It is sometimes more useful to think of orbital motion in terms of
the energy of a celestial body rather than in terms of the forces
that are acting on it. In particular, we will concern ourselves with
kinetic energy - which represents the energy associated with a
body due to its motion - and (gravitational) potential
energy - which represents
the energy possessed by a system by virtue of the relative positions of
its component parts.
If we imagine a satellite of mass m a distance r
from the centre of its planet of mass M (M>>m)
moving at a speed v.
Then using Newton's law of gravitation and equating it to the centripetal
force, we find:
GMm/r^{2} = mv^{2}/r.
The kinetic energy of the satellite, E_{k}, is given by
E_{k} = ½mv^{2} = GMm/2r,
which shows that E_{k} is always positive. The
gravitational potential energy, E_{p} is given
by
E_{p} = - GMm/r.
E_{p} is always negative
because gravitational forces are attractive and
E_{p}=0 when the
satellite is at an infinite distance away from the planet.
The total energy of the satellite, E, is then given by
E = E_{k} + E_{p}
= - GMm/2r.
The total energy of the satellite is hence negative, which means
that the satellite will never be able to escape from the gravitational
pull of the planet.
The law of conservation of energy tells us that, in the absence
of forces other than gravitational forces, the sum of the kinetic energies
and mutual potential energy of the bodies in an isolated system remains
constant, i.e.
E = constant.
This means that
as the satellite moves in its orbit around the planet, its kinetic and
potential energy may be interchanged, but the total energy of its orbit
remains constant. At periapsis, when the satellite is at its closest to the
planet and hence moving at its fastest, the kinetic energy component of the
total energy is at its greatest and the gravitational potential energy is
at its minimum (i.e. its most negative value). As the satellite climbs out
of the gravitational potential well of the planet towards apoapsis, gaining
potential energy as it does so, it is slowing down and hence losing kinetic
energy.
We are now in a position to consider the variety of orbits that
correspond to different values of the total energy.
Let us suppose that a particle is launched from P in a
direction at right angles to the line SP from the Sun
at S, as shown in
Figure 39.
figure 39:
A family of orbits around the Sun of different total energy.
If the launching speed is close to zero, the energy of the particle
is significantly negative and the particle follows an almost
straight line towards S; strictly, it would be an elliptical
path with an extremely small width and a major axis only very slightly
longer than SP.
At a slightly higher launching speed, the energy of the particle
would be less negative and the orbit would resemble an
ellipse. The launching point P would be the aphelion. At some
higher, but still negative, value of the energy, the orbit would be a
perfect circle with S at the centre. A still further increase
of energy would lead to an elliptical orbit once again, but now P
is the perihelion. Ultimately, a situation
is reached where the total energy is precisely zero. The orbit
in this case is a parabola. The particle moves further and further from the
Sun, approaching infinite distance with vanishingly small speed.
Any further increase in launching speed gives the particle a
positive energy and produces a trajectory that is a hyperbola - the
particle now approaches infinity with a significant speed.
Hence it can be seen that orbits with a negative
total energy are either circular or elliptical in shape. These are
known as bound orbits. An orbiting body in a bound orbit
cannot escape unless energy is supplied to make the total energy
positive. When the total energy of an orbit is positive, it is
known as an unbound orbit, and it is parabolic or hyperbolic
in shape. A particle in a parabolic or hyperbolic orbit will never
return to the Sun.
It can be seen from Figure 39 that,
near P, it is quite difficult to tell whether the orbit
of a particle is a circle, an ellipse, a parabola or a hyperbola.
This can be a problem when astronomers attempt to analyse the orbits
of comets and asteroids; the bound orbit of a comet
with a total energy only slightly less than zero is almost
indistinguishable from that of a hyperbolic orbit of small positive
energy if the comet is visible only when it penetrates the inner
regions of the solar system.
We can now put together all that we have learnt about celestial
mechanics to calculate an orbit of a particle given some initial
conditions. Suppose that a particle of mass m is launched
with a velocity v_{0} from a point P, at a
distance r_{0} from the centre of force F
(of mass M), as shown in Figure 40.
How do we deduce the size, shape, and orientation of the subsequent
orbit?
figure 40:
Calculating an orbit from initial conditions.
The first thing to do is to test whether the total energy is
positive or negative (or perhaps, by chance, zero). Only if it
is negative will we have a bounded orbit, and we shall limit our
attention here to such cases, i.e. to closed elliptical (or
circular) orbits. From the values of v_{0} and
r_{0} we know the total energy of the particle:
E = ½mv^{2}_{0} -
GMm/r_{0} = - GMm/2r_{0}.
But, recalling from our study of the
ellipse
that the semi-major axis, a, of an elliptical
orbit is equal to the mean distance of the particle from the centre
of force, and given that the total energy of the orbit is conserved,
we can write
E = - GMm/2a.
Thus we know the distance a.
Next, we can use the fact that the directions of v and r are
orthogonal at periapsis (r_{1} and
v_{1}) and apoapsis (r_{2} and
v_{2}) to write the scalar magnitude of the
angular momentum, L, which is conserved, as
L = mv_{1}r_{1} =
mv_{2}r_{2}
or
1/r_{1} = mv_{1}/L,
1/r_{2} = mv_{2}/L.
We can insert the value of 1/r as defined by either of these
equations in the expression for the total energy at apoapsis or periapsis:
½mv^{2}_{1} -
GMm^{2}v_{1}/L = - GMm/2a.
This quadratic, when solved, has as its roots the value of
v_{1} in terms of the other (known) quantities in the
equation. v_{2} can be found in a similar manner
and it is then a simple matter to deduce r_{1}
and r_{2} using the angular momentum equation given above.
This then fixes the eccentricity of the ellipse, through the relations
r_{1} = a(1-e),
r_{2} = a(1+e).
Finally, the orientation of the major axis, relative to which the
initial position vector makes an angle
_{0}, is
determined through the
polar equation of an ellipse derived earlier.
r_{0} = a (1 - e^{2}) /
(1 + e cos).
Thus the orbit is completely specified.
Up to now, we have been concerned with the orbits of two bodies
acting under the influence of their mutual gravitational attraction.
When Newton formulated and solved this two-body problem and successfully
applied it to problems involving the orbits of the planets around the Sun
and the Moon around the Earth, he owed his success to two fortunate
circumstances. First, the planetary masses are all small compared
to the Sun's mass, just as the masses of the satellites are small
compared to the masses of their planets. Hence the mutual
gravitational attractions of the planets are small in comparison to the
Sun's force of attraction on each planet. Second, the diameters of the
planets are small compared with their distances from each other and from
the Sun. Newton realized, of course, that for accurate work in
celestial mechanics, the attraction of the other planets and their
shapes and internal constitutions had to be
taken into account. Solutions to this many-body problem have
occupied the minds of the most eminent astronomers and mathematicians
for the last three centuries. This is a vast and complicated
subject and lies well outside the scope of the present course. For further
reading, see Chapter 14 of Roy and Clarke.