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The 'eccentric anomaly' ... by the formula = ), then solve the ... where is the hyperbolic eccentric anomaly. This equation ...
A radial hyperbolic trajectory is a non-periodic trajectory on a straight line where the relative speed of the two objects always exceeds the escape velocity. There are two cases: the bodies move away from each other or towards each other. This is a hyperbolic orbit with semi-minor axis = 0 and eccentricity = 1.
The true anomaly is the angle labeled in the figure, located at the focus of the ellipse. It is sometimes represented by f or v. The true anomaly and the eccentric anomaly are related as follows. [2] Using the formula for r above, the sine and cosine of E are found in terms of f :
McNaught has a hyperbolic orbit but within the influence of the inner planets, [9] is still bound to the Sun with an orbital period of about 10 5 years. [3] Comet C/1980 E1 has the largest eccentricity of any known hyperbolic comet of solar origin with an eccentricity of 1.057, [10] and will eventually leave the Solar System.
For the hyperbolic case, there is a formula similar to the above giving the elapsed time as a function of the angle (the true anomaly in the elliptic case), as explained in the article Kepler orbit. For the parabolic case there is a different formula, the limiting case for either the elliptic or the hyperbolic case as the distance between the ...
The true anomaly is usually denoted by the Greek letters ν or θ, or the Latin letter f, and is usually restricted to the range 0–360° (0–2π rad). The true anomaly f is one of three angular parameters (anomalies) that defines a position along an orbit, the other two being the eccentric anomaly and the mean anomaly.
where M is the mean anomaly, E is the eccentric anomaly, and is the eccentricity. With Kepler's formula, finding the time-of-flight to reach an angle (true anomaly) of from periapsis is broken into two steps: Compute the eccentric anomaly from true anomaly
The eccentric anomaly is also not defined for parabolic and hyperbolic trajectories, and instead the parabolic anomaly or hyperbolic anomaly are used. [ 3 ] True anomaly at epoch ( ν 0 {\displaystyle \nu _{0}} ) — angle that represents the real angular displacement of the orbiting body at the epoch time, taking into account the varying speed ...