From: Anthony Jackson <[EMAIL PROTECTED]>
Zan Lynx wrote:
> You are absolutely certain that a hyperbolic solar orbit with aphelion
> preceding Earth so that Earth captures it at its slowest point would
> always exceed LEO?
For Earth, and assuming we're talking about something that actually ends
up at an altitude adequate to do a re-entry, yes.
For the benefit of those interested, there are some rules of thumb about
orbital mechanics that make guesstimating about these kinds of scenarios
possible with fairly simple math.
* measure distance for gravity purposes center-to-center
* gravity follows the inverse square law for distance: 2 times farther
it is at 1/4 strength, 3 times farther 1/9 strength, 1/2 as far it is 4
times stronger, etc
* gravity is linear stronger as planets get more massive - the earth is
80 times more massive than the earth, so gravity is 80 times stronger
for the same distance (center to center that is)
* orbital speed also follows the inverse square law for distance -
doubling the orbit radius means 1/4 the speed in miles per second -
again, radius is measured center to center, even low earth orbit is over
4000 miles from the earth's center
* escape velocity is always 40% faster than orbital velocity (multiply
by square root of 2 for exact ratio) - this means that escape velocity
also decreases to 1/4 speed if the distance from the planet center doubles
* escape velocity is also maximum falling velocity for the planet, if
the item had started indefinitely far away and had fallen toward earth
under influence of earth's gravity only
* this maximum falling speed is why the item orbiting the sun that would
happen to meet the earth ends up meeting it moving so fast - their
orbits around the sun are about the same speed, but it falls toward the
earth for a zillion miles an gets most of that maximum falling distance
plus whatever head start it had that let it approach the earth to
begin with = greater than escape velocity
* the combination being greater than escape velocity is why asteroids
and such don't just end up captured like in a lot of stories - they just
escape again for the most part
* fall-toward-and-then-escape-again doesn't happen in a straight line -
faster-than-escape velocity orbits always make a curve called a
hyperbola, and it is the same thing as a gravity slingshot maneuver - it
always leaves the planet as fast as it arrived with respect to the
planet and tiny bits of thrust can make large changes in the approach
and departure angles that define the hyperbola, so in the sense of
getting around the sun and to other planets it can add or subtract some
or all of the slinging planet's orbital velocity around the sun
* speeds higher than orbital velocity but below escape make for an
elliptical orbit rather than a circular one - going that bit faster
makes the far side of the orbit higher. of course since that side is
farther away, orbital velocity there is lower, but the object in
elliptical orbit is going that tiny bit less still, so that it falls in
toward the planet, eternally oscillating between these two extremes. the
degree faster and slower controls how eccentric (how oblong) the ellipse is
--
Jeff Wilson - [EMAIL PROTECTED]
< http://www.io.com/~jwilson >
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