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5 Kepler’s Laws
5.1 Introduction
Throughout human history, the motion of the planets in the sky was a mystery: why did some planets move quickly across the sky, while other planets moved very slowly? Even two thousand years ago it was apparent that the motion of the planets was very complex. For example, Mercury and Venus never strayed very far from the Sun, while the Sun, the Moon, Mars, Jupiter and Saturn generally moved from the west to the east against the background stars (at this point in history, both the Moon and the Sun were considered “planets”). The Sun appeared to take one year to go around the Earth, while the Moon only took about 30 days. The other planets moved much more slowly. In addition to this rather slow movement against the background stars was, of course, the daily rising and setting of these objects. How could all of these motions occur? Because these objects were important to the cultures of the time, even foretelling the future using astrology, being able to predict their motion was considered vital. The ancient Greeks had developed a model for the Universe in which all of the planets and the stars were each embedded in perfect crystalline spheres that revolved around the Earth at uniform, but slightly different speeds. This is the “geocentric”, or Earth-centered model. But this model did not work very well–the speed of the planet across the sky changed. Sometimes, a planet even moved backwards! It was left to the Egyptian astronomer Ptolemy (85 − 165 AD) to develop a model for the motion of the planets (you can read more about the details of the Ptolemaic model in your textbook). Ptolemy developed a complicated system to explain the motion of the planets, including “epicycles” and “equants”, that in the end worked so well, that no other models for the motions of the planets were considered for 1500 years! While Ptolemy’s model worked well, the philosophers of the time did not like this model–their Universe was perfect, and Ptolemy’s model suggested that the planets moved in peculiar, imperfect ways. In the 1540’s Nicholas Copernicus (1473 − 1543) published his work suggesting that it was much easier to explain the complicated motion of the planets if the Earth revolved around the Sun, and that the orbits of the planets were circular. While Copernicus was not the first person to suggest this idea, the timing of his publication coincided with attempts to revise the calendar and to fix a large number of errors in Ptolemy’s model that had shown up over the 1500 years since the model was first introduced. But the “heliocentric” (Sun- centered) model of Copernicus was slow to win acceptance, since it did not work as well as the geocentric model of Ptolemy. Johannes Kepler (1571 − 1630) was the first person to truly understand how the planets in our solar system moved. Using the highly precise observations by Tycho Brahe ( − 1601) of the motions of the planets against the background stars, Kepler was able to formulate three laws that described how the planets moved. With these laws, he was able to predict the future motion of these planets to a higher precision than was previously possible.
Many credit Kepler with the origin of modern physics, as his discoveries were what led Isaac Newton (1643 − 1727) to formulate the law of gravity. Today we will investigate Kepler’s laws and the law of gravity.
5.2 Gravity
Gravity is the fundamental force governing the motions of astronomical objects. No other force is as strong over as great a distance. Gravity influences your everyday life (ever drop a glass?), and keeps the planets, moons, and satellites orbiting smoothly. Gravity affects everything in the Universe including the largest structures like super clusters of galaxies down to the smallest atoms and molecules. Experimenting with gravity is difficult to do. You can’t just go around in space making extremely massive objects and throwing them together from great distances. But you can model a variety of interesting systems very easily using a computer. By using a computer to model the interactions of massive objects like planets, stars and galaxies, we can study what would happen in just about any situation. All we have to know are the equations which predict the gravitational interactions of the objects. The orbits of the planets are governed by a single equation formulated by Newton:
Fgravity =
GM 1 M 2
R^2
A diagram detailing the quantities in this equation is shown in Fig. 5.1. Here Fgravity is the gravitational attractive force between two objects whose masses are M 1 and M 2. The distance between the two objects is “R”. The gravitational constant G is just a small number that scales the size of the force. The most important thing about gravity is that the force depends only on the masses of the two objects and the distance between them. This law is called an Inverse Square Law because the distance between the objects is squared, and is in the denominator of the fraction. There are several laws like this in physics and astronomy.
Figure 5.1: The force of gravity depends on the masses of the two objects (M 1 , M 2 ), and the distance between them (R).
