In orbital mechanics, the eccentric anomaly is an angular parameter that defines the position of a body that is moving along an elliptic Kepler orbit. The eccentric anomaly is one of three angular parameters ("anomalies") that define a position along an orbit, the other two being the true anomaly and the mean anomaly.
Consider the ellipse with equation given by:
where a is the semi-major axis and b is the semi-minor axis.
For a point on the ellipse, P = P(x, y), representing the position of an orbiting body in an elliptical orbit, the eccentric anomaly is the angle E in the figure. The eccentric anomaly E is one of the angles of a right triangle with one vertex at the center of the ellipse, its adjacent side lying on the major axis, having hypotenuse a (equal to the semi-major axis of the ellipse), and opposite side (perpendicular to the major axis and touching the point P′ on the auxiliary circle of radius a) that passes through the point P. The eccentric anomaly is measured in the same direction as the true anomaly, shown in the figure as . The eccentric anomaly E in terms of these coordinates is given by:
and
The second equation is established using the relationship
which implies that sin E = ±y/b. The equation sin E = −y/b is immediately able to be ruled out since it traverses the ellipse in the wrong direction. It can also be noted that the second equation can be viewed as coming from a similar triangle with its opposite side having the same length y as the distance from P to the major axis, and its hypotenuse b equal to the semi-minor axis of the ellipse.
The eccentricity e is defined as:
From Pythagoras's theorem applied to the triangle with r (a distance FP) as hypotenuse:
Thus, the radius (distance from the focus to point P) is related to the eccentric anomaly by the formula
With this result the eccentric anomaly can be determined from the true anomaly as shown next.
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.
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In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the longest semidiameter or one half of the major axis, and thus runs from the centre, through a focus, and to the perimeter. The semi-minor axis (minor semiaxis) of an ellipse or hyperbola is a line segment that is at right angles with the semi-major axis and has one end at the center of the conic section.
In celestial mechanics, a Kepler orbit (or Keplerian orbit, named after the German astronomer Johannes Kepler) is the motion of one body relative to another, as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space. A Kepler orbit can also form a straight line. It considers only the point-like gravitational attraction of two bodies, neglecting perturbations due to gravitational interactions with other objects, atmospheric drag, solar radiation pressure, a non-spherical central body, and so on.
In astrodynamics, the orbital eccentricity of an astronomical object is a dimensionless parameter that determines the amount by which its orbit around another body deviates from a perfect circle. A value of 0 is a circular orbit, values between 0 and 1 form an elliptic orbit, 1 is a parabolic escape orbit (or capture orbit), and greater than 1 is a hyperbola. The term derives its name from the parameters of conic sections, as every Kepler orbit is a conic section.
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