Thiele-Innes to Campbell Orbital Elements¶

Cite as

Faria, J. P. (2026). Thiele-Innes to Campbell orbital elements (kima Research Notes). Zenodo.

DOI: 10.5281/zenodo.23165531
License: CC BY 4.0

If you use kima, please also cite Faria et al. (2018) and, for astrometric fitting, Baycroft, Faria & Delisle (2026).

BibTeX

@misc{surname2026thiele,
  author    = {Faria, J.~P.},
  title     = {Thiele-Innes to Campbell orbital elements},
  year      = {2026},
  publisher = {Zenodo},
  doi       = {10.5281/zenodo.23165531},
  url       = {https://doi.org/10.5281/zenodo.23165531}
}

Author: J. P. Faria
License: CC BY 4.0

Introduction¶

An orbit can be parametrised by several equivalent sets of orbital elements. This note describes the conversion between the Campbell (geometric) elements $(a, i, \Omega, \omega)$, and the Thiele-Innes (TI) elements $(A, B, F, G)$, and vice versa. The first set is easier to interpret physically and geometrically, while the second makes an astrometric orbital model (or part of it) linear, and is the form in which, e.g., the Gaia mission publishes astrometric orbits.

The TI elements are named after Thiele (1883)¹, who introduced the idea (although not in the form presented here), and Innes (1926)² who developed the method further (see van den Bos 1926³). The Campbell elements are, to the best of the author's knowledge, named after William Wallace Campbell$^*$. The inversion presented here follows closely Appendix A of Halbwachs et al. (2023)⁴, who in turn follow Binnendijk (1960)⁶; the geometric interpretation and the numerical remarks are additions.

Note that only the four elements describing the size and orientation of the orbit are involved. The eccentricity $e$, the orbital period $P$ and the time of periastron $T_p$ are the same in the two parametrisations and are not affected by the conversion.

The Thiele-Innes elements¶

Let $E$ be the eccentric anomaly and define the position of the body in the orbital plane (in units of $a$), with $X$ pointing towards periastron and $Y$ perpendicular to it, in the direction of motion: $$ X = \cos E - e, \qquad Y = \sqrt{1 - e^2}\,\sin E . $$

The offsets of the body's position on the plane of the sky, towards the north ($\Delta\delta$) and towards the east ($\Delta\alpha^*$), are then

\[ \begin{aligned} \Delta\delta &= A\,X + F\,Y \\ \Delta\alpha^* &= B\,X + G\,Y \end{aligned} \qquad\Longleftrightarrow\qquad \begin{pmatrix} \Delta\delta \\ \Delta\alpha^* \end{pmatrix} = \underbrace{\begin{pmatrix} A & F \\ B & G \end{pmatrix}}_{\mathbf{M}} \begin{pmatrix} X \\ Y \end{pmatrix} . \]

The TI elements are the four entries of a $2\times2$ matrix $\mathbf{M}$ which maps the orbital plane onto the plane of the sky. Because the sky position is linear in $A, B, F, G$ (once $P$, $e$, and $T_p$ are fixed), they are very convenient for orbit fitting, whereas the angles $i$, $\Omega$ and $\omega$ appear non-linearly.

Campbell $\rightarrow$ Thiele-Innes¶

In terms of the Campbell elements, the TI elements are (see Binnendijk 1960⁶; Heintz 1978⁵, Halbwachs et al. 2023⁴):

\[ \begin{aligned} A &= a (\cos \omega \cos \Omega - \sin \omega \sin \Omega \cos i) \\ B &= a (\cos \omega \sin \Omega + \sin \omega \cos \Omega \cos i) \\ F &= -a (\sin \omega \cos \Omega + \cos \omega \sin \Omega \cos i) \\ G &= -a (\sin \omega \sin \Omega - \cos \omega \cos \Omega \cos i) \end{aligned} \]

where the Campbell elements are defined as follows:

$a$ (often called $a_0$ for a photocentric orbit) is the semi-major axis of the orbit. It is in the same units as the TI elements, often AU or mas.
$i$ is the inclination of the orbit: the angle between the orbital plane and the plane of the sky. By convention, $i \in [0, \pi/2]$ when the orbital motion is in the direct sense (counterclockwise on the sky, i.e. in the direction of increasing position angle), and $i \in [\pi/2, \pi]$ when the motion is retrograde (clockwise).
$\Omega$ is the position angle of the ascending node, measured from north through east. Without radial velocities there is no way to tell which of the two nodes is the ascending one, so the node taken as "ascending" is, by convention, the one on the side where right ascension increases. Therefore, $\Omega \in [0, \pi]$.
(When radial velocities are available the convention changes, see below.)
$\omega$ is the argument of periastron, measured in the orbital plane starting from the ascending node defined above, in the sense of the orbital motion.
(Note that Halbwachs et al. call $\omega$ the "periastron longitude".)

