Re the tower-jump-to-orbit maneuver, or the minimum altitude for
zero-rocket-assist orbital insertion:
Spike Jones (email@example.com)
Wed, 01 Mar 2000 20:16:31 -0800 wrote:
>If a skyhook existed, one could climb sufficiently
>high on it, then let go of the cable and fall into orbit. I just did a
quick back of the envelope calc
>on that and it looks like if you climb one Earth diameter up the cable and
let go, one would fall
>into a minimal orbit.
>It was my impression that if one wanted a reasonably circular orbit, the
only altitude one could
>jump into orbit from is the altitude of goesynchronous orbit. With any
lower altitude, you need a
>lot more sideways velocity for a circular orbit then sitting at the top of
a tower that tall gives
I emailed Darin a little note, to wit:
>Spike didn't say that the orbit was circular. <snip>... the orbit is
>You're right, that a jump from a geosynchronous-orbit-tall tower will give
you a circular orbit. >In fact, all you have to do is raise your feet off
the platform, cause, hey, you and the platform
>are already in orbit. No jumping to it.
Then spike replied (to Darin, not to me):
>That's a good question, one that I want to work on. If one went up about
an earth diameter on a
>cable and let go, what would be the shape of that orbit? Also, I have
sharpened my calcs a bit
>and found that the whole question of climbing a cable and letting go to
get into a minimal orbit
>might be wrong: I found a flaw in my reasoning today.
So I puzzled over it for a week, and found no flaw in spike's reasoning,
or, to be more precise, no flaw in his conclusion. Quite elegant, actually.
According to Kepler and Newton all planetary orbits are elliptical (a
circle being a special case of an ellipse), with the center of the orbited
planet located at one of the foci. If one is considering orbits around the
earth, then the earth is at one focus. If one then imagines an earth-sized
sphere adjacent to the earth, its center defines a second focus, and the
furthest out point on the earth-sized sphere--the point diametrically
opposite the point of contact between the earth-sized sphere and the real
earth--can be chosen to define one point--the extreme end--of an ellipse.
Then, by symmetry the other end of the ellipse just grazes the atmosphere
at the far side of the real earth. Now, by Newton and Kepler, this ellipse
is a valid earth orbit, and the condition of it grazing the atmosphere of
the real earth makes it the minimum viable orbit for the condition of
climbing the tower and just letting go. Thus, one comes up with the
earth-diameter high tower that spike derived. Stepping off the tower at
that height corresponds to entering the orbit at the apogee. Just grazing
the earth at the other end is the perigee. Nifty.
Best, Jeff Davis
"Everything's hard till you know how to do it."
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