Question

from the book feynmans lost lecture by david goodstein what are the key steps in feynman's original derivation of kepler's first law from the law of universal gravitation, using only high-school algebra and trigonometry. draw some key diagrams to clarify the important steps.

Answer 1

kepler's first law from the law of universal gravitation

1. Since the planets move on ellipses (Kepler's 1st Law), they are continually accelerating, as we have noted above. As we have also noted above, this implies a force acting continuously on the planets.
2. Because the planet-Sun line sweeps out equal areas in equal times (Kepler's 2nd Law), it is possible to show that the force must be directed toward the Sun from the planet.
3. From Kepler's 1st Law the orbit is an ellipse with the Sun at one focus; from Newton's laws it can be shown that this means that the magnitude of the force must vary as one over the square of the distance between the planet and the Sun.
4. Kepler's 3rd Law and Newton's 3rd Law imply that the force must be proportional to the product of the masses for the planet and the Sun

Newton's law of gravitation

By Newton's second law, the gravitational force that acts on the planet is:

where is the mass of the planet and has the same value for all planets in the solar system. According to Newton's third Law, the Sun is attracted to the planet by a force of the same magnitude. Since the force is proportional to the mass of the planet, under the symmetric consideration, it should also be proportional to the mass of the Sun, . So

where is the gravitational constant.

The acceleration of solar system body number i is, according to Newton's laws:

where is the mass of body j, is the distance between body i and body j, is the unit vector from body i towards body j, and the vector summation is over all bodies in the world, besides i itself.

In the special case where there are only two bodies in the world, Earth and Sun, the acceleration becomes

which is the acceleration of the Kepler motion. So this Earth moves around the Sun according to Kepler's laws.

If the two bodies in the world are Moon and Earth the acceleration of the Moon becomes

So in this approximation the Moon moves around the Earth according to Kepler's laws.

In the three-body case the accelerations are

These accelerations are not those of Kepler orbits, and the three-body problem is complicated. But Keplerian approximation is the basis for perturbation calculations.