It is possible to determine
the mass of a gravitating body by considering the motion
of any of its orbiting satellites.The analysis is simplified if we assume
the following:
The mass of each body
acts as though concentrated at its centre.
Orbits are nearly circular.Eccentricity
is close to zero.
The only force acting
is a central one that obeys an inverse square law.
For a general motion
we have the conic:
l = r ( 1 + ecosq
)
Polar Equation
for Conic Section
.
Here l is the length
of the semi-latus rectum and e is a value between 0 and 1 termed
the 'eccentricity'. Note that
if 0< e < 1 then
1 +ecosq
is never zero so that r is always finite.The conic is an ellipse.
if e = 1 then as q
approaces p
then r must increase without limit. The conic is a parabola.
if e >1 then 1 +ecosq=0
when
cos q
= -1/e . The conic is a hyperbola.
Clearly for orbital motion
we must restrict ourselves to the case 0< e < 1 which is elliptical
motion.
.
M = Gravitating body
m = satellite
r,q
= position of satellite
l = semi-latus rectum
.
ELLIPTICAL
MOTION
For the general position
r,q,
we see that if e = 0 then r = l = constant for all q
and the
orbit is circular, simplifying the analysis.
Analysis of Motion We start with two
basic laws:
F = - mv2
r
Circular motion of
m about M
Gravitational
attraction over distance r
F = - GmM
r2
and the standard circular
motion results:
v = rw
w =2p
T
v2
= 4p2r2
T2
from these it is easily
demonstated that the time period of the orbit and the radius
are related by:
T2
= 4p2
r3 GM
Kepler's Third
Law
From
a simple plot of T2 against r3 the mass of a planet
may be determined from the gradient.
