The relationship between the altitude of an anti-radial burn on apogee

I recently conducted an experiment using Kerbal Space Program to analyze the effect of burn altitude on orbital properties. Unfortunately im a bit of a rookie when it comes to orbital dynamics so I'd appreciate if someone could explain the data I've been getting.

Details of experiment: An anti-radial burn of 50m/s will be conducted at heights of (0%, 20%, 40%, 60%, 80%, and 100%) of orbital altitude. The starting orbit has a semi-major axis of 30,000,000 m, inclination of 0, and eccentricity of 0.9. After conducting the burn, changes to the Apogee (main focus of experiment), perigee, inclination, period, and eccentricity will be measured.

The problem is that I cannot seem to find a mathematical relationship online between altitude of the burn and apogee and I cannot explain the data I have found. It is periodic by nature, but the relationship is very odd.

These are the resultant apogees from my experiment. Note: The x-axis labels are as follows (1=0%, 2=20%, 3=40%, 4=60%,5=80%,6=100%,7=80%, 8=60%, 9=40%, 10=20%, 11=0%. and 1-5 are while ascending and 7-11 are while descending.)

The mathematical relationship is the orbit equation

$$r={\ell^2\over m^2\gamma}{1\over1+e \cos{\theta}}$$

Decomposing the angular momentum $\ell=mvr\sin{\alpha}$ it can be expressed as

$$r={\gamma\,{(1+e \cos{\theta})}\over v^2\sin^2{\alpha}}$$

Where the height of the apoapsis will be

$${\gamma\,{(1-e)}\over v^2\sin^2{\alpha}}$$

In order to figure out the effects, we can analyze what the burn changes in each component:

• change in the angle of the velocity vector $\alpha$ - if the spaceship is ascending, the sine of the angle increases, if the spaceship is descending, the sine component decreases. The change will be larger when the burn is relatively big compared to orbital velocity.
• change (increase) in the velocity component $v$ - since an anti-radial burn is perpendicular to the original velocity vector, the lower the initial velocity, the bigger the increase (both absolutely and relatively)
• eccentricity $e=\sqrt{1+{2E\ell^2\over m^3\gamma^2}}$ will generally increase when burning anti-radial when descending, and decrease when burning while ascending, as the momentum term dominates. If neither ascending nor descending, eccentricity would increase slightly due to increasing speed.
• angular momentum $\ell=mvr\sin{\alpha}$ will generally increase when the ship moves slowly near the apoapsis or when the ship is ascending, decrease otherwise, based on the analysis of its components above
• orbital energy $E$ is the sum of potential and kinetic energy, and will increase, as kinetic energy is $mv^2/2$ and will increase due to $v$ increasing (the higher $v$ is, the larger the increase in energy), potential energy remains unchanged at the moment of the burn
• note that for elliptical orbits $E$ is negative, so increasing $E$ means changing towards zero, increasing $\ell$ reduces $e$
• $\gamma$ is constant for a specific body

The angle component will be the dominant one for burns smaller than the orbital velocity, except when the spaceship is moving horizontally (the sine function changes slowly around π/2).

The empirical results seem to support this:

• when the spaceship is ascending, an anti-radial burn will lower the apoapsis (divide by a larger sine of the angle of velocity vector)
• when the spaceship is descending, an anti-radial burn will raise the periapsis (divide by a smaller sine of the angle of velocity vector)
• when the spaceship is flying level, an anti-radial burn would raise the apoapsis slightly due to increasing eccentricity (through increasing velocity)
• the first two effects will be larger when the ship moves slower (is higher), due to the angle changing more. This is clearly visible in the graph.
• I'm not sure if the third will be larger or smaller when moving slower, as, on one hand, the velocity increases more, on the other the energy gains are lower when moving slower. Empirically they look similar.

(Though, for the record, if you care about the apoapsis height, it's most efficient to burn prograde or retrograde at the periapsis.)