The formulae and "givens" used here come from this site, now located at world-building course, which has been not only immensely helpful (and easy for this amateur to use), but extremely interesting in all its disussion. Some of the formulae as given by Prof. (Ms.) Viau differ slightly in format from other sources, but apparently are just variant (easier!) ways of saying the same thing. (I do know enough Math to understand that a = b/2 is the same as 2a = b.)
Note: for pi, I use 3.14159. Kilometres are converted to miles with 0.6214.
I think these are pretty much established and set in stone (at least to my satisfaction).
Equatorial circumference: 44832.573 km. (27858.961 mi), R=7135.332 km.
Like the earth, Cindu is slightly flattened at its poles (98.413 km less in circumf.), therefore-- Polar circumf: 44687.16 km. (27768.6 mi.) or R=7112.19 km. (this is correct for the "li" unit of measure and cannot easily be changed)
Average radius= 7123.76 km., Ave. circumf.= 44759.87 km. (for comparison: Earth polar Radius=6357 km, Eq.R=6378, average 6367.5)
Therefore Equatorial Rc/Re = 1.121 which will be used in the caluculations below; average Rc/Re= 1.119 or 1.12 also
Cindu's sun (called lero) is a G-0 star, temp. approx 6000K, Luminosity 1.36, Mass 1.05, Radius 1.13 (where Earth's G-2 Sun is 1 for these values); life of ±9.18 billion yrs; equivalent orbit to support life 1.22 AU, which means Cindu must be approx. 113.5 million miles away. (I seem to have adopted a figure of 1.2155 AU, or 113,041,500 miles.)
Gravity of Cindu: Given(assumed): densityC 5.l (densityE given 5.5-- i.e. density in grams/cm3)
G = Dc/De * Rc >>>> 5.1/5.5 * 1.121 = 1.0395 (where Earth Gravity= 1) Cindu's gravity is approx. 4% greater than Earth's.
Mass of Cindu: Given: Earth's Mass= 23
M = volume * density, i.e (4pi*r^3)/3 * 5.1 >>>> (4*3.14159*1.121^3)/3 * 5.1 = (17.7022/3 = 5.9007) * 5.1 = 30.094, rounded to 30 -- so Mass(Cindu) relative to Earth's is 30/23 = 1.304 Cindu's mass is approx 30% greater than Earth's
Orbit/Length of year. Kepler's 3rd Law: (length of year)^2 = AU^3 >>>(Yc)^2 = (1.2155)^3 >>>& Y^2 = 1.79582, the square root of which is: 1.34008
365 * 1.34008 = 489.13 Edays per Cindu year, or (*24) 11,739 hrs. Cindu's day is 25hr 18min (25.3), therefore 11739/25.3= 463.9968, or 464 Cdays. (1 Cindu hour = 75.9min or 1.265 Ehr, 1 Earth hour = .79 HRc)
The Cindu year is 1.34 Earth year; an Earth year is .7463 (.75) Cindu year.
464 days divides evenly into 16 months of 29 days each-- each month (which always begins on the same day) contains 4 seven-day weeks, plus an un-numbered "mid-month day" between the 14th and 15th.
[IRRELEVANT COMMENT/BOAST: Back in 1976 when I first conceived Cindu, I simply posited a priori that it was bigger than Earth but still earth-like, had a hotter sun, therefore had to be a bit further away; I then assumed that, of course, its year would be longer, and I think I pulled 464 days out of thin air. Imagine my surprise, when I found Prof. Viau's site almost 25 years later, that these "guesses" were right-on!!!]
The Cindu day is divided into 20 hours, of 50 mins of 100 secs. The day begins at midnight, which is 0100, midday is 1100; at the equinoxes, the sun rises approx. 0600, sets approx.1600.
However, the Cindu year is not precisely 464*20hrs (9280 Chr); in fact it is 9283 hr 8 mins. (9283.16) or 464.158 days, which means that an extra day accumulates every 6-1/3 years; thus there are 3 leap years in every 19-year period, at 6-7-6 year intervals. It is now year 754 on Cindu; the last leap year was 751, the next one will be 758.
The information here is more tentative....
There are two moons, Vuruna and Lalap.
[COMMENT: The names come from legend: Vuruna was a fair royal maiden who fell in love with a handsome but disreputable commoner, a drunkard-- his true name is lost, the epithet Lalap means 'to stagger'. They were cast out of their community, doomed to wander forevermore. Because of this, it would be desirable if Lalap's orbit could be irregular, "staggering"-- could it be tumbling in space?; or, that Vuruna and Lalap are seen as being in constant pursuit of each other, trying but always failing to re-unite (it may be they reunite briefly whenever they eclipse/occultate each other)]
(For comparison with the following, bear in mind that Earth's moon has a radius of approx. 1738 km. and circumference ±10864 km. or 6790 mi.; it is approx. 384,385 km=238,800 mi. from the Earth. Its density is given as 3.34.)
1. VURUNA is larger than Earth's moon, and more distant.
[COMMENTS: (a) One idea I had about Vuruna: Cindu is the 2nd planet in its system, perhaps much earlier it was the 3rd; Vuruna was then the 2nd planet. Some catastrophe knocked it out of its orbit, and it was captured by Cindu. A Knowledgeable Person, however, said this was highly unlikely if not impossible. (b) Since Vuruna is quite large; could it have an atmosphere? water? could it support life? Furthermore, does it rotate on its own axis, or, like our moon, always present the same face toward Cindu? These are matters I haven't really thought about. Incidentally, Kash and Gwr astronauts, with the aid of the Aliens, have walked on Vuruna.]
