Using the precise data collected by Tycho Brahe, Johannes Kepler carefully analyzed the positions in the sky of all the known planets and the Moon, plotting their positions at regular intervals of time. From this analysis, he formulated three laws, which we address in this section.
Kepler’s First Law
The prevailing view during the time of Kepler was that all planetary orbits were circular. The data for Mars presented the greatest challenge to this view and that eventually encouraged Kepler to give up the popular idea. Kepler’s first law states that every planet moves along an ellipse, with the Sun located at a focus of the ellipse. An ellipse is defined as the set of all points such that the sum of the distance from each point to two foci is a constant. (Figure) shows an ellipse and describes a simple way to create it.
Figure a shows an x y coordinate system and an ellipse centered on the origin with foci f 1 on the left and f 2 on the right, both on the x axis. Focus f 1 is also labeled M. A point above focus f 2 is labeled m. The right triangle formed by f 1, f 2, and m is shown in red. Figure b shows a similar ellipse, with the sun shown and labeled as M and as Sun at f 1. A planet mass m is shown above f 1, at a vertical distance r from f 1. The location where the ellipse intersects the horizontal axis on the left is labeled as point A, and the location where the ellipse intersects the horizontal axis on the right is labeled as point B.
Figure 13.16 (a) An ellipse is a curve in which the sum of the distances from a point on the curve to two foci
(
f
1
and
f
2
)
is a constant. From this definition, you can see that an ellipse can be created in the following way. Place a pin at each focus, then place a loop of string around a pencil and the pins. Keeping the string taught, move the pencil around in a complete circuit. If the two foci occupy the same place, the result is a circle—a special case of an ellipse. (b) For an elliptical orbit, if
m
≪
M
, then m follows an elliptical path with M at one focus. More exactly, both m and M move in their own ellipse about the common center of mass.
For elliptical orbits, the point of closest approach of a planet to the Sun is called the perihelion. It is labeled point A in (Figure). The farthest point is the aphelion and is labeled point B in the figure. For the Moon’s orbit about Earth, those points are called the perigee and apogee, respectively.
An ellipse has several mathematical forms, but all are a specific case of the more general equation for conic sections. There are four different conic sections, all given by the equation
α
r
=
1
+
e
cos
θ
.
The variables r and
θ
are shown in (Figure) in the case of an ellipse. The constants
α
and e are determined by the total energy and angular momentum of the satellite at a given point. The constant e is called the eccentricity. The values of
α
and e determine which of the four conic sections represents the path of the satellite.
An x y coordinate system and an ellipse centered on the origin with foci f 1 on the left and f 2 on the right, both on the x axis, are shown. Focus f 1 is also labeled M. A point on the ellipse in the first quadrant is labeled m. The horizontal segment connecting the foci f 1 and f 2, and the segment connecting f 1 and m are shown in red. The angle between those segments is labeled Theta.
Figure 13.17 As before, the distance between the planet and the Sun is r, and the angle measured from the x-axis, which is along the major axis of the ellipse, is
θ
.
