<p>So I know this is a loose proof and a lot of assumptions are made, but I do not understand how these assumptions lead to the quantities we obtain so I will be asking questions at each step:</p>
<p>start with the Virial theorem <span class="math-container">$$2T = U$$</span> then define <span class="math-container">$$T = \frac{1}{2} M_{gal} V^2_{characteristic} , U = \frac{-G M_{gal}^2}{R_{gal}}$$</span></p>
<ul>
<li>what is <span class="math-container">$V_{characteristic}$</span> the velocity of ?</li>
<li>why is the potential defined this way? I know for gravitational potential we have <span class="math-container">$U = \frac{-G m_1 m_2}{R_{gal}}$</span> but why is <span class="math-container">$m_1 =m_2$</span> here? what does this mean physically?</li>
</ul>
<p>later in the proof a surface brightness is defined if we were to take &quot;all spirals have the same surface brightness at their centers&quot; to be <span class="math-container">$I_0 = \frac{L}{\pi R_{gal}^2}$</span></p>
<ul>
<li>How was the relationship between the luminosity and surface brightness obtained ? and how does our assumption get us to this relation?</li>
<li>I am assuming <span class="math-container">$R_{gal}$</span> is the radius of the galaxy , why is it used in this relation?</li>
</ul>

CC BY-SA 4.0
2022-10-11T16:46:15.483Z
Where is this proof you’re quoting from?
Peter Erwin
CC BY-SA 4.0
2022-10-11T21:46:44.157Z
can be found here : http://user.astro.columbia.edu/~gbryan/G9001_2004/g9001_notes4.pdf
realanswers
<galaxy><velocity><definition>
Deriving the Tully-Fisher relation from the virial theorem
realanswers
# Deriving the Tully-Fisher relation from the virial theorem

(Asked by: username_0 on 2022-10-10)

So I know this is a loose proof and a lot of assumptions are made, but I do not understand how these assumptions lead to the quantities we obtain so I will be asking questions at each step:

start with the Virial theorem $$2T = U$$ then define $$T = \frac{1}{2} M\_{gal} V^2\_{characteristic} , U = \frac{-G M\_{gal}^2}{R\_{gal}}$$

* what is $V\_{characteristic}$ the velocity of ?
* why is the potential defined this way? I know for gravitational potential we have $U = \frac{-G m\_1 m\_2}{R\_{gal}}$ but why is $m\_1 =m\_2$ here? what does this mean physically?

later in the proof a surface brightness is defined if we were to take "all spirals have the same surface brightness at their centers" to be $I\_0 = \frac{L}{\pi R\_{gal}^2}$

* How was the relationship between the luminosity and surface brightness obtained ? and how does our assumption get us to this relation?
* I am assuming $R\_{gal}$ is the radius of the galaxy , why is it used in this relation?

**Tags:** galaxy, velocity, definition

> **Comments:**
> **username_1**: Where is this proof you’re quoting from?  
> **username_0**: can be found here : http://user.astro.columbia.edu/~gbryan/G9001_2004/g9001_notes4.pdf  

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## 0 answers

