Chapter 4. Newton's Laws of Gravitation

4.1 Introduction

The orbits of planets, moons, and spacecrafts are all possible due to gravity.

The goals of this module: After completing this exercise, you should be able to:

1. Explain how the force of gravity depends on mass and distance.
2. Describe what Newton's law of gravity teaches us about orbits.
3. Compare your weight on different planets.

In this module you will explore:

1. How the gravitational force depends on mass and distance.
2. How the shapes of different orbits are predicted by Newton's formulation of the law of gravity.
3. How the interplay between velocity and gravity results in stable orbits.
4. How gravitational acceleration differs from the force of gravity.

Why you are doing it: Gravity is one of the fundamental forces in the Universe. It is responsible for the orbits of all objects and keeps us safely bound to our Earth. Understanding Newton's law of gravity is key to understanding many problems in astronomy.

4.2 Background

Newton's Law of Universal Gravitation

Newton's law of universal gravitation is used to calculate the masses of astronomical objects, their orbits, their rotational and orbital precessions, satellite trajectories, etc. It is one of the most important and most widely used laws in astronomy!

Newton's law of gravity is remarkable because it is truly universal: it describes the motions of everyday objects in numerous situations on Earth (such as a baseball being thrown or an acorn falling to the ground) as well as the detailed interactions of orbiting objects everywhere in the Universe. It is simply amazing that a physical law formulated in the late 17^th century applies today in the same form in such a wide variety of circumstances.

This powerful law of gravitation is quite simply a description of how the mutual attraction between any two objects depends on their masses and their separation. In this activity, we will first study how the force of gravity depends on mass and then how it depends on the distance between the two objects. We will then examine how the law applies to orbital mechanics.

4.3 Newton's Law of Gravitation - Proportional to Mass

Newton's law of gravitation states that the force of attraction between any two objects is directly proportional to the product of their masses. We can write this mathematically as follows:

where F is the force of gravity, and m₁ & m₂ are the masses of the two objects. The symbol \( \propto \) means "is proportional to." The animation below illustrates this dependence. Change the mass of the left ball to investigate how the force of gravity depends on mass. See what happens to the gravitational force when you double the mass, then triple the mass, etc.

4.4 Newton's Law of Gravitation - Inversely Proportional to Distance

The law of gravitation states that the force of gravity between any two objects is inversely proportional to the square of the distance between them. This means that the force of gravity becomes increasingly smaller as the objects move farther apart. Because gravity depends on the square of the distance, it becomes smaller very quickly as two objects move apart. Mathematically, we can write this as:

Where r is the distance between the centers of the two objects. In the animation below, we can see how this dependence works. Change the separation between the balls to see how the force of gravity depends on the distance between two objects. See what happens when you double the distance, then triple the distance, etc.

The farther apart two objects become, the weaker the attraction between them becomes. This is shown dramatically in the following figure:

4.5 Newton's Law of Gravity - Combining Mass and Distance

Now let's put together what we have learned. We know that the force of gravity is proportional to the product of the two masses. We also know that it is inversely proportional to the square of the distance. Combining these, we can write:

G is simply a constant (a number) that allows us to exchange the proportionality symbol for an equals sign in the equation. Change the mass of the left ball and the distance between the balls in the animation below to see how varying both the mass and the distance of separation affects the force of gravity between two objects. Investigate what happens as you change the values of the variables.

4.6 Comparing Different Planets

While we technically feel some gravitational attraction from every other object in the entire Universe, by far the strongest effect on us comes from the Earth itself. Earth is a very massive, very close object; the previous animation shows that those two facts mean the Earth exerts a very strong gravitational force on us. It is the Earth's gravity that keeps us from flying off into space. The strength of the gravitational force on us is simply known as our weight. In other words, F_⊕ = Weight. (⊕ is the symbol for Earth.) This means that our weight depends on the mass of the Earth, our own mass, and how far we are from the center of the Earth. So, losing weight is as easy as climbing a very tall mountain: when we increase the distance between us and the center of the Earth, the force of gravity we feel, and therefore our weight, decreases! In the same way, if we were to travel to a different planet then our weight would change depending on how strong the force of gravity is on that planet.

Type your weight on Earth in the box above to see what you would weigh on the other planets.

4.7 Relation to Speed

Gravity is what keeps spacecraft, planets, stars, and even galaxies in orbit! What makes something orbit another object? The following animation demonstrates the answer. An object will be launched from above the North Pole, with an increasing speed on each attempt, starting with zero speed on attempt A, and increasing the speed from B to F. Click on play.

4.8 Orbiting The Sun

To see how these concepts relate to planets orbiting around the Sun, look at the figure below:

The planet falls toward the Sun while moving forward, resulting in a stable orbit.

4.9 Conic Sections

We have talked about circular and elliptical orbits so far. By studying the form of the law of gravity, Newton discovered that orbits can have other shapes as well. These shapes are all known as conic sections. Conic sections are simply slices that can be cut from a cone.

Notice how slicing the cone in different ways results in different orbital shapes.

4.10 Kepler's Third Law Revised

Newton's laws give us insight into why Kepler's empirical (i.e., not mathematically derived) laws are true. Through his work, Newton was able to explain why the planets travel on elliptical orbits (Kepler's first law); it is because of the \( \frac{1}{ r^{2} } \) dependence of the law of gravity. He showed that Kepler's second law (a line joining the planet and the Sun sweeps out equal areas in equal times) is due to the fact that the Sun's gravitational force pulls an orbiting body radially towards itself. Newton also derived (and corrected!) Kepler's third law from the law of universal gravitation. Kepler's third law states that the square of the orbital period (P) is proportional to the cube of the orbit's semi-major axis (a). Newton demonstrated that the correct version of Kepler's third law is:

The reason that Kepler's law works so well in our solar system is that m₂, the mass of a planet, is completely insignificant compared to m₁, the mass of the Sun. When the two masses are comparable, Newton's correction becomes very important, as shown in the animation below. Here the blue and red objects represent stars. Play the animation.

In our solar system the planets are dominated by the Sun's gravity. When stars are in orbit around each other, their masses are closer in value so we see them orbiting quite differently than planets around the Sun.

4.11 Quick Check Quiz

Indepth Activity: Newton's Laws of Gravitation