You must specify what kind of clock you will use.
The atomic clock is very sensitive to acceleration because internally works with a receptor of the signals originated by the vibration frequency of the atom of Cesium. This receptor counts a certain number of vibrations and make click to set a second. Acceleration will cause an intercepting problem and by such the atomic clock will show malfunction.
When the atomic clock is exposed to almost zero gravity, the entire spaceship or satellite -including the clock- will suffer of changes. It has been proved hundreds of times that traveling from ground zero to outer space causes changes in matter and energy behavior.
Example, astronauts standing 6 months in the Space Station return "older" this is to say, with illnesses found in people older than the current age of the astronauts (this fact destroys completely the fallacies obtained from Relativity) and another example, liquid metals that can't be mixed by any means on ground zero, these can be mixed in outer space.
The atomic clock was calibrated on ground zero and will suffer of a continued malfunction as long as it stays traveling surrounding earth. The rate of erroneous data will be regular.
If you use a mechanical or digital (battery/electric) clock, you might need of years in order to detect the difference in nanoseconds because these clocks can't be calibrated to such a level.
If you use a sand clock or a water clock, well, will be obvious that their malfunction will give you disparate results.
So, not all the clocks malfunction the same when exposed to outer space.
Which one is the clock you will select in your thought experiment?