# Properties of Disk Galaxies: Kinematics

Spiral galaxies typically show flat rotation curves. Dark Matter!

The luminosity of a spiral galaxy correlates with its rotation velocity: the Tully-Fisher Relationship or or, in magnitudes  • Is there a physical basis for the Tully-Fisher relationship?
• What are these two quantities (L and Vcirc) tracing?

 First, remember what determines the circular velocity: so that we don't know the mass of a galaxy, but we know its luminosity, so let's make up a quantity called the total mass-to-light ratio (which includes everything: stars, gas, dark matter): now remember that surface brightness is luminosity over area: or, solving for R: OK. Now, mass is mass: so equate our two mass expressions: substitute in for R: and solve for L: Whew! So Tully-Fisher works if surface brightness times total (not stellar) mass-to-light-ratio squared is constant. In other words, the stars and the dark matter are somehow linked.

Why would that be true?
We don't understand it, but it seems to work!
But this tells us something fundamental about how galaxies formed. Any model for galaxy formation must explain the Tully-Fisher relationship.

OK, so let's look at the Tully-Fisher relationship for nearby galaxies using different wavelengths:

 B (Blue) Tully Fisher R (Red) Tully Fisher H (Infrared) Tully Fisher X-axis: ~2Vcirc Y-axis: absolute B magnitude X-axis: ~2Vcirc Y-axis: absolute R magnitude X-axis: ~2Vcirc Y-axis: absolute H magnitude slope: -8.0 alpha: 3.2 scatter: 0.25 mag slope: -8.8 slope: 3.5 scatter: 0.25 mag slope: -11.0 alpha: 4.4 scatter: 0.19 mag

Question: Why would the relationship change depending on what wavelength you look at?

## The Baryonic Tully-Fisher Relationship

(Figures from McGaugh 2005)

 Let's look at a different version of the classic Tully-Fisher relationship: Blue luminosity versus circular speed for a sample of spiral galaxies. We see the linear relationship, with a decent bit of scatter. Blue luminosity (LB) versus circular speed (Vf) Now let's use the colors and luminosities of the galaxies, along with stellar population models, to work out the mass of all the stars in each galaxy. If we plot that on a TF-like diagram, the scatter is much less for massive galaxies, but the low mass galaxies don't fit. But there's more stuff than just stars in a galaxy -- we haven't accounted for gas. Low mass spirals are preferentially more gas-rich, so we are missing a lot of their mass. Stellar disk mass (M*) versus circular speed (Vf) If we define the total baryonic mass of the galaxy by adding both stars and gas together, we get any extremely tight relationship over orders of magnitude in mass! There is a basic, fundamental relationship between the amount of normal (baryonic) mass in a spiral galaxy and the speed at which they rotate. This is a huge constraint on models of dark matter and galaxy formation. Baryonic disk mass (Md) versus circular speed (Vf)