Thiele−Innes elements

The Thiele−Innes elements, named after Thorvald N. Thiele (1838–1910) and Robert T.A. Innes (1861–1933), are auxiliary quantities that can be used to calculate the relative apparent position of components of a binary star system.

Orbital elements

Apparent orbit of the binary star system alpha Centauri from 1975 to 2050. (ρ,ϑ) and (x,y) are indicated. The stellar disks are not to scale. (North is up; a telescope would probably show the image upside down, so with North down.)

The orbit of a binary star and the relative position at any moment of its components (specifically the position of the weaker component B relative to the brighter component A) are characterized by a set of seven orbital elements. These elements are (as an example numerical values are given for alpha Centauri):

symbol	orbital element	alpha Centauri
P	period of revolution	79.910 ± 0.011 years
a	semi-major axis (in arcseconds)	17″.570 ± 0″.022
i	inclination between plane of true orbit and plane of its projection on the celestial sphere	79°.205 ± 0°.041
Ω	ascending node: position angle of intersection line between true and apparent orbits	204°.850 ± 0°.084
T	instant (Julian year) of the stars' passage through the periastron	1875.660 ± 0.012
e	numerical eccentricity of the orbit	0.51790 ± 0.00076
ω	angle between ascending node and periastron, counted in the direction of motion	231°.650 ± 0°.076

Definition of the Thiele-Innes elements

The Thiele-Innes elements are constants that describe how the real orbit in space is projected onto the celestial sphere. They are functions of the orbital elements a, Ω, ω, and i (the so-called classical or Campbell elements), and are defined as follows:

A = a · (cos ω cos Ω − sin ω sin Ω cos i)

B = a · (cos ω sin Ω + sin ω cos Ω cos i)

F = a · (−sin ω cos Ω − cos ω sin Ω cos i)

G = a · (−sin ω sin Ω + cos ω cos Ω cos i)

For alpha Centauri one finds from the orbital elements above:

A = 8″.8076

B = 6″.9231

F = −13″.3613

G = −3″.9378

From elements to position

The time (t)-dependent part of the Kepler orbit of the binary star follows from the three remaining orbital elements, e, T and P. First one calculates the mean anomaly M(t) at a time t:

M = (t − T) · 360°/P (modulo 360°)

Next, find the eccentric anomaly E, such that E = M + e sin(E) (Kepler's equation).

Definition of the rectangular coordinates X and Y for the Kepler orbit of a binary star (Thiele-Innes formalism)

From E and the orbital eccentricity e follow the rectangular coordinates within the Kepler orbit:

X = cos(E) − e

Y = √(1−e²) · sin(E)

The apparent position (x,y) of the double star, in rectangular coordinates, as seen from Earth, can then be calculated by means of:

x = A·X + F·Y

y = B·X + G·Y

(positive x is North, positive y is East).

This can be elegantly written as a simple matrix multiplication:

${\binom {x}{y}}={\begin{pmatrix}A&F\\B&G\end{pmatrix}}{\binom {cos(E(t))-e}{{\sqrt {1-e^{2}}}.sin(E(t))}}$

From x and y follow the phase angle ϑ (theta; measured counterclockwise in degrees from North) and the apparent distance ρ (rho; in arc seconds):

ρ = √ [x² + y²]

ϑ = arctan(y/x) (modulo 360°) (if x>0), or

arctan(y/x) + 180° (if x<0);

if x=0, then ϑ = 90° (if y>0) or ϑ = 270° (if y<0)

A calculation

Example: alpha Centauri. At the beginning of the year 2010 (t = 2010.0) one finds:

M = 245°.211

E = 224°.436

X = cos(E) − e = −1.23193

Y = √(1−e²) · sin(E) = −0.59891

x = AX + FY = −2″.848

y = BX + GY = −6″.170

ρ = √ [x² + y²] = 6″.796

ϑ = arctan(y/x) + 180° = 245°.222

Popularity

Using the Thiele-Innes method has long been attractive because of the fact that the Thiele-Innes elements are constants that need to be evaluated only once; thereafter, in an age before the modern computer, a time series of E(t), and thus a full series of positions (ρ,ϑ) could be calculated efficiently.