Local Group timing argument

The dynamics of M31 and the MW can be used to estimate the total mass in the LG. From the Doppler shifts of spectral lines, one can measure the radial velocity of M31 with respect to the MW^6.5,

$$v=-118{\hbox{\rm km s$^{-1}$}}\,.$$ (6.7)

The negative sign means that Andromeda is moving toward the MW. This may be surprising, given that most galaxies are moving apart with the general Hubble flow. The fact that Andromeda is moving toward the MW is presumably because their mutual gravitational attraction has halted, and eventually reversed their initial velocities. Kahn and Woltjer pointed out in 1952 that this leads to an estimate of the masses involved.

Since M31 and the MW are by far the most luminous members of the LG we can neglect in the first instance the others, and treat the two galaxies as an isolated system of two point masses. Since M31 is about twice as bright as the MW, and given that they are so similar, it is presumably also about twice as massive. If we further assume the orbit to be radial, then Newton's law gives for the equation of motion

$${d^2r\over dt^2} = -{GM_{\rm total}\over r^2}\,,$$ (6.8)

where $M_{\rm total}$ is the sum of the two masses. Initially, at $t=0$, we can take $r=0$ (since the galaxies were close together at the Big Bang).

The solution can be written in the well known parametric form

$$r = {R_{\rm max}\over 2} (1-\cos\theta)$$

$$t = \left({R^3_{\rm max}\over 8\,G\,M_{\rm total}}\right)^{1/2}\, (\theta-\sin\theta)\,.$$ (6.9)

The distance $r$ increases from 0 (for $\theta=0$) to some maximum value $R_{\rm max}$ (for $\theta=\pi$), and then decreases again. The relative velocity

$$v = {dr\over dt} = {dr\over d\theta}/{dt\over d\theta} = \left({2... ...}\over R_{\rm max}}\right)^{1/2} \left({\sin\theta\over 1-\cos\theta}\right)\,.$$ (6.10)

The last three equations can be combined to eliminate $R_{\rm max}$, $G$ and $M_{\rm total}$, to give

$${v\,t\over r} = {\sin\theta\,(\theta-\sin\theta)\over (1-\cos\theta)^2}\,.$$ (6.11)

$v$ can be measured from Doppler shifts, and $r\approx 710{\hbox{\rm kpc}}$ from Cepheid variables. For $t$ we can take the age of the Universe. Current estimates of $t$ are quite accurate^6.6, but even using ages of the oldest MW stars, $t\sim 15{\hbox{\rm Gyr}}$ gives an interesting result. Using these numbers, we can solve the previous equation (numerically) to find $\theta=4.32$ radians, assuming M31 is on its first approach to the MW^6.7.

Substituting yields $M_{\rm total}\approx 3.66\times 10^{12}\hbox{$M_\odot$}$, and hence for the MW mass, $M\approx M_{\rm total}/3$

$$M\approx 1.2\times 10^{12}\hbox{$M_\odot$}\,,$$ (6.12)

comfortably higher than our lower limit Eq.(6.6).

Since the luminosity of the MW (in the V band) is $1.4\times 10^{10}\hbox{$L_\odot$}$, the corresponding mass-to-light ratio for the MW is around $\Gamma=100$. Furthermore, the estimate of $M$ is increased if the orbit is not radial, or M31 and the MW have already had one (or more) pericenter passages since the Big Bang.

If all stars in the MW and M31 were solar mass stars, we would expect $\Gamma=1$. Now even in the solar neighbourhood, most stars are less massive than the sun, and so the mass-to-light ratio for the stars is about 3 or so^6.8. So the very large mass inferred from the LG dynamics strongly corroborates the evidence from rotation curves and Oort's constants, that most of the mass in the MW (and presumably also in M31) is dark.

From these numbers, we can also estimate the extent $R_\star$ of such a putative dark halo. If the the circular velocity $V_c=220{\hbox{\rm km s$^{-1}$}}$ out to $R_\star$, then from $V_c^2=GM/R_\star$ we find

$$R_\star={GM\over V_c^2} \approx {G\,10^{12}\hbox{$M_\odot$}\over (220{\hbox{\rm km s$^{-1}$}})^2} \approx 100{\hbox{\rm kpc}}\,.$$ (6.13)

If, as is more likely, the rotation speed eventually drops below 220km s$^{-1}$, then $R$ is even bigger. Hence the extent of the dark matter halo around the MW and M31 is truly enormous. Recall that the size of the stellar disk is of order 20kpc or so, and the distance to M31 $\sim 700{\hbox{\rm kpc}}$. So the dark matter haloes of the MW and M31 may almost overlap.