True anomaly

Formulas

From state vectors

For elliptic orbits, the true anomaly ν can be calculated from orbital state vectors as:

${\displaystyle \nu =\arccos \frac{\mathbf{e}\cdot \mathbf{r}}{\left|\mathbf{e}\right|\left|\mathbf{r}\right|}}$

(if r ⋅ v < 0 then replace ν by 2π − ν)

where:

• v is the orbital velocity vector of the orbiting body,
• e is the eccentricity vector,
• r is the orbital position vector (segment FP in the figure) of the orbiting body.

Circular orbit

For circular orbits the true anomaly is undefined, because circular orbits do not have a uniquely determined periapsis. Instead the argument of latitude u is used:

${\displaystyle u=\arccos \frac{\mathbf{n}\cdot \mathbf{r}}{\left|\mathbf{n}\right|\left|\mathbf{r}\right|}}$

(if r_z < 0 then replace u by 2π − u)

where:

• n is a vector pointing towards the ascending node (i.e. the z-component of n is zero).
• r_z is the z-component of the orbital position vector r

Circular orbit with zero inclination

For circular orbits 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:

${\displaystyle l=\arccos \frac{r_x}{\left|\mathbf{r}\right|}}$

(if v_x > 0 then replace l by 2π − l)

where:

• r_x is the x-component of the orbital position vector r
• v_x is the x-component of the orbital velocity vector v.

From the eccentric anomaly

The relation between the true anomaly ν and the eccentric anomaly ${\displaystyle E}$ is:

${\displaystyle \cos \nu =\frac{\cos E-e}{1-e\cos E}}$

or using the sine and tangent:

${\displaystyle \begin{array}{cc}\sin \nu & =\frac{\sqrt{1-e^2}\sin E}{1-e\cos E}\\ \tan \nu =\frac{\sin \nu }{\cos \nu }& =\frac{\sqrt{1-e^2}\sin E}{\cos E-e}\end{array}}$

or equivalently:

${\displaystyle \tan \frac{\nu }{2}=\sqrt{\frac{1+e}{1-e}}\tan \frac{E}{2}}$

so

${\displaystyle \nu =2\arctan \left(\sqrt{\frac{1+e}{1-e}}\tan \frac{E}{2}\right)}$

Alternatively, a form of this equation was derived by R. Broucke and P. Cefola that avoids numerical issues when the arguments are near ${\displaystyle \pm \pi }$, as the two tangents become infinite. Additionally, since ${\displaystyle \frac{E}{2}}$ and ${\displaystyle \frac{\nu }{2}}$ are always in the same quadrant, there will not be any sign problems.

${\displaystyle \tan \frac{1}{2}(\nu -E)=\frac{\beta \sin E}{1-\beta \cos E}}$ where ${\displaystyle \beta =\frac{e}{1+\sqrt{1-e^2}}}$

so

${\displaystyle \nu =E+2\arctan \left(\frac{\beta \sin E}{1-\beta \cos E}\right)}$

From the mean anomaly

The true anomaly can be calculated directly from the mean anomaly ${\displaystyle M}$ via a Fourier expansion:

${\displaystyle \nu =M+2\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k}\left[\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{J}_n(-ke){\beta }^{|k+n|}\right]\sin kM}$

with Bessel functions ${\displaystyle J_n}$ and parameter ${\displaystyle \beta =\frac{1-\sqrt{1-e^2}}{e}}$.

Omitting all terms of order ${\displaystyle e^4}$ or higher (indicated by ${\displaystyle \mathcal{O}\left(e^4\right)}$), it can be written as

${\displaystyle \nu =M+\left(2e-\frac{1}{4}{e}^3\right)\sin M+\frac{5}{4}{e}^2\sin 2M+\frac{13}{12}{e}^3\sin 3M+\mathcal{O}\left(e^4\right).}$

Note that for reasons of accuracy this approximation is usually limited to orbits where the eccentricity ${\displaystyle e}$ is small.

The expression ${\displaystyle \nu -M}$ is known as the equation of the center, where more details about the expansion are given.

Radius from true anomaly

The radius (distance between the focus of attraction and the orbiting body) is related to the true anomaly by the formula

${\displaystyle r(t)=a\frac{1-e^2}{1+e\cos \nu (t)}}$

where a is the orbit's semi-major axis.

In celestial mechanics, Projective 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 in the projective space.

