Plot of the orbits of the Gliese 581 system.
Orbit data taken from
Mayor et al. (2009), arXiv:0906.2780v1 [astro-ph.EP]
"The HARPS search for southern extra-solar planets XVIII. An Earth-mass planet
in the GJ 581 planetary system"
The configuration is given in Table 2. The eccentricities of the inner two
planets (e and b) are fixed to zero. Planetary positions are given for JD
2453152, which is close to the first RV observation by Udry et al.
(http://cdsarc.u-strasbg.fr/ftp/cats/J/A+A/469/L43/gl581.dat)
The semimajor axes in the paper are given to only 1 significant figure for the
inner three planets, so I have recalculated the semimajor axes from Kepler's
Third Law using a stellar mass of 0.31 times solar. The semimajor axes used are
thus:
Planet e: 0.0285 AU
Planet b: 0.0406 AU
Planet c: 0.0730 AU
Planet d: 0.2181 AU
To represent an orbit with given values of semimajor axis (a), eccentricity (e),
argument of periastron (omega), period (P) and time of periastron (T):
The semiminor axis (b) is given by
b = a*sqrt(1-e^2)
The position of the center of the ellipse relative to the star is
x = -a*e, y=0 (for periastron in the direction of positive x-axis)
The distance from the star to the point of periastron is
q = a*(1-e)
The rotation is given by -omega, since SVG rotations are defined in a clockwise
sense, as opposed to orbital elements which are given in an anticlockwise
sense.
The mean anomaly (M) at time t is given by
M = (t-T)/P * 360 degrees and constrained to the range [0,360) degrees
This can be converted to the eccentric anomaly (E) by iterative solution of the
Kepler equation:
M = E - e sin E
The r and theta coordinates of the planet are then derived from the eccentric
anomaly:
r = a*(1-e*cos(E))
theta = acos((cos(E)-e)/(1-e*cos(E))) for the case E<180 degrees
theta = 360-acos((cos(E)-e)/(1-e*cos(E))) otherwise
This can then be converted to Cartesians for SVG:
x = r*cos(theta + omega)
y = -r*sin(theta + omega) |