Thiele−Innes elements

For (amateur) astronomers who want to observe double stars, it is useful to know how far apart the two components of the double star are: the narrower the distance, the more difficult it is to see the two stars separated. Available tables usually only give the separation at one single moment.^[4] But as the two stars of a binary revolve around each other, their apparent distance varies continually. The Thiele-Innes formalism is an efficient method to calculate how the distance, and the so-called position angle, or orientation of the line connecting the two stars, vary over time.

The procedure essentially consists of two steps:

• One step is to calculate the relative position^[5] of the stars in their real mutual orbit at a given time.

• The other step is to project this position onto the celestial sphere. This second step is performed by means of the Thiele-Innes elements.

The efficiency of the Thiele-Innes method stems from the fact that the Thiele-Innes elements that are needed for the second step have to be evaluated only once.

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^[6]):

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^[7]), and are defined as follows:^[8]

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

Summarize Timeline Fact Check

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^[9]).

This can be elegantly written as a simple matrix multiplication:

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

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)

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

Historically, the Thiele-Innes elements simplified various calculations regarding the orbits of binary stars.^[10] 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.^[11]