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Basics of Celestial Mechanics

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Kepler's Third Law

The Law of Harmonies states that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. Mathematically,

T^2 \propto a^3

where

T

is the orbital period and

a

is the semi-major axis.

Kepler's First Law

Also known as the Law of Ellipses, it states that all planets move in elliptical orbits with the Sun at one focus. This implies that planet-Sun distance varies throughout the year.

Kepler's Second Law

Known as the Law of Equal Areas, it states that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. This means that a planet travels faster when it is closer to the Sun and slower when it is farther away.

Newton's Law of Universal Gravitation

Newton's Law states that every point mass attracts every other point mass by a force acting along the line intersecting both points. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between the points:

F = G \frac{m_1 m_2}{r^2}

, where

F

is the gravitational force,

G

is the gravitational constant,

m_1

and

m_2

are the masses, and

r

is the distance.

Conservation of Angular Momentum

In celestial mechanics, angular momentum is conserved in a system where there is no external torque. This principle implies that a celestial body will spin faster as it collapses or moves towards conservation of its orbital angular momentum when in an elliptical orbit.

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