A Simpler Orbital Period Estimate

  It's better to include the sinusoidal terms with the linear terms when doing a least squares fit as the following example shows. Again, Gadbury's 1672 Sun positions were used. It's difficult to tell if the period has changed over the elapsed time. The value used for T appears to be stable, yielding the itself back for the estimate of the period. For comparison, the value given in the Explanatory Supplement to the Astronomical Almanac is 365.2421897 days.

The result varies with the number of sinusoidal terms used for the fit. I have some doubts about this fit even though it gave the minimum error.

A Caution on the Extracting Process

  The extraction of the Sun's mean motion will not always work if one does not know the precise form of the sinusoidal motion and the calculation is not done over the approximate period. However, using cosines and sines for the approximate period and half that seems to work fairly well. Below 301 days were used instead of 366. The first day was t=0.

More Detail on the Orbital Period Calculation & a Filter Matrix

  The two step least squares process of extracting the Sun's mean motion from the position data can be reduced to the multiplication of the positions by a single matrix, P.

The linear and non-linear functions can be represented by two matrices, f and g.

Repeated multiplication of the modified positions by P moves us closer to the approximate mean motion and the resulting period can be calculated.

The four decimal place accuracy of the previous result for the period may have been a coincidence but the uncertainty appears to be in the 3rd decimal place. The nonlinear portion of the positions no longer appears to be tilted.

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.

Orbital Elements for the Year 1495

  I was able to get a good fit for Regiomontanus' positions for the Sun for the year 1495 using Legendre polynomials to "smooth" the data. The difference between the two is consistent with rounding off to the nearest minute.

Again the minimum for daily changes in the positions for the smoothed data were used to determine the position of apogee and an ellipse was then fit to the daily changes in position.

The rms deviation between the fitted ellipse and the smoothed positions of the Sun was about 1.5 minutes so the original positions may not have been solely based on observations. The orbital elements are similar to those for 1494 but with a better value for the length of the year.

I tried to do a set of elements for the year 1500 but the difference between the smoothed and original positions showed relatively large deviations near Feb 28.

Supplemental (Apr 9): The fitted elliptical orbit was chosen so as to minimize the rms deviation for the interval from 20 to 320 days after the beginning of the year to make the fit more precise. The use of an eccentric circle may have may have introduce a systematic error into the Sun's positions which in turn may have affected the eccentricity of the elliptical orbit.

29 Feb 1504 Lunar Eclipse

  Christopher Columbus may have had a copy of Regiomontanus' Ephemerides with him in 1504 when he predicted the lunar eclipse for the natives in Jamaica to get them to continue with their provisions while he and his crew were shipwrecked. The Wikipedia article on the eclipse asserts that he had an almanac by the Portuguese astronomer Abraham Zacuto and in another place that he had Regiomontanus' Calendarium. With the rivalry between Portugal and Spain at the time it is unlikely that they would be sharing information concerning navigation. The information in the Calendarium is less detailed than that in the Ephemeris and the position given for the Sun differs somewhat. The first three columns of the Calendarium contain the day of the month, a letter indicating the day of the week and a column with days as designated in a Roman calendar. The table of regions for the Ephemeris and Calendarium both indicate a 0^h 0^m offset for Nuremberg so that may be the meridian that was used.

Orbital Elements for the Year 1494

  It's not clear when Regiomontanus produced his Ephemerides for 1494-1506 or the location for which it was computed. The ephemeris for 1494 was used to compute a set of orbital elements and presumably the positions of the Sun were calculated for noon which was the practice since Ptolemy's time. The ephemerides of the Sun were based on observations but comparisons of the positions for the months December and January with the fit show some distortion. As with the orbital elements for 1672 the daily changes were used to determine the time and position of the Sun's apogee.

The fit of the Sun's positions for 1494 gave the following elements.

The position of perigee is a few degrees earlier than for 1672 but occurs about the same time of year. The eccentricity, e, is also slightly larger too but still less than the value for the eccentricity in Ptolemy's time.

Regiomontanus died while working on calendar reform in Rome in 1476.

Supplemental (Apr 7): I used Legendre polynomials to fit the daily positions of the Sun and then fit a Keplerian orbit to the daily changes in fit's position for the Sun. The fit of the Legendre polynomials is worse at the beginning and end of the year. The central part of the curve is a good match to the elliptical orbit.

Supplemental (Apr 8): I was checking the length of the year 1494 in Regiomontanus' Ephemerides and noticed that there were only 365 days present where it should have been designated as bisextilis, i.e., a leap year. But February, 1494 in the Ephemerides does not have 29 days.  The year 1500 is properly designated as bisextilis and February, 1500 does have 29 days.