<p>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).</p>
<p>So why does the velocity dispersion rise as a function of radius for <span class="math-container">$R\lesssim r_{200}$</span> 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.</p>
<p>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. <a href="https://arxiv.org/abs/1302.0288" rel="nofollow noreferrer">Ludlow et al. 2013</a>).</p>
<p>In particular, if the system accreted material rapidly at recent times, it will have more mass at higher radii.</p>
<p>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 <span class="math-container">$r$</span> in a spherical system,
<span class="math-container">$$v_\mathrm{circ}=\sqrt{GM(r)/r},$$</span>
where <span class="math-container">$M(r)$</span> is the mass enclosed within the radius <span class="math-container">$r$</span> (not the total mass). <span class="math-container">$M(r)$</span> always increases with <span class="math-container">$r$</span>. If the mass of the system is concentrated toward the center, <span class="math-container">$M(r)$</span> grows slowly with <span class="math-container">$r$</span>, and potentially <span class="math-container">$v_\mathrm{circ}$</span> decreases with radius. If the system has a lot of mass at higher radii, however, then <span class="math-container">$v_\mathrm{circ}$</span> could potentially increase with radius (this happens specifically if <span class="math-container">$M(r)$</span> grows more quickly than <span class="math-container">$r$</span>).</p>
<p>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 <span class="math-container">$v_\mathrm{circ}$</span> should decrease with radius. While the velocity dispersion is more complicated than <span class="math-container">$v_\mathrm{circ}$</span>, it scales similarly.</p>

