Kepler’s Laws of Planetary Motion

What do you think? What does the term weightless mean to you? Have you ever observed someone in a weightless environment? If so, when? How did their weightless environment differ from a normal environment? When asking students to express their ideas, you might try one of the following methods. (1) You could ask them to write their answers in their notebook and then discuss them. (2) You could ask them to first write their ideas and then share them with a small group of 3 or 4 students. At that time you can have each group present their consensus idea. This can be facilitated with the use of whiteboards for the groups. The most important aspect of eliciting student’s ideas is the acceptance of all ideas as valid. Do not correct or judge them. You might want to ask questions to help clarify their answers. You do not want to discourage students from thinking about these questions and just waiting for the correct answer from the teacher. Thank them for sharing their ideas. Misconceptions are common and can be dealt with if they are first expressed in writing and orally. Most students will probably refer to the astronauts and the videos they have seen of them over the years. Some may know of the plane used to practice being weightless. Some may say in outer space you are weightless. You might ask them “where” in outer space you need to be for this to occur.

Kepler’s Laws Johannes Kepler built his ideas on planetary motion using the work of others before him. Nicolaus Copernicus and Tycho Brahe Kepler’s laws are covered in more detail on the next two slides.

Kepler’s Laws All planets move in elliptical orbits with the Sun at one of the focal points. A line drawn from the Sun to any planet sweeps out equal areas in equal time intervals. The square of the orbital period of any planet is proportional to cube of the average distance from the Sun to the planet.

Kepler’s Laws, cont. Based on observations made by Brahe Newton later demonstrated that these laws were consequences of the gravitational force between any two objects together with Newton’s laws of motion

Kepler’s Laws Kepler’s first law Kepler’s second law Orbits are elliptical, not circular. Some orbits are only slightly elliptical. Kepler’s second law Equal areas are swept out in equal time intervals. Be sure students interpret the diagram correctly. How does t1 relate to t2 ? (They are equal). How does A1 relate to A2 ? (They are equal.) How does the speed of a planet close to the sun compare to the speed of a planet that is far from the sun? (Planets travel faster when they are closer to the sun. In this example, the distance on the left is greater than the distance on the right, and the times are equal, so the speed on the left is greater.)

Kepler’s Laws Kepler’s third law Relates orbital period (T) to distance from the sun (r) Period is the time required for one revolution. As distance increases, the period increases. Not a direct proportion T2/r3 has the same value for any object orbiting the sun At this point, you may wish to have students work through the derivation mentioned in the Quick Lab on page 249 of the Student Edition. This will help them see the connection between Newton’s law of universal gravitation and Kepler’s laws. The web site: http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=9.0 allows students to visualize the first two laws. If you choose to show the first law, you see an elliptical orbit with the lengths measured from each focal point. The sum of these lengths is constant by the definition of an ellipse. Choosing the 2nd law allows you to see the equal areas swept out in equal amounts of time. The simulations also allows the user to change other parameters. These simulations can be downloaded. The home web site is: http://www.phy.ntnu.edu.tw/ntnujava/index.php

Kepler’s 3rd Law equation  

Period and Speed of an Object in Circular Orbit The dimensional analysis recommended on this slide is a good practice that provides a refresher on the units from previous chapters and reinforces the use of appropriate units. Using SI units, prove that the units are consistent for each equation shown above.

Kepler Law Video https://www.youtube.com/watch?v=92ROKc-BwDc

Classroom Practice Problems A large planet orbiting a distant star is discovered. The planet’s orbit is nearly circular and close to the star. The orbital distance is 7.50  1010 m and its period is 105.5 days. Calculate the mass of the star. Answer: 3.00  1030 kg What is the velocity of this planet as it orbits the star? Answer: 5.17  104 m/s This problem will provide good practice for the students. After they work on it for few minutes, guide them along. One way to solve this problem is to take the equation for period and square both sides. Make sure they square both the 2 and the pi (42). Then they can substitute values and solve for m. Another possible error is the use of days for T. For the velocity calculation, most will probably use the equation on the previous slide. After some time, point out that they could have obtained the velocity without the mass by using the distance divided by the time (or circumference divided by period). This will reinforce the validity of the equations shown on the previous slide.

Weight and Weightlessness Bathroom scale A scale measures the downward force exerted on it. Readings change if someone pushes down or lifts up on you. Your scale reads the normal force acting on you.

Apparent Weightlessness Elevator at rest: the scale reads the weight (600 N). Elevator accelerates downward: the scale reads less. Elevator in free fall: the scale reads zero because it no longer needs to support the weight. Remind students that the scale reading is the normal force. In the second case, when an elevator starts to move downward, you feel “lighter” for a brief moment. (After this, the elevator returns to a constant speed, and the scale reading goes back up to its initial value.) Ask students what would happen if the person in the free-falling elevator held an apple out in front of his face and let go. (The apple would remain in the same spot because it is falling with the same acceleration as him, 9.81 m/s2.) Students may have had similar experiences on amusement park free-fall rides.

Apparent Weightlessness You are falling at the same rate as your surroundings. No support force from the floor is needed. Astronauts are in orbit, so they fall at the same rate as their capsule. True weightlessness only occurs at great distances from any masses. Even then, there is a weak gravitational force. Remind students that orbiting means falling at the same rate that Earth curves away from you (while moving sideways), so you never get any closer.

Classroom Exercise Make a sketch showing the path of Earth as it orbits the sun. Describe the motion of Earth as it follows this path. Describe the similarities and differences between the path and motion of Earth and that of other planets. The paths should be slightly elliptical. Some planets are more elliptical than others. The planet moves fastest when nearest the sun and slower at the greatest distance. (The reason for this can be seen from the web site recommended for Kepler’s laws. The force does not act perpendicular to the velocity. It has a component causing acceleration and deceleration at various point on the ellipse.)