# Velocity Dispersion

by shounakbhatta
Tags: dispersion, velocity
 P: 272 Hello, I have several question in this thread which is related to velocity dispersion as well as some other areas of galaxy formation and super massive black holes (SMBH) (1) The sigma is used to calculate the speed of the star at the edge of the galaxy and there is a direct relationship with the SMBH. The heavier the black hole, the faster the stars move around. The M Sigma relation and the Sigma -- are they the same? (2) The velocity dispersion measures the mass of a galaxy, by determining the kinetic energy and finally the potential energy of a stable system through 'virial theorem'. Is that correct? (3) The M Sigma relation when plotted on a graph shows that the black hole mass M is directly proportional to the stellar velocity dispersion right? (4) The numerical figure of M σ relation states: M/10^8M ~~....... What does '~~' means? Thanks, -- Shounak
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P: 5,196
 Quote by shounakbhatta Hello, I have several question in this thread which is related to velocity dispersion as well as some other areas of galaxy formation and super massive black holes (SMBH) (1) The sigma is used to calculate the speed of the star at the edge of the galaxy and there is a direct relationship with the SMBH. The heavier the black hole, the faster the stars move around.
Sort of, yeah. What velocity dispersion means is literally the dispersion, or spread, in the distribution of the stellar velocities. In other words, if you plot histogram of all the stellar velocities for all the stars in this galaxy, that distribution has some width σ, and this width is the velocity dispersion. So it really tells you how spread out the stellar velocities are around the mean velocity.

 Quote by shounakbhatta The M Sigma relation and the Sigma -- are they the same?
I don't understand what you are asking.

 Quote by shounakbhatta (2) The velocity dispersion measures the mass of a galaxy, by determining the kinetic energy and finally the potential energy of a stable system through 'virial theorem'. Is that correct?
Yeah, that's pretty much correct. The virial theorem expresses the energy conditions required for a system to be dynamically stable. So, a more massive galaxy can have a larger velocity dispersion, and yet still remain a gravitationally bound system, because it has a deeper gravitational potential well, so even though some stars are moving faster, they still don't escape.

So, velocity dispersion is kind of an observational proxy for galaxy mass, and what the M-sigma relation is really saying is that SMBH mass correlates with host galaxy mass, suggesting that the evolution and growth of the galaxy with time is somehow related to the evolution of its central black hole.

 Quote by shounakbhatta (3) The M Sigma relation when plotted on a graph shows that the black hole mass M is directly proportional to the stellar velocity dispersion right?
Not sure. This isn't my area of expertise, and to be honest, I don't really remember. Are you sure it isn't some sort of power law, rather than being linear? Maybe you were looking at a log-log plot?

 Quote by shounakbhatta (4) The numerical figure of M σ relation states: M/10^8M ~~....... What does '~~' means? Thanks, -- Shounak
It's not clear to me what symbol you are talking about. Is it this one: ≈ ? If so, this symbol ≈ means "approximately equal to."
 P: 272 Thank you very much. Yes the '~' is what you told, it is 'approximately equals to.'
 P: 272 Velocity Dispersion Well, I have few more questions: Hubble classification divides galaxies into the following forms: (1) Elliptical (2) Spiral (3) Lenticular (4) Other irregular galaxies Our Milky Way is considered to be a spiral galaxy. Now,is there any relation between the type of galaxy with the formation of SMBH? I mean to say that does M-Sigma relation determines the galaxy size and SMBH between any type of galaxies mentioned above? Secondly, what is a Local group? Going through Wikipedia I found that galaxies have been classified into spiral, elliptical ,irregular galaxies, dwarf elliptical, dwarf spheroidal. Now M110 and M32, denotes they are 'satellite of Andromeda'. Andromeda is a galaxy, I understand, but what does 'satellite of Andromeda' means?
 Admin P: 21,866 Satellite galaxies are small galaxies that are gravitationally bound to a larger galaxy, e.g., our own Milky Way Galaxy or Andromeda Galaxy. The Large and Small Magellanic Clouds are satellite galaxies of the Milky Way. Satellite galaxies tend to be dwarf or irregular galaxies. For example - http://en.wikipedia.org/wiki/Dwarf_spheroidal_galaxy List of MW satellite galaxies - http://www.astro.uu.se/~ns/mwsat.html A presentation on MW dward satellites - http://www.astro.uni-bonn.de/~mwhalo...t37-kroupa.pdf
 P: 272 Ok, thanks for the reply. That means the small galaxies are controlled by the main host galaxy? So Andromeda's satellite are those which are bound to Andromeda and Andromeda in turn is controlled by Milky Way? Is that so? -- Shounak
P: 21,866
 Quote by shounakbhatta Ok, thanks for the reply. That means the small galaxies are controlled by the main host galaxy? So Andromeda's satellite are those which are bound to Andromeda and Andromeda in turn is controlled by Milky Way? Is that so? -- Shounak
I wouldn't say that the Andromeda galaxy is 'controlled' by the MW. The gravitational attraction is mutual.

