Surface Brightness: Hubble Profile | Infinite Luminosity?

[Logarythmic]

Homework Statement

Show that the Hubble profile of surface brightness

I(r) = I_0 \left(1+\frac{r}{R}\right)^{-2}

leads to an infinite total luminosity, while the law

I = I_0 exp[-(r/a)^{1/4}],

with a a constant, does not.
Here I_0 and R are constants and r is the distance from the centre. The scale length R is typically around 1 kpc.2. The attempt at a solution
I have no clue. How can I relate total luminosity with surface brightness?

 

[JeffKoch]

I'd suggest starting with units. What are the units of surface brightness and luminosity?

 

[Logarythmic]

Well, I think surface brightness is luminosity per arcsecond squared so

I \propto L

but that doesn't make sense, does it?

 

[JeffKoch]

What are the units of luminosity, and what are the units of surface brightness? Thinking about that may help you see how they're related.

 

[Logarythmic]

But what IS total luminosity? Absolute luminosity is energy per second and apparent luminosity is energy per second per unit area. I don't know if it should be the total luminosity emitted or the total luminosity observed.

 

[JeffKoch]

I would read the question as asking about the total power radiated into all space. From the units, looks like you need to integrate out the area, yes?

 

[Logarythmic]

How do you mean?

 

[JeffKoch]

If I understand the question, the apparent "surface" brightness must have units of power per unit area per steradian. This is a line integral through the volume, so assuming spherical symmetry an Abel transform gives you the local emissivity in units of power per unit volume per steradian. An integral over all space and angles gives you the luminosity in units of power. Does this help?

 

[Logarythmic]

The book mentions nothing about Abel transforms and I have never worked with those so I don't think that is the way to do it.

 

[JeffKoch]

O.K., perhaps I'm not understanding the question, the assumptions and equations provided to you, or the level of detail expected in your answer.

Some other background that might help is here:
http://www.ph.qmul.ac.uk/pog/chapters/PoG%20Chapter%202%20v3.03_2005.pdf
http://www.astro.rug.nl/~ahelmi/galaxies_course/class_VII-E/ellip-06.pdf
http://www.journals.uchicago.edu/AJ/journal/issues/v122n4/201253/201253.text.html