Today you will be using a computer program called “Planets and Satellites” by Eugene Butikov to explore Kepler’s laws, and how planets, double stars, and planets in double star
Figure 5.2: Four types of curves can be generated by slicing a cone with a plane: a circle, an ellipse, a parabola, and a hyperbola. Strangely, these four curves are also the allowed shapes of the orbits of planets, asteroids, comets and satellites!
Figure 5.3: An ellipse with the major and minor axes identified.
Exercise #2: In the ellipse shown in Fig. 5.5, two points (“P 1 ” and “P 2 ”) are identified that are not located at the true positions of the foci. Repeat exercise #1, but confirm that P 1 and P 2 are not the foci of this ellipse. (2 points)
Figure 5.4: An ellipse with the two foci identified.
Figure 5.5: An ellipse with two non-foci points identified.
Now we will use the Planets and Satellites program to examine Kepler’s laws. It is possible that the program will already be running when you get to your computer. If not, however, you will have to start it up. If your TA gave you a CDROM, then you need to insert the CDROM into the CDROM drive on your computer, and open that device. On that CDROM will be an icon with the program name. It is also possible that Planets and Satellites has been installed on the computer you are using. Look on the desktop for an icon, or use the start menu. Start-up the program, and you should see a title page window, with four boxes/buttons (“Getting Started”, “Tutorial”, “Simulations”, and “Exit”). Click
Now let’s put the Initial Velocity down to a value of 1.0. Run the simulation. What is happening here? The orbit is now a circle. Where are the two foci located? In this case, what is the distance between the focus and the orbit equivalent to? (4 points)
The point in the orbit where the planet is closest to the Sun is called “perihelion”, and that point where the planet is furthest from the Sun is called “aphelion”. For a circular orbit, the aphelion is the same as the perihelion, and can be defined to be anywhere! Exit this simulation (click on “File” and “Exit”).
Exercise #4: Kepler’s Second Law: “A line from a planet to the Sun sweeps out equal areas in equal intervals of time.” From the simulation window, click on the “Second Law” after entering the Kepler’s Law window. Move the Initial Velocity slide bar to a value of 1.2. Hit Go. Describe what is happening here. Does this confirm Kepler’s second law? How? When the planet is at perihelion, is it moving slowly or quickly? Why do you think this happens? (4 points)
Look back to the equation for the force of gravity. You know from personal experience that the harder you hit a ball, the faster it moves. The act of hitting a ball is the act of applying a force to the ball. The larger the force, the faster the ball moves (and, generally, the farther it travels). In the equation for the force of gravity, the amount of force generated depends on the masses of the two objects, and the distance between them. But note that it depends on one over the square of the distance: 1/R^2. Let’s explore this “inverse square law” with some calculations.
- If R = 1, what does 1/R^2 =?
- If R = 2, what does 1/R^2 =?
- If R = 4, what does 1/R^2 =?
What is happening here? As R gets bigger, what happens to 1/R^2? Does 1/R^2 de- crease/increase quickly or slowly? (2 points)
The equation for the force of gravity has a 1/R^2 in it, so as R increases (that is, the two objects get further apart), does the force of gravity felt by the body get larger, or smaller? Is the force of gravity stronger at perihelion, or aphelion? Newton showed that the speed of a planet in its orbit depends on the force of gravity through this equation:
V =
√ (G(Msun + Mplanet)(2/r − 1 /a)) (2)
