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Jupiter Time |
Date: Thu, 18 Jul 2002 17:24:54 -0400 (EDT)
From: "Christopher M. Hirata" <chirata@Princeton.EDU>
To: time-sig@chapters.marssociety.org
Subject: [Mars_Time-sig] Jupiter Calendar
[ to Mars Society Time list, from "Christopher M. Hirata" <chirata@Princeton.EDU> ]
Forgive me for changing the subject, but I thought I'd throw in some comments related to Tom Gangale's calendars for the Jovian satellites.
Before I go on, let me mention that we should be trying to develop calendars for all of the potentially habitable celestial bodies, not just Mars. In doing so, I think we will be led to ask (and maybe answer) some presently unasked questions about what makes a good calendar and what does not. (I have candidacy exams in October but after that I'll be able to do some work on this.)
As Tom appears to have discovered, calendars for the Jovian (or, for that matter, Saturnian, Uranian, or Neptunian) satellite system suffer from the difficulty of having too many cycles to reconcile. We have been considering calendars for Mars, in which our only two cycles are the sol and the Martian year. (Well, a couple other cycles such as the 26-month synodic cycle governing Earth-Mars launch windows have been suggested, but these don't seem to be the primary discussion points.) In the case of Jupiter, we have a different solar day on each moon.
The first point to realize is that Io, Europa, and Ganymede are locked into a Laplace resonance as a result of tidal evolution. The Laplace resonance forces the satellites to move around Jupiter such that their longitudes satisfy:
L(Io) - 3*L(Europa) + 2*L(Ganymede) = 180 degrees
which, in terms of orbital periods (this relation holds for either the sidereal or solar periods!)
1/T(Io) + 2/T(Ganymede) = 3/T(Europa)
[Incidentally, it is this Laplace resonance that causes Jupiter's rotational energy to be changed into heat energy in the interior of the Galileans, thereby powering the volcanoes of Io and the (probable) melting of ice in Europa's subsurface.]
This means that if we know where Europa and Ganymede are, we know where Io is. For example, when Europa and Ganymede are in conjunction, Io is on the other side of Jupiter. Thus, any calendar that works for Europa and Ganymede simultaneously will also work for Io.
Getting the same calendar to work for Europa and Ganymede is a bit of a challenge. Tom uses the near 1:2 ratio of their orbital periods to establish "circads" (1/8 of a Ganymede day or 1/4 of a Europa day.) Unfortunately, the circads are not the same length, which causes a problem when we try to make this work for Io. So I'm going to make a radical proposal here. I suggest that we base a Jovian calendar on the following two cycle lengths:
7.166386 days = mean solar day of Ganymede 437.64 days = period of precession of Ganymede's perijove relative to the Sun
The latter cycle can probably serve the same role for Jovian colonists as the terrestrial year does for us. (After all, the Romans had one year of roughly that length.) The ratio of these cycles is roughly 61.07:1. What they allow us to do is to write the joviocentric elongations of Io, Europa, and Ganymede in terms of two angles, alpha and beta:
elongation of Io = 4*alpha + 3*beta + 180 degrees elongation of Europa = 2*alpha + beta elongation of Ganymede = alpha
where alpha advances through one cycle (360 degrees) every 7.166386 days and beta advances through one cycle every 437.64 days. Since the Jovian satellites are tidally locked to Jupiter, the elongation of a particular satellite directly translates into where on that satellite it is day, and where on that satellite it is night.
There are still many mathematical details left to be filled in here -- such as how to divide up Ganymedean days (alpha) and "years" (which are actually precession periods, beta) -- as well as some aesthetic issues. There may also be better ways to make use of the Laplace resonance. But we should think seriously about the resonance, because it does tie together cycles on three of the Jovian moons. And, if we wait a few more billion years, there's a good chance Callisto will also become tied up in the resonance.
Christopher Hirata chirata@princeton.edu Princeton Univ. Physics Dept. http://www.princeton.edu/~chirata/ AIM, MSN: C3twentyfive Yahoo: go4tni
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