1. Orbits and Eccentricity
More with Ellipses in Sec. 8.2b

2. Orbits and Eccentricity
Many moons ago, it was discovered that many celestial
bodies (for example, those orbiting the sun) followed
elliptical paths…
Perihelion – point closest to the sun in such an orbit
Aphelion – point farthest from the sun in such an orbit
The shape of an ellipse (including these orbital paths)
is related to its eccentricity…

3. Orbits and Eccentricity
a – c
a + c
Center
Sun
at focus
Semimajor Axis
Orbiting
Object

4. Definition: Eccentricity of an Ellipse
The eccentricity of an ellipse is
where a is the semimajor axis, b is the semiminor axis, and c is
the distance from the center of the ellipse to either focus.
What is the range of possible “e” values for an ellipse?
What happens when “e” is zero?
 A CIRCLE!!!

5. Practice Problems The semiminor axis is only 0.014% shorter
The Earth’s orbit has a semimajor axis
and an eccentricity of
Calculate and interpret b and c.
Semiminor Axis
The semiminor axis is only 0.014% shorter
than the semimajor axis…

6. Practice Problems The Earth’s orbit is nearly a perfect circle, but
The Earth’s orbit has a semimajor axis
and an eccentricity of
Calculate and interpret b and c.
Aphelion of Earth:
Perihelion of Earth:
The Earth’s orbit is nearly a perfect circle, but
the eccentricity as a percentage is 1.67%; this
measures how far off-center the Sun is…

7. Other Types of Practice Problems
Prove that the graph of the equation is an ellipse, and
find its vertices, foci, and eccentricity.
Put into standard form:

8. Other Types of Practice Problems
Prove that the graph of the equation is an ellipse, and
find its vertices, foci, and eccentricity.
Standard Form:
Vertices:
Eccentricity:
Foci:

9. Other Types of Practice Problems
Write an equation for the given ellipse.
(–4, 5)
Center:
C(–4, 2)
(0, 2)
Semimajor, semiminor axes:
Standard Form: