Gaia Data Release 1 Documentation release 0
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is a fixed orthogonal matrix (the transpose of the matrix A G defined in Vol. 1, Eq. 1.5.11 of ESA 1997). R i (θ) is the 3 × 3 matrix representing a rotation of the coordinate frame by the angle θ about axis i. Since A G is orthogonal, the inverse transformation to Equation 3.9 is r ICRS = A G r Gal . (3.12) Given (α, δ), application of Equation 3.7 and Equation 3.9 gives the galactic position in Cartesian coordinates. Some care should be exercised when converting the Cartesian coordinates to spherical (l, b) in order to avoid quadrant ambiguity and numerical inaccuracy near the poles. Recommended formulae (e.g. Ch. 4 in van Altena 2012) use the four-quadrant inverse tangent (atan2 or similar) available in all high-level programming languages: l = atan2(Y Gal , X Gal ) , b = atan2 Z Gal , X 2 Gal + Y 2 Gal . (3.13) Note that Equation 3.13 works also for vectors that are not of unit length. 3.1.7.1.2 Transformation of proper motion The transformation of the proper motion components (µ α∗ , µ δ ) to (µ l∗ , µ b ) (where µ l∗ = µ l cos b) requires the use of the four auxiliary column matrices p ICRS = − sin α cos α 0 , q ICRS = − cos α sin δ − sin α sin δ cos δ (3.14) and p Gal = − sin l cos l 0 , q Gal = − cos l sin b − sin l sin b cos b . (3.15) Geometrically, p ICRS and q ICRS represent unit vectors in the directions of increasing α and δ, respectively, expressed by their Cartesian components in ICRS. Similarly, p Gal and q Gal are unit vectors in the directions of increasing l and b, respectively, expressed by their Cartesian components in the galactic system. The Cartesian components of the so-called proper motion vector can now be written in ICRS as µ ICRS = p ICRS µ α∗ + q ICRS µ δ , (3.16) and in the galactic system as µ Gal = p Gal µ l∗ + q Gal µ b . (3.17) These column matrices transform exactly as any other Cartesian vector, namely µ Gal = A G µ ICRS (3.18) and µ ICRS = A G µ Gal . (3.19) Applying Equation 3.14, Equation 3.16, and Equation 3.18 therefore gives the Cartesian proper motion vector in the galactic system, from which the components along l are b are obtained by means of Equation 3.17, using the orthogonality of p Gal and q Gal : µ l∗ = p Gal µ Gal , µ b = q Gal µ Gal . (3.20) For completeness we give also the corresponding calculation in ICRS: µ α∗ = p ICRS µ ICRS , µ δ = q ICRS µ ICRS . (3.21) 159 3.1.7.1.3 Error propagation The statistical errors associated with the astrometric parameters α, δ, , µ α∗ , and µ δ are given by the standard uncertainties σ α∗ , σ δ , σ , σ µα∗ , and σ µδ together with the correlation coe fficients ρ(α, δ), ρ(α, ), etc. For notational convenience we may number the five parameters 0, 1, . . . , 4; thus σ 0 = σ α∗ , ρ 01 = ρ(α, δ), etc. Let e ≡ e 0 e 1 e 2 e 3 e 4 = ∆α∗ ∆δ ∆ ∆µ α∗ ∆µ δ (3.22) be a vector containing the errors, that is the di fferences between the measured and true astrometric parameters, expressed in mas or mