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Appendix A: General error propagation

Although the propagation of errors is discussed in many textbooks (see, for example, Brandt 1999; Bevington & Robinson 2003), we find it instructive for the subsequent discussion to give a brief exposition of this technique.

In the context of the error propagation, it is convenient to represent the astrometric parameters by a vector a of length 6. All information on the standard errors, σ, of the parameters and correlations between them is contained in the 6 × 6 variance-covariance matrix C with the elements of the latter being (A.1)where ρ_ik is the correlation coefficient of ith and kth parameter.

If vector of the parameters a₀ undergoes a transformation giving new vector, a = f⁽a₀⁾, small variations in the parameters are related as (A.2)In matrix form this can be written: (A.3)where J is the Jacobian matrix of the transformation: (A.4)evaluated at the point a₀. Now let Δa be the difference between the estimated and true parameter vectors. If the estimate is unbiased, then E(Δa₀) = 0, where E is the expectation operator, and the covariance matrix of a₀ is given by , with the prime denoting matrix transposition. It follows from (A.3) that a is also unbiased, to the first order in the errors, and that its covariance is given by (A.5)This equation is the basis for the error propagation discussed below.

If the inverse function f^-1 exists, then it is possible to transform the data set [a,C] back to the original form , and the two representations can be regarded as equivalent from the point of view of information content. A necessary condition for this is that |J_f| ≠ 0, in which case . The transformations discussed here satisfy this condition.

A simple example:

to illustrate the general error propagation technique using the Jacobian, we give below some very simplified formulae. We emphasize that they should not be used for actual calculations, but are only given for illustration. The simplistic formulae for transforming a celestial position over the epoch difference t are (A.6)It is useful to point out that in this equation the proper motion in right ascension does not contain the factor cosδ. This is not a good physical model of how the stars move on the sky: in general it describes a curved, spiralling motion towards one of the poles, whereas real (unperturbed) stars are expected to move along great-circle arcs. Although the difference with respect to the rigorous model (Sect. 5) is often very small, it becomes significant over long time intervals or for stars near the celestial poles. In this model, the changes in the proper motion components and in the parallax are neglected and the Jacobian matrix for the epoch transformation is then: (A.7)The inverse transformation is obtained by reversing the sign of t. It is easily verified that the resulting matrix is indeed the inverse of Eq. (A.7).

The covariance matrix for the six astrometric parameters at epoch t are obtained from (A.5); this yields in particular for the variances in position: (A.8)(with all quantities in the right members referring to the initial epoch). Here the notation means the coefficient of correlation between the astrometric parameters x and y.

Finally, let us consider an extreme case of very large epoch difference. Putting formally t → ∞, we find that (A.9)while σ_ϖ, σ_μα, σ_μδ, and σ_μr are unchanged. Similarly, after direct calculations, we obtain the limiting forms of all nine correlation coefficients affected by the transformation: (A.10)Although the terms in the right-hand sides of Eqs. (A.9) and (A.10) refer to the initial epoch, we do not show it explicitly because these quantities remain unchanged.

Thus all information about initial covariances of the positions becomes less significant as the epoch difference increases and vanishes in the long run. Similar arguments hold for the rigorous propagation, except that they cannot be demonstrated so easily.

Initialization of C₀:

the initial covariance matrix C₀ must be specified in order to calculate the covariance matrix of the propagated astrometric parameters C. Available astrometric catalogues seldom give the correlations between the parameters, nor do they usually contain radial velocities. Absence of the correlations does not create any problems for the error propagation since all the off-diagonal elements of C₀ are just set to zero, but the radial velocity is crucial for the rigorous propagation. While the Hipparcos and Tycho catalogues provide the complete first five rows and columns of C₀, this matrix must therefore be augmented with a sixth row and column related to the initial radial proper motion μ_r0. If the initial radial velocity v_r0 has the standard error σ_vr0 and is assumed to be statistically independent of the astrometric parameters in the catalogue, then the required additional elements in C₀ are (A.11)(ESA 1997; Michalik et al. 2014). If the radial velocity is not known, it is recommended that v_r0 = 0 is used, together with an appropriately large value of σ_vr0 (set to, for example, the expected velocity dispersion of the stellar type in question), in which case [C₀]₆₆ in general is still positive. This means that the unknown perspective acceleration is accounted for in the uncertainty of the propagated astrometric parameters.

It should be noted that strict reversal of the transformation (from T to T₀), according to the standard model of stellar motion, is only possible if the full six-dimensional parameter vector and covariance is considered.

