Unbound Orbits

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 rp=ar_p = a, the length of the semi-latus rectum is p=2ap = 2a. The equation of the parabolic trajectory is then:

r=2a1+cosθ(3.3.1)\tag{3.3.1} r = \frac{2a}{1 + \cos \theta}

The angular momentum of such a trajectory is h=2μah = \sqrt{2 \mu a} from eqn. (3.1.10)(3.1.10). 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 rr from the focus, is equal to the escape velocity (if it was moving in a circle of radius rr):

v=2μr(3.3.2)\tag{3.3.2} v = \sqrt{\frac{2\mu}{r}}

The velocity at the point of closest approach is

vp=2μrp=2μa(3.3.3)\tag{3.3.3} v_p = \sqrt{\frac{2\mu}{r_p}} = \sqrt{\frac{2\mu}{a}}

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 aa. 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

rp=(a)(e1)=a(1e)(3.3.4)\tag{3.3.4} r_p = (-a) (e - 1) = a(1-e)

The semi-latus rectum is

p=(a)(e21)=a(1e2) p = (-a)(e^2 - 1) = a(1 - e^2)

The equation of the hyperbolic trajectory is

r=a(1e2)1+ecosθ r = \frac{a(1-e^2)}{1 + e \cos \theta}

Both of which are identical to the formulas for ellipses. We see that the true anamoly can only take values between cos1(1e)-\cos^{-1} \left(\frac{-1}{e}\right) and cos1(1e)\cos^{-1} \left(\frac{-1}{e}\right). This wasn’t an issue earlier, because ee was less than 1 for a bound orbit. Note that the bound for a parabola is (π,π)(-\pi, \pi)

The angular momentum of the system is h=μa(1e)h = \sqrt{\mu a (1 - e^)}, and the total energy is given by ε=μ2a\varepsilon = -\frac{\mu}{2a}. Note that the energy is positive, due to aa being negative. The velocity of the body at a distance rr from the focus is given by

v2=μ(2r+1a)(3.3.5)\tag{3.3.5} v^2 = \mu \left( \frac{2}{r} + \frac{1}{a} \right)

The velocity at the point of closest approach is

vp=μ(2rp+1a)=μa(1+e1e)(3.3.6)\tag{3.3.6} v_p = \sqrt{\mu \left( \frac{2}{r_p} + \frac{1}{a} \right)} = \sqrt{\frac{\mu}{a} \left( \frac{1+e}{1-e} \right)}

The velocity at infinity is

v=μa(1e1+e)(3.3.7)\tag{3.3.7} v_\infty = \sqrt{\frac{\mu}{a} \left( \frac{1-e}{1+e} \right)}

Variants of Kepler’s Equation

In a parabolic trajectory, Barker’s equation relates the time of flight tt to the true anamoly θ\theta. If the time of periapsis is τ\tau, then

tτ=2a3μ(tanθ2+13tan3θ2)(3.3.8)\tag{3.3.8} t - \tau = \sqrt{\frac{2a^3}{\mu}} \left( \tan \frac{\theta}{2} + \frac{1}{3} \tan^3 \frac{\theta}{2} \right)

In a hyperbolic trajectory, the hyperbolic Kepler’s equation relates the mean anamoly MM to the hyperbolic anamoly HH:

M=esinhHH(3.3.9)\tag{3.3.9} M = e \sinh H - H

where the mean anamoly is defined as M=μa3(tτ)M = \sqrt{\frac{\mu}{-a^3}} (t - \tau). The true anamoly and hyperbolic anamoly are related by

tanθ2=e+1e1tanhH2(3.3.10)\tag{3.3.10} \tan \frac{\theta}{2} = \sqrt{\frac{e + 1}{e - 1}} \tanh \frac{H}{2}

Radial trajectories

A radial parabolic trajectory is a degenerate case of an parabolic trajectory, where the eccentricity ee 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 ee 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.