Planetary Motion

• In the previous projectile motion example, we used the gravitational force with the expression $F=mg$ and gravitational acceleration as being constant near the Earth's surface.
• However, the gravitational force between masses is most generally given by Newton's law of universal gravitation:
  $$F=G\frac{m_1m_2}{r^2}$$
  Here, $G=6.6743\times 10^{-11}~m^3 kg^{-1} s^{-2}$ is called the universal gravitational constant. The force is attractive and along the direction connecting the two masses.
• This expression should be used when studying the motion of planets and moons.
• Let's study the motion of a planet (mass $m$ )moving under the gravitational force of the Sun (mass $M$). If we take the sun at the origin, the vector expression of the force acting on the planet would be:
  $$\vec{F}=-G\frac{Mm}{r^3}\vec{r}$$

• Since the orbit of the planet will be at a plane (2D), the position vector $\vec{r}$ and accordingly the acceleration vector $\vec{a}$ would have two components as:
  

  $$\vec{r} = x\hat{i}+y\hat{j}$$

  $$\vec{a} = \frac{d^2x}{dt^2}\hat{i}+\frac{d^2y}{dt^2}\hat{j}$$

  
  
• Newton's 2$^{nd}$ law as $\vec{a}=\vec{F}/m$ and also velocity expressions for the x- and y-components:
  

  $$\frac{d^2x}{dt^2}=-G\frac{M}{r^3}x ~~~\&~~~ \frac{dx}{dt}= v_x$$

  $$\frac{d^2y}{dt^2}=-G\frac{M}{r^3}y ~~~\&~~~ \frac{dy}{dt}= v_y$$

  
  
• Now, we have a set of equations.