Kepler's Laws, Gravity, Escape Velocity and Orbital Motion

Kepler's Three Laws

Kepler's three laws are:

• First Law. The planets move in elliptical orbits around the Sun, which is located at one of the foci of the ellipse.
• Second Law. The straight line connecting the planet to the Sun (the radius vector) sweeps out equal areas in equal times. In other words, the areal velocity is constant. For example, if t₁ = t₂, then A₁ = A₂. This means that planets do not move at the same speed at all points of the orbital path: they move faster near perihelion (the point nearest the Sun) and more slowly near aphelion (the farthest point).
• Third Law. The squares of the orbital periods of planets are proportional to the cubes of their mean distances from the Sun: T² = K · r³.

Gravitational Field Intensity (g)

The intensity of the gravitational field, g, is the magnitude that defines the gravitational field from the dynamic point of view and can be regarded as the force that acts on a unit mass placed at the point of observation. That is, the force F on a mass m is given by: F = m g.

Gravitational Potential

Define gravitational potential (V) at a point as the potential energy acquired by a unit mass placed at that point. In other words, V is the potential energy per unit mass at the chosen location.

Escape Velocity

Escape velocity. The escape speed is the speed that must be given to a body of mass m so that it can escape the gravitational pull of a planet of mass M and radius R. Since the launch is performed in a conservative field, the mechanical energy is conserved. At infinity the final velocity is zero and the potential energy is also zero.

Gravitational Field on Earth

Gravitational field on Earth. The Earth's gravitational field is the disturbance it produces around it by being a mass that attracts other masses. The terrestrial intensity is the force with which the Earth attracts a unit mass at a point.

Mass of Earth: 5.98 × 10²⁴ kg.
Radius of Earth: 6.67 × 10⁶ m.

Weight of a Body

Weight of a body. The weight P of a body is the force with which the Earth attracts the mass. The weight at any point is radial and directed toward the center of the Earth.

The weight of a body at some distance from the Earth's surface can:

• Cause the object to fall to the Earth's surface with a known acceleration of gravity, which we denote by g.
• Hold the object or a satellite in orbit around Earth. In this case the weight acts as the centripetal force necessary to describe a closed orbit.

Gravitational Potential Energy

Gravitational potential energy. It is the work done by the gravitational field to move the mass m from a given point to infinity.

Earth's Gravitational Potential

Earth's gravitational potential. At a point in the gravitational field, it is the work done by the field to move a unit mass from that point to infinity.

Orbital (Circular) Velocity

Orbital velocity. The orbital (circular) velocity for a circular orbit of radius r around the Earth is the linear speed of the satellite. For a circular orbit this speed is derived from centripetal balance and is commonly given by v = sqrt(GM/r), where M is the Earth's mass and G the gravitational constant.

Period of Revolution

The period of revolution. The period is the time it takes a satellite to describe a complete orbit.

Translational Mechanical Energy

Translational mechanical energy. The mechanical energy of a satellite in orbit around Earth is the sum of its kinetic energy plus its potential energy. The total (mechanical) energy is negative for bound orbits and equals half the value of the potential energy (with a negative sign). The negative sign corresponds to closed orbits, typical of objects that do not have enough energy to escape Earth's gravity.