where “r” is the radial distance of the planet from the Sun, and “a” is the mean orbital radius (the semi-major axis). Do you think the planet will move faster, or slower when it is closest to the Sun? Test this by assuming that r = 0.5a at perihelion, and r = 1.5a at aphelion, and that a=1! [Hint, simply set G(Msun + Mplanet) = 1 to make this comparison very easy!] Does this explain Kepler’s second law? (4 points)
Start-up the Third Law simulation and hit Go. You will see that the inner planet moves around more quickly, while the planet in the larger ellipse moves more slowly. Let’s set-up the math to better understand Kepler’s Third Law. We begin by constructing the ratio of of the Third Law equation (P^2 = C × a^3 ) for an arbitrary planet divided by the Third Law equation for the Earth:
P (^) P^2 P (^) E^2
C × a^3 P C × a^3 E
In this equation, the planet’s orbital period and average distance are denoted by PP and aP , while the orbital period of the Earth and its average distance from the Sun are PE and aE. As you know from from your high school math, any quantity that appears on both the top and bottom of a fraction can be canceled out. So, we can get rid of the pesky constant “C”, and Kepler’s Third Law equation becomes:
P (^) P^2 P (^) E^2
a^3 P a^3 E
But we can make this equation even simpler by noting that if we use years for the orbital period (PE = 1), and Astronomical Units for the average distance of the Earth to the Sun (aE = 1), we get:
P (^) P^2 1
a^3 P 1
or P (^) P^2 = a^3 P (5)
(Remember that the cube of 1, and the square of 1 are both 1!) Let’s use equation (5) to make some predictions. If the average distance of Jupiter from the Sun is about 5 AU, what is its orbital period? Set-up the equation:
P (^) J^2 = a^3 J = 5^3 = 5 × 5 × 5 = 125 (6)
So, for Jupiter, P 2 = 125. How do we figure out what P is? We have to take the square root of both sides of the equation:
√ P 2 = P =
125 = 11. 2 years (7)
The orbital period of Jupiter is approximately 11.2 years. Your turn:
If an asteroid has an average distance from the Sun of 4 AU, what is its orbital period? Show your work. (2 points)
In the Third Law simulation, there is a slide bar to set the average distance from the Sun for any hypothetical solar system body. At start-up, it is set to 4 AU. Run the simulation, and confirm the answer you just calculated. Note that for each orbit of the inner planet, a small red circle is drawn on the outer planet’s orbit. Count up these red circles to figure out how many times the Earth revolved around the Sun during a single orbit of the asteroid. Did your calculation agree with the simulation? Describe your results. (2 points)
If you were observant, you noticed that the program calculated the number of orbits that the Earth executed for you (in the “Time” window), and you do not actually have to count up the little red circles. Let’s now explore the orbits of the nine planets in our solar system. In the following table are the semi-major axes of the nine planets. Note that the “average distance to the Sun” (a) that we have been using above is actually a quantity astronomers call the “semi-major axis” of a planet. a is simply one half the major axis of the orbit ellipse. Fill in the missing orbital periods of the planets by running the Third Law simulator for each of them. (3 points)
Table 5.1: The Orbital Periods of the Planets
Planet a (AU) P (yr) Mercury 0.387 0. Venus 0. Earth 1.000 1. Mars 1. Jupiter 5. Saturn 9.54 29. Uranus 19.22 84. Neptune 30.06 164. Pluto 39.5 248.
Notice that the further the planet is from the Sun, the slower it moves, and the longer it takes to complete one orbit around the Sun (its “year”). Neptune was discovered in 1846, and Pluto was discovered in 1930 (by Clyde Tombaugh, a former professor at NMSU). How
Figure 5.6: A diagram of the definition of the center of mass. Here, object one (M 1 ) is twice as massive as object two (M 2 ). Therefore, M 1 is closer to the center of mass than is M 2. In the case shown here, X 2 = 2X 1.
window of the simulation.
