Tolman Surface Brightness Test -

[_Andreas]

Tolman Surface Brightness Test -- need help!

Richard Tolman derived that in an expanding universe with any arbitrary geometry, the surface brightness of a set of identical objects will decrease by (1+z)^4.

I have two questions:

1) The surface brightness is the same as the flux (W/m^2), right?
2) My understanding so far has been that the received flux decreases with the distance squared, so reading that it will decrease by a factor of (1+z)^4 is a bit confusing. Is it perhaps a factor that is ignored when the distance from Earth is relatively small (because z is so small)?

Need help urgently!

 

[_Andreas]

Oh, and another question. Some sources say the SB will decrease by (1+z)^2 instead of (1+z)^4 in an expanding universe. Which is correct?

 

[_Andreas]

I posted this in the astrophysics section yesterday, but I think it's more of a cosmological question, so I decided to post it here too.

 

[_Andreas]

I've received answers to the first two questions already, so no need for anyone to bother with them. The question in my second post is still unanswered, though.

 

[hellfire]

The surface brightness is:

$$S = \frac{f}{\pi \delta^2}$$

The angular diameter $\delta$ is defined:

$$\delta = \frac{D}{d_A}$$

With $d_A$ the angular diameter distance. Thus, for the surface brightness you get:

$$S = \frac{f d^2_A}{\pi D^2}$$

The angular diameter distance relates to the luminosity distance as $d_A = d_L / (1+z)^2}$:

$$S = \frac{f d^2_L}{\pi D^2 (1+z)^4}$$

And the measured bolometric flux is:

$$f = \frac{L}{4 \pi d_L^2}$$

Finally:

$$S = \frac{L}{4 \pi^2 D^2 (1+z)^4}$$

Which is equal to:

$$S(z) = \frac{S(z = 0)}{(1+z)^4}$$

So the decrease is with the fourth power.

 

[_Andreas]

Thanks!

What I'm trying to do is getting a grasp of the basics of a Tolman SB test. Please correct me if I'm wrong:

To be able to do a Tolman SB test you first have to observe a couple of identical objects at different redshifts. "Identical" here means objects of the same spectral type and luminosity class. Then you measure their apparent areas and their apparent magnitudes, and from these two parameters you get their surface brightnesses. Now, if the universe is expanding, the surface brightness of an object at, say, z = 0,4 should differ by a factor of 1,4^4 to an identical one at z~0 (that is, the ratio of SB (z~0) to SB (z = 0,4) should equal approx. 1,4^4).

But all of the above is of course highly simplified. For example, the observed SB isn't the "real" SB, correct? To find the "real" SB you have to correct for, among other things, interstellar extinction and interstellar reddening.

Any thoughts?

 

[_Andreas]

"Then you measure their apparent areas and their apparent magnitudes, and from these two parameters you get their surface brightnesses."

I'm thinking of the relation S=m+2,5logA here, where A is the apparent area.

 

[_Andreas]

Since I don't know how to contact a moderator, I have to say it here instead: I'd really, really like some feedback on my two latest posts, so I've decided to move them to the "Homework&Coursework Questions" section. You're free to close this thread.

 

[hellfire]

I don't think interstellar extinction is a real problem for this test, because the models of interstellar medium are pretty accurate and should make it possible to estimate it. However, there must be other problems especially like the definition and measurement of surface brightness for which a radius has to be identified and defined for the whole sample of galaxies. I think that this issue should make it very difficult to handle with the resolution at high redshifts as well as possible evolution of galaxy sizes and surface brightness and therefore definition of the sample, etc. Googling for Tolman surface brightness test I have seen some arxiv entries of published papers. These may give you a feeling about the real problems.

 

[_Andreas]

Why would the evolution of galaxy sizes matter?

 

[hellfire]

The variation of surface brightness with $1/(1+z)^4$ assumes objects that are equal in luminosity and size. If the size or the luminosity of the galaxy sample vary with redshift, then there is an additional component for the variation of the surface brightness with $z$.