From: Lee Corbin (lcorbin@tsoft.com)
Date: Wed Feb 12 2003 - 21:00:30 MST
Damien writes
> [mailto:owner-extropians@extropy.org]On Behalf Of Damien Broderick
> Sent: Wednesday, February 12, 2003 10:12 AM
> Earth spins counterclockwise or East (looking down from the N pole). The
> geostationary lump orbits in the equatorial plane counterclockwise, matching
> exactly. It also might start off (but not necessarily, for angular momentum
> reasons Spike announced) with one face tide-locked to the Earth.
>
> Imagine it's a small globe and it starts getting squeezed into an egg/oval
> with long axis normal to the surface; this lengthens until the whole lump is
> a huge cable with a double bell-curve cross section. I assume it's
> progressively forced to precess East until the long axis of the lengthening
> egg is tangent to the Earth.
>
> Why? Because the lower tip of the egg is moving at 5500 km/hr but while it
> remains lower than the core body it's circling the Earth in an orbit where
> it only needs to move at 4000 or 3000 km/hr, so it's kicked upwards in an
> easterly direction. Meanwhile, the upper tip is going too slowly, so it
> drags behind to the west.
Good description.
I suggest considering the force of gravity on the Eastern-most
(i.e. lower) segment, its present velocity must slow by and by,
or it will gain the same height as the center of the body at
geosynchronous orbit. So I see the bottom part presently
falling into line (between the upper geosynchronous part and the
center of the Earth). [But see below]
> I first imagined the lower part swinging all the way up and over, and the
> upper part vice versa (except that it's longer, to compensate for the
> lessened gravity farther from the Earth, which makes matters even more
> difficult), and then swinging on, with the rotation slowing but just maybe
> still moving fast enough as it neared the ground to < ahem > tear your head
> off if you incautiously got in its way. :)
We need to be more precise about initial conditions, I'm
afraid. First, a good picture. If O represents the Earth,
then the initial state of the small satellite can be rep-
resented by a period, as in this diagram:
O .
The satellite is a LONG way away. The dynamics get very
complicated if it begins to extend projections both towards
the Earth (down or left in the diagram) and towards the
moon (up or right). But let us suppose that we have spun
up the satellite in the right way as Spike suggested. Then
as the arms become longer, it loses angular momentum about
its center. I guess that in the last few passes past Earth,
the lower arm is moving very slowly. Maybe it could be
grabbed as it whips through the atmosphere.
But note that in this scenario there has been enough time---
the arms have grown gradually enough---so that they remain
straight (i.e. are not curved).
My remarks at the top of this post, as perhaps yours too,
refer to the scenario where we are presented with a fait
accompli: extensions from the satellite already sweep
down half way to Earth and as far in the other direction,
and, they have the velocity of the satellite, and there
is no increase in angular momentum except that which will
arise as a result of the initial placement. What will
happen? Like you say, a tendency to assume the shape
of an integral sign will occur, with the lower parts
attempting to complete a whole orbit sooner than the
higher parts, because, at first, they all have the same
velocity.
If the extensions are much smaller, maybe I can cope with
them: suppose that each arm is just 440 miles long, or
about a fiftieth of the Earth-Geosynch distance.
O .
If the arms are suddenly snapped into place, and they
start to gradually drift below-East and above-West
(the former leading and the latter trailing the
satellite), then you may be right and this would still
impart so much angular momentum to the satellite that
it would begin to rotate, arms and all.
Lee
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