CC BY-SA 4.0
2023-06-22T18:15:59.110Z
The red lines don't indicate "high cluster mass", they indicates more massive _galaxies_ within the cluster(s).
Peter Erwin
CC BY-SA 4.0
2023-06-22T18:18:13.877Z
"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.
Peter Erwin
CC BY-SA 4.0
2023-06-22T18:20:19.603Z
@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
CC BY-SA 4.0
2023-06-22T18:21:04.177Z
@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
CC BY-SA 4.0
2023-06-22T18:27:54.490Z
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>
Peter Erwin
CC BY-SA 4.0
2023-06-22T18:30:45.133Z
Regarding my first comment: these are velocity-dispersion profiles for clusters. But they break this down into "velocity dispersion of cluster, computed using all galaxies ('All galaxies')" and "velocity dispersion computing using just galaxies with stellar masses in specific bins". The red line is the computation using only very massive galaxies.
Peter Erwin
CC BY-SA 4.0
2023-06-22T18:36:12.487Z
@PeterErwin 1. Yes but the efficiency of dynamical friction is proportional to mass. Less than ~1/1000 the mass of the cluster (depending on accretion time) and they won't sink over the age of the universe. 2. Particularly massive subhalos? Anyway this is all kind of beside the point because even the recently accreted material that will sink hasn't yet, so it's contributing mass at large radii. And massive systems that sink do so by trading energy with the background, so other material gets elevated.
Sten
CC BY-SA 4.0
2023-06-22T18:41:09.707Z
@PeterErwin I'll remove the mention of cluster mass if I misunderstood the article! It sounds like the connection to merging still stands. I definitely stand by the "recently accreted material remains near the outskirts of the system" claim since I've looked pretty closely at the average behavior of accreted particles in simulations, and this claim holds for them.
Sten
CC BY-SA 4.0
2023-06-22T18:50:55.080Z
Your supposition is that galaxies at large radii are in largely tangential orbits. If this were true, then the dispersion profile would be increasingly biased towards tangential dispersion as you go out in radius. But modeling of observed clusters show increasing _radial_ anisotropy (more radial orbits) as you go out in radius. Galaxies at large radii are in more radial orbits, not more tangential orbits.
Peter Erwin
CC BY-SA 4.0
2023-06-22T18:51:20.823Z
See, e.g., the discussion in [this talk](https://adlibitum.oats.inaf.it/biviano/orbits.pdf).
Peter Erwin
CC BY-SA 4.0
2023-06-22T18:51:56.563Z
@PeterErwin Galaxies aren't in largely tangential orbits. Where did I suppose that? (The mention of the circular velocity is just to give an intuition for how orbital velocities scale.)
Sten
CC BY-SA 4.0
2023-06-22T18:55:02.113Z
How else am I to interpret your statement that recently accreted material "has too much energy and angular momentum to sink to the center"? You're saying they cannot have primarily radial orbits. (I think "sink to the center" and our references to dynamical friction are confusing things: infalling galaxies are primarily _falling_, not sinking.)
Peter Erwin
CC BY-SA 4.0
2023-06-22T18:59:55.093Z
@PeterErwin A particle in the host contributes mass (on average) to a radius range between its apocenter and pericenter. The apocenter and pericenter are comparatively stable, aside from adiabatic contraction and dynamical friction. Material that fell in earlier has lower apocenter and pericenter. So it contributes mass to lower radii. When I say "outskirts" I don't literally mean $R_{200}$. I mean that the new material contributes mass to radii that are, on average, higher than the radii contributed to by older material.
Sten
CC BY-SA 4.0
2023-06-22T19:06:14.900Z
@PeterErwin That is what leads to the conclusion that a high recent accretion rate means that the system's mass is (relatively) weighted toward the outside of the system.
Sten
CC BY-SA 4.0
2023-06-22T19:34:18.017Z
I think the re-edited version is OK, now, since it no longer seems to be suggesting that recently accreted material _can't_ fall into the central regions. (It _does_, but we agree that it probably spends most of its time near apocenter.)
Peter Erwin
CC BY-SA 4.0
2023-06-22T19:38:10.817Z
Now the only problem is that your answer doesn't really address the question of why the _right-hand_ (non-merging) profiles look the way they do -- in particular, why the RH profile is flatter when only massive galaxies are counted vs becoming steeper when less massive galaxies are included.
Peter Erwin
CC BY-SA 4.0
2023-06-22T20:00:10.393Z
