Goals: The student will|
- Become familiar with graphs of the form r = F(φ) in polar coordinates (r, φ), in particular with the circle, ellipse and other conic sections.
- Will understand the role of the semimajor axis and eccentricity in determining the nature of an ellipse.
- Will understand that the fixed point in any planetary system is its center of gravity. (This is used in a later section in deriving the rocket principle).
Terms: Focus (of ellipse), eccentricity, semi-major axis, orbital elements, mean anomaly (qualitatively only), center of gravity.
Stories and extras: Method used in the search for other planetary systems, and the recently discovered planetsary system of Upsilon Andromeda.
Guiding questions and additional tidbits
(Suggested answers in parentheses, brackets for comments by the teacher or "optional")
Start the lesson by reviewing polar coordinates. Then point out that in cartesian coordinates (preceding lesson) the ellipse is symmetric--around the axes and around their crossing point.
The motion of a planet of a satellite in an elliptic orbit is not symmetric--the center of motion, the focus, is shifted towards one end, and the planet's distance from it goes up and down each orbit, like a wave.
It suggests that cartesian coordinates are not the most suitable for handling orbits--and that, in fact, turns out to be the case. Polar coordinates, centered on the focus of the ellipse, are more suitable. As will be seen, the wave-like dependence of the distance is then clearly seen as a result of the wave-like variation of the cosine of the polar angle.
Then present section 11b of "Stargazers," up to "Refining the First Law". Discuss the material, guided by the questions below, and conclude with "Refining the First Law" and its questions, as listed further below.
[Optional section, for students familar with trigonometry]
-- If a cartesian system (x,y) has the same origin as a polar system, and the reference line from which φ is measured in the polar system is the x axis--given (r,φ) of a point, what are its (x,y)?
-- Given (x,y), what relation will give r? [Optional: and what function of φ can you express?]
r2 = x2 + y2.
[tan φ = y/x]
[Note in this section we denoted the polar angle by f, not by φ, because that is the notation used in the study of orbits. We will use capital F(f) for "a function of f" that is, "some expression involving f". ]
-- How can you express the equation of a line in polar coordinates?
r = F(f) is one way.
[More generally, any expression F(r,f) involving (r,f)) can be used, for example in the form F(r,f) = a, where a is some number or zero. The important thing is that at least over some range, each f has some corresponding value of r--sometimes, more than one value.
For instance,the line
3 r sin(f) – 2 r cos(f) = 10
What sort of line is it? Don't panic! It is just a straight line, as you easily see once you convert it to cartesian coordinates, using the relations developed earlier for the (r, φ) notation.
-- What is F(f) in the equation of a circle?
A constant number R, equal to the radius. Then r = R for any value of f.
[This is similar to the straight line y = 3. On that line, the value of y is the same for any x, and therefore the value of x does not have to appear.]
-- The equation of an ellipse is r = a(1-e2)/(1 + e cos f). What is e here?
A number between 0 and 1, known as the eccentricity of the ellipse.
-- For what value of f is r smallest, and for what value is it biggest?
Smallest for f=0, cos f =1; biggest for f=180°, cos f = -1.
--Using the identity (1-e2) = (e-1)(e+1), this may also be written
r = a(e-1)(e+1)/(1 + e cos f)
[The class should know this identity, or else, it may be proven on the board.]
Using that form, what are the biggest and smallest value of r?
A student or the teacher derive these on the board: biggest a(1+e), smallest a(1-e).
[Optional: Draw on the board an ellipse, around its focus, and on it, show the biggest and smallest r.
If this is the elliptic orbit of an Earth satellite, the smallest and biggest distances are know as perigee and apogee, where "gee" stands for "geo", the Earth.
For planets orbiting the Sun, these are called the perihelion and aphelion, from "helios", the Sun.]
-- What then is the total width of the ellipse along its major axis?
It is the sum of the smallest and biggest distances:
a(1+e) + a(1-e) = 2a.
-- In an orbit following Kepler's laws, what is meant by "the orbital elements"?
They are quantities used to define an orbit.
-- Which orbital elements do you know so far?
The semi-major axis a and the eccentricity e
-- A circle is also an ellipse. What can you say about its (a,e)?
a is the radius, e is zero.
--Which of these elements appear in Kepler's 3rd law?
The semi-major axis a is the precise meaning of the "mean distance" we used earlier. The square of the orbital period T is proportional to the third power (cube) of the semi-major axis.
--How many orbital elements define the motion of a satellite or planet, and what do you know about them?
Six elements exist: a and e define the orbital ellipse, the mean anomaly M is an angle that defines the satellite's position along that ellipse and 3 additional angles define the orientation of the orbital ellipse in 3-dimensional space.
[We can extend the equation of an ellipse to e = 1 or even e > 1, but must be careful. Suppose we draw a series of ellipses, of increasing e, all with the same minimum distance a(1-e). Then as e approaches 1, the semimajor axist grows to be larger and larger.
With e=1, we have something infinite (a) multiplying something that is zero (1-e). We thus need to rewrite the equation in the form
r = p/(1 + e cosf)
With e = 1 this gives a parabola, with e>1, a hyperbola.]
--Kepler claimed that planets orbited the center of the Sun. In a "double star" where two stars of comparable size are bound together by gravity, which of them serves as a center?
Neither--each rotates around its commen center of gravity.
[Teacher may mention that Mizar, the last star on the handle of the Big Dipper, is a double star. Good binoculars or even very sharp eyes (given a dark clear sky!) can distinguish its two components.]
-- How do astronomers search for planets around other stars?
Planets are too dim and too close to their suns to be seen by today's telescopes. However, each such sun also moves, around the joint center of gravity of it and of its planets. For instance, with just one planet, the Sun is always on the opposite side of the center of gravity.
Therefore, if astronomers see a slight wobble in the position of a distant star, they may deduce from it the existence and motion of its planets.