Calculating Orbital Eccentricity
Lesson 3 of 4
Objective: SWBAT define eccentricity and calculate the eccentricity of various objects in the solar system using an algebraic equation.
This lesson is aligned to NGSS HS-ESS1-4: Use mathematical or computational representations to predict the motion of orbiting objects in the solar system. Similarly, it is also aligned to the following Common Core standards: MP.2, MP.4, HSN-Q.A.2, HSA-CED.A.2, HSA-CED.A.4 as it explicitly requires distinct mathematical reasoning, modeling, and the derivation of mathematical and scientific relationships.
This class begins with students entering the room, taking out their Earth Science Reference Tables [ESRTs], and starting work on that day's 'Warm Up' activity. Students always work silently and independently before we collectively review the activity each day.
While the activity usually varies (ranging from multiple choice questions, short performance tasks, or free-response questions), I always try to ensure that the 'Warm Up' involves problems that can clear up potential misconceptions or identify learning gaps, which is often based on the classroom assessment data from previous lessons. For this lesson, I picked three multiple choice questions pertinent to the current unit on Astronomy that some students had struggled with.
First, it is important to have all materials prepared and ready for this laboratory activity. (Note: usually, I have a set student who has the classroom job of passing out all necessary materials to students during the 'Warm Up' activity.)
Materials: (each student receives one of each)
- 1 sheet of blank paper (8.5 x 11 inches)
- Pencil (students should not use pens, as they often don't work well when using the drawing compass)
- Approximately 22 cm of string (have these already cut!)
- A thick piece of cardboard (pieces of cardboard can be purchased, but it's usually just as easy to cut up pieces of cardboard boxes)
- A drawing compass
- 2 push pins
- Metric ruler
Introduction: [Note: Please refer to resource below for student note sheets, as they will be periodically writing down information during this section.]
I usually introduce the activity by having all of the laboratory materials ready under a document camera. I then set up one push pin in the center of my piece of paper, loop the string around it, and draw a circle. I then very directly ask students, "What shape did I just draw?"
Generally, they're all able to correctly identify that I drew a circle.
Then, I repeat the procedure, but this time, I use two push pins in the center of the piece of paper, drawing an ellipse. After constructing it, I ask students the same question: "What shape did I just draw?"
Here, responses tend to be mixed. Often, students will guess things like a circle or oval. Then I introduce the concept of an ellipse by explaining that this shape, while nearly circular, is not actually a circle. It is what scientists call an ellipse. Then, I pose the question: "What makes a circle different from an ellipse?"
Again, answers tend to vary, but students generally are able to identify that the circle has one center point (focus), while the constructed ellipse has two (foci). I then clarify that these "center points" are actually called foci.
After defining foci, I then introduce the relationship between ellipses and orbital shapes. I state that: "All of the objects that orbit the Sun may appear circular to us, but they actually all have elliptical orbits. All of these objects that orbit the Sun, including the Earth, orbit in an elliptical shape, with the Sun at one of the foci. [Note: This is Kepler's First Law, although this will not be introduced until a later lesson]
I then ask, "If the Sun it at one of the foci, what is usually at the other foci?"
I select a variety of responses, but usually students are quick to understand that the other orbital foci is just a point in space - there is nothing there.
I then point them to a sample image (refer to Laboratory Introduction resource) showing an illustration of an elliptical orbit, having students note the two foci. Then, I point them to the image below (refer to Laboratory Introduction resource), which has an elliptical orbit with two additional parts labeled, the "distance between foci" and the "length of major axis". We then collectively define the major axis as the "distance between orbits that pass through both foci."
"But the interesting part is, we can actually see how elliptical each planet's orbit is. By using the same algebraic equation that astronomers use, we can figure out the orbital eccentricity of each planet. The eccentricity is actually how elliptical, or circle-like, a planet's orbit is. We can calculate it using an equation. The equation itself is listed below (refer to Laboratory Introduction resource), but it is actually a ratio between the distance between the two foci and the length of the major axis (HSA-CED.A.2; HSA-CED.A.4)."
Then, we do a quick example:
"Let's look at the ellipse I drew for you earlier in class. I can use my metric ruler to measure the distance between foci, which is the numerator ("d") in my eccentricity equation. Then, I can also use my metric ruler to measure the length of the major axis, which is the denominator ("L") in my eccentricity equation. Then, I can use my calculator (Note: Have students round to the thousandths place when entering these numbers into the calculator) to determine "e", which is the orbital eccentricity.
Usually, I start this activity collectively and proceed slowly through each step, gradually releasing students to work when it looks as if they're performing the procedure correctly. Some students may struggle with knotting the string, so allot extra time as necessary (other students can also help here!). It will also help to remind them to show their work and write in units [Note: the eccentricity, or "e" value, actually does not have any units].
- Take your piece of string and tie the ends together to form a loop. [Note: It is okay if students have slightly different string lengths.]
- On plain white paper draw a straight line lengthwise down the middle of the paper.
- Near the center of this line, draw two dots 3 centimeters apart.
- Placing the paper on a piece of cardboard, put a thumbtack in each dot (focus).
- Loop the string around the thumbtack's metal base and draw the ellipse by placing your pencil inside the loop as shown in the image (refer to example).
- Label this ellipse #1.
- Measure the distance (in centimeters) between the thumbtack holes (foci). This is “d”. Record this on your Report Sheet.
- Measure the length of the major axis (L) (in centimeters) and record this on the Report Sheet (HSN-Q.A.2).
- Move each tack out 1 centimeter and draw a new ellipse. Label it #2 and measure and record d and L.
- Move each tack out another 1 centimeter and draw another ellipse. Label it #3 and measure and record d and L.
- Move each tack out another 1 centimeter and draw another ellipse. Label it #4 and measure and record d and L.
- Place a dot in the exact middle of the first two foci. Using a drawing compass with a pencil, construct a circle. Place the point of the compass in the center dot. Extend the compass along the major axis so the pencil touches ellipse #1. This will be the radius of the circle you are to draw.
- Using the given equation in your Introduction, calculate the eccentricity (e) of each of the five figures. Show all work on your Report Sheet.
The laboratory discussion can and should be started as soon as students have completed the laboratory sections (including showing all work and units!). Generally the key idea they need to leave this with, beyond defining what orbital eccentricity actually looks like, is to determine the relationship between the variables in the equation and the shapes of eccentricity (CCSS MP.2 & MP.4).
The lab data should show that as the distance between foci increases, the eccentricity values should also increase, which results in a more oblong, less "circle-like" shape. As the orbital eccentricity values get closer to zero, the shape approaches a circle. This is the reason that Ellipse #5 is actually a circle, as they should calculate the eccentricity as zero here.
Also, given the time constraints in your classroom, it may be necessary to extend this lesson into an alternate day and/or have them complete some sections for homework (as long as the laboratory section is completed, students should need no additional materials to complete the discussion section). Either way, there should be at least a brief, whole-class debrief on both the laboratory exercise and the summative learning as a result. I usually ask various students to summarize the key points and concepts emphasized in the lab.