<p>Calculation of solar eclipses can be done using <strong>Besselian elements</strong>. 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:</p>
<ul>
<li>X and Y: the coordinates of the center of the shadow in the fundamental plane</li>
<li>D: the direction of the shadow axis on the celestial sphere</li>
<li>L1 and L2: the radii of the penumbral and umbral cone in the fundamental plane</li>
<li>F1 and F2: the angles which the penumbral and umbral shadow cones make with the shadow axis</li>
<li><span class="math-container">$\mu$</span>: the ephemeris hour angle</li>
</ul>
<p>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 <span class="math-container">$t_0$</span>. The polynomial expension is of the form, for a Besselian element a:</p>
<p><span class="math-container">$$a = a_0 + a_1\times t + a_2\times t^2 + a_3\times t^3$$</span></p>
<p>(a third order expansion is enough in general), with <span class="math-container">$t = t_1 - t_0$</span>, <span class="math-container">$t_0$</span> being the Terrestrial Dynamical Time (TDT) to the nearest hour of the instant of greatest eclipse.</p>
<p><strong>Sources:</strong></p>
<ul>
<li><a href="http://eclipse.gsfc.nasa.gov/SEmono/reference/explain.html" rel="nofollow noreferrer">NASA webpage about solar eclipse</a></li>
<li><a href="http://eclipse.gsfc.nasa.gov/SEcat5/beselm.html" rel="nofollow noreferrer">Besselian elements of solar eclipse</a></li>
</ul>

CC BY-SA 4.0
2013-09-26T12:58:28.137Z
2024-04-16T20:11:31.000Z
2024-04-16T20:11:31.000Z
MBR
<p>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?</p>

CC BY-SA 4.0
2019-03-09T20:08:08.903Z
Check this: https://www.eso.org/public/outreach/eduoff/aol/market/collaboration/eclipse99/projects/solecl-2d.htm. There are a short explanation and conditions to occur eclipse.
User123
CC BY-SA 4.0
2024-04-17T01:48:11.073Z
An article on how to compute Besselian Elements is here:
https://www.celestialprogramming.com/eclipsesGeneratingBesselianElements/generatingBesselianPolynomials.html
Greg Miller
<solar-eclipse><lunar-eclipse>
What is the formula to predict lunar and solar eclipses accurately?
eminemence
# What is the formula to predict lunar and solar eclipses accurately?

(Asked by: username_0 on 2013-09-26)

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?

**Tags:** solar-eclipse, lunar-eclipse

> **Comments:**
> **username_1**: Check this: https://www.eso.org/public/outreach/eduoff/aol/market/collaboration/eclipse99/projects/solecl-2d.htm. There are a short explanation and conditions to occur eclipse.  
> **username_2**: An article on how to compute Besselian Elements is here: https://www.celestialprogramming.com/eclipsesGeneratingBesselianElements/generatingBesselianPolynomials.html  

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## 1 answers

### Answer by username_3 on 2013-09-26. Score: 12 | Accepted answer

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 expansion 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:**

* [NASA webpage about solar eclipse](http://eclipse.gsfc.nasa.gov/SEmono/reference/explain.html)
* [Besselian elements of solar eclipse](http://eclipse.gsfc.nasa.gov/SEcat5/beselm.html)


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