Did Ptolemy Have a Time Interval Table or Formula?

  The time interval formula is all that is needed to compute the times of other events if one has a known position and time for the Sun like Ptolemy's autumnal equinox.

Did Ptolemy a table of time differences from the time of perigee that allowed him to compute the time for passage between two longitudes? Ptolemy's anomaly table may be such a device. Ptolemy gave the angular position of perigee so one can compute the angular distance from perigee for a given longitude. Using the anomaly table in reverse one can determine the corresponding mean motion and the time interval from perigee. Doing this for two longitudes allows one to determine the time it would take to go from one to the other. So if the date and time of one longitude is known one can compute the date and time of another.

Supplemental (Feb 18): The Julian dates assume UT which is Greenwich Mean Time but the latitude of Alexandria is about 30° E so it is 1/12th of a day ahead of UT and this means that one needs to subtract 0.083 from the times above to correct them.

Ptolemy's Ecliptic Longitude Formula & Event Julian Dates

  The longitude formula assumed in the last blog gives incorrect ecliptic longitudes and we need to be more careful about the Julian dates of the equinoxes, solstices and perigee. The times for the quadrant passages were correct since they involved 2nd differences.

Since the time of the autumnal equinox is known, the Julian dates of the vernal equinox and solstices can be computed. The epoch is chosen to be the beginning of the year 132 and is given for comparison.

The anomaly is measured from the time of perigee, t_p, and this longitude is given by Ptolemy. Since the time of the autumnal equinox is also given we can determine the date and time of perigee.

We now have the correct longitude formula and can compute the true anomaly for a set of times of the year.

This data can then be used to fit an ellipse to Ptolemy's anomaly. As one can see there is a systematic error of about half a degree for the residuals of the linear formula involving the true anomaly and its rate of change but this is below the 1 minute of arc precision that Ptolemy appears to have been working at.

The values of σ and τ allow us to compute the Keplerian elements as before.

Keplerian Elements From Ptolemy's Anomaly for the Sun

  Ptolemy gave a table of the Sun's anomaly that was accurate to about 1 arc minute in his Almagest along with other data from which we can make an estimate of the Sun's "orbit."
We start with a figure of the equinoxes and solstices and the positions of perigee and apogee in the plane of the ecliptic. In Toomer, Ptolemy's Almagest, p. 153, Ptolemy states that the apogee was 24.5° in advance of the summer solstice which would place it at Gem 5.5° in his twelve 30° divisions of the ecliptic measured from the vernal equinox. Similarly the perigee is at Sag 5.5°.

The positions of perigee and apogee were determined by the nonuniform rate of motion of the Sun being slowest at apogee and fastest at perigee. He also gives Hipparchus' values for  the time it took to go from the vernal equinox to the summer solstice (94½ days) and the time to go from the summer solstice to the autumnal equinox (92½ days). He says that his observations gave similar values and from a fit of his anomaly we see that this is so.

The time it takes to traverse a quadrant is greatest for the quadrant containing the apogee and least in the quadrant containing perigee. The value for λ₀ was chosen so that the autumnal equinox occurs at 132 Sep 25 2pm. Ptolemy's anomaly only has sin terms in it and is accurate to about 1 arc minute as the residuals for the fit show.

We can use Ptolemy's ecliptic longitude to compute an "orbit" for the Sun in the same way that we did using the longitude obtained with the MICA data. The epoch chosen was the beginning of the year, 132 Jan 1, noon, and we can use the the longitude to estimate the angle and time of perigee, then calculate the true anomaly, ν, at intervals of one day.

Using the linear relation between the cosine of ν and the square root of its rate of change we can estimate σ and τ which in turn allows us to estimate the eccentricity, e, and the other Keplerian elements. The residuals for fit of the formula are a few minutes of arc.

One can find Ptolemy's value for obliquity of the ecliptic on p. 63 of Toomer's book. The summary of the estimated Keplerian elements is as follows.