The projective anomaly is usually denoted by ${\displaystyle \theta }$ and is usually restricted to the range 0 - 360 degrees (0 - 2 ${\displaystyle \pi }$ radians).

The projective anomaly ${\displaystyle \theta }$ is one of four angular parameters (anomalies) that defines a position along an orbit, the other three being the eccentric anomaly, the true anomaly, and the mean anomaly.

In the projective geometry, circles, ellipses, parabolae, and hyperbolae are treated as the same kind of quadratic curves.

Projective parameters and projective anomaly

An orbit type is classified by two project parameters ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ as follows,

• circular orbit ${\displaystyle \beta =0}$
• elliptic orbit ${\displaystyle \alpha \beta <1}$
• parabolic orbit ${\displaystyle \alpha \beta =1}$
• hyperbolic orbit ${\displaystyle \alpha \beta >1}$
• linear orbit ${\displaystyle \alpha =\beta }$
• imaginary orbit ${\displaystyle \alpha <\beta }$

where

${\displaystyle \alpha =\frac{(1+e)(q-p)+\sqrt{(1+e{)}^2(q+p{)}^2+4e^2}}{2}}$

${\displaystyle \beta =\frac{2e}{(1+e)(q+p)+\sqrt{(1+e{)}^2(q+p{)}^2+4e^2}}}$

${\displaystyle q=(1-e)a}$

${\displaystyle p=\frac{1}{Q}=\frac{1}{(1+e)a}}$

where ${\displaystyle \alpha }$ is semi major axis, ${\displaystyle e}$ is eccentricity, ${\displaystyle q}$ is perihelion distance, and ${\displaystyle Q}$ is aphelion distance.

Position and heliocentric distance of the planet ${\displaystyle x}$, ${\displaystyle y}$ and ${\displaystyle r}$ can be calculated as functions of the projective anomaly ${\displaystyle \theta }$ :

${\displaystyle x=\frac{-\beta +\alpha \cos \theta }{1+\alpha \beta \cos \theta }}$

${\displaystyle y=\frac{\sqrt{{\alpha }^2-{\beta }^2}\sin \theta }{1+\alpha \beta \cos \theta }}$

${\displaystyle r=\frac{\alpha -\beta \cos \theta }{1+\alpha \beta \cos \theta }}$

Kepler's equation

The projective anomaly ${\displaystyle \theta }$ can be calculated from the eccentric anomaly ${\displaystyle u}$ as follows,

• Case : ${\displaystyle \alpha \beta <1}$

${\displaystyle \tan \frac{\theta }{2}=\sqrt{\frac{1+\alpha \beta }{1-\alpha \beta }}\tan \frac{u}{2}}$

${\displaystyle u-e\sin u=M={\left(\frac{1-{\alpha }^2{\beta }^2}{\alpha (1+{\beta }^2)}\right)}^{3/2}k(t-T_0)}$

• case : ${\displaystyle \alpha \beta =1}$

${\displaystyle \frac{s^3}{3}+\frac{{\alpha }^2-1}{{\alpha }^2+1}s=\frac{2k(t-T_0)}{\sqrt{\alpha ({\alpha }^2+1{)}^3}}}$

${\displaystyle s=\tan \frac{\theta }{2}}$

• case : ${\displaystyle \alpha \beta >1}$

${\displaystyle \tan \frac{\theta }{2}=\sqrt{\frac{\alpha \beta +1}{\alpha \beta -1}}\tanh \frac{u}{2}}$

${\displaystyle e\sinh u-u=M={\left(\frac{{\alpha }^2{\beta }^2-1}{\alpha (1+{\beta }^2)}\right)}^{3/2}k(t-T_0)}$

The above equations are called Kepler's equation.

Generalized anomaly

For arbitrary constant ${\displaystyle \lambda }$, the generalized anomaly ${\displaystyle \mathrm{\Theta }}$ is related as

${\displaystyle \tan \frac{\mathrm{\Theta }}{2}=\lambda \tan \frac{u}{2}}$

The eccentric anomaly, the true anomaly, and the projective anomaly are the cases of ${\displaystyle \lambda =1}$, ${\displaystyle \lambda =\sqrt{\frac{1+e}{1-e}}}$, ${\displaystyle \lambda =\sqrt{\frac{1+\alpha \beta }{1-\alpha \beta }}}$, respectively.

• Sato, I., "A New Anomaly of Keplerian Motion", Astronomical Journal Vol.116, pp. 2038-3039, (1997)