Earth's Gravitational Field and Kepler's Laws of Planetary Motion

Earth's Gravitational Field

The gravitational field of the Earth is the disturbance that occurs in the space surrounding a body having mass. The intensity of the gravitational field at a point in space is the force with which the Earth attracts a unit mass located at that point. The weight of a body is the force with which the Earth attracts it. The gravitational potential energy of a mass m at a point in Earth's gravitational field is the work performed by the gravitational field to move the mass m from that point to infinity. The gravitational potential at a point in Earth's gravitational field is the work performed by the gravitational field to move the unit mass from that point to infinity.

• Period of revolution: Time it takes a satellite to describe a complete orbit.
• Escape velocity: The velocity a body must acquire to escape Earth's gravitational attraction.
• Mechanical energy: A satellite in orbit around the Earth will have mechanical energy that is the sum of its kinetic energy and its gravitational potential energy.

Kepler's Laws of Planetary Motion

First Law

All planets move in elliptical orbits with the Sun at one focus. We can deduce that the orbits are flat from the direction of conservation of angular momentum of the planets. Gravitational forces are central forces; their direction is the radius. Therefore, the moment of these forces about the center is zero, and the angular momentum of a planet is constant.

Second Law

The line joining a planet to the Sun sweeps out equal areas in equal times. This law follows from the conservation of the modulus of angular momentum of the planets. The modulus of angular momentum can be expressed as: |L| = |r x mv| = rmv sin(Y) = rmv = rm ds/dt. The swept area is a circular sector = dA = rds/2 = r^2dY/2. Comparing them, we obtain |L| = 2m dA/dt. Since L is constant, so will the ratio dA/dt. This ratio is called the sector velocity and measures the speed at which areas are swept.

Third Law

The square of the time period of a planet is directly proportional to the cube of the mean distance of the planet to the Sun: T^2 = Cr^3. We have obtained the expressions of the orbital velocity, v, and the period of revolution, T, of a satellite: v = sqrt(GM/r), T = 2πr/v. To derive the relation between the period of revolution and the radius of the orbit, we replace the expression of v in the expression of T squared: T^2 = 4π^2r^3/GM. Thanks to this law, we can determine the masses of the planets that have at least one satellite whose period of revolution and orbital radius are known.