Kepler's equation
With the help of Kepler 's equation can be obtained in the Kepler problem solving tasks. In particular, the instantaneous angular distance ( true anomaly T) of a celestial object P can be determined from the Z periapsis its Keplerian orbit as a function of time t.
In the most common elliptical Keplerian orbit, the procedure is as follows: In a radius of ellipse the celestial object corresponding point X with so-called eccentric anomaly E and a fictitious point Y, t the end of the uniform passage of time are simulated and the so-called mean anomaly M, introduced, ie M t. The application of the second Kepler 's law leads to a relationship - the Kepler equation - between these two anomalies E and M, with which, finally, the true anomaly T can be found as a function of the mean anomaly M and the time t.
- 2.1 True anomaly
- 2.2 orbital radius
- 2.3 web speed
Derivation of Kepler's Equation
Mean anomaly
The uniformly passing time can be illustrated on a circular path with a constant angular velocity of the motion of a fictitious body. It requires a radius as an auxiliary circle around the Kepler ellipse ( orbit ) on which the fictitious body Y circulates down. Y is at the time as well as the real object in the periapsis, and has the same round-trip time.
The instantaneous position of the point Y as the angle ( angle of all of the following are represented by radians) indicated in the sub -circuit in relation to the center C periapsis Z and designated as the average anomaly M:
Where U is the orbital period, and is the constant angular velocity. In time, the celestial object t0befindet in periapsis ( Periapsiszeit ), where it has the smallest distance to its center of gravity S.
The Kepler problem is the computational application of the second Kepler 's law, that is an indication of the position of the celestial body P ( true anomaly ) at a predetermined time ( mean anomaly ). Accordance with this law of equal size at equal time intervals swept by the line connecting the center and sky train body surfaces are easier to handle on the detour via the radius calculated in elliptic geometry. There is an affinity, assign the sub-areas of the ellipse proportional circle sectors that are easy to compute intermediate radius and ellipse.
According to the second Kepler 's law, the proportion of elliptical Teifläche on the ellipse is equal to the circular sector at the perimeter. In the same period of the driving beam of the body P sweeps in relation exactly the same area as the radius vector of the point Y:
Is the semi-major axis of the ellipse, and the radius of the circumscribed circle is the semi-minor axis of the ellipse. reflects the affinity between radius and ellipse. The latter is with reciprocal value of this ratio in each parallel to the minor axis of the " upset " radius.
Eccentric anomaly
Due to the semi-minor axis parallel projection of the point P on the perimeter of the auxiliary point X, the angle was called the center C to Z periapsis of Kepler eccentric anomaly E is produced. The affinity established the following relationship:
After insertion of equation (2 ) into Equation (3) follows:
Kepler equation
Using equation (4) the desired relation between the eccentric anomaly ( point X) and the mean anomaly ( point Y ) is found indirectly. The direct connection is formed by the steps of:
When the driving beam in a period U travels the angle and turns over the area, so it sweeps up to the time the angle and a smaller by a factor of area:
The analogous consideration for the driving beam over the angle gives:
The area CXZ consists of the sub-areas and CXS SXZ:
The Teifläche CXS is a straight -limited triangle with the base and the height:
E is the numerical eccentricity of the ellipse, which indicates the distance between the center and the focal point in relation to the semi-major axis a.
The partial surface SXZ is according to equation ( 4) equal to the surface CYZ, whose value is given in equation (5).
By substituting the equations ( 6), ( 8) and ( 5) from equation (7):
This eventually results in the Kepler equation:
Solution of Kepler 's equation
The Kepler equation is not in closed form by the eccentric anomaly resolvable. Examples to identify with her from the mean anomaly are:
1 The size can be considered as zero point of the function of the Kepler equation:
2 A more stable, but slower converging method is based on the Banach fixed point theorem:
3 can alternatively be approximated for small eccentricity:
Solving some tasks in the Kepler problem
True anomaly
For a celestial body on a Keplerian orbit is indicated for the time or for the associated mean anomaly of the site or the true anomaly. With the help of the Kepler equation, the eccentric anomaly is first detected ( see above). From the latter follows the true anomaly by any of the following relationships:
Or
Here is the linear eccentricity of the orbit ellipse. To solve for is a case distinction between necessary and each.
- The denominator of the second formula is just the distance of the object in the sky to the focal point:
- The formulas can easily be converted to or and the result is:
And
There are still numerous other relationships that have been developed in the long history of celestial mechanics between the true anomaly, the eccentric anomaly and the mean anomaly. In particular, can the true anomaly - without going through the Kepler equation - directly from a special differential equation to calculate what is for numerical approximation methods of interest.
Orbital radius
With the direction of the true anomaly of a heavenly body is provided on its Keplerian orbit for a time t. The Relevant distance - the orbital radius - can be calculated as follows:
Web speed
The variation of the true anomaly corresponds to the angular velocity with respect to the Gravizentrum. Thus, the normal component of the velocity follows directly from
The radial velocity is the change in the orbit radius over time:
The web speed or orbital velocity then follows to
Easier it is to the train speed over the hodograph from the face set derived
From this follow the minimum and maximum velocity in Apozentrum and pericentre an elliptical path