Useful Equations

Diameter=$2R$ Circumference=2$\pi R$ $v=\frac{\mbox{distance}}{\mbox{time}}$

$\alpha ($angular diameter $)=\frac{\mbox{diameter}}{\mbox{distance}}57.3{% {}^{\circ }\hspace{0.3in}}$distance (pc)= $\frac{1}{\alpha _{parallax}(^{\prime \prime })}$

$\theta(arcsec)=0.25\frac{\lambda(\mu m)}{D(m)}$      $\lambda f=c\hspace{0.5in}E=h f\hspace{0.5in}$ $% F=\sigma T^{4}$

$L=F\cdot 4\pi R^{2}=4\pi \sigma R^{2}T^{4}\hspace{0.5in}$In solar units: $% L=R^{2}T^{4}$

Brightness $\propto \frac{L}{d^{2}}\hspace{0.3in}\lambda _{peak}=\frac{0.0029m}{T}\hspace{0.2in}\frac{\lambda _{obs}}{% \lambda _{true}}=1+\frac{v}{c}$ ($v<<c$) $\hspace{0.5in}$
$z=\frac{\lambda_{observed}-\lambda_{true}}{\lambda_{true}}=\frac{\sqrt{1+\frac{v}{c}}}{\sqrt{1-\frac{v}{c}}} -1$ $F=\frac{GMm}{r^{2}}\hspace{0.1in}F=m\frac{v^{2}}{r}=ma$

perihelion= $a(1-e)\hspace{0.1in}$ aphelion= $a(1+e)\hspace{0.1in}$

$F=\frac{GMm}{r^{2}}\hspace{0.1in}F=m\frac{v^{2}}{r}=ma$ $a=\frac{v^{2}}{r}=\frac{GM}{r^2}$ $F_{tidal}=\frac{2GMmR}{r^{3}}$

$P^{2}(yr)=\frac{a^{3}(AU)}{M(M_{\odot })}\hspace{0.3in}P=2\pi \sqrt{\frac{% r^{3}}{GM}}$ $v=\sqrt{\frac{GM}{r}}\hspace{0.3in}v_{escape}=\sqrt{\frac{2GM}{r}}$ fraction remaining $= \left ( \frac{1}{2}\right )^{t/T}$
Condition for escape: $6\times v_{moleculuar} \ge v_{escape}$

$v=\sqrt{\frac{GM}{r}}\hspace{0.5in} \hspace{0.5in}v_{escape}=\sqrt{\frac{2GM}{r}}$

$v_{escape}=11.2\;$km/s $\sqrt{\frac{m(M_{Earth})}{R(R_{Earth})}} \hspace{0.5in}v_{molecular}=0.157$ km/s $\sqrt{\frac{T(K)}{mass(m_{H})}}$

$m-M=5\log{\frac{D}{10~pc}}\hspace{0.5in} D=10~pc \times 10^{(m-M)/5}$
$M=4.85-2.5\log{L (L_\odot)}\hspace{0.5in} L(L_\odot)=10^{-0.4(M-4.85)}$

$R_{Schwarzschild}=M (M_{\odot})\times 3~ km$ $v_{recession}=H_0~ d$ age $=\frac{970}{H_0}$ billion yrs