Moon Motion Relative to the Earth-Sun Line

  I was looking at the Moon's motion relative to the line through the centers of the Earth and the Sun and found what appears to be a Lissajous curve for the direction Moon.

This track has a three dimensional structure.

This plot of the focus and point of view shows the observer's relative position to the Moon's orbit.

The colors of the eye positions indicate the color of the lenses for the 3D glasses. The corresponding curves have the opposite colors since they are blocked by a lens so the eye will see a black line with the proper relative position.

Edit (Aug 28): Corrected the horizontal axis label in the 3D image.

Supplemental (Aug 28): Here's another view with a large black dot showing the relative position of the Moon for noon on the day of the eclipse. Remember there's a change of scale for the vertical axis. Note the Moon's orbit is inclined relative to the ecliptic by about 5 degrees and the sine of 5 degrees is about 0.087.

The Moon appears to have been below the ecliptic on the day of the eclipse.

Supplemental (Aug 29): I was concerned by the relatively low position of the Moon position on the plot for the day of the Aug eclipse but we are viewing the Moon's orbit from the plane of the ecliptic. Viewing from below the ecliptic and bring the Moon closer to the Earth-Sun line.

Creating a 3D Plot of Data

  One may ask how the 3D plot of the Moon's position was created. First one needs to convert distance, RA and DEC data into x, y & z coordinates. Then one needs to select a point for the focus of attention, f, and the relative observer position and the positions of eyes.

The direction vectors associated with spherical coordinates were used to determine a frame of reference for each eye. To keep track of everything it helps to run a check on the intersection of the lines of sight of each eye, f '.

After translating the original x, y, z positions one gets the following 3D image.

The left eye is the cyan curve and the right eye is red.

The Moon's Motion in 2017

  Here's a 3D plot of the Moon's orbit for 2017. I tried to lighten the red a little to reduce the residual red traces. The apparently vertical sides are due to the stretched vertical axis. The plane of the Moon orbit appears to be rotating about the to & fro axis. It's easiest to view in three dimensions if you track the black lines that you see with red/cyan 3D glasses on. Move towards and away from the screen to find the best viewing distance.

Here's the same with the axes and gridlines removed.

The small dots are daily noon positions and the large dot is the last position, New Year's Day 2018. The the horizontal axis was reduced slightly to increase the right margin.

Lunar Orbit for Month of Aug 2017 Eclipse

  Here's an exaggerated 3D view of the Moon's orbit for the month of the Aug 2017 eclipse. The point at the origin is noon on the day of the eclipse. The units for the axis are angles given in radians. Red-cyan 3D glasses are required for viewing.

The Moon appears to have been deflected by its close approach to the line through the centers of the Earth and the Sun.

Eclipse Prediction

  There are rumors that there will be an eclipse next week. How do we know this is so? We can start with HORIZONS data for the daily noon positions of the Sun and Moon in geocentric equatorial coordinates for the year 2017. Next we convert RA and DEC into the spherical angles φ and θ in radians to determine the angular separation of the Sun and Moon.

The first plot is Δφ and the jags occur because Δφ is always increasing and we only need to know the relative angular distance between them and when it crosses the line Δφ=0. The second plot tells us when Δθ=0. But for an eclipse to occur both conditions need to be satisfied at approximately the same time. So we look through our table of Δφ and Δθ for nearly simultaneous crossings. The occurs twice in 2017 on or about Feb 2 and Aug 21.

The eclipse on Feb 26th has already occurred so we'll focus on the Aug 21 eclipse candidate. Another plot gives us a better picture of when both Δφ and Δθ cross the horizontal axis.

It's difficult to tell exactly when the Sun and Moon will be closest together since they are moving at different rates but a calculation indicates the minimum separation is just under half a degree. The directions of the Sun and Moon are e_S and e_M respectively.

This looks promising since the Moon's paralax, the shift in angle for moving from the subsolar position on the Earth surface to the Earth's limb, is about 0.0179 rad and adding the apparent angular radius of the Moon, 0.0043 rad, and the angular size of the Sun, 0.0047 rad, we get a maximum allowable separation of 0.0268 rad. Adding a box to indicate these bounds to a plot of Δθ vs. Δφ indicates that there will indeed be an eclipse at this time.

The point on the curve closest to the origin is towards the end of the eclipse which would explain the relatively later time.

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.

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.