# Thread: galaxy -- mass enclosed as a function of distance.

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## galaxy -- mass enclosed as a function of distance.

Does anyone know a general formula for computing the mass of a galaxy interior to a point at a given distance, given given the galaxy's total mass?

Knowing that galaxies have flat rotation curves, one way I can think of is to re-write the circular velocity formula to solve for M: v=sqrt(GM/r) --> M=v2r/G. So for any galaxy, the mass interior to a given r could be a function of v and r.

I imagine this is just a rough estimate, as the flat rotation curves are not perfect. Is there more widely accepted formula?

2. I think you are on the right track. If the dark matter has a density that falls off like 1/r2, then you get your result. That density falloff is associated with "isothermal" models of dark matter, so there's some theoretical expectation for it. I don't know why one would expect the dark matter to be isothermal, but it's easy to show that an ideal gas with constant temperature will balance gravity if its density falls off like 1/r2.

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Originally Posted by tony873004
Does anyone know a general formula for computing the mass of a galaxy interior to a point at a given distance, given given the galaxy's total mass?

Knowing that galaxies have flat rotation curves, one way I can think of is to re-write the circular velocity formula to solve for M: v=sqrt(GM/r) --> M=v2r/G. So for any galaxy, the mass interior to a given r could be a function of v and r.

I imagine this is just a rough estimate, as the flat rotation curves are not perfect. Is there more widely accepted formula?
I don't think there is any simple formula for this, as the galaxy mass calculation is rather complex requiring elliiptical integrals and/or numerical methods for solution. Here is a recent paper that derives the mass distribution for spiral galaxies based on the rotation curves and Newtonian gravity only. It assumes a thin disk shape. Figure 2 in the paper shows the falloff of mass density with radius for this case, and may be of use to derive mass inside a given radius.

http://arxiv.org/PS_cache/arxiv/pdf/...803.0556v1.pdf

TomT

4. Did I have an answer to this that disappeared? I pointed out that the above paper treats the mass/light ratio as if it was completely unconstrained, which is logically equivalent to introducing dark matter even as it claims it doesn't. The only difference is that their dark matter is in the disk, but it makes more sense for dark matter to be spherically symmetric if it is isothermal and does not come from the gas or the stars.

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Originally Posted by Ken G
Did I have an answer to this that disappeared?
Hi Ken,
Evidently the site was shut down for a while due to maintenance. That may have something to do with your disappearing post.

I pointed out that the above paper treats the mass/light ratio as if it was completely unconstrained, which is logically equivalent to introducing dark matter even as it claims it doesn't. The only difference is that their dark matter is in the disk, but it makes more sense for dark matter to be spherically symmetric if it is isothermal and does not come from the gas or the stars.
One other difference besides the "dark matter" being in the disk, is that they consider the "dark matter" to be baryonic, i.e. dust, grains, gases, and plasma that we haven't detected yet. Their reason for disputing the validity of the M/L ratio seems to be that M/L doesn't consider energy level effect on light emission, or, in other words, they seem to be questioning whether the galaxy is isothermal. Their justification for this position is that an edge on view of a typical spiral galaxy shows a dark band through the center of the disk, and interpret this opague area as cooler mass in the outer part of the disk.

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Thanks, Ken and Tom for the replies and the link. In the 4th line of section 1.2 under "Historical Background", I see they have the exact formula I derived by isolating M from the circular velocity formula.

From that paper, formula 1 is V(r)=1-e^(-r/Rc). And knowing that V(r)=sqrt(GM/r), I think I can set these equations equal to each other: 1-e^(-r/Rc) = sqrt(GM/r). Now I can isolate M. M = r*(1-e^(-r/Rc)^2/G. I don't know exactly how they arrive at Rc (a description of the various radii of the cores of different galaxies), but I see they use 0.015 for the Milky Way and values upto 0.1 for other galaxies.

I think this gives me enough to play around with. Thanks again.

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I just noticed the date on that paper: March 4, 2008. You guys are certainly on top of things!

