From: Adam Stephanides <adamsteph@earthlink.net>
Subject: (whorl) Re: Green's orbit
Date: Sun, 09 Apr 2000 17:47:12 

Apologies in advance to those bored by "techie stuff"; you can just skip
this message.

Kieran Mullen wrote:

[I wrote:] 
> > Furthermore, as someone
> > else once pointed out, if Blue and Green's orbits brought them into
> > conjunction as frequently as every six years, they would not be stable:
> > the repeated perturberations from each other's gravity would pull them
> > into new orbits.
> >
> 
>     Hmm... I don't think this is correct.  It is true that if their periods were
> exactly integer multiples, you would have a problem (like the Shepherd Moons and
> the Cassini divisions of Saturn's rings).  But if they are not integer multiples
> you should be ok, I think.  (After all, the gaps in Saturn's rings do not deny
> the existence of the rings themselves!)

After reading your reply, I realized I didn't really know why
"resonances" (to use the technical term) had the effects they did.  So I
did a little research, beginning with _Newton's Clock: Chaos in the
Solar System_ by Ivars Peterson, and going on to what non-technical
articles dealing with the gaps in the asteroid belt (which seem to be
better understood, and are a better analogy to our hypothetical
situation) I could find in my local academic library.  What I found out
is that the situation is much more complex than I had realized.

In the first place, the picture I had in my head of repeated
gravitational tugs gradually pulling the asteroid out of orbit is
completely wrong.  In the case of the 3/1 gap in the asteroid belt
(where the period of the asteroid is a third that of Jupiter's), where
the dynamics are the simplest and best understood, in numerical
simulations an asteroid can spend up to a million years in a
near-circular orbit and then jump to an eccentric orbit crossing
Mars's.  Moreover, to explain why there is a gap at the 2/1 resonance,
you need to take into account the gravitational effects of Saturn as
well.  And at the 3/2 resonance, there is actually a cluster of
asteroids, not a gap.  So resonances with small numbers don't
automatically imply unstable orbits.

On the other hand, there are no asteroids with periods equal or near to
six-sevenths of Blue's (which Green's is most likely to be if it has an
"ordinary" orbit).  And a numerical simulation of 3000 randomly chosen
orbits between the 2/1 gap and Jupiter's orbit yielded no surviving
orbits in that region.  On the other other hand, the fact that in OBW we
have two bodies of comparable sizes, rather than an asteroid and
Jupiter, may make a difference.  

So I would sum up the findings of my research as follows:

1) The fact that Blue and Green reach conjunction every few years does
not, in itself, imply that their orbits are unstable; but

2) The fact that Green's period would have to be close to six-seventh of
Blue's probably means that its orbit would be unstable; but

3) I don't really know.

Does anybody know a good celestial mechanic?  If not, I may make
inquiries on sci.astro.

Finally, a minor point.  If Blue and Green do have conjunctions every
six years, it has to be exactly every six years, for reasons given in my
reply to mantis below.  But, as I said earlier, they may not.

mantis wrote:

> Model 1: Two Planets (Earth and Venus), with standard Bode-Titus style
> orbits and a comparatively wide separation (significant fraction of an
> Astronomical Unit).  (Problems: to see one as a looming disk from the
> surface of the other, the world has to be much closer than this suggests.
> Possible solution: ditch Bode-Titus! <g>)

Not just the appearance argues against Bode-Titus, but the tides: after
all, Earth gets no tides from Venus.  So Green has to be quite a bit
closer.  But ditching Bode-Titus shouldn't be a problem: it's just a
numerical regularity with no known theoretical explanation, and it
doesn't even work for Nepture (or Pluto either, but there are arguments
that Pluto isn't really a planet).

At first I thought Kepler's Third Law would be a problem, but when you
do the calculations, it isn't (at least I don't think so).  Kepler's
Third Law says that the ratio of the square of a planet's period to the
cube of its semimajor axis (the average of its closest distance to the
sun and its furthest distance away from the sun) is constant for the
planets in a given system.  Green seems to be hotter than Blue (the
jungles), so its orbit is inside Blue's and its period is smaller.  If
its period is six-seventh of Blue's, then its semimajor axis is about .9
as long as Blue's.  If the distance between Blue and Green at
conjunction is insignificant compared to the distance between Blue and
the Short Sun, and if we assume that blue's orbit is nearly circular,
then Green's distance when closest to the Short Sun is about .8 its
distance when furthest from the Short Sun, which doesn't seem too
eccentric to allow life to evolve on Green (by comparison, Venus's
average distance from the Sun is .72 times Earth's distance from the
Sun).  On the other hand, if Green's period is five-seventh that of
Blue's, then under the same assumptions its closest approach to the
Short Sun must be .6 times its furthest distance, which does seem
excessive to me (though again, I'm not an expert).

So neither Bode-Titus nor Kepler argue against "ordinary" planetary
orbits for Blue and Green.  Stability may or may not, as I discuss
above; and I still think that Horn's narrative implies that Green moves
visibly across the sky, which is another problem for this model.

--Adam

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