Tully-Fisher: Blog 18, Worksheet 6.1, Problem 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: \[L \sim v_{max}^4 \], 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:

Source

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

The \(v_{max} \sim σ \) assumption is a pretty safe one to make because σ is basically the width of the possible velocities distribution. It has the same units as v_{max} and will always be off just by a constant.

In the previous problem, we derived \[M \approx \frac{σ^2R}{ G}\] a relationship that can be rearranged to show \[ v_{max}^2 \sim σ^2 \approx \frac{GM}{ R} \] So as v increases as mass increases and radius decreases.

(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 (M/L).

This part is really just setting up definitions for part c: \[ <I> =   \frac{L}{R^2} \] can be rearranged to \[ R = \left(\frac{<I>}{L} \right)^{\frac{1}{2}} \] and \[ x = \frac{M}{L}\] which is \[ M = \frac{x}{L}\]

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

\[v_{max}^2 \sim \frac{M}{R} = \frac{ \frac{x}{L}}{\left(\frac{<I>}{L} \right)^{\frac{1}{2}}}= x <I>^{\frac{1}{2}} L^{\frac{1}{2}} \] Dropping the constants gives: \[L \sim v_{max}^4 \]

(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 (λcen ~ 445 nm, blue light), this relation is approximately: \[M_B = -10 log \left( \frac{v_{max}}{ km/s} \right) + 3\] 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?
\[M_B = -10 log \left( \frac{v_{max}}{ km/s} \right) + 3\] \[M_B = -10 log \left( \frac{400}{ km/s} \right) + 3\] \[M_B \approx -23 \] We know the distance formula from last week and can use it to find our answers: \[ d = 10^{\frac{m-M + 5}{5}} \] \[ d = 10^{\frac{13 - (-23) + 5}{5}} = 1.6 \times 10^8 \: pc \]