College Stellar Astrophysics
EQUATIONS
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INFORMATION
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$$ \large
\frac{m_1}{m_2} = \frac{r_2}{r_1} = \frac{a_2}{a_1} $$
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\( {m_\#} = mass \hspace{4 pt} of \hspace{4 pt} star \hspace{4 pt} \# \hspace{4 pt} (M_{sun}) \\ {r_\#} = distance \hspace{4 pt} from \hspace{4 pt} the \hspace{4 pt} center \hspace{4 pt} of \hspace{4 pt} mass \hspace{4 pt} \\ \hspace{11 pt} to \hspace{4 pt} star \# (A.U.) \\ {a_\#} = semi-major \hspace{4 pt} axis \hspace{4 pt} of \hspace{4 pt} orbit \hspace{4 pt} (A.U.) \) |
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$$ \large
\frac{m_1}{m_2} = \frac{v_2}{v_1} $$
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\( {m_\#} = mass \hspace{4 pt} of \hspace{4 pt} star \hspace{4 pt} \# \hspace{4 pt} (M_{sun}) \\ {v_\#} = radial \hspace{4 pt} velocity \hspace{4 pt} of \hspace{4 pt} star \hspace{4 pt} \# \hspace{4 pt} (km \hspace{2 pt} s^{-1})\) |
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$$ \large
\frac{{m_2}^3 \hspace{2 pt} sin^3 i}{(m_1 + m_2)^2 pt} = \frac{{v_1}^3 T}{2 \pi G} $$
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\( {m_\#} = mass \hspace{4 pt} of \hspace{4 pt} star \hspace{4 pt} \# \hspace{4 pt} (kg) \\ \textbf{i} = inclination \hspace{4 pt} angle \hspace{4 pt} (deg) \\ {v_\#} = radial \hspace{4 pt} velocity \hspace{4 pt} of \hspace{4 pt} star \hspace{4 pt} \# (m \hspace{2 pt} s^{-1}) \\ \textbf{T} = orbital \hspace{4 pt} period \hspace{4 pt} (s) \\ \textbf{G} = gravitational \hspace{4 pt} constant \hspace{4 pt} \\ \hspace{7 pt} (6.67384 x 10^{-11 pt} \hspace{2 pt} m^3 \hspace{2 pt} kg^{-1} \hspace{2 pt} s^{-2})\) |
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$$ \large
M_J = {\left(\frac{5kT}{G \mu m_p} \right )}^{\frac{3}{2}} \hspace{2 pt} \left(\frac{3}{4 \pi \rho_0} \right ) ^{\frac{1}{2}} $$
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\(\\ \textbf{M_J} = Jeans \hspace{4 pt} Mass \hspace{4 pt} (kg) \\ \textbf{k} = Boltzmann's \hspace{4 pt} constant \\ \hspace{5 pt} (1.38 \hspace{2 pt} x \hspace{2 pt} 10^{-23} J \hspace{2 pt} K^{-1}) \\ \textbf{T} = temperature \hspace{4 pt} (K) \\ \textbf{G} = gravitational \hspace{4 pt} constant \\ \hspace{9 pt} (6.67 \hspace{2 pt} \hspace{2 pt} 10^{-11} \hspace{2 pt} m^3 \hspace{2 pt} kg^{-1} \hspace{2 pt} s^{-2}) \\ {\mu} = mean \hspace{4 pt} molecular \hspace{4 pt} weight \hspace{4 pt} \\ \textbf{m_p} = mass \hspace{4 pt} of \hspace{4 pt} proton \hspace{4 pt} (kg) \\ {\rho_0} = density \hspace{4 pt} of \hspace{4 pt} cloud \hspace{4 pt} (kg \hspace{2 pt} m^{-3})\) |
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$$ \large
R_J = \sqrt{\frac{15kT}{4\pi G \mu m_p \rho_0}} $$
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\(\\ \textbf{R_J} = Jeans \hspace{4 pt} Length \hspace{4 pt} (m) \\ \textbf{k} = Boltzmann's \hspace{4 pt} constant \\ \hspace{5 pt} (1.38 \hspace{2 pt} x \hspace{2 pt} 10^{-23} J \hspace{2 pt} K^{-1}) \\ \textbf{T} = temperature \hspace{4 pt} (K) \\ \textbf{G} = gravitational \hspace{4 pt} constant \\ \hspace{9 pt} (6.67 \hspace{2 pt} \hspace{2 pt} 10^{-11} \hspace{2 pt} m^3 \hspace{2 pt} kg^{-1} \hspace{2 pt} s^{-2}) \\ {\mu} = mean \hspace{4 pt} molecular \hspace{4 pt} weight \hspace{4 pt} \\ \textbf{m_p} = mass \hspace{4 pt} of \hspace{4 pt} proton \hspace{4 pt} (kg) \\ {\rho_0} = density \hspace{4 pt} of \hspace{4 pt} cloud \hspace{4 pt} (kg \hspace{2 pt} m^{-3})\) |