Vuruna's radius is 4304 km, C = 27042 km or 16804 mi.; its R is thus .6748 Re and .6042Rc
Its density (assumed/given) is 3.3. (Almost the same as our moon's density-- so of similar ~same composition?)
Its gravity (Dv/De * Rv) >>>> 3.3/5.5 *.6748 = .405
(compared to Earth's Gravity=
OR: compared to Cindu: 3.3/5.1 *.6042 = .391 [is this valid???]
Its mass (using the Vol * Dens formula above): 4pi*r^3/3 * 3.3>>>> 4.2474 which is .185 the mass of Earth (23), or .1425 the mass of Cindu (30)
Roche's Limit: the minimum distance (in planetary radii, measuring from center to center) a body must be from its planet to avoid being pulled apart by gravity. The Formula is: 2.423 *R(planet)*Dens(planet) / Dens(moon)
For Vuruna this is: 2.423 * 1.121 * 5.1 / 3.3 = 4.198. Therefore from the center of Cindu to the Center of Vuruna, there must be at least 4.198 Radii/Cindu or 29954 km. In fact, Vuruna is much farther away (just as, for Earth's moon, Roche's limit is 3.99 radii, but it is in fact approx. 60 radii distant).
(QUESTION: given the size of Vuruna, is this OK? It is certainly beyond the Roche limit, but should it be farther away?)
Viau gives this formula for the orbit of a moon: T = 1.4 * sq.rt of r^3/Mass(planet) where T= time in hours, r is the average distance in radii of the planet (Cindu); Mass-Cindu is 1.304.
[CONFESSION: I wanted Vuruna to orbit Cindu in as close as possible to 1 month (29 Cindu days)-- 29 days = 580 Cindu hours *1.265 = 733.7 Earth hours, which I used for T, and then worked backwards to find r, which turned out to be 71.01545 (I suppose I could have been even more precise, but figuring out cube roots by hand is a chore); that figure, times the Cindu R= 7135.332 km gives the 506,718 km figure above. QUESTION: Am I wrong to want the orbital time = 1 month? My reason is, that as presently conceived, Cindu's New Year always begins on the same day, the vernal equinox of the N. hemisphere. Is this realistic, an amazing coincidence, or am I pushing things???]
Anyway, r^3/1.304 (358144.7/1.304) = 274650.844, whose sq.rt. is 524.0714 * 1.4 = 733.7 Earth hours; 733.7 *.79= 579.6, call it 580 Cindu hours, which is 29 Cindu days. Ta-da!!
ALBEDO: more reflective than our Moon (no definite figure yet; I don't understand "albedo")
ANGLE OF VIEW: how big does Vuruna appear to be, viewed from Cindu? The formula is given as: Radius(moon)/Distance(planet-moon) = Tan theta/2
Thus: 4304/499,583 >>>> 0.008615 *2 = Tan(theta) 0.017224 (say 0.017)-- thus, according to the table at Viau, Vuruna would cover 1º of sky, viewed from Cindu (Earth's moon, 0.5º). [PROBLEM: I would have liked for the angle to be closer to 2º or even greater; I'm not sure what would need to be adjusted to achieve that. Given the desired 29-day orbit, the distance, I think, can't be changed; I'm not sure what a larger radius would do-- I tried Rv=5107 km and even 6000 km, but those did not increase the angle appreciably (it came out .02....-- in the Viau table, tan .017 yields 1º, tan .035 yields 2º). I somehow suspect it would not be a good idea to have Vuruna be the size of the Earth or larger....]
2. LALAP is much smaller, and of irregular shape, rather like an Idaho potato (length a bit greater than width, circumference varies in places over the length). It may be a chunk broken off either Cindu or Vuruna in some ancient catastrophe, or it may be a captured asteroid [if these ideas are even feasible].
Tentatively, I posit its average radius as 1012 km, Circ. ±6358.6 km= 3951.2 mi; Density 3 or so-- could be more, especially if it's an ancient chunk of Cindu or an asteroid (??); with these figures, Roche limit is ±4.6(*7135.332 km); I haven't done the other calculations because these figures can be changed, depending on----
my questions as to its location: (1) Does it orbit Vuruna, in rather close (or far??) orbit? Presumably when it is between V and C, the orbit would be stretched somewhat toward Cindu.
(2) Could it simply orbit Cindu, much closer in than V, perhaps near Roche's limit? In this case, its orbit might be stretched a bit toward Vuruna when it's between C and V. (I'm sure option 1 or 2 would be easiest to deal with.)
(3) Could it have a more elliptical orbit, in which it comes close to Cindu (i.e. V. is one axis of the ellipse, the other is somewhere in the space between C - V...is this possible? Interesting-- it would appear to get bigger as it approached Cindu...) But what would prevent Lalap from falling onto Cindu?
(4) Could it have such an irregular orbit (sort of a figure-8?) that sometimes (mostly?) it orbits C, but regularly switches over to orbit V, then back again to orbit C, and so on??? I would have no idea how to figure that out.
Reflectivity: it might be brighter than Vuruna; if it rotates, perhaps it has variable brightness (one side, or large areas, darker than the other-- is that possible? Same question would apply to Vuruna, if it rotates) ======================================
That's about it. Any comments, advice, questions, corrections etc. are welcome, indeed, earnestly requested!! -- e-mail to me at: rfmilly(at)msn(dot)com
Thanks for your attention. Roger Mills