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I need to know if a satellite with onboard power can maintain a geo-synchronous orbit at, for example, 45 degree latitude?

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    $\begingroup$ It is simply impossible. Orbits don't work like that. $\endgroup$ – A. C. A. C. Nov 8 '17 at 18:47
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    $\begingroup$ A powered vehicle could travel in such a path, but I wouldn't describe it with the words "satellite" or "orbit". $\endgroup$ – Keith McClary Nov 8 '17 at 19:27
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Because the potential is central, the motion of the satellite is necessarily in a plane with the Earth at its centre. This therefore excludes any constant latitude other than the Equator.

Geo-stationary orbits will remain over the same points above the equator. This orbit will take 23 hrs 56 mins to complete.

Geo-synchronous orbits will also take 23 hrs 56 mins to complete, and the orbit will be along a full great circle, although not of constant latitude, unless it is along the Equator.

[Hat tip to @Joshua for the distinction between sideral and solar day.]

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    $\begingroup$ Geo-stationary and geo-synchronous orbits take 23 hrs 56 mins to complete, approximately. Earth rotates in about 23 hrs 56 mins, not 24 hrs (a sidereal day vs. a solar day). $\endgroup$ – NeutronStar Nov 8 '17 at 19:25
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    $\begingroup$ @Joshua yes this is an important distinction. I made the change. $\endgroup$ – ZeroTheHero Nov 8 '17 at 19:59
  • $\begingroup$ @ZeroTheHero you were corrected, but you've chosen to keep the phrasing "Obviously, ..."? Clearly wasn't obvious ;) $\endgroup$ – CDCM Nov 8 '17 at 23:26
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    $\begingroup$ @CDCM corrected for a 2nd time. :D $\endgroup$ – ZeroTheHero Nov 9 '17 at 0:00
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First about terminology.

Geosynchronous orbit [1] is any orbit with period matching Earth sidereal day. Now you can have such orbit with any inclination and with any eccentricity, no problem. Geostationary orbit [2] is the one for which the satellite stays over the single point at the surface. Generally geosynchronous orbits are not geostationary. Only the circular and equatorial geosynchronous orbit is geostationary. I assume that you are actually referring to the latter.

The simple reason why you can't have true inclined geostationary orbit is in the word "inclined". The whole orbit lays in the plane that goes through the center of the earth. The inclination means that this plane is at an angle with equatorial plane. As consequence the sattelite will neccessarily change the latitude going both over northern and southern hemispheres.

If you try to use the engine to keep your spacecraft over the same region you will have to burn it constantly compensating gravity pull. The constant force you need to provide will depend on your altitude and latitude but to understand what you basically doing imagine that you try to keep your sattelite over the north pole. You will then have to stop the sattelite completely and constantly fire your engine downwards. The moment you turn off the engine you will simply fall on the north pole...

What you can do in real life is to use the orbit at which the sattelite will be approximately stationary for majority of the time. You can then put a small constellation of sattelites on such orbits that will substitute each other and provide in this way some imitation of the geostationary sattelite.

You can do that with the so-called Tundra orbit [3] named after Russian military satellites. This orbit is geosynchronous i.e. having the period of one sidereal day but not geostationary. It's highly elliptical, so that most of the time the satellite stays in the apogee. The resulting ground track is something like this figure of eight (from here),

The Tundra Orbit

The small loop in the northern hemisphere here actually represents the apogee where the sattelite stays the majority of the day. The wider loop will be passed much faster.

Similar option is the Molniya orbit [4] (named after Russian satellites again) that's also highly elliptic orbit but with period of half of sidereal day. Each day it stays for several hours approximately over one region in the western hemisphere and for several hours over the region in the eastern hemisphere. Again the wide paths in the souther hemisphere actually take only a couple of hours. (image from here)

The Molniya Orbit

Of course this is not the ideal solution but the working one, used in real life.

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