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Appendix A: Scaled modelling of kinematics (SMOK)

A formalism called SMOK is introduced in this paper to facilitate a rigorous manipulation of small (differential) quantities in the celestial coordinates. It is reminiscent of the “standard” or “tangential” coordinates in classical small-field astrometry (e.g., Murray 1983; van Altena 2013), using a gnomonic projection onto a tangent plane of the (unit) celestial sphere, but extends to three dimensions by adding the radial coordinate perpendicular to the tangent plane. This simplifies the modelling of perspective effects.

Figure A.1 illustrates the concept. In the vicinity of the star let c be a comparison point fixed with respect to the solar system barycentre (SSB). As shown in the diagrams:

• 1.

  The barycentric motion of the star is scaled by the inverse distance to c, effectively placing the star on or very close to the unit sphere.

• 2.

  Rectangular coordinates are expressed in the barycentric [ p_cq_cr_c ] system with r_c pointing towards c, and p_c, q_c in the directions of increasing right ascension and declination.

The first point eliminates the main uncertainty in the kinematic modelling of the star due to its poorly known distance. The second point allows us to express the scaled kinematic model in SMOK coordinates a, d, r that are locally aligned with α, δ, and the barycentric vector.

Fig. A.1 Two steps in the definition of SMOK coordinates. In the top diagram the motion of an object in the vicinity of the fixed point c is modelled by the function b(t) expressed in the barycentric [ xyz ] system. A scaled version of the model is constructed such that the scaled c is at unit distance from the solar system barycentre (SSB). In the bottom diagram new coordinate axes [ pcqcrc ] are chosen in the directions of increasing right ascension, declination, and distance, respectively, at the comparison point (αc,δc) being the projection of c on the unit sphere.

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Up to the scale factor | c | ^-1 discussed below, the SMOK coordinate system is completely defined by the adopted comparison point (α_c,δ_c) using the orthogonal unit vectors (A.1)[ p_cq_cr_c ]is the “normal triad” at the comparison point with respect to the celestial coordinate system (Murray 1983)¹⁰. We are free to choose (α_c,δ_c) as it will best serve our purpose, but once chosen (for a particular application) it is fixed: it has no proper motion, no parallax, and no associated uncertainty. Typically (α_c,δ_c) is chosen very close to the mean position of the star.

The motion of the star in the Barycentric Celestial Reference System (BCRS) is represented by the function b(t), where b is the vector from SSB to the star as it would be observed from the SSB at time t. The scaled kinematic model s(t) = b(t) | c | ^-1 is given in SMOK coordinates as (A.2)and can in turn be reconstructed from the SMOK coordinates as (A.3)a, d, r are dimensionless and the first two are typically small quantities (≲ 10^-4), while r is very close to unity.

The whole point of the scaled kinematic modelling is that s(t) can be described very accurately by astrometric observations, even though b(t) may be poorly known due to a high uncertainty in distance. This is possible simply by choosing the scaling such that | s(t) | = 1 at some suitable time. This works even if the distance is completely unknown, or if it is effectively infinite (as for a quasar).

The scale factor is | c | ^-1 = ϖ_c/A, where ϖ_c is the parallax of c and A the astronomical unit. The measured parallax can be regarded as an estimate of ϖ_c.

In the following we describe some typical applications of SMOK coordinates.

Appendix A.1: Uniform space motion

The simplest kinematic model is to assume that the star moves uniformly with respect to the SSB, that is (A.4)where b_ep is the barycentric position at the reference epoch t_ep, and v is the (constant) space velocity. The scaled kinematic model expressed in the BCRS is (A.5)where (A.6)and (A.7)are constant vectors. The uniform motion can also be written in SMOK coordinates as (A.8)The six constants a(t_ep), d(t_ep), r(t_ep), ȧ, ḋ, ṙ are the kinematic parameters of the scaled model; however, to get the actual kinematics of the star we also need to know ϖ_c.

Appendix A.2: Relation to the usual astrometric parameters

Choosing (α_c,δ_c) to be the barycentric celestial coordinates of the star at t_ep, and ϖ_c equal to the parallax at the same epoch, we find (A.9)where μ_{α ∗}, μ_δ are the tangential components of the barycentric proper motion at the reference epoch t_ep, and μ_r is the “radial proper motion” allowing one to take the perspective effects into account. μ_r is usually calculated from the measured radial velocity and parallax according to Eq. (1).

