View Full Version : Keplerian 'best fit' vs. 'mean' elements

2014-Apr-25, 10:16 AM
Since a year I have developed an amateur interest in celestial mechanics - for a better understanding how 'the world goes round'. During my autodidactic studies I found E. M. Standish's paper Keplerian Elements for Approximate Position of the Major Planets (http://iau-comm4.jpl.nasa.gov/keplerformulae/kepform.pdf). I have programmed the schedule with the given 6 constant pairs for our planet earth and tested it vs. MICA (http://aa.usno.navy.mil/software/mica/micainfo.php). Between 1800 and 2050 the heliocentric longitude \lambda =\arctan\frac{y'}{x'}+\varpi deviates from the corresponding MICA output less than 0.006 degrees - very impressing to me (MICA-Input: Earth, Heliocentric Ecliptic, Positions, Mean Ecliptic and Equinox of J2000.0).

Now for my two questions:
1. What is the difference between best fit Keplerian elements and mean Keplerian elements? Up to now I just understood them as different names for the same because the best fit math is the way to find the optimal coefficients of an ansatz. And these coefficients (I thought) are called the mean coefficients. But Standish warns: 'Such elements are not intended to represent any sort of mean.'

2. To check my understanding of the model I tried to calculate the time T_{VE} of the vernal equinox of a certain year - without succes. My first (and only) guess was that \lambda is always equal to 180 degrees at T_{VE}. But that is not the case. At T_{VE} (read from MICA) of the years {1800, 1900, 2000, 2050} I calculate \lambda={182.8, 181.4, 180.0, 179.3} degrees by Standish and MICA tells me the same values (same input path as above). Would anyone be so kind to give me a hint how to gain T_{VE} from the Kepler model? What is the criteria for T_{VE} in terms of the model variables and constants? The math like solving nonlinear equations is not my problem. I am also familiar with the meaning of coordinates, anomalies etc.. My deficit is the ansatz.

Kind regards