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\[ \cos x=\sum_{k=0}^{\infty}\frac{(-1)^k}{(2k)!}x^{2k} \]

The homomorphism \(f\) is injective if and only if its kernel is only the singleton set \(e_G\), because otherwise \(\exists a,b\in G\) with \(a\neq b\) such that \(f(a)=f(b)\).

$$
\cos x=\sum_{k=0}^{\infty}\frac{(-1)^k}{(2k)!}x^{2k}
$$

Eclipse Prediction with MathJax

This document demonstrates the use of MathJax to format equations for predicting eclipses.

1. Synodic Month

The synodic month represents the time between successive new moons. It’s computed as follows:

\[ T_s = \frac{1}{\frac{1}{T_m} - \frac{1}{T_e}} \]

where:

  • \( T_s \) is the synodic month (about 29.53059 days),
  • \( T_m \) is the Moon’s sidereal orbital period around Earth (27.321661 days),
  • \( T_e \) is Earth’s sidereal orbital period around the Sun (365.25636 days).

2. Draconic (Nodal) Month

The draconic month is the period it takes the Moon to return to one of its nodes. Eclipses can occur only when the Moon is near a node:

\[ T_d = 27.2122 \text{ days} \]

This period is important for predicting eclipse timings.

3. Eclipse Season

Eclipses can occur when the Sun is near a node within a certain angular range, calculated by:

\[ T_{\text{season}} = \frac{1}{\frac{1}{T_d} - \frac{1}{T_s}} \]

This results in an eclipse season every 173.31 days, where at least one solar or lunar eclipse is possible.

4. Anomalistic Month

The anomalistic month is the time for the Moon to return to perigee (the point closest to Earth), which impacts whether a solar eclipse will be total or annular:

\[ T_a = 27.55455 \text{ days} \]

5. Saros Cycle

The Saros cycle is a period after which similar eclipses will recur with almost identical geometry. It is approximately given by:

\[ T_{\text{Saros}} = 18 \times T_y + 10 + \frac{1}{3} \text{ days} \]

where \( T_y \) represents the tropical year (365.2422 days). This means similar eclipses occur roughly every 18 years, 11 days, and 8 hours.

6. Angular Separation for Eclipse Occurrence

For an eclipse to occur, the angular separation between the Moon and the Sun, as viewed from Earth, must fall within a critical limit:

\[ \Delta \theta = \left| \theta_{\text{Sun}} - \theta_{\text{Moon}} \right| < \theta_{\text{limit}} \]

where:

  • \( \theta_{\text{Sun}} \) and \( \theta_{\text{Moon}} \) are the Sun’s and Moon’s angular positions relative to Earth,
  • \( \theta_{\text{limit}} \) is the maximum separation within which an eclipse is possible (approximately \( 15^\circ \)).

7. Spherical Trigonometry for Celestial Positioning

Precise eclipse predictions require calculating the exact positions of the Sun, Moon, and Earth using spherical trigonometry:

\[ \cos(\theta) = \sin(\delta_1) \sin(\delta_2) + \cos(\delta_1) \cos(\delta_2) \cos(\alpha_1 - \alpha_2) \]

where:

  • \( \theta \) is the angle between two celestial objects,
  • \( \delta_1 \) and \( \delta_2 \) are their declinations (angular distance from the celestial equator),
  • \( \alpha_1 \) and \( \alpha_2 \) are their right ascensions (angular distance measured along the celestial equator).

Summary

Combining these cycles—synodic, draconic, anomalistic months, and the Saros cycle—with precise geometry allows for accurate eclipse prediction.