The Tully-Fisher relation in spirals

The circular velocity as function of radius - the rotation curve - in spirals tends to be flat, i.e. the rotation velocity is independent of radius sufficiently far out in the disk. We argued in section 5.2 that this was evidence for the presence of dark matter in spiral disks, since it implies lots of mass in the outer parts, whereas we see very little light there. Figure10.2 compares the measured rotation velocity $V_c$ (written $W$ in the figure) with the total absolute magnitude $M=-2.5\log(L)$+constant, for a sample of spiral galaxies. Brighter spirals (more negative $M$) rotate faster (larger $W$). Since the figure shows there to be a linear relation between $M$ and $V_c$, it implies a power-law relation between the intrinsic luminosity $L$ and the circular velocity $V_c$, called the Tully-Fisher relation:

$$L\propto V_c^\alpha , \alpha\approx 4 .$$ (10.1)

The slope $\alpha$ of this relation depends somewhat on which band is being used (i.e., does $L$ refer to a V -band, or an I -band luminosity).

Figure 10.2: R-band absolute magnitude $M$ of spiral galaxies as function of the maximum rotational velocity $W=V_c$ (in km s$^{-1}$). Brighter spiral galaxies (more negative $M$; recall that magnitude $M$ is related to luminosity $L$ as $M=-2.5\log(L)$+constant) rotate faster.

What is the origin of this relation? Remember that the circular velocity $V_c$ is determined by the enclosed mass,

$$V_c^2 = {G M\over R} .$$ (10.2)

Let us introduce a global mass-to-light ratio, [M/L], as

$$\hbox{[M/L]}\equiv {M\over L} ,$$ (10.3)

then

$$V_c^2\propto \hbox{[M/L]} {L\over R} .$$ (10.4)

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The mass-to-light ratio [M/L] will depend on the type of stars in the galaxy (which determines $L$), and the amount of dark matter (which mainly determines $M$). The surface brightness $I$ is the luminosity per unit area, $I=L/(\pi R^2)$, or solving for $R$

$$R\sim \left({L\over I}\right)^{1/2} .$$ (10.5)

Combining the last two equations gives

$$L\propto {V_c^4\over\hbox{[M/L]} I} .$$ (10.6)

So the Tully-Fisher relation is reproduced, with $\alpha=4$, when the surface brightness $I$ and mass-to-light ratio [M/L] are independent of the luminosity of the galaxy. Remark that this is not a derivation of the Tully-Fisher relation: we are just trying to understand what is required for spirals to follow this relation. The fact that [M/L] and $I$ are independent of $L$ implies that somehow stars and dark matter are closely linked, or in other words: the star formation history is largely determined by the mass of the dark halo of the galaxy.