Rotation curve and dark matter

[Ranku]

Do all galaxies with dark matter halo have flat rotation curve?

 

[Vanadium 50]

What have you looked into and what part of it didn't you get?

Further, if you tell that there is extra matter by the flat(ter) rotation curve, aren't you asking a tautological question? And if it's not tautological, maybe you should explain what you mean more.

 

[DaveC426913]

We infer dark matter from the gravitational effects we observe.

Edit: Ah! That's the word that was on the tip of my brain: tautological.

 

[Ranku]

What have you looked into and what part of it didn't you get?

Further, if you tell that there is extra matter by the flat(ter) rotation curve, aren't you asking a tautological question? And if it's not tautological, maybe you should explain what you mean more.

Since flat rotation curve is usually mentioned in relation to spiral galaxies, I was wondering if it’s true of all types of galaxies with a dark matter halo.

 

[malawi_glenn]

I was wonder if it’s true of all types of galaxies with a dark matter halo.

I think you are asking if also non-spiral galaxies "need" dark matter as a halo to explain their properties?

 

[Ranku]

I think you are asking if also non-spiral galaxies "need" dark matter as a halo to explain their properties?

That would be also another way of putting it.

 

[malawi_glenn]

That would be also another way of putting it.

Because "having a dark matter halo" is not something we know that any galaxy have.
It is something we have thrown into explain the galaxies seemingly anomalous behavior from Newtonian and Einsteinian gravity.

 

[Ranku]

Because "having a dark matter halo" is not something we know that any galaxy have.
It is something we have thrown into explain the galaxies seemingly anomalous behavior from Newtonian and Einsteinian gravity.

Then that goes back to my original question as to is dark matter halo unique to spiral galaxies, and if so why. Since dark matter is present in much greater proportion to matter, we’d expect it to accumulate around all types of galaxies as well.

 

[malawi_glenn]

Have you done any searches regarding this question yourself? What did you find?

here are some articles you can read
https://astrobites.org/2013/04/04/do-elliptical-galaxies-have-dark-matter-halos/
https://www.researchgate.net/publication/361212481_Thesis_2020

 

[Maarten Havinga]

While it appears that ranku is all OK with the dark matter paradigm, I warn against this way of naming asymptotically flat rotation curves tautological with dark matter haloes.

Because it simply ignores and discredits all the work done on MOND, especially when the question is about ALL dark matter haloes (it is not easily explicable in Lambda-CDM why all galaxies with rotation curves have the same asymptotic flatness).

 

[malawi_glenn]

I warn against this way of naming asymptotically flat rotation curves tautological with dark matter haloes.

Yeah there is no equivalence between dark mater haloes and flat galactic rotation curves.

Because it simply ignores and discredits all the work done on MOND,

I would not call it "discredits". MOND researchers discredits Particle DM research so.
And IIRC, the "amount" of MOND research is just a fraction of all Particle DM research.
But that is a for different thread I pressume.

 

[Maarten Havinga]

Ignoring someone's work right at its area of relevance is discrediting that work IMO. MOND never ignored DM research, MOND researchers just had a different opinion.

 

[davicle]

Because "having a dark matter halo" is not something we know that any galaxy have.
It is something we have thrown into explain the galaxies seemingly anomalous behavior from Newtonian and Einsteinian gravity.

Because "having a dark matter halo" is not something we know that any galaxy have.
It is something we have thrown into explain the galaxies seemingly anomalous behavior from Newtonian and Einsteinian gravity.

I am probably asking a question which has been asked before, but I have only just joined the forum.
I have looked at sketch curves showing the actual and "expected" galaxy rotation curves. My question is - on what basis do astronomers expect the curve that they say they expect. And what theoretical justification is there for this basis? What equation are they using to calculate the orbital velocity as a function of distance?

 

[Vanadium 50]

My question is - on what basis do astronomers expect the curve that they say they expect.

They see the stars, which tells how much matter is there.

 

[Ibix]

My question is - on what basis do astronomers expect the curve that they say they expect.

You can measure luminosity of different parts of the galaxy and that gets you an estimate of the mass distribution. Then you plug that into Newtonian gravity and equate the gravitational force to the centripetal force to get a predicted rotation rate as a function of distance from galactic center.

 

[malawi_glenn]

My question is - on what basis do astronomers expect the curve that they say they expect. And what theoretical justification is there for this basis? What equation are they using to calculate the orbital velocity as a function of distance?

It's just Newtons gravitational law, assuming axis-symmetrical mass distribution.
https://www.astro.umd.edu/~richard/ASTRO620/QM_chap5.pdf

 

[davicle]

Thanks for this - it was most helpful. But on reading this it raised more questions than answers.
Newton's equation strictly applies to two point masses M and m separated by a distance d, which in theory is accurately known. But when one of the masses M is diffuse e.g. a galaxy, you run into a problem of just what is d. The above text attempts to get round this problem by assuming (I can only put it like this as no justification or reference is given in the text) that only the mass of the galaxy inside the orbiting star's orbit gravitationally acts on the star. Surely the whole mass of the galaxy acts on the star, not just the mass inside the star's orbit.

