Solid angle (in steradians)
$$ \Omega=\frac{\textrm{area}}{r^2} $$
(solid angle of spherical cap: $2\pi (1-\cos \theta) \approx \pi \theta^2$ where $\theta$ is the angle of the circumference from the center)
https://en.wikipedia.org/wiki/Radiometry#Radiometric_quantities
Energy:
$$ dE=B_\nu \cos \theta_1 \,dA_1\,d\nu \,d\Omega_1 \,dt $$
Cosine term is due to Lambert’s cosine law, where the specific intensity
Converting from directional quantities to nondirectional quantities:
$$ I=\frac{\int B_{\nu}\cos\theta_1 dA_1 d\Omega_1 d\nu}{A_2}=\frac{B_{\nu}\pi R_{\odot}^2\Omega_1 \Delta \nu}{A_2}=B_{\nu}\pi R_{\odot}^2\Delta \nu/r^2 $$
In general, directional quantities such as radiative intensity can be converted to non-directional quantities such as radiant exitance by integrating over all angles. Usually this is a hemisphere so its just multiplied by $\pi$
To get stefan-boltzmann law, integrate over all frequencies to get
$$ I=\sigma T^4R_{\odot}^2/r^2 $$
$\pi$ disappears because $B_\nu$ is directional but the stefan-boltzmann law isn’t
Wien’s displacement law: $\lambda = 2.898\times 10^{-3}\mathrm{mK}/T$
| Name | Definition | Units | Symbol |
|---|---|---|---|
| Luminosity/Radiant flux | total power emitted (use stefan-boltzmann law: $L=A\epsilon \sigma T^4$ | $\mathrm{W}$ | $L$ |
| Radiant exitance | radiance integrated over a hemisphere. **For Lambertian emitters it is equal to $M_e=\pi \mathrm{sr} \, I_{e, \Omega} | ||
| =\epsilon \sigma T^4$** | $\mathrm{Wm^{-2}}$ | $M_e$ | |
| Radiance | spectral radiance but integrated over all wavelengths: $I_{e,\Omega}=\epsilon \sigma T^4/\pi$. This is also known as etendue. | $\mathrm{W m^{-2 }sr^{-1}}$ | $I_{e,\Omega}$, $I$ |
| Spectral exitance | Spectral radiance integrated over all angles | $\mathrm{W m^{-2} Hz^{-1}}$ | $M_{e,\nu}$ |
| Spectral radiance | Use Planck’s Law: $B_\nu=2h\nu^3c^{-2}\left(\exp \left(\frac{h\nu}{k_BT}\right)-1\right)^{-1}$. The integral of the visible area is $\pi R_{Sun}^2$ as that is the projection of the sphere onto a plane at right angles to the receiver (the limb of the Sun) (see spectral radiance) | ||
| When $h\nu \ll k_B T$, you get the Rayleigh-Jeans law: $B_{\nu }(T)=2\nu ^{2}k_{\text{B}}Tc^{-2}$ | $\mathrm{W m^{-2} Hz^{-1} sr^{-1}}$ | $B_\nu$, $I_\nu$ | |
<aside> 💡 EMISSIVITY = ABSORPTIVITY (kirchhoff’s law) albedo (reflectivity) + absorptivity + transmittance = 1
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