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.

Extending the Range of tan(θ) and atan(t)

The tangent and inverse tangent function in Mathcad only work as one would like them to for multiples of θ=π/2. If one computes θ'=atan[tan(θ)] one will see that the difference Δθ=θ'-θ=±kπ/2 for some integer k.

One has to keep track of which quadrant θ is in and redefine both functions in order to get θ'=θ. The simplest solution that I could find is shown below.

A recheck shows that everything works out as one would want.

This this should work for any value of θ since the new definitions avoid values close to where the inverse tangent function is discontinuous, i.e., θ=±π/2.

Edit (Mar 14): I did a little more testing and found a missing case in the earlier program so I replaced the image formerly posted with programs containing the newer version. There was a problem with just θ=-π/2 for the extended range. Bugs like that can be very difficult to find.

The Eccentricity of the Earth's Orbit for the Next Year

  I just got through computing the Earth's orbit for the coming year using an update of the method found in Ptolemy's Almagest and got a reasonably good value. The times used for the equinoxes and solstices are found in HM Nautical Almanac Office, Planetary and Lunar Coordinates 2001-2020.

As one can see from the lengths of the seasons, the value for Δt is largest for the 2nd quadrant so the aphelion occurs in summer and perihelion in winter. One can see that the orientation of the Earth's orbit has changed somewhat since Ptolemy's time.

When doing orbit calculations it is necessary to convert the eccentric anomaly, E, to the true anomaly, θ, and and vice versa and it is easiest to work with the complete circle or the interval [0°, 360°]. I wrote some Mathcad programs, tan2π(θ) and atan2π(t), to help do this.

I haven't tested them extensively but they seem to work ok. They have to work together. If you look carefully you will see that I used complex numbers to extend the range of the useful interval.

Supplemental (Mar 12): I haven't checked to see if multiplying a tangent by a constant will affect results any but off the top of my head the range of the tangent is [-∞, ∞] and the sign of the number probably won't change while the magnitude will so the sector or "step" used will probably remain the same. Any resulting error is likely to be small.

tan180 and atan180

  The orbit calculation in the last blog used some nonstandard tangent and inverse tangent functions defined on the interval [0,π] or [0°,180°]. They reduce the conversion of the eccentric anomaly to the true anomaly to a single step and simplify the definition of the true anomaly, θ(t).

Both definitions are one-to-one as the following check shows.

Using the Times for Quadrant Passage to Fit an Elliptical Orbit

  The times of occurrence of the Equinoxes and Solstices provide enough information to determine some of the Keplerian orbital elements. The Equinoxes and Solstices, which are the Cardinal Points of the Celestial Sphere, are separated by 90° and we can use Ptolemy's times for quadrant passage to illustrate the process. The eccentric circle (still used today) and the elliptical orbit are shown in the figure below as well as the arcs W, X, Y and Z for the quadrants and we need to find the unknown times it takes for the true anomaly, θ, to travel through arcs Y and Z. It is fairly easy to compute the eccentric anomaly, E, and the true anomaly for some arbitrary time if the eccentricity, e, is known so we can start by assuming e = 0.02 and see what happens.

We can use a trick found in Ptolemy to find the time, t_P, it takes for the Sun to travel from the Autumnal Equinox to perigee by searching for the value of t_P which makes the difference in the true anomaly between the Summer Solstice and the Vernal Equinox 90°. The time, t'_P, it takes to go from perigee to the Winter Solstice can be found by also setting the sum of the two perigee angles equal to 90°. The difference in angles can again be used to find the time, t_Z, it takes to travel through arc Z on the ellipse.

Ptolemy gives the time it takes to go from the Autumnal Equinox to the Vernal Equinox as 178¼ days. The assumed value of e does not necessarily make the sum of times  t_Y and t_Z equal to this but we can search for a value of e that does and e = 0.0206 works quite well. With the times known we can compute their correspond true anomalies and we get a value about ½° off from Ptolemy's result of 65½° for the angle between the Autumnal Equinox and perigee. We can use t_P to compute the date of perigee and that of apogee a half year later. Here is a compilation of the results and some checks.

The value of the eccentricity for this orbit is very close to that found using just the fitted formula for Ptolemy's anomaly table. So we have shown that a knowledge of the times of the Equinoxes and Solstices can be used to compute an orbit for the Sun and indirectly for the Earth. More generally the times and angular positions of any three points can be used to compute an orbit so the circumstances of solar and lunar eclipses can also be used.

Some Procedures Used to Find the Equinoxes & Solstices

  A nonstandard interpolation function was used to find the time when the data crossed a certain value and consists of two parts. The first step is to employ a function called crossings which identified the set of indices, X, where one data point was below the value and the next above, or vice versa. The parameter sgn is need to specify whether ascending or descending crossings are needed.

Linear interpolation was then used to estimate the time when the data would reach the given value.

This procedure was used to find the midday values of the azimuth and altitude of the Sun. It produces some small errors so a linear fit was used to find the times of the equinoxes and solstices.