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Originally Posted by tony873004
I just noticed the date on that paper: March 4, 2008. You guys are certainly on top of things!
Hi tony873004,
It was a coincidence that I learned of the paper at the same time you posted your question. There are a large number of papers that calculate galaxy mass without recourse to dark matter spheres. You might call this method a minority view, as it says the usual M/L interpretation is not accurate. The majority doesn't agree with this, but the method does some include some tools (equations) that may be useful to your calculation.
TomT

9. Originally Posted by TomT
Their reason for disputing the validity of the M/L ratio seems to be that M/L doesn't consider energy level effect on light emission, or, in other words, they seem to be questioning whether the galaxy is isothermal. Their justification for this position is that an edge on view of a typical spiral galaxy shows a dark band through the center of the disk, and interpret this opague area as cooler mass in the outer part of the disk.
Yes, but that's nonsense. The mass/light ratio is controlled by the stellar distribution. The baryonic mass of a galaxy is in its stars (baryons make stars copiously), and the light comes from the stars, so looking at gas and dust temperatures seems like a total red herring to me. If you want the mass/light ratio to be a variable in your model (which they do), you have to be able to explain why your galaxy makes different stellar distributions where you want it to. It's just trading one uncertainty (the presence of dark matter even though there's no direct evidence for it) for another (why the stellar distributions are different even though there's no direct evidence they are). At least dark matter is also needed on cluster and cosmological scales.

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Originally Posted by Ken G
Yes, but that's nonsense. The mass/light ratio is controlled by the stellar distribution. The baryonic mass of a galaxy is in its stars (baryons make stars copiously), and the light comes from the stars, so looking at gas and dust temperatures seems like a total red herring to me. If you want the mass/light ratio to be a variable in your model (which they do), you have to be able to explain why your galaxy makes different stellar distributions where you want it to. It's just trading one uncertainty (the presence of dark matter even though there's no direct evidence for it) for another (why the stellar distributions are different even though there's no direct evidence they are). At least dark matter is also needed on cluster and cosmological scales.
Didn't mean for this to get into a discussion of the merits of dark matter, and the discussion of clusters and cosmological scales should probably be part of another thread. But for the present discussion, what is the opague band seen in the edgewise view of a galaxy? The current paper claims it is cooler mass within the outer part of the disk. If so, how do we know what form of mass it is, or whether or not it is a significant part of the galaxy total.

11. Originally Posted by TomT
But for the present discussion, what is the opague band seen in the edgewise view of a galaxy?
It is a bunch of dust of fairly negligible mass. If these authors think they can put the mass of a galaxy in that dust, they will first have to rewrite all that is known about dust opacity.
The current paper claims it is cooler mass within the outer part of the disk.
The current paper is really out to lunch on that point-- obviously it is cooler mass within the outer part of the disk, every astronomer knows this. The issue, which they give no support for whatsoever, is whether or not it could be a significant mass component. That solution would have been tried long before dark matter halos, these authors are only about 30 years behind the times on that score. Indeed, that is the issue that I predict will make this paper unpublishable in its present form (note that it is only a preprint at the moment).
If so, how do we know what form of mass it is, or whether or not it is a significant part of the galaxy total.
Because baryons make stars, and that's the whole basis of a mass to light ratio in the first place. It is like these authors have simply decided to discard a significant fraction of galactic astronomy all because they don't like dark matter halos.

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Originally Posted by Ken G
It is a bunch of dust of fairly negligible mass. If these authors think they can put the mass of a galaxy in that dust, they will first have to rewrite all that is known about dust opacity.The current paper is really out to lunch on that point-- obviously it is cooler mass within the outer part of the disk, every astronomer knows this. The issue, which they give no support for whatsoever, is whether or not it could be a significant mass component. That solution would have been tried long before dark matter halos, these authors are only about 30 years behind the times on that score. Indeed, that is the issue that I predict will make this paper unpublishable in its present form (note that it is only a preprint at the moment).
Because baryons make stars, and that's the whole basis of a mass to light ratio in the first place. It is like these authors have simply decided to discard a significant fraction of galactic astronomy all because they don't like dark matter halos.
Hi Ken,
I reviewed our discussion on the Ken Nicholson paper on this subject, for old times sake.

http://www.bautforum.com/questions-a...ulation-5.html

I thought your post #150 in that thread, the last one on p.5, summed up the M/L vs dark matter issue pretty well. I wouldn't be too hard on the current paper. It is a similar calculaton to the Nicholson one.

TomT

13. Yes, it reminds me of the Nicholson paper too. Indeed they should probably have just cited Nicholson, unless it couldn't get published either. The point is, there is nothing wrong with saying "here's the mass distributon that works if it's all in a disk", especially if that calculation was somehow not possible in the past when the initial studies of diskshaped galaxes were done. The problem is in pretending that this solves the problems on the grounds that the new M/L ratio is plausible. Dark matter is all about M/L, it's the M without the L. So one cannot say "I've solved the dark matter problem-- I just let M/L be unconstrained" as if this is somehow a new idea. Astronomers have to work within the constraints they infer on M/L, so one cannot say anything about the baryonic mass distribution until one has a plausible model for M/L. That was also Nicholson's problem. I can't say how we know M/L for baryons, but I can say that it has to be addressed by anyone who would claim a different M/L for baryons without suggesting how it could be so, because if one thinks baryons make stars (which we do think), then M/L is constrained, and one is led to looking at non-baryonic ways of increasing M/L, but they should be in a halo not a disk.

