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Encyclopedia > Eccentric anomaly

The eccentric anomaly is the angle between the direction of periapsis and the current position of an object on its orbit, projected onto the ellipse's circumscribing circle perpendicularly to the major axis, measured at the centre of the ellipse. In the diagram below, it is E (the angle zcx). This article is about several astronomical terms (apogee & perigee, aphelion & perihelion, generic equivalents based on apsis, and related but rarer terms. ... In physics, an orbit is the path that an object makes, around another object, whilst under the influence of a source of centripetal force, such as gravity. ... In mathematics, an ellipse (from the Greek for absence) is a plane algebraic curve where the sum of the distances from any point on the curve to two fixed points is constant. ...

sets out the points and labels used in the present explanation of Keplers equation File links The following pages link to this file: Keplers laws of planetary motion Mean anomaly Eccentric anomaly True anomaly Categories: GFDL images ... sets out the points and labels used in the present explanation of Keplers equation File links The following pages link to this file: Keplers laws of planetary motion Mean anomaly Eccentric anomaly True anomaly Categories: GFDL images ...

In astrodynamics eccentric anomaly E can be calculated as follows: Astrodynamics is the study of the motion of rockets, missiles, and space vehicles, as determined from Sir Isaac Newtons laws of motion and his law of universal gravitation. ...

$E=arccos {{1-left [ mathbf{r} right ] / a} over e}$

where:

• $mathbf{r},!$ is the orbiting body's position vector (segment sp),
• $a,!$ is the orbit's semi-major axis (segment cz), and
• $e,!$ is the orbit's eccentricity.

The relation between E and M, the mean anomaly, is: In astrodynamics or celestial dynamics orbital state vectors (sometimes State Vectors) are vectors of position () and velocity () that together with their time () ( epoch) uniquely determine the state of an orbiting body. ... In geometry, the term semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolas. ... (This page refers to eccitricity in astrodynamics. ... In the study of orbital dynamics the mean anomaly is a measure of time, specific to the orbiting body p, which is a multiple of 2π radians at and only at periapsis. ...

$M = E - e cdot sin{E}.,!$

For small values of e (e < 0.6627434) this equation can be solved iteratively, starting from E0 = M and using the relation $E_{i+1} = M + e,sin E_i$. The first few terms of the expansion in powers of e are: In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ...

• $E_1 = M + e,sin M$
• $E_2 = M + e,sin M + frac{1}{2} e^2 sin 2M$
• $E_3 = M + e,sin M + frac{1}{2} e^2 sin 2M + frac{1}{8} e^3 (3sin 3M - sin M)$.

For references on details of this derivation, as well as other more efficient methods of solution, see Murray and Dermott (1999, p.35). For a derivation of the limiting value of e see Plummer (1960, section 46).

The relation between E and T, the true anomaly, is: In astronomy, the true anomaly (, also written ) is the angle between the direction z-s of periapsis and the current position p of an object on its orbit, measured at the focus s of the ellipse (the point around which the object orbits). ...

$cos{T} = {{cos{E} - e} over {1 - e cdot cos{E}}}$

or equivalently

$tan{T over 2} = sqrt{{{1+e} over {1-e}}} tan{E over 2}.,$

The relations between the radius (position vector magnitude) and the anomalies are:

$r = a left ( 1 - e cdot cos{E} right ),!$

and

$r = a{(1 - e^2) over (1 + e cdot cos{T})}.,!$

Johannes Keplers primary contributions to astronomy/astrophysics were his three laws of planetary motion. ... In the study of orbital dynamics the mean anomaly is a measure of time, specific to the orbiting body p, which is a multiple of 2π radians at and only at periapsis. ... In astronomy, the true anomaly (, also written ) is the angle between the direction z-s of periapsis and the current position p of an object on its orbit, measured at the focus s of the ellipse (the point around which the object orbits). ...

## References

• Murray, C. D. & Dermott, S. F. 1999, Solar System Dynamics, Cambridge University Press, Cambridge.
• Plummer, H.C., 1960, An Introductory treatise on Dynamical Astronomy, Dover Publications, New York. (Reprint of the 1918 Cambridge University Press edition.)

Results from FactBites:

 Anomaly - LoveToKnow 1911 (402 words) Mean R anomaly is the anomaly which the body would have if it moved from the pericentre around F with a uniform angular motion such that its revolution would be completed in its actual time (see Orbit). Eccentric anomaly is defined thus: - Draw the circumscribing circle of the elliptic orbit around the centre C of the orbit. In the ancient astronomy the anomaly was taken as the angular distance of the planet from the point of the farthest recession from the earth.
 Wikinfo | Anomaly (324 words) The eccentric anomaly is observed from the centre of the ellipse, using a projection of the planet's position onto the circumscribing circle; and In spacecraft operations, an anomaly is an event that results in an unexpected and unwanted event that may cause a negative impact to the state of health of the spacecraft. The Ararat anomaly, an object on Mount Ararat clamed to be Noah's Ark.
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