In the following formulas, \(ν\) is in Hz, \(\lambda\) in μm, \(\sigma\) in cm-1, and \(T\) in \(K\). The rest of the constants are given below.
\[L_{v}=\frac{2 h v^{3}}{c^{2}} \frac{1}{e^{h v / k T}-1} \mathrm{W} \mathrm{m}^{-2} \mathrm{sr}^{-1} \mathrm{Hz}^{-1}\]
\[v_{\text {peak}}=\frac{a_{3} k}{h} T \quad \mathrm{Hz} \hspace{3cm} L_{v, \text { peak }}=\frac{2 a_{3}^{3} k^{3}}{h^{2} c^{2}} \frac{1}{e^{a_{3}}-1} T^{3} \quad \mathrm{W} \mathrm{m}^{-2} \mathrm{sr}^{-1} \mathrm{Hz}^{-1}\]
\[L_{v}^{P}=\frac{2 v^{2}}{c^{2}} \frac{1}{e^{h v / k T}-1} \quad \text { photon } \mathrm{s}^{-1} \mathrm{m}^{-2} \mathrm{sr}^{-1} \mathrm{Hz}^{-1}\]
\[v_{\text {peak}}^{P}=\frac{a_{2} k}{h} T \quad \mathrm{Hz} \hspace{3cm} L_{v, \text { peak }}^{P}=\frac{2 a_{2}^{2} k^{2}}{h^{2} c^{2}} \frac{1}{e^{a_{2}}-1} T^{2} \quad \text { photon } \mathrm{s}^{-1} \mathrm{m}^{-2} \mathrm{sr}^{-1} \mathrm{Hz}^{-1}\]
\[L_{\lambda}=\frac{2 \times 10^{24} h c^{2}}{\lambda^{5}} \frac{1}{e^{10^{6} h c / 2 k T}-1} \mathrm{W} \mathrm{m}^{-2} \mathrm{sr}^{-1} \mu \mathrm{m}^{-1}\]
\[\lambda_{\text {peak }}=\frac{10^{6} h c}{a_{5} k T} \quad \mu \mathrm{m} \hspace{3cm} L_{\lambda, \text { peak }}=\frac{2 a_{5}^{5} k^{5}}{10^{6} h^{4} c^{3}} \frac{1}{e^{a_{5}}-1} T^{5} \quad \mathrm{W} \mathrm{m}^{-2} \mathrm{sr}^{-1} \mu \mathrm{m}^{-1}\]
\[L_{\lambda}^{P}=\frac{2 \times 10^{18} c}{\lambda^{4}} \frac{1}{e^{10^{6} h c / 2 k T}-1} \text { photon } \mathrm{s}^{-1} \mathrm{m}^{-2} \mathrm{sr}^{-1} \mu \mathrm{m}^{-1}\]
\[\lambda_{\text {peak }}^{P}=\frac{10^{6} h c}{a_{4} k T} \quad \mu \mathrm{m} \hspace{3cm} L_{\lambda, \text { peak }}^{P}=\frac{2 a_{4}^{4} k^{4}}{10^{6} h^{4} c^{3}} \frac{1}{e^{a_{4}}-1} T^{4} \quad \text { photon } \mathrm{s}^{-1} \mathrm{m}^{-2} \mathrm{sr}^{-1} \mu \mathrm{m}^{-1}\]
\[L_{\sigma}=2 \times 10^{8} h c^{2} \sigma^{3} \frac{1}{e^{100 h c \sigma / k T}-1} \quad \mathrm{W} \mathrm{m}^{-1} \mathrm{sr}^{-1}\left(\mathrm{cm}^{-1}\right)^{-1}\]
\[\sigma_{\text {peak}}=\frac{a_{3} k}{100 h c} T \quad \mathrm{cm}^{-1} \hspace{3cm} L_{\sigma, \text { peak }}=\frac{200 a_{3}^{3} k^{3}}{h^{2} c} \frac{1}{e^{a_{3}}-1} T^{3} \quad \mathrm{W} \mathrm{m}^{-2} \mathrm{sr}^{-1}\left(\mathrm{cm}^{-1}\right)^{-1}\]
