**Details: series
expansion, Newton's method**

**Movie
of
elliptical motion**

**Plots
of
elliptical Kepler motion**

**Circumgerence
of an ellipse**

**
**E. Zinner: Astronomie, Alber, Freiburg/München 1951.

In 1609 Kepler published his work Between 1617 and 1621 Kepler wrote The term The mean anomaly M is the angular
distance from perihelion which a (fictitious)
planet would have if it moved on the circle of
radius a with a Operating with radians Kepler's equation
is: or, using degrees: The equation can be derived from
Kepler's second law. The value of M at a given time is easily
found when the eccentricity e and the eccentric
anomaly E are known. The problem is to find E
(from which the position of the planet can be
computed) when M and e are known. The true anomaly (symbol φ) is the
angular distance of the planet from the perihelion
of the planet, as seen from the Sun. For a
circular orbit, the mean anomaly and the true
anomaly are the same. The difference between the
true anomaly and the mean anomaly is called the The form is preset to: The results, as shown in the figure
below, are:

*Astronomia
Nova*, containing the first (and the second)
law of planetary motion:

*Planets move in elliptical orbits with the sun
at one focus.* *Epitome
Astronomiae Copernicanae*, the first
astronomy textbook based on the Copernican model.
Kepler introduced what is now known as *Kepler's
equation* for the solution of planetary
orbits, using the eccentric anomaly E, and the
mean anomaly M. *anomaly* (instead of *angle*),
which means irregularity, is used by astronomers
describing planetary positions. The term
originates from the fact that the observed
locations of a planet often showed small
deviations from the predicted data.
*constant* angular velocity
and with the same orbital period T as the real
planet moving on the ellipse. By definition, M
increases linearly (uniformly) with time. **E(t) - e*sin[E(t)] = M(t)****E(t) -
(180°/π)*e*sin[E(t)] = M(t)**

Kepler's equation cannot be solved algebraically.
It can be treated by an iteration
methods. One of them is Newton's method, finding
roots of

Equation of
Center C: **JavaScript using
Newton's method:**eccentricity e=0.5

mean anomaly M=27° or t/T=0.075.true anomaly phi=75.84°

eccentric anomaly E=48.43°

An example of a series expansion is:
For small eccentricities the mean anomaly
M can be used as an initial value E
(c) 2006-2013 J. Giesen
Updated: 2013, Aug 07 |