Tully-Fisher: the Second Law

Previously I noted how we teach about Natural Law, but we no longer speak in those terms. All the Great Laws are already know, right? Surely there can’t be such things left to discover!

That rotation curves tend towards asymptotic flatness is, for all practical purposes, a law of nature. It is tempting to leap straight to the interpretation (dark matter!), but it is worth appreciating the discovery for itself. It isn’t like rotation curves merely exceed what can be explained by the stars and gas, nor that they rise and fall willy-nilly. The striking, ever-repeated observation is an indefinitely extended radial range with near-constant rotation velocity.

The rotation curves of galaxies over a large dynamic range in mass, from the most massive spiral with a well measured rotation curve (UGC 2885) to tiny, low mass, low surface brightness, gas rich dwarfs.

New Laws of Nature aren’t discovered every day. This discovery should have warranted a Nobel prize for Vera Rubin and Albert Bosma. If only we were able to see it in those terms three decades ago. Instead, we phrased it in terms of dark matter, and that was a radical enough idea it has to await verification in the laboratory. Now the prize will go to some experimental group (should there ever be a successful detection) while the new law of nature goes unrecognized. That’s OK – there should be a Nobel prize for a verified laboratory detection of non-baryonic dark matter, should that ever occur – but there should also be a Nobel prize for flat rotation curves, and it should have been awarded a long time ago.

It takes a while to appreciate these things. Another well known yet unrecognized Law of Nature is the Tully-Fisher relation. First discovered as a relation between luminosity and line-width (figure from Tully & Fisher 1977), this relation is most widely known for its utility in measuring the cosmic distance scale.

The original Tully-Fisher relation.

At the time, it gave the “wrong” answer (H₀ ≠ 50), and Sandage is reputed to have suppressed its publication for a couple of years. This is one reason astronomy journals have, and should have, a high acceptance rate – too many historical examples of bad behavior to protect sacred cows.

Besides its utility as a distance indicator, the Tully-Fisher relation has profound implications for physical theory. It is not merely a relation between two observables of which only one is distance-dependent. It is a link between the observed mass and the physics that sets the flat velocity.

The stellar mass Tully-Fisher relation (left) and the baryonic Tully-Fisher relation (right). In both cases, the x-axis is the flat rotation velocity measured from resolved rotation curves. In the right panel, the y-axis is the baryonic mass – the sum of observed stars and gas. The latter appears to be a law of nature from which galaxies never stray.

The original y-axis of the Tully-Fisher relation, luminosity, was a proxy for stellar mass. The line-width was a proxy for rotation velocity, of which there are many variants. At this point it is clear that the more fundamental variables are baryonic mass – the sum of observed stars and gas – and the flat rotation velocity.

I had an argument – of the best scientific sort – with Renzo Sancisi in 1995. I was disturbed that our then-new low surface brightness galaxies were falling on the same Tully-Fisher relation as previously known high surface brightness galaxies of comparable luminosity. The conventional explanation for the Tully-Fisher relation up to that point invoked Freeman’s Law – the notion (now deprecated) that all spirals had the same central surface brightness. This had the effect of suppressing the radius term in Newton’s

V² = GM/R.

Galaxies followed a scaling between luminosity (mass) and velocity because they all had the same R at a given M.

By construction, this was not true for low surface brightness galaxies. They have larger radii at fixed luminosity (representing the mass M). That’s what makes them low surface brightness – their stars are more spread out. Yet they fall smack on the same Tully-Fisher relation!

Renzo and I looked at the result and argued up and down, this way and that about the data, the relation, everything. We were getting no closer to understanding it, or agreeing on what it meant. Finally he shouted “TULLY-FISHER IS GOD!” to which I retorted “NEWTON IS GOD!”

It was a healthy exchange of viewpoints.

Renzo made his assertion because, in his vast experience as an observer, galaxies always fell on the Tully-Fisher relation. I made mine, because, well, duh. The problem is that the observed Tully-Fisher relation does not follow from Newton.

But Renzo was right. Galaxies do always fall on the Tully-Fisher relation. There are no residuals from the baryonic Tully-Fisher relation. Neither size nor surface brightness are second parameters. The relation cares not whether a galaxy disk has a bar or not. It does not matter whether a galaxy is made of stars or gas. It does not depend on environment or pretty much anything else one can imagine. Indeed, there is no intrinsic scatter to the relation, as best we can tell. If a galaxy rotates, it follows the baryonic Tully-Fisher relation.

The baryonic Tully-Fisher relation is a law of nature. If you measure the baryonic mass, you know what the flat rotation speed will be, and vice-versa. The baryonic Tully-Fisher relation is the second law of rotating galaxies.