A geometric picture¶

To gain some intuition on the sines and cosines above, note that the transformation from the orbital plane to the plane of the sky can also be represented by the matrix product

\[ \mathbf{M} = \begin{pmatrix} A & F \\ B & G \end{pmatrix} = a \; \underbrace{\begin{pmatrix} \cos\Omega & -\sin\Omega \\ \sin\Omega & \cos\Omega \end{pmatrix}}_{R(\Omega)} \underbrace{\begin{pmatrix} 1 & 0 \\ 0 & \cos i \end{pmatrix}}_{\text{tilt}} \underbrace{\begin{pmatrix} \cos\omega & -\sin\omega \\ \sin\omega & \cos\omega \end{pmatrix}}_{R(\omega)} . \]

From right to left:

$R(\omega)$ rotates the periastron direction onto the line of nodes (the line where the orbital plane intersects the sky plane).
The tilt leaves the component along the line of nodes unchanged, and foreshortens the perpendicular component by $\cos i$.
$R(\Omega)$ rotates the line of nodes to its position angle on the sky.

In particular, a circle of radius $a$ in the orbital plane is mapped to an ellipse on the sky with semi-axes $a$ and $a|\cos i|$.

Thiele-Innes $\rightarrow$ Campbell¶

Any $2\times2$ matrix can be split in two pieces: one that acts as a rotation combined with a scaling, and one that acts as a reflection combined with a scaling. For $\mathbf{M}$, this split can be written

\[ \mathbf{M} = \underbrace{\begin{pmatrix} c_1 & -s_1 \\ s_1 & c_1 \end{pmatrix}}_{\text{rotation} \times p} + \underbrace{\begin{pmatrix} c_2 & s_2 \\ s_2 & -c_2 \end{pmatrix}}_{\text{reflection} \times q} \qquad\text{with}\qquad \begin{aligned} c_1 &= (A + G)/2, & s_1 &= (B - F)/2, \\ c_2 &= (A - G)/2, & s_2 &= (B + F)/2 . \end{aligned} \]

This is useful because the tilt matrix can also be split in the same way,

\[ \begin{pmatrix} 1 & 0 \\ 0 & \cos i \end{pmatrix} = \frac{1 + \cos i}{2} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + \frac{1 - \cos i}{2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \]

and sandwiching each piece between $R(\Omega)$ and $R(\omega)$ gives a rotation by $\omega + \Omega$ for the first piece, and a reflection involving $\omega - \Omega$ for the second. Carrying this out, we find

\[ \begin{aligned} A + G &= 2p \cos(\omega + \Omega), &\qquad B - F &= 2p \sin(\omega + \Omega), \\ A - G &= 2q \cos(\omega - \Omega), &\qquad B + F &= -2q \sin(\omega - \Omega), \end{aligned} \]

with $$ p = a\,\frac{1 + \cos i}{2}, \quad q = a\,\frac{1 - \cos i}{2} . $$

In summary:

the sum $\omega + \Omega$ is the angle of the rotation part,
the difference $\omega - \Omega$ is the angle of the reflection part
the sizes $p, q$ of the two parts carry $a$ and $i$.

Semi-major axis and inclination¶

The sizes $p$ and $q$ can be read off from $\mathbf{M}$, and since $a = p + q$ and $\cos i = (p - q)/(p + q)$, we can obtain both $a$ and $i$. Binnendijk (1960)⁶ arranges the calculation through the intermediate variables

\[ u = \frac{A^2 + B^2 + F^2 + G^2}{2} \qquad \text{and} \qquad v = AG - BF . \]

These have a simple meaning: $u = p^2 + q^2$ and $v = p^2 - q^2 = a^2 \cos i$ (the determinant of $\mathbf{M}$, which is unchanged by the rotations and equals $a^2\cos i$ because of the tilt).

