Draft:Projective Anomaly

Template:AfC submission 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 the $\theta$ and is usually restricted to the range 0 - 360 deg (0 - 2 $\pi$ rad).

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

In the projective geometry, a circle, ellipse, parabola, and hyperbola are treated as the same kind of quadratic curves.

projective parameters and projective anomaly

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

• circular orbit $\beta =0$

• elliptic orbit $\alpha \beta <1$

• parabolic orbit $\alpha \beta =1$

• hyperbolic orbit $\alpha \beta >1$

• linear orbit $\alpha =\beta$

• imaginary orbit $\alpha <\beta$

where

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

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

$q=(1-e)a$

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

where $\alpha$ is semi-major axis,$e$ iseccentricity, $q$ is perihelion distance、$Q$ is aphelion distance.

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

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

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

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

Kelper's equation

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

• Case : $\alpha \beta <1$

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

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

• case : $\alpha \beta =1$

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

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

• case : $\alpha \beta >1$

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

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

The above equations are called Kepler's equation.

Generalized anomaly

For arbitrary constant $\lambda$, thegeneralized anonaly $\mathrm{\Theta }$ is related as

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

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

References

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

See also

• two body problem

• mean anomaly

• eccentric anomaly

• true anomaly

• Kepler's equation

• projective geometry