Astrophysics Radiative Transfer & Radiometry Cheat Sheet

1. Terminology Cross-Reference: Astrophysics vs. Standard Radiometry

A common source of confusion is the clash between traditional engineering radiometry terms and astronomical conventions.

Quantity (Astrophysics)	Symbol	SI Units	Radiometry (SI / Optics) Term	Notes
Specific Intensity	$I_{ν}$	$W \cdot m^{- 2} \cdot {sr}^{- 1} \cdot {Hz}^{- 1}$	Spectral Radiance ( $L_{ν}$ )	Invariant along a ray in vacuum; represents surface brightness.
Integrated Intensity	$I$	$W \cdot m^{- 2} \cdot {sr}^{- 1}$	Radiance ( $L$ )	$I = \int_{0}^{\infty} I_{ν} d ν$ .
Monochromatic Flux	$F_{ν}$	$W \cdot m^{- 2} \cdot {Hz}^{- 1}$	Spectral Irradiance / Radiant Exitance	Net energy per unit area per unit frequency: $F_{ν} = \int I_{ν} \cos θ d Ω$ .
Bolometric Flux	$F$	$W \cdot m^{- 2}$	Irradiance / Radiant Exitance ( $E, M$ )	Total flux over all frequencies ( $F = σ T^{4}$ for a blackbody)[cite: 1]. Follows $1 / d^{2}$ at distance $d$ .
Luminosity	$L$	$W$ (or $erg \cdot s^{- 1}$ )	Radiant Power / Radiant Flux ( $Φ$ )	Total intrinsic energy emitted per unit time ( $L = 4 π R^{2} σ T^{4}$ ).
Specific Energy Density	$U_{ν}$ [cite: 1]	$J \cdot m^{- 3} \cdot {Hz}^{- 1}$ [cite: 1]	Spectral Energy Density	Energy per unit volume per frequency: $U_{ν} = \frac{1}{c} \int_{4 π} I_{ν} d Ω$ .
Total Energy Density	$U$ [cite: 1]	$J \cdot m^{- 3}$ [cite: 1]	Radiant Energy Density ( $w, u$ )	$U = \int_{0}^{\infty} U_{ν} d ν = a_{B} T^{4}$ [cite: 1].

Caution: In standard radiometry, "Radiant Intensity" ( $I_{e, Ω}$ ) is defined as power per unit solid angle ( $W \cdot {sr}^{- 1}$ ) for point sources. In astrophysics, "Intensity" almost always refers to Specific Intensity (power per unit area per unit solid angle).

2. Local Medium Parameters (Emission & Absorption)

Quantity	Symbol	SI Units	Mathematical Definition / Meaning
Volume Emissivity	$J_{ν}$ [cite: 2]	$W \cdot m^{- 3} \cdot {Hz}^{- 1}$ [cite: 2]	Total energy emitted per unit time, volume, and frequency across all $4 π$ steradians[cite: 2].
Emission Coefficient	$j_{ν}$ [cite: 2]	$W \cdot m^{- 3} \cdot {sr}^{- 1} \cdot {Hz}^{- 1}$ [cite: 2]	Power per unit volume, frequency, and solid angle[cite: 2]. If isotropic: $j_{ν} = \frac{J_{ν}}{4 π}$ .
Linear Absorption Coefficient	$α_{ν}$ [cite: 2]	$m^{- 1}$	Attenuation rate per meter along path $s$ : $d I_{ν} = - α_{ν} I_{ν} d s$ [cite: 2].
Mass Absorption Coefficient (Opacity)	$κ_{ν}$	$m^{2} \cdot {kg}^{- 1}$	Opacity per unit mass: $α_{ν} = ρ κ_{ν}$ (where $ρ$ is mass density).
Optical Depth	$τ_{ν}$	Dimensionless	Integrated opacity along line of sight: $d τ_{ν} = α_{ν} d s$ .
Source Function	$S_{ν}$	$W \cdot m^{- 2} \cdot {sr}^{- 1} \cdot {Hz}^{- 1}$	Ratio $S_{ν} \equiv \frac{j_{ν}}{α_{ν}}$ . In LTE, $S_{ν} = B_{ν} (T)$ (Planck function).

3. Fundamental Governing Equations

Equation of Radiative Transfer

\frac{d I_{ν}}{d s} = j_{ν} - α_{ν} I_{ν} ⟺ \frac{d I_{ν}}{d τ_{ν}} = S_{ν} - I_{ν} [cite: 2]

Optically thin without absorption ( $α_{ν} = 0$ ):[cite: 2] $I_{ν} = \int_{path} j_{ν} d s = \frac{J_{ν}}{4 π} L$
Optically thick in thermal equilibrium (LTE): $I_{ν} \to S_{ν} = B_{ν} (T)$

Isotropic Radiation Relations

For an isotropic radiation field ( $I_{ν} = const$ over solid angle):

U_{ν} = \frac{4 π}{c} I_{ν} ⟹ I_{ν} = \frac{c}{4 π} U_{ν}

Hemispherical Surface Flux

The one-way net flux emerging from an isotropic boundary (e.g., blackbody surface) into a hemisphere:

F_{ν} = \int_{0}^{2 π} d ϕ \int_{0}^{π / 2} I_{ν} \cos θ \sin θ d θ = π I_{ν}

F = π I = \frac{c}{4} U = \frac{c a_{B}}{4} T^{4} = σ T^{4} [cite: 1]