Question

Why planet revolve in an elepticle orbit

Answer 1

Planets are attracted to the Sun or big star in the middle. The gravitational force causes the attraction between them.

Initially of the planet is stationary, or traveling along the line joining the planet and Sun, the planet will collide Sun. Planets have components of velocity tangential to the line joining them to the Sun. So they tend to move away with a centripetal acceleration given by gravity force.

  So planets change their direction continuously at each point on their orbits. But they travel still away from Sun.  Mathematically if this situation is solved, we get the ellipse as the path of the planet.

  It is a differential equation of 2nd degree and 1st order, we need to solve to get the result.   The elliptical path is the result when there is no other force acting on the system.  Kepler derived it.

        m d²r/dt² = - GMm/r² + m r ω²

=====     Detailed Solution   ======

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.

Answer 2

because of gravitational force provided by sun.