LightSail 2's Behavior in Orbit

  The Planetary Society's LightSail 2 is to go into orbit aboard SpaceX's Falcon Heavy later this year if all goes well. The initial orbit will be a circular orbit 720 km above the Earth's surface. The light sail will face the Sun as it moves away from it and the plane of the sail will lie along the direction of a line from the Sun as it moves toward it. The effect of switching back and forth between its boost and cruise phases will be a slow but steady increase in the satellite's orbital energy.

The following figure defines the plane of the orbit with the x-axis pointing towards the Sun. The unit vector n indicates the direction of the acceleration caused by the light pressure on the sail during the boost phase. The equations below indicate the specific radial and angular forces acting on light sail as it orbits the Earth. μ is constant GM for the Earth.

The set of equations above are difficult to solve analytically but it's not too difficult to do a numerical calculation. The following calculations used 1 second steps in time. While in cruise phase the satellite coasts in a Keplerian orbit. The following table gives values for the selected points of the first orbit. The units for time, angle and radius are seconds, radians and meters. The highest and lowest points of the orbit are the apogee r_a and perigee r_p. The cruise phase was allowed to continue past the switching point at θ=2π in order to get the values for the perigee. Interpolation was used to get a better estimate of the values.

From this table we can compute some of the orbital elements for the Keplerian orbit of the cruise phase. There is a gain in energy during the boost phase.

The light sail will spiral away from a ballistic object in the same initial circular orbit since their angular separation will increase over time in addition to the radial separation.

The separation viewed from the perspective of revolutions shows the alternating crossing of the axes over time.

Supplemental (Jul 28): Two additional plots to clarify relative positions. The first is Δr=r-r₀ where r₀ is the radius of the circular orbit.

The second is the angular separation of the light sail from a ballistic object in the original circular orbit.

Edit (Jul 28): Found an error in the first two plots and removed them.

Edit (Jul 28): Found an error in the Δθ plot. It was a dumb mistake. You can't subtract radians from degrees. Replaced plot. Found a minor error in the spiral plot calculation which didn't affect the result much. Left plot as it was. I used 1 minute steps for the longer time period used in all the plots so there is a little additional error present.