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.
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.
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