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Homework Help: Planck function expressed in Wavenumber?

  1. Nov 19, 2009 #1
    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?


    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. jcsd
  3. Nov 19, 2009 #2
    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 [itex]r[/itex] by a portion of the surface of the sphere whose area, [itex]A[/itex], equals [itex]r^2[/itex] "

    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
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