General DescriptionKepler's Laws apply to elliptical orbits involving only two bodies. When three bodies are involved we have to make some changes and approximations if we wish to use simple orbital mechanics. In the case shown below, a space probe is sent into orbit around the Sun heading toward the planet. This orbit follows an elliptical orbit around the Sun. The planet does not interact with the probe until it approaches close enough for the planet's gravity to be stronger than that of the Sun's. This region is called the "Sphere of Influence". |
Inside this region, the probe is in orbit around the planet rather than the Sun. The probe entered the sphere of influence with more than the escape velocity of the planet, and as a result will escape the sphere of influence. While near the planet the path will be a hyperbolic one rather than elliptical and its center of motion will be the planet rather than the Sun. Once the probe leave the planet's sphere of influence it will again be in an elliptical orbit around the Sun - but different from that when it approached the planet. |
The Gravitational Force varies in proportion to the Mass of the attracting body and inversely with the square of the distance to the body. The Sun is much more massive than any of the planets and its gravity dominates the Solar System. Only quite near the planets, does the planetary gravity become stronger than that of the Sun.The first reaction is to determine the point between Sun and Planet where their gravitational forces cancel out. This would be OK if we were not considering a moving objects. Since there is orbit acceleration as well as a gravitational one, these have to be considered also. Laplace derived the following for the Radius of the Sphere of Influence
r_{P} = D_{SP}
[M_{P}/M_{Sun}]^{2/5}
The Sun is 1047 times more massive than Jupiter
(M_{Sun}/M_{J} = 1047). The distance between the two is 5.02 AU
(AU = Astronomical Units = 150 million km). So
r_{J} = (5.2 x
1.5x10^{8}km) [1/1047]^{2/5} = 48.3 million km
The most
distant satellites of Jupiter are about 1/2 this distance away from the planet.
Jupiter's radius is 73,500 km
so in terms of the planets radius,
R_{Jupiter}: r_{J} = 657 R_{Jupiter}
Earth:
Earth is 1/333,000 as massive as the Sun and only 1 AU away.
It's sphere of influence is thus:
r_{E} = 1.5x10^{8}km
[1/333,000]_{2/5} = 927,000 km
= 2.4 x distance from Earth to the
Moon
= 145 R_{Earth} (145 times the radius of the Earth)
In order to properly describe the motion of the probe with in the
sphere of influence of the planet, we must know its velocity with respect to the
planet. The motion in the solar sysem is known relative to the Sun and we must
transform the velocity relative to the Sun, v_{o} to one relative to the
planet, v. We must know the velocity of the planet, v_{p}, relative to
the sun. We must also know where the space probe enters the sphere of influence.
The diagram to the right illustrates the parameters involved.
The velocity
relative to the planet is given by subtracting the planet's velocity from the
space probe's.
v = v_{o} - v_{p}
Since velocity has direction and magnitude, we need to determine both. The result of a little trigonometry givesSince the space probe enters the sphere of influence with a net
velocity inward, the probe will have a velocity relative to the planet greater
than the escape velocity of the planet. This produces a hyperbolic orbit within
the sphere of influence. All orbits (elliptical, parabolic, and hyperbolic) are
described geometrically by,
r = a ( e^{2} - 1) /( e cos f + 1)
WhereWhen the probe is very far from the planet (entering the sphere of
influence), then it is at its maximum angle , f_{m}.
e cos f_{m} + 1 = 0
and this gives a relationship with
the eccentricity of the orbit.
e = -1/cos f_{m}
The impact parameter, b, is the distance
from the planet that the probe would have it travelled in a straight line inside
the sphere of influence. This parameter along with the velocity, v, determines
the deflection angle, 2q.
b = [GM/v^{2} ] cot q
M = mass of the planetThese relationships will be important in deternining the effect of the planet on the space probes orbit. One useful property of the orbits is that they are symmetric about the semi-major axis. The velocity of infall before perijov (closest point of approach to Jupiter) will have the same magnitude as the outgoing velocity at the same distance from perijov. We will use this in our example.
Let's set up a realistic example of using Jupiter to increase the
energy of a space probe as it flys by Jupiter.
We will have to do this in
several separate steps in order to not get to confused.