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Clearly this is a hypothetical question.

Say we bring a star baseball player into NASA, prep them appropriately for a mission in space, and fly them up to the International Space Station. They go on a spacewalk with a baseball, and at the apoapsis (highest, slowest point in the orbit) throw it retrograde as hard as they can. Could they decelerate the baseball enough that its periapsis (lowest, fastest point in the orbit) dips into Earth's atmosphere enough to de-orbit the ball over time?

(Let's assume that the ball must de-orbit within about 10 years or less. 10,000 years is too long. Also, let's neglect any loss in mobility that a space suit might cause.)

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  • $\begingroup$ wait, what about if you HIT the ball. So a clean pitch to Bo Jackson on his best day ... $\endgroup$ – Fattie May 5 '16 at 19:25
  • $\begingroup$ an interesting question unrelated to orbital mechanics is, would a baseball pitcher be faster in vacuum? does air resistance slow down baseball pitches much? what would the top pitcher's speed be on the moon? $\endgroup$ – Fattie May 5 '16 at 19:27
  • $\begingroup$ @JoeBlow Yes, air resistance has a huge effect on pitch speed as does gravity. Mitigating those will allow the ball to maintain its initial velocity for a much longer time. It will also be easier to throw a ball less gravitationally inclined. $\endgroup$ – cat May 5 '16 at 20:14
  • $\begingroup$ @JoeBlow Yes, air resistance has a big effect. This article claims that a baseball pitch slows down by about 9% during its flight. Of course, trying to pitch while wearing a space suite might be... difficult. $\endgroup$ – David Richerby May 6 '16 at 1:09
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You don't need to throw the ball!

At the altitude of the ISS the atmosphere is thick enough that it loses 50-100m of altitude every day due to the drag. At that rate over your ten year timescale the ISS would lose 180 to 360km. When you take into account the increased drag at lower altitudes ten years is enough to bring the ISS crashing to a fiery end.

So just put the ball in your pocket and wait.

Actually this is just as well because the orbital velocity of the ISS is a bit over 17000 miles per hour and the record speed for throwing a baseball (not in a spacesuit!) is only a shade over 100 mph.

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    $\begingroup$ Ok, good point. How much sooner than the ISS would the ball burn up if we threw it? $\endgroup$ – Martin Carney May 4 '16 at 16:31
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    $\begingroup$ @MartinCarney: That would be a very difficult calculation. It might even stay in orbit longer as I bet the drag to mass ratio of the ISS is higher than that of a ball. $\endgroup$ – John Rennie May 4 '16 at 16:35
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    $\begingroup$ @JohnRennie If we model the baseball as a sphere... Oh. $\endgroup$ – hBy2Py May 4 '16 at 18:32
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    $\begingroup$ @EmilioPisanty: Or additionally, would the ball de-orbit more quickly if you threw it behind you, or if you threw it downward? $\endgroup$ – Ellesedil May 4 '16 at 19:23
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    $\begingroup$ @Ellesedil Throwing the baseball down does not add energy, so all it does is change the ellipticity of the orbit while keeping the major axis constant. Depending on whether you're on the apogee or perigee, it will tend to make the orbit either more or less elliptical, with more elliptical orbits tending to de-orbit faster through their increased drag at perigee. Now, as to whether the de-orbit time increases compared to a backwards throw, though... $\endgroup$ – Emilio Pisanty May 4 '16 at 20:22
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John Rennie already gave the practical answer considering the atmosphere, noting that without doing anything objects near the ISS will deorbit quickly from drag. But that's letting reality get in the way of a good physics problem. I'll show that while a human can't send a ball crashing into the surface in one orbit, they can come close.

The ISS is listed as having a typical orbital velocity of $v = 7.66\times10^3\ \mathrm{m/s}$. For a test particle in orbit around the Earth, gravitational parameter $GM_\oplus = 3.98\times10^{14}\ \mathrm{m^3/s^2}$, a circular orbit (which we'll assume for concreteness and simplicity) will have a semimajor axis (i.e. radius) of $$ a_0 = \frac{GM_\oplus}{v^2} = 6.79\times10^6\ \mathrm{m}. $$ It will have a specific energy of $\epsilon_0 = -GM_\oplus/2a_0 = -v^2/2$, and a specific angular momentum of $h_0 = a_0v$.

Your first instinct1 might be to throw the ball down with speed $\Delta v$. This will add velocity perpendicular to its current velocity, so the change in energy is simple: $\epsilon_\mathrm{down} = \epsilon_0 + \Delta\epsilon_\mathrm{down}$, $\Delta\epsilon_\mathrm{down} = (\Delta v)^2/2$. Because the added velocity is along the radial direction, angular momentum does not change: $h_\mathrm{down} = h_0$.

