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

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

$\nu \in \left[1, 2000\right]$

where

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

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

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

$\nu \in \left[1, 2000\right]$

$\nu = 1$ $B_\nu = 1.77\times 10^{8}$

Along the curve

②

Domain of the variable — blue

\nu_{\min} = 1,\quad \nu_{\max} = 2000

$\nu \in \left[1, 2000\right]$

$\nu = 1$ $\nu = 1$

Along the curve

③

Response curve — green

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

$\nu \in \left[1, 2000\right]$

$\nu = 1$ $B_\nu = 1.77\times 10^{8}$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}B_\nu}{\mathrm{d}\nu}

$\nu \in \left[1, 2000\right]$

$\nu = 13.57$ $B_\nu^{\prime} = 4.31\times 10^{9}$

Along the curve

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: $\nu$ , $T$ $\rightarrow$ $B_\nu$ . Integrating B_ν gives σ T⁴/π. The ultraviolet catastrophe dies in the exponential.

Purpose

Purpose: compute

B_\nu

from

\nu

T

in Astrophysics via

B_\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

\nu = 500.000\,\mathrm{THz}

T = 5772.000\,\mathrm{K}

, the governing relation

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

yields

B_\nu = 2.930e+12\,\mathrm{MJy/sr}

. A family of Planck curves. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Frequency \nu500 THzTemperature T5772 K

Outputs

Radiance B_\nu2930158710127.351 MJy/sr

Reading speed

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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