Sunday, August 13, 2017

The Sun's Apparent Motion in the Plane of the Ecliptic

  I redid the series of fits for the Sun's apparent position this time in the plane of the Ecliptic. The primary motion is of course a Keplerian ellipse. The horizontal and vertical axes are the major and minor axes and at the beginning of the year the Sun in near perigee on the right and moves upwards. The units are AU.

The Keplerian elements for the fit are as follows.

The residuals of this fit form a rose curve which appears to be due to solar pulls and torques acting on the Moon's orbit. Again the units are AU.

The residuals of the second fit are down to μAUs and more random in appearance. There's an odd step in the direction of increasing perigee at the end of the year. Could the Earth be slowing down and spending more time at perihelion? What effect would that have on global warming?

Fits can produce some deviations when all the error isn't accounted for.

Wednesday, August 9, 2017

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.

Wednesday, August 2, 2017

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.

Tuesday, August 1, 2017

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.