Exercise 5: Binary Star systems. We now want to set-up some special binary star orbits to help you visualize how gravity works. This requires us to access the “Input” window on the control bar of the simulation window to enter in data for each simulation. Clicking on Input brings up a menu with the following parameters: Mass Ratio, “Transverse Velocity”, “Velocity (magnitude)”, and “Direction”. Use the slide bars (or type in the numbers) to set Transverse Velocity = 1.0, Velocity (magnitude) = 0.0, and Direction = 0.0. For now, we simply want to play with the mass ratio. Use the slide bar so that Mass Ratio = 1.0. Click “Ok”. This now sets up your new simulation. Click Run. Describe the simulation. What are the shapes of the two orbits? Where is the center of mass located relative to the orbits? What does q = 1.0 mean? Describe what is going on here. (4 points)
Ok, now we want to run a simulation where only the mass ratio is going to be changed. Go back to Input and enter in the correct mass ratio for a binary star system with M 1 = 4.0, and M 2 = 1.0. Run the simulation. Describe what is happening in this simulation. How are the stars located with respect to the center of mass? Why? [Hint: see Fig. 5.6.] (4 points)
Finally, we want to move away from circular orbits, and make the orbit as elliptical as possible. You may have noticed from the Kepler’s law simulations that the Transverse Velocity affected whether the orbit was round or elliptical. When the Transverse Velocity = 1.0, the orbit is a circle. Transverse Velocity is simply how fast the planet in an elliptical orbit is moving at perihelion relative to a planet in a circular orbit of the same orbital period. The maximum this number can be is about 1.3. If it goes much faster, the ellipse then extends to infinity and the orbit becomes a parabola. Go back to Input and now set the Transverse Velocity = 1.3. Run the simulation. Describe what is happening. When do the stars move the fastest? The slowest? Does this make sense? Why/why not? (4 points)
The final exercise explores what it would be like to live on a planet in a binary star system–not so fun! In the “Two-Body and Many-Body” simulations window, click on the
See if you can find the value of q at which larger values cause the planet to “stay home”, while smaller values cause it to (eventually) crash into one of the stars (stepping up/down by 0.01 should be adequate). (2 points)
Ok, reset q = 0.5, and now let’s adjust the Planet–Star Distance. In the Settings window, set the Planet–Star Distance = 0.1 and run a simulation. Note the outcome of this simulation. Now set Planet–Star Distance = 0.3. Run a simulation. What happened? Did the planet wander away from its parent star? Are you surprised? (4 points)
Astronomers call orbits where the planet stays home, “stable orbits”. Obviously, when the Planet–Star Distance = 0.24, the orbit is unstable. The orbital parameters are just right that the gravity of the parent star is not able to hold on to the planet. But some orbits, even though the parent’s hold on the planet is weaker, are stable–the force of gravity exerted by the two stars is balanced just right, and the planet can happily orbit around its parent and never leave. Over time, objects in unstable orbits are swept up by one of the two stars in the binary. This can even happen in the solar system. In the Comet lab, you can find some images where a comet ran into Jupiter. The orbits of comets are very long ellipses, and when they come close to the Sun, their orbits can get changed by passing close to a major planet. The gravitational pull of the planet changes the shape of the comet’s orbit, it speeds up, or slows down the comet. This can cause the comet to crash into the Sun, or into a planet, or
cause it to be ejected completely out of the solar system. (You can see an example of the latter process by changing the Planet–Star Distance = 0.4 in the current simulation.)
5.5 Summary
(35 points) Please summarize the important concepts of this lab. Your summary should include:
- Describe the Law of Gravity and what happens to the gravitational force as a) as the masses increase, and b) the distance between the two objects increases
- Describe Kepler’s three laws in your own words, and describe how you tested each one of them.
- Mention some of the things which you have learned from this lab
- Astronomers think that finding life on planets in binary systems is unlikely. Why do they think that? Use your simulation results to strengthen your argument.
Use complete sentences, and proofread your summary before handing in the lab.
5.6 Extra Credit
Derive Kepler’s third law (P^2 = C × a^3 ) for a circular orbit. First, what is the circumference of a circle of radius a? If a planet moves at a constant speed “v” in its orbit, how long does it take to go once around the circumference of a circular orbit of radius a? [This is simply the orbital period “P”.] Write down the relationship that exists between the orbital period “P”, and “a” and “v”. Now, if we only knew what the velocity (v) for an orbiting planet was, we would have Kepler’s third law. In fact, deriving the velocity of a planet in an orbit is quite simple with just a tiny bit of physics (go to this page to see how it is done: http://www.go.ednet.ns.ca/∼larry/orbits/kepler.html). Here we will simply tell you that the speed of a planet in its orbit is v = (GM/a)^1 /^2 , where “G” is the gravitational constant mentioned earlier, “M” is the mass of the Sun, and a is the radius of the orbit. Rewrite your orbital period equation, substituting for v. Now, one side of this equation has a square root in it–get rid of this by squaring both sides of the equation and then simplifying the result. Did you get P^2 = C × a^3? What does the constant “C” have to equal to get Kepler’s third law? (5 points)
5.7 Possible Quiz Questions
- Briefly describe the contributions of the following people to understanding planetary motion: Tycho Brahe, Johannes Kepler, Isaac Newton.
- What is an ellipse?
- What is a “focus”?
- What is a binary star?
- Describe what is meant by an “inverse square law”.
- What is the definition of “semi-major axis”?