Here are some notes on the MW, Andromeda and other galaxies of the Local Group.

ttp://csep10.phys.utk.edu/astr162/lect/gclusters/localg.html

Local Group Galaxies (derived from Mateo 1999)
http://ned.ipac.caltech.edu/level5/Mateo/table1.html

The Local Group of Galaxies
http://messier.seds.org/more/local.html

The Andromeda Galaxy is the largest member of the Local Group, bigger even than our Milky Way.
http://spiff.rit.edu/classes/phys230...cal_group.html

The observed properties of dwarf galaxies in and around the Local Group
https://www.astrosci.ca/users/alan/N..._Database.html

http://www.ast.cam.ac.uk/~mike/local.html

Stardate (McDonald Observatory) has a statement that Milky Way may be a little smaller but more massive than M31.
http://stardate.org/astro-guide/btss...es/local_group

Estimates of the masses of MW and Andromeda are found in N.W. Evans (Oxford), M.I. Wilkinson (Cambridge), The Mass of the Andromeda Galaxy, http://arxiv.org/abs/astro-ph/0004187, April 2000
"We find the halo of Andromeda has a mass of about 12 x 10^11 solar masses, together with a scalelength of roughly 90 kpc and a predominantly isotropic velocity distribution. For comparison, our earlier estimate for the Milky Way halo is 19 x 10^11 solar masses."

However on the BBC Universe site, there is:
 . . . . Although the largest, the Andromeda Galaxy may not be the most massive, as recent findings suggest that the Milky Way contains more dark matter and could be the most massive in the grouping. The 2006 observations by the Spitzer Space Telescope revealed that M31 contains one trillion (1012) stars: at least twice the number of stars in the Milky Way galaxy, which is estimated to be 200–400 billion. The Andromeda Galaxy is estimated to be 7.1×1011solar masses. In comparison a 2009 study estimated that the Milky Way and M31 are about equal in mass, while a 2006 study put the mass of the Milky Way at ~80% of the mass of the Andromeda Galaxy. The two galaxies are expected to collide in 3.75 billion years, eventually merging to form a giant elliptical galaxy. . . . .
Such numbers are subject to change with better/more observations and newer (ostensibly better or improved) models.

Milky Way (notes and some discussion of rotation curve)
http://abyss.uoregon.edu/~js/ast122/lectures/lec25.html

The Dynamics of the Galaxies in the Local Group
Roeland van der Marel (STScI), Main collaborators: Jay Anderson, Gurtina Besla, Nitya Kallivayalil, Tony Sohn
http://www.stsci.edu/~marel/talks/LGcolo1.pdf
P: 21
 Quote by shounakbhatta Hello, I have several question in this thread which is related to velocity dispersion as well as some other areas of galaxy formation and super massive black holes (SMBH) (3) The M Sigma relation when plotted on a graph shows that the black hole mass M is directly proportional to the stellar velocity dispersion right? -- Shounak
It is a close to linear relation on a log-log plot.
see plot on wikipedia http://en.wikipedia.org/wiki/M-sigma_relation
by the way, the validity of M-sigma relation being applicable to all galaxies is still under debate, there has been recent findings that this relation isn't applicable to BCGs (Brightest cluster galaxies) that reside within the center of galaxy clusters.

