Astronomy 101
Forces, Orbits, and Energy
What You Need to Know

The subject material we are now covering in class is difficult to take in all at once. It is essential that you see the material, if you are to appreciate how the astronomer determines how the mechanism of the universe operates. However, it is equally imperative that we not get bogged down in the physics and that we get on to astronomy. This is a brief summary of the salient points of Newton's Laws.

Newton formulated 3 laws of motion:

  1. The law of inertia. This had been stated earlier by Galileo, among others. This law can be summarized as: An object at rest tends to remain at rest; an object in motion tends to remain in uniform linear motion. In the absence of forces, masses do not accelerate (change their velocity). Note that velocity is a vector; it incorporates both magnitude and direction. A chance in the direction of motion requires an acceleration, and hence a force.
  2. The rate of change of motion (the acceleration) is proportional to the force acting on the body. F = m a provides the relation between the strength and direction of the force F acting on a mass m, and the resulting acceleration a. Note that the mass m is defined by this relation.
  3. For every action, there is an equal and opposite reaction.

I. Forces

Newton's Laws describe how matter reacts to forces. Gravity is a force. All matter in the universe feels an attractive force to all other matter in the universe. The gravitational force is described by the expression

Fg = G m M / d2 .

This equation is a algebraic expression describing the gravitational force Fg between 2 objects with masses m and M, separated by a distance d. G is a constant, that is, a number (G=6.7x10-8 cm3 / gm / sec). M, m, and d are variables. They stand for something. d can be replaced by any expression, such as R or foo, so long as it is remembered what the variable stands for. This is the basis of algebra: arithmetic uses numbers (e.g., 1+3=?); algebra uses variables (e.g., X+Y=4, solve for X as a function of Y). To evaluate the strength of the gravitational force between two objects, you multiply their masses by the number G and divide by the square of the distance between the masses.

By Newton's second law, a force produces an acceleration, or a change in velocity. The algebraic expression for this is F=ma. The force F is given by the product of the mass of the object m and the acceleration a. If any two of these quantities are known, the third can be determined. Using these expressions, the gravitational acceleration, the change in velocity of an object with mass m due to the gravitational force, is described by the expression

F = ma = G m M / d2 .
Since the mass m is on both sides of the equation, it cancels out, and one can simplify the expression to
a = G M / d2 .
That m is not to be seen in this equation means that all masses accelerate at the same rate under the influence of a particular gravitational force. Galileo was right; Aristotle was wrong. In the absence of drag due to air, a feather and a ton of lead will fall at the same rate.

If you are m and M represents the mass of the Earth, this acceleration a represents your downward acceleration due to gravity. Your weight is the upward force exerted on the soles of your feet (if you're standing) by the surface of the earth to prevent you from falling towards the center of the Earth. The gravitational force down and your weight (an upwards force) balance and you do not accelerate. You are in equilibrium. If instead you are actually falling, you have no weight (although your mass does not change).

Note that there is nothing to prevent me from using M to represent your mass, and m to represent the mass of the Earth. In that case, the variable a would represent the acceleration of the Earth due to your mass. This is small, but real. This is an example of Newton's third law - of equal and opposite reactions. The gravitational force attracting you to the center of the Earth is the same magnitude (but opposite in direction) to the gravitational force attracting the center of the Earth to you. While the forces are the same strength, the accelerations are different by the ratio of the masses (F = ma). Your acceleration is some 1021 times that experienced by the Earth because of your gravity.

II. Orbits

An orbit is merely the trajectory followed by a mass under the influence of the gravity of another mass. Gravity and Newton's laws explain orbits. The gravitational force between two bodies causes an acceleration. This can take the form of a change in the speed (the magnitude of the velocity) or in the direction of motion. In a circular orbit (or for any particle travelling is a circular path) the acceleration is given by the expression

a=V2/d ,
where V represents the velocity of the object and d represents the radius of the orbit. This is the centrifugal force you feel when you turn a corner at high speed: because of Newton's first law, you want to keep going in a straight line. The car seat exerts a force on you to keep you within the car as it turns.

This acceleration in orbit is the gravitational acceleration, so we can equate the two expressions. This gives

V2/d = G M / d2
which is equivalent to saying
V = (GM/d)1/2 .
(note: (GM)1/2 is the square root of the product of G and M) This gives us the velocity of a body in a circular orbit. In low Earth orbit, orbital velocities are about 17,500 miles per hour. The expression for orbital velocity is a very important equation. It says that if we know the orbital velocity and the radius of the orbit, then we can determine the mass of the central object. This is the only way to determine the masses of stars and planets.

