I've just found out that a negative specific heat capacity is possible. But I have been trying to find an explanation for this with no success.
Negative heat capacity would mean that when a system loses energy, its temperature increases. How is that possible in the case of a star? Musn't there be a source of energy to increase the temperature of any system?
Answer
Consider a satellite in orbit about the Earth and moving at some velocity $v$. The orbital velocity is related to the distance from the centre of the Earth, $r$, by:
$$ v = \sqrt{\frac{GM}{r}} $$
If we take energy away from the satellite then it descends into a lower orbit, so $r$ decreases and therefore it's orbital velocity $v$ increases. Likewise if we add energy to the satellite it ascends into a higher orbit and $v$ decreases.
This is the principle behind the negative heat capacity of stars. Replace the satellite by a hydrogen atom, and replace the Earth by a large ball of hydrogen atoms. If you take energy out then the hydrogen atoms descend into lower orbits and their velocity increases. Since we can relate velocity to temperature using the Maxwell-Boltzmann distribution this means that as we take energy out the temperature rises, and therefore the specific heat must be negative.
This is all a bit of a cheat of course, because you are ignoring the potential energy. The total energy of the system decreases as you take energy out, but the decrease is accomplished by decreasing the potential energy and increasing the kinetic energy. The virial theorem tells us that the decrease of the potential energy is twice as big as the increase in the kinetic energy, so the net change is negative.
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