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In celestial mechanics, mean anomaly is a parameter relating position and time for a body moving in a Kepler orbit. It is based on the fact that equal areas are swept in equal intervals of time by a line joining the focus and the orbiting body (Kepler's second law).

The mean anomaly increases uniformly from 0 to 2\pi radians during each orbit. However, it is not an angle. Due to Kepler's second law, the mean anomaly is proportional to the area swept by the focus-to-body line since the last periapsis.

The mean anomaly is usually denoted by the letter M, and is given by the formula:

 M =  n \, t =  \sqrt{\frac{ G( M_\star \! + \!m ) } {a^3}} \,t

where n is the mean motion, a is the length of the orbit's semi-major axis, M_\star and m are the orbiting masses, and G is the gravitational constant.

The mean anomaly is the time since the last periapsis multiplied by the mean motion, and the mean motion is 2\pi radians divided by the duration of a full orbit.

The mean anomaly is one of three angular parameters ("anomalies") that define a position along an orbit, the other two being the eccentric anomaly and the true anomaly. If the mean anomaly is known at any given instant, it can be calculated at any later (or prior) instant by simply adding (or subtracting) \sqrt{\frac{ G( M_\star \! + \!m ) } {a^3}} \,\delta t where \delta t represents the time difference. The other anomalies can hence be calculated.

Formulas[edit]

The mean anomaly M can be computed from the eccentric anomaly E and the eccentricity e with Kepler's Equation:

M =  E - e \cdot \sin E

To find the position of the object in an elliptic Kepler orbit at a given time t, the mean anomaly is found by multiplying the time and the mean motion, then it is used to find the eccentric anomaly by solving Kepler's equation.

It is also frequently seen:

M =  M_0 + nt,

Again n is the mean motion. However, t, in this instance, is the time since epoch, which is how much time has passed since the measurement of M0 was taken. The value M0 denotes the mean anomaly at epoch, which is the mean anomaly at the time the measurement was taken.

See also[edit]

References[edit]

  • 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.)

Original courtesy of Wikipedia: http://en.wikipedia.org/wiki/Mean_anomaly — Please support Wikipedia.
This page uses Creative Commons Licensed content from Wikipedia. A portion of the proceeds from advertising on Digplanet goes to supporting Wikipedia.

73 news items

 
reportingclimatescience.com
Wed, 15 Apr 2015 06:52:30 -0700

Click to enlarge. NASA GISS global surface temperature anomaly map for March 2015 showing particularly high anomalies across North America, northern Europe, Eurasia, China and the Arctic. Note: Gray areas signify missing data. Note: Ocean data are not ...

WSU News

WSU News
Thu, 26 Mar 2015 14:17:59 -0700

This winter – December to February – was the warmest since 1991-92, with a mean anomaly of +3.5 degrees. The 2014-15 winter was not only one of the mildest winters on record, but also one of the warmest seasons, relative to normal, in recent history.
 
reportingclimatescience.com
Sun, 15 Feb 2015 04:45:00 -0800

Click to enlarge. Global temperature anomaly map for January 2015 showing particularly high anomalies in Siberia, China, the west of North America and the Arctic. Note: Gray areas signify missing data. Note: Ocean data are not used over land nor within ...

reportingclimatescience.com

reportingclimatescience.com
Sun, 15 Mar 2015 06:52:30 -0700

Click to enlarge. NASA GISS global surface temperature anomaly map for February 2015 showing particularly high anomalies across northern Europe, Eurasia, China and the Arctic. Note: Gray areas signify missing data. Note: Ocean data are not used over ...

RealClimate

RealClimate
Tue, 06 Jan 2015 07:02:18 -0800

That means you need fewer data points to make a good estimate of the global value. The 2 \sigma uncertainty in the global mean anomaly on a yearly basis are (with the current network of stations) is around 0.1ºC in contrast that to the estimated ...
 
GPS World magazine
Wed, 11 Mar 2015 14:15:40 -0700

The procedure took somewhat longer than that for the first satellite as it also involved changing the mean anomaly of the satellite to be about 180° away from that of the first satellite. The locations of the satellites in the Galileo constellation are ...

The Guardian

The Guardian
Wed, 18 Mar 2015 18:09:30 -0700

Norfolk Island will no longer be a semi self-governing territory, after the commonwealth government announced that residents will have to pay tax and their government will be replaced with a local council. In exchange, islanders will receive Medicare ...

The Guardian (blog)

The Guardian (blog)
Mon, 05 Jan 2015 18:13:58 -0800

... above average. This general trend is in line with preliminary estimates for global temperatures for 2014 from the World Meteorological Organisation, which suggest 2014 is likely to be the warmest year on record: Global temperature mean anomaly ...
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