In this section, I first review how incorporating gravity changes the physics of thermal systems and discuss the type of systems well-approximated by this treatment. I then outline two examples of successful evolution equations. Finally, I discuss some of the ‘paradoxical’ features.
How the situation changes with gravity
Gravitational forces are negligible in the terrestrial thermal systems with which we are familiar. But in extraterrestrial systems such as galaxies, this assumption is of course no longer justified. Unlike the local collisions and forces in an ideal gas, the gravitational force is long-range; the range of the dominant interaction is large relative to the spatial size of the system. Consequently, the forces on a given star are not only due to its nearest neighbours, but include a contribution from the large scale structure of the stellar system. Indeed, if the density of stars is spatially constant (cf. Fig. 1), the gravitational force exerted on a given star (by the rest of the stellar system) at the apex is the same from the patch of stars of solid angle \(d\Omega \) surrounding it at distance \(r_1\), as from the patch at distance \(r_2\). Clearly if the distribution of stars were exactly spherical, there would be no net force on this star. However, if the density of stars falls off more slowly in one direction, then only this very global feature of the entire stellar system will be responsible for the force on our star. This contrasts sharply with the forces experienced by a molecule in gas, which come only from its nearest neighbours and thus is a much more local feature.
The gravitational potential \(V \sim \frac{1}{r}\) is asymptotically zero; and this dominates the behaviour of SGS due to (i) its infinite range and (ii) the fact that a (potentially infinite) amount of energy can be released as two point particles get arbitrarily close together, as seen in Fig. 2.
Here, we primarily focus on the gravitational n-body case where stellar systems are treated as collections of n point masses. Unlike ideal gases, the total energy is not even approximately the sum of the kinetic energy of the constituents since the (negative) gravitational potential energy must be included. The Hamiltonian for such a system of n ‘particles’ of equal mass m is thus:
$$\begin{aligned} H(\mathbf {q,p})=\sum \limits _{i=1}^n \frac{\mathbf {p_i}^2}{2m}-\frac{1}{2}\sum \limits _{i=1}^n\sum \limits _{i\ne j} \frac{Gm^2}{|\mathbf {q_i}-\mathbf {q_j}|} \end{aligned}$$
(1)
Whilst this is an idealisation, it provides a very successful description of elliptical galaxies (\(10^{11}\) stars) and globular clusters, i.e. spherical gravitationally bound systems of about \(10^5\) stars, which both contain very little interstellar medium (dust and gas).
Of course, for some systems we cannot ignore hydrodynamics—namely when interstellar dust and gas are relevant. And for some systems general relativity cannot be ignored. For example, this applies when black holes are present, and when the cosmological structure i.e. curvature of space on very long length scales, cannot be ignored, such as in the dynamics of clusters of galaxies.
Indeed, I should make an obvious and more general disclaimer: whilst Newtonian thermal physics can be used in galactic dynamics describing extraterrestrial systems it is (unsurprisingly) far from the whole story. Nevertheless, models based on the simple Hamiltonian (1) have had some venerable successes: cf. Sect. 2.2.
Successes
I shall sketch two approaches, the first assuming stars do not ‘collide’, the second allowing for collisions. To model these gravitating systems, the broad idea is to find a probability density function f in phase space and consider its evolution.
Modelling a stellar system to be collisionless requires the approximation that no ‘encounters’ occur. An encounter occurs when two stars are so close as to cause a gravitational perturbation, altering their orbits. (Collisions involving physical contact between stars are exceedingly rare and can be ignored in most models.)
The star’s orbit is then approximated by assuming the total mass of the system is smoothly distributed instead of concentrated in point-like stars. This ‘collisionless’ (encounter-less) approximation holds for certain systems, in particular: for globular clusters and elliptical galaxies (containing about \(10^{10}\) stars) since, for timescales less than the relaxation time, stellar encounters are unimportant except at their centres (Binney and Tremaine 1987).
Here, the relaxation time is proportional to the number of stars and the time taken for a star to cross the galaxy (the crossing time). After the relaxation time the star’s actual velocity differs from the smooth gravitational field case and its orbit will deviate from the smooth field model by an amount of the order of its original velocity.
