Could NASA put an artificial radar telescope satellite in geosyncronous orbit around Mars to help locate dangerous asteroids on a trajectory that would place them on a path to strike Earth?
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1$\begingroup$ As far as Areostationary orbits are concerned, see: What factors would make station-keeping of an Areostationary Mars satellite necessary? and also Is it possible to deploy “geostationary” Mars orbiter? $\endgroup$– uhohCommented Aug 6, 2019 at 4:06
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$\begingroup$ Welcome to Space! I improved your title so it better matches the body of your question (upon which uhoh based his answer). Also, although this is not a duplicate question, the old title might have falsely led some readers to think so. Nice question! $\endgroup$– DrSheldonCommented Aug 6, 2019 at 4:26
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5$\begingroup$ Why an orbit around Mars, why not around Earth? $\endgroup$– UweCommented Aug 6, 2019 at 7:52
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4$\begingroup$ Even if a satellite in Mars orbit might be useful to help locate asteroids, what possible advantage could there be to making it areostationary? $\endgroup$– prlCommented Aug 6, 2019 at 22:03
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Could NASA put an artificial radar telescope satellite in geosyncronous orbit around Mars to help locate dangerous asteroids on a trajectory that would place them on a path to strike Earth?
The "Could X happen?" or "Could X be done?" questions almost always collect comments like "it depends on how much (list of resources) you have."
But in this case, given (only) a billion dollars and ten years this could be done without any unheard of or super-advanced technology. Just a lot of hard work, planning, money, time, and effort.
This is the wrong place for it
By far the largest number of NEOs with a chance of hitting Earth are those with regular orbits around the Sun. As @MarkAdler's answer to Why has the Earth-Sun libration point L1 been chosen over L2 for NEOCam to detect new NEOs? points out you would really like to have an infrared sensing telescope some place closer to the Sun than Earth so that it can look outwards (away from the Sun) and see the objects warmed by the Sun against a cold sky.
Radar is lame, let the Sun "paint" the objects for you instead
At 1 AU the Sun "lights up" every square meter of your target with 1360 Watts of power; for free!
At an observing distance $R$ the infrared signal power that you receive from of the object will scale as $$\frac{A_T}{R^2}$$
where $A_T$ is the area of the telescope's aperture. However, if you are using radar, the radar signal power you'll receive will scale as $$\frac{A_1 A_2}{R_1^2 R_2^2} \approx \frac{A_1 A_2}{R^4}$$
where $A_1, A_2$ are the areas of the transmit and receive dishes and $R_1, R_2$ are the distances from the object to the transmitter and receiver.
$R^{-4}$ drops off a lot faster than $R^{-2}.$
Even with the most powerful radar systems on Earth, we only see astroids at distances of a few million to a few tens of millions of kilometers at best.
There's no way you could build a radar system larger than Earth's largest radar in orbit around Mars for a billion dollars in ten years.
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1$\begingroup$ Of course, it might be desirable to detect NEOs before they reach 1AU, but the concept stands. $\endgroup$ Commented Aug 7, 2019 at 3:56
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$\begingroup$ @RolandHeath that is true. If you like you can ask a new question about how much harder it is to detect an asteroid at 2 AU or 3 AU compared to 1 AU. I could address that quantitatively. $\endgroup$– uhohCommented Aug 7, 2019 at 5:12