GRAVITY: KEPLER'S AND NEWTON'S LAWS

Planetary Orbits: Circles or Ellipses?

Planets orbit the sun, but what is the shape of the orbital path and how does the planet move along this orbital path?

Two closed paths are the circle and the ellipse.

These closed curves are of the same family of curves; all that distinguished them from one another is how elongated (flattened) they are. This flattening is called eccentricity. A circle has e = 0, or zero eccentricity (they are perfectly round). Ellipses have varying degrees of eccentricity, somewhere between slightly greater than zero and slightly less than one ( 0 < e < 1 ).

The parts of a circle are basically a center, a radius, and a circumference. The circumference is the distance around the circle. The radius is the distance from the center to the circumference. The parts of an ellipse are two focus points (called foci, "fo-sigh"), the major axis and the minor axis. As eccentricity increases the ellipse becomes more flattened.

Kepler's Laws of Planetary Motion

It has long been thought that the circle was the orbital path of planets. That wisdom was proved incorrect.

Kepler's 1st Law: The orbital path of planets are ellipses. The sun is situated at one of the foci.

Two important terms are: aphelion is the farthest point from the sun in the orbit; perihelion is the closest point from the sun in the orbit. (both intersect the major axis).

Kepler's 2nd Law:. A line from a planet to the sun sweeps out equal areas in equal times.

This crazy statement is really about the speed at which a planet orbits the sun. When a planet is at perihelion, it is moving fastest along the orbital path; when planet is aphelion it is moving slowest. That is, planets do not move along their orbital path with constant speed; their speed actually changes as they orbit the sun. Kepler's second law is a statement about the conservation of angular momentum.

Newton's Laws of Motion

All matter has mass. Mass is a measure of how matter resists forces that would change the present motion of the matter. This is called inertia.

The concept of inertia compels us to consider Newton's Laws of Motion.

1st is the Law of Inertia. If no force is acting on an object, then the object will stay in its present state of motion no matter what that motion is. If the object is at rest... then it will stay at rest. If it is moving at 10 km/s due south, it will stay that way--- unless a force acts upon it to change its motion.

2nd is the Law of Acceleration. If a force is acting on an object, its state of motion will change such that the object will be compelled to move in the direction of the force. If the force is in the current direction of motion, the object simple increases in speed (accelerates) in its current direction. If the force is perpendicular to the motion of the object, the object will follow a curved path as it is compelled to move off in the direction of the force, after which it will then simply accelerate in the direction of the force.

EXAMPLE: You have a rock on the end of a short rope. You twirl the rock around your head. At any instant of time, the rock is moving with a velocity in a given direction, and because of inertia, it wants to keep moving in that direction at that instant. But, you are applying a force on the rock through the rope and the force is directed along the direction of the rope, toward the center of the circle. This force compels the rock to change its direction of motion toward the center of the circle (Newton's 2nd Law). But its inertia keeps it from falling into the center. So, the circular motion of the rock is the combination of the Law of Inertia (rock wants to keeps going the direction its in) and the Law of Acceleration (rock is compelled to move toward the center of the circle). Now, you let go of the rope, what does the rock do? It flies off in the direction and with the speed it had at the instant you let go (Newton's 1st Law).

Newton's Laws of Gravity and Orbits

All masses attract one another. All masses.

The force of the attraction is called gravity. The force of gravity has a very specific and well known behavior. The greater the masses, the greater the gravity (mutual attraction). The smaller the distance separating the masses, the greater the gravity. And, of course the converse is true. The smaller the masses, the weaker the gravity. The larger the distance between the masses, the weaker the gravity. (We worked all this out in class lecture.)

Just like the inverse square law for light... gravity is also an inverse square law..... two examples.... (for details, see the table in the notes).

If two masses are separated by a distance 2 times greater than their original distance, the gravity gets 4 times weaker. Imagine it. Think about it.

If two masses are separated by a distance 1/2 of their original distance, the gravity gets 4 times stronger. Imagine it. Think about it.

EXAMPLE combining Laws of Motion and Gravity: You have a cup of coffee in the passenger seat of your car and you are driving 60 mph straight down the highway. Gravity is forcing the coffee downward into your seat and the seat is forcing the coffee upward in a perfect balance so that the net force on the coffee is zero. Suddenly a jerk pulls in front of you and you slam on the break (apply a sideways force). Your car slows down, but your cup of coffee, has nothing to stop it from continuing to move 60 mph down the road, so it doesn't feel any force applied to it- so it continues at its current speed and direction (Law of Inertia). Since your car slowed down, the coffee catches up to your dashboard and collides with it- a ha ha- now it feels a force! The coffee rejoins the speed of your car. But alas, there is now no seat for it to rest on to balance the downward force of gravity. Oh crap. The force of gravity compels the coffee to move toward the center of the earth and you are left with a big smelly mess on your car floor.

So, what is an orbit? In short an orbit of a smaller mass object around a (fixed) higher mass object is a delicate balance between (1) the perpetual falling of the small mass object toward the center of high mass object due to the force of gravity, and (2) the inertia of the orbiting object because of its instantaneous transverse (sideways) velocity of motion, which keeps it wanting to move sideways relative to its downward fall. Its much like the rock you twirled on the end of the rope.