AST-05

Stefan–Boltzmann luminosity

L = 4π R² σ T⁴. Blackbody luminosity of a star.

Reading speed

RadiationStefan–Boltzmann

Governing equation

L=4\pi R^2\sigma T^4

$T \in \left[2500, 40000\right]$

where

R: Radius (R_\odot)
T: Temperature (K)
L: Luminosity (L_\odot)

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.

00:0 / 00:08

①

Governing law — red

L=4\pi R^2\sigma T^4

$T \in \left[2500, 40000\right]$

$T = 2500$ $L = 0.035$

Along the curve

②

Domain of the variable — blue

T_{\min} = 2500,\quad T_{\max} = 40000

$T \in \left[2500, 40000\right]$

$T = 2500$ $T = 2500$

Along the curve

③

Response curve — green

L=4\pi R^2\sigma T^4

$T \in \left[2500, 40000\right]$

$T = 2500$ $L = 0.035$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}L}{\mathrm{d}T}

$T \in \left[2500, 40000\right]$

$T = 2735.8$ $L^{\prime} = 7.43\times 10^{-5}$

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-05 — Stefan–Boltzmann luminosity) is the form associated with Stefan–Boltzmann. Working symbols: $R$ , $T$ $\rightarrow$ $L$ . σ = 5.67×10⁻⁸ W/m²K⁴. The HR diagram's main sequence is this plus structure.

Purpose

Purpose: compute

L

from

R

T

in Astrophysics via

L=4\pi R^2\sigma T^4

L = 4π R² σ T⁴. Blackbody luminosity of a star. Use it when a real astrophysics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

R = 1.000\,\mathrm{R_odot}

T = 5772.000\,\mathrm{K}

, the governing relation

L=4\pi R^2\sigma T^4

yields

L = 1.000\,\mathrm{L_odot}

. A glowing sphere, an L bar. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Radius R1 R_\odotTemperature T5772 K

Outputs

Luminosity L1.000 L_\odot

Reading speed

Watch on YouTube

Free library

Full library

Free PDF / open book

YouTube channels

00:0 / 00:08

Narration of this film

A glowing sphere, an L bar.

σ = 5.67×10⁻⁸ W/m²K⁴. The HR diagram's main sequence is this plus structure.

Reading speed

Watch on YouTube