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$$ \large t_{ff} = \frac{1}{2} \sqrt{ \frac{R^3}{GM} } $$
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\(\\ {t_{ff}} = free - fall \hspace{4 pt} timescale \hspace{4 pt} (s) \\ \textbf{R} = radius \hspace{4 pt} of \hspace{4 pt} star \hspace{4 pt} (m) \\ \textbf{G} = gravitational \hspace{4 pt} constant \\ \hspace{9 pt} (6.67 \hspace{2 pt} \hspace{2 pt} 10^{-11 pt} \hspace{2 pt} m^3 \hspace{2 pt} kg^{-1 pt} \hspace{2 pt} s^{-2 pt}) \\ \textbf{M} = mass \hspace{4 pt} of \hspace{4 pt} the \hspace{4 pt} star \hspace{4 pt} (kg)\) |
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$$ \large
E = f Mc^2 $$
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\(\\ \textbf{E} = output \hspace{4 pt} energy \hspace{4 pt} (J) \\ \textbf{f} = fraction \hspace{4 pt} of \hspace{4 pt} mass \hspace{4 pt} converted \hspace{4 pt} to \hspace{4 pt} energy \\ \textbf{M} = mass \hspace{4 pt} of \hspace{4 pt} the \hspace{4 pt} star \hspace{4 pt} (kg) \\ \textbf{c} = speed \hspace{4 pt} of \hspace{4 pt} light \hspace{4 pt} (3.00 \hspace{2 pt} x \hspace{2 pt} 10^8 \hspace{2 pt} m \hspace{2 pt} s^{-1 pt})\) |
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$$ \large \tau_{nuc} = \frac{ f \epsilon M c^2}{L} $$
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\( \tau_{nuc} = nuclear \hspace{4 pt} timescale \hspace{4 pt} (s) \\ \textbf{f} = fraction \hspace{4 pt} of \hspace{4 pt} nuclear \hspace{4 pt} fuel \\ \hspace{14 pt} available \\ \epsilon = efficiency \hspace{4 pt} of \hspace{4 pt} mass \\ conversion \\ \textbf{M} = mass \hspace{4 pt} of \hspace{4 pt} star \hspace{4 pt} (kg) \\ \textbf{c} = speed \hspace{4 pt} of \hspace{4 pt} light \hspace{4 pt} (3.00 \hspace{2 pt} m \hspace{2 pt} s^{-1}) \\ \textbf{L} = luminosity \hspace{4 pt} (kg \hspace{2 pt} m^2 \hspace{2 pt} s^{-3} \) |
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$$ \large \tau = \int_{0 pt}^{X} n \hspace{2 pt} C_{ext} \hspace{2 pt} dx $$
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\(\\ {\tau} = optical \hspace{4 pt} depth \\ \textbf{n} = particle \hspace{4 pt} density \hspace{4 pt} (m^{-3})\\ {C_{ext}} = true \hspace{4 pt} extinction \hspace{4 pt} cross\\ \hspace{18 pt} section \hspace{4 pt} (m^{2})\\ \textbf{x} = depth \hspace{4 pt} (m) \) |
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$$ \large
R_{Sch} = \frac{2GM}{c^2} $$
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\(\\ {R_{Sch}} = Schwarzchild \hspace {4 pt} radius \hspace {4 pt} (km)\\ \textbf {G} = gravitational \hspace {4 pt} constant \hspace {4 pt} \\ \hspace{8 pt}(6.67384 \hspace{2 pt} x \hspace{2 pt} 10^{-11} \hspace{2 pt} m^3 \hspace{2 pt} kg^{-1} \hspace{2 pt} s^{-2}) \\ \textbf{M} = mass \hspace {4 pt} of\hspace {4 pt} the\hspace {4 pt} object \hspace {4 pt} (kg) \\ \textbf{c} = speed \hspace {4 pt} of \hspace {4 pt} light \hspace {4 pt}(km/s) \) |
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