Squiggle Math……………sure, why not

We are bounded in a nutshell of Infinite space: Blog Post #18, Worksheet # 6.1, Problem #4: Squiggle Math……………sure, why not

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 \(v_{max}\) 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 \(v_{max} \sim \sigma\)(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?

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

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

(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 (\(\lambda_{cen} \sim 445 nm\) , 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?

(a) As established in the previous problem, “in observational astronomy, finding the velocity of any one, singular, small point, is both incredibly difficult, nigh impossible, as well as inefficient. It is much more useful to measure the combined velocity scatter of a system and use this value interchangeably in velocity equation to describe the total velocity of a large enough system.” And under this understanding, we can see why \(\sigma \) makes a good approximation for \(v_{max}\) for any large/diverse enough system. And now substituting \(v_{max}\) back into the final iteration of the virial theorem to find the relation of speed and mass , we get:   \[ v_{max}^2 \approx \frac{G (M) }{R} , \] taking away the constants, \[ v_{max}^2 \approx \frac{M}{R} , \] indicating a proportional relationship to M, and an inversely proportional relationship to R.

(b) This step merely establishes the proportions we will need in a bit. But taking the opportunity, we’ll explain the nature of squiggle math. This process bases itself not on actual numbers and cannot be used to find a definitive answer for any problem, rather it just states proportions and says one thing changes as the other changes, essentially. So if the problem describes how \[ \langle I \rangle = L/R^2 , \] so \[\frac{L}{R^2}\] and this is a proportion. In squiggle math, this would be represented by \[ L \sim R^2\] indicating the existence of this proportion. Another ratio given is (M/L), which means \(\frac{M}{L}\) which in squiggle math is \[M \sim L\]. These relationships are useful to understand the general physics of what is occurring in a problem, taking us to understand some key relationships in nature in very simple terms.

(c) Now, we can start using squiggle math to find the Tully-Fisher Relation, a simple representation of how two distinct physical properties of galaxies interact. Using what we have from part (a) and (b), we have: \[ v_{max}^2 \approx \frac{M}{R} , \] \[ L \sim R^2 , \] \[M \sim L ,\] and now these can be rearranged and combined. \[ v_{max}^2 \sim \frac{M}{R} , \] \[ v_{max}^2 R \sim M , \] \[ (M \sim v_{max}^2 R)^2 , \] \[ (M)^2 \sim v_{max}^4 R^2 , \]  which makes it easily changeable with a bit a squiggle math and the equations we had found just before, we have: \[ (L)^2 \sim v_{max}^4 L , \]  \[ L \sim v_{max}^4  , \]  which is exactly the Tully-Fisher Relation.

(d) For this problem, the Tully-Fisher Relation was already expanded on for us, and made into an equation that gives the absolute magnitude of a galaxy moving at a particular speed for a particular wavelength of light ( 445 nm in this case). This equation is: \[M_B = -10 \log\left(\frac{v_{max}}{km/s}\right) +3 , \] and using this with the distance modulus we used previously in Cepheid and Supernovae problems (http://ay17-rcordova.blogspot.com/2015/10/close-star-distant-star-near-starthat.html) \[M_{absolute} = M_{apparent} - 5 \times (\log_{10} (Distance )-1 ) ,\] we can find the distance of the galaxy once given the apparent magnitude, which is 13 in this problem.

Substituting the known value of 400 km/s rotational velocity, we have: \[M_B = -10 \log\left(\frac{400 km/s}{km/s}\right) +3 , \] \[M_B = -10 \log(400) +3 , \] \[M_B = - 23. \] Now that we have the Absolute and apparent magnitudes, we can solve for the distance to this galaxy. \[M_{absolute} = M_{apparent} - 5 \times (\log_{10} (D)-1 ) ,\] rearrange to solve for D, \[ 5 \log_{10} (D)- 5  = M_{apparent} - M_{absolute},\] \[  \log_{10} (D)  =\frac{ M_{apparent} - M_{absolute} + 5}{5},\] \[  D  =10^{\frac{ M_{apparent} - M_{absolute} + 5}{5}},\] and start plugging in: \[  D  =10^{\frac{ 13 - (-23) + 5}{5}},\]\[  D  =10^{8.2} ,\] \[  D  = 1.58 \times 10^8 pc ,\] \[  D  = 1.58 \times 10^5 kpc. \] The actual distance to the Spiral Galaxy.