A further assumption is made that the galaxy's gravitationally active mass is always placed at the centre of the galaxy. This has to be done in order to deduce d. The only case I know where this can be shown to be true is for a point mass outside a sphere, where the gravitational centre of the sphere acts at the centre of mass of the sphere regardless of distance. But it is not true for any other shape I have looked at - ring, rod, planar rectangle or circular disc.

Can anyone shed any further light on theoretical justificationfor the above "assumptions"?

 

[Ibix]

Can anyone shed any further light on theoretical justificationfor the above "assumptions"?

You can easily generalise Newton's equations for diffuse masses. It's not usually done in introductory physics courses because it formally requires vector calculus, but the concept isn't difficult. Essentially you divide the diffuse mass up into a lot of small masses and calculate the total force from treating each small mass as a point source. That's only an approximation, but the approximation gets better if you divide the mass up into more smaller masses. And you can consider the limiting case as you divide the mass up into infinitely many infinitely small sources (that's where calculus comes in) to make the error in the approximation go to zero.

Once you've done that, it does indeed turn out that the only mass that matters is the mass inside the star's orbit because the gravitational effect of matter outside its orbit turns out to add up to a net zero. And the matter inside (assuming cylindrical symmetry, which is reasonable for a galaxy) turns out to have the same gravitational effect as it would if it were all concentrated at a point at the center. So the assumptions you're talking about aren't assumptions - they are actually rigorously derived results.

 

[ohwilleke]

Do all galaxies with dark matter halo have flat rotation curve?

In a dark matter particle paradigm there are dark matter halos that are inferred in galaxies (but not all galaxies), galaxy clusters, and also in intergalactic media.

There are a few rare dwarf galaxies that do not seem to have dark matter halos (which the dark matter paradigm usually attributes to tidal stripping from a nearby larger galaxy), and some elliptical galaxies with inferred dark matter halos indistinguishable from not having one statistically (typically the most spherical elliptical galaxies), in part because the uncertainties are large.

A "flat rotation curve:" is a well-defined concept in a spiral galaxy.

But, the concept of a "rotation curve" is not necessarily a well defined concept in other kinds of galaxies such as irregularly shaped "dark matter dominated" low surface brightness dwarf galaxies and elliptical galaxies and galaxy clusters where dark matter is inferred based upon the mass to light ratio (which compares the two normalized with a typical star having a mass and light output of 1.0), and/or from gravitational lensing of photons.

Also, of course, a "flat rotation curve" in a spiral galaxy is not actually perfectly flat, this term is just a rough approximation. The inner core of a spiral galaxy typically has lower rotational speeds that build up rapidly to an asymptotic flat level in most of the outer part of the spiral galaxy and it isn't quite a straight line even in the outer galaxy. See, e.g., this diagram:

Further, typically in the very outer fringes of a spiral galaxy, there are some stars rotating beyond the distance from the center shown on a flat rotation curve chart, but it isn't resolved well enough with the small number of stars there to accurately establish a rotation speed in that galactic fringe.

Finally, keep in mind that a lot of the data in the observations used to build rotation curve diagrams isn't terribly precise (as is typical of astronomy observations of galaxies generally). There are quite a few aspects of an observation like determining the extent to which you are looking down from directly above it or edge on or somewhere in between that inject lots of uncertainty into the measurements.

 

[DaveC426913]

Once you've done that, it does indeed turn out that the only mass that matters is the mass inside the star's orbit because the gravitational effect of matter outside its orbit turns out to add up to a net zero. And the matter inside (assuming cylindrical symmetry, which is reasonable for a galaxy) turns out to have the same gravitational effect as it would if it were all concentrated at a point at the center.

Does the matter outside the star's orbit have to be homogenous in density for it to sum to net zero? Or at least cylindrically symmetrical? Or does that matter?

 

[Ibix]

Does the matter outside the star's orbit have to be homogenous in density for it to sum to net zero?

Good point - it needs to be cylindrically symmetric (or close enough for jazz, in practice).

 

[ohwilleke]

I am probably asking a question which has been asked before, but I have only just joined the forum.
I have looked at sketch curves showing the actual and "expected" galaxy rotation curves. My question is - on what basis do astronomers expect the curve that they say they expect. And what theoretical justification is there for this basis? What equation are they using to calculate the orbital velocity as a function of distance?

The expected curve is based upon visible matter only using Newtonian gravity (i.e. F=GM/r²) and a reasonable model of the distribution of the visible matter in the galaxy based upon observations made (usually described in a formula that is discussed in the methods section of the paper doing the calculation). There is some calculus that goes into it, but that is the basic concept.

