LOG#197. The 3 laws of the galaxies.


3…Only 3…And 3 laws…3 laws are everywhere. You do know the 3 laws of newtonian mechanics. You do know (I am sure you do) the 3 laws of the robotics. Or the 3 laws of thermodynamics. And you sure do know as well the 3 laws of Kepler describing celestial motion. All is three. And you are three? And old Minbari joke, I am sorry…Zathras would say:

This short post is going to tell you the amazing 3 laws of the galactic motion, as state to S. McGaugh, whom I dedicate this humble post…After all, the version of the 3 laws is due to him.

The 3 laws of the (spiral-like) galaxies

1st. Rotation curves tend towards asymptotic flatness. Mathematically speaking, this law can be stated as follows:

    \[\boxed{\lim_{R\rightarrow\infty} V_r=\mbox{constant} }\]

Remark: No theory so far – just data. No one can denies the flatness of rotation curves, a discovery that should have deserved a Nobel prize for Vera Rubin. Too late!

2nd. Baryonic mass scales as the fourth power of rotation velocity (Baryonic Tully-Fisher).

    \[\boxed{M_b\propto V_f^4}\]

Remark: This law can “always” be interpreted in terms of dark matter (with sufficient fine-tuning). How? It is quite simple. Begin with Newton and Kepler! You should expect in galaxies the same behavior than those seen in planets, shouldn’t you? Then

    \[F_g=F_c\rightarrow V^2=G_N\dfrac{M}{R}\]

For a disk spiral-like galaxy, define the surface density \Sigma=M/R^2. Then, squaring the newtonian law, you get


Split \Sigma=\Sigma_b+\Sigma_{DM}. Since the velocity remains CONSTANT (according to the previous 1st law of the galaxies), and you do know that baryons follow the newtonian (keplerian) motions, we are forced to assume that \Sigma_{DM}>>\Sigma_{b}. This fact is usually taken from granted from different sources, but it also a follow-up of assuming that dark matter (DM) does exist!

Remark (II): For elliptical galaxies, the (baryonic?) Faber-Jackson relation takes the place of Tully-Fisher’s. It is a relationship between luminosity and velocity dispersion. It can derived as follows

i) The gravitational potential energy of any elliptical galaxy

    \[U_g=-\alpha \dfrac{M^2}{R}=-G_N\dfrac{M^2}{R}\]

shares a link with the kinetic energy for an elliptic galaxy


if velocity is related to 1D dispersion \sigma via V^2=3\sigma^2, since


ii) Use the virial theorem, 2E_k+U_g=0 to get




iii) Assuming M/L\sim constant, a fixed mass to light ratio, and where L is the luminosity, you get

    \[R\propto \dfrac{LG_N}{\sigma^2}\]

If you also assume that the brightness B=L/4\pi R^2 is constant, then

    \[L\propto 4\pi\left(\dfrac{LG_N}{\sigma^2}\right)^2B\]

and therefore, substituting,

    \[L\propto \dfrac{\sigma^4}{4\pi G_N^2B}\]

i. e.,


Q. E. D. Note that we get the universal Faber-Jackson relation if and only if M/L and the brightness L\propto IR^2 is the same for ALL the elliptical galaxies. Quite an statement. Also, sometimes people prefer to write L=C\sigma^\gamma, where \gamma is a free parameter close to 4, the “ideal” Faber-Jackson case.

3rd. Gravitational force correlates with baryonic surface density.

    \[\boxed{-\dfrac{\partial \Phi}{\partial R}\propto \sigma_b^{1/2}}\]

Indeed, the baryonic surface density correlates with acceleration. Furthermore, the Renzo’s rule states, that: “When you see a feature in the light, you see a corresponding feature in the rotation curve.”

Remark: it might stem more naturally from a universal force law.

These 3 laws are completely general for disk galaxies. There is no exception to these laws. There are complementary statements for elliptical galaxies too.

Let me add some stunning comments made by McGaugh in his talks:

  1. The Tully-Fisher relation is not well understood in the context of dark matter. There are many hand-waving models, none of which are completely satisfactory.
  2. One expects, from basic physics, that variations in the distribution of baryonic mass should have an impact on the Tully-Fisher relation (lines). They do not (data).
  3. The residuals from Tully-Fisher are nearly to totally imperceptible, depending weakly on the choice of circular velocity measured. This causes a fine-tuning problem which is generic to any flavor of dark matter…The contribution of the baryonic and dark matter to any given point along the rotation curve must be finely balanced, like a see-saw. As the baryonic contribution increases with baryonic surface density, the dark matter contribution decreases. The two components know intimately about each other…This is the reason of Renzo’s rule and a higher fact (Sancisi 1995, private communication, see also Sancisi 2004, IAU 220, 233): “The distribution of mass is coupled to the distribution of light”.

