Concepts: Kepler's laws and Newton's law of gravitation
Gravitation22 min readLesson 2 of 6
2.1 Kepler's three laws
Kepler stated three laws from the observed motion of the planets. The first says the path of a planet is an ellipse, and the sun sits at one of its two foci. The second says that the radius drawn from the sun to the planet covers the same area in the same time, wherever the planet happens to be. The third connects the period to the size of the orbit: the square of the period varies as the cube of the semi-major axis.
The second law is a statement about angular momentum. The gravitational force on the planet acts along the line toward the sun, so it produces no torque about the sun, and the angular momentum of the planet about the sun stays constant. The areal velocity is half the angular momentum per unit mass, so equal areas in equal times follows directly. The figure shows an ellipse with the sun at one focus and the shaded area swept out in a small time interval.
The third law relates the period to the size of the orbit, and it is the law that lets a period be predicted from a radius. For circular orbits the derivation is short: the gravitational force supplies the centripetal force, so G M m over r squared equals m v squared over r, and substituting v as 2 pi r over T gives T squared proportional to r cubed. The constant of proportionality depends on the mass of the central body, which is why the law holds for all planets around the same sun with the same constant.
2.2 Newton's law of gravitation
Newton's law states that every particle attracts every other particle with a force proportional to the product of the masses and inversely proportional to the square of their separation. The force is central and attractive, and it obeys the third law: the force on one body due to the other is equal and opposite to the force on the other due to the first. The constant G is universal, with the value 6.67 times 10 to the power minus 11 in SI units.
The inverse square form is what produces Kepler's laws. The equal-area law comes from the force being central, and the period law from the inverse square magnitude. Newton tested the inverse square law against the Moon itself, treating the Moon as a body falling toward the Earth. The Moon is about sixty Earth radii away, so the acceleration there should be smaller than the surface value by a factor of about 3600. Multiplying that small acceleration by the square of the orbital period should return the orbital radius, and it did. That agreement between an Earth-bound value of g and the motion of a distant body was the decisive check.
For a system of several masses the forces add as vectors, by superposition. The force on a mass due to several others is the vector sum of the individual forces, each computed as if the other masses were absent. This lets the field of a complicated distribution be built from the fields of its parts.
2.3 Worked examples
The force between two 1 kg masses separated by 1 m is G, about 6.67 times 10 to the power minus 11 N. That is tiny, which is why the gravitational attraction between everyday objects is not noticed unless one of the masses is planetary. The gravitational force between the Earth and a 1 kg body at the surface is about 9.8 N, and dividing by the electron mass and the like only shows how weak gravity is at small scales.
The ratio of the gravitational force to the electrostatic force between two protons is about 10 to the power minus 36, which shows that gravity dominates on astronomical scales only because large bodies are electrically neutral. The ratio is worth remembering as a demonstration of how much weaker gravity is than the electric force between two charged particles.
A further point about the law is that the two forces of a pair are always equal and opposite, so the accelerations they produce are inversely proportional to the masses. The heavy body barely moves while the light one accelerates toward it, which is why the Earth does not visibly move toward a falling stone. A useful exercise is to compare the accelerations of two masses pressed together under their mutual gravity. The forces are equal and opposite, so the accelerations are inversely proportional to the masses, exactly as in any third law pair. The heavy body barely moves while the light one accelerates toward it, which is why the Earth does not visibly move toward a falling stone.
Common mistake
The force between two bodies depends on the distance between their centres, not on the radii of the bodies, and it is an inverse square in that distance. Substituting the diameter or the surface separation for the centre separation is a common error.