As in Boltzmann’s treatment of a dilute gas, we define a probability density function \(f(\overrightarrow{r},\overrightarrow{v},t)\) where \(f(\overrightarrow{r},\overrightarrow{v},t) d^3rd^3v\) gives the probability at t of finding a star in volume \(d^3r\) around r with velocity within \(d^3v\) of v. Since we assume all N stars have the same probability density function (and are stochastically independent of each other), this function is defined in a 6-dimensional phase space, rather than the 6N-dimensional phase space of the entire set of N stars.
The collisionless Boltzmann equation gives this function’s evolution;
$$\begin{aligned} \frac{\partial f}{\partial t} +[f,H]=0 \end{aligned}$$
(2)
where H is given by Eq. 1. Note that the collisionless Boltzmann equation is nonlinear as the gravitational potential \(\Phi (x,t)\) depends on the distribution of stars’ masses, \(f(\overrightarrow{r},\overrightarrow{v},t)\).
We can define the entropy
$$\begin{aligned} S=-N\int f(\mathbf {r},\mathbf {v},t) \ln f(\mathbf {r},\mathbf {v},t)d^3rd^3v. \end{aligned}$$
(3)
To look at the evolution of a stellar system over timescales longer than the relaxation time, in which encounters between stars must be considered, we need what is (usually) called the Fokker-Planck approximation. The encounter operator, \(\Lambda [f]\), gives the difference of the probability that a star is scattered into and out of a volume of phase space in a given time interval. Equation 2 becomes
$$\begin{aligned} \frac{\partial f}{\partial t} +[f,H]= \Lambda [f]. \end{aligned}$$
(4)
To sum up: the collisionless Boltzmann and Fokker-Planck equations have proven to be empirically successful evolution equations for the systems described at the end of Sect. 2.1.Footnote 1
Unusual features
However, the extension of thermal physics to SGS is far from seamless. There are a wide array of problems surveyed in Callender (2011): of which I will consider only three.
(1) Strong interactions Firstly, functions, such as energy and entropy, are often not additive or extensive for SGS. For an ideal gas the total energy E is the kinetic energy K, whereas for gravitating systems the (negative) potential energy U contributes: \(E=K+U\). Functions such as energy and entropy are usually additive: the energy of a combined system A + B is just the sum of the energy of A and the energy of B. Usually the Hamiltonian of the joint system is \(H_{AB}= H_A + H_B + H_{int}\), but it can be approximated by \(H_{AB}= H_A + H_B\). (So strictly speaking, the energy is additive iff there are no interactions, i.e. \(H_{int}=0\)). However, a SGS will not have even approximately additive functions since the neighbouring stars do not contribute the majority of the influence on a particular star (Cf. Fig. 1). That is, the interaction Hamiltonian, \(H_{int}\not \approx 0 \). The physical reason for this can be seen in Figs. 3 and 4, showing how putting together two ‘boxes’ of gravitating stars alters both boxes: the long-range attractive forces result in ‘clustering’ or ‘clumping’ not seen for ideal gases (or indeed real gases in terrestrial settings, which are well described by zero or only short-range forces between constituents). For these gases, short-range potentials are dominant—adding two boxes of gases does not alter the systems in such a dramatic way, since the systems only interact at their boundary.
As a consequence, variables such as energy and entropy are usually taken to be extensive. Here, a variable is called ‘extensive’ if it depends linearly on the size of, i.e. the number of constituents in, the system (e.g. mass, internal energy, volume)Footnote 2 and is called ‘intensive’ if independent of system size (e.g. density, pressure). The energy of a subsystem is proportional to the volume, whereas interactions between subsystems are proportional to their interface boundary’s surface area and are, therefore, of a smaller order of magnitude, provided the subsystems are big enough. So strictly speaking, even for short-range potentials, entropy and energy are only extensive in the thermodynamic limit. But although this is a matter of degree, there is still a contrast of principle with SGS. For energy and entropy are not extensive for gravitating systems, no matter how large the system.Footnote 3
(2) Putting in energy reduces the temperature Gravitating systems can have a very unusual property: negative heat capacity. The heat capacity (at constant volume) is the amount of energy required to raise the temperature by one degree at constant volume;
$$\begin{aligned} C_V= \frac{\partial E}{\partial T}\bigg |_{V}. \end{aligned}$$
(5)
When the system is in virial equilibrium (where \(2K+U=0\)), the total energy is negative (\(E=K+U\), so \(E=-K\), where K is by definition positive). From the equipartition theorem , we have \(K=\frac{3}{2}Nk_BT\). This implies \(E= -\frac{3}{2}Nk_B T\) and thus \(C_V=-\frac{3}{2}Nk_B\): the heat capacity is negative. If the system gives out energy, the temperature will increase. If you put energy into a system, the temperature goes down. Indeed, unusual!
