# Orbital Mechanics Formulas
:::info
\begin{align}
G &= 6.67\times10^{-11}\\
m_E &= 5.98\times10^{24}(kg)
\end{align}
:::
$(3.58)$
$$h=rv_\theta=r^2 \dot \theta$$

$(3.59)$
$$Area = \frac{(r\Delta \theta)(r)}{2}\\
\frac{dArea}{dt}=\frac{1}{2}(r^2 \dot \theta)$$

$(3.70)$
$$p(\theta)=\frac{1}{r(\theta)}=\frac{Gm_s}{h^2}+A\;cos\theta$$$where\ A\ is\ a\ constant$

$(3.71)$
$$
e = \frac{r}{d}=\frac{r}{C-r\;cos\theta}\\
\frac{1}{r}=\frac{1}{e\,C}+\frac{1}{C}\;cos\theta$$$Compare\ with\ 3.70$
$$\Rightarrow e=\frac{Ah^2}{Gm_s}$$

$(3.73)$
$$\frac{1}{r}=\frac{Gm_s}{h^2}+A\;cos\theta$$$differentiate$
\begin{align}
\dot r=A\,r^2\,\dot\theta sin\theta\\
\Rightarrow v_r=A\,r\,v_\theta \, sin\theta
\end{align}
$(3.76)$
$$
A= \frac{v_r}{r\;v_\theta\;sin \theta}
$$$Plug\ in\ (3.73)$
$$
\frac{1}{r}-\frac{Gm_s}{h^2}=\frac{v_r}{r\;v_\theta\;sin \theta}\;cos\theta$$\begin{align}
tan\theta
&=\frac{v_r}{r\;v_\theta\;(\frac{1}{r}-\frac{Gm_s}{h^2})}\\
&=\frac{v_r}{v_\theta-\frac{Gm_s}{h}}
\end{align}

$(3.79)$
$$
\frac{1}{r_p}=\frac{Gm_s}{h^2}+A\\
\frac{1}{r_a}=\frac{Gm_s}{h^2}-A\\
\Rightarrow A=\frac{1}{2}(\frac{1}{r_p}-\frac{1}{r_a})
$$

$(3.83)$
\begin{cases}
\dfrac{1}{r_p}=\dfrac{Gm_s}{h^2}+A\\
e=\dfrac{Ah^2}{Gm_s}
\end{cases}\begin{gather}
\frac{1}{r_p}=A\;(\frac{1+e}{e})\\
A = \frac{e}{r_p(1+e)}\\
\Rightarrow h=\sqrt{Gm_s r_p(1+e)}=r_p^2 \; \dot \theta
\end{gather}\begin{align}
v_p&=r_p \dot \theta = \frac{h}{r_p}\\
&=\left[\frac{Gm_s(1+e)}{r_p}\right]^{\frac{1}{2}}
\end{align}

$(3.85)$
\begin{cases}
\dfrac{1}{r_p}=A\;(\dfrac{1+e}{e})=\dfrac{Gm_s}{h^2}(1+e)\\
\dfrac{1}{r_a}=A\;(\dfrac{1-e}{e})=\dfrac{Gm_s}{h^2}(1-e)
\end{cases}