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AST-18

Planck radiance B_ν

B_ν = 2 h ν³ / c² / (e^{hν/kT} − 1). Specific intensity of a blackbody.

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RadiationPlanck

Governing equation

Bν=2hν3c2(ehν/kT−1)−1B_\nu=\dfrac{2h\nu^3}{c^2}(e^{h\nu/kT}-1)^{-1}

where

\nu
Frequency (THz)
T
Temperature (K)
B_\nu
Radiance (MJy/sr)

Lecture brief

Historical brief

Hubble expansion, Jeans collapse, Eddington luminosity, Bondi accretion and Stefan–Boltzmann stars are the first astrophysical budgets. The sheets scale a star, a cloud and a horizon. This sheet (AST-18 — Planck radiance B_ν) is the form associated with Planck. Working symbols: ν\nu, TT →\rightarrow BνB_\nu. Integrating B_ν gives σ T⁴/π. The ultraviolet catastrophe dies in the exponential.

Purpose

Purpose: compute BνB_\nu from ν\nu, TT in Astrophysics via Bν=2hν3c2(ehν/kT−1)−1B_\nu=\dfrac{2h\nu^3}{c^2}(e^{h\nu/kT}-1)^{-1} B_ν = 2 h ν³ / c² / (e^{hν/kT} − 1). Specific intensity of a blackbody. Use it when a real astrophysics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given ν=500.000 THz\nu = 500.000\,\mathrm{THz}, T=5772.000 KT = 5772.000\,\mathrm{K}, the governing relation Bν=2hν3c2(ehν/kT−1)−1B_\nu=\dfrac{2h\nu^3}{c^2}(e^{h\nu/kT}-1)^{-1} yields Bν=2.930e+12 MJy/srB_\nu = 2.930e+12\,\mathrm{MJy/sr}. A family of Planck curves. Move a slider: the numbers are this situation, not a canned story.

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Inputs

Outputs

  • Radiance B_\nu2930158710127.351 MJy/sr
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AST-18 · spectrum
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Narration of this film

A family of Planck curves.

Integrating B_ν gives σ T⁴/π. The ultraviolet catastrophe dies in the exponential.

Reading speed

Watch on YouTube