Appendix A.3: Differential operations

Uniform space motion does not map into barycentric coordinates α(t), δ(t) that are linear functions of time. The non-linearity derives both from the curvilinear nature of spherical coordinates and from perspective foreshortening depending on the changing distance to the object. Both effects are well known and have been dealt with rigorously by several authors (e.g., Eichhorn & Rust 1970; Taff 1981). The resulting expressions are non-trivial and complicate the comparison of astrometric catalogues of different epochs. For example, approximations such as (A.10)cannot be used when the highest accuracy is required. By contrast, the linearity of Eq. (A.8) makes it possible to write (A.11)to full accuracy, provided that the same comparison point is used for both epochs. (Strictly speaking, the same scale factor must also be used, so that in general r(t₂) − r(t₁) = (t₂ − t₁)ṙ ≠ 0.) If the position at the reference epoch coincides with the comparison point used, the resulting ȧ, ḋ are the looked-for proper motion components according to Eq. (A.9); otherwise a change of comparison point is needed (see below).

Appendix A.4: Changing the comparison point

Let (α₁,δ₁) and (α₂,δ₂) be different comparison points with associated triads [ p₁q₁r₁ ] and [ p₂q₂r₂ ]. If a₁(t), d₁(t), r₁(t) and a₂(t), d₂(t), r₂(t) describe the same scaled kinematics we have by Eq. (A.3) (A.12)Thus, given a₁(t), d₁(t), r₁(t) one can compute s(t) from the first equality in Eq. (A.12), whereupon the modified functions are recovered as (A.13)This procedure can be applied to s(t) for any particular t as well as to linear operations on s such as differences and time derivatives.

Appendix A.5: Epoch propagation

An important application of the above formulae is for propagating the six astrometric parameters (α₁,δ₁,ϖ₁,μ_{α ∗ 1},μ_δ1,μ_r1), referring to epoch t₁, to a different epoch t₂. This can be done in the following steps:

• 1.

  Use (α₁,δ₁) as the comparison point and compute [ p₁q₁r₁ ] by Eq. (A.1). At time t₁ the SMOK parameters relative to the first comparison point are a₁(t₁) = d₁(t₁) = 0, r₁(t₁) = 1, ȧ₁ = μ_{α ∗ 1}, ḋ₁ = μ_δ1, ṙ₁ = μ_r1.

• 2.

  Calculate s(t₁) and ṡ using Eqs. (A.6)–(A.7).

• 3.

  Calculate s(t₂) by means of Eq. (A.5). Let s₂ = | s(t₂) | be its length (close to unity).

• 4.

  Calculate r₂ = s(t₂) /s₂ and hence the second comparison point (α₂,δ₂) and triad [ p₂q₂r₂ ].

• 5.

  Use Eq. (A.13) to calculate the SMOK parameters at t₂ referring to the second comparison point. For the position one trivially gets a₂(t₂) = d₂(t₂) = 0 and r₂(t₂) = s₂. For the proper motion parameters one finds , , and .

• 6.

  The astrometric parameters at epoch t₂ are α₂, δ₂, ϖ₂ = ϖ₁/s₂, μ_{α ∗ 2} = ȧ₂/s₂, μ_δ2 = ḋ₂/s₂, μ_r2 = ṙ₂/s₂.

This procedure is equivalent to the one described in Sect. 1.5.5, Vol. 1 of The Hipparcos and Tycho Catalogues (ESA 1997).

Appendix B: The HIPPARCOS Catalogue

This Appendix describes the calculation of relevant quantities from the new reduction of the Hipparcos Catalogue by van Leeuwen (2007b). Data files were retrieved from the Strasbourg astronomical Data Center (CDS) in November 2013 (catalogue I/311). These files differ slightly from the ones given on the DVD published along with the book (van Leeuwen 2007a), both in content and format, as some errors have been corrected. The data needed for every accepted catalogue entry are:

• the five astrometric parameters (α,δ,ϖ,μ_{α ∗},μ_δ);

• the 5 × 5 normal matrix N from the least-squares solution of the astrometric parameters (for a 5-parameter solution this equals the inverse of the covariance matrix C);

• the chi-square goodness-of-fit quantity Q for the 5-parameter solution of the Hipparcos data;

• the degrees of freedom ν associated with Q.

