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This article deals with what an observer placed on the surface of a planet will see overhead.
The moon has no atmosphere, so its sky is always black. However, in the daytime it will be impossible to see stars unless the observer is well shielded from the sunlight, direct or reflected from the ground.
The Sun for the lunar observer is the same as for the terrestrial one, only it is somewhat brighter (and colored pure white) due to lack of atmospheric scattering and absorption. The lunar day lasts a month. The invisible side of the Moon is not "dark" — it receives as much sunlight as the side visible from Earth.
The most interesting feature of the Moon's sky is, of course, the Earth, spinning against the black backdrop of space. Its visible diameter (1.9°) is four times the diameter of the Moon as seen from the Earth. The Earth will show phases just like the Moon for the terrestrial observer, but they will be opposite: when the terrestrial observer sees the full Moon, the lunar observer will see "new Earth", and vice versa. The Earth's albedo is much higher than that of the Moon, so the full Earth will glow ~130 times brighter that the full Moon does for the terrestrial observer.
The Earth can be seen only on the visible hemisphere of the Moon, still the Earth is not completely fixed on the lunar sky: due to librations, it performs a complex wobbling movement. Near the boundary of the visible and invisible hemispheres of the Moon, the Earth will be sometimes below the horizon, and sometimes above it. Additionally, because the Moon's orbit is eccentric, the Earth's apparent size in the sky will vary slightly, by about 5% either way (between 1.8° and 2.0°).
The Earth and the Sun sometimes meet on the lunar sky, which causes an eclipse. On the Earth, one then sees a lunar eclipse, where the Moon gets into the Earth's shadow; but on the Moon, one sees the Sun go behind the Earth — so for the lunar observer it is a solar eclipse. As the apparent diameter of the Earth is four times more than that of the Sun, the Sun can hide behind the Earth for hours. The Earth's atmosphere then becomes visible as a reddish ring.
The terrestrial solar eclipses, on the other hand, are difficult to notice for lunar observers, because the Moon's shadow nearly tapers out at the Earth's surface. All one can see is a small darkened spot on the full Earth's disk.
So whenever an eclipse of some sort is occurring on the Earth, an eclipse of another sort is occurring on the Moon. In short, eclipses occur for both Earth and Lunar observers whenever the two bodies and the Sun align in a straight line.
Mars has only a thin atmosphere, however it is extremely dusty and lots of light is scattered. The sky is thus quite bright during the daytime and stars are not visible.
The Sun on the Mars is seen 1.6 times smaller (0.35°) than on Earth, so it sends 2.5 times less light.
Mars has two natural satellites: Phobos and Deimos. Phobos has one-third to one-half the angular diameter of the Sun, but Deimos will be barely more than a dot (only 2' angular diameter). The apparent motion of Phobos will be in reverse, due to its fast orbital motion: it will rise in the west and set in the east. Phobos orbits so close (in a low-inclination equatorial orbit) that it cannot be seen north of 70.4°N or south of 70.4°S latitude; high-latitude observers will also notice a decrease in Phobos' apparent size, the additional distance being non-negligible. Phobos' apparent size varies by up to 45% as it passes overhead, due to its proximity to Mars' surface. For an equatorial observer, for example, Phobos would be about 0.14° upon rising and swell to 0.20° by the time it reaches the zenith.
Deimos rises in the east and sets in the west, like a "normal" moon, although its appearance is star-like. Its brightness would vary between Venus' and Vega's (as seen from Earth). Being relatively close to Mars, it cannot be seen from Martian latitudes greater than 82.7°. Finally, the orbital period of Deimos of about 30.5 hours exceeds the Martian solar day of about 24.5 hours by such a small amount that it takes 2.7 days between rising and setting for an equatorial observer.
The Earth is well seen from Mars as a double star (the Moon seen as a fainter companion). The maximum visible distance between the Earth and the Moon will be about 25′ (at conjunction of the Earth and the Sun — for the terrestrial observer it will be opposition of Mars and the Sun).
See also: Time and date and astronomy on Mars, Transit of Phobos from Mars, Transit of Deimos from Mars
As seen from Phobos, Mars is 6400 times larger and 2500 times brighter than the full Moon as seen from Earth, taking up a full 1/4 of the width of a celestial hemisphere.
As seen from Deimos, Mars is 1000 times larger and 400 times brighter than the full Moon as seen from Earth, taking up a full 1/11 of the width of a celestial hemisphere.
Mercury has no atmosphere, like the Moon, so its sky is always black. The day here is 176 terrestrial days (exactly 2 mercurian years). The Sun's motion across the sky, however, is not uniform because of Mercury's relatively large eccentricity. At certain points on Mercury's surface, an observer would be able to see the Sun rise about halfway, then reverse and set, then rise again; all within the same Mercurian day. This is because approximately four days prior to perihelion, Mercury's orbital velocity exactly equals its rotational velocity, such that the Sun's apparent motion ceases. At perihelion, Mercury's orbital velocity exceeds the rotational velocity, thus the Sun appears to retrograde. Four days after perihelion, the Sun's normal apparent motion resumes.
The visible diameter of the Sun on Mercury is 2.5 larger than on Earth, so it is more than 6 times brighter.
The second brightest object on the mercurian sky is Venus. It is here much brighter than for terrestrial observers. The cause for this is that when Venus is closest to Earth, it is between the Earth and the Sun, so we see its dark side. In fact, when Venus is brightest in the Earth's sky, we see only a narrow crescent of it.
For a Mercurian observer, on the other hand, Venus is closest when it is in opposition to the Sun and shows its full disk. The apparent magnitude of Venus should then be approximately −8.
The Earth and the Moon are also well visible, their apparent magnitude being about −6 and −1 respectively. The maximum apparent distance between the Earth and the Moon will be about 15′.
All other planets will be visible just like they do on Earth, being only somewhat less bright in opposition.
The zodiacal light for Mercurial observers should be much more prominent than for terrestrial ones.
Suppose an observer sits on 1 Ceres, the largest asteroid. Surprisingly, he will not be able to see any other asteroid without a telescope! The asteroid belt is in fact very sparsely populated. Occasional "close approaches" do occur, but they are spread out over eons.
For someone who sits on one of the Jupiter's moons, the most prominent feature of the sky will be, of course, Jupiter. For an observer on Io, the Jupiter's apparent diameter will be about 20° (40 times the visible diameter of our Moon). Amazingly enough, since the Galilean moons of Jupiter are spin-locked to the planet, Jupiter always appears in the same spot in the sky of these worlds! (This same effect applies to our moon, and was first pointed out in a remarkable piece by the Dutch astronomer and physicist Christiaan Huygens). Observers on the far side of the Galilean satellites would never see Jupiter.
For observers on extrasolar planets, the constellations will be quite different. The Sun will be visible to the naked (human) eye only for distances below 20–25 parsecs.