@PeterErwin I think that's not what the original poster was asking, since they only mentioned radius and not mass. I interpreted "why should the velocity dispersion becomes smaller as more and more galaxies are included with increasing radius?" as mistakenly assuming that the velocity dispersion includes all radii below $r$ as opposed to just the particular radius bin. Come to think of it, it's worth addressing that explicitly...
Sten
CC BY-SA 4.0
2023-06-25T20:38:54.313Z
@Sten I see indeed my original question implied I was including al galaxies below R instead of at diff radius, which was wrong. Funny enough the mass profile and "orbital" speed at diff radius is what I'm really getting at. You said "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)."  --- to be continued in next comment.
ABC
CC BY-SA 4.0
2023-06-25T20:39:51.537Z
@Sten That would be my interpretation too, but doesn't this contradict the fact galaxies and galaxy clusters are not like our solar system because of the dark matter, and  the NFW mass profile first increases as r^2 and at larger radius ln(r). Even though there's no net rotation, each individual galaxy still orbit more or less circularly around virialized region obeying sqrt(GM/r), ignoring infall. So the orbital speed should first increase then decrease, shouldn't it?
ABC
CC BY-SA 4.0
2023-06-26T08:15:32.670Z
@ABC You are right that according to the paradigm established in simulations, the velocity dispersion should rise at small radii. I suspect that for the non-merging clusters, this rise is at sufficiently small radii that these plots and/or data simply do not resolve it. For example, the window function is Gaussian with width $0.2r_{200}$ (I think, they confusingly say $R_\mathrm{max}$ without defining it anywhere else), which would mean that any feature roughly smaller than that would be erased.
Sten
CC BY-SA 4.0
2023-06-29T05:03:15.933Z
@Sten Thank you for confirming. So when looking at the rotational curves of spiral galaxies at large radius, some of them rise slightly some of them are more or less flat. Does that mean their profile deviates from the NFW profile, according to which the curve should decrease (not as much as without dark matte of course)?
ABC
CC BY-SA 4.0
2023-06-29T12:05:18.347Z
@ABC NFW circular velocities increase up to some multiple of the scale radius and then decrease. So rotation curves that are flat over a wide range of radii don't correspond to NFW. But the rotation curve arises from the (quadrature) sum of the circular velocity profiles set by the baryonic and dark matter components. The flat part of the rotation curve arises where the baryon-driven circular velocity profile is decreasing while the dark matter-driven circular velocity profile is increasing, so that their (quadrature) sum is constant. (Note: these comments now concern galaxies, not clusters.)
Sten
CC BY-SA 4.0
2023-06-30T05:32:28.193Z
@Sten, that makes sense. Interesting what you said. So it is true the NFW profile describes dark matter only. I always wondered why we don't have or use a profile that includes both baryons and dark matter? Even though baryons are not important at large radius it contributes more than dark matter in the core region right (as seen in the rotational curve for different components that I think you were referring to)? Aren't we underestimating mass by integrating NFW profile only?
ABC
CC BY-SA 4.0
2023-06-30T11:24:31.960Z
@ABC NFW profile (approximately) describes systems that consist only of collisionless material. It's not even accurate in general for the dark matter component of galaxies, because the baryons backreact on the dark matter. It's just fairly accurate at cluster scales (where the baryons can be approximated as collisionless) or for very small halos (where the baryons contribute negligibly). The reason for its prominence is that it's fairly universal in such systems. For galaxies there isn't such universality.
Sten
CC BY-SA 4.0
2023-07-06T03:10:15.347Z
@Sten Got it thank you!
ABC
CC BY-SA 4.0
2023-06-22T07:08:34.253Z
2023-06-22T20:25:34.163Z
2023-06-22T20:25:34.163Z
Sten
<p>In the paper <a href="https://doi.org/10.1093/mnras/sty2379" rel="nofollow noreferrer">10.1093/mnras/sty2379</a>, there are plots showing the velocity dispersion as a function of radius.</p>
<p><a href="https://i.sstatic.net/oVaV0l.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/oVaV0l.png" alt="velocity dispersion profiles" /></a></p>
<p>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?</p>
<p>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...</p>