The astrometric parameters at the Hipparcos reference epoch J1991.25 are directly taken from the fields labelled RArad, DErad, Plx, pmRA, and pmDE in the main catalogue file hip2.dat. Units are [rad] for α and δ, [mas] for ϖ, and [mas yr^-1] for μ_{α ∗} and μ_δ. It is convenient to express also positional differences (such as SMOK coordinates a and d) and positional uncertainties in [mas]. The elements of N thus have units [mas^-2 yr^p], where p = 0, 1, or 2, depending on the position of the element in the matrix.

The calculation of N, Q, and ν is described hereafter in some detail as the specification of C deviates in some details from the published documentation. Clarification on certain issues was kindly provided by van Leeuwen (priv. comm.).

The number of degrees of freedom is (B.1)where N_tr is the number of field transits used (label Ntr in hip2.dat) and n is the number of parameters in the solution (see below; most stars have n = 5). The goodness-of-fit given in field F2 is the “gaussianized” chi-square(Wilson & Hilferty 1931)(B.2)computed from Q, the sum of the squared normalized residuals, and ν. For “good” solutions Q is expected to follow the chi-square distribution with ν degrees of freedom (Q ~ χ²(ν)), in which case F₂ approximately follows the standard normal distribution, F₂ ~ N(0,1). Thus, F₂> 3 means that Q is “too large” for the given ν at the same level of significance as the + 3σ criterion for a Gaussian variable (probability ≲ 0.0044)¹¹. Given F₂ from field F2, and ν from Eq. (B.1), it is therefore possible to reconstruct the chi-square statistic of the n-parameter solution as (B.3)We also introduce the square-root of the reduced chi-square, (B.4)which is expected to be around 1.0 for a “good” solution (see further discussion below). u is sometimes referred to as the standard error of unit weight (Brinker & Minnick 1995).

The catalogue gives the covariance matrix in the form of an upper-diagonal “weight matrix” U such that, formally, C = (U′U)^-1. This inverse exists for all stars where a solution is given. (For the joint solution we actually need the normal matrix N = U′U, see below.) For solutions with n = 5 astrometric parameters there are n(n + 1)/2 = 15 non-zero elements in U. For some stars the solution has more than five parameters, and the main catalogue then only gives the first 15 non-zero elements, while remaining elements are given in separate tables.

Let U₁, U₂, ..., U₁₅ be the 15 values taken from the fields labelled UW in hip2.dat. The matrix U is computed as (B.5)Here f_i, i = 1...n, are scaling factors which for the CDS data must be calculated as (B.6)where u is given by Eq. (B.4) and σ_· are the standard errors given in fields e_RArad through e_pmDE of hip2.dat. Equation (B.6) applies to data taken from the CDS version of the catalogue (I/311). For catalogue data on the DVD accompanying the book (van Leeuwen 2007a), scaling factors f_i = 1 apply, although those data are superseded by the CDS version.

The 5 × 5 matrix N = U′U computed using the first five rows and columns in U, as given in Eq. (B.5), contains the relevant elements of the normal matrix for any solution with n ≥ 5. Thus, for solutions with n> 5 there is no need, for the catalogue combination, to retrieve the additional elements of U from hip7p.dat, etc. The situation is different when the covariance matrix is needed: it is then necessary to compute the full n × n normal matrix N before C = N^-1 can be computed.

The normal matrix N computed as described above incorporates the formal uncertainties of the observations; as described in van Leeuwen (2007a) these are ultimately derived from the photon statistics of the raw data after careful analysis of the residuals as function of magnitude, etc. If the adopted models are correct we expect the F₂ statistic to be normally distributed with zero mean and unit standard deviation, and the standard error of unit weight, u, to be on the average equal to 1. In reality we find (for solutions with n = 5) that their distributions are skewed towards higher values, especially for the bright stars where photon noise is small and remaining calibration errors are therefore relatively more important. To account for such additional errors the published standard errors σ_{α ∗}, etc., in hip2.dat include, on a star-by-star basis, a correction factor equal to the unit weight error u obtained in its solution. This is equivalent to scaling the formal standard errors of the data used in the solution by the same factor. In order to make the computed normal matrix, covariance matrix, and goodness-of-fit statistics consistent with the published standard errors it is then necessary to apply the corresponding corrections, viz.: (B.7)For the catalogue combination we use N_corr and Q_corr whenever u> 1, but N and Q if u ≤ 1.

© ESO, 2014