I’ve tried to find a nice neat formula that I might use for a science fiction story that would calculate the apparent size of Jupiter in the sky, while standing on the surface of one of the Jovian moons. (Any will do; it’s the formula I’m hoping for, not the specific answer to any one moon. It needn’t even be Jupiter particularly.)

Also, I notice that most of Jupiter’s moons have little or no axial tilt; is this because they are tidally locked to the planet in synchronous orbit? If Jupiter had a moon not tidally locked, could it in theory have an axial tilt? (I’m guessing here, but it’d probably end up tilted in the same plane as its revolution around the planet after a while, right?)

Basically, I’m using the GURPS mechanics to design a star system with habitable moons orbiting a gas giant. I just wanted to get some of the details straight.

I get ~19 degrees too. For compairson, the moon seen from Earth subtends an angle of about 0.54 degrees. Jupiter really looms in the Io sky, as you might expect.

Without knowledge of the math involved, I seem to recall that the primary of a satellite just outside Roche’s limit (and therefore the maximum possible observed size) would subtend an apparent angle of about 30 degrees – which doesn’t sound like much, until you realize that, e.g., as the trailing edge of a rising planet clears the horizon, the leading edge is already a third of the way to straight overhead.

So, using a list of the major Jovian moons and refiguring the Io result, we get a little over 4 degrees for Callisto, so it would be around 8 times the diameter of the full moon… interesting.

I’ve been listening to ‘The Rowan’ by Anne McCaffrey, which has a lot of action taking place on a domed Callisto base and referring to seeing Jupiter hanging there impressively in Callisto’s sky. Then again, there are also references to Earth and the outer galaxy being regularly ‘blocked’ by Jupiter on a nearly daily basis, which absolutely doesn’t make sense considering Callisto’s 16 day orbital period. Oh well.

Here’s the research question: Of all known natural satellites, which one has the largest apparent object in the sky?

Obvious guesses would include the sun over Mercury, or Jupiter over its lowest moon, but others might be the real winner. We don’t need a large planet, if the satellite’s orbit is low enough.

I thought that Jupiter’s magnetic field is so strong that it has a high level of radiation around it? (seem to recall reading this in an article challenging the basic premise of Heinlein’s Farmer in the Sky?)

Yep, they’re both the opposite of cosine, in the same way that multiplication is the opposite of division. The first one is pronounced “inverse cosine”.

Name Usual notation Definition Domain of x for real result Range of usual principal value
arcsine y = arcsin(x) x = sin(y) −1 to +1 −π/2 ≤ y ≤ π/2
arccosine y = arccos(x) x = cos(y) −1 to +1 0 ≤ y ≤ π
arctangent y = arctan(x) x = tan(y) all −π/2 < y < π/2
arccotangent y = arccot(x) x = cot(y) all 0 < y < π
arcsecant y = arcsec(x) x = sec(y) −∞ to −1 or 1 to ∞ 0 ≤ y < π/2 or π/2 < y ≤ π
arccosecant y = arccsc(x) x = csc(y) −∞ to −1 or 1 to ∞ −π/2 ≤ y < 0 or 0 < y ≤ π/2

There are two possible categories of winner: You’ve either got a gravitationally-bound object, which must therefore be outside the Roche limit, or you’ve got a structurally-bound object, which can be as close as it damn well pleases.

In the first case, the best results would of course be for something which is as close to the Roche limit as possible, and we’ll want the limit to be as close to the planet as possible, in terms of the planetary radius. This means that we want as large a ratio as possible between the moon’s density and the planet’s (i.e., the moon should be much more dense than the planet). This situation of maximum density ratio probably occurs for one of the gas giant planets, and they all have enough moons that there’s probably one very close to the Roche limit.

In the second case, where we’re ignoring the Roche limit, we must necessarily look at a very small object. In which case, we need a rigorous definition of what we’re calling a “moon” or “satellite”. Would a ring particle count, for instance? The gas giants’ rings get pretty close to the surface. And what about man-made objects?

For GURPS purposes the magnetosphere can be ignored. The trick is placing an epistellar gas giant at a habitable distance from the star. Too close, and the sun’s gravity strips the excess satellites out of the gas giants’ orbit; too far, and it’s bitterly cold beyond the snow line.

They’re fun mechanics, but you really need to put it into a spreadsheet for it all to make sense.

I’m still debating the axial tilt, though. I suppose in theory you could have the gas giant’s axis at 90 degrees to the ecliptic; the moons could revolve around the gas giant on that 90 degree plane. If the moons also had a 90 degree axial tilt, they would have a relatively steady day-night cycle based only on their own rotation (that is, they wouldn’t be eclipsed by the transit of the gas giant).

GURPS = Generic Universal Role Playing System. It’s a rule set for the RPG (role-playing game) of the same name. GURPS is much more inclusive than simple Dungeons and Dragons, which spends almost all its time concentrated in fantasy/medieval/magical scenarios. The GURPS book on space has a nice chapter on designing a solar system, calculating mass and diameter and gravity of the various satellites, and so on.

It is a set of rules designed to allow you to create your own tabletop roleplaying game (think Dungeons and Dragons) without re-inventing the wheel entirely. It allows reality-checks, which are ways to incorporate real-world physics into the game in a standard fashion, hence this thread.