Blackbody Radiation: Solving Introduction to Cosmology Eq. #25

[4everphysics]

Hello. I am trying to study "Introduction to Cosmology" by Barbara Ryden,
but I am stuck with an equation from chapter two, and I have no idea how to figure out this. If you can suggest me a reading material or can explain the equation to me, that would be wonderful.

It is page 20 of the book, chapter 2, equation # 25.

It says:

The energy density of photons in the frequency range f -> f + df is given by the blackbody function

ε(f) df = ((8*∏*h)/(c^3)) ((f^3 * df)/(exp(hf/kT) -1))
and what is that exp?

then it goes...
The peak in the blackbody function occurs at hf_peak =(approx) 2.82kT. Integrated over all frequencies, equation #25 yields a total energy density for blackbody radiation of
ε_γ = α * T^4, ( I wrote "_" to mean subscript).
where
α = ((∏^2)/(15))((k^4)(h^3 * c^3)) <this h is actually h with the dash

the definition of 'α' and the equation#25 just came out of nowhere, and I am very clueless.
And they don't look anything like the blackbody equation that I know:
P_rad = σεAT^4 where ε is emissivity and σ is the Stefan-Boltzmann constant.

Thank you for your help..


Kyle Lee

 

[Nabeshin]

Hello. I am trying to study "Introduction to Cosmology" by Barbara Ryden,
but I am stuck with an equation from chapter two, and I have no idea how to figure out this. If you can suggest me a reading material or can explain the equation to me, that would be wonderful.

It is page 20 of the book, chapter 2, equation # 25.

It says:

The energy density of photons in the frequency range f -> f + df is given by the blackbody function

ε(f) df = ((8*∏*h)/(c^3)) ((f^3 * df)/(exp(hf/kT) -1))
and what is that exp?

Exp is the exponential function, i.e. \exp ( h f/kT ) = e^{hf/kT}

then it goes...
The peak in the blackbody function occurs at hf_peak =(approx) 2.82kT. Integrated over all frequencies, equation #25 yields a total energy density for blackbody radiation of
ε_γ = α * T^4, ( I wrote "_" to mean subscript).
where
α = ((∏^2)/(15))((k^4)(h^3 * c^3)) <this h is actually h with the dash

the definition of 'α' and the equation#25 just came out of nowhere, and I am very clueless.
And they don't look anything like the blackbody equation that I know:
P_rad = σεAT^4 where ε is emissivity and σ is the Stefan-Boltzmann constant.

Thank you for your help..


Kyle Lee

The definition of constants can be a little confusing, but they're just that: constants. Emissivity is taken to be one (perfect black body, after all), so integrating over all frequencies you just get the expression Ryden quotes. You're welcome to do it yourself, if you like.

These are the most basic property of black bodies, have you studied them at all before? This should be pre-requisite knowledge for Ryden's book.

 

[4everphysics]

Thank you so much for your help.
I only have finished Halliday Resnick Walker's Fundamentals of Physics.
I am not sure if that is enough for the prereq? I sure have never seen
the energy density expression in HRW though..

so do you mean if I integrate the element "ε(f) df" over all the frequency,
I get the equation P_rad = σεAT^4?

Thank you for your help.
Sincerely

 

[clamtrox]

so do you mean if I integrate the element "ε(f) df" over all the frequency,
I get the equation P_rad = σεAT^4?

That's right. The integral is slightly involved, but you should be able to do it by expanding 1/(1-exp(-hf/kT)) (note the minus sign in the exponential, it's not a typo) as a geometric series, integrating each term of the series separately, and then you probably need to look up some value for the Riemann zeta function

 

[Nabeshin]

Thank you so much for your help.
I only have finished Halliday Resnick Walker's Fundamentals of Physics.
I am not sure if that is enough for the prereq? I sure have never seen
the energy density expression in HRW though..

so do you mean if I integrate the element "ε(f) df" over all the frequency,
I get the equation P_rad = σεAT^4?

Thank you for your help.
Sincerely

It's important to note that the two constants \sigma and a are related, but not the same. Specifically, a= \frac{\sigma}{4 c}.

In Ryden, what she's using is the energy density integrated over all solid angles. A lot of the time we instead use the power per solid angle in a frequency interval instead, so it's easy to get confused here. So, you will not get the familiar expression P= \sigma T^4, since you're not integrating a power at all, but rather an energy density. But again, if you do the integral with the zeta functions and all, you will get all the right factors and reproduce Ryden's result.