How do I calculate evection and variation for the moon in my simple solar system model?
I am building a model of a simple, fictional sun-planet-moon solar system (not the real Sol-Terra-Luna system).
Purpose: to calculate the timing of astronomical events like eclipses and moon phases, and the positions of the sun and moon in the sky of the planet at a given time.
Question: how accurate would this model be compared to a simulation using realistic gravity?
In the model, the sun-planet-moon barycenter is located at the origin of the reference frame, with the plane of the ecliptic as the $xy$-plane and the orbital axes parallel to the $z$-axis. All orbits are circular and in the plane of the ecliptic. At $t = 0$ the three bodies are colinear, with the moon located between the sun and the planet. 380
The sun and the planet-moon barycenter both orbit the sun-planet-moon barycenter in exactly 360 local days; the planet and the moon both orbit the planet-moon barycenter in approximately $\frac{360}{13}$ (≈27.7) local days, with one synodic month (time between full/new moons) being exactly 30 local days. One local day is 75,325.8 Earth seconds (approximately 21 Earth hours).
In the model: - The center of the sun has coordinates (-SunBarycenterDistance * cos($t$), -SunBarycenterDistance * sin($t$)) - The center of the planet has coordinates (PlanetOrbitRadius * cos($t$) + PlanetBarycenterDistance * cos(13$t$), PlanetOrbitRadius * sin($t$) + PlanetBarycenterDistance * sin(13$t$)) - The center of the moon has coordinates (PlanetOrbitRadius * cos($t$) - MoonOrbitRadius * cos(13$t$), PlanetOrbitRadius * sin($t$) - MoonOrbitRadius * sin(13$t$))
Where - SunBarycenterDistance is the distance between the sun-planet-moon barycenter and the center of the sun, calculated based on the masses of the sun and the planet-plus-moon subsystem: (PlanetBarycenterDistance * (MassPlanet + MassMoon)) / (MassSun + MassPlanet + MassMoon) - PlanetOrbitRadius is the distance between the planet-moon barycenter and the sun-planet-moon barycenter, calculated based on the mass of the sun and the orbital period of the planet: cbrt((TimePlanetOrbit² * G * MassSun) / (4π²)) - PlanetBarycenterDistance is the distance between the planet-moon barycenter and the center of the planet, calculated based on the masses of the planet and the moon: (MoonOrbitRadius * MassMoon) / (MassPlanetPlusMoon) - MoonOrbitRadius is the distance between the planet-moon barycenter and the center of the moon, calculated based on the mass of the planet and the orbital period of the moon: cbrt((TimeMoonOrbit² * G * MassPlanet) / (4π²)) - G is the gravitational constant
I have been informed that in reality, the orbits of the planet and moon about their shared barycenter would not be perfectly circular, and the speed of their orbits about their shared barycenter would not be constant, due to the gravitational influence of the star. Assuming masses and distances are tuned to produce the orbital periods I specified, how much deviation would there be in a real physics simulation from the mathematically-perfect, uniform circular motion of my model? Is there a convenient way to account for this deviation? 380