If a satellite is shut down, will it eventually reach the high temperatures of the exosphere?

Question

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.

Answer 1

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.

That would be true if both were approximately black bodies. The equilibrium temperature would be the same. But if (for example) the gas is a poor IR emitter, it might have to reach a much higher temperature to be in thermal equilibrium.

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, ...

... or the solid objects and the gas molecules do not radiate energy at the same rate when at the same temperature.

Solid objects are often excellent IR emitters, but several gases are nearly transparent.

Answer 2

There's no need to speculate here: it is the usual experience of satellite operators that an unpowered satellite in low Earth orbit runs cold, typically ~0 Celsius. That's the temperature at which solar heating balances radiational cooling in the infrared.

The exosphere is, for practical purposes, transparent to infrared and visible radiation. This means it can neither absorb nor radiate heat away in the infrared or visible range. It can, however, absorb ultraviolet and soft x-ray radiation from the Sun. The Sun is quite variable at these wavelengths: during the maximum of the 11 year solar activity cycle, the exosphere is hotter than it is at the minimum.