- #1
cliowa
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The situation is this: Two space vehicles A and B are doing a race in space. A is in front of B and they are both in orbit around the earth. For simplicity let this orbit be a circle (i.e. neglectable eccentricity). Now, B wants to get past A, that is, B wants to cross the "line" connecting the center of the Earth and A in his orbit. That means, he'll catch up a higher angular velocity \omega. B has two choices: he can break or he can accelerate (both in tangential dierction). What should B do?
For simplicity ignore all details: B won't lose any mass, and he'll stay in an (approximately) circular orbit.
One approach would be this: B could brake instantaneously and thereby lower his angular momentum L=mrv=mr^{2} \omega, which would get him behind and then make him "fall" to a lower orbit, picking up speed. Looking at \omega as a function of L, one gets: \omega(L)=C\cdot L^{-3}, C being some constant involving the masses etc. but not the radii. This would lead to argue: If L goes down (braking) \omega must go up and B will pass by A.
Now, I'm not so sure about the validity of this argument, as there are several critical points. First of all: If B brakes instantaneously, i.e. his speed v decreases, so does his L (that's ok so far). But this also means, that \omega would decrease. How could I tell that the increase of \omega caused by falling towards the Earth is bigger than this decrease?
Second: Are the simplifications made here ignoring important aspects or even not valid at all? This whole low eccentricity thing might spoil our argument.
Thanks for any comments. Best regards...Cliowa
For simplicity ignore all details: B won't lose any mass, and he'll stay in an (approximately) circular orbit.
One approach would be this: B could brake instantaneously and thereby lower his angular momentum L=mrv=mr^{2} \omega, which would get him behind and then make him "fall" to a lower orbit, picking up speed. Looking at \omega as a function of L, one gets: \omega(L)=C\cdot L^{-3}, C being some constant involving the masses etc. but not the radii. This would lead to argue: If L goes down (braking) \omega must go up and B will pass by A.
Now, I'm not so sure about the validity of this argument, as there are several critical points. First of all: If B brakes instantaneously, i.e. his speed v decreases, so does his L (that's ok so far). But this also means, that \omega would decrease. How could I tell that the increase of \omega caused by falling towards the Earth is bigger than this decrease?
Second: Are the simplifications made here ignoring important aspects or even not valid at all? This whole low eccentricity thing might spoil our argument.
Thanks for any comments. Best regards...Cliowa
