Computing Position As A Function of Time
Kepler used his two first laws for computing the position of a planet as a function of time. His method involves the solution of a transcendental equation called Kepler's equation.
The procedure for calculating the heliocentric polar coordinates (r,θ) to a planetary position as a function of the time t since perihelion, and the mean motion n = 2π/P, is the following four steps.
- 1. Compute the mean anomaly
- 2. Compute the eccentric anomaly E by solving Kepler's equation:
- 3. Compute the true anomaly θ by the equation:
- 4. Compute the heliocentric distance r from the first law:
The important special case of circular orbit, ε = 0, gives simply θ = E = M. Because the uniform circular motion was considered to be normal, a deviation from this motion was considered an anomaly.
The proof of this procedure is shown below.
Read more about this topic: Kepler's Laws Of Planetary Motion
Famous quotes containing the words position, function and/or time:
“The word “jealousy” is often used as if it were synonymous with envy; but I think the distinction worth preserving. Jealousy is predominantly concerned with the fear of loss of something one possesses, envy with the wish to own something another possesses. Othello suffers from the fear that he has lost Desdemona’s love. Iago suffers from envy of the position held by Cassio, to which he feels entitled.”
—Anthony Storr (b. 1920)
“The information links are like nerves that pervade and help to animate the human organism. The sensors and monitors are analogous to the human senses that put us in touch with the world. Data bases correspond to memory; the information processors perform the function of human reasoning and comprehension. Once the postmodern infrastructure is reasonably integrated, it will greatly exceed human intelligence in reach, acuity, capacity, and precision.”
—Albert Borgman, U.S. educator, author. Crossing the Postmodern Divide, ch. 4, University of Chicago Press (1992)
“A big man has no time really to do anything but just sit and be big.”
—F. Scott Fitzgerald (1896–1940)