Through research and observation, scientists have constructed accurate models of the shape, weight, and orbital mechanics of the sun, moon, and earth. We also know, for the most part, the present positions and velocities of these three bodies. Applying Newton’s Laws (and other gravitational physics) to these models, scientists can extrapolate the positions of the bodies to figure out when an eclipse will occur — these calculations are accurate to within a minute for eclipses that will occur a hundred years from now.
The basics of prediction of eclipses are based upon the Saros cycle, discovered by early Babylonians. The Saros cycle is a period of time equal to approximately 6,585.3 days. That works out to be 18 years, 11 days 8 hours, but an eclipse actually occurs every sar, which is half of a Saros, or about every 9 years, alternating between full lunar and solar eclipses every nine.
The calculation is much more complex than that, because the gravitational pull of the Earth, Sun, and Moon all must be factored in, as well as rotational speeds and distances from one another, but the Saros cycle has held true for the last several thousand years as a predictor of full eclipses, although it is important to note that many partial eclipses occur in the midst of a Saros.
Both the NASA websites I’ve cited give a lot of helpful (although math-heavy) information that goes into the calculation you never hear about when the news channels announce a coming eclipse.
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