Orbits of Planets are Ellipses



Assume a planet with mass M orbiting the Sun with mass S.  Conservation of energy can be written as

 

 (1) P.E. + K.E. = E

 

   1/2 Mv2  - GMS/R = E

 

The velocity v along the orbit is the derivative dS/dt.  Hence,

 

v2 = (dS/dt)2 , which can be written as

 

(2) v2 = (dS/dt)2 = (dR/dt)2 + R2(dq/dt)2

 

by noticing from the diagram above that (dS)2 = (dR)2 + R2(dq).

 

Using conservation of angular momentum L = I w = MR2 dq/dt = constant, we can substitute dq/dt = L/MR2 in equation (2) for v2.  Then equation (1) for conservation of energy becomes

 

E = 1/2 (L2/(MR4)) [ (dR/dq)2 + R2 ]  - G (MS/R)

 

Rearranging terms gives

 

[ (dR/dq)2 + R2 ] = (2EMR4/L2)  + (2GSM2R3/L2)

 

The derivative dR/dq  thus can be written as

 

(3) dR/dq = R[(2EM/L2) R2 + 2 GSM2/L2 R - 1 ] 1/2

 

defining combinations of constants p = L2/GSM2 and (e2 - 1)= 2EL2/(G2S2M3)  equation (3) can be written as

 

(4) dR/dq = R [(e2 - 1)/p2 R2   +  (2/p) R - 1 ] 1/2

 

The differential equation (4) above has a solution that can be verified by differentiation:

 

(5) R = p /(1 + e cos (q + c))

 

This is the equation for an ellipse with the origin at one focus (the sun).  The constant of integration c is zero if q = 0 for Rp at perihelion and q = p for Ra at aphelion.

 

Rp = p/(1+e)

 

Ra = p/(1-e)

 

Since the semi-major axis of the ellipse is 2a = Rp + Ra, we have

 

p = a (1- e2)

 

 

Consider four different situations for the ratio of the magnitude of potential energy to kinetic energy and what they imply for the eccentricity e

 

(1)         P.E. > K.E.

 

The total E is negative and since

 

e2 = 2EL2/(G2S2M3) + 1 , e2 < 1 and the path of the planet is an ellipse with semimajor axis a and semiminor axis c = a(1- e2)1/2

 

(2)         P.E. = K.E.

 

When E = 0, the eccentricity  e is zero, describing a parabola.  This is the condition for escape.

 

(3)         P.E. < K.E.

E > 0 , the eccentricity  e is greater than 1 and the path is hyperbolic

 

(4)         When eccentricity e is zero:

 

e = 2EL2/(G2S2M3) + 1 = 0

 

or E = - G2S2M3/2L2

 

This is the minimum value (largest negative value) that E (total energy) can take for any orbital path.   The corresponding K.E. is the minimum value of K.E. for any possible conic section. The orbit is a circle.