# Intuitive understanding of the virial radius/mass

I'm currently trying with no avail to understand the intricacies that define the virial radius or mass in a way that is different from the actual galactic radius/mass.

I understand it is derived in relation to the critical density of the Universe, such that:

$\rho_c(z) = \frac{3H^2(z)}{8\pi G} \approx\frac{3H_0^2}{8\pi G}[\Omega(1+z)^3 + \Omega_\Lambda]$

$M_{vir} = \frac{4\pi}{3}\rho_c(1+\Delta_{vir})r^3_{vir}$

So if an area is overdense it will lead to accretion of mass for the dark matter halo. My issue however is is the concept of actually if it is not true mass contained within the virial mass, what is it? Is it a fraction of the mass out to a certain radius of the halo? Is it an amount that must be reached such that it can accrete surrounding matter?

The basic problem of defining the mass of a galaxy is that the density decreases from the center and out, but never reaches zero. Hence, there is no well-defined radius at which to stop counting particles, so when you say "the actual mass", this concept doesn't really exist.

Nevertheless, there several definitions that do make physical sense.

### Virial mass

The virial mass $M_\mathrm{vir}$ is defined as the mass contained within the virial radius $r_\mathrm{vir}$, which in turn corresponds to the radius of a spherical region which is sufficiently overdense wrt. the background density that it will detach from the Hubble expansion and start collapsing to form a structure. This region will then virialize, such that roughly the particles inside $r_\mathrm{vir}$ will be gravitationally bound to each other, and particles outside will not.

The value of the overdensity $\Delta_\mathrm{vir}$ depends on the cosmological model. In an Einstein-de Sitter Universe, it takes the value $\sim 178$ (at high redshifts), but since reality (and $N$-body simulations) differ somewhat from a spherical collapse in an EdS Universe, often the value 180 or even 200 is used.

### Overdensity mass

In principle, the virial mass is the gravitationally bound mass, but it is roughly equal to the mass you'll get if you sum all particles from the center and outwards, until the average density inside is equal to $\sim200$ times the average density of the Universe. To avoid confusion, you also sometimes use the term $M_{180}$ or $M_{200}$ for the mass inside the radius $r_{180}$ or $r_{200}$. Note also that various authors use various definitions of the density with respect to which to calculate the overdensity — some use the total mass density (i.e. dark matter + baryons), some use the total energy density (i.e. including radiation and dark energy), and some use the critical density $\rho_\mathrm{c}$ (which is equal to the former in a flat universe). At high redshifts (but not too high), where dark energy and radiation can be neglected, the difference is not large, though. I think the original definition is from Bryan & Norman (1998), who use $\rho_\mathrm{c}$.

Observationally, you can't see the total mass directly, since most of it is dark. For galaxy clusters, you often use $M_{500}$ which is a smaller total mass than $M_{200}$, since you stop counting at higher density. Because galaxy/cluster densities drop roughly as power laws as you move away from the center, the difference between $M_{200}$ and $M_{500}$ is not huge, however.

### Half-light mass

An often-used measure of the size of a galaxy (or cluster) is the half-light — or effective — radius, which is defined as the radius within which half of the light is emitted. But this is much smaller than $r_{200}$, typically only 1-2% (e.g. Kravtsov 2013). The reason is simply that what we see as a beautiful galaxy is only the small part in the center of a much larger halo of dark matter (and hot gas emitting only diffuse X-rays).

The bottom point is that there is no such thing of an actual mass; it depends on what you need the mass for, and how you obtain it.

• I see! I think the small connection I was missing is the point you mentioned when describing the Virial mass itself. That definition makes sense in that it overcomes expansion at this mass. Thank you very much, this clears it up nicely. As a side point, is there a connection between the virial radius and the turn around radius? – Wrothschild May 21 '18 at 9:45
• @Wrothschild I don't remember the details, but I don't think there can be 1:1 correspondence, since the turnaround radius depends quite sensitively on the cosmology, whereas the virial radius is given mainly by the total mass inside the collapsed region. – pela May 21 '18 at 12:39