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[[File:Eccentric and true anomaly.PNG|thumb|The true anomaly of point ''P'' is the angle ''<math> \theta</math>''. Also shown is the eccentric anomaly of point ''P'', which is the angle ''E''. The center of the ellipse is point ''C'', and the focus is point ''F''. The radial position vector '''r''' is taken from the focus ''F'', not from the center of coordinates ''C''. Auxiliary circle has radius ''a''; minor auxiliary circle has radius ''b''.]]
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In [[celestial mechanics]], '''true anomaly''' is an angular [[parameter]] that defines the position of a body moving along a [[Keplerian orbit]]. It is the angle between the direction of [[periapsis]] and the current position of the body, as seen from the main focus of the [[ellipse]] (the point around which the object orbits).  
 
The true anomaly is usually denoted by the [[Greek alphabet|Greek letters]] <math>\,\nu</math> or <math>\,\theta</math>, or the [[Latin script|Latin letter]] <math> f </math>.
 
The true anomaly is one of three angular parameters ("anomalies") that define a position along an orbit, the other two being the [[eccentric anomaly]] and the [[mean anomaly]].
 
==Formulas==
===From state vectors===
For elliptic orbits '''true anomaly''' <math>\nu\,\!</math> can be calculated from [[orbital state vectors]] as:
 
:<math> \nu = \arccos { {\mathbf{e} \cdot \mathbf{r}} \over { \mathbf{\left |e \right |} \mathbf{\left |r \right |} }}</math> &emsp;&emsp;(if <math>\mathbf{r} \cdot \mathbf{v} < 0</math> then replace <math>\nu\ </math> by <math>2\pi-\nu\ </math>)
 
where:
* <math> \mathbf{v}\,</math> is [[orbital velocity vector]] of the orbiting body,
* <math> \mathbf{e}\,</math> is [[eccentricity vector]],
* <math> \mathbf{r}\,</math> is [[orbital position vector]] (segment ''fp'') of the orbiting body.
 
====Circular orbit====
For [[circular orbit]]s the true anomaly is undefined because circular orbits do not have a uniquely determined periapsis. Instead one uses the [[argument of latitude]] <math>u\,\!</math>:
 
:<math> u = \arccos { {\mathbf{n} \cdot \mathbf{r}} \over { \mathbf{\left |n \right |} \mathbf{\left |r \right |} }}</math> &emsp;&emsp;(if <math>\mathbf{n} \cdot \mathbf{v} >0</math> then replace <math>u\ </math> by <math>2\pi-u\ </math>)
 
where:
* <math> \mathbf{n} </math> is vector pointing towards the ascending node (i.e. the z-component of <math> \mathbf{n} </math> is zero).
:
 
====Circular orbit with zero inclination====
For [[circular orbit]]s with zero inclination the argument of latitude is also undefined, because there is no uniquely determined line of nodes. One uses the [[true longitude]] instead:
 
:<math> l = \arccos { r_x  \over { \mathbf{\left |r \right |}}}</math> &emsp;&emsp;(if <math> v_x > 0\ </math> then replace <math>l\ </math> by <math>2\pi-l\ </math>)
 
where:
* <math>r_x \,</math> is x-component of [[orbital position vector]] <math>\mathbf{r}</math>,
* <math>v_x \,</math> is x-component of [[orbital velocity vector]] <math>\mathbf{v}</math>.
 
===From the eccentric anomaly===
The relation between the true anomaly <math>\,\nu</math> <!--''&nu;''--> and the [[eccentric anomaly]] ''E'' is:
 
:<math>\cos{\nu} = {{\cos{E} - e} \over {1 - e \cdot \cos{E}}}</math>
 
or equivalently
 
:<math>\tan{\nu \over 2} = \sqrt{{{1+e} \over {1-e}}} \tan{E \over 2}.</math>
 
Therefore
 
:<math> \nu  = 2 \, \mathop{\mathrm{arg}}\left(\sqrt{1-e} \, \cos\frac{E}{2} , \sqrt{1+e}\sin\frac{E}{2}\right)</math>
 
where <math>\operatorname{arg}(x, y)</math> is the polar argument of the vector <math>\left(x, y\right)</math> (available in many programming languages as the library function [[atan2|atan2(y, x)]] in [[Fortran]] and [[MATLAB]], or as ArcTan(x, y) in [[Wolfram Mathematica]]).
 
===Radius from true anomaly===
The radius (distance from the focus of attraction and the orbiting body) is related to the true anomaly by the formula
 
:<math>r = a\cdot{1 - e^2 \over 1 + e \cdot \cos\nu}\,\!</math>
 
where ''a'' is the orbit's [[semi-major axis]] (segment ''cz'').
 
==See also==
* [[Kepler's laws of planetary motion]]
* [[Eccentric anomaly]]
* [[Mean anomaly]]
* [[Ellipse]]
* [[Hyperbola]]
 
==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.)
 
{{orbits}}
 
[[Category:Orbits]]

Latest revision as of 11:33, 20 December 2014

For the offense, you might contain Gunboats which can simply shoot at enemy safeguarding coming from a range and Landing Work which you must refill when you train pieces for example Rifleman, Heavy, Zooka, Warrior and Fish tanks. To your village defenses, you might offer structures like Mortar, Machine Gun, Sniper Tower, Cannon, Flamethrower, Mine, Tank Mine, Boom Cannon and Drive Launcher to assist we eradicate enemies.

In view that explained in the extremely Clash of Clans' Kin Wars overview, anniversary hoa war is breach into a couple phases: Alertness Day and Entertainment Day. Anniversary image lasts 24 hours as well as means that you could certainly accomplish altered things.

Gemstones are known as your games primary forex. The Jewels are often purchase resources along with speeding up numerous powerful tasks. The Jewels can also be was considered to buy bonus items. Apart from that, this may also let the leader alleged any undesired debris to be able to obtain a much more gems. Players definitely will obtain Gems through concluding numerous tasks or it could be that using the clash of clans identify available online.

Take note of how money your teen would be shelling out for playing games. These kinds towards products aren't cheap and then there is truly the option of getting yourself much more add-ons in the game itself. Establish month-to-month and twelve-monthly restrictions on the size of money that is likely to be spent on applications. Also, have conversations as well as the youngsters about viewing your spending habits.

Keep your game just some possible. While car-preservation is a good characteristic, do not count about it. Particularly, when you the 1st time start playing a game, you may not provide any thought when the game saves, which might result in a decrease of significant info down the line. Until you thoroughly grasp the sport better, time and again save yourself.

Should really you perform online multi-player game titles, don't neglect the strength of color of voice chat! A mic or headset is a very simple expenditure, and having most of the capability to speak to your fellow athletes makes a lot of positive factors. You are within a position to create more influential connections with the gaming community and stay some far more successful institution person when you perhaps may be able connect out loud.

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