(3) The gravothermal catastrophe Thirdly, there is the infamous gravothermal catastrophe (Lynden-Bell et al. 1968). To explain this, let us consider in general terms which evolutions are entropically favourable. Whether a process (such as expansion) increases entropy depends on whether the phase space volume increases. Thus, for example, expansion of an ideal gas is entropically favoured since it increases the volume available. Ceteris paribus, the hotter the system the higher its entropy as more momentum states are available (due to the increased kinetic energy). So whether an expansion of a self-gravitating system increases or decreases entropy depends on how the competing factors affect the phase space volume (Wallace 2010). An increased volume means more spatial states but results in a decreased number of momentum states as the kinetic energy has decreased, since work is done against the attractive gravitational field.
Turning now to SGS: when the density contrast between the edge and centre of a SGS is great enough, we conceptually divide the system into a uniform core and a uniform halo, each in virial equilibrium. If a small amount of heat is transferred to the envelope from the core, the core’s kinetic energy decreases, making it favourable for the core to contract (as \(U=2E\), E has decreased so U is more negative). Since the core has negative heat capacity, losing energy increases the temperature. The core decreases in entropy but this is more than offset by the expansion and cooling of the halo.Footnote 4 The heat flow and contraction increases the temperature gradient between the core and envelope and thus the process of heat transfer from the core to the halo is self-perpetuating.
The gravitational potential, \(V\sim \frac{1}{r}\), being unbounded from below as \(r\rightarrow 0\), means that this collapse would appear to continue without end. For an infinite amount of potential energy can be released by moving two particles closer and closer together, as seen in Fig. 2. Consequently, it seems that there are no equilibrium states. No equilibrium will be reached since, according to the gravitational potential, the core can keep contracting indefinitely becoming infinitely dense.
Is this gravothermal collapse observed? Here we meet a familiar philosophical theme: that singularities in one theory can signify the breakdown of that theory, and often signal some features of the successor theory (Berry 2002; Batterman 2001)—so that idealisations taking some quantity to infinity can play a key role in inter-theory relations. More generally, physics consists of models which have a limited domain of applicability; if you push any model of physics far enough it will break down. As Feynman quips: “When you follow any of our physics too far, you find it always gets into some kind of trouble” (Feynman et al. 1964, §28.1). The same point is made in the literature about SGS: Hut says “whenever a theory predicts the occurrence of singularities, it has been a sign that other physical effects, which have been overlooked, will kick in before actual infinities are reached” (Hut 1997).
But to return the question of gravothermal collapse: indeed, as Hut says, other physical effects eventually kick in. Globular clusters undergo this gravothermal collapse, albeit over a period of tens of millions of years. Agreed: in a globular cluster, the formation of hard binaries provides the core with an energy source (Spitzer and Ostriker 1997, p. 363): nevertheless, once exhausted gravitational collapse will continue. Another instance is a contracting gas cloud (that ultimately will form stars) where the heat is emitted as electromagnetic radiation (due to the presence of an interstellar medium which is absent from globular clusters). In the case of stars, fusion processes provide the energy source to resist gravitational collapse but eventually this energy source runs out. In this case, gravitational collapse resumes until another effect (dependent on the star’s mass) kicks in. For example: for stars of around 10 solar masses, collapse continues until a supernova occurs leaving a neutron star in which the degeneracy pressure (a consequence of the Pauli exclusion principle) resists the attractive force of gravity (Phillips 2013).
But I will not need more details about these “additional physical effects”. For this paper, the main point of all these other effects is that they involve various theories and subdisciplines of physics such as hydrodynamics, quantum theory—and statistical mechanics (cf. Sects. 4 and 5).Footnote 5