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1) Is the rover itself powered on during the cruise to Mars?
2) If so, does that power come from the MMRTG on the rover, or is power supplied to the rover by the cruise stage?
3) If not, would the heat from the MMRTG be enough to keep the rover alive in space?
I think that the paper called "In-Flight Performance of the Mars Science Laboratory Spacecraft Cruise Phase Thermal Control Systems" (https://doi.org/10.2514/6.2012-3517) can provide some insights on this topic.
According to the above paper, during cruise mode, most of the sub-systems are off. The most active sub-system is the thermal control.
While a majority of the Entry, Descent, Landing and Surface-specific hardware subsystems are dormant during the nine-month transit from Earth to Mars, all of the spacecraft’s thermal control systems are active and are relied upon to maintain hardware within allowable temperature limits. The thermal control subsystem is designed to support routine cruise phase activities such as spacecraft subsystem monitoring, health checks, and operational maintenance. Additionally, the thermal control system must accommodate less frequent but more significant changes to the spacecraft state such as attitude turns, trajectory correction maneuvers, guidance and navigation system calibrations, and EDL phase preparations.
One of the main tasks of the thermal control subsystem is to waste the heat created by the Multi-Mission Radioisotope Thermoelectric Generator (MMRTG). The MMRTG create about 2000W and the heat is removed using a cooling liquid that transports the heat to the spacecraft surface.
Looking at the images on the paper and here (https://mars.nasa.gov/MPF/mpf/mpfcruise.html) we can see that the spacecraft includes large solar array. I think that the rover active sub-system are getting power from the rover battery and the battery is charged using the solar panels.
When the Rover battery state-of-charge is lowered or raised, the resultant disturbance in the spacecraft power bus necessitates a state change in the number of power generating solar array segments.
