Wednesday, January 27, 2016

Derivation of the Ellipse Fit Formula

  One can derive the linear formula for the angular velocity fit by starting with the first two of Kepler's laws. The constant coefficients can be solved for using linear least squares.

The error that results from fitting the ecliptic longitude with the linear function plus some Fourier terms (δ below) is mainly due to the motion of the Earth about the Earth-Moon barycenter. Using the barycenter as the frame of reference would probably result in less error. The plot covers 3 Julian years and one sees there is a little more than 12 months per year per Julian year.

The coefficients for the λ fit in radians are as follows.

Looking at the error in the computed longitude can see how perturbation theory might have gotten its start.

Determining Orbital Elements From Observations

  It has been reported in the news lately that there are hints of the existence of a large but unobserved 9th planet so it might be useful to look at how one can determine an orbit directly from observations. This is not the approach that one usually finds in astronomy books where one goes from orbital elements to positions. The Sun's apparent motion in the heavens is the simplest example and we can try to determine it's motion relative to the Earth. In lieu of observations we can use the positions of the Sun in the geocentric coordinate frame from an astronomical almanac. I used the US Naval Observatory's MICA computer almanac to generate a set of daily positions at noon for the years 2000-2003. These can be expressed as latitude, θ, and longitude, φ.

  From these coordinates on can determine the directions of the Sun, er, for the time differences, Δt, from noon on Jan 1, 2000. These directions also allow us find a matrix, P, which will enable us to determine the inclination of the ecliptic, ι, and the ecliptic longitude of the Sun, λ, for given Δt. B is the basis for the coordinates in the ecliptic reference frame. One has to adjust the longitudes computed using the angle function to get the approximate linear increase shown in the plot.

  We can separate out the mean motion of the Sun along the ecliptic by fitting a linear function of time plus a simple Fourier series using linear least squares. Since the longitude is not a linear function of the mean daily motion, n, we have to search for a value which minimizes the variance, V.  We can then let M be the mean ecliptic longitude which is the linear part of λ.

  The nonlinear portion can be used to determine the time and longitude of perigee where the difference between the two longitudes is zero. The longitude of perigee is found by interpolation. The longitude measured from perigee is the true anomaly, ν.

It can be shown that the square root of the time rate of change of the true anomaly is a simple sinusoidal function of the true anomaly. This allows us to use the true anomaly to estimate the eccentricity of the orbit.

To determine the equation for the ellipse in the plane of the orbit and semimajor axis, a, we have to make the assumption that the mean radius of the orbit is 1 AU. The other parameters can be found by using standard formulas in the astronomy books.

The values of the Keplerian elements found compare well with those of the Earth's orbit.