Equations and Spheres


As I mentioned last week, some of the resources we can utilize in space are orbital dynamics themselves.

The Clohessy-Wiltshire equations describe how a small ‘chaser’ spacecraft will behave in the reference frame of a larger target in a near-circular orbit. Fortunately, these conditions do a good job describing a small inspection satellite crawling the surface of the ISS. The equations describe the chaser’s dynamics in the reference frame fixed to the target’s center of mass.

The Clohessy-Wiltshire equations are traditionally written in Cartesian Coordinates:

    \[\ddot{r}_x = 3n^2x+2n\dot{y}\]

    \[\ddot{r}_y = -2n\dot{x}\]

    \[\ddot{r}_z = -n^2z\]

Where n is a constant of the target’s circular orbit \sqrt{\frac{\mu}{R^3}}

Is it possible for this chaser to ‘stick’ to the target? Yes, if you’re in the right place. The CW equations are not new, so one would expect this question to have a well-established answer. However, without a contactless actuator allowing the chaser to safely maneuver close to the target, you didn’t want satellites anywhere near each other, let alone ‘sticking’ to one another. web search history Where does this relative attractive acceleration occur? Wherever the radial component of the relative acceleration is negative – that is, orbital mechanics cause the chaser to accelerate towards the target.
The radial component of acceleration in spherical coordinates is complex if the chaser is moving in the target frame (that is, \dot{x},\dot{y} \neq 0. However, if the radial acceleration of a stationary chaser on the surface of a spherical target of radius r is

    \[\ddot{\rho} = n^2 r\left( 3 \sin^2{\theta}\cos^2{\phi}-\cos^2{\theta} \right)\]

This leads to the condition that if 3\tan^2{\theta}\cos^2{\phi} \leq 1 a stationary chaser will constantly accelerate towards the target.

Where does this acceleration occur? Darker blue areas on this plot show negative/inward radial accelerations.

acceleration surface

This initial exploration leaves a lot of unanswered questions: What happens when these forces are mapped onto the surface of a real target like the ISS? Is it possible for an inspection satellite to take advantage relative velocity to safely enter the red ‘repulsive’ zones? If so, what would those trajectories look like?