Fundamental Laws of Gravitation and Planetary Motion

Kepler's Laws of Planetary Motion

Johannes Kepler, utilizing the precise astronomical measurements made by Tycho Brahe—especially concerning the distance of Mars from the Sun—concluded that planetary trajectories are not circular but elliptical.

Kepler's First Law: The Law of Orbits

All planets move in elliptical orbits with the Sun located at one focus.

Kepler's Second Law: The Law of Areas

The radius vector connecting the Sun and a planet sweeps out areas that are directly proportional to the time interval spent.

Kepler's Third Law: The Law of Periods

The squares of the orbital periods ($T^{2}$) are directly proportional to the cubes of the semi-major axes ($a^{3}$) of the respective orbits.

Newton's Law of Universal Gravitation

Galileo came to the conclusion that the speed of falling bodies is proportional to time, and the distance traveled is proportional to the square of the time. The law of universal gravitation can be stated as: The force of attraction between two masses is proportional to the product of the masses and inversely proportional to the square of the distance that separates them.

Cavendish's achievement is said to be the 'weighing of the Earth' because, in determining the value of $G$, he could determine the mass of Earth. ($G = 6.67 \times 10^{ -11}$ Nm$^{2}$/Kg$^{2}$)

The Gravitational Field Concept

A region of space is said to contain a gravitational field if a test mass placed within it experiences a gravitational force.

Gravitational Field Strength ($g$)

Gravitational field strength is defined as the force experienced per unit mass. Its unit is N/kg, which is dimensionally equivalent to m/s$^{2}$.

The gravitational field is a central force, since its direction always passes through the mass creating the field. To visualize a gravitational field, we define the field lines (or lines of force) that must satisfy the following criteria:

1. Lines originate at infinity and terminate at the masses.
2. The lines are drawn so that the direction of the gravitational field is tangent to them at each point.
3. The lines are drawn so that their density is proportional to the field strength.
4. The field lines cannot intersect each other, as the field can only have one direction at any given point.

Gravitational Potential Energy

The gravitational force is conservative, which means that the work done between two points does not depend on the path followed, but only on the starting and ending points.

Gravitational potential energy ($U$) is the energy possessed by a body found inside a gravitational field. The conventional origin of the gravitational potential energy is infinity, because it is the only point where the masses do not interact.

The gravitational potential energy at a point is the work done by the gravitational force to move the mass ($m$) from that point to infinity. All central forces are conservative and therefore support the definition of potential energy.