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:
“Will women find themselves in the same position they have always been? Or do we see liberation as solving the conditions of women in our society?... If we continue to shy away from this problem we will not be able to solve it after independence. But if we can say that our first priority is the emancipation of women, we will become free as members of an oppressed community.”
—Ruth Mompati (b. 1925)
“Advocating the mere tolerance of difference between women is the grossest reformism. It is a total denial of the creative function of difference in our lives. Difference must be not merely tolerated, but seen as a fund of necessary polarities between which our creativity can spark like a dialectic.”
—Audre Lorde (1934–1992)
“I never knew anyone yet who got up at six who did anything more useful between that time and breakfast than banging a tennis ball up against the side of the house, waiting for the more civilized members of the party to get up.”
—Robert Benchley (1889–1945)