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.

Orbital Elements for the Year 1672

  John Gadbury's Ephemerides of the Celestial Motions for 1672 set a new benchmark for data on the motions of the planets. The positions are given daily for noon London mean time. I was able to "smooth" the data to get positions for the Sun to seconds of arc accuracy. The round off errors indicate that the data as presented is accurate to the nearest minute.

I was able to get the period by interpolating the data to find the length of the year, T, and dividing 360° by it to get the mean motion, n. The daily changes in the positions allowed the estimation of the positions of apogee and perigee for the orbit and an eccentricity of e=0.018 gave a good fit for these changes.

Here's the executive summary. The dates are given in the "Old Style" Julian calendar which was used until about 1750 in England and its American colonies. The dates in the Gregorian calendar added 10 days so the corresponding date for the apogee in the Gregorian calendar would be Jun 28.  Apogee now occurs about Jul 3rd or 4th and perigee on about Jan 2nd or 3rd. The change is quite noticeable over this period of time.

An act of Parliament made the creation of the Nautical Almanac and Astronomical Ephemeris an annual occurrence starting in 1866. Rheticus published an ephemeris for 1551 and Regiomontanus did an ephemeris for 1494-1506. The United States' Almanac, 1843-1845 is credited to Henry Wadsworth Longfellow.

The 4 Quarters for the Year 1650

  The Four Quarters or Seasons are given in early American almanacs such as Uriah Oakes, An Almanack for the Year of Our Lord 1650,

 The 4 Quarters of this year. 1650

  The Vernall or Spring Quarter begins when [the] Sun enters into [the]
first degree of the signe Aries, which is this year [the] tenth of march
56 min past 3 in the morning making then equal day & night.
  The Estivall or Summer Quarter begins at such time as the Sun enters into 
Cancer, which this year is upon [the] 11 of June 46 m. after 7 in the 
morning: making then our longest day (viz. 15 hours & 8 min) & shortest 
night (viz. 8 hours 52 m.)
  The Autumnall or Harvest Quarter begins when the Sun enters into the sign 
Libra, which this year is upon [the] 12 of Septem: 33 m. after 10 at night, 
making then equall day & night.
  The Hybernall or Winter Quarter begins when [the] Sun enters into 
Capricorn, which this year is the 12 of December 9 min. past 9 before noon, 
making then our shortest day (viz. 8 hours & 52 min.) & longest night, 
(viz. 15 hours 8 min.)

Note the dates are given in Old Style which indicates the old Julian calendar used before the switch to New Style or the Gregorian calendar in which the dates appear to be 10 days later.

A Perihelion Problem

As shown previously, given the lengths of the seasons one can find an approximate value for the eccentricity, e, of the Earth's elliptical orbit.

One can also estimate the time and position of the Earth's perihelion by using Kepler's equation to compute true anomalies of the angular distance of the Earth from perihelion. All we know are differences in positions and times so we start with the time it takes for the Earth to go from the Vernal Equinox to the Summer Solstice, an angle of 90° in 92.753 days. We can assume a value for the time it will take to go from perihelion to the equinox, then use Kepler's equation to compare values obtained for a single step with that it takes for two steps by plotting their difference. The two values should be equal so their difference should be zero.

The data can be put in tabular form and we can use interpolation to when this difference is actually zero.

A solve block confirms the time it takes to go from perihelion to the equinox and we can calculate the corresponding true anomaly. We find it takes 76.146 days to move through an angle of 76.911°. The position of perihelion is then 283.089° for 2015 from which we can determine its date and time since those of the equinox are known.

The computed time of perihelion is off by a day from that given in HM Planetary & Lunar Coordinates 2001-2020 and one has some doubts about the complete accuracy of Kepler's elliptical orbits. Gravitational perturbations by the Sun, Moon and planets might be an explanation. One can see the difficulties that Ptolemy might have had with his determination of the lengths of the seasons if they varied that much. And, the ellipses are probably more accurate than his eccentric circle.

Supplemental (Mar 18): There's a round off error of ±½ minute in the times of the equinoxes and solstices that can affect results somewhat. It's not just perturbations.

Supplemental (Mar 18 23:48): The computed perihelion time is very close to the perihelion time of the Earth-Moon barycenter. The Moon tugs the Earth about a little altering the point in its path that is closest to the Sun.