CC BY-SA 4.0
2023-06-22T20:05:07.847Z
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
<redshift><galaxy-cluster><radial-velocity><dispersion>
Why does the velocity dispersion within a galaxy cluster decrease as a function of radius?
ABC
# Why does the velocity dispersion within a galaxy cluster decrease as a function of radius?

(Asked by: username_0 on 2023-06-21)

In the paper [10.1093/mnras/sty2379](https://doi.org/10.1093/mnras/sty2379), there are plots showing the velocity dispersion as a function of radius.

[![velocity dispersion profiles](https://i.sstatic.net/oVaV0l.png)](https://i.sstatic.net/oVaV0l.png)

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...

**Tags:** redshift, galaxy-cluster, radial-velocity, dispersion

> **Comments:**
> **username_1**: 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.  

---

## 1 answers

### Answer by username_1 on 2023-06-22. Score: 2 | Accepted answer

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](https://arxiv.org/abs/1302.0288)).

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.

> **Comments:**
> **username_2**: The red lines don't indicate "high cluster mass", they indicates more massive _galaxies_ within the cluster(s).  
> **username_2**: "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.  
> **username_1**: @username_2 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.)  
> **username_1**: @username_2 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?  
> **username_2**: 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>  
> **username_2**: Regarding my first comment: these are velocity-dispersion profiles for clusters. But they break this down into "velocity dispersion of cluster, computed using all galaxies ('All galaxies')" and "velocity dispersion computing using just galaxies with stellar masses in specific bins". The red line is the computation using only very massive galaxies.  
> **username_1**: @username_2 1. Yes but the efficiency of dynamical friction is proportional to mass. Less than ~1/1000 the mass of the cluster (depending on accretion time) and they won't sink over the age of the universe. 2. Particularly massive subhalos? Anyway this is all kind of beside the point because even the recently accreted material that will sink hasn't yet, so it's contributing mass at large radii. And massive systems that sink do so by trading energy with the background, so other material gets elevated.  
> **username_1**: @username_2 I'll remove the mention of cluster mass if I misunderstood the article! It sounds like the connection to merging still stands. I definitely stand by the "recently accreted material remains near the outskirts of the system" claim since I've looked pretty closely at the average behavior of accreted particles in simulations, and this claim holds for them.  
> **username_2**: Your supposition is that galaxies at large radii are in largely tangential orbits. If this were true, then the dispersion profile would be increasingly biased towards tangential dispersion as you go out in radius. But modeling of observed clusters show increasing _radial_ anisotropy (more radial orbits) as you go out in radius. Galaxies at large radii are in more radial orbits, not more tangential orbits.  
> **username_2**: See, e.g., the discussion in [this talk](https://adlibitum.oats.inaf.it/biviano/orbits.pdf).  
> **username_1**: @username_2 Galaxies aren't in largely tangential orbits. Where did I suppose that? (The mention of the circular velocity is just to give an intuition for how orbital velocities scale.)  
> **username_2**: How else am I to interpret your statement that recently accreted material "has too much energy and angular momentum to sink to the center"? You're saying they cannot have primarily radial orbits. (I think "sink to the center" and our references to dynamical friction are confusing things: infalling galaxies are primarily _falling_, not sinking.)  
> **username_1**: @username_2 A particle in the host contributes mass (on average) to a radius range between its apocenter and pericenter. The apocenter and pericenter are comparatively stable, aside from adiabatic contraction and dynamical friction. Material that fell in earlier has lower apocenter and pericenter. So it contributes mass to lower radii. When I say "outskirts" I don't literally mean $R_{200}$. I mean that the new material contributes mass to radii that are, on average, higher than the radii contributed to by older material.  
> **username_1**: @username_2 That is what leads to the conclusion that a high recent accretion rate means that the system's mass is (relatively) weighted toward the outside of the system.  
> **username_2**: I think the re-edited version is OK, now, since it no longer seems to be suggesting that recently accreted material _can't_ fall into the central regions. (It _does_, but we agree that it probably spends most of its time near apocenter.)  
> **username_2**: Now the only problem is that your answer doesn't really address the question of why the _right-hand_ (non-merging) profiles look the way they do -- in particular, why the RH profile is flatter when only massive galaxies are counted vs becoming steeper when less massive galaxies are included.  
> **username_1**: @username_2 I think that's not what the original poster was asking, since they only mentioned radius and not mass. I interpreted "why should the velocity dispersion becomes smaller as more and more galaxies are included with increasing radius?" as mistakenly assuming that the velocity dispersion includes all radii below $r$ as opposed to just the particular radius bin. Come to think of it, it's worth addressing that explicitly...  
> **username_0**: @username_1 I see indeed my original question implied I was including al galaxies below R instead of at diff radius, which was wrong. Funny enough the mass profile and "orbital" speed at diff radius is what I'm really getting at. You said "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)."  --- to be continued in next comment.  
> **username_0**: @username_1 That would be my interpretation too, but doesn't this contradict the fact galaxies and galaxy clusters are not like our solar system because of the dark matter, and  the NFW mass profile first increases as r^2 and at larger radius ln(r). Even though there's no net rotation, each individual galaxy still orbit more or less circularly around virialized region obeying sqrt(GM/r), ignoring infall. So the orbital speed should first increase then decrease, shouldn't it?  
> **username_1**: @username_0 You are right that according to the paradigm established in simulations, the velocity dispersion should rise at small radii. I suspect that for the non-merging clusters, this rise is at sufficiently small radii that these plots and/or data simply do not resolve it. For example, the window function is Gaussian with width $0.2r_{200}$ (I think, they confusingly say $R_\mathrm{max}$ without defining it anywhere else), which would mean that any feature roughly smaller than that would be erased.  
> **username_0**: @username_1 Thank you for confirming. So when looking at the rotational curves of spiral galaxies at large radius, some of them rise slightly some of them are more or less flat. Does that mean their profile deviates from the NFW profile, according to which the curve should decrease (not as much as without dark matte of course)?  
> **username_1**: @username_0 NFW circular velocities increase up to some multiple of the scale radius and then decrease. So rotation curves that are flat over a wide range of radii don't correspond to NFW. But the rotation curve arises from the (quadrature) sum of the circular velocity profiles set by the baryonic and dark matter components. The flat part of the rotation curve arises where the baryon-driven circular velocity profile is decreasing while the dark matter-driven circular velocity profile is increasing, so that their (quadrature) sum is constant. (Note: these comments now concern galaxies, not clusters.)  
> **username_0**: @username_1, that makes sense. Interesting what you said. So it is true the NFW profile describes dark matter only. I always wondered why we don't have or use a profile that includes both baryons and dark matter? Even though baryons are not important at large radius it contributes more than dark matter in the core region right (as seen in the rotational curve for different components that I think you were referring to)? Aren't we underestimating mass by integrating NFW profile only?  
> **username_1**: @username_0 NFW profile (approximately) describes systems that consist only of collisionless material. It's not even accurate in general for the dark matter component of galaxies, because the baryons backreact on the dark matter. It's just fairly accurate at cluster scales (where the baryons can be approximated as collisionless) or for very small halos (where the baryons contribute negligibly). The reason for its prominence is that it's fairly universal in such systems. For galaxies there isn't such universality.  
> **username_0**: @username_1 Got it thank you!  

---