MOdified Newtonian Dynamics (MOND) is a controversial subject for many people. However, from the phenomenological viewpoint, it works “too well” excepting some critical counter-examples. It could hint a missing code or regime in our understanding of gravity or kinematics at some (big, infrared in terms of theoretical physicists like me) distances. An idea I have suggested here  and one year ago (in 2016, at IARD), is that maybe some new principle for gravity and motion is acting over large scales. With minimal (maximal) acceleration, and a maximal (minimum) length, you have the different asymptotical laws than those in Kepler or Newton cases. Imagine a mass bounded by gravity in which there are some type of background and minimal (maximal) acceleration a_0, Newton’s second law provides:




If you square it,  you get


If you plug that G_N^2<<1, a_0^2<<1, you obtain the MONDian law


Remark: You could fit the minimal acceleration to the cosmological constant, via a_0=c\sqrt{\Lambda}. Is there any other physically appealing definition?

Imagine now you include a Hooke law term as well. Of course it is related to the cosmological constant term (but it has opposite sign to normal Hooke law from springs!). You could think the cosmic hookean term as certain class of maximal length term. By duality, some type of maximal tension/minimal tension. Mimicking the above arguments:

    \[\dfrac{v^2}{R}=\dfrac{G_NM}{R^2}+a_0+\Lambda R\]

    \[v^2=\dfrac{G_NM}{R}+a_0R+\Lambda R^2\]

Square it…

    \[\boxed{v^4=\dfrac{G_N^2M^2}{R^2}+a_0^2R^2+\Lambda^2 R^4+2G_NMa_0+2G_NMR\Lambda+2a_0\Lambda R^3}\]

I suspect that in the limit where you neglect G^2_N, a_0^2, \Lambda^2<<1, you get a correction to MOND. Interestingly, this correction to MOND are two terms proportional to the cosmological constant, that become more and more important as R grows. This generalized MOND must be provided by some quanta of spacetime…The darkons, because they are associated to dark matter and dark energy. Can the galactic motion be the first true hint of long-scale features of quantum gravity? It is a possibility I have never heard written with these words. I think so. It is a cunning and striking possibility!

Let me finish with more great S. McGaugh epic words:

Logical possibilities in the battle Dark Matter vs. MOND.

  • ΛCDM is fine; puzzling observations will be explained by complicated feedback processes.
  • MOND gets predictions right because there is something to it — dark matter doesn’t exist.
  • We have no clue what is going on.

Let me point out additional thoughts on this (nonsense?) competition and struggle:

  1. It could be true BOTH that there is “some dark matter particle” and a new kinematics/dynamics behind universal rotation curves.
  2. We need complementary approaches and more experiments to decide.
  3. Quantum gravity (QG) is hidden but it could happen that the dark stuff is related to it as well. I will not tell that QG is the only cause of flat rotation curves but it is an additional idea. What if DM are “heavy gravitons” modifying our notions of inertia and gravity at long distances? It can not be rejected at all this option!

PLOT HOLE: Not even 3? What about 0th laws? Or 4th or additional laws? Well, of course they can be stated! In fact, S. McGaugh has himself sketched a new law (read his blog entry https://tritonstation.wordpress.com/2016/09/26/the-third-law-of-galactic-rotation/, an references therein), the radial acceleration relationship for galaxies. It reads:


and where g_0=a_0\sim 10^{-10}m/s^2 is a scale that pervades the galaxies…And MOND. Maybe a QG bat-signal?

May the 3 laws of the galaxies be with you!

P. S.: What about elliptical and other exotic galaxies? Interesting issue indeed. I have no enough data for a significative answer in THIS post. However, the 3 laws stated as above, are UNIVERSAL. Don’t forget it before going into elliptical or exotic galaxies. Dwarf galaxies, however, are important, much more than elliptical galaxies.

P. S. (II): Read Stacy McGaugh and his work here http://astroweb.case.edu/ssm/!

P. S. (III): Atom fans here? No way, Kepler laws hold at atomic level more or less (“keplerian quantum mechanics” played an interesting rolo in the origin of quantum mechanics itself, via Bohr-Sommerfeld rules!)

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