132 AD Orbital Elements

  I computed the Sun's "orbital elements" for the 132 AD HORIZONS data and got the following results:

The HORIZONS' osculating elements for 132 Jan 1 gives an eccentricity of about 0.018 which is a slightly better match to the 0.020 value obtained from Ptolemy's anomaly. It would have been interesting to see what Ptolemy's values for the lengths of the seasons were. The obliquity of the Ecliptic, ι, is close to the "observed" differences in the solstice meridian altitudes. Using the equation for the secular changes in the obliquity found on p. 114 of USNO Explanatory Supplement to the Astronomical Almanac I got a value of 23° 40' 43.77".

Supplemental (Mar 7): I checked some of the elements using Newcomb's formulas for the secular variations and got the following results for the eccentricity, angle of perigee and obliquity of the ecliptic:

     e = 0.01745
     π = 250° 56' 58.33"
     ι = 23° 40' 44.62"

The angle of perigee, λ_p or π, indicates that it is in Sagittarius at ♐10.9° which is not far from Ptolemy's ♐5½°. I don't know how accurate Newcomb's formulas are since he doesn't specify a range for their accuracy. Nutation might account for some of the differences.

Characterizing Ptolemy

Judging by the simplicity of his observations we could ask if Ptolemy was a naive impressionist? Did he assume that what appears to be true is actually true? The counter to that is that all measurements have a context associated with them and this is the simplest system to use. There are advantages for uniformity over space and time which produces a consistent set of data to work with. Perhaps it would be best to judge him solely as an observer.

He tells us that the Sun's motion is not entirely uniform and presents an anomaly based on the accepted times for the lengths of the seasons. His epicycle theory implies an eccentricity for the Earth's orbit of about 1/50 which can be compared to the modern value of about 1/60. He gives values for the times of the equinoxes and solstices which tells us about the length of the year and an estimate of the time when the Earth and the Sun are closest together. Although the epicycle theory seems a little absurb today it is now common practice to use Fourier series to represent periodic changes in time. He presents a fairly good picture of the Earth's relative motion to the Sun for his time.

Ptolemy systematize the current understanding of astronomy and prepared the way for future advances which the later increase in the precision of measurement brought about. We now know of secular changes, changes from one century to the next, in the orbits of the planets. Ancient observations are useful today in that they provide a check for modern theories of the motion of the planets throughout the ages.

An Astronomy Tutorial

  I've been skimming through a hundred year old tutorial on Astronomy by Barlow & Bryan, Elementary Mathematical Astronomy, which touches on some relevant topics for observing the equinoxes and solstices and the study of Ptolemy's observations.

determining the meridian line

latitude by observation

longitude by observation

Obliquity of the Ecliptic

Lengths of Morning and Afternoon

precession of the equinoxes

First Point of Aries

Motion of the Apse Line

Effect of Precession on the Climate

correction for refraction

correction for latitude

personal equation

elimination of personal equation

132 AD Summer Solstice Calculation

The determination of the time of the summer solstice in 132 AD has an added complication since the Sun's altitude at noon passes through a maximum at this time of year.

The changes in altitude are nearly linear and the time at which change is zero can be used to find the time of the solstice. Using central differences gives a better estimate of the peak value.

The calculation is similar to that used to find the time of the equinox.

A plot shows that this procedure gives a good estimate for the time of the peak.

The times of the solstices are not affected by the latitude of the place of the observation but those of the equinoxes are so the length of the seasons, the interval for the quadrant passages, are subject to error. The computed seasonal intervals may vary from year to year so an averaged value may work better. This may be why Ptolemy used Hipparchus' seasonal lengths for his anomaly.

Ptolemy would have had no problem doing the interpolations since differences in times are proportional to the differences in altitude for the equinoxes and second differences for the solstices. He would also have been familiar with finding maxima and minima since Hero of Alexandria had shown in the 1st Century AD that light travels the shortest path between two points.

Ptolemy appears to have done his observations in Alexandria I am not aware of the exact location. The value for the latitude of Alexandria suggests a location somewhat south of the location for the Museum which were the coordinates assumed for Alexandria. Ptolemy's altitude for the Celestial Equator would be subject to observation errors for his latitude and that of the direction of his Zenith. There is also a slight correction for geodetic latitude that he would have ignored as well as that for atmospheric refraction. If we emulate Ptolemy correctly it may be possible to correct his time for the Autumnal Equinox. The time of day would probably be within two or three hours of the time that Ptolemy gave. As we saw calendar creep complicates determining what he meant by the date given, Athyr 7.

Supplemental (Mar 1): On p. 134 of Toomer, Ptolemy's Almagest, Ptolemy states that the error in the observed times of the solstices and equinoxes may have been up to 1/4 of a day. From our calculations an error of 1/24th degree produced a 2 hour error. Three random errors of about 1/24th degree with 3 standard deviation bounds would result in a error of 3√3 times that or about 10 hours and 4 sources of error of error would result in 6 times that or 12 hours of error. With the Vernal Equinox moving along the ecliptic the length of the seasons change over long periods of time. I don't recall Ptolemy stating how he determined the time of day of an equinox or a solstice. In both cases it may have been by direct observation either of the crossing of the assumed equator or of the altitude of the Sun respectively.