# How to calculate eclipses the heliocentric way?

It is my understanding that modern methods for calculating the time and location of solar and lunar eclipses (e.g. NASA website https://eclipse.gsfc.nasa.gov/eclipse.html) are integrating equations of motions to determine the positions of the Earth, Sun and Moon relative to each other. The models may include other solar system objects (e.g. Jupiter) and treat objects not as point-like but extended to improve accuracy and prediction range into the past and future (several millennia).

This is quite sophisticated, but hard to follow for people unfamiliar with Newtonian mechanics and numerical methods. This made me think about a more accessible method to compute eclipses. If I make less stringent requirements for accuracy, say,

• Prediction range: +/- 5 years from today,
• Point of maximum eclipse: Within 500km of true location.

Is it then possible to compute positions of Earth/Moon using a heliocentric model where they move periodically in ellipses, possibly with apsidal/nodal precession and enough other periodic perturbations to achieve the desired accuracy? In other words, add a few vectors parameterized by time to get the positions of the Earth and Moon in a coordinate system where the Sun is at the origin. (Which CS would be most useful?)

Ideally, one would plug in a reference position/date, some solar system constants like sidereal year, sidereal day, synodic month, etc, a date and then get the 2 positions.

(Computing where the shadows intersect the Earth or Moon is not considered part of this question, just the centers of Earth and Moon.)

I've been researching this for months now and ordered a book on astronomical algorithms, but could not determine whether I'm trying something stupid...

• As the answers to this recent question explain, lunar theory is complicated. The plane of the Moon's orbit precesses with a very short period (around 18.6 years), and that plane is vital for eclipse predictions. But that doesn't mean that you have to numerically integrate equations of motion. There are equations using trig functions that give pretty good approximations. – PM 2Ring Jun 18 '20 at 18:33