Kepler’s Laws and Newton’s Law of Universal Gravitation
Kepler’s Laws
Kepler’s laws are empirical laws, enunciated in the seventeenth century to describe the motion of planets around the Sun:
1st Law (Law of Orbits)
Planets move in elliptical orbits with the Sun at one of the two foci. As the force is central, the angular momentum remains constant: L = R x M · v = constant.
2nd Law (Law of Areas)
A line joining a planet and the Sun sweeps out equal areas during equal intervals of time. This implies that the linear speed is greater when the planet is closer to the Sun. This law is equivalent to the conservation of angular momentum of the planet with respect to the Sun.
The area of the parallelogram described by the vectors r and dr is given by the magnitude of the vector product of both. Thus, the area of the triangle will be:
3rd Law (Law of Periods)
The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. One consequence is that the linear velocity of the planets is not constant but depends on the orbital radius: a planet moves faster the smaller its orbit.
Kepler’s laws were later theoretically demonstrated thanks to Newton’s Law of Universal Gravitation. Equating the centripetal force with the gravitational force yields:
Newton’s Law of Universal Gravitation
Enunciated by Newton in the seventeenth century, this law explained all known gravitational effects at the time, including the movement of planets, tides, and falling objects. It governs the interactions between any two masses.
The law states: Every body in the universe attracts every other body with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. Mathematically:
- F is the gravitational force between two bodies of masses m1 and m2, considered point masses at their centers.
- r is the distance between the centers, and ur is a unit vector pointing from the body exerting the force to the body experiencing it. The minus sign indicates that the force is attractive.
- G is the gravitational constant, experimentally measured in 1785 by Henry Cavendish using a torsion balance. Its value is:
Knowing the value of G allowed for the determination of the mass of the Sun and the planets.
The equation of gravitational force applies equally to both masses. For example, the Earth’s attraction on the Moon is equal and opposite to the Moon’s attraction on Earth. For any body on the surface of a star, the following must be fulfilled in vector form:
This allows us to calculate g0 on the star’s surface.
For a set of particles, the gravitational force on each particle is the vector sum of the forces produced by the other particles.
Gravitational Potential Energy
The gravitational force, being conservative, has an associated gravitational potential energy function, Ep, such that the work done by the force between two points equals the decrease in this potential energy. The work of the conservative force is:
The total work is the sum of the elementary work, found through integration:
Therefore, the gravitational potential energy of a particle of mass m1 at a distance r from another mass m2 is:
Here, we assume the potential energy at infinity is zero. The negative sign arises because gravity is always attractive, and an external force is needed to move a body from a field to infinity at constant speed.
The gravitational potential energy associated with two point masses separated by a distance r equals the work done by the gravitational force to separate them infinitely (rB = ∞).
For a system with more than two masses, the total gravitational potential energy is the sum of the potential energies of all distinct pairs of masses. For example, for a three-mass system:
Due to gravity, bodies tend to fall spontaneously into regions of lower potential energy.
Potential Energy Near Earth’s Surface
The gravitational force acting on a body of mass m is its weight, F = –m · g j. Considering a constant value of g near Earth, the work done by the weight force when the body moves vertically from point A to B is:
Thus, the potential energy at a height h is:
Where we have chosen the zero-energy level at h = 0.
If a body of mass m on Earth’s surface rises to a height h, the change in potential energy can be expressed as:
