Date: Thu, 28 Mar 2002 15:20:38 -0600
Subject: Re: (urth) Blue moon
From: Adam Stephanides 

on 3/25/02 5:19 PM, Roy C. Lackey at rclackey@stic.net wrote:

Thanks for digging these figures up!  They indeed present serious
difficulties. 

>>> IIRC, we are told at the beginning of OBW that conjunction is two years
>>> away. We are also told that at its closest approach, at conjunction, that
>>> Green is some 35,000 leagues (105,000 miles) from Blue. Conjunction took
>>> place while Silkhorn was in Gaon. The war between Blanko and Soldo took
>>> place only a matter of weeks after Silkhorn left Gaon. During that war,
>>> Sfido told Incanto that Gagliardo had calculated that Green was then over
>>> 80,000 leagues (about 250,000 miles) from Blue. Hide, at about that same
>>> time, told Incanto that Horn had been gone for three years. Therefore, at
>>> that time, conjunction was one year past. Therefore, in the year since
>>> conjunction, Green had diverged only about 145,000 miles from Blue. We are
>>> told that conjunction occurs every six years.Therefore, at the mid-point
>>> between conjunctions, Green would have to be at its greatest distance from
>>> Blue. Orbital velocities tend to remain fairly constant, afaik. So, based
> on
>>> the numbers above, Green would never be more than about 540,000 miles from
>>> Blue.

This last bit doesn't follow.  It's true that orbital velocities tend to
remain fairly constant if the orbit is approximately circular.  But it
doesn't follow that the rate at which the distance between two planets
changes remains fairly constant (and indeed, it doesn't).  But still, your
numbers do indeed rule out the idea of Blue and Green having independent
orbits around the Short Sun, I think.

Suppose that they do have independent orbits.  Conjunction is every six
years, so six Blue years equal either five or seven Green years.  Hence the
time between conjunction and Gagliardo's observation is either five-sixth or
seven-sixth of a Green year.  In either case, the distance between
conjunction and the position Green is in at Gagliardo's observation (call it
position A) is the distance Green travels in one-sixth of a Green year.

Now at conjunction Green is 105,000 miles from Blue.  At position A Green is
250,000 miles from Blue.  Hence position A cannot be more than 355,000 miles
from conjunction (actually, we could get an even better estimate, but it's
not necessary).  If Green's orbit is approximately circular, then simple
geometry (think a regular hexagon) shows that Green's distance from the
Short Sun itself can't be more than 355,000 miles, with Blue's distance not
much more: that is, less than twice the distance from the Earth to the Moon.

This is impossible.  Even if we posit that the Short Sun is a very small,
dim star, the numbers don't work.  Mantis's formula for the orbital period
of a satellite was:

Orbital period (in days) for moon = .0074 * sqrt (R^3/M), where M is the
combined mass of the two bodies in Earth masses.

Applying this to Blue and the Short Sun, and assuming Blue's year to be
close in length to an Earth year, we find that for Blue to be less than
twice the distance from the Short Sun that the Moon is from the Earth, and
yet have an orbital period twelve times as long, the total mass of Blue and
the Short Sun would have to be a small fraction of the Earth's mass.

As mantis points out, when orbits are not circular orbital velocity is not
constant, so by assuming that Green's orbit is eccentric we can push Green
and Blue farther out.  But doing so raises other, insuperable problems.  Say
we want to push Green at conjunction, and hence Blue, out to 35 million
miles.  This is probably still too close, assuming the Short Sun is
approximately like our own (Earth is 83 million miles from the sun, iirc),
but it's convenient to work with, and more than sufficient for our purposes.
If Green is 35 million miles from the Short Sun at its farthest point, its
orbit must be at least 70 million miles in length.  Since Green goes 355,000
miles in a sixth of a year, it is covering 1/200 of its orbit during this
period.  In other words, Green's orbital velocity near conjunction is about
1/33 of its average velocity.  As mantis says, planets sweep out equal areas
in equal times.  This means, if I'm not mistaken, that

orbital velocity at perihelion     distance from sun at apohelion
------------------------------  =  ------------------------------
orbital velocity at apohelion      distance from sun at perihelion

Hence, Green's distance from the Short Sun at perihelion can be no more than
1/33 of its distance at apohelion, or approximately one million miles.  The
complex flora on Green could not possibly survive such an eccentric orbit.

What about combining a reasonably eccentric orbit with a small, dim, Short
Sun?  Some very rough calculations (which I won't bother to write down)
suggest that that won't work either.  Even if Green is twice as far at
apohelion as it is at perihelion, this gets the Short Sun's mass up to more
than Earth's mass, if that.  (It might be objected that if Blue were Ushas,
than the Short Sun might be the "bare" White Fountain.  But what would have
caused the sun to shed over 99% of its mass, without rendering Ushas and
Lune uninhabitable?)

Mantis's outer Green can't help us either.  That would mean that Green was
nearest to the Short Sun at conjunction, meaning that it was going fastest
at that point, not slowest, which just makes things worse.

So if these figures are correct, Green and Blue can't have independent
orbits.  But I still don't see how it's possible for Green to be a satellite
of Blue and have "conjunction" every six years: i.e. a six-year "month."
For that matter, if Green's "orbital period" around Blue is six times as
long as Blue's orbital period around the Short Sun, in what sense can Green
be said to be a satellite of Blue at all?

I see four possibilities:

1) Green is in one of the "strange orbits" me and mantis discussed some time
back (the discussion should be in the archives somewhere).

2) The time indications are only approximate, and the time between
conjunction and Gagliardo's observation was really much less than a year.
Are there any mentions of the seasons?  Or has somebody put together a
chronology for TBOTSS?  If one was posted here, I've forgotten.

3) Gagliardo was talking through his hat, and his distances bear no
relationship to reality.

4) Wolfe didn't care about the astronomical plausibility of his "solar
system," any more than he did about the physiological plausibility of his
inhumi reaching escape velocity unaided.

I'm leaning more and more towards 4).

--Adam


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