(11a) Ellipses and Kepler's First Law |
Part of a high school course on astronomy, Newtonian mechanics and spaceflight
by David P. Stern
This lesson plan supplements: "Ellipses & Kepler's 1st Law," section #11a-b: on disk Skepl1st.htm and Skepl1st.htm , on the web
http://www.phy6.org/stargaze/Skepl1st.htm http://www.phy6.org/stargaze/Skepl1stA.htm
"From Stargazers to Starships" home page and index: on disk Sintro.htm, on the web |
Goals: The student will
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 planetary system of Upsilon Andromeda. "Alien Planets" (Skepl1stA.htm) gives further details about such searches, and how the transit of a planet in front of a star could give clues about whether the planet is a gas giant or icy or rocky. 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?]
[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?
[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?
[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-e^{2})/(1 + e cos f). What is e here?
-- For what value of f is r smallest, and for what value is it biggest?
--Using the identity (1-e^{2}) = (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.] 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.
-- What then is the total width of the ellipse along its major axis?
-- In an orbit following Kepler's laws, what is meant by "the orbital elements"?
-- Which orbital elements do you know so far?
-- A circle is also an ellipse. What can you say about its (a,e)?
--Which of these elements appear in Kepler's 3rd law?
--How many orbital elements define the motion of a satellite or planet, and what do you know about them?
[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?
-- How do astronomers search for planets around other stars?
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.
-- How can astronomers find whether a planet is habitable?
If the planet "transits" in front of the star, the dimming is proportional to the cross-sectional area of the planet, telling about the radius of the planet. From the wobble of the star's position we have the planet's mass, so these two give an estimate of the planet's density, allowing gaseous planets and solid ones to be told apart. |
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Author and Curator: Dr. David P. Stern
Mail to Dr.Stern: stargaze("at" symbol)phy6.org .
Last updated: 12.17.2001