There is a certain amount of art in addition to science that goes into estimating the distribution of the visible matter in a galaxy in a reasonable way.

 

[davicle]

You can easily generalise Newton's equations for diffuse masses. It's not usually done in introductory physics courses because it formally requires vector calculus, but the concept isn't difficult. Essentially you divide the diffuse mass up into a lot of small masses and calculate the total force from treating each small mass as a point source. That's only an approximation, but the approximation gets better if you divide the mass up into more smaller masses. And you can consider the limiting case as you divide the mass up into infinitely many infinitely small sources (that's where calculus comes in) to make the error in the approximation go to zero.

Once you've done that, it does indeed turn out that the only mass that matters is the mass inside the star's orbit because the gravitational effect of matter outside its orbit turns out to add up to a net zero. And the matter inside (assuming cylindrical symmetry, which is reasonable for a galaxy) turns out to have the same gravitational effect as it would if it were all concentrated at a point at the center. So the assumptions you're talking about aren't assumptions - they are actually rigorously derived results.

Thanks very much for this. The calculus involved in dealing with a graded density circular disc must be complicated but I would like to see it and hopefully understand it. Could you give me a link to a paper where the derivation is explained step by step?

 

[Ibix]

Thanks very much for this. The calculus involved in dealing with a graded density circular disc must be complicated but I would like to see it and hopefully understand it. Could you give me a link to a paper where the derivation is explained step by step?

Google 'shell theorem'. It's a one liner if you know Gauss' Theorem, and with high school calculus I don't think it's particularly difficult, just messy.

 

[snorkack]

Newton's equation strictly applies to two point masses M and m separated by a distance d, which in theory is accurately known. But when one of the masses M is diffuse e.g. a galaxy, you run into a problem of just what is d. The above text attempts to get round this problem by assuming (I can only put it like this as no justification or reference is given in the text) that only the mass of the galaxy inside the orbiting star's orbit gravitationally acts on the star. Surely the whole mass of the galaxy acts on the star, not just the mass inside the star's orbit.

A further assumption is made that the galaxy's gravitationally active mass is always placed at the centre of the galaxy. This has to be done in order to deduce d. The only case I know where this can be shown to be true is for a point mass outside a sphere, where the gravitational centre of the sphere acts at the centre of mass of the sphere regardless of distance. But it is not true for any other shape I have looked at - ring, rod, planar rectangle or circular disc.

Can anyone shed any further light on theoretical justificationfor the above "assumptions"?

You can easily generalise Newton's equations for diffuse masses. It's not usually done in introductory physics courses because it formally requires vector calculus, but the concept isn't difficult. Essentially you divide the diffuse mass up into a lot of small masses and calculate the total force from treating each small mass as a point source. That's only an approximation, but the approximation gets better if you divide the mass up into more smaller masses. And you can consider the limiting case as you divide the mass up into infinitely many infinitely small sources (that's where calculus comes in) to make the error in the approximation go to zero.

Once you've done that, it does indeed turn out that the only mass that matters is the mass inside the star's orbit because the gravitational effect of matter outside its orbit turns out to add up to a net zero. And the matter inside (assuming cylindrical symmetry, which is reasonable for a galaxy) turns out to have the same gravitational effect as it would if it were all concentrated at a point at the center. So the assumptions you're talking about aren't assumptions - they are actually rigorously derived results.

I am inclined to support davicle here, not Ibix.
Gauβ Law and other derivations only show that external mass is irrelevant in case of spherical symmetry - NOT in case of axial symmetry.
Consider a mass inside a spherically symmetric shell. Shell Theorem requires the gravity inside to be zero.
If you look at the masses then consider alternately integrating over pairs of opposing spatial angles.
Since the shell has uniform surface density, a small angle as viewed by inside observer has the mass proportional to the square of distance from observer to shell. But the attraction of gravity is inversely proportional to square of distance. Therefore, the attractive force to a distant but large mass of a section of shell is exactly equal and canceled by the attractive force of a smaller but nearby mass of the section of shell on the opposite side, for every pair of opposing directions and therefore in total.
This derivation will NOT be applicable in case of a ring! The mass of a section of a ring in a given angle is proportional to the first power of distance, not square. The attraction of gravity is still inversely proportional to square of distance - which means that the nearby section of ring will exercise stronger attraction than the distant opposite section. Stated otherwise, if the ring is a band of a shell, with constant width, the observer inside the ring will see the near section as the wider band of sky, and only the centre of the band, opposite the narrow band of the far section, cancels out.
Since a disc outside an observer is describable as a series of concentric rings, its gravity also should not cancel out.