Most of the changes in orientation of the elliptical orbit are due to the rotation of the equatorial reference frame in the plane of the ecliptic. One would expect some of the difference in the value for the eccentricity e to be due to Ptolemy's longitude and the use of an eccentric circle but some of it may be due to perturbations also. The astronomical unit, AU, used here is based on the elliptical orbit for Ptolemy's data but a change in the mean motion would suggest a change in the mean radius since the areal velocity is fairly constant.

edit (Feb 17): Corrected the date and time for the autumnal equinox but the formula assumed for the longitude above proved to be incorrect. See the next blog for more details.

What Does Linear Least Squares Do?

  The equations for the linear least squares coefficients are derived from the assumption that the sum of the square of the errors is a minimum. This may not be the actual case as the following example shows. The result is slightly biased with the fit line (dotted) being slightly oblique to the true line (solid).

A calculation shows that for the fit the 1st moment of the errors, δ'', is zero while this is untrue for the assumed errors, δ. This result is quite general and can be proven assuming the equations for the fit.

One can also show the unexpected result that the average of the errors is also zero starting from the first of the fit equations.

Both conclusions are due to the original assumption that the sum of the square of the errors is a minimum and may be in conflict with the true values. For random errors the conclusions are approximately true and are not in conflict with the use of linear least squares for an estimate of the coefficients.

Residuals for the Solar Longitude Fit

  The residuals for the longitude fit in the last blog have arc second accuracy. Here are the final correction for the longitude fit and resulting residuals.

The periodic sin function present appears to be associated with the Moon orbital motion and can be removed leaving a long term residual.

This may be due to planetary perturbations and/or errors in the fit. The lunar synodic period works quite well with time measured in days. The fit of Sun's motion with the lunar disturbance of the Earth's position added appears to have about 10 arc second accuracy.

Finding the Mean Motion by Averaging

  One does not have to use least squares to find the mean motion of the Sun along the ecliptic. The 3 years of data for the ecliptic longitude obtained from MICA give a fairly good estimate of the mean motion, n. One can interpolate to find the time it takes to move an integral multiple of 360° then divide to estimate the mean motion.

Using this estimate of the mean motion it's a simple task to find the nonuniform motion.

The longer the interval of time the better the estimate of the mean motion as can be seen in the following plot. Notice that the difference between the successive averages and the original estimate returns to zero at the end of the years.

Judging by Ptolemy's value of the mean motion found in Toomer, Ptolemy's Almagest, p. 140, it hasn't changed much in 2000 years.

Ptolemy's ecliptic longitude was measured from the Vernal Equinox. To find the mean motion in a fixed reference frame like J2000.0 one has to subtract the precession rate of the Equatorial reference frame. A similar numerical value is used to define the astronomical unit (AU) which is the radius of a circular orbit that has this value for its mean motion. The AU is a trifle smaller than the Earth mean distance from the Sun.

A Problem With Least Squares Fits and a Fix

  One sometimes encounters a problem when using least squares to find the best fit of a formula to a set of data. A simple example is fitting a line to the Sun's ecliptic longitude as a function of time.

Notice that the nonuniform motion, δ, slopes slightly downward when one would expect just a sinusoidal function. The problem is that least squares warped the error raising it on the left and lowering it on the right to reduce the variance. If one could fit the smaller nonuniform motion first one might avoid the problem but one has to fit the linear portion first to estimate the mean motion, n. What one really wants to do is separate the uniform motion from the nonuniform motion. This can be done by successive residual fits that alternates between finding uniform and nonuniform motion in the residuals. The function correct_α does 10 iterations of this here.

The nonuniform motion is level after the correction and the variance, V, is slightly larger. Here are the programs that were used to do the corrections.

The first function, Δα, fits the nonuniform motion in a set of residuals then uses an new residual to extract some uniform motion from it. The second function, correct_α, repeats this process nitr times.