Subject: (urth) Blue moon

From: Roy C. Lackey <rclackey at stic.net>
Date: Mon, 25 Mar 2002 23:19:59 +0000

This occurred to me last night, and I took it to mantis first today, and he
said to post it all to the list, so here it is:

>You should post this to the list!
>
>>You seem to like to take a sliderule to things, so consider this
>>observation. I'm about 3/4 of the way through IGJ, in my re-read of the SS
>>series. While I am not fond of the idea of Urth=Blue/Lune=Green, and my
>>observation doesn't compel that reading, if I'm correct, then Blue and
Green
>>are linked in a way that I can only account for if one is orbiting the
>>other. In other words, Blue and Green can not be independent bodies
orbiting
>>the sun in the manner of, say, Mars, Earth, and Venus.
>>
>>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.
>
>Velocities are a little tricky in that they do change: it is "equal area in
>equal time," with the "area" in this case being the triangle created by
>planet at point 1, point 2, and the primary star.  Line between point 1 and
>point 2 being the segment of the orbit.  The world does speed up at closest
>encounter and slow down at furthest point.
>
>So anyway, we are given that the closest is 105,000 miles, and you are
>saying that based on the movement in one year the maximum separation is
>540,000 miles; the average would be 322,500 miles.
>
>>I'm no expert on celestial mechanics, but I find it hard to believe that
two
>>planets, effectively in the same orbital plane and the same distance from
>>the sun, could exist that close together without the planets orbiting each
>>other, or one body being a satellite of the other. No? Where am I wrong?
Is
>>Green Blue's moon?
>
>I'm no expert, but I do like to play with these things.  In the past I
>wondered online whether it could possibly be that weird "dosey-do" trick
>that some of the Jovian moons do--two worldlets in the same orbit, they
>swing by each other and exchange orbits: the slightly lower/faster one
>changing places with the higher/slower one.  This is a boggling thing in
>the real world!
>
>It doesn't seem to match Blue/Green, because it would happen every year or
>less.
>
>Today, with your new numbers, I'm wondering if Green is in a weird orbit
>between Blue and Blue's "trojan point."  Because otherwise it is very
>difficult to see how a normal satellite could take six years to come so
>close and then back off again. (No, the trojan point idea is impossible:
>the trojan point is where forces are cancelled out--it is not a gravity
>point, and as such it has no attraction)
>
>In dealing with satellites it is often handy to use "planetary radii."  If
>we give Blue a radius of 3960 miles (ala Earth), then hey!
>
>Roy's Green Orbit
>105,000 miles = 26.51 radii
>322,500 miles = 81.44 radii
>540,000 miles = 136.36 radii
>
>A large moon is expected to orbit an earthlike planet between 10 and 80
>radii, so this looks quite promising!  (FWIW the Roche Limit is around 2.89
>radii: that point where worlds are torn up into confetti.)
>
>Orbital period (in days) for moon = .0074 * sqrt (R^3/M)
>
>Where R = orbit in thousands of miles
>Where M = sum of planet and moon in Earth masses.
>
>Figuring that Green is equal to Blue, then let M = 2.
>
>This gives an answer of . . .
>
>30 days.
>
>6 years = 2191.5 days
>
>We can solve this for M, which would then give us the mass of Green!
>
>Shot in the dark: M = 1.5 . . . 35 days.
>
>Shot in the dark: M = 1.16 (like Earth + Mars) . . . 40 days.
>
>Third shot: M = 1 . . . 42.86 days
>
>Well, there's that old problem again!  If Green is a satellite of Blue,
>orbiting at 322,500 miles, it should complete an orbit in 43 days!
>
>Maybe you should post all of this to the list--I didn't think I'd get so
>wrapped up in it so quickly.
>
>=M=

-Roy

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From: Adam Stephanides <adamsteph at earthlink.net>
Date: Thu, 28 Mar 2002 21:20:38 +0000

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

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

>>> the numbers above, Green would never be more than about 540,000 miles from

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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From: Roy C. Lackey <rclackey at stic.net>
Date: Fri, 29 Mar 2002 05:44:31 +0000

>This is impossible. [snip]

Huh? I don't follow that. Gagliardo's numbers don't say anything about the
length of either body's solar orbit that I can see. His numbers indicate
that at your position A the two bodies have _diverged_ by about 145,000
miles since conjunction. In that one year interval both bodies have also
circumnavigated the sun. Two years later Blue and Green will have attained
maximum separation, a distance af about 540,000 miles, then start gradually
getting closer together until the next conjunction.

In other words, for the two bodies to be in independent orbits, those orbits
would have to be almost congruent--almost, but not quite. Think of the two
bodies as twin planets, traveling almost in tandem around the sun. The
attraction of one body for the other causes minor changes in their orbital
velocities, which average out over a six-year cycle, but account for the
fluctuation in distance between the two bodies during the cycle. Or am I
missing something so obvious I can't see it?

>I'm leaning more and more towards 4).

That seems to be where most here who have an opinion about it are leaning. I
would tend to agree--but, Wolfe could have avoided the whole thing by
invoking some kind "magic" and just left out those numbers, which certainly
weren't necessary to the plot.

-Roy

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From: Adam Stephanides <adamsteph at earthlink.net>
Date: Fri, 29 Mar 2002 20:04:41 +0000

on 3/28/02 11:44 PM, Roy C. Lackey at rclackey at stic.net wrote:

> From: Adam Stephanides <adamsteph at earthlink.net>
> Date: Thursday, March 28, 2002 3:19 PM

I was treating Blue and Green as analogous to Earth and Venus: completely
independent orbits which remain fixed (i. e. don't oscillate).  What you
describe is more like mantis's twin moons chasing each other.

But I think it's probably hopeless to try and figure out what, if anything,
Wolfe had in mind.

--Adam

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