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:
“Men have an extraordinarily erroneous opinion of their position in nature; and the error is ineradicable.”
—W. Somerset Maugham (1874–1966)
“No one, however powerful and successful, can function as an adult if his parents are not satisfied with him.”
—Frank Pittman (20th century)
“One of the greatest faults of the women of the present time is a silly fear of things, and one object of the education of girls should be to give them knowledge of what things are really dangerous.”
—Ellen Henrietta Swallow Richards (1842–1911)