Then $(u + v)(u - v) = 4p^2q^2$, so that $u + \sqrt{(u+v)(u-v)} = (p+q)^2 = a^2$, and

\[ \boxed{ a = \sqrt{u + \sqrt{(u + v)(u - v)}} } \qquad\qquad \boxed{ i = \arccos \frac{v}{a^2} } \]

Note that the sign of $v$ (the orientation of the mapping) tells whether the motion is direct ($v > 0$) or retrograde ($v < 0$), and the $\arccos$ returns $i \in [0, \pi]$ as required.

Numerical considerations

The expression for $i$ is exact but ill-conditioned for nearly face-on orbits ($i \approx 0$ or $\pi$), because $\arccos$ is very flat near $\pm 1$ and rounding can even push $v/a^2$ outside $[-1, 1]$. Using $p$ and $q$ directly:

\[ \begin{aligned} p = \tfrac12\sqrt{(A+G)^2 + (B-F)^2} \qquad & a = p + q \\ q = \tfrac12\sqrt{(A-G)^2 + (B+F)^2} \qquad & i = 2\arctan\sqrt{q/p} \end{aligned} \]

avoids this. Here, the expression for $i$ follows from $\tan^2(i/2) = (1-\cos i)/(1+\cos i) = q/p$.
Halbwachs et al. (2023) use a similar half-angle formulation.

The angles $\omega$ and $\Omega$¶

The angles are obtained from the two angles of the rotation and reflection parts:

\[ \begin{aligned} \omega + \Omega &= \operatorname{atan2}\big(B - F,\; A + G\big) \\ \omega - \Omega &= \operatorname{atan2}\big(-(B + F),\; A - G\big) \end{aligned} \]

which are the angles of the vectors $(A+G,\, B-F) = 2p\,(\cos(\omega+\Omega), \sin(\omega+\Omega))$ and $(A-G,\, -(B+F)) = 2q\,(\cos(\omega-\Omega), \sin(\omega-\Omega))$. Since $p$ and $q$ are non-negative, the signs of the two components fix the quadrant unambiguously.

Beware of the arctan and the sign

Binnendijk's formulas are often written as $\omega + \Omega = \arctan\frac{B-F}{A+G}$ and $\omega - \Omega = \arctan\frac{B+F}{G-A}$. These are correct as ratios (they equal the tangent of the angle), but a plain $\arctan$ discards the quadrant, so these are only preliminary estimates; the quadrant must then be fixed from the signs of the numerator and denominator (Halbwachs et al. 2023). In the second expression both $B+F$ and $G-A$ carry the opposite sign to $\sin(\omega-\Omega)$ and $\cos(\omega-\Omega)$ respectively, so naively passing them to atan2 gives an angle that is off by $\pi$. Using the forms above, the signs are already absorbed.

The two angles are then recovered as

\[ \boxed{\omega = \frac{(\omega+\Omega) + (\omega-\Omega)}{2}} \qquad\qquad \boxed{\Omega = \frac{(\omega+\Omega) - (\omega-\Omega)}{2}} \]

The $\pi$ ambiguity¶

The sum $\omega + \Omega$ and the difference $\omega - \Omega$ are only known modulo $2\pi$, so $\omega$ and $\Omega$ are only known modulo $\pi$, and not independently: shifting both by $\pi$, $$ (\omega, \Omega) \;\longrightarrow\; (\omega + \pi,\; \Omega + \pi), $$ leaves $A$, $B$, $F$ and $G$ unchanged (every term in their definitions is a product of one function of $\omega$ and one of $\Omega$, and each flips sign). Geometrically, this is the freedom of choosing which end of the line of nodes is called the ascending node: swapping the two nodes changes $\Omega$ by $\pi$, and since $\omega$ is measured from the ascending node, it changes by $\pi$ too.

The TI elements cannot resolve this degeneracy, so the convention $\Omega \in [0,\pi]$ is used. In practical calculations: since atan2 returns angles in $(-\pi, \pi]$, the half-difference above lies in $(-\pi, \pi]$. If it is negative, we add $\pi$ to both $\Omega$ and $\omega$. Finally, we wrap $\omega$ into $[0, 2\pi)$ by adding or subtracting $2\pi$ as needed.

