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 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):
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).
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:
( x y ) = ( A F B G ) ( c o s ( E ( t ) ) − e 1 − e 2 . s i n ( E ( t ) ) ) {\displaystyle {\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 Historically, the Thiele-Innes elements simplified various calculations regarding the orbits of binary stars. Using the Thiele-Innes method has also long been attractive because of the fact that the elements A, B, F, and G 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.
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