Overview

Preparation and Materials

Objectives

• use a computational model to discover possible answers to a question about a natural phenomenon.

• practice accurately observing and recording data from a scientific experiment.

• communicate and defend a scientific argument while collaborating with other students.

• describe accurately the motion of the planets.

• describe elliptical orbits.

Standards

• NSES content standard (9-12 A) on science as inquiry, including:
  • Use technology and mathematics to improve investigations and communications.
    
  • Communicate and defend a scientific argument.

• NSES content standard (9-12 D) on origin and evolution of the universe.

• NSES content standard (9-12 E) on understandings about science and technology.

• DoDDs Science 9 objectives, including:
  • Use of computational models.
    
  • Use careful systematic observation and data collection to obtain valid information.
    
  • Relate force, motion, energy and power.

Activities

1. If the students have not previously studied the solar system, be sure the students understand the properties of the planets. Two sites which cover this with great detail using the latest images are Solar Views and Nine Planets.

2. If the students have not studied why the Earth orbits the sun, consider doing the Earth-Sun lesson before this one.

3. Make the following points about the solar system:
   1. The distance between the planets is much greater than the size of the planets themselves.
   2. The planets do not move in exact circles.
   3. The sun contains 98% of the mass of the solar system, and is the dominant object in controlling the motion of the other objects in the solar system.
   4. If we are to accomplish anything in science, it is extremely important that we are careful observers.

4. With the monitor displayed so that the students can see it, set the Scale of the model, which is found under the Galaxy menu, to Solar System.
5. Make sure that the Galaxy Setup is for a spherical galaxy of 2 stars. Other options do not matter, as we are going to change them. Generate a new galaxy. Open the star list by selecting Show List under the Galaxy menu. Change the options for the first object, and give it the mass of the Sun (330000 earth masses) and position it in the center of the coordinate system. Have it be stationary. You might want to start off by checking to see of your students can correctly state that an unmoving object in the center of the coordinate system is represented by (x,y,z)=(0,0,0) and (vx,vy,vz)=(0,0,0).
6. If the students have not learned about ellipses, familiarize them with the terms semimajor axis, focus, eccentricity, perihelion, and apihelion.

7. Use the information from the table below to help you fill in the information for the planets. For the second object, give it a mass for Mercury, and position it on the x axis at the perihelion distance (r=a*(1-e)). Give it some guess for the initial velocity. A good first guess might be to take 2*pi*a as a "order of magnitude" number for the distance Mercury travels around the sun, and divide it by the time it takes in days which can be calculated from the table below. The direction of the velocity should be tangential to the position. This will be the y direction.

8. Have the students save the model as a starting point for each trial.

9. Have the students notice that the shape of the orbit changes depending on what they use of an initial velocity.

10. Leaving the star list open, run a model from the beginning until almost halfway through the first orbit. Stop the model, and step through one timestep at a time until the halfway point is reached (z=0 on Mac, y=0 on Windows). The star list will show the x value of Mercury. Compared to the r at apihelion (r=a*(1+e)), does the planet's observed position overshoot or undershoot the expected value of the planet's position at perihelion?

11. Have the students adjust the initial velocity, running models with different initial velocities until they find a perihelion velocity which gives the proper apihelion distance for Mercury's orbit. (The initial velocity that worked best for me was 0.03375.) The student can then determine from the star list what the velocity is of the planet at apihelion.

12. Once the students have the perihelion velocity, the next step is to determine what this velocity is in the reference frame of the Earth's orbit about the sun. For students who have not studied trigonometry, they may consider using the helper spreadsheet.
13. The values of the position and velocity at perihelion are now known for a coordinate system with perihelion occurring on the -x axis. The perihelion longitude in a standard reference frame for all planets in the solar system can be found in the tables below. In the standard frame, we can express the object at perihelion in either x, y, and z coordinates, or in terms of distance (r), angle around the plane of the solar system (longitude), or angle up off of the plane of the solar system (latitude). At perihelion, the latitude is at a maximum, and is given by the inclination of the orbit. We can express the x, y, and z coordinates as
    • x_peri=r*cos(perihelion longitude)*sin(90-inclination)
    • y_peri=r*cos(90-inclination)
    • z_peri=r*sin(perihelion longitude)*sin(90-inclination)

14. The velocity can be determined quite easily. We know the tangential velocity at perihelion. Since the latitude (angle off of the plane of the solar system) is at a maximum at perihelion, the planet is not moving up or down in the y (z on Windows) direction at that exact point in time (vy_peri=0). The x and z coordinates of the velocity can be chosen such that the velocity is at right angles to the position.
    • vx_peri=-v_tangential*sin(perihelion longitude)
    • vy_peri=0
    • vx_peri=v_tangential*cos(perihelion longitude)

15. Once the students have used the model to determine the perihelion velocity, and have used either a helper spreadsheet or the above trigonometry to then translate the position into the standard reference frame of the solar system, they still don't know WHEN the planet is at perihelion. The tables below give the last time each planet was observed to be in perihelion. We would like to be able to find out where the planet is at a specific time, perhaps January 1st, 2000. If the student can determine the number of days between the desired date and the known date from the table, the student can then run the model from the known date (last perihelion) to the desired date. If the known date is more recent than the desired date, use a negative value for the time step to run the model backwards.

16. NOTE: To ensure accuracy, you might keep the info window open. A good check on the accuracy of this simulation is to see that the energy is conserved. If the time step is too large, errors will quickly accumulate, resulting in a change in the energy. If the energy of the model is changing, you know your time step is too large.

17. For a large class, consider breaking up into groups so that different people can determine the exact location and velocity of the planets on January 1st, 2000. When the class has completed their individual tasks, they then can combine their data and have a complete model of the solar system, starting from January 1st, 2000.

Tables

Discussion of the Simulation

Ask the students to discuss the motion of the planets. Why is it that the planets have almost the same plane of motion, but not exactly. What might that tell us about the formation of the solar system?

Discussion of Observation

Before the Copernican revolution, mankind thought that the Sun and all of the planets orbited the Earth. A series of scientists noted that it made more sense if the planets and the Earth orbited the Sun. As scientists, we want to be able to observe, understand, and predict. Newton's law of gravity (the same thing GalaxSee models) was the first time anyone had ever been able to not just predict the motion of the planets, but to do so with a simple explanation of why it might happen (gravity).

One of the key observations that scientists were trying to explain was the retrograde motion of planets. From viewers on Earth, planets would move in one direction across the sky, but occassionally backtrack for a period of time, and then start forward in their original direction of motion. The planets were evened named because of this wandering across the sky, the word planet literally means "wanderer".

Assign them to write a clear and accurate report of what they observed. Emphasize that it is important that they know what software was used, and what parameters were set. Be sure to go through the setup procedure again so that they can record this information.

Collaboration

Extensions

• Further Experimentation

  Have the students determine whether there are any planets reasonably close together in the night sky. The planets that are observable with the naked eye are Venus, Mars, Jupiter, and Saturn. Check your local star charts for times to observe.

  Have students try to use their model to predict how far apart Jupiter and Saturn will be as viewed from Earth. Have them try to actually observe this in the night sky.

• Thinking Harder

  Pluto was originally found by a scientist who claimed that he knew where to look by a wobble in Neptune's orbit. How massive would pluto have to have been for this to be true?