From: "Roy C. Lackey" 
Subject: (urth) Blue moon
Date: Mon, 25 Mar 2002 17:19:59 -0600

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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