# What is the formula to predict lunar and solar eclipses accurately?

A number of ancient civilizations had devised methods to predict exact dates and times of such eclipses, marking them as important events. Hence I assume the predictions were based on calculations, which should be quite easy to do now. So what is the exact formula to predict the exact date and time(this is optional, but desirable) of lunar and solar eclipses? Also how to calculate if the solar eclipse will be visible from a particular location or not?

• Since no one has mentioned it: en.wikipedia.org/wiki/Saros_%28astronomy%29 – user21 Dec 28 '15 at 4:21
• solar eclipse can only occur when there is a new moon. it happens because the moon will become in between the sun and earth, creating a shadow across earth. – user14055 Aug 31 '16 at 7:32
• Check this: eso.org/public/outreach/eduoff/aol/market/collaboration/…. There are a short explanation and conditions to occur eclipse. – User123 Mar 9 '19 at 20:08

## 2 Answers

The NASA sites have some very useful resources for this I will list them below:

Lunar Eclipses

This Link has an index for all lunar eclipses from -1999 to +3000, predominantly a statistics page but also has this page that contains how to calculate when lunar eclipses are.

There is more than one formula depending on which time frame you are trying to look in.

This is the formula for eclipses between the year 2005 and 2050:

$$\Delta T = 62.92 + 0.32217 * t + 0.005589 * t^2$$

Where:
$$y = year + (month - 0.5)/12$$
$$t = y - 2000$$

Solar Eclipses

This Link has an index like above but for all of the solar eclipses from -1999 to +3000.

This link has the formula for calculating solar eclipses. This is the formula for between 2005 and 2050:

$$\Delta T = 62.92 + 0.32217 * t + 0.005589 * t^2$$

Where:
$$y = year + (month - 0.5)/12$$
$$t = y - 2000$$

• Do you have the same formula repeated twice in your answer? – user21 Mar 2 '16 at 16:38
• $\Delta T$ is the difference in seconds between terrestrial time and UT, not the time of an eclipse. – Mike G Sep 22 '16 at 23:17

Calculation of solar eclipses can be done using Besselian elements. The basic idea is to compute the motion of the Moon's shadow on a plane that crosses the Earth's center. Then, the shadow cone of the Moon can be projected on the Earth surface. The Besselian elements are the following:

• X and Y: the coordinates of the center of the shadow in the fundamental plane
• D: the direction of the shadow axis on the celestial sphere
• L1 and L2: the radii of the penumbral and umbral cone in the fundamental plane
• F1 and F2: the angles which the penumbral and umbral shadow cones make with the shadow axis
• $\mu$: the ephemeris hour angle

what you have to do now is to compute the variation of these parameters, which are time-dependent. It happens that it can be done using polynomial expensions for a given reference time $t_0$. The polynomial expension is of the form, for a Besselian element a:

$$a = a_0 + a_1\times t + a_2\times t^2 + a_3\times t^3$$

(a third order expension is enough in general), with $t = t_1 - t_0$, $t_0$ being the Terrestrial Dynamical Time (TDT) to the nearest hour of the instant of greatest eclipse.

Sources: