Properties of ellipses
• An ellipse can be drawn by stretching a string around two pins and pulling a pencil around, as shown in the
diagram. Notice that the two foci (singular: focus) of the ellipse are the locations of the two pins. The long axis of
the ellipse is called the major axis; half this length is the semimajor axis. The short axis is the minor axis.
• The shape of a particular ellipse depends on its eccentricity,
which describes how much longer the major axis is than the
o A very eccentric ellipse has a major axis that is
much longer than its minor axis, giving the
ellipse a very "stretched out" oval shape. A
perfect circle has zero eccentricity, meaning no difference between the major and
minor axes. Kepler's first law allows a planetary orbit to be an ellipse with any
• Kepler's first law tells us about orbital shape, it does not provide information about
• None of the planets has a perfectly circular orbit, which means that all planets (including
Earth) are closer to the Sun on one side of their orbit than on the other. The Sun's off-
center position arises because it is located at a focus of each planet's elliptical orbit, rather
than at the center of the ellipse.
• Kepler's second law states that as a planet orbits the Sun, it sweeps out equal areas in
o same ideas holds for any object orbiting the Sun: An object must move faster when
it is closer to the Sun and slower when it is farther from the Sun.
• Kepler's third law states that a planet's orbital period, p, is related to its average
(semimajor axis) orbital distance, a, according to the mathematical relationship p2=a3.
o From the relationship p2=a3, it follows that planets closer to the Sun must orbit at
higher average speeds than planets farther from the Sun. For example, Venus must
orbit the Sun faster than Earth, because Venus is closer to the Sun.
o Kepler’s third law tells us that the orbital period of the planet is related to its
average distance from the star, but not to the planet’s mass. This is actually an
approximation of Newton's derivation of the Kepler's third law. If the planet's mass
is deemed insignificant compared to the mass of the star the planet orbits, then we
arrive at Kepler's third law. Since planets' masses are generally many orders of
magnitude less than the mass of the stars they orbit, this is a valid approximation in
most cases. • Aphelion is the point at which an object orbiting the Sun is farthest from the Sun.
• An ellipse is a type of oval that happens to be the shape of bound orbits. An ellipse can be
drawn by moving a pencil along a string whose ends are tied to two tacks; the locations of
the tacks are the foci (singular, focus) of the ellipse.
• A focus is one of two special points within an ellipse that lie along the major axis; these
are the points around which we could stretch a pencil and string to draw an ellipse. When
one object orbits a second object, the second object lies at one focus of the orbit.
• Eccentricity is a measure of how much an ellipse deviates from a perfect circle. It is
defined as the center-to-focus distance divided by the length of the semimajor axis.
• Perihelion is the point at which an object orbiting the Sun is closest to the Sun.
• A semimajor axis is half the distance across the long axis of an ellipse; in your textbook, it
is usually referred to as the average distance of an orbiting object, abbreviated a in the
formula for Kepler's third law.
• If you are in an elevator that is accelerating upward, you will feel a downward force while
the elevator moves upward with increasing speed
• In an elevator, acceleration occurs when your speed is increasing either upward or
downward. When it is increasing upward, you will feel a force pressing you to the floor.
o Suppose you are in an elevator. As the elevator starts upward, its speed will
increase. During this time when the elevator is moving upward with increasing
speed, your weight will be greater than your normal weight at rest
o Increasing speed means acce