# Why does the velocity dispersion within a galaxy cluster decrease as a function of radius?

In the paper 10.1093/mnras/sty2379, there are plots showing the velocity dispersion as a function of radius. Focusing on the right-hand plot, non-merging normal clusters, intuitively why should the velocity dispersion be smaller for galaxies that are farther from the center?

This would imply that the distribution of redshifts in a galaxy cluster is narrower in the outer regions of the cluster compared to the core. That seems counter-intuitive...

• When you say "why should the velocity dispersion becomes smaller as more and more galaxies are included with increasing radius?", it sounds like you are assuming that $\sigma_P(R)$ is the velocity dispersion of galaxies below $R$. If I understood that correctly, I should point out that actually $\sigma_P(R)$ is the velocity dispersion of galaxies "at" $R$ (with a window function given by equation 2). So a decreasing $\sigma_P(R)$ just means more distant galaxies orbit more slowly.
– Sten
Jun 22 at 20:05

Clusters are velocity dispersion-supported, not rotation-supported. Their net rotation is minimal. So when the velocity dispersion drops at larger radii, that really means that orbital velocities as a whole are going down with radius. That's expected for a system where the mass is concentrated toward the center (e.g. the Solar System).

So why does the velocity dispersion rise as a function of radius for $$R\lesssim r_{200}$$ in systems that are merging (left-hand panel), while it falls for systems that are not merging (right-hand panel)? A primary principle responsible for this effect is as follows.

For collisionless systems accreting in a cosmological context, material is approximately stratified, such that recently accreted material remains (on average) at higher radii than material that was accreted earlier. This is because material that accreted later has more energy and angular momentum than that accreted earlier. Consequently, the mass at each radius is connected to how rapidly the system was accreting at the time that its overall size was (something proportional to) that radius (e.g. Ludlow et al. 2013).

In particular, if the system accreted material rapidly at recent times, it will have more mass at higher radii.

Merging systems are biased toward having a high recent accretion rate. Hence, they tend to have more mass at higher radii. How this connects to orbital velocities can be understood on an approximate level by thinking about the velocity of a circular orbit at the radius $$r$$ in a spherical system, $$v_\mathrm{circ}=\sqrt{GM(r)/r},$$ where $$M(r)$$ is the mass enclosed within the radius $$r$$ (not the total mass). $$M(r)$$ always increases with $$r$$. If the mass of the system is concentrated toward the center, $$M(r)$$ grows slowly with $$r$$, and potentially $$v_\mathrm{circ}$$ decreases with radius. If the system has a lot of mass at higher radii, however, then $$v_\mathrm{circ}$$ could potentially increase with radius (this happens specifically if $$M(r)$$ grows more quickly than $$r$$).

Conversely, non-merging systems are biased toward having a low accretion rate, so they tend to have less mass at higher radii. So for them, it makes sense that $$v_\mathrm{circ}$$ should decrease with radius. While the velocity dispersion is more complicated than $$v_\mathrm{circ}$$, it scales similarly.

• The red lines don't indicate "high cluster mass", they indicates more massive galaxies within the cluster(s). Jun 22 at 18:15
• "recently accreted material remains near the outskirts of the system. It has too much energy and angular momentum to sink to the center" -- recently accreted material is in the outskirts mostly because it simply hasn't had time to fall into the inner regions yet, not because it can't. Jun 22 at 18:18
• @PeterErwin Only true for particularly massive subhalos, which sink due to dynamical friction. That's not most material. (It's perhaps most of the visible material but not most of the gravitating material.)
– Sten
Jun 22 at 18:20
• @PeterErwin Regarding the first comment, maybe I'm confused about what this article is doing. Are their velocity dispersion profiles even looking at the velocity dispersion of the cluster (as I assumed) or of the constituent galaxies?
– Sten
Jun 22 at 18:21
• 1. All the halos will experience dynamical friction. 2. How do you explain ram-pressure stripping of infalling spirals, if they're supposed to stay in the outskirts> Jun 22 at 18:27