Found an Odd Error for a Fit of the Sun's Position

  I was preparing for the solar eclipse later this month, doing a fit of the Sun's relative position from the center of the Earth, and the fit didn't turn out as I expected. A linear least squares fit of the Sun's position for the functions indicated resulted in the following relative differences between the fit and the calculated positions.

The sinusoidal function is mainly due to the Moon's pull on the Earth but there appears to be another component present. How can one explain the displacement of mean error from zero? It turns out there are some Chebyshev polynomials present.

This polynomial series generates a displacement of the following form given in AU.

After subtracting this and the displacement due to the pull of the Moon the following error remains.

The error can be measured in micro AU (μAU). For comparison the Earth moves about the Sun at a mean rate of 2π/365.24=0.0172AU/day=12μAU/min. It probably wouldn't hurt to get accurate measurements of the start and end eclipse times.

Essentially the same error or perhaps correction is present in both MICA and HORIZONS data for the Sun's position relative to the Earth.

With Light Sail Acceleration in the Direction of Sunlight Preferred

  One can study various modes of light sail operation to see how it will perform. The parasitic reduction of perigee seems to be associated with acceleration in the direction of sunlight and requires more energy for insertion into a higher circular orbit. The following mode of operation prefers acceleration in the direction of sunlight. The attitude of the sail is directed as follows.

The relative change in the apogee of the sail increases with time but the perigee also decreases even if a small 10 second step is used in the calculation.

Again, more Δv is required to put the light sail into a circular orbit than for a ballistic transfer orbit. More time would be required to compensate for the loses when the Δv is in the proper direction orbit insertion.

If nature favors some modes of light sail operation over others the preferred mode would make the desired orbital changes in the least possible time. It may require a little R & D to determine the optimal solution. The rate of the gain in altitude at apogee is greatest for α=0 when the light sail moves in the direction of sunlight. If the drop in perigee isn't compensated for light sail won't climb out of the Earth's gravity well before it encounters atmospheric drag. So one will have to do the climb stepping from one circular orbit to a higher one.

The Effect of Changing the Step Size on the Light Sail's Orbits

  Since the light sail's orbit appeared to show signs of a deteriorating perigee I tried playing with the attitude of the sail relative to the direction of sunlight to get relatively more angular acceleration. In the figure below the direction on sunlight is to the left. The light sail's acceleration is along its normal n which is at an angle α relative to the horizontal axis. The changes to the equations of motion are also shown.

The angle of the sail was set to α=π/2-θ when it was above the x-axis and moving away from the Sun and α=π/2 when it was below in order to restrict the acceleration some. The behavior was sensitive to the size of the step so Δt was set to 10 seconds. This gave better results for the behavior of Δr=r-r₀.

It still drifts away from a ballistic object in the original circular orbit due to changes in the eccentricity and the period of the orbit.

Changing the step size resulted in a closer match of the light sail's Δv's with the transfer orbit  Δv's for insertion into the higher circular orbit.

So it appears that a light sail can make changes to a transfer orbit. Unfortunate, the apogee is on the sunward half of the orbit and one can't use sunlight to accelerate towards the sun to go into a higher circular orbit. This is where an auxiliary propulsion system would be needed if one wanted to do orbit changes. A flyby mission would not require orbit insertion.

Supplemental (Aug 1): After the passage of half a year the sunlight will be in the opposite direction (to the right in the first figure) and the Δv will be directed properly for insertion into the higher circular orbit. It may be possible for a light sail to slowly work its way out of a gravity well.

LightSail 2 Needs to Compensate for the Decrease in Perigee Periodically

  One can compare the energy change in LightSail 2's orbit that with that required to go into a transfer orbit. After a number of revolutions however the perigee of the light sail starts to drop. This loss of energy has to be compensated for if one desires to go into a circular orbit at the light sail's apogee.

In the plot above the blue curve indicates the velocity change Δv required to go from a point of the light sail's orbit into a circular orbit. The solid red line is the total Δv required to go from the original circular orbit into a circular orbit at the apogee of the transfer orbit. The dashed red line indicates just the Δv required to go from the apogee of the transfer orbit into a circular orbit. Initially the orbit injection Δv's of the light sail match up with the Δv's for the transfer orbit. After about six revolutions more Δv is required to compensate for the drop in perigee. The implication appears to be that the orbit of the light sail needs to be periodically corrected to a circular orbit to keep the light sail from becoming parasitic. The major deficiency of the light sail is that the direction of its thrust is limited. It's lack of angular acceleration requires an auxiliary propulsion system for some orbit changes.

Supplemental (Aug 1): The deviations in the Δv's for the light sail when going to the higher circular orbit are affected by the step size. Here the step size was Δt=60 seconds. Compare next blog.

How Long Would It Take LightSail 2 To Escape From the Earth?

 One can compute LightSail 2's gain in energy over time and estimate the average rate at which it will gain energy.

This average rate appears to be linear and allows one to estimate how long it would take for the light sail to acquire enough energy to escape from the Earth's gravity well. The answer is 6.4 years.

The model used was overly simplified neglecting the Earth's shadow and didn't take into account the need to raise the height of perigee to avoid atmospheric drag. As the Earth moves about the Sun in its orbit the direction of sunlight will slowly change.

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.

Refractive Index of Air as a Function of Wavelength

  In 1908 Rentschler published a study on the refractive index of a gas for different wavelengths in the Astrophysical Journal. I used his data for air in Table II to find the coefficients of a Cauchy formula to fit the data. One can also use the NIST calculator to get data for air and do another fit.

The Rentschler data appears to be for dry air. If one computes the index of refraction for the CRC Handbook data using the given wavelengths for air and computed wavelengths for the Balmer series for hydrogen the plot appears to be displaced somewhat. Note that the humidity of the air also affects the index of refraction. I used the ratio of the wavelength in air to 5000Å for the formula so it would be easier to substitute wavelengths given in nanometers (nm) using 500/λ with the same coefficients.