<p>I am not an expert, but it seems that the fundamental plane is a relation among characteristic quantities of the galaxy, showing a correlation that is the analogous of the Tully-Fisher relation for spiral galaxies.</p>

<p>You can start from the virial theorem: 
$\frac{M}{R}\sim v^2$</p>

<p>($M$ mass contained in the radius $R$, $v$ velocity dispersion of the stars, also indicated with $\sigma$ in these cases)
meaning that the stars behave like an isothermal sphere.</p>

<p>Assuming that all galaxies have the same mass-to-light ratio
$L/M$, and that all galaxies have the same surface brightness</p>

<p>$\Sigma=L/R^2$</p>

<p>(with $L$ luminosity), you obtain the Tully-Fisher relation:</p>

<p>$L\sim v^4$</p>

<p>Of course, all galaxies do not have the same surface brightness.
So if you take $\Sigma=L/R^2$ and substitute that into the virial theorem (keeping the mass-to-light ratio assumption), then a relation between luminosity, surface brightness, and velocity dispersion is obtained:</p>

<p>$L \sim \sigma^4 \Sigma^{-1}$</p>

<p>You can plot the stars distributions in elliptical galaxy according to this relation, and will note that the stars do not distribute randomly, but are more concentrated along the fundamental plane (more or less the same happens for the HR diagram):</p>

<p><img src="https://i.sstatic.net/y7Qrg.gif" alt="enter image description here"></p>

<p>(Here $R_e$ is the so called <a href="http://en.wikipedia.org/wiki/Effective_radius">effective radius</a>).</p>

<p>For completeness, another illuminating picture from <a href="http://www.astro.virginia.edu/class/whittle/astr553/Topic07/t7_FP_projections.html">here</a>:</p>

<p><img src="https://i.sstatic.net/vuvdn.gif" alt="enter image description here">
References <a href="http://www2.astro.psu.edu/users/rbc/a480/lec10n.pdf">here</a> and <a href="http://star-www.st-and.ac.uk/~spd3/Teaching/AS3011/AS3011_6.pdf">here</a>.</p>

CC BY-SA 3.0
2014-09-24T03:08:55.717Z
Thank you! You've been a star explaining this. Especially the first diagram. Really helped me visualize the fundamental plane better!
Hari Sheldon
CC BY-SA 3.0
2014-09-24T03:15:50.577Z
@HariSheldon, you are very welcome. I am happy that the answer worked. If you like, please accept the answer.
Py-ser
CC BY-SA 3.0
2014-09-23T11:21:23.490Z
2014-09-23T11:26:31.677Z
2014-09-23T11:26:31.677Z
Py-ser
<p>An alternate explanation rather than from wikipedia is preferable.</p>

<galaxy><astrophysics><luminosity>
What is the Fundamental plane for Elliptical Galaxies?
Hari Sheldon
# What is the Fundamental plane for Elliptical Galaxies?

(Asked by: username_0 on 2014-09-23)

An alternate explanation rather than from wikipedia is preferable.

**Tags:** galaxy, astrophysics, luminosity

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## 1 answers

### Answer by username_1 on 2014-09-23. Score: 9 | Accepted answer

I am not an expert, but it seems that the fundamental plane is a relation among characteristic quantities of the galaxy, showing a correlation that is the analogous of the Tully-Fisher relation for spiral galaxies.

You can start from the virial theorem:
$\frac{M}{R}\sim v^2$

($M$ mass contained in the radius $R$, $v$ velocity dispersion of the stars, also indicated with $\sigma$ in these cases)
meaning that the stars behave like an isothermal sphere.

Assuming that all galaxies have the same mass-to-light ratio
$L/M$, and that all galaxies have the same surface brightness

$\Sigma=L/R^2$

(with $L$ luminosity), you obtain the Tully-Fisher relation:

$L\sim v^4$

Of course, all galaxies do not have the same surface brightness.
So if you take $\Sigma=L/R^2$ and substitute that into the virial theorem (keeping the mass-to-light ratio assumption), then a relation between luminosity, surface brightness, and velocity dispersion is obtained:

$L \sim \sigma^4 \Sigma^{-1}$

You can plot the stars distributions in elliptical galaxy according to this relation, and will note that the stars do not distribute randomly, but are more concentrated along the fundamental plane (more or less the same happens for the HR diagram):

![enter image description here](https://i.sstatic.net/y7Qrg.gif)

(Here $R\_e$ is the so called [effective radius](http://en.wikipedia.org/wiki/Effective_radius)).

For completeness, another illuminating picture from [here](http://www.astro.virginia.edu/class/whittle/astr553/Topic07/t7_FP_projections.html):

![enter image description here](https://i.sstatic.net/vuvdn.gif)
References [here](http://www2.astro.psu.edu/users/rbc/a480/lec10n.pdf) and [here](http://star-www.st-and.ac.uk/~spd3/Teaching/AS3011/AS3011_6.pdf).

> **Comments:**
> **username_0**: Thank you! You've been a star explaining this. Especially the first diagram. Really helped me visualize the fundamental plane better!  
> **username_1**: @HariSheldon, you are very welcome. I am happy that the answer worked. If you like, please accept the answer.  

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