 Quote by shounakbhatta Well, I have few more questions: Hubble classification divides galaxies into the following forms: (1) Elliptical (2) Spiral (3) Lenticular (4) Other irregular galaxies Our Milky Way is considered to be a spiral galaxy. Now,is there any relation between the type of galaxy with the formation of SMBH? I mean to say that does M-Sigma relation determines the galaxy size and SMBH between any type of galaxies mentioned above?
Some papers in this area consider the relation between the buldge stellar dispersion and the SMBH mass, it also works well in most cases. The concept backing this is generally the same -- the virial theorem. But in the case of Spirals, the whole system isn't as "messed up" as in the case of Ellipticals.
The M-sigma relation as i know it, usually only works well for Ellipticals if one considers the stellar dispersion of the whole galaxy.
 P: 272 Hello Astrofan, Thanks once more for your insightful reply. Actually, I came a little bit later to the Faber-Jackson and Tully-Fisher relations. As you have pointed out above rightly, the m sigma relation works out properly with elliptical galaxies, but NOT in case of spiral galaxies. But isn't that the Tully Fisher relation works out for Spiral galaxies? Faber Jackson for Elliptical galaxies? One more thing L is directly proportional to sigma^4. Is this the eqn.for Tully Fisher? What is the eqn.for Faber Jackson relation? Somehow, Faber-Jackson & Tully-Fisher are related. Is that both depicts a proportional relation? Your answers would help me out. Thanks.
P: 21
 Quote by shounakbhatta Hello Astrofan, But isn't that the Tully Fisher relation works out for Spiral galaxies? Faber Jackson for Elliptical galaxies? One more thing L is directly proportional to sigma^4. Is this the eqn.for Tully Fisher? What is the eqn.for Faber Jackson relation? Somehow, Faber-Jackson & Tully-Fisher are related. Is that both depicts a proportional relation? Your answers would help me out. Thanks.
Hi there,

Here you're mixing stuff, neither Faber-Jackson nor Tully-Fisher has any direct link to SMBH masses. They are as you know, just relations between stellar motion and the luminosity of the whole galaxy.
Tully–Fisher relation discusses the "luminosity-stellar rotation velocity"
http://en.wikipedia.org/wiki/Tully%E...isher_relation
http://www.daviddarling.info/encyclo..._relation.html
Faber-Jackson relates the "luminosity-stellar dispersion"
http://en.wikipedia.org/wiki/Faber%E...ckson_relation

I have to say that there is some connection between the Tully–Fisher relation and the BH mass, but it is not so trivial.

Please refer to this paper if you are interested in some more relations that are being used to estimate BH masses http://adsabs.harvard.edu/abs/2012MNRAS.424..224H
The introduction part should quite suffice to clear up some confusion.

Although we have some idea as to how these relations should be valid in theory, but still note that these relations are still more or less empirical and might as well break down for various extermes.
 P: 272 Yes, you are very right. I was thinking that as the M-sigma relation helps to determine the speed of the star, then as heavier the black hole, the more faster the stars move, hence the M-sigma relation has direct relation with the BH. As you have said that there is some relation with the Tully-Fisher model with the BH, what is that? I have one more thing, that I need to ask you. The equation for the Faber-Jackson relation: L is directly proportional to sigma^4. What is the exact equation for Tully-Fisher? Some say L is directly proportional to V^4. Is both Tully-Fisher and Faber-Jackson plotted, gives linear?
 P: 272 Hello, what is the exact equation form for Faber-Jackson and Tully-Fisher relation? Wikipedia states: L is directly proportional sigma^4, but that it is in Faber-Jackson. Can anybody please explain it?

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