What is the minimum altitude required for a Cubesat*-like object to orbit around the Earth? Could you initiate an orbit inside the Earth's atmosphere?
*A CubeSat is a 10 cm (1 liter) cube with a mass of no more than 1.33 kilograms.
The answer varies with atmospheric density (due to varying solar activity), with satellite geometry and mass, and with attitude. But for a typical 3U cubesat, the minimum altitude for a circular orbit to complete at least one revolution is approximately 150 km.
My colleagues and I collected the following orbital data from a cubesat we were operating:
The vertical jumps on the graph are where the orbit model was updated with new ranging measurements. We last made contact with it at 145 km altitude, about 40 minutes prior to its final destruction.
Here's a quick database exercise based on TLE data for the FLOCK 3U Cubesat constellation. I took the last published elset for each of the 28 objects and calculated the apogee & perigee:
Perigee: 162.6 +- 13.2 km
Apogee: 175.1 +- 14.8 km
Lots of caveats here. Tracking can get sketchy near decay, and sensor coverage can vary alot from rev to rev. So it's not clear how many revs were left for these objects. I'd look at this as a rough upper bound. pericynthion's data is based on last operator contact, so will be more accurate.
An aside, but something that threw me for a bit when looking at this was the launch date. Event though the FLOCK satellites went up on an Antares in Feb of 2014, they were deployed from ISS. So they were assigned the international designator (1998-067*) and launch date (1998-11-20) for ISS. Something to keep in mind when working with this data. The catalog numbers are sequential, the rest of the stuff can surprise you.
Theoretically - You could orbit at any height, as long as the lowest point of your orbit doesn't hit the Earth...
Your problem is that an orbit within the atmosphere has quite a bit of drag (minor understatement) so it will decay really quickly, and with a lot of spectacular light and sound.
For a Cubesat, and in fact any satellite you want to complete at least one orbit, you are really going to have to be outside the atmosphere.