## Sunday, December 02, 2007

### Rotational Curves of Galaxies

Stars and other objects that are bound to spiral galaxies rotate around a common center. The rotation velocity of the stars is such that the orbits are stable. The required velocity for this depends on the attractive force acting on the star, which results from the matter content in the galaxy. The larger the force, the higher the velocity has to be for a stable orbit.

Measuring the velocities of stars as a function of their distance to the galaxy's center therefore allows to draw conclusions about the matter distribution inside the galaxy. The plot below shows an example of the velocity, v in km/s, as a function of the distanceto the center, R in kpc, for the galaxy NGC 3198

Naively one would expect one can estimate the rotation curves as follows. Most of the matter we see is located in the center of a galaxy. The gravitational field is weak enough so one can use the Newtonian limit. A star in the outer arms should thus roughly follow an orbit on which the attractive gravitational force balances the centripetal force. Requiring both to be equal one finds that the square of the stars velocities should drop with the inverse of the distance to the center:

If you look at the plot above however, you see that this is not what we observe. Instead, the velocity seems to become constant towards the outer regions. And the above example is not a single case. If you look at Begeman's paper, you will find many similar looking curves for spiral galaxies (for more recent measurements, see e.g. astro-ph/0107326).

The observations can be explained by assuming a significant amount of non-visible (dark) matter which is distributed through the galaxy. This dark matter is today the most widely accepted - though not the only - explanation for the rotational curves. The challenge here is that the ratio of dark matter to visible matter (the mass-to-light ratio, commonly denoted M/L) depends on the type of galaxy. E.g. globular clusters show little or no evidence for dark matter.

For other experimental evidence, see also our previous post on Dark Matter.

This post is part of our 2007 advent calendar A Plottl A Day.

1. Dear Bee and Stefan,

The Tully-Fisher relation has always fascinated me. It seems to me to hold some clue about galactic structure and dark matter.

Quoting from Wikipedia

"Tully-Fisher relation, published by astronomers R. Brent Tully and J. Richard Fisher in 1977, is an empirical relationship between the intrinsic luminosity or stellar mass of a spiral galaxy and its velocity width (the amplitude of its rotation curve). The luminosity is the amount of light energy emitted by the galaxy per unit time; it can be measured using the galaxy's apparent brightness when the distance to the galaxy is known. The velocity width is measured via the width or shift of spectral lines and the Doppler effect.

The quantitative relationship between luminosity and velocity width is a function of the wavelength at which the luminosity is measured, but roughly speaking, the relation has luminosity proportional to velocity to the fourth power."

-- Tully-Fisher implies a that there is a relationship between luminosity and mass of the spiral galaxy, and that seems to me to imply that a spiral galaxy has a rather definite structure in terms of regular and dark matter; one simply cannot put together spirals from any old mix of regular and dark matter (or else, how would the Tully-Fisher relationship come about?).

It is past breakfast time, but spiral galaxies come like pancake mix, I think, if you add too much or too little water, something goes wrong, and you don't get a spiral.

All this is very vague, but if you have any insights, do share!

2. Thank you very much for this simple post, which clarifies things and also raises a problem in my mind. I shouldn't really be commenting as I'm not much good at entering controversial discussions, but I'm surprised if F_G should be proportional to 1/r^2

This is because F_G = mMG/r^2 where M is the mass of the galaxy contained out to radius r.

Hence M is dependent on r, e.g. if the spiral galaxy has a uniform density and is shaped like a cylinder, the effective value of M to use in the formula for F_G is going to be

M = {Pi}*{r^2}*h*{rho, density}

where h is the thickness of the galaxy (height of the disc).

Hence, F_G would then be

F_G = m{Pi}*{r^2}*h*{rho, density}G/r^2

which might be independent of radius, because the two r^2 terms cancel. (Obviously, it might not be strictly independent of radius if the galaxy's thickness and/or density varies as a function of r.)

As a result, you then get

F_C = F_G

(v^2)/r ~ 1

(taking F_G as constant so F_G = 1)

Hence

v ~ r^0.5

This is the opposite conclusion to that which you reach in this post, where you find that the theory says

v ~ 1/r^0.5

I'm curious whether anyone has actually made very detailed calculations of what the velocity curves should be, allowing for density variation with radius and for variation in the thickness of the galaxy with radius.

