I'm working on a short story which I've decided to set at the south pole of Saturn's moon Enceladus. I thought I would share with you a few paragraphs from my notes.
How big is Saturn in Enceladus's sky? According to Wikipedia, Enceladus orbits 237948 km from Saturn and Saturn is 108728 km in diameter (pole to pole). Popping these two figures into the angular diameter calculator at http://rechneronline.de/sehwinkel/angular-diameter.php
tells us that it is 25.7 degrees wide -- bigger than your spread hand at arm's length (about 20 degrees). On Earth, the full moon is 0.5 degrees wide -- smaller than your finger tip -- so Saturn is 50 times wider than that. The rings would barely appear as a line, because Enceladus orbits within the outermost E Ring and the rings are less than a kilometer thick (but the rings' shadow is visible on the planet's face, changing with Saturn's 29.5-year orbital period).
Enceladus's orbital period is 32.9 hours and it is tidally locked, keeping one face turned toward its primary at all times. Unlike Earth's moon, it does not librate (wobble). Its axial tilt is zero and the inclination of its orbit relative to Saturn is very near zero. This means that as seen from the south pole of Enceladus Saturn sits on the horizon, with its south pole uppermost and the line of the rings horizontal (but this line is so fine as to be nearly invisible, and it's probably below the horizon anyway -- however, when seen from anywhere other than the pole, the E ring in which Enceladus is embedded may appear as something like a Milky Way). Because Enceladus is tidally locked, Saturn does not move in the sky at all, but it does go through phases along with Enceladus's day, with a complete cycle every 32.9 hours. You can always tell what time it is on Enceladus by looking up at Saturn (if you happen to be at a place on the moon where Saturn is visible).
The shadow of Saturn falls across the rings during the Saturnian equinoxes (every 15 years). When this is happening, Enceladus experiences a solar eclipse every day. For how much of Saturn's year does this occur? Saturn's axial tilt is 26.73 degrees, which means that the sun rises 27 degrees above the horizon at the solstice. Since Saturn is 26 degrees wide in Enceladus's sky, 13 degrees of that is above the horizon (as see from the pole), and the sun rises at most 27 degrees above the horizon, that implies that these eclipses occur during (very roughly) half of Saturn's year: those periods, near the equinox, when the sun is less than 13 degrees above the horizon as seen from the pole. That's about seven years out of every fifteen.
Around the solstices, when the sun is higher in the sky than 13 degrees, there are no eclipses. The eclipses begin with a brief blip each day (the sun appears to graze the top of Saturn in the sky) and get longer and longer as the equinox approaches, maxing out at about two and a half hours (26 degrees / 360 = 0.07, times Enceladus's 32.9-hour day = 2.37 hours) -- these maximum eclipses occur at the equinox, when the sun as seen from Enceladus appears to be in Saturn's ring plane. Saturn's equinox is also Enceladus's equinox (axial tilt zero), so the period of maximum eclipses is also the time when, as seen from the pole, the sun drops below the horizon and is not seen again for 15 years (or reappears after a 15-year absence). The period of eclipses lasts for (very roughly) 4 years of increasingly long eclipses before the sun vanishes and 4 years of decreasing eclipse length after it reappears, with seven years of no eclipses in between.
All that being said, sunlight at Saturn is only 1% of what we see on Earth, so whether the sun is in the sky or not, human eyes would perceive the scene as near-perfect blackness. My astronaut main character will need an image-enhancing faceplate.ETA:
Dr. Plotka writes to say: "1% of the Earth's sunlight is plenty for seeing things. The sun is very very bright, and we don't need anything like that much light. Specifically, full sunlight on Earth is up to 100 kilolux, so on Enceladus it would be around 1 kilolux. Which is about the same as TV studio lighting, and twice as bright as a well-lit office."ETA 2:
Rob French says: "I don't think you meant to say that the shadow of Saturn falls across the rings at the equinox. Maybe the shadow of Enceladus? The shadow of Saturn falls across the rings the entire year."
My reply: I wasn't clear on what I was trying to do there. The real question I was trying to answer was: how common are eclipses on Enceladus? Or, to turn the problem around, for how much of Saturn's year does the planet's shadow on the rings reach all the way to Enceladus? It does at the equinox, obviously, when the shadow and the rings are coplanar, but for how much of the year on either side of the equinox does that remain true? The answer is that Saturn's shadow reaches Enceladus for about half of Saturn's year.