Outside of visible light, a good way to observe the rest of the Universe is with infrared light. Titan’s atmosphere is relatively transparent to some infrared wavelength bands, as can be seen from Cassini-spacecraft pictures like the one inlined here: Cassini: Mission to Saturn: Peering Through Titan’s Haze. It shows a false-color picture of Titan in 1.3, 2.0, and 5.0 microns wavelength, and one can easily see surface details.
So with infrared observations, one can do much of the astronomy that I’d described earlier without leaving Titan’s surface.
Radio observations open up some additional possibilities.
One may be able to observe the radio emissions of Saturn’s auroras, and likely the radio emissions of the Sun and Jupiter. Even if one could only observe the Sun and Jupiter, one could infer Saturn’s presence from Saturn’s eclipses of them.
One would also be able to observe radio sources outside of the Solar System, and infer Saturn’s presence from eclipses of them also. But it would be much harder to detect Saturn’s rings or other moons in that fashion — since they cover much less of the sky, it will be much more improbable. Saturn’s rings are nearly in the plane of Titan’s orbit, so an eclipse by them is as improbable as an eclipse by one of Saturn’s larger moons.
Turning from passive to active radio astronomy, we consider radar (“radio detection and ranging”). Saturn’s round-trip time is about 8 seconds, so there may not be much motivation to search for it by radar unless one could already see the planet in the visible or the infrared.
But once one tries to see Saturn with radar, one will make a remarkable discovery. One will be able to see Saturn’s rings, but Saturn itself will be very difficult. Even if one does not see Saturn, one will get evidence of its presence from the farthest parts of the rings apparently being absent, from Saturn eclipsing those parts.
Once one sees Saturn’s rings, one may want to look for other objects, and one may quickly find Saturn’s inner moons. One may also measure their sizes by this means, though Rhea, the largest of these moons, has a radius round-trip time of only 5 milliseconds.
The outer moons will be more difficult. Hyperion is usually around Saturn’s distance from Titan, but it gets closest well away from Saturn’s direction. Iapetus has a round-trip time of about 24 seconds, and it may be hard to find it unless one was looking for very distant moons. There are farther moons than Iapetus, but they are also smaller, adding to the difficulty in finding them.
It will be much harder to measure direction than distance with radar, because radio waves have much longer wavelengths than visible light or infrared light. One could get some direction information by using more than one receiver, with three collinear receivers needed for an unambiguous direction. But Titan’s radius travel time is 8.6 milliseconds, so one won’t get much accuracy there. But radar observations will nicely complement optical ones, since they fill in each others’ gaps.
Looking outside the Saturn system, one will be able to do radar astronomy on the Sun, Mercury, Venus, the Earth, the Moon, Mars, and Jupiter’s four big moons. Jupiter itself shares with Saturn a low radar reflectivity.
One can also look for Doppler shifts, and as one observes the rings, one would notice that they have orbit velocities ranging from 16 km/s at their outer edge to 20 km/s at their inner edge. That would demonstrate that they are a swarm of small objects rather than large solid objects.
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