This is something that was mentioned briefly on a cosmology course I did, but only about a minute was spent on it before the lecturer went on to other matters, so I didn't ever get the chance to investigate the details of the theory. I don't doubt the carefully collected evidence for the real velocity versus radius curves, just the theoretical interpretation.

Has anyone who claims that this is evidence of dark matter actually been able to make it a quantitative claim by estimating the quantity of dark matter actually involved? I expect that would be hard because we can only see each galaxy from one angle, so it would be hard to estimate the density and thickness of galaxies, unless estimates are made from generalized models of galaxy shape.

(As background information in elementary physics to justify my argument above, Newton geometrically proved that for any symmetrical distribution of matter, the effective gravitational mass is that within the radius r for the purpose of calculating gravity forces at radius r. E.g., if you were half way down to the middle of the Earth, the gravity there would be that due to just the mass of the planet which is within the radius that you are at, and not the total mass of the Earth. The gravitational effects from matter at bigger radii around you, simply cancels out as far as you are concerned.)

3. Hi Nige,

Thanks for your comment, which raises an interesting point. As I said in the above post "Naively one would expect one can estimate the rotation curves as follows. Most of the matter we see is located in the center of a galaxy." I.e. the estimate I give above assumes that the mass is distributed similarly to the visible matter in the Galaxy, in which case the two-liner should be a good approximation for the stars in the outer regions.

I'm curious whether anyone has actually made very detailed calculations of what the velocity curves should be, allowing for density variation with radius and for variation in the thickness of the galaxy with radius.

Modifying the matter density \rho to be a different function is exactly how dark matter is used to explain the observed curves. The point is that this modified \rho is not to be expected from the visible matter content.

Has anyone who claims that this is evidence of dark matter actually been able to make it a quantitative claim by estimating the quantity of dark matter actually involved?

Yes, but details depends on the galaxy. I currently don't have a good reference, but I think its somewhere around 80%. The numbers are confirmed by data from gravitational lensing (one can infer the total mass from the strength of the light deviation, which is stronger than what the visible matter could achieve). See also our earlier post on Dark Matter.

I expect that would be hard because we can only see each galaxy from one angle, so it would be hard to estimate the density and thickness of galaxies,

Again, you can measure the brightness of the galaxy and infer whether what you see has the right distribution to give the observed rotation curves. The answer is no. Besides this, Galaxies shapes are not completely arbitrary but are rather similar for certain classes, so you can do statistics and estimate their shape.

Best,

B.

4. Dear Arun:

Indeed, Tully-Fisher is an interesting relation. I think it will turn out to be very useful in understanding dark matter, or whatever else causes the rotational curves. I would expect that a successful explanation should also result in the correct Tully-Fisher law. I have no good insights into this, except that I think it should be paid more attention to. Best,

B.

5. Ooh, this is going to be an educational month :D.

I have an easy question about dark matter calculations. As far as I understand, they're (almost?) always done with Newtonian gravity---because, as you say, it should be weak enough that it doesn't matter. My question is, has anyone ever sat down and calculated things with GR, just to confirm that we're not missing something? Or, instead, is it so incredibly obvious that nothing new would be gained... and if so, could you try to strengthen my conviction in this matter?

6. Hi Domenic,

... has anyone ever sat down and calculated things with GR, just to confirm that we're not missing something?

You may be interested in the argument of Cooperstock and Tieu (astro-ph/0507619) and the controversy this has spurred. In my understanding of the debate, it seems that GR cannot be made responsible for the rotational curves.

Best, Stefan

7. Hi Domenic:

The Newtonian limit is a good approximation as long as curvature is small, and velocities are much smaller than the speed of light. The curvature scales roughly like M/R^3, i.e. it is typically strong close by the center of a black hole. (But *not* necessarily at the horizon. This is a common mistake many people make: if the black hole is large, the curvature at the horizon can be arbitrarily small.)

Also, we sit in an outer arm of a spiral galaxy much like the ones observed and the curvature from the galaxy's mass distribution is indeed negligible, and velocities are much smaller than the speed of light. Given that one can take the Newtonian limit, compute the Next-to-Newtonian corrections and convince oneself that they are small for the system under consideration, I don't know whether somebody has done a full GR calculation. It sounds to me a bit like computing an arrow's flight by using full Special Relativity and Quantum Field theory. Given the progress in numerical possibilities I'd think it was possible though.