Given $\epsilon$ and $h$, we can calculate the corresponding $a$ and $e$ (eccentricity) according to $$ a = -\frac{GM_\oplus}{2\epsilon}, \qquad e = \sqrt{1-\frac{h^2}{GM_\oplus a}}. $$ From there it's a simple matter to find the perigee according to $$ r_\mathrm{per} = (1-e) a. $$ Plugging in numbers for $\Delta v = 100\ \mathrm{mph} = 44.7\ \mathrm{m/s}$, this gets us $$ r_\mathrm{per,down} = 6.75\times10^{6}\ \mathrm{m}. $$

However, you can do better by throwing directly backward. This is the most efficient way to lower perigee. In this case, the new specific kinetic energy is $k_\mathrm{back} = (v-\Delta v)^2/2$, meaning the energy changes by $\Delta\epsilon_\mathrm{back} = k_\mathrm{back} - v^2/2$. The angular momentum also changes in this case: $h_\mathrm{back} = a_0 (v-\Delta v)$. Plugging in numbers gets us $$ r_\mathrm{per,back} = 6.64\times10^6\ \mathrm{m}. $$

Now it turns out the highest point on Earth is Chimborazo, with an elevation above the center of the Earth of $r = 6.38\times10^6\ \mathrm{m}$. Thus you could not force the ball to hit any part of the Earth with with velocity. However, the effect is not negligible. If we look at values of $r_\mathrm{per}/r$, we started out at $1.064$ and got to either $1.059$ (throwing down) or $1.040$ (throwing backward).

How far into the atmosphere does throwing backward get you? According to this tool for the NRLSISE-00 atmosphere model, the density of the atmosphere going from an altitude of $415\ \mathrm{km}$ to $259\ \mathrm{km}$ increases by a factor of about $30$. Thus whatever drag the ISS experiences can be increased quite a bit with just a small change to the orbit.

atmosphere density as a function of altitude

How fast would you need to throw the ball to deorbit it in the absence of an atmosphere? If we throw backward, our old $a = a_0$ will be our new apogee. We want our new perigee to be the altitude $r$. Solving $r_\mathrm{apo,per} = (1 \pm e) a$ for $a$ tells us we want a new semimajor axis of $a = (a_0+r)/2$. Backtracking through $\epsilon = -GM_\oplus/2a$ this tells us what the new energy is and thus what the change in velocity must be (since we can only change kinetic energy with an instantaneous impulse). The answer is $120.\ \mathrm{m/s} = 268\ \mathrm{mph}$. This relatively small number compared to the orbital velocity is a reflection of just how close low-Earth orbit is to the surface compared to the radius of Earth.


1Unless you've played Kerbal Space Program.

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    $\begingroup$ If one was in a circular orbit with altitude H, and wished to throw the ball with the smallest delta-V that would cause its perigee to be L, would throwing it backward be best? It's possible to change the perigee of an orbit without changing the energy if one also changes the apogee and vice versa, but I don't know if that would require more or less delta-V. $\endgroup$ – supercat May 5 '16 at 14:50
  • $\begingroup$ @supercat Perhaps, but I'm limiting myself to one impulse. $\endgroup$ – user10851 May 5 '16 at 20:07
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On a practical note, there was a space walk a few years back when they replaced a failed ammonia pump that was the size of a refrigerator. The astronaut simply gave it a swift push away from the station knowing that it would soon deorbit fast enough that it wouldn't be a collision hazard.

PS- If you have Kerbal Space Program, this would be a fun thing to test.

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    $\begingroup$ If you're talking about the ammonia pump failure on the ISS in 2010, that pump was returned safely via the space shuttle to NASA on Earth for inspection. $\endgroup$ – pentane May 4 '16 at 20:44
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    $\begingroup$ KSP actually led me to ask this question. I had some cases where separating my re-entry stage from the last engine + fuel tank would drop my re-entry pod's apoapsis too low, causing it to blow up due to reentry heating, instead of just aerobraking over a few passes. $\endgroup$ – Martin Carney May 4 '16 at 22:16
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    $\begingroup$ ...Which now that I think about it, doesn't quite apply the same on Kerbin as it does on Earth - Kerbin is 1/10 the size of Earth, and their atmospheres differ quite a bit too. $\endgroup$ – Martin Carney May 4 '16 at 22:48
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    $\begingroup$ @pentane As important as it is to replace the pump. In the longer term it is also very important to postmortem why the pump failed, for when they can't replace the pump (on Mars). $\endgroup$ – Aron May 5 '16 at 5:02
  • $\begingroup$ @pentane I just read that paper. I don't know why I did. But it was super cool! $\endgroup$ – corsiKa May 5 '16 at 14:58

protected by Qmechanic May 5 '16 at 10:47

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