Question

Why does planet moves only in eliptical orbit?

Answer 1

A planet moves (revolves) in an orbit around a bigger planet or a Sun like star.  If the planet (moon) has a velocity perpendicular to the line joining it and the bigger planet or Sun, then its orbit is always an ellipse. Circle is a special case of ellipse.  

    The planet or satellite or moon wants to go straight due to its inertia (momentum). But bigger planet or the Sun tries to pull it towards. So a middle path is taken. This happens at each point on the trajectory. So the planet ends up in a closed path.  We will show that it is actually an ellipse only.

This is stated as Kepler's gravitation law. We can derive the path for a moon m or planet going around a bigger planet or star M using calculus. The only force F present is the gravitational attraction force GMm/r².  

Here r(t) = distance between m and M.
         ω= dФ/dt = angular velocity.  
         md²r(t) / dt² - m r ω² is the centripetal force.
         L = angular momentum (vector) = m r² ω
         T = Torque = r X F = 0 as vector r and F are collinear.
         T = dL/dt = 0.   Hence  L is a constant.
          => ω = L/(mr²)      --- (1)

We have simple and short method as follows. The equation of motion when r(t) is not constant is:

   Force = - GMm/r² =  m d²r/dt² - m r ω²            --- (2)
Substitute for ω from (1):    
                - GM / r² = d² r/dt²  -  L²/(m²r³)              --- (3)

This is a differential equation of 2nd degree in r and t. To solve this, 
   Let  r = 1/u    ---- (4)
=>   dr/dt  = -1/u² * du/dt = -1/u² * du/dФ  * dФ/dt
=>            =  -1/u² * du/dФ * ω =  -1/u² * L/(mr²) * du/dФ
=>   dr/dt = -L/m  * du/dФ             --- (5)
Now, again,    d²r/dt²  = -L/m * d²u/dФ² * dФ/dt 
                                  = - L²/(m²r²) * d²u / dt²         --- (6)

Substitute (6) & (4) in  (3) to get:
          - GM u² = - L²u²/m² * d²u/dt² - L²u³ /m²
      =>  GM m² /L² - u = d²u/dt²       ---- (7)

This is a simple Ordinary differential equation ODE in 2nd degree. We know this corresponds to SHM.  The solution to this is:

        u = GMm²/r + A Cos ωt,       where  ω² = L²/(GMm²)
   
 Hence,  1/r  = GMm² + A Cos ω t is the solution.       --- (8)
                               A = constant

The standard equation in polar coordinates for an ellipse is:
       1/r =  (a/b²) (1 + e cos Ф)  --- (9)
                a = semimajor axis,        b = semiminor axis
                e = eccentricity         Ф = angle of radius wrt axis
                b²/a = semi latus rectum (focal chord perpendicular to axis)

Comparing (8) and (9) we can say that any planet or mass attracted by a central force and having a velocity (linear) orbits in an elliptical orbit.