Nevertheless, I agree it is worth pointing out that if M/L really could be much higher for baryons than we think it could be, there are disklike solutions to the galactic rotation curves. I think if Nicholson had said that, he might have been able to get his work published (and I suggested that to him but I don't know if he tried it).

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Originally Posted by Ken G
Yes, it reminds me of the Nicholson paper too. Indeed they should probably have just cited Nicholson, unless it couldn't get published either. The point is, there is nothing wrong with saying "here's the mass distributon that works if it's all in a disk", especially if that calculation was somehow not possible in the past when the initial studies of diskshaped galaxes were done. The problem is in pretending that this solves the problems on the grounds that the new M/L ratio is plausible. Dark matter is all about M/L, it's the M without the L. So one cannot say "I've solved the dark matter problem-- I just let M/L be unconstrained" as if this is somehow a new idea. Astronomers have to work within the constraints they infer on M/L, so one cannot say anything about the baryonic mass distribution until one has a plausible model for M/L. That was also Nicholson's problem. I can't say how we know M/L for baryons, but I can say that it has to be addressed by anyone who would claim a different M/L for baryons without suggesting how it could be so, because if one thinks baryons make stars (which we do think), then M/L is constrained, and one is led to looking at non-baryonic ways of increasing M/L, but they should be in a halo not a disk.

Nevertheless, I agree it is worth pointing out that if M/L really could be much higher for baryons than we think it could be, there are disklike solutions to the galactic rotation curves. I think if Nicholson had said that, he might have been able to get his work published (and I suggested that to him but I don't know if he tried it).
Hi Ken,
I have a couple more questions/thoughts on M/L ratio.

(1) Is the L in M/L visible light only, or does L mean electromagnetic radiaton in general? If L means the latter, does it cover all wavelengths, or a limited portion of the spectrum? For example, does it include xrays? Does it include the radiation emitted by gases? Does it include the radiation emitted by dust or other non star bodies?

(2) You mention that baryons make stars. Are all the baryons in a galaxy tied up in stars, or are stars constantly being formed? I think the answer is the latter. So what is the fraction of baryons in a galaxy that are not included in the star mass yet? Are these baryons detectable? Do they emit detectable EM and are they therefore included in the M/L ratio?

(3) The other part of the question is that if more than the visible part of the spectrum is used, how is an "M" assigned to each source and how much uncertainty is in this? Then all the "M"s have to be somehow incorporated into an M representative of the galaxy. Is this what has been done to get the M/L ratio?

I'm not requesting anyone to provide answers to these questions. The answers to many of them probably aren't known to anyone. I raise them just to illustrate that there must be uncertainty in the value of M/L, and I wonder if we have a good handle on how big the uncertainty is.

TomT

15. Originally Posted by tony873004
Does anyone know a general formula for computing the mass of a galaxy interior to a point at a given distance, given given the galaxy's total mass?

Knowing that galaxies have flat rotation curves, one way I can think of is to re-write the circular velocity formula to solve for M: v=sqrt(GM/r) --> M=v2r/G. So for any galaxy, the mass interior to a given r could be a function of v and r.

I imagine this is just a rough estimate, as the flat rotation curves are not perfect. Is there more widely accepted formula?
You're correct. Vogt et al (2004) provided this version of the formula:

M(r) = 2.3265 x 10^5 V^2(r) r

where (r) represents the Mass inside or rotational velocity at the radius "r" from the center of the galaxy.

Using the MOND formula, the mass is:

M=V^4/Ga0

where "a0" is the mond acceleration constant.

Using this formula masses are calculated as:

M = V^4/1.59 x 10-2

The two equations will give similar masses.

Using NGC 4548 which has a cepheid distance of 16.2 Mpc (HKP final report) as an example:

The 2MASS K-band radius for this galaxy is 3.52' so the radius is 16.6 kpc. The rotational velocity for the galaxy is 215 (+/-20) km s-1 (Cornell University SFI++ database).

So using standard dynamics one gets a mass of 1.79x 10^11 solar masses.

Using MOND you only need the rotational velocity and one gets 1.34 x 10^11 solar masses.

As an interesting sidelight, using the standard Newtonian formulation, the mass calculated is dependent upon the radius. Radius can be defined in a number of ways, but often the radius is simply taken as the radius at which the surface brightness in the B-band drops to 25 mag arc sec^-2. At a given rotational velocity, lower mean surface brightness galaxies have larger diameters than higher mean surface galaxies. Thus when calculating masses, a lower SB galaxy actually has a larger calculated mass. Whether or not there is any physical reason why a lower mean SB galaxy should be more massive is a curiosity that would be worth investigating.

