If we consider Mars, with its relatively low gravitational acceleration and low density atmosphere, I had always assumed that if we have a single-stage-to-orbit launch vehicle on the surface of Mars that is required to achieve a stable circular orbit upon engine cutoff, the most fuel efficient method would be to perform a single engine burn in order to minimize exposure to gravitational losses. However, would it somehow be beneficial (in terms of minimum fuel use to get to orbit) to actually have, for example, 2 burns with a coast phase in between them? Does anyone know if any papers have been written regarding the best number of burns for a SSTO vehicle to reach orbit so as to minimze fuel use?
Orbits are closed, so you will always return to the last altitude at which you shut down your engines. If that place was in the atmosphere, or close enough to the surface of an airless body that small perturbations would result in an impact, then you are not in orbit. Bottom line: if you want to maximize efficiency by expending as much propellant as possible close to the surface, then you have no choice but to do a second burn at some point to raise the periapsis.
The most efficient way to get into orbit off of an airless body (i.e., not Mars) would be a short, high thrust, very close to horizontal burn right at the surface, a coast halfway around the planet, and lastly a periapsis raise maneuver at apoapsis. The periapsis raise maneuver would be quite small compared to the launch maneuver. E.g. ~100 m/s as compared to several km/s.