Teacher Notes



Elliptical Orbits
Publication No. 13918
IntroductionJohannes Kepler observed that the planets orbit the Sun in elliptical orbits—not circular orbits as originally thought. As a consequence, Kepler successfully determined the orbiting properties of any object. With the Elliptical Orbit Overhead Platform, show your students the elliptical shape of a planet’s orbit around the Sun and discuss Kepler’s laws of planetary motion. Concepts
MaterialsElliptical Orbit Overhead Platform*
Nuts and bolts, 2 sets* Overhead projector Overhead transparency pen* Paper towels Rubber feet, 1 sheet of 4* Ruler String* Washer* *Materials included in kit. Safety PrecautionsPlease follow normal classroom safety guidelines. DisposalMaterials may be saved for future use. Prelab Preparation
Procedure
{13918_Procedure_Figure_3}
Teacher Tips
Correlation to Next Generation Science Standards (NGSS)^{†}Science & Engineering PracticesDeveloping and using modelsUsing mathematics and computational thinking Disciplinary Core IdeasMSESS1.B: Earth and the Solar SystemHSESS1.B: Earth and the Solar System Crosscutting ConceptsStability and changeScale, proportion, and quantity Systems and system models Performance ExpectationsHSPS21: Analyze data to support the claim that Newton’s second law of motion describes the mathematical relationship among the net force on a macroscopic object, its mass, and its acceleration. DiscussionLong before Isaac Newton (1642–1727) “discovered” gravity and developed his laws of motion, Johannes Kepler (1571–1630) observed the planets and formulated his own laws of motion. In 1543, just before his death, Nicolaus Copernicus (1473–1543) proposed that the Sun was the center of the solar system and that the planets orbit the Sun in a circular path. In 1609, Kepler published a book entitled Law of Planetary Periods. In his manuscript, Kepler presented observations made by the wellrespected astronomer Tycho Brahe (1546–1601), as well as his own observations and calculations, to show that the planets do not travel in perfect circular orbits. Instead, they travel in oblong circular paths known as ellipses. An ellipse is an elongated circle that has two separate points of rotation, or foci, instead of one central point (see Figure 4). {13918_Discussion_Figure_4}
The “oblongness” of the ellipse is known as its eccentricity. The eccentricity (e) is calculated as the ratio of the distance between the foci to the length of the major axis of the ellipse (Equation 1).
{13918_Discussion_Equation_1}
f is the foci separation {13918_Discussion_Figure_5}
The eccentricities of most of the planets are very close to zero, so on a small, classroomscale model, it is difficult to see the difference between a circular orbit and an elliptical orbit of a planet. Mercury, Pluto and Halley’s comet are exceptions (see Table 1). These orbiting bodies have larger eccentricities and therefore follow more elongated paths. Table 1 {13918_Discussion_Table_1}
*AU is an abbreviation for the Astronomical Unit which is equal to the Earth’s average distance to the Sun (1.50 x 1011 m). As a consequence of the conservation of angular momentum, the orbital speed of a planet is inversely proportional to its distance from the Sun. Therefore, as a planet’s distance from the Sun increases, its orbital speed decreases. The closer a planet is to the Sun, the faster it orbits. A planet’s fastest and slowest orbiting speeds occur at the perihelion and aphelion, respectively. Kepler noticed this change in the orbiting speed of the planets (from Brahe’s observations) and discovered that the area of space swept by a planet in a given amount of time is always the same, no matter where the planet is in its orbit. Kepler called this the Law of Equal Areas (see Figure 6). {13918_Discussion_Figure_6}
This law follows from the basic property of an ellipse observed in step 7 and shown in Figure 7—for any point on the ellipse, the sum of the distance from the two foci is constant.
{13918_Discussion_Figure_7}
The area of space produced by arc A in Figure 6 is equal to the area of space of arc B. The sum of distances a and b is equal to the sum of distances c and d.Kepler’s observations also allowed him to formulate an equation for the period of revolution of the planets. Kepler calculated that the square of a planet’s period of revolution is proportional to the “radius” of the elliptical orbit cubed (Equation 2). An ellipse does not have a true “radius,” but it can be shown that onehalf the length of the major axis (also known as the semimajor axis) is equal to the average separation of a planet from the Sun as it travels in its elliptical orbit. Therefore, the semimajor axis is used as the “radius” of the ellipse. {13918_Discussion_Equation_2}
T is the planet’s period of revolution Referenceshttp://ssd.jpl.nasa.gov/ (accessed November 2003) Recommended Products