Appendix B: Elements of the Jacobian matrix

This appendix gives explicit formulae for the 36 partial derivatives constituting the Jacobian matrix needed to calculate the covariance matrix of the propagated parameters according to Eq. (A.5). In what follows, we introduce symbols χ to designate the partial derivatives of the logarithm of the velocity factor: It can be seen from Eq. (74) that Similarly, the logarithmic differential of the time factor can be written as where, as it follows from Eqs. (76)–(78), The quantities X, Y, and Z are defined by the Eqs. (75), (54) and (55). We give below, for reference, these quantities explicitly: It is convenient to eliminate dlnf_D and dlnf_V from the expressions for the differentials of the proper motions (68) and (70), replacing them by dlnf_T and the differentials of the astrometric parameters. To simplify following formulae, we introduce special designations for the coefficients of dlnf_T in dμ and dμ_r, respectively: We, moreover, show how the partial derivatives of the propagated positions with respect to the initial radial proper motion may be expressed in terms of the propagated proper motions. As it has been noted, the term proportional to the propagated barycentric position u in Eq. (69) is not significant because it is normal to both p and q. However, keeping the first item in this term, u dlnf_D, and using the Eq. (72) for dlnf_D, we can write the derivative as Substituting Eq. (44) for u and making use of the propagation of the proper motion given by Eq. (86), we find that Taking the dot products with p and q, we finally get the formulae for J₁₆ and J₂₆ given below, respectively.

The elements of the Jacobian matrix are given hereafter.

Appendix C: Elements of the Jacobian matrix neglecting light-time effects

This appendix gives explicit formulae for the 36 partial derivatives constituting the Jacobian matrix of the propagated astrometric parameters for the case when light-time effects are not taken into account. The following formulae can be obtained either by a direct differentiation of the corresponding equations in Sect. 5.5, or more easily by putting f_T = f_V = 1 and τ_A = 0 in the derivatives in Appendix B. The elements given below are equivalent to the elements given in Vol. 1, Sect. 1.5.5 of the Hipparcos and Tycho catalogues (ESA 1997). In that publication, the radial proper motion μ_r is denoted ζ, and the distance factor f_D is denoted f.

Appendix D: Approximate formulae for the light-time effects

In this appendix we derive approximate formulae for the effects of the light-time on the propagated astrometric parameters. These formulae should not be used for the actual propagation, but only to estimate the significance of the effects.

It is clear from Sect. 5.3 that the light-time effects are determined by the scaling factors in time and velocity, f_T and f_V. Since these factors are very close to unity, it is useful to introduce two small quantities, ε_T and ε_V, which can be regarded as small parameters of the employed formalism: (D.1)Since ε_T and ε_V are zero at t = 0, it is convenient to represent them as an explicit functions of time. Expanding Eqs. (54) and (57) in a Taylor series in time and keeping the first-order terms, we find that (D.2)i.e. ε_V = 2ε_T.

As the next step, we express the propagated astrometric parameters as linear functions of ε_T and ε_V by a series expansion to the first order. We denote the approximate quantities calculated neglecting the light-time effects, that is for ε_T = ε_V = 0, with a tilde. Substituting f_T from Eq. (D.1) to the definition of the distance factor (43), we get (D.3)It follows from Eq. (44) that the propagated barycentric position can be written as (D.4)and formula (45) gives the propagated parallax (D.5)Expansion of the product , which appears in the formula of the propagated proper motion (50), to the first order in ε_T and ε_V gives . Since ε_T and ε_V are of the same order-of-magnitude, and , we can omit the second term to get (D.6)The propagated proper motions then become Putting in Eqs. (D.4), (D.5), (D.7), and (D.8), we readily obtain the effects of the light-time on the astrometric parameters (D.9)We note the following relations between the effects (D.10)It is instructive to express the effects in terms of the physical parameters, including the effects in velocity: (D.11)These relations lead to important conclusions about the behaviour of the effects. The effects on the position and parallax are quadratic functions of time, while the effects on the proper motion and velocity increases linearly with time. This confirms the conclusion drawn from the numerical calculations shown in Fig. 2. All the effects are roughly proportional to the third power of the space velocity, while the dependence on distance is different for the velocities (b^-1), position and proper motions (b^-2), and parallax (b^-3).

For the practical estimation of the effects we give the following formulae for the position, and velocity,

Appendix E: Applicability of the uniform rectilinear model

In this appendix we briefly consider the conditions under which stellar motion may be regarded as uniform. A uniform motion implies absence of acceleration. In practice, however, accelerated motion may be treated as uniform if observable effects of the acceleration are negligible compared to the required astrometric accuracy. The effect of a constant acceleration a on the barycentric position of a star during a timespan t is Δb ≃ at²/ 2. The corresponding change in the angular position θ of the star is Δθ ≃ a_⊥t²/ (2b), where a_⊥ is the tangential component of the acceleration. The motion may be regarded as uniform if | Δθ | ≪ σ_θ, the required astrometric accuracy in angular position after time t. For the proper motion, we similarly have the condition | Δμ | ≪ σ_μ, where Δμ ≃ a_⊥t/b. The former (positional) criterion is usually stricter since t is typically much greater than 2σ_θ/σ_μ.

The acceleration along the line of sight, a_∥ (taken to be positive when directed away from the SSB), causes a change in parallax by Δϖ ≃ − Aa_∥t²/ (2b²), where A is the astronomical unit, and in radial velocity by Δv_r ≃ a_∥t. If a_⊥ and a_∥ are of similar magnitudes, we find that | Δϖ | is smaller than | Δθ | by a factor A/b ≪ 1, so the effect in parallax is never a limitation. On the other hand, under fairly realistic assumptions it may happen that the acceleration effect is more important in radial velocity than in position.

