Gravothermal catastrophe: looking for simple explanation

Basically, it's a consequence of negative heat capacity. Gravitationally bound systems can (often do) behave such that adding energy results in reducing temperature, and vice versa. You can understand this intuitively if you consider a simple two-body system: adding energy to the system causes the orbits to expand, and bodies on larger orbits move at lower speeds; thus, even if the orbits are highly eccentric, the average speed of the two bodies goes down when energy is added, and this generalizes to systems of many particles, where the average velocity determines the temperature of the system.

If a system with negative heat capacity comes into contact with a large thermal reservoir at higher temperature, it will absorb heat from that reservoir... and get colder. And thus continue absorbing heat until the reservoir is exhausted- it cannot come to equilibrium.

If the higher-temperature reservoir is another gravitationally bound system (in particular, if it is a subset of the same gravitationally bound system), then you have the conditions for a gravothermal catastrophe. This is because a high-temperature, negative-heat capacity system in contact with a cold sink will give up heat until it is entirely exhausted, getting hotter, rather than colder, as it does so. So, when you have a high-temperature gravitational system and a low-temperature gravitational system in contact, you end up with all of the energy being transferred out of the hot section, which contracts, and into the cold section, which expands, and probably becomes unbound.

This tends to happen in old globular clusters and bright elliptical galaxies, where the cores can get hot, start transferring kinetic energy to the outer regions, and collapse.

Consider the virial theorem, which says (ignoring complications like rotation and magnetic fields) that twice the summed kinetic energy of particles ($K$) in a collection of particles (could be the gas in a star, could be stars in a star cluster) plus the (negative) gravitational potential energy ($\Omega$) equals zero. $$ 2K + \Omega = 0$$

Now you can write down the total energy of the system as $$ E_{tot} = K + \Omega$$ and hence from the virial theorem that $$E_{tot} = \frac{\Omega}{2},$$ which is negative.

If we now remove energy from the system, for instance by allowing the gas to radiate away energy or by allowing some of the stars in the cluster to escape, such that $\Delta E_{tot}$ is negative, then we see that $$\Delta E_{tot} = \frac{1}{2} \Delta \Omega$$

So $\Omega$ becomes more negative - which is another way of saying that the star or the cluster of stars is attaining a more collapsed configuration.

Oddly, at the same time, we can use the virial theorem to see that $$ \Delta K = -\frac{1}{2} \Delta \Omega = -\Delta E_{tot}$$ is positive. i.e. the kinetic energies of particles in the gas (and hence their temperatures) actually become hotter. In other words, the gas has a negative heat capacity. But a hotter temperature means more radiation or that more stars can escape, and if the energy losses continue, then so does the collapse - this is the gravothermal catastrophe.

This process is ultimately arrested in a star by the onset of nuclear fusion. This replaces the radiative losses with nuclear energy. In the case of a star cluster the collapse can be halted by the formation of close binary systems. These can keep the core hot by transferring energy from the binary orbit to stars in the core.