# Planck function expressed in Wavenumber?

1. Nov 19, 2009

### nordmoon

1. The problem statement, all variables and given/known data

I am checking some equations for my simulation and are looking at the Planck function. My question involves the constant used for the Planck function expressed in Wavenumbers. I have found this expression for the function (http://pds-atmospheres.nmsu.edu/education_and_outreach/encyclopedia/planck_function.htm) but I don't understand where the unit comes from for the first Planck constant alpha1 = 2 h c^2 = 1.191066 · 10^-5 mW · m-2 · steradian-1/cm-4? I would like to understand this before I use it. There is another expression for the first Planck constant including pi but has a different unit.

I guess I am looking for a Planck function expression which uses wavenumbers, but it would be nice to see how alpha 1 is obtained.

2. Relevant equations

Planck function dependent on wavenumber w (cm-1)

B(T,w) = (alpha1 w^3)/[exp(alpha2 w/T) - 1]

alpha1 = 2 h c^2 = 1.191066 · 10^-5 mW · m-2 · steradian-1/cm-4 <-- how?

when

h is Planck's constant (6.62620 · 10^-34 Joule second)
c is the speed of light (2.99793 · 10^8 m/second)

I have found another source that

alpha1 = 2 pi h c^2 = 3.741 771 18(19) × 10−16 W·m²

3. The attempt at a solution

Putting in the numbers I obtain:

alpha1 = 2 h c^2 = 1.19107 * 10 ^-12 W/cm^2 = 1.19107 * 10 ^-16 W/m^2

This is not, 1.191066 · 10^-5 mW · m-2 · steradian-1/cm-4 .. where does the unit steradian-1/cm-4 come from?

I might add that I did find a few papers using this number, C1 = 1.191062£10^-12 W cm^2 /sr, but it still does not explain the origin of the two unit values. (http://www.iop.org/EJ/article/1674-...quest-id=2a611551-3e23-4cae-9cb4-58c619d41cbe) I couldn't find the original when looking at references..

Last edited by a moderator: Apr 24, 2017
2. Nov 19, 2009

### jdwood983

From my astronomy sources, " A steradian is defined as the http://en.wikipedia.org/wiki/Solid_angle" [Broken] subtended at the center of a sphere of radius $r$ by a portion of the surface of the sphere whose area, $A$, equals $r^2$ "

Steradians are commonly used in astronomy and astrophysics (where Planck's constant is frequently used as well). Outside of astronomy/astrophysics, I do not know where else one might find the unit used.

Last edited by a moderator: May 4, 2017