There is no mysterious force which keeps bodies in orbit. In fact, bodies in orbit are continuously falling. What keeps them in orbit is their sideways velocity. The force of gravity changes the direction of the motion by enough to keep the body going around in a circular orbit. You can look at it in the following way: as the body falls, the object it falls towards keeps moving to the side so the falling body never gets to it. As an aside, an astronaut in orbit is weightless because he (or she) is continuously falling. Weight is the force exerted by the surface of the Earth to counteract gravity. The Earth, the Sun, and the Moon have no weight! Your weight depends on where you are - you weigh less on the top of a mountain than you do in a valley; your mass is not the same as your weight.

Newton's theory, also called Newtonian mechanics, is a theory of masses and how they act under the influence of gravity. Through using this theory we can determine the masses of objects without having to go to them. We can observe the universe; it is not necessary to experiment with it.

III. Deriving Kepler's Third Law

Kepler's Third Law states that P2 = d3 for the planets, where P is the orbital period measured in years and d is the distance from the Sun, measured in Astronomical Units (AU). Clearly this is true for the Earth, where P=1 year and d=1 AU.

Earlier we derived an equation for the velocity of an object in a circular orbit about a mass M,

V = (GM/d)1/2 .
The circumference of a circular orbit with a radius d is 2πd. The velocity (or more correctly, the speed) of an object is the distance it travels divided by the time it takes, so the orbital velocity is
Vorb = 2πd/P
since the period P is the time it takes to go once around the orbit. Equating these two velocities, we find that
(GM/d)1/2 = 2πd/P
Squaring both sides gives
GM/d = 4π2d2/P2
which simplifies to
P2 = (4π2/GM)d3
Note that this is Kepler's third law, P2 = d3, with the complication of constant of proportionality. What is this constant? The numerical value of 4π2/GM, where M is the mass of the Sun, is 2.96 x 10-25 in units of seconds2 per cm3. Realizing that there are 31 million seconds in one year, and the Astronomical Unit is 1.5 x 1013 cm, the numerical value of the proportionality constant 4π2/GM is 1 in units of years and Astronomical Units, as Kepler posited.

So Kepler's third law can be derived from Newton's laws of motion. Newton's laws, plus the law of Gravity, actually constitute a theory that makes broad sweeping predictions, while Kepler's laws are empirical laws that simplify the observed motions of the planets without explaining why. This generalization of Kepler's third law, P2 = (4π2/GM)d3, works not only in our Solar System, but everywhere in the universe!

To come up with Kepler's second law requires a diversion into the concept of energy.

IV. Energy

Another important concept is that of energy. There are two kinds of energy we are concerned with now: kinetic energy (abbreviated K) and potential energy (abbreviated U). Kinetic energy is energy of motion, and is evaluated using the equation

K = 1/2 m V2 ,
where m is the mass of the body and V is its velocity. Kinetic energy can never be negative. The potential energy is energy that is available to the object, but is currently not being used. The potential energy available due to gravity is given by the expression
U = -G m M / d ,
where m and M are the masses of the 2 objects and d is the separation. Note the similarity between this expression and that for the gravitational force. By convention, potential energy is negative. As a body falls due to gravity, its potential energy decreases and its kinetic energy increases. Energy is conserved, so the sum of K and U must stay constant.

V. Orbits and Energy

In an orbit, conservation of energy can be used to explain Kepler's second law. When the distance to the Sun decreases, the potential energy decreases as well. This must be compensated for with an increase in kinetic energy, which means the orbital velocity must be fastest when closest to the Sun.

The potential energy is given by the expression

U = -G m M / d .
In any bound orbit, the total energy (K+U) is less than zero, because there is more potential energy than kinetic energy. In a circular orbit, with the distance d a constant, the kinetic energy is 1/2 the potential energy. If the kinetic energy is large enough, it can overcome the gravitational attraction and the object can escape the gravitational pull. This happens when the total energy is zero, or when the kinetic energy is as large as the potential energy. You can use this fact to calculate the velocity necessary to escape from orbit. This is the escape velocity required to send astronauts to the Moon, send the Galileo spacecraft to Jupiter, or send the Voyager spacecraft out of the Solar system. The escape velocity is
Vesc = (2G M / d)1/2 .
Note that this is only 40% greater than the circular orbital velocity. Vesc from the Earth is about 25,000 miles per hour.

The point of going through this exercise is to illustrate the elegance and the power of Newton's theory. It is a true triumph of the human intellect. It permits Earthbound observers to determine the mass of the universe, based on observations of distances (from parallaxes), periods (from timing), and velocities (which we'll talk about when we discuss light). I do not expect you remember these equations, much less be able to derive the results outlined above. However, I do expect you to understand the concepts involved. You need to know Newton's laws. You need to understand the concepts of forces, orbits, and energy. We will use them repeatedly as we take the measure of the universe.