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 --