User:Nedim Ardoğa/Math

Let
 * G=Gravitational constant
 * M=Solar mass
 * R= Distance between the Sun and the Earth
 * x=Distance from the Earth
 * r=Distance from the Sun (r=R-x)
 * T=Period of the revolution
 * t=Time variable

a=acceleration of Falling

Kepler's Third Law

 * Gravitational force for unit mass.... $$ G \cdot \frac{M}{R^2} $$
 * Centrifugal force for unit mass.......  $$ \frac {v^2}{R} $$
 * Period of revolution......................   $$T^2= \frac{4 \cdot \pi ^2 \cdot R^3}{G\cdot M}$$

Acceleration

 * $$ v=\frac{dx}{dt}=- \frac{dr}{dt}\qquad \Rightarrow \qquad dt=-\frac{dr}{v}$$
 * $$ v=\int_0^T{a \quad dt} = \int_0^T{\frac{G \cdot M}{r^2}}dt= - \int_R^r{\frac {G \cdot M }{r^2 \cdot v}}dr \qquad \Rightarrow \qquad \frac {dv}{dr}=-\frac{G \cdot M}{r^2\cdot v}$$
 * $$\int_ 0^V{v \quad dv}=-\int_R^r{\frac {G\cdot M }{r^2 }}dr$$
 * $$ \frac{v^2}{2}=G \cdot M \cdot (\frac{1}{r}-\frac{1}{R})$$
 * $$v= \sqrt{2\cdot G \cdot M \cdot \frac{R-r}{r\cdot R}} \qquad \Rightarrow \qquad \frac{dr}{dt} = \sqrt{2\cdot G \cdot M \cdot \frac{R-r}{r\cdot R}}$$
 * $$\sqrt{\frac {R\cdot r}{R-r}} dr=\sqrt{G\cdot M} dt \qquad \Rightarrow \qquad \int{\sqrt{\frac {R\cdot r}{R-r}}} dr=\int{\sqrt{G\cdot M}} dt$$

Let :$$ r=R\cdot (\sin (\phi))^2 \qquad \Rightarrow \qquad  \frac=2\cdot R\cdot\sin(\phi)\cdot\cos(\phi)$$


 * $$ -\sqrt{2\cdot G\cdot M}\cdot t= \sqrt R \int\limits_{\pi}^{\beta} \sqrt { \frac{R \cdot (\sin (\phi))^2}{R\cdot (1-(sin (\phi))^2) }}\cdot 2\cdot R\cdot\sin(\phi)\cdot\cos(\phi) d\phi$$


 * $$=2\cdot R^{3/2}\int (sin(\phi))^2 d\phi =2\cdot R^{3/2}(\frac{\phi}{2}-\frac{\sin (2\cdot \phi)}{4}) $$
 * $$=2\cdot R^{3/2} (\frac{\beta}{2}-\frac{\sin(2\cdot\beta)}{4}-\frac{\pi}{4})$$

For r=0 : $$-\sqrt{(2\cdot G\cdot M)}\cdot t=2\cdot R^{3/2}\frac{(-\pi)}{4} $$
 * $$t^2=\frac{{\pi}^2 \cdot R^3}{8\cdot G\cdot M}$$

Free fall time wrt period
Free fall time t can be expressed in terms of Period T;
 * $$\frac{t^2}{T^2} =\frac{{\pi}^2 \cdot R^3}{8\cdot G\cdot M} \cdot\frac{G\cdot M}{4 \cdot {\pi}^2 \cdot R^3} $$


 * $$ t= \frac {T}{\sqrt {32}}$$

Free fall time wrt constant acceleration fall time
Supposing that the acceleration a0 is constant;
 * $$R= \frac {a_0 \cdot {t_0}^2}{2} \qquad \Rightarrow \qquad  R= \frac{G \cdot M}{2 \cdot R^2 } \cdot {t_0}^2  $$


 * $${t_0}^2= 2 \cdot \frac{R^3}{G \cdot M}$$


 * $$\frac {t^2}{{t_0}^2}=\frac{{\pi}^2 \cdot R^3}{8\cdot G\cdot M} \cdot \frac {G \cdot M}{2 \cdot R^3}= \frac {{\pi}^2}{16}$$


 * $$ t=\frac{\pi \cdot t_0}{4}$$