I ran a small simulation and determined that this produces no significant error. Basically, loosing one cold day in the northern hemisphere is balanced by loosing a hot day in the southern.
Computing the Tilt of the Earth
The model uses Kepler's equation correctly ... except to find the angle of the Perihelion (the closest distance between the Earth and Sun). For that, they used the wrong equation. As a result, their calculations are off by one day.
|Method||MA Angle of Perihelion||Vernal |
|Model-E||0.030551958||1.75049825||Mar 19/20||Jun 20/21||Sep 21/22||Dec 20/21|
|Correct||0.046738227||2.67790317||Mar 20/21||Jun 21||Sep 22/23||Dec 21/22|
It isn't so much that the model gives the wrong results (a one day error should be insignificant) as it is that they used the wrong equation. If they can't get something this simple correct, then what about the rest of the model?
|TA||True anomaly||Angle seen from the Sun with respect to the Perihelion|
|EA||Eccentric anomaly||Projection of the TA angle onto a circle|
|MA||Mean anomaly||Angle on a circle - 2Pi * ( day/365 )|
|e||Eccentricity||Relationship between a circle and an ellipse|
MA = [ (day of year/days per year) * 360 degrees ] - MA of the Perihelion MA = EA - e * Sin(EA) Kepler's equation Tan(TA/2) = Sqrt[(1+e)/(1-e)] * Tan (EA/2)
The problem is that the model used the True Anomaly (TA) of the Vernal Equinox directly in Kepler's equation without first converting the TA to the EA (using the third equation above). Logically, this is a pretty big error. Practically, it produces only a one day error ... for the Earth.
It should be noted that these subroutines could be used in other programs for other purposes. When applied to other planets, or perhaps the Moon, this error could be very significant.
Reference with details and proofs related to these equations.