Re: Gravity and the Sun

From: Randy Poe (poespam-trap_at_yahoo.com)
Date: 08/19/04 

Date: 19 Aug 2004 09:30:33 -0700

"JM Albuquerque" <jm.aREMOV.E@sapo.pt> wrote in message news:<2ohuksFb04c0U1@uni-berlin.de>...

> The problem is that Earth speed is a fundamental parameter
> in order that Earth orbit be stable.

Not true. Where you're going wrong is in not knowing
the celestial mechanics of non-circular orbits, such as
those followed by comets and very eccentric earth satellites.

> An object stopped at Earth distance from the Sun will fall to the
> Sun along a straight line and will not have a "circular" orbit.

That's true.

> Earth mass and its kinetic energy is fundamental.
> Without the right mass and the right speed the Earth will not
> orbit the Sun, but will either go away or fall straight to the Sun.

That's not true. If a satellite in circular orbit gives itself
a little boost with the onboard engines, it does not go
straight away or fall straight to earth. Instead what it has
done is kick itself into an elliptical orbit.

Your intuition is correct in telling you that when the
conditions for circular orbit are not met, the radius must
change. You are leading yourself astray in terms of what
happens next. Conservation of angular momentum turns out
to be mathematically equivalent to Kepler's equal area
law: the area swept out per unit time is a constant. Thus,
the satellite gets faster and faster as it falls inward,
eventually reaching closest approach and highest velocity.
Then it swings around and starts falling outward again,
slowing down in the meantime. There is a corresponding
point of farthest distance where the satellite moves
slowest.

This isn't the only possibility. Given enough energy, a
satellite can have a hyperbolic (open) orbit and never
return. Such an orbit will still have the property that
there is a very fast-moving point of closest approach.

Some pedant will point out that parabolic orbits are
also possible. These are at the boundary between ellipse
and hyperbola in energy.

> At this point it is clear that if the Sun gravity is switched off
> during a given period of time the Earth will go away and when
> the Sun gravity is switched on again the new orbit is no longer
> a stable orbit. The new orbit will have excess radius, or else
> a speed deficit.
>
> Angular momentum must be conserved.

Angular momentum is conserved in elliptical and hyperbolic
orbits.

> So, when gravity is switched on again we have already seen
> that Earth speed remains the same, but with a longer radius,
> so that the usual balance between the "gravity force" and the
> "centrifugal force" doesn't balance anymore.

And so it's not circular, but a different shape.

> - The Earth will spiral away forever because the angular
> momentum is conserved (same inertia), but the force of
> gravity decreases inversely proportional to distance.

Nope. Conservation of angular momentum is NOT "same inertia".
Linear momentum is not conserved in the presence of an
external force. F = dp/dt.

          - Randy

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