We do not consider here the acceleration caused by stellar or planetary companions, which affects specific objects in a very specific way and may be very important. Indeed, as emphasized in the introduction, one of the objectives of the uniform rectilinear hypothesis is precisely to enable the detection of such cases. Rather, we need to consider accelerations that affect all, or most of, the stars and which could therefore potentially render the model invalid as a general basis for high-precision astrometric analyses. The most important such acceleration is caused by the large-scale gravitational field of the Galaxy, i.e. the curvature of Galactic stellar orbits.

At the arbitrary point b in the Galaxy (relative to the SSB) the acceleration vector can be estimated as a = −∇ψ, where ψ is some suitable model of the Galactic potential (Binney & Tremaine 2008). It should be recalled that the uniform rectilinear model refers to the motion of stars relative to the SSB, and that the SSB itself is subject to some acceleration a(0). The observable effects must therefore be evaluated for the differential acceleration Δa = a(b) − a(0), and the quantities a_∥ and a_⊥ discussed above are therefore the components of Δa along and perpendicular to the line of sight⁶. In a smooth potential both components vanish as b → 0.

Rather than using a (rather uncertain) global potential model, however, it is more illuminating to analyse the differential effects based on a few relatively well-determined structural Galactic parameters. We assume an axisymmetric potential in galactocentric cylindrical coordinates (R,z) and consider separately the acceleration components in the Galactic plane (along R) and perpendicular to it (along z). To avoid confusion with the b denoting a star’s distance from the SSB, we subsequently use B to denote Galactic latitude, and L for the longitude.

Acceleration in the Galactic plane:

in the axisymmetric approximation the acceleration in the Galactic plane is directed towards the Galactic centre and of magnitude a = V(R) /R², where V(R) denotes the circular velocity as radial distance R. The Sun is currently located close to the Galactic plane at a radius R₀ ≃ 8.4 kpc from the Galactic centre, where the circular velocity is V₀ ≡ V(R₀) ≃ 254 km s^-1 (Reid et al. 2009). The expected acceleration at the location of the Sun is therefore m s^-2.

The effects in position and proper motion are proportional to a_⊥/b, which in a smooth potential become distance-independent for sufficiently small b, that is in the solar neighbourhood. It is interesting to derive the corresponding local approximations for the acceleration components. This can be done in complete analogy with the well-known derivation of the Oort formulae for the radial and tangential velocities of circular motions in terms of the Oort constants A and B (e.g. Binney & Merrifield 1998). With a(R) denoting the acceleration towards the Galactic centre at radius R, we find (E.1)where L is the Galactic longitude of the star, as seen from the Sun, and (E.2)are constants analogous to A and B in the Oort formulae. Using a(R) = V(R) /R² we can in fact express E and F in terms of the Oort constants as (E.3)In Eq. (E.1) we take a_⊥ to be positive in the direction of increasing L. Since the Galactic rotation curve is nearly flat (A + B = 0), we have F ≃ 0 and , where Ω₀ = V₀/R₀ ≃ 9.8 × 10^-16 s^-1 is the circular angular velocity at the Sun. In the solar neighbourhood, the effect of the curvature of Galactic orbits on the position after time t can therefore be estimated as (E.4)thus negligible at the 1 μas precision for time intervals up to 100 yr. The corresponding effect on the radial velocity is (E.5)where we have again assumed a flat rotation curve.

Beyond the solar neighbourhood, e.g. at distances of the order of R₀ from the Sun, the differential acceleration is of the order of the solar acceleration, or . The astrometric effects, being proportional to , are therefore of the same order of magnitude as computed above for the solar neighbourhood.

Acceleration perpendicular to the Galactic plane:

in the solar neighbourhood, the component of the acceleration perpendicular to the Galactic plane, at distance z above the plane, is approximately given by a(z) = −2πGΣ(z), where is the surface density within ± z of the Galactic plane. Within a few hundred pc from the Sun we can assume an approximately constant mass density ρ₀, yielding a(z) ≃ − Kz where K = 4πGρ₀ is the square of the angular frequency of the oscillations in z. The acceleration relative to the SSB follows the same formula if z is interpreted as the vertical coordinate of the star relative to the Sun, that is, z = bsinB. For the components of a(z) perpendicular to and along the line of sight we readily find (E.6)where a_⊥ is positive in the direction of increasing B. Using ρ₀ ≃ 0.1 M_⊙ pc^-3 (Holmberg & Flynn 2000), we have K ≃ 5.7 × 10^-30 s^-2, and the accumulated effects in position and radial velocity after time t can then be estimated as (E.7)and (E.8)

These approximations are valid for distances b up to a few hundred pc, beyond which the effects may be considerably smaller.

The effects of the acceleration perpendicular to the Galactic plane are therefore more important than the radial acceleration, which simply reflects the shorter oscillation period in the z direction, 2πK^{− 1/2} ≃ 84 Myr, compared to the circular period 2π/ Ω₀ ≃ 200 Myr. However, the general conclusion is that Galactic accelerations are negligible at micro-arcsecond accuracy over time periods of at least 50 yr.

© ESO, 2014