First, Earth is not the only planet in the solar system. And if one tries to find the answer accurately enough, that person will have to take into account the gravitational effects of other planets. The most important is Jupiter, and its typical effect on the orbit of the Moon is 10-5 – large enough that Brown for example had to include it in his table.
The next effect is that the Earth is not just a point mass, or even a precise sphere. Its rotation causes it to swell at the equator, and that affects the Moon’s orbit on the 10-6 floors.
The orbit around the Earth ultimately depends on the full and mass distribution gravity field of Earth (is what Sputnik-1 nominally mapped) —and both this, and the reverse effect from the Moon, are in levels 10-8. At levels 10-9 then there will be tidal strain effects (“solid tide”) On Earth and moon, as well as words Attractive red shift and other generalist relativistic phenomena.
To predict the position of the Moon as accurately as possible, one must ultimately have at least several models for these various effects.
But there is a much more immediate problem to be solved: one has to know the initial conditions of the Earth, the Sun and the Moon, or in other words, one must know as accurately as possible the position and speed. them at a particular time. And convenient enough, now there’s a really good way to do it, because Apollo 11, 14 and 15 all remaining laser reflection on the moon. And by precisely determining how long it would take a laser pulse from Earth to go one round to these reflectors, it is now, in fact, possible to measure the position of the Moon to millimeters accurately.
Okay, so how do the modern analogues of the Babylonian cremators actually work? Inside they are solving the equations for all the important celestial bodies in the solar system. They perform symbolic preprocessing to make their digital jobs as easy as possible. And then they directly solve the differential equations for the system, appropriately inserting models for things like the distribution of mass on Earth.
They start from specifically measured initial conditions, but then they continually insert new measurements, trying to correct the parameters of the model to optimally reproduce all the measurements they have. It is very similar to one Typical machine learning tasks—With the training data here are solar system observations (and generally consistent with least squared).
But, OK, so there’s a model that one can run to find something like the position of the Moon. But one doesn’t want to have to do it explicitly every time one needs to get results; instead, in reality, one just wants to store a large table of pre-computed results and then do something like interpolation to get any concrete results one needs. And indeed that’s how it’s done today.
How is it really completed
Back to the 1960s NASA began to directly solve the differential equations for the motion of the planets. It was more difficult to deal with the Moon, but in the 1980s that was handled in a similar way. Ongoing data from things like lunar reflectors have been added and all available historical data is also inserted.
The result of all of this is JPL Development Ephemeris (JPL DE). In addition to the new observations being used, the basic system is updated every few years, usually to get what’s needed for some spacecraft to travel to some new location in the solar system. (The newest is DE432, built to The Pluto.)