Online material

Appendix A: Apparent orbit and astrometric signature

The astrometric signature of a planetary or brown dwarf companion corresponds to the size of the apparent orbit of the primary star, i.e. on the projection of the true orbit into the tangential plane. Thus, the conversion of the semi-major axes of the true orbit into angular units only provides an upper limit for the observable astrometric signature of any given companion; the true observable astrometric signature might be considerably smaller, depending mainly on the eccentricity, the longitude of the periastron and the inclination. In the most extreme case, the semi-major axis of the apparent orbit corresponds to the semi-minor axis of the true orbit only, and the semi-minor axis of the apparent orbit could even be identical to zero.

The size of the apparent orbit, i.e. its semi-major and semi-minor axis, can be calculated from the orbital elements of the true orbit. Note that most textbooks on double stars cover only the opposite problem, namely the reconstruction of the true orbit from the observed apparent orbit of a visual double star, for which there exist numerous graphical and analytical methods (see e.g. Heintz 1978).

The Kowalsky method as formulated by Smart (1930) uses the following general quadratic equation for a conic section to describe the apparent orbit: (A.1)where the five parameters P, Q, R, S and T are determined by the five orbital elements semi-major axis a, eccentricity e, longitude of periastron ω, ascending node Ω, and inclination i. The equation describes an ellipse if P and Q have the same sign and are not equal. If R does not equal zero the ellipse is rotated in the given coordinate system. If S or T do not equal zero the ellipse is offset from the zero point of the coordinate system. According to Smart (1930), the coefficients in Eq. (A.1) are given by (A.2)where A, B, F and G denote the familiar Thiele-Innes constants: (A.3)

Note that our scaling factor is different from the one used by Smart (1930), because it is convenient for our purposes if the absolute term in Eq. (A.1) equals  − 1.

In order to determine the semi-major and semi-minor axes of the apparent orbit, we perform a principal axes transformation. We determine the eigenvalues λ₁ and λ₂ of the matrix ℳ (A.4)which describes the binary quadratic form A.1, as Inserting the eigenvalues into the characteristic equation, we obtain (A.7)which corresponds to Eq. (A.1) in a rotated and shifted coordinate system. From Eq. (A.7) we immediately determine the semi-major axis a_app and the semi-minor axis b_app of the apparent orbit as (A.8)The astrometric signature α then directly corresponds to the semi-major axis of the apparent orbit, a_app, converted to angular units with the help of the parallax ϖ: (A.9)Astrometric measurements are sometimes only performed in one dimension at a time; examples are Hipparcos or PRIMA (Delplancke 2008). In the case of PRIMA the observing direction is given by the baseline orientation, which is flexible to a certain degree. Thus, it might be advantageous for scheduling purposes to know the direction where most of the astrometric signal can be expected, i.e. the orientation of the apparent orbit. The rotation which leads from Eqs. (A.1) to (A.7) is characterized by the rotation angle ϕ, which can be obtained from (A.10)If the sign of R and P − Q is taken into account, this returns an angle ϕ in the range between 0 and 360 ° corresponding to the position angle of the semi-major axis of the apparent orbit.

© ESO, 2011