Blog #18, Worksheet #6.1, Problem #4, Tully-fisher

4) Over time, from measurements of the photometric and kinematic properties of normal galaxies, it became apparent that there exist correlations between the amount of motion of objects in the galaxy and the galaxy’s luminosity. In this problem, we’ll explore one of these relationships. Spiral galaxies obey the Tully-Fisher Relation: \[L \propto v^4\]where L is total luminosity, and vmax is the maximum observed rotational velocity. This relation was initially discovered observationally, but it is not hard to derive in a crude way: 

(a) Assume that \(v_{max} \thickapprox σ\) (is this a good assumption?). Given what you know about the Virial Theorem, how should \(v_{max} \)relate to the mass and radius of the Galaxy?

We can assume that \(v_{max} \thickapprox σ\) because the mean velocity scatter will of course be lower than the maximum observed velocity. However, because the grouping of stars are roughly similar, the maximum observed velocity will not be much higher than the mean unless there is an anomaly. Thus it will not be too far off to approximate the scatter velocity as the vmax of the system.

We can use knowledge of the Virial Theorem to then relate mass and radius to the vmax using what we found from problem #3. \[M = \frac{\sigma ^2 R}{G}\] Thus \[\sigma = (\frac{GM}{R})^{\frac{1}{2}} \thickapprox v_{max}\]

(b) To proceed from here, you need some handy observational facts. First, all spiral galaxies have a similar disk surface brightnesses \(I = \frac{L}{R^2}\) Freeman’s Law. Second, they also have similar total mass-to-light ratios \(\frac{M}{L}\).

(c) Use some squiggle math (drop the constants and use \(\thickapprox \) instead of =) to find the Tully-Fisher relationship.

We will use general proportionality to find the Tully-Fisher relation. We know from part one that \[v_{max}^2 \propto \frac{M}{R} \] We know from Freeman's law that \[R \propto L^{\frac{1}{2}}\]We also know that \(M \propto L\) from the mass-to-light ratio. Plugging in this info, we get \[v_{max}^2 \propto \frac{L}{L^{\frac{1}{2}}}\]If we square this, we get to the Tully-Fisher relation \[L \propto v^4\]

(d) It turns out the Tully-Fisher Relation is so well-obeyed that it can be used as a standard candle, just like the Cepheids and Supernova Ia you saw in the last worksheet. In the B-band \(λ_{cen} \thickapprox 445 nm \) blue light, this relation is approximately: \[M_B = -10 \, log(\frac{v_{max}}{km/s}) + 3 \]

Suppose you observe a spiral galaxy with apparent, extinction-corrected magnitude B = 13 mag. You perform longslit optical spectroscopy (ask a TF what that is), obtaining a maximum rotational velocity of 400 km/s for this galaxy. How distant do you infer this spiral galaxy to be?

We first solve for \(M_B \) and obtain \[M_B = -10 \, log(\frac{400}{km/s}) + 3 = -23.02\]We then use the distance modulus equation we obtained ob a previous worksheet which we gives us the relationship of distance to apparent and absolute magnitude. \[d = 10^{\frac{m - M_B + 5}{5}} \]giving us \[d = 10^{\frac{13 + 23 + 5}{5}} \]and a final answer of \[d = 10^{8.2} \, parsecs = 1.58 x 10^8 \, parsecs \]