this because it was causing more propellants to be used to maintain Skylab's correct attitude (orientation) in space (at least this is what I've been told...it would be nice to find a reference to back this claim up).

From this answer which shows the following screen shot from page 120 in this large, 436 page PDF of NASA Technical Memorandum NASA TM-X 64817: MSFC Skylab Attitude and Pointing Control System Mission Evaluation:

enter image description here

Question: How many times would two astronauts have to run around on this track to turn Skylab by 10 arc minutes around its long axis?

The question is a bit challenging and may be difficult to answer, depending on the orientation of the long axis with respect to the orbital plane and direction of motion. If necessary to simplify answering, one can choose it to be perpendicular to the orbital plane.

Classic NASA Film - Skylab - #4, jogging

Source: Classic NASA Film - Skylab - #4

Clarifications that were captured in comments which have since been moved:

  • This question is about rotation, it’s not about the oscillation. Rotation around the long axis by 10 arc minutes changes the direction of the other two axes by that amount.

  • When the astronauts stop “rotating” the spacecraft will also stop rotating, but it will have been rotated by a small amount.

  • Assume that the two astronauts want to do this on purpose and have disabled or adjusted the attitude control system so as not to fight or resist this rotation.

  • Assume that the astronauts are running in the same direction spaced by 180 degrees, opposite each other.

  • If it takes a long time, there may be some complexities depending on if Skylab is currently operating in an Earth-oriented or Sun-oriented attitude (for more see the linked NASA Technical Memorandum). You may make some simplifying assumptions if it helps.

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$
    – called2voyage ♦
    May 8, 2019 at 14:07
  • $\begingroup$ key clarifications in those comments have been added back into the question. $\endgroup$
    – uhoh
    May 9, 2019 at 1:36

2 Answers 2

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To create the oscillation around the Z axis, they'd have to run around once.

The oscillation in the X axis would be created with each footfall.

Skylab was about 1000 times as massive as an astronaut. Estimating the astronauts centre of mass to be at about three quarters of the circumferance, and the distribution of Skylab's mass to be skewed to around seven eigths from the centre, then:

The acceleration to set two astronaut rotating is three halves, by eight over seven thousands, so the astronaut would have to go around $\frac{1750}{3}$ times for each rotation of Skylab we only want Skylab to go around a sixth of a degree, so the astronaut would go around $\frac{1750}{3(6\times360)}$ times in the time Skylab took to rotate 10 arc-minutes - or about a quarter time around.

oops, it's r-squared, not r. So that makes it five eighths of a circuit.

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  • $\begingroup$ Okay but I've asked about a rotation, not an oscillation. $\endgroup$
    – uhoh
    May 6, 2019 at 10:27
  • $\begingroup$ "Question: How many times would two astronauts have to run around on this track to turn Skylab by 10 arc minutes around its long axis?" I'm asking exactly this question. The oscillations are interesting and I encourage you or anyone else to ask a new question about it. $\endgroup$
    – uhoh
    May 6, 2019 at 10:37
  • $\begingroup$ @uhoh In my interpretation, your question title is about orientation, which is not oscillation and also not rotation. :-) $\endgroup$
    – peterh
    May 6, 2019 at 12:55
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    $\begingroup$ @peterh orientations are usually described as a sequence of rotations. $\endgroup$
    – uhoh
    May 6, 2019 at 12:56
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    $\begingroup$ The use of “oscillation” is infelicitous, but the calculation (within its approximations) is right. Running then stopping will cause Skylab to rotate the other way then stop after some angular displacement. $\endgroup$
    – Bob Jacobsen
    May 9, 2019 at 1:54
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tl;dr: I get good agreement with @JCRM's estimate!

Moments of inertia found in NASA TM X-64746 Skylab Attitude Control and Angular Momentum Desaturation with One Double-Gimbaled Control Moment Gyro also found archived here:

                                   Command/Service Module
                                      undocked  docked
Minimum Moment-of-Inertia       Ix     0.893     0.992      x 10^6 kg m^2
Intermediate Moment-of-Inertia Iy      3.813     6.168      x 10^6 kg m^2
Maximum Moment-of-Inertia      Iz      3.882     6.245      x 10^6 kg m^2

I assume that it is Ix that corresponds to rotation about the "long axis".

Starting with a diameter of 6.6 meters, I estimate the track radius to be 2.8 meters and the pair of astronauts' effective centers of mass to be 0.9 meters higher at 1.9 meters. To do it right you'd have to integrate $r^2 \frac{dm(r)}{dr} dr$.

I get the moment of inertia for a pair of 65 kg astronauts to be 470 kg m^2, which is about 2,110 times smaller than the moment of inertia of the ISS around the x or long axis.

note: Thanks to @JCRM for noting that it's necessary to use the moments of inertia when the Command/Service module is docked. That's how the astronauts got there in the first place!

10 arc minutes is 2,160 times smaller than a complete circle.

So to rotate Skylab by 10 arc minutes, the two astronauts would have to "jog" only 0.98 of one lap.

Skylab's moments of inertia

Skylab's moments of inertia

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  • $\begingroup$ X was the cylindrical axis $\endgroup$
    – user20636
    May 9, 2019 at 7:02
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    $\begingroup$ @BobJacobsen moment of inertia must be defined relative to some specific center of rotation. In the answer above, I've calculated the moment of inertia of a pair of astronauts rotating about Skylab's central axis whereas in your reference I'm guessing it's the moment of inertia of a single human about their own center of mass. Apples and oranges. $\endgroup$
    – uhoh
    May 9, 2019 at 13:33
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    $\begingroup$ Bob's source does act as a source for an average weight of Mercury era American airmen being 75 kg $\endgroup$
    – user20636
    May 9, 2019 at 13:48
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    $\begingroup$ it also gives their center of mass as 0.98 m above the floor when standing (well, 31 inches down a 69.4 inch body), $\endgroup$
    – user20636
    May 9, 2019 at 13:55
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    $\begingroup$ @BobJacobsen now I won't be able to sleep until I can get to the bottom of this ;-) Ah, I think what you are talking about will indeed agree with the integral, and it is my collapsing the astronauts to points that "looses" some of the moment that can be recovered by considering the moments of each astronaut about their centers plus the moment of the centers around the axis. I think I've got it now. $\endgroup$
    – uhoh
    May 9, 2019 at 13:55

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