Unbound Orbits
We now move on to unbound orbits. These are of much less importance for our concerns, so we won’t talk too much about them. However, unlike what you might think, they aren’t unphysical; examples include comets originating from the Oort cloud, or perhaps even having interstellar origin. You can check this wiki page for particular examples.
As we proved in chapter 3.1, trajectories having energy greater than or equal to zero are unbound. We also saw that these trajectories have eccentricities greater than or equal to 1, and thus are parabolic or hyperbolic in nature. We’ll briefly go over both of them. You should at least have some knowledge of these conic sections before proceeding further.
Parabolic trajectory
A parabola is a conic section with eccentricity equal to 1. The vertex point is the point of closest approach and is closest to the focus. If the distance to the focus from the vertex of the parabola is , the length of the semi-latus rectum is . The equation of the parabolic trajectory is then:
The angular momentum of such a trajectory is from eqn. . Because the total energy of a parabolic trajectory is zero, the potential energy must cancel out the kinetic energy. Using this we get that the velocity at any point, at a distance from the focus, is equal to the escape velocity (if it was moving in a circle of radius ):
The velocity at the point of closest approach is
Hyperbolic trajectory
A hyperbola is a conic section with eccentricity greater than 1. The vertex point is the point of closest approach and is closest to the focus. Let the semi-major axis of the hyperbola be . By convention, we define it to be a negative quantity. This is done so that the equations match with those of an elliptical orbit. The distance of vertex from focus is
The semi-latus rectum is
The equation of the hyperbolic trajectory is
Both of which are identical to the formulas for ellipses. We see that the true anamoly can only take values between and . This wasn’t an issue earlier, because was less than 1 for a bound orbit. Note that the bound for a parabola is
The angular momentum of the system is , and the total energy is given by . Note that the energy is positive, due to being negative. The velocity of the body at a distance from the focus is given by
The velocity at the point of closest approach is
The velocity at infinity is
Variants of Kepler’s Equation
In a parabolic trajectory, Barker’s equation relates the time of flight to the true anamoly . If the time of periapsis is , then
In a hyperbolic trajectory, the hyperbolic Kepler’s equation relates the mean anamoly to the hyperbolic anamoly :
where the mean anamoly is defined as . The true anamoly and hyperbolic anamoly are related by
Radial trajectories
A radial parabolic trajectory is a degenerate case of an parabolic trajectory, where the eccentricity is still equal to 1, however the angular momentum of the system is zero. The trajectory is a straight line, and the two bodies move away from each other, the velocity tending to zero as they get infinity far away from each other. It is classified as an parabolic trajectory since the total energy of the system is zero.
A radial hyperbolic trajectory is a degenerate case of an hyperbolic trajectory, where the eccentricity is equal to 1 and the angular momentum of the system is zero. The trajectory is a straight line, and the two bodies move away from each other, having a finite velocity at infinity. It is classified as an hyperbolic trajectory since the total energy of the system is positive.