With the MOND formula, the mass is independent of the radius of the galaxy, and only depends upon the rotational velocity. Since the calculated radius depends upon the distance, the standard mass calculations have an additional uncertainty introduced by distance errors. On the other hand, the MOND masses have an additional uncertainty introduced by uncertainty in the MOND acceleration constant (a0) because the constant's value is empirically derived.

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Originally Posted by dgruss23
You're correct. Vogt et al (2004) provided this version of the formula:

M(r) = 2.3265 x 10^5 V^2(r) r

where (r) represents the Mass inside or rotational velocity at the radius "r" from the center of the galaxy.

Using the MOND formula, the mass is:

M=V^4/Ga0

where "a0" is the mond acceleration constant.

Using this formula masses are calculated as:

M = V^4/1.59 x 10-2

The two equations will give similar masses.

Using NGC 4548 which has a cepheid distance of 16.2 Mpc (HKP final report) as an example:

The 2MASS K-band radius for this galaxy is 3.52' so the radius is 16.6 kpc. The rotational velocity for the galaxy is 215 (+/-20) km s-1 (Cornell University SFI++ database).

So using standard dynamics one gets a mass of 1.79x 10^11 solar masses.

Using MOND you only need the rotational velocity and one gets 1.34 x 10^11 solar masses.

As an interesting sidelight, using the standard Newtonian formulation, the mass calculated is dependent upon the radius. Radius can be defined in a number of ways, but often the radius is simply taken as the radius at which the surface brightness in the B-band drops to 25 mag arc sec^-2. At a given rotational velocity, lower mean surface brightness galaxies have larger diameters than higher mean surface galaxies. Thus when calculating masses, a lower SB galaxy actually has a larger calculated mass. Whether or not there is any physical reason why a lower mean SB galaxy should be more massive is a curiosity that would be worth investigating.

With the MOND formula, the mass is independent of the radius of the galaxy, and only depends upon the rotational velocity. Since the calculated radius depends upon the distance, the standard mass calculations have an additional uncertainty introduced by distance errors. On the other hand, the MOND masses have an additional uncertainty introduced by uncertainty in the MOND acceleration constant (a0) because the constant's value is empirically derived.
Hi Dave,

Very interesting calculations. Can you provide the results from these 2 methods for the galaxy NGC3198. The reason I ask is that, if so, we can compare them to those of Nicholson and van Albada. Nicholson uses a method with all the mass in the galactic disk, and van Albada use a method with the disk mass based on M/L ratio, and additional mass in spherical shells around the galaxy.

The results that would be interesting to compare to are:

NGC 3198 mass: Nicholson 1*10^11 msun. van Albada 1.5*10^11 msun.

TomT

17. Originally Posted by TomT
Hi Dave,

Very interesting calculations. Can you provide the results from these 2 methods for the galaxy NGC3198. The reason I ask is that, if so, we can compare them to those of Nicholson and van Albada. Nicholson uses a method with all the mass in the galactic disk, and van Albada use a method with the disk mass based on M/L ratio, and additional mass in spherical shells around the galaxy.

The results that would be interesting to compare to are:

NGC 3198 mass: Nicholson 1*10^11 msun. van Albada 1.5*10^11 msun.

TomT
The 2MASS K-band radius for NGC 3198 is 3.97' (the B-band D25 diameter is 3.08'). The cepheid distance is 13.8 Mpc (HKP) and the rotational velocity from the Cornell database is 155 km s-1.

The Newtonian mass is 8.89 x 10^10 solar masses.

The MOND mass is 3.63 x 10^10 solar masses.

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Originally Posted by dgruss23
The 2MASS K-band radius for NGC 3198 is 3.97' (the B-band D25 diameter is 3.08'). The cepheid distance is 13.8 Mpc (HKP) and the rotational velocity from the Cornell database is 155 km s-1.

The Newtonian mass is 8.89 x 10^10 solar masses.

The MOND mass is 3.63 x 10^10 solar masses.
Dave, Thanks for the data and references.

This gives 4 results for NGC 3198 mass:

MOND .36*10^11 solar masses
Newtonian .89*10^11 solar masses (Vogt et al) and 1.0*10^11 (Nicholson)
M/L based w/dark matter spheres 1.50*10^11 solar masses (van Albada)

I see there are 2 follow on papers where Vogt et al analyze M/L effects. Maybe these will clarify some of the questions on what is included in M/L.