\[L_{\sigma}^{P}=2 \times 10^{6} c \sigma^{2} \frac{1}{e^{100 h c \sigma / k T}-1} \text { photon } \mathrm{s}^{-1} \mathrm{m}^{-2} \mathrm{sr}^{-1}\left(\mathrm{cm}^{-1}\right)^{-1}\]
\[\sigma_{\text {peak }}^{P}=\frac{a_{2} k}{100 h c} T \quad \mathrm{cm}^{-1} \hspace{3cm} L_{\sigma, \text { peak }}^{P}=\frac{200 a_{2}^{2} k^{2}}{h^{2} c} \frac{1}{e^{a_{3}}-1} T^{2} \quad \text { photon } \mathrm{s}^{-1} \mathrm{m}^{-2} \mathrm{sr}^{-1}\left(\mathrm{cm}^{-1}\right)^{-1}\]
\[L=\frac{2 \pi^{4} k^{4}}{15 h^{3} c^{2}} T^{4} \quad \mathrm{W} \mathrm{m}^{-2} \mathrm{sr}^{-1}\]
\[M=\frac{2 \pi^{5} k^{4}}{15 h^{3} c^{2}} T^{4} \quad \mathrm{W} \mathrm{m}^{-2}\]
\[L^{P}=\frac{4 \zeta(3) k^{3}}{h^{3} c^{2}} T^{3} \quad \text { photon } \mathrm{s}^{-1} \mathrm{m}^{-2} \mathrm{sr}^{-1}\]
\[M^{P}=\frac{4 \pi \zeta(3) k^{3}}{h^{3} c^{2}} T^{3} \quad \text { photon } \mathrm{s}^{-1} \mathrm{m}^{-2}\]
\begin{array}{rl}
\int_{\sigma}^{\infty} L_{\sigma^{\prime}} d \sigma^{\prime}=2 \frac{k^{4} T^{4}}{h^{3} c^{2}} \sum_{n=1}^{\infty}\left(\frac{x^{3}}{n}+\frac{3 x^{2}}{n^{2}}+\frac{6 x}{n^{3}}+\frac{6}{n^{4}}\right) e^{-n x} & \mathrm{W} \mathrm{m}^{-2} \mathrm{sr}^{-1} \newline
\int_{\sigma}^{\infty} L_{\sigma^{\prime}}^{P} d \sigma^{\prime}=2 \frac{k^{3} T^{3}}{h^{3} c^{2}} \sum_{n=1}^{\infty}\left(\frac{x^{2}}{n}+\frac{2 x}{n^{2}}+\frac{2}{n^{3}}\right) e^{-n x} & \text { photons } \mathrm{s}^{-1} \mathrm{m}^{-2} \mathrm{sr}^{-1} \newline
\text {where } x=\frac{100 h c \sigma}{k T}
\end{array}
\[T^{\prime}=T \sqrt{\frac{c-u}{c+u}}\] (\(u\) is relative velocity of source, \(u > 0\) indicating recession)
\begin{align}
& h=6.6260693 \times 10^{-34} \quad \mathrm{W} \mathrm{s}^{2} \quad & Planck's \; constant \newline
& c=2.99792458 \times 10^{8} \quad \mathrm{m} / \mathrm{s} \quad & speed \; of \; light \newline
& k=1.380658 \times 10^{-23} \quad \mathrm{J} / \mathrm{K} \quad & Boltzmann's \; constant \newline
& \zeta(3)=1.202056903159594 \quad & Apéry's \; constant, \; Riemann \; Zeta \; function \newline
& a_{2}=1.59362426004 \quad & solution \; to \quad 2\left(1-e^{-x}\right)-x=0 \newline
& a_{3}=2.82143937212 \quad & solution \; to \quad 3\left(1-e^{-x}\right)-x=0 \newline
& a_{4}=3.92069039487 \quad & solution \; to \quad 4\left(1-e^{-x}\right)-x=0 \newline
& a_{5}=4.96511423174 \quad & solution \; to \quad 5\left(1-e^{-x}\right)-x=0
\end{align}