Virialization or Thermalization?

In the literature on gaseous halos, there's a lot of confusion about two concepts that are closely related but distinct: virialization and thermalization. People often use the first term when they really mean the second. However, virialization is a global property of the entire system, including both the dark matter and the baryonic matter. Thermalization, on the other hand, is a property of just the baryonic matter and usually pertains to a local subset of the entire system. This article tries to clarify how the concepts differ.

Virialization

The virial theorem, as applied in astronomy, pertains to a self-gravitating system of particles. It’s useful because it allows us to determine relationships between integral quantities describing the overall system without knowledge of the system's structural details, which can be very complex.

It is usually derived from an expression for the total moment of inertia of the entire system of particles. As the system settles into a steady-state configuration, the time derivative of its total moment of inertia approaches zero. Setting it exactly to zero leads to the virial theorem relating total kinetic energy ($E_{\rm kin}$) to total gravitational energy ($E_G$), often expressed as

A virialized self-gravitating system is therefore one that is close enough to a steady state to approximately satisfy the virial theorem. Real cosmological halos don't quite satisfy the virial theorem because they are continually accreting new matter, meaning that the halo's overall moment of inertia is still changing with time.

This is where judicious use of the virial theorem's often-neglected surface pressure term can come in handy. Infalling matter is insignificant at small radii in a cosmological halo, where local conditions can be close to a steady state. If you place an imaginary spherical boundary at a particular radius $r$, the momentum flux of particles moving through it to larger radii balances the momentum flux of particles moving through it to smaller radii. It's almost as if the particles are being reflected off of the boundary, and you can calculate what the pressure on that boundary would be if that were actually happening.

Accounting for the effects of the boundary pressure gives the following modified form of the virial theorem, which applies to the subset of the system within the imaginary boundary:

where $P_{\rm b}$ is the effective boundary pressure at $r_{\rm b}$. To calculate the boundary pressure correction, you need a momentum distribution function for the particles at $r_{\rm b}$. That's an easy correction to make for a singular isothermal sphere but not so easy to make in general.

Also, you can't make $r_{\rm b}$ too large without reaching a radius at which the amount of infalling matter starts to produce significant departures from the steady-state assumption used to derive the virial theorem. The term virial radius is often used to refer to that transitional radius, but there's no consensus definition of it, because the transition from infall to a steady state is a gradual function of $r$. People therefore tend to select a definition of $r_{\rm b}$ suitable for their own particular investigation.

Once a virial radius $r_{\rm vir}$ has been defined, one can integrate over all of the particles within it to find a total kinetic energy $E_{\rm kin} (<r_{\rm vir})$. The result can be expressed in terms of a mean squared particle speed:

It is then tempting to define a corresponding virial temperature via

But notice what's going on here. The left hand side of the equation assumes that the mean mass per particle is $\mu m_p$, which describes the gaseous matter but not the majority of the matter in a cosmological halo. And the right side of the equation gives the specific energy for everything inside of $r_{\rm vir}$, which is not necessarily the same as the specific energy of matter in a thin shell at $r_{\rm vir}$.

The singular isothermal sphere with an isotropic velocity dispersion is a special case in which $v_{\rm rms}$ at $r_{\rm vir}$ is identical to $v_{\rm rms}$ everywhere else inside of $r_{\rm vir}$. That makes it particularly easy to calculate the surface pressure term in the virial theorem and to derive a virial temperature, $T_{\rm vir} = GM(<r) \mu m_p / 2kr$, that is independent of radius but technically assumes that all the matter is baryonic.

In an NFW potential well [1], however, the quantity $E_{\rm kin} (<r) / M(<r)$ depends on $r$ and is not determined by conditions local to the shell at radius $r$. For a given value of $GM(<r) / r$, the quantities $E_{\rm kin} (<r)$ and $E_G (<r)$ both depend on the concentration of the halo. In a potential well of greater NFW concentration, the absolute value of $E_G(<r)$ will tend to be greater for a given $M(<r)$ because the mass is generally closer to $r=0$.

Thermalization

Hopefully it's clear by now that virialization is a global property of a self-gravitating system that is converging toward a steady configuration, and it necessarily includes both the gas and the dark matter. However, many people in the field use the word virialization when speaking about how non-thermal gas motions relax toward hydrostatic equilibrium.

That particular use of the term causes a lot of confusion, because the virial theorem is a statement about the relationship between kinetic energy and potential energy, not thermal energy and potential energy. In the context of a cosmological halo, the dark matter never truly thermalizes because two-body collisions between dark-matter particles are essentially non-existent. There's no way for the velocity dispersion of those particles to become isotropic in the way that the velocities of baryonic gas particles do.

As a result, there are really two distinct steps in the relaxation of the baryonic medium toward a steady-state configuration. The first is virialization, in which infalling blobs of matter are gravitationally scattered through violent relaxation, which does not change the specific kinetic energy of the matter by a large factor. The second is thermalization, in which the bulk kinetic energy of the gas dissipates into thermal energy, and it depends on small-scale transport processes, not large-scale gravitational scattering.

The end point of thermalization is hydrostatic equilbrium, which satisfies

with $P$ and $\rho$ applying only to the baryonic gas. If all the gas pressure is thermal and the potential well is spherical, this condition can be expressed as

But most illuminating form of the hydrostatic equilibrium equation is

where $T_\phi(r) \equiv G M(<r) \mu m_p / 2kr$ is a local gravitational temperature.

That last form is my favorite because of how it clarifies the relationship between gas temperature and the gravitational potential. As you can see, the gravitational temperature has been defined so that it's identical to the virial temperature of a singular isothermal sphere, which is the only configuration in which $G M(<r)/r$ is constant with radius. And the constant of proportionality between $T$ and $T_\phi$ is determined by the logarithmic slope of the pressure gradient. If that slope is –2, as it would be for dark matter configured as a singular isothermal sphere, then $T = T_\phi$. But keep in mind that gas in a cosmological halo does not necessarily end up with the same pressure profile as the dark matter because its collisional properties are different.

More importantly, if there are significant gas motions that have not yet thermalized, one needs to add inertial and acceleration terms to the momentum equation:

Notice that if $\langle \partial \mathbf{v} / \partial t \rangle \ll \langle \mathbf{v} \cdot \nabla \mathbf{v} \rangle \sim \nabla \phi$, then bulk gas motions can be virialized without being thermalized. One example is a system of cold ($T \ll T_\phi$) discrete clouds on ballistic orbits that rarely cross paths.

In a spherical steady-state system, inclusion of the inertial term gives

where the average indicated by angled brackets is over time and solid angle within a shell at radius $r$. Rewriting this expression to express steady-state gas temperature in terms of other quantities gives

There are now two things entering into the relationship between $T$ and $T_\phi$. In addition to the pressure-gradient factor, there is also a correction term that depends on bulk kinetic energy.

The origin of that correction term has little to do with virialization. It comes from separating kinetic energy of gas particles into isotropic and anisotropic momentum components when inserting the concept of thermal pressure into the fluid equations.

The whole scenario can also be described in terms of particle distribution functions governed by the Boltzmann Equation. Then the distinction between the pressure gradient term and the inertial term in the momentum equation would vanish. Instead, a stress-tensor divergence term would represent them both. And we'd go back to conceiving of virialization simply in terms of total kinetic energy, without separating it into thermal and non-thermal parts.

So that's why it's important to use language that clearly distinguishes between (global) virialization and (local) thermalization. Otherwise, confusion ensues.