TomT

19. Originally Posted by TomT
(1) Is the L in M/L visible light only, or does L mean electromagnetic radiaton in general?
All forms of light. I believe what dominates is ultraviolet and ultraviolet reprocessed into infrared in the immediate environment of the star, but visible also contributes a lot.

For example, does it include xrays?
It doesn't matter, the contribution is negligible inside a galaxy. In contrast, galaxy clusters are permeated with intergalactic "coronae" which emit most of their light in the X-rays, so getting M/L ratios on the galaxy-cluster scale is a very different process than within a galaxy.

Does it include the radiation emitted by gases?
Yes, but that's also neglible unless it is directly traceable to a nearby star, in which case the light depends on the star not the gas.

Does it include the radiation emitted by dust or other non star bodies?
Same as for the gas.
(2)So what is the fraction of baryons in a galaxy that are not included in the star mass yet?
It is small, given current estimates. I can't speak to the reliability of those estimates because I have only passing knowledge of them, but I do say that anyone who wishes to contradict those observationally supported estimates must be able to directly address the question. The point is, "dark" baryons are everyone's most obvious solution to dark matter-- yet that possibility has long since been discarded, even in galaxies.
Are these baryons detectable? Do they emit detectable EM and are they therefore included in the M/L ratio?
There was some consideration of cold molecular hydrogen as a dark matter candidate, but my impression was, that was ruled out. The problem is, you can make it not emit light by cooling it down, but you have to explain why it is that cold, and you also have to explain the amount of absorption it produces against background sources. Maybe it's possible, but it has to be addressed.

(3) The other part of the question is that if more than the visible part of the spectrum is used, how is an "M" assigned to each source and how much uncertainty is in this?
The key here is the "Salpeter initial mass function", which is our best attempt to identify the mass distribution of stars, which then connects to the light they make. I just don't know enough about how that gets determined-- masses of stars pretty much inescapably revolve around binary systems, no pun intended.

Then all the "M"s have to be somehow incorporated into an M representative of the galaxy. Is this what has been done to get the M/L ratio?
Yes, the Salpeter approach has been applied to a lot of environments, and some deviations are found, but in general there seems to be a consistency. The problem is that the L comes from a different population of stars than the M, so the process is fraught with peril, and that's why I don't know enough about it to comment.
I raise them just to illustrate that there must be uncertainty in the value of M/L, and I wonder if we have a good handle on how big the uncertainty is.
I couldn't say, I can only say that anyone who wishes to argue an M/L that is outside what is generally viewed as reasonable had better be ready to provide those answers.

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Originally Posted by Ken G
All forms of light. I believe what dominates is ultraviolet and ultraviolet reprocessed into infrared in the immediate environment of the star, but visible also contributes a lot.

It doesn't matter, the contribution is negligible inside a galaxy. In contrast, galaxy clusters are permeated with intergalactic "coronae" which emit most of their light in the X-rays, so getting M/L ratios on the galaxy-cluster scale is a very different process than within a galaxy.

Yes, but that's also neglible unless it is directly traceable to a nearby star, in which case the light depends on the star not the gas.

Same as for the gas.
It is small, given current estimates. I can't speak to the reliability of those estimates because I have only passing knowledge of them, but I do say that anyone who wishes to contradict those observationally supported estimates must be able to directly address the question. The point is, "dark" baryons are everyone's most obvious solution to dark matter-- yet that possibility has long since been discarded, even in galaxies.
There was some consideration of cold molecular hydrogen as a dark matter candidate, but my impression was, that was ruled out. The problem is, you can make it not emit light by cooling it down, but you have to explain why it is that cold, and you also have to explain the amount of absorption it produces against background sources. Maybe it's possible, but it has to be addressed.

The key here is the "Salpeter initial mass function", which is our best attempt to identify the mass distribution of stars, which then connects to the light they make. I just don't know enough about how that gets determined-- masses of stars pretty much inescapably revolve around binary systems, no pun intended.

Yes, the Salpeter approach has been applied to a lot of environments, and some deviations are found, but in general there seems to be a consistency. The problem is that the L comes from a different population of stars than the M, so the process is fraught with peril, and that's why I don't know enough about it to comment.
I couldn't say, I can only say that anyone who wishes to argue an M/L that is outside what is generally viewed as reasonable had better be ready to provide those answers.
Hi Ken G,
Thanks for taking the time to respond to my barrage. As I said, I really wasn't expecting you to respond to it all. I like to investigate these topics as more of a side interest and hobby, so will be fun to look into all this.
TomT

21. Feel free to report back on your findings, I'm sure this is an area most of us don't know much about.

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