Formulae

https://youtu.be/d05oyVJpv24

Reduced Mass #

$$ \mu =\frac{m_1m_2}{m_1+m_2} $$

Angular Momentum #

$$ J=mr^2\theta' $$

Energy #

$$ E=\frac{1}{2}mr’^2+\frac{J^{2}}{2mr^{2}}+V=constant $$

Areal velocity #

Area swept out by the radius vector in time dt: $$ \frac{1}{2}r^2θ’=\frac{J}{2m}=constant $$ general property taking place under central force.

Differential equation of an orbit #

$$ \begin{align} u=1/r\\ \frac{du^2}{d\theta ^2}+u+\frac{m}{J^2u^2}f(\frac{1}{u})=0 \end{align} $$

Kepler’s Laws #

1. Law of elliptical orbits $$e=\frac{J^2A}{mK}=\sqrt{1+\frac{2EJ^2}{mK^2}}$$ $$\frac{l}{r}=1+e\cos(\theta-\theta’_{constant});\\ l=e/A$$ where $f(r)=-K/r^2$. $$E=-K/2a$$
   1. Perihelion: minimum ‘r’
      1. $r_1=\frac{l}{1+e}$
   2. Aphelion: maximum ‘r’
      1. $r_2=\frac{l}{1-e}$
2. Sweeps equal areas in equal δT
3. Period of motion $$T=2\pi a^{3/2} \sqrt{\frac{m}{K}}$$

Semimajor axis $a=\frac{r_1+r_2}{2}=\frac{l}{1-e^2} \Rightarrow l=a(1-e^2)$

So, $$ \begin{align} r=\frac{a(1-e^2)}{1+e\cos(\theta-\theta_{constant})}\\ r_1=a(1-e)\\ r_2=a(1+e) \end{align} $$

Velocity and time period for circular motion #

$$ v=R\sqrt{\frac{g}{r}} $$

$$ T=\frac{2\pi r}{v} $$