 

[davicle]

I can see why a point mass inside a sphere has no net gravitational attraction from any mass of the sphere "outside" that point, and is only acted on by that mass of the sphere inside. But my own doodlings have shown that this certainly does not happen with a disc. What does happen is that the mass of the galaxy outside the star's orbit acts to pull the star away from the galaxy's centre. This is more than counteracted by the mass inside the star's orbit which acts to pull the star towards the centre, so the star keeps in orbit. But the more mass is heaped outside the star's orbit (I'm thinking of dark matter here) the more is the pull away from the centre, the less net gravitational pull it feels and the slower the star orbits. This is in direct conflict with what the effect of dark matter is supposed to be - speeding up the star's velocity from the "expected" slow velocity.

 

[Bandersnatch]

This is in direct conflict with what the effect of dark matter is supposed to be - speeding up the star's velocity from the "expected" slow velocity.

If you were to model DM as having the same type of mass distribution as luminous matter, sure. But collisionless matter cannot form a disc, as the particles have a hard time losing energy or exchanging momenta.
So it has to remain in a roughly spherical distribution, with the vast majority of particles streaming through the central area on nearly degenerate orbits. That's why it's referred to as dark mater 'halo'.
I.e. to make a quick-and-dirty model of the effect of DM on the orbital velocities of your luminous matter, you should assume spherical distribution with density increasing with decreasing radius.

 

[ohwilleke]

If you were to model DM as having the same type of mass distribution as luminous matter, sure. But collisionless matter cannot form a disc, as the particles have a hard time losing energy or exchanging momenta.
So it has to remain in a roughly spherical distribution, with the vast majority of particles streaming through the central area on nearly degenerate orbits. That's why it's referred to as dark mater 'halo'.
I.e. to make a quick-and-dirty model of the effect of DM on the orbital velocities of your luminous matter, you should assume spherical distribution with density increasing with decreasing radius.

That is basically the NFW distribution. But, the NFW distribution is far from the best fit to the data for halo distribution models. The best fit DM halo mass distributions are not spherically symmetric, although they aren't as compressed as the luminous matter distribution.

 

[Bandersnatch]

far from the best fit to the data

Oh, sure. But it's not terrible if one just wants to get an idea of what's going with their doodling. I wasn't even thinking of the NFW profile for this purpose. More like, assume $\rho (r)$ is inversely proportional to r and see what happens to the velocities. Then assume $1/{r^2}$, see what happens. Then maybe constant density, and so on. Spherical symmetry makes this easy to do.

 

[snorkack]

If you were to model DM as having the same type of mass distribution as luminous matter, sure. But collisionless matter cannot form a disc, as the particles have a hard time losing energy or exchanging momenta.
So it has to remain in a roughly spherical distribution, with the vast majority of particles streaming through the central area on nearly degenerate orbits.

Why should vast majority of particles stream through the central area, i. e. have low periapse requiring low individual angular momentum? In case of no collisions, would there be any mechanism to clear out high periapse orbits?

 

[strangerep]

[...] my own doodlings have shown that this certainly does not happen with a disc. [...]

FYI, you longer need to "doodle" with these sorts of nontrivial computations...

I just posted a thread about Jo Bovy's new interactive book on Galactic Dynamics. Among many other things, it covers detailed derivations for various galactic geometries, e.g., thin disk, thick disk, and others.

 

[Bandersnatch]

Why should vast majority of particles stream through the central area, i. e. have low periapse requiring low individual angular momentum? In case of no collisions, would there be any mechanism to clear out high periapse orbits?

Hey, thanks for this question. I've been trying to recall the argument for that, and I'm increasingly certain I've just pulled it out of my lower back at some point in the past and then started believing it.
Because as you say, it shouldn't be the case, should it? And in any case, the density being inversely dependent on the radius doesn't require 'the vast majority' of DM to be going through the central region. The vast majority of the mass should actually be in the halo, given the approx. 1/r density profile, since the volume obviously grows faster with r.

 

[snorkack]

Because as you say, it shouldn't be the case, should it? And in any case, the density being inversely dependent on the radius doesn't require 'the vast majority' of DM to be going through the central region. The vast majority of the mass should actually be in the halo, given the approx. 1/r density profile, since the volume obviously grows faster with r.

The entire mass is in the halo for the simple reason that the mass is infinite - 1/r density obviously means that the density diverges to infinity at r=0, while the mass diverges to infinity with r².
But note that for a spherically symmetric distribution of masses on circular orbits, the said spherical distribution is completely free and arbitrary.
Whereas eccentric orbits provide constraints on the relative densities of different shells. On the assumption that the orbits are incommensurate periods and the particles have spread uniformly around their orbits, we can figure out instant mass distribution from distribution of orbits.
If all orbits are high eccentricity, then every single particle spends most of time in the halo moving slowly, and a small fraction of time moving fast through the centre. But since the particles coming from different directions get together in the centre, what is the net result? Which distribution of orbits gives 1/r density?