Best,

B.

8. Hi all,

just to add a few data about NGC 3198, the galaxy whose rotation curve is shown in the plot: It is located in the constellation Big Dipper - here is a photo. It's apparent diameter in visible light is 8.5 arcmin, and at a distance of about 14.5 Mpc, this corresponds to a radius of 18 kpc. For comparison, the Milky Way has a diameter of about 15 kpc, so this is pretty similar. The distance of the Sun to the galactic centre is about 8 kpc.

The point is, the "flat part" of the rotation curve is already quite far out, and there, the relevant mass of visible matter inside the orbit of a star around the galactic centre hardly changes any more with the distance to the centre.

Best, Stefan

9. Here's how I saw the Cooperstock & Tieue paper in 2005 (you will find the comment on cosmicvariance.com somewhere), written differently here.

The non-Newtonian component of the Cooperstock-Tieu model is the term induced by the angular momentum of the galaxy.

The metric outside a compact body of mass M and angular momentum S is
roughly, in spherical coordinates:

ds^2 = (1- 2M/r) dt^2 - dr^2/(1- 2M/r) - r^2 ( dÎ¸^2 + sinÎ¸^2 dÏ†^2)
+ ( 4S sinÎ¸)/r^2 r sinÎ¸ dÏ† dt

(Misner, Thorne, Wheeler 19.13)

The last term is non-Newtonian and resembles the term with the function N from the paper. A rough estimation of its value at the periphery of a galaxy shows it to be much smaller (by v/c, where v is an average stellar velocity and by geometric factors - most of the mass contributes very
little to the angular momentum) than the Newtonian potential M/r.

I translate the main claim of the paper to be that the periphery of the galaxy is highly non-Newtonian, and motion is dominated by the angular momentum term. The non-linearity claim I translate to the claim that the frame dragging term accumulates as one moves from the center of the galaxy to the periphery until it dominates the Newtonian potential.

____

That is, the leading Newtonian term is M/r.

The leading angular momentum term is S/r^2 ~ a Mvr/r^2 = a M/r v; where a is a geometric factor. Just remember v is expressed in terms of light speed and so in terms of MKS units is v/c.

Hmm, staring at it now, I'm not as sure as I was back then then this is necessarily negligible in effect.

10. Dear Arun,

Stupid question, I didn't read the paper: So the claim is the angular momentum of the disk grows significantly outside the central region in some unintuitive way, such that in the outer regions it has summed up to makes a non-negligible effect? Best,

B.

11. Hi Stefan and Bee, another great ponderous posting! If two people are at a childrens playground that has a "roundabout", one can experience the observational implications of this interesting post. One person (A) goes to the centre of the roundabout, whilst the other (B) stays at the outer edge. A see's B rotating at a faster rate than B see's A, for fact of relative positions about a centre of mass? Imagination is needed by B (because of the fact, B would need to imagine A was rotating whilst B was static) in order to arrive at the same conclusions as A at the centre of the roundabout.

Now interestingly we observe all galaxies from afar, from way beyond the outside edge of every single galaxy, whilst our location within our milky way should act as a location point of an equal distribution zone.

Could it be that our position within our galaxy, affects our ability to "observe" other galaxies, in a sense we are always at the extreme "outer-edge" of every other galaxy we observe, our line-of-sight is really an extension of a specific "extended arm" for spiral galaxies?

Back to the roundabout, if one adds another person (C) around the middle, in between A and B, then asks him/her who has the greatest "OBSERVED" velocity, A at the centre or B at the outer edge, i'm certain that B would "appear" to have the greatest, relative to C, because of the backgound space behind B would be the gauging factor in observations.

When one looks into the centre of galaxies, from within that galaxy, then one cannot impose a "background" space that is infact not there?

12. I didn't read the paper: So the claim is the angular momentum of the disk grows significantly outside the central region in some unintuitive way, such that in the outer regions it has summed up to makes a non-negligible effect?

Dear Bee, I'll have to look up the paper to tell you precisely what's there - it has been a while.

