STEM 699 2014 Derivation of Kepler’s Laws from Newton’s Laws Kepler’s Laws 1. A planet orbits the sun in an elliptical path with the sun at one of the foci. 2. The area swept out by a planet radially from the sun over a fixed period of time is constant. (Hence the planets vary their speed as orbit the sun; planets travel faster when they are closer to the sun.) 3. The square of the orbital period is proportional to the cube of the semimajor axis of the orbit. Newton’s Laws 1. (Newton’s Second Law) 2. (Newton’s Law of Universal Gravitation) Suppose a point of mass is located at the origin (0, 0) and a point of massis located at . We assume (for time being) that the point of mass is fixed and doesn’t move. Let denote the vector , denote the length of this r M r vector (i.e., ), be the unit vector in the direction of (so ). Then the gravitation force that r exerts on is given by where is a universal constant, independent of , , and G r. Putting these two equations together we get or equivalently Angular Momentum of planet about the origin is conserved Suppose a point mass is located at the origin (0, 0) and a point massis located at . Suppose that at all times, there is a central force acting on (i.e., a force always in the direction of =. Note is the position of mass m By law of universal relative to r r M. ’s gravitation, gravity is such a force. Let be the velocity vector of. Define the angular momentum of to be . Then Proof. where is the acceleration. The first term is 0, because a . As for the second term, a is in the direction of so because , it is also 0. r A corollary of this result is that the motion of must lie in a single plane determined by the normal vector , .
STEM 699 2014 Derivation of Kepler’s Laws from Newton’s Laws Kepler’s Laws 1. A planet orbits the sun in an elliptical path with the sun at one of the foci. 2. The area swept out by a planet radially from the sun over a fixed period of time is constant. (Hence the planets vary their speed as orbit the sun; planets travel faster when they are closer to the sun.) 3. The square of the orbital period is proportional to the cube of the semimajor axis of the orbit. Newton’s Laws 1. (Newton’s Second Law) 2. (Newton’s Law of Universal Gravitation) Suppose a point of mass is located at the origin (0, 0) and a point of massis located at . We assume (for time being) that the point of mass M is fixed and doesn’t move. Let r denote the vector , r denote the length of this vector (i.e., ), be the unit vector in the direction of r (so ). Then the gravitation force that exerts on is given by where G is a universal constant, independent of , , and r. Putting these two equations together we get or equivalently Angular Momentum of planet about the origin is conserved Suppose a point mass is located at the origin (0, 0) and a point massis located at . Suppose that at all times, there is a central force acting on (i.e., a force always in the direction of r =. Note r is the position of mass m relative to M. By ’s law of universal gravitation, gravity is such a force. Let be the velocity vector of. Define the angular momentum of to be . Then Proof. where a is the acceleration. The first term is 0, because . As for the second term, a is in the direction of r so because , it is also 0. A corollary of this result is that the motion of must lie in a single plane determined by the normal vector , .
Derivation_of_Keplers_Laws_2014.docx