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5. A derivation of the conicsection equation. |
Now it is time for us to become a little bit more specific in describing the trajectories that a planet will follow in its orbit. So far we have shown that the orbit of two planets will lie in a plane and the center of mass of the system will travel with a constant velocity. We will now show that the motion of an orbiting body will follow the path of a conic section. Once this is done we will review just what is meant when we say "conic section."
The first thing that we should do to make our analysis of the system a bit simpler is to switch to a polar coordinate system with the large mass at the origin. Secondly we will allowFrom the geometry above we can see that the radial vector can be rewritten as this equation.
our coordinate system to rotate with the orbiting body. To facilitate this request we will designate a unit vector that lies in the direction of the radial vector ur. We will also designiate another unit vector in the direction of increasing q. We will refer to this vector as uq. Figure 5.1 Figure 5.2 Furthermore, we can track the motion of our two unit vectors with the following two equations.
Now, if we take note of the fact that the direction of increasing q is really just the derivitive of the positional vector then we have these two crucial identities:
Our next step is to compute the velocity and acceleration vectors and express them in term of our unit vectors.
and
If we consider an arbitrary force acting on mass m we would likely be motivated to break it into the components that match our coordinate system. This consideration yields
Applying Newton's second law (F=ma) to equations [5.5] and [5.6] and considering the vector components separately leads us to
This gives us a set of differential equations that describe the motion of a body under the influence of any force that occurs in the plane of motion. However, the force that we are studying is far more specific than that. Because we are limiting the scope of our examination to a two body universe, we only need to consider the force that lies on the line r that connects the two bodies. Since this leads us to assume that no forces exist in any other direction, we can also assume that Fq=0. This will compell us to rewrite the first of equations [5.7] as
Now if we multiply through by r and apply the inverse of the product rule for derivitives we will obtain
Upon integration we arrive at the very interesting relationship
where h is taken to be some positive constant. This relationship prove to be very useful in our future ventures toward deriving the equation for conic sections. However, its real power lies in the fact that this is actually a statement of the conservation of angular momentum which is a form of Kepler's second law for orbital motion.
Now we have established some rules for how a body will behave under the influence of a single radial force. Let us now recall that this force is specifically a gravitational force. Recall Newton's law for gravitation:
This leads us to rewrite the second of equations [5.7] as
Now, if we induldge ourselves by taking a few steps away from intuition, we can put this equation into a very solvable form. We will introduce z = 1/r and we will use differential notation to rewrite all t in terms of q.
and
Once forms of equations [5.10], [5.13], and [5.14] are substituted into equation [5.12] and 1/z is substituted for r, we obtain
which leads to
For all you Differential Equations geeks this should come as quite a relief. This linear differential equation bears a striking resemblence to a undamped and forced harmonic oscillator equation. If we think about it that really makes perfect sense. Since we are neglecting any frictional forces acting on the orbiting body the motion will be undamped. Furthermore, we know that the body is being "forced" centripitally by the central body. Now we can easily show that this equation has eigen values of
which leads to
Combining these results leads to
This reduces the solution of the ODE to
In order to solve for the constants A and B we will rely on the geometry of the situation and choose a convenient initial condition accordingly. If we choose the minimum value for r to occur when theta is equal to zero we will make our lives much easier. If r is at a minimum then z will be at a maximum. This implies that the first derivitive of z will be equal to zero and the second derivitive will be negative when theta equals zero. This will imply that A = 0 and B > 0. Figure 5.3
Finally, if we replace 1/r for z in this expression, solve for r, then do a little manipulation we will find ourselves with the following equation
This is the long sought after equation for a conic section. The parameter e = Bh2/k is known as the eccentricity. The value of the eccentricity parameter determines the specific conic section to be traced. Figure 5.4 identifies the type of curve associated with all possible
values of e. Figure 5.4
