$$ \vec{F}{21}=-\frac{Gm_1m_2}{r^3{12}}\vec{r}{12}=-\frac{Gm_1m_2}{r^2{12}}\hat{r}_{12} $$

• Where $r_{12}$ is the magnitude of $\vec{r}{12}$ and $\hat{r}{12}$ is the unit vector pointing in the direction of the displacement

$$ \vec{r}{12}=-\vec{r}{12} \qquad \qquad \vec{F}{12}=-\vec{F}{21} $$

Gravitational Potential Energy

$$ W_{grav}=-\int_{R}^{\infin}\frac{GmM_E}{r^2}\,\text{d} r=-\frac{GmM_E}{R}=U(R) $$

• Where $U$ is the potential energy, $R$ is the radius from the center of the earth, $M_E$ is the math of the earth and $m$ is the mass of an object

Kepler’s laws

Kepler's Laws

Simulation of Kepler's laws

<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/a3edc2fc-ee9d-4092-a9e3-dde54c70cbac/1st_law.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/a3edc2fc-ee9d-4092-a9e3-dde54c70cbac/1st_law.png" width="40px" /> $1^{\text{st}}$ Law: all planets move in elliptical orbits, with t he sun at one focus of the ellipse

• $a$ is the semi major axis, $e$ is the eccentricity, $S$ and $S'$ are the foci of the ellipse </aside>

<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/745730a9-19f8-4fb2-9b8b-3e878789e7bd/2nd_law.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/745730a9-19f8-4fb2-9b8b-3e878789e7bd/2nd_law.png" width="40px" /> $2^{\text{nd}}$ Law: A line that connects a planet to the Sun sweeps out equal areas in equal times

</aside>

<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/eb2014e7-bda0-49af-b3dc-b3bbc9bd9b50/3rd_law.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/eb2014e7-bda0-49af-b3dc-b3bbc9bd9b50/3rd_law.png" width="40px" /> $3^{\text{rd}}$ Law: The square of the period of any planet about the sun is proportional to the cube of the planet’s mean distance from the sun

$$ \begin{aligned} T^2&=\frac{4\pi^2}{GM}r^3 \\ T^2&\propto r^3 \end{aligned} $$

</aside>

Potential energy and force

$$ \begin{aligned} F_r&=-\frac{\text{d} U}{\text{d} r}\\ &=-\frac{\text{d}}{\text{d} r}\left(-\frac{GM_Em}{r}\right)=-\frac{GM_Em}{r^2}\,. \end{aligned} $$

• The minus sign indicates that the force is radially inward

Escape speed

$$ v_{\text{esc}}=\boxed{\sqrt{\frac{2GM}{R}}} $$

Gravitational potential energy for many particles

The potential energy (or also called binding energy) of a system of particle is defined as:

$$ U=-G\left( \frac{m_A m_B}{r_{AB}} + \frac{m_A m_C}{r_{AC}} + \frac{m_C m_B}{r_{CB}} \right) $$

Satellites

$$ \begin{aligned} v&=\sqrt{\frac{GM_E}{r}} \\ E_P&=-\frac{GM_Em}{r} \\ E_k&=\frac{1}{2}m\frac{GM_E}{r} \\ E_m&=E_p+E_k \\ &=\frac{1}{2}E_p=-E_k \end{aligned} \\ \text{in the case of an eleptical orbit with }a \\ \text{as the semi major axis then:} \\ E_m=-\frac{GM_Em}{2a} $$