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For elliptical orbits it can also be calculated from the periapsis and apoapsis since = and = (+), where a is the length of the semi-major axis. = + = / / + = + where: r a is the radius at apoapsis (also "apofocus", "aphelion", "apogee"), i.e., the farthest distance of the orbit to the center of mass of the system, which is a focus of the ellipse.
Halley's calculations enabled the comet's earlier appearances to be found in the historical record. The following table sets out the astronomical designations for every apparition of Halley's Comet from 240 BC, the earliest documented sighting. [7] [166] In the designations, "1P/" refers to Halley's Comet; the first periodic comet discovered.
Also shown are: semi-major axis a, semi-minor axis b and semi-latus rectum p; center of ellipse and its two foci marked by large dots. For θ = 0°, r = r min and for θ = 180°, r = r max. Mathematically, an ellipse can be represented by the formula:
is the semi-major axis, is the standard gravitational parameter. Conclusions: For a given semi-major axis the specific orbital energy is independent of the eccentricity. Using the virial theorem we find: the time-average of the specific potential energy is equal to
a is the orbit's semi-major axis; G is the gravitational constant, M is the mass of the more massive body. For all ellipses with a given semi-major axis the orbital period is the same, regardless of eccentricity. Inversely, for calculating the distance where a body has to orbit in order to have a given orbital period T:
Semi-major axis (a) — half the distance between the apoapsis and periapsis. The portion of the semi-major axis extending from the primary at one focus to the periapsis is shown as a purple line in the diagram; the rest (from the primary/focus to the center of the orbit ellipse) is below the reference plane and not shown.
is the length of the semi-major axis, is the standard gravitational parameter. Conclusions: For a given semi-major axis the specific orbital energy is independent of the eccentricity. Using the virial theorem to find: the time-average of the specific potential energy is equal to −2ε
where is the semi-major axis, the semi-minor axis. Kepler's equation is a transcendental equation because sine is a transcendental function, and it cannot be solved for algebraically. Numerical analysis and series expansions are generally required to evaluate .