Summary of the algorithm¶

Compute $p$, $q$ (or $u$, $v$) from $A, B, F, G$, then $a$ and $i$.
Compute $\omega + \Omega = \operatorname{atan2}(B - F, A + G)$ and $\omega - \Omega = \operatorname{atan2}(-(B + F), A - G)$.
Take the half-sum and half-difference to get $\omega$ and $\Omega$.
If $\Omega < 0$, add $\pi$ to both $\Omega$ and $\omega$; then wrap $\omega$ to $[0, 2\pi)$.

Degenerate cases¶

Face-on orbits ($i = 0$ or $\pi$): either $q$ or $p$ vanishes, so one of the two angles above is undefined (0/0). Only $\omega + \Omega$ (for $i=0$) or $\omega - \Omega$ (for $i=\pi$) is determined, and $\omega$ and $\Omega$ individually are meaningless. For near face-on orbits, their uncertainties will be large.
Circular orbits ($e \to 0$): the periastron is undefined, and so is $\omega$ (as well as $T_p$). This is a property of the orbit itself and is independent of the conversion, but it makes the TI elements and the derived $\omega$ poorly constrained.

Combining with radial velocities¶

The $\Omega \in [0, \pi]$ convention above is a consequence of using astrometry alone. In the radial-velocity convention, the ascending node is the one where the star moves away from the observer, and this removes the ambiguity: $\Omega$ can then take any value in $[0, 2\pi)$.

When a combined astrometric + RV solution is available, the astrometric $(\omega, \Omega)$ and the spectroscopic $\omega_1$ must be reconciled by choosing, between $(\omega, \Omega)$ and $(\omega + \pi, \Omega + \pi)$, the pair that is consistent with $\omega_1$. Which one is correct also depends on whether the photocentre follows the star whose velocities are measured (then $\omega \approx \omega_1$) or its companion (then $\omega \approx \omega_1 \pm \pi$), see Appendix B of Halbwachs et al. (2023). This is a piece of information that neither dataset contains on its own.

Orbital playground¶

The figure below lets you explore the conversion between Thiele-Innes and Campbell elements, the degeneracy that arises with astrometry alone, and how radial velocities resolve it.

An interactive version is available at the kima documentation.

thiele_innes_widget

Implementation Notes¶

In kima, the conversions from Thiele-Innes to Campbell elements (and vice versa) are implemented in two functions from kima.pykima.analysis:

kima.pykima.analysis.campbell_to_thiele_innes(a, i, Ω, ω)
kima.pykima.analysis.thiele_innes_to_campbell(A, B, F, G, ω_ref=None)

The latter function will resolve the ω ambiguity when a reference value is provided, e.g., from a combined astrometry + RV fit.

Acknowledgements¶

The author used Claude (Sonnet 5.5, Anthropic) to assist in the preparation of this note: to review the mathematical derivations and the reference list, to suggest the geometric interpretation of the inversion, to proofread the text, and to write the interactive figure. The author checked the content and takes responsibility for any remaining errors.

References¶

Thiele, T. N. Neue Methode zur Berechung von Doppelsternbahnen. AN 104, 245 (1883)
NASA ADS, DOI ↩
Innes, R. Orbital Elements of Binary Stars. Circular of the Union Observatory Johannesburg, 68, 355 (1926)
NASA ADS
(note: this is the same reference as 3. below; Innes wrote Sections 2 and 4 of that article.) ↩
van den Bos. Orbital Elements of Binary Stars. Circular of the Union Observatory Johannesburg, 68, 352 (1926)
NASA ADS ↩
Halbwachs et al. Gaia Data Release 3: Astrometric binary star processing. A&A 674, A9 (2023)
arXiv, DOI ↩↩
Heintz, W. D. Double Stars. Geophysics and Astrophysics Monographs (1978)
NASA ADS ↩
Binnendijk, L. Properties of Double Stars: A Survey of Parallaxes and Orbits. University of Pennsylvania Press (1960)
NASA ADS (ISBN10: 125844464X, ISBN 13:9781258444648) ↩↩↩