Friday, May 13, 2016
The Orbital Period for Gadbury's 1672 Data
When trying to fit observations of the Sun's position to an elliptical orbit one needs to know the mean motion or the period. But when working with a set of data points subject to error this can be a little difficult but not impossible. We can use Gadbury's 1672 positions for the Sun to illustrate the process. To separate the linear and sinusoidal parts of the motion we first do a least squares linear fit of the ecliptic longitude, θ. The matrix f contains two column vectors with the values of the first being all equal to 1 and the values of the second are equal to the row index, k. The difference between the observed and fit values is δ. As mentioned before the least squares fit can skew the fit to get a lower value for the sum of the squares of the error so there is still some mean motion in δ. We can get around this by fitting a series of sinusoidal curves and subtract this from the original data to get a more linear curve θ'.
Repeating this process slowly extracts the linear mean motion and the slope of the line converges to a fixed value. This slope can be used to compute the period, T, of the orbit since the Sun moves through 360° in this time. The value found is very close to that of the tropical year which is the length of time it takes to go from one Spring Equinox to the next.
You may have noticed that a value for the semi-major axis of the orbit was missing from our set of orbital elements but this can be computed using Kepler's Third Law. One can fit the curve above to a constant term plus an exponential curve if the decay rate is known and search for the rate that produces the least error for the fit.