Finding a constant of proportionality (Astro)

[lquinnl]

Homework Statement

OK so I'm doing a past exam paper as some revision:

The central galaxy in the Perseus cluster has an X-ray spectrum in wavelength units
which is well described by the power law

F_\lambda \propto \lambda^{-2} .

If the spectrum in frequency units is described by

F_v \propto v^\beta,

calculate \beta .

Homework Equations

Obviously we have v = \frac{c}{\lambda}

The Attempt at a Solution

so if v = \frac{c}{\lambda}

then does that mean that \beta = -3

I feel like I am really missing the point here. Any help would be greatly appreciated.

Thanks in advance!

 

[voko]

How did you get -3?

 

[lquinnl]

Because lambda is to the power of -2 and lambda is related to frequency as v = c lambda^{-1} therefore adding an extra -1

does that make sense or is it total *@~% ?

 

[voko]

I really cannot follow this logic. How does this "adding" come about?

 

[lquinnl]

Because

λ^-1 λ^-2 = λ^-3

Am I on the right lines?

 

[voko]

No, you are not. You really need to think how one could go from wavelength to frequency in a formula.

 

[lquinnl]

Looking at this again I am not convinced of my reasoning. I have this:

F_\lambda \propto \lambda^{-2} .



F_v \propto v^{-\beta} .



F_v \propto v^{-\beta} \propto {\frac{c}{\lambda}}^{-\beta} .

Any chance of a poke in the right direction, if you know?

 

[voko]

You have a formula that contains \lambda^{-2}.

You need to express this formula via \nu. How would you do it?

 

[lquinnl]

You have a formula that contains \lambda^{-2}.

You need to express this formula via \nu. How would you do it?

F_\lambda \propto \lambda^{-2} \propto (v^{-1})^{-2} \propto v^{-3}

v \propto \lambda^{-1}

F_v \propto v^{\beta} \propto \lambda^{-1} \propto (v^{-3})^{-1} \propto v^{-4}

so \beta = 4

?

P.s. Thanks for all this help.

 

[voko]

F_\lambda \propto \lambda^{-2} \propto (v^{-1})^{-2}

This part is correct.

But (\nu^{-1})^{-2} is NOT \nu^{-3}. I note that you made a similar mistake in another derivation. What is the proper result of such an operation?

 

[lquinnl]

But (\nu^{-1})^{-2} is NOT \nu^{-3}. I note that you made a similar mistake in another derivation. What is the proper result of such an operation?

OF COURSE! I feel so stupid today, really need to wake up!

(v^{-1})^{-2} = v^{2}

so

F_v \propto v^{\beta} \propto \lambda^{-1} \propto (v^{2})^{-1} \propto v^{-2}

and

\beta = 2

Am I Right?

 

[lquinnl]

Or have I messed up still, regarding a sign?

 

[voko]

But (\nu^{-1})^{-2} is NOT \nu^{-3}. I note that you made a similar mistake in another derivation. What is the proper result of such an operation?

OF COURSE! I feel so stupid today, really need to wake up!

(v^{-1})^{-2} = v^{2}

This is correct. But what you do then puzzles me. You have just proved that since

\lambda^{-2} \propto \nu^{2}

we have

F \propto \lambda^{-2} \propto \nu ^{2}

 

[lquinnl]

What I'm saying is that

F_\lambda \propto \lambda^{-2} \propto v^{2}

from v = \frac{c}{\lambda} we have v \propto \lambda^{-1}

so

F_v \propto \lambda^{-1}

now i think maybe this should be how i follow:so

F_v \propto v^{\beta} \propto (\lambda^{-1})\propto(v^{2})^{-1}

 

[voko]

Again, you just got the expression in terms of v^{2}. While you know it should be in terms of v^{\beta}. What does this mean to you?

 

[lquinnl]

Again, you just got the expression in terms of v^{2}. While you know it should be in terms of v^{\beta}. What does this mean to you?

So if F_v \propto v^{\beta} \propto v^{-2}

\beta = -2

 

[voko]

Why is this MINUS 2? Just two messages above it was PLUS 2.

 

[lquinnl]

I thought it should be minus 2 from this statement:

F_v \propto v^{\beta} \propto (\lambda^{-1})\propto(v^{2})^{-1}[/QUOTE]

 

[voko]

I think you are very confused. Let me explain the whole thing to you.

When something (A) is proportional something else (B), then A = CB, where C does not contain B in any way.

In your case, you have F proportional to \lambda^{-2}, which means F = k\lambda^{-2}

You are asked to find out to what degree of \nu that is proportional. This is done by expressing \lambda via \nu, which is \lambda = c\nu^{-1}, so you end up with F = k(c\nu^{-1})^{-2} = kc^{-2}\nu^{2}, which simply means that F \propto \nu^{2} and \beta = 2.

 

[lquinnl]

THANK YOU VERY MUCH!

I feel like I have done an incredible job of over-complicating this problem!

I think I was getting confused by the fact we have F_v and F_lambda.

So it was just a case of expressing wavelength as frequency and amalgamating the powers.

Seems incredibly straightforward now!

Thanks again