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 Mv^{2} - GMS/R = E

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

v^{2} =
(dS/dt)^{2}
, which can be written as

(2) v^{2} =
(dS/dt)^{2
}= (dR/dt)^{2} + R^{2}(dq/dt)^{2}

by noticing from
the diagram
above that (dS)^{2 }= (dR)^{2} + R^{2}(dq).

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

E = 1/2 (L^{2}/(MR^{4}))
[ (dR/dq)^{2 }+
R^{2} ]
- G (MS/R)

Rearranging terms
gives

[ (dR/dq)^{2 }+ R^{2} ] = (2EMR^{4}/L^{2}) + (2GSM^{2}R^{3}/L^{2})

The derivative dR/dq thus can be
written as

(3) dR/dq^{ }= R[(2EM/L^{2)} R^{2} + 2 GSM^{2}/L^{2}
R - 1 ] ^{1/2}

defining
combinations of
constants p = L^{2}/GSM^{2} and
(e^{2} - 1)= 2EL^{2}/(G^{2}S^{2}M^{3}) equation (3) can be written as

(4) dR/dq^{ }= R [(e^{2} - 1)/p^{2}
R^{2 } +
(2/p) R - 1 ] ^{1/2}

^{ }

(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 R_{p} at perihelion and q = p
for R_{a}
at aphelion.

R_{p} =
p/(1+e)

R_{a} =
p/(1-e)

Since the
semi-major axis of
the ellipse is 2a = R_{p} + R_{a}, we have

p = a (1- e^{2})

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

e^{2} = 2EL^{2}/(G^{2}S^{2}M^{3})
+ 1 , e^{2} < 1 and the path of the
planet is an
ellipse with semimajor axis a and semiminor axis c = a(1- e^{2})^{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 = 2EL^{2}/(G^{2}S^{2}M^{3})
+ 1 = 0

or
E = - G^{2}S^{2}M^{3}/2L^{2}

^{ }

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.