ORBITAL MOTION

Armed with his laws of motion, Newton attacked the problem of orbital motion (and hence Kepler's 3 laws).

Newton first proposed that the force which holds the planets in orbit is gravity. (Newton is actually the one who postulated the existence of some form of universal gravitation between massive bodies.) Gravity is a central force, which means that it is directed toward the "mass."



The expression for gravity is

Gravity = - G M m / R²,

where the minus sign means that gravity is attractive and G is a constant number. In our Universe it has a certain value which we can measure in the laboratory. The G is referred as the Gravitational Constant.

Now, how can we understand orbital motion?

On the Earth, if I throw a ball, gravity pulls the ball to the ground.



The Earth exerts a downward force on the ball which accelerates the ball toward the ground at a rate of 9.8 (meters per second) per second. This rate of increase of v is independent of the mass of the ball and how hard I throw the ball outward. This implies that the distance the ball travels from my feet depends upon how hard I throw the ball horizontally and that how long it takes to reach the ground is independent of how hard I throw the ball out horizontally. If I am 4.9 meters tall, the ball will take 1 second to reach the ground, regardless of how hard I throw it horizontally (unless I am Superman).

For orbital motion, we must take account of the fact that the surface of the Earth is not flat, it is curved. (The Earth is spherical in shape.)



The surface of the Earth falls away from the ball while the ball falls toward the center of the Earth. If you throw the ball out just right, the ball and the surface of the Earth fall together and allow the ball to go into orbit. For a circular orbit around the Earth, a speed of



is required.

KEPLER'S 3RD LAW OF PLANETARY MOTION

A planet of mass m moves at speed v in a circular orbit of size R around a star of mass M. The Centrifugal Force felt by the planet is given by

mv²/R

For an object in a circular orbit, the centrifugal force must be balanced by the force of gravity. That is, we require that
Centrifugal Force + Gravitational Force = 0

===> mv²/R = GMm/R² ===> v² = GM/R

To put this is a more familiar form, note that speed = v = circumference of the orbit/period

===> v = 6.28 R / P

so that

6.28² R² / P² = GM/R

===> P² proportional to R³

This is Kepler's 3rd law!!

Kepler's 3rd Law of Planetary is simply a consequence of Newton's Laws of Motion and the form for the Law of Universal Gravitation.

KEPLER'S 2ND LAW OF PLANETARY MOTION

Introduce the linear momentum

Linear Momentum = mass x velocity

The amount of linear momentum an object has does not change in the absence of outside forces; the linear momentum is a conserved quantity. For spinning motion, we define an angular momentum as

angular momentum = mass x velocity x distance from the "spin" axis .

The angular momentum for objects moving in gravitational fields does not change if the objects are not acted upon by an outside "force" (torque).

So, because the angular momentum of an object remains constant, if the object moves closer to and then farther from the "attractor", its speed must change too. How does the speed change? The answer is simple since

mass x velocity x R = constant

as R changes. This implies that v must change in the opposite sense to R.

Kepler's 2nd Law of Planetary Motion is thus simply an alternative way to state the Law of Conservation of Angular Momentum!