If a satellite is shut down so that it neither generates additional heat from whatever internal energy source it might have, nor is able to dissipate heat by an active cooling system, will it eventually reach the high temperatures of its surrounding rarefied molecules in the exosphere? Wikipedia (in the thermosphere article) claims that the rate at which the heat would be transferred to the object is much lower than the rate at which it would dissipate heat by thermal radiation:
The highly attenuated gas in this layer can reach 2,500 °C (4,530 °F) during the day. Despite the high temperature, an observer or object will experience low temperatures in the thermosphere, because the extremely low density of the gas (practically a hard vacuum) is insufficient for the molecules to conduct heat. A normal thermometer will read significantly below 0 °C (32 °F), at least at night, because the energy lost by thermal radiation would exceed the energy acquired from the atmospheric gas by direct contact.
Unfortunately, there is no citation for this claim. To me, it still doesn't make sense, because the object would be subject to the same environment that maintains the exosphere hot. In other words, if the object is able to emit thermal radiation at a higher rate than the influx of heat, so would the surrounding molecules.
To me there seems to be only two possibilities. Either Wikipedia is wrong and the thermometer would eventually read a higher temperature, or solid objects are somehow not able to absorb energy from the Sun at the same rate as the gas molecules or ions in the exosphere, which seems more likely.