As I recall:

C & T have an ansatz for the metric of a cylindrically symmetric (coordinates r, z, Ï†, t ) rotating mass distribution which has a term like

N(r,z) dÏ† dt

and as I recall it, N is crucial to rotation curve. I don't remember whether the paper discusses it, but it seems clear to my befogged eyes that N can only arise from angular momentum.

Just to get a handle on things, I asked what would the compact object that approximates the source that a body at the periphery of the galaxy would see, and how does its angular momentum contribution to the metric compare to the mass term? (ought to be small, I think).

Anyway, C&T write out the equations of motion and are able to write the solution for a metric with a general source as a infinite series of Bessel functions.

The rotation curve for such a metric is numerically curve-fitted with an observed galactic rotation curve to obtain coefficients of the first several terms in this series.

Now you have the particular metric for a galaxy, you use the Einstein equation to obtain the source.

Lo and behold, the source tracks well with the visible stellar mass in the galaxy; no dark matter is needed.

The problem with the metric that C&T obtain is as follows - we need it to be an even function of z. It turns out to depend on |z| rather than z^2, and there is a discontinuous derivative at the equatorial plane which seems to imply a thin sheet of high density mass in the equatorial plane. I think it is also not clear whether they have included the mass of this sheet in the computed mass of the galaxy. i.e., in other words, the "dark matter" could be this equatorial plane density singularity.

If C&T have included the mass of the equatorial sheet then such a sheet is not necessarily fatal to their idea.

I've been meaning for the longest time to carefully work it out for myself, simply as a means of "getting into the groove". Somehow that is not happening. :(

I hope I've been a bit clearer than mud.

13. Hi Bee,

Many thanks for your reply, especially about the central bulge in galaxies. If all the mass is assumed to be in the middle, the predicted variation of velocity with radius is

v ~ 1/r^0.5

while if all the mass is assumed to be uniformly distributed in a disc shape (like a coin), then the distribution is

v ~ r^0.5.

If the facts are between these two extremes, then v might very well be independent of radius, without requiring dark matter. It looks to me as if the implications about dark matter are extremely sensitive to assumptions made about how visible mass is distributed inside a spiral galaxy.

Using a discrepancy between the theory and observation to deduce the how the abundance of darm matter varies in the galaxy is therefore likely to be extremely sensitive to inaccuracies in the theory. I don't think discrepancies between theory and observation are always a good reason for modifying a theory (that was what Ptolemy did when he just added new epicycles to fix problems in his theory of the Earth-centred universe, instead of checking the original theory to see if it is wrong).

I'm going to have to try to track down all the literature on the theoretical predictions and interpretations of this. (At present, I don't even know who first "predicted" the velocity curve of a galaxy, or whether that came before or after observations, and before or after the Friedmann critical density suggested the possibility of dark matter.)

There must be some dark matter around in the form of neutrinos and there is some evidence from astronomy for dark matter effects, but I want to see extraordinary evidence for the extraordinary claim that 80% or more of the universe is stuff that nobody has seen in the lab. It's worse than epicycles to accept it without seeing extremely rigorous evidence, and a discrepancy between a theoretical prediction and observation is not strong evidence unless the theoretical prediction is proved to be correct.

The latest media story about an astronomer claiming that a cosmic void is "evidence" for the multiverse demonstrates how wishful think can bias groupthink.

The only convincing estimate of dark matter I have seen is that based on the discrepancy between the observed matter in the universe and the critical density in the Friedmann-Robertson-Walker metric of general relativity, but there are still issues there. For example, it's assumed widely that quantum gravity only departs from general relativity on small scales where quantum effects become important. However, when you think about exchange radiation (gravitons) in quantum gravity, because the universe isn't static but masses are receding from each other (hence redshifts of received radiation), the gravitons received will be redshifted and that means they'll carry less energy, reducing the strength of gravity between masses which are receding from one another at relativistic velocities (i.e., reducing the gravitational coupling constant G over cosmological sized distances). If this effect is true (if gravity is a quantum field theory involving exchange of gravitons between receding gravitational charges - masses - in the universe), the whole Friedmann (et al.) application of general relativity to cosmology breaks down, because general relativity will then only valid for intermediate distances where distances are neither too small nor too big.

Best,
Nige

14. Hi Nige:

It looks to me as if the implications about dark matter are extremely sensitive to assumptions made about how visible mass is distributed inside a spiral galaxy.

The point is the visible distribution of mass can be measured. It is, as I said above, mostly in the center of the galaxy. There are of course some uncertainties attached to these measurements, but the visible matter is most definitely not uniformly distributed throughout the disk. And no, the results are not 'extremely' sensitive on this distribution. Also, besides being in conflict with observation one can't just distribute matter somehow one also has to argue why it can be distributed so.

I don't think discrepancies between theory and observation are always a good reason for modifying a theory [...] I'm going to have to try to track down all the literature on the theoretical predictions and interpretations of this.

Sure. We've all been there, done that. You might want to have a look at the repeatedly mentioned earlier post on Dark Matter where I list other evidence besides the rotational curves, as well as references.

There must be some dark matter around in the form of neutrinos and there is some evidence from astronomy for dark matter effects,

Sure. Neutrinos have been tried, black holes, brown dwarfs, other things. Neutrinos don't work quite the right way roughly because they are relativistic and don't clump as you need them to. Presently favoured candidates are weakly interacting massive particles (WIMPs). Best,

B.

15. Hi Paul:

I am not sure I understand your question. Our motion in the outer spiral arm of our galaxy does of course affect observations, in the sense that our motion towards some galaxies and away from some others contributes to the redshift. This effect can however be subtracted from the data. (Besides this we have of course all the stars from the Milky Way hanging in front of us, which probably can be quite annoying if one wants to look at galaxies a Gpc away). Best,

B.

16. Hi Bee,

If you would like to see a (IMHO) beautiful correlation involving *all* virialized stellar systems, look at

http://xxx.lanl.gov/pdf/astro-ph/9910541

sp., figure 1.

The plot is rotated in the "kappa" space (see eqn. 7 of the paper), the reasons for that are in the paper.

That paper has about 2 citations or so. That's an example of what a lack of advertisement can do to (again IMHO) a nice scientific result.

Basically, we have tried to understand what happens when you fit various stellar systems into a *two* component virial theorem, that is, formed by dark matter + baryonic matter. The result is a "curved" fundamental plane, as can be seen in figure 1 (the curve is a projection of this plane in a 3 dimensional parameter space), in which *all* systems can be fit into.

Until then, no one ever explained the so called
“cosmic metaplane”, that is, an ensemble of interrelated fundamental planes involving all virialized systems. In our paper, it is shown that these "planes" can be thought of as *one* plane alone. Notice that globular clusters lay in the place where dark matter is zero (horizontal part of the curve). Figure 2 has many implications, that are still under work.

Best,
Christine

17. BTW that paper was published in ApJ 528, L5 (2000).

18. Hi Christine:

B.

19. Great, it's been a lot of time since my collaborators and I worked on that paper (I worked on it nursing my baby at the computer at the same time!).

I wrote:

"it is shown that these "planes" can be thought of as *one* plane alone."

Well, not exactly, let me correct myself. More appropriately, the 2-VT predicts an asymptotic tilt (with a fixed value, which is interesting) relatively to the 1-component
virial theorem, and also a "curvature" of the virial plane. All stellar systems can be reasonably fit into such an scheme.

The k1 parameter is basically mass; the k2 is (mean surface brightness)^3 x M/L; and the k3 parameter includes a correction to the M/L ratio given that we add a term to the gravitational energy of the system due to the interaction of the two components (hence differs from the k3 parameter defined by Burstein et al. 1997, in which the usual one-component virial theorem is used).

Perhaps a 3VT could do even better? dark matter, stars and gas, separately. I don't know.

Best,
Christine

20. Hi Stefan and Bee,

I am afraid I have to point out an incorrectness here:

The graph you are showing has actually been obtained by measuring the gas velocity (HI), not that of the stars (as you are suggesting).
To my knowledge the conclusion of a flat rotation curve is indeed practically exclusively based on the observation of gas, primarily the 21cm line of hydrogen, but also optical observations of HII-regions (e.g. Vera Rubin's first observations). The point here is that gas can easily be accelerated by rotating magnetic fields during times when it is ionized, and this should result in a flat rotation curve (see my page Galactic Rotation Curves and the Dark Matter Myth for more). The motion of stars is obviously unaffected by this, and thus the stars and the gas are not co-rotating (the latter being the vital assumption for the 'dark matter' conclusion).

Thomas

21. Hi Thomas,

thanks for your comment. Indeed, data are sometimes more subtle than simple texbook explanations of them. In this case, the rotational curve taken from the Begelman paper was measured with the radio emission of the "forbidden" 21 cm hyperfine interaction line of neutral hydrogen, which probes the distribution of gas in the galaxy. However, the paper by McGaugh, Rubin and de Blok we have mentioned, for example, explicitly compares the the 21 cm rotational profiles with the profiles measured in the visible HÎ± line, and finds a good agreement.

As far as I know, HÎ± emissions come predominantly from stars, or at least, they can be used to estimate the contribution coming from stars by comparing the relative strength of HÎ± to other lines in the hydrogen spectrum - do you agree to that? But then..

and thus the stars and the gas are not co-rotating

there should be observational evidence for this from observations of the rotational dynamics of the Milky Way? Can you give a reference? And if there is a differenential rotation between gas and stars, what are the relative masses involved, and is this really sufficient to "explain way" the rotational curve puzzle? Are there other publications about this?

As a general remark, as interesting as the points you are raising might be to gain a deeper understanding of the physics involved, this is not the place to promote your own pet theories about everything. We have neither the time, nor the patience, to go into details, and to be sincere, when looking at the long list of well-established theories and models in physics you want to myth-bust, I am not sure at all if a discussion is worth the effort. Sorry, but your self-promotion borders to abusing our hospitality. Please stop doing that.

Best, Stefan

22. Hi Stefan,

First of all, it was not my intention to promote my own theories on your blog, but primarily to point out some misconceptions that your article appears to propagate (I am sure you agree that science should be based on balanced information and not on omissions or incorrect information). I have linked to my own website only for the same reason as you are linking to other sites, namely to provide more information for the interested reader.

In this sense, please note that HÎ± (emission-) lines are actually also produced by gas (recombination of ionized hydrogen in HII-regions), not by stars (see this article where it is mentioned more explicitly than in the reference you gave).
The regions responsible for the HÎ± emission may be geographically more associated with stars than the 21cm emission, but whilst being ionized, the gas atoms could still be crucially affected by any galactic magnetic field (-rotation) .

Thomas

23. If we assume that the rotational curve effect occurs in our solar system as well, is it reasonable to expect that plotting the rotational velocities of the planets should show some discrepancies from the naive application of newtonian gravity or even einstein's theory of gravity? Since the planets are right in our backyard, could really accurate measurements not be made to reveal this effect?
Thank you for the great job you are doing with this blog! The site is outstanding! There is a LOT to learn from it.

24. Hi Anonymous,

thanks for the cheerful words!

The idea to look closely at the orbits of the planets in the Solar System to see if there are some deviations from general relativity and Newtonian gravity is very sensible, in principle.

However, to my knowledge, no such discrepancies have been found yet - any new effects may be just too faint to be detected with today's precision.

For example, NASA's Jet Propulsion Lab can account for all known orbital data of the planets (their coordinates in the sky, or ephemerides, as you can access them at their HORIZONS web interface) with Newtonian gravitation plus the appropriate relativistic corrections. This may be so because the orbits are just not known yet precisely enough to squeeze out information about possibly existing deviations.

Perhaps it will be possible to search for discrepancies once the outer planets have been visited by spacecraft which allow a very precise determination of the planet's orbit via their telemetry data from orbits around the planets.

Best, Stefan

25. Should galaxy space be quasi 2d
then gravity will fall off linearly
and star velocity won't depend on the distance from the center.
Simple enough.

26. Michael: Yes, we all know that the derivative of log is 1/r. As it happens though, we live in 3 dimensional space.

27. Hi Stefan,

When I was trying to find out about calculations of ancient eclipses some years ago (to match with literary sources), I remember reading something that we need to construct a much more reference inertial frame for our solar system in order to get to the next level of precision in this solar system orbit measurements and calculations; and that work for that was ongoing. Do you happen to know anything about this?

Best,
-Arun