Book contents
- Frontmatter
- Contents
- Preface
- Using your personal computer for astronomy
- DEFAULT: default value input routnine & YESNO: ‘Y’ or ‘N’ input routine
- MINSEC: converts between decimal hours/degrees and minutes/seconds form
- JULDAY: calendar date to Julian day number since 1900 January 0.5
- CALDAY: Julian day number since 1900 January 0.5 to calendar date
- TIME: converts between local civil and sidereal times
- EQHOR: converts between equatorial and horizon coordinates
- HRANG: converts between right ascension and hour angle
- OBLIQ: calculates the value of the obliquity of the ecliptic
- NUTAT: finds corrections for nutation in longitude and obliquity
- EQECL: converts between equatorial and ecliptic coordinates
- EQGAL: converts between equatorial and galactic coordinates
- GENCON: converts between any of the coordinate systems
- PRCESS1: approximate precession of equatorial coordinates & PRCESS2: rigorous precession of equatorial coordinates
- PARALLX: converts between geocentric and apparent position
- REFRACT: calculates the effect of atmospheric refraction
- RISET: finds the circumstances of rising and setting
- ANOMALY: solves Kepler's equation for elliptical motion
- SUN: finds the ecliptic coordinates of the Sun
- SUNRS: finds the circumstances of sunrise and sunset
- PELMENT: returns the orbital elements of the major planets
- PLANS: finds the position of a planet
- MOON: finds the position and parallax of the Moon
- MOONRS: finds the circumstances of moonrise and moonset
- MOONNF: finds the times of new and full moon
- ECLIPSE: finds the circumstances of lunar and solar eclipses
- DISPLAY: displays an eclipse in graphical form
- ELOSC: finds positions from osculating elliptical elements
- RELEM: converts elliptic orbital elements from one epoch to another
- PCOMET: finds the position of a comet from parabolic elements
- PFIT: finds parabolic elements from observations & EFIT: finds elliptical elements from observations
- List of variables
- Bibliography
- Index
- PROGRAMS AVAILABLE ON DISK
ANOMALY: solves Kepler's equation for elliptical motion
Published online by Cambridge University Press: 17 February 2010
- Frontmatter
- Contents
- Preface
- Using your personal computer for astronomy
- DEFAULT: default value input routnine & YESNO: ‘Y’ or ‘N’ input routine
- MINSEC: converts between decimal hours/degrees and minutes/seconds form
- JULDAY: calendar date to Julian day number since 1900 January 0.5
- CALDAY: Julian day number since 1900 January 0.5 to calendar date
- TIME: converts between local civil and sidereal times
- EQHOR: converts between equatorial and horizon coordinates
- HRANG: converts between right ascension and hour angle
- OBLIQ: calculates the value of the obliquity of the ecliptic
- NUTAT: finds corrections for nutation in longitude and obliquity
- EQECL: converts between equatorial and ecliptic coordinates
- EQGAL: converts between equatorial and galactic coordinates
- GENCON: converts between any of the coordinate systems
- PRCESS1: approximate precession of equatorial coordinates & PRCESS2: rigorous precession of equatorial coordinates
- PARALLX: converts between geocentric and apparent position
- REFRACT: calculates the effect of atmospheric refraction
- RISET: finds the circumstances of rising and setting
- ANOMALY: solves Kepler's equation for elliptical motion
- SUN: finds the ecliptic coordinates of the Sun
- SUNRS: finds the circumstances of sunrise and sunset
- PELMENT: returns the orbital elements of the major planets
- PLANS: finds the position of a planet
- MOON: finds the position and parallax of the Moon
- MOONRS: finds the circumstances of moonrise and moonset
- MOONNF: finds the times of new and full moon
- ECLIPSE: finds the circumstances of lunar and solar eclipses
- DISPLAY: displays an eclipse in graphical form
- ELOSC: finds positions from osculating elliptical elements
- RELEM: converts elliptic orbital elements from one epoch to another
- PCOMET: finds the position of a comet from parabolic elements
- PFIT: finds parabolic elements from observations & EFIT: finds elliptical elements from observations
- List of variables
- Bibliography
- Index
- PROGRAMS AVAILABLE ON DISK
Summary
The starting point for calculations involving a body in an elliptical orbit is often the mean anomaly, AM. This is the angle moved by a fictitious body in a circular orbit of the same period as the real body, the angle being reckoned in the same sense as the direction of motion of the real body from the point of closest approach (the periapsis).The quantity needed is the true anomaly, AT, which measures the angle moved by the real body since periapsis, and it is related to AM through the eccentric anomaly, AE, by Kepler's equation.
AE - (EC × SIN(AE)) = AM,
where EC is the eccentricity of the orbit. Unfortunately, this equation is not easily solved, but the solution can be approximated by a trigonometric expansion called the equation of the centre.If EC is less than about 0.1 and high precision is not required, the first term of the expansion may suffice, giving
AT = AM + (2 × EC × SIN(AM)),
where AT and AM are expressed in radians.
For more accurate work, the equation must be solved explicitly for AE, and then AT calculated from
TAN(AT/2) = ((1 + EC)/(1 - EC))0.5 × TAN(AE/2).
The routine given here solves Kepler's equation by an iterative method in which an approximate solution for AE is repeatedly refined until the error between (AE - (EC x SIN(AE))) and AM is less than a given error (10-6 radians).
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- Astronomy with your Personal Computer , pp. 111 - 114Publisher: Cambridge University PressPrint publication year: 1990