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While sending a space probe to orbit around a gas giant is often heard of, are there plans to send a space telescope to orbit around gas giants, so that it can operate far away from the sun, or even see the outer solar system clearer?

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No, Jupiter is simply too far away (Distance = less data transmitted), it's a hard environment (Lots of radiation), and relatively little is accomplished by sending it out so far. But I do know of at least one serious proposal to send a telescope to Mars. A bare bones spy satellite (Telescope only really) was donated to NASA, and a proposal called Mars Orbiting Space Telescope (MOST). The idea is to send the telescope to Mars, where it could take photos of Mars, and take photos of deep space. It was tossed around almost 3 years ago, no idea what the current status is of it, but it seems likely it isn't going anywhere. A complete study of what might be done can be found at the Planetary Society.

There are two advantages of a telescope at Mars. The first is that it could get high resolution pictures of Mars. The second is that it would give a wider baseline, allowing us to determine where an object is with much higher precision than we can today, both inside and outside of the Solar System. I would imagine that we could extend the parallax distance from 1000 parsecs to perhaps 2000 or more. It would not be particularly useful for looking at dim objects, to keep the costs down, but would be great for looking at the gas giants, for instance.

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  • $\begingroup$ Pushing a telescope like Gaia to the outer Solar System, but not to any planet, should've been a good idea after having completed the primary mission of at least three orbits at its Lagrange point. Its parallax precision would've multiplied. Even if only for selected samples as the data transfer is reduced. Mars' baseline is a modest improvement to 3.3 compared to Earth's 2 AU. At Jupiter's distance it is 11, if it could survive 5 years in half an orbit out there. $\endgroup$ – LocalFluff Nov 11 '15 at 11:45
  • $\begingroup$ It's true that Mars isn't that far away, I would hope that simultaneous measurements would help, especially to exactly place solar system objects. $\endgroup$ – PearsonArtPhoto Nov 11 '15 at 21:38
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If you're going to send a telescope far away, you shouldn't put it in orbit around a planet.

When a telescope orbits a planet, the planet blocks part of the sky, limiting your observations. It also reflects light, limiting your observations some more. In most orbits, the telescope travels through the planet's shadow cone, which leads to temperature swings.
So these days, space telescopes are sent to Lagrange points instead (the James Webb telescope will be parked at the Earth-Sun L2 point, for example).

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  • $\begingroup$ There are pros and cons both ways. James Web is an infrared, which is why it makes sense to send it to a Lagrange point near Earth. Part of the value of sending a telescope outside of Earth's boundaries is the ability to explore different parts of the Solar System. $\endgroup$ – PearsonArtPhoto Nov 11 '15 at 21:37
  • $\begingroup$ It might also be useful in the hunt for near-Earth asteroids/objects that are inside of Earth's orbit (i.e., when Mars is on the opposite side of the sun). We currently have very limited capabilities for looking at objects inside of our orbit because of that pesky bright disk in the sky ;P $\endgroup$ – honeste_vivere Jan 13 '16 at 14:54
  • $\begingroup$ Wouldn't a Mars-orbiting telescope have the same problem? $\endgroup$ – Hobbes Jan 13 '16 at 15:23

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