# Surface Brightness: Hubble Profile | Infinite Luminosity?

• Logarythmic
In summary, the Hubble profile of surface brightness leads to an infinite total luminosity, while the law I = I_0 exp[-(r/a)^{1/4}], with a a constant, does not.

## 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?

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

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

$$I \propto L$$

but that doesn't make sense, does it?

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.

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.

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?

How do you mean?

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?

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.

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 [Broken]
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 [Broken]

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## 1. What is surface brightness?

Surface brightness is a measure of the amount of light coming from a given area on the surface of an object. It is typically expressed in units of magnitude per square arcsecond.

## 2. How is surface brightness related to the Hubble profile?

The Hubble profile is a mathematical model used to describe the surface brightness distribution of galaxies. It follows a specific mathematical formula and is often used to study the structure and properties of galaxies.

## 3. What is the concept of infinite luminosity in relation to surface brightness?

Infinite luminosity refers to the idea that the surface brightness of an object would continue to increase indefinitely as you move closer to its center. This is a theoretical concept and does not exist in reality, but it helps us understand the behavior of surface brightness profiles.

## 4. How does surface brightness affect our understanding of galaxies?

Surface brightness is an important factor in studying the structure and properties of galaxies. It can reveal information about their age, size, and mass distribution. It also allows us to compare and classify different types of galaxies.

## 5. Can surface brightness be used to measure the distance to a galaxy?

Yes, surface brightness can be used as one of the methods to estimate the distance to a galaxy. This is because the apparent brightness of an object decreases with distance, and by measuring its surface brightness, we can calculate its distance based on the inverse square law of light.