Eclipse Prediction

  There are rumors that there will be an eclipse next week. How do we know this is so? We can start with HORIZONS data for the daily noon positions of the Sun and Moon in geocentric equatorial coordinates for the year 2017. Next we convert RA and DEC into the spherical angles φ and θ in radians to determine the angular separation of the Sun and Moon.

The first plot is Δφ and the jags occur because Δφ is always increasing and we only need to know the relative angular distance between them and when it crosses the line Δφ=0. The second plot tells us when Δθ=0. But for an eclipse to occur both conditions need to be satisfied at approximately the same time. So we look through our table of Δφ and Δθ for nearly simultaneous crossings. The occurs twice in 2017 on or about Feb 2 and Aug 21.

The eclipse on Feb 26th has already occurred so we'll focus on the Aug 21 eclipse candidate. Another plot gives us a better picture of when both Δφ and Δθ cross the horizontal axis.

It's difficult to tell exactly when the Sun and Moon will be closest together since they are moving at different rates but a calculation indicates the minimum separation is just under half a degree. The directions of the Sun and Moon are e_S and e_M respectively.

This looks promising since the Moon's paralax, the shift in angle for moving from the subsolar position on the Earth surface to the Earth's limb, is about 0.0179 rad and adding the apparent angular radius of the Moon, 0.0043 rad, and the angular size of the Sun, 0.0047 rad, we get a maximum allowable separation of 0.0268 rad. Adding a box to indicate these bounds to a plot of Δθ vs. Δφ indicates that there will indeed be an eclipse at this time.

The point on the curve closest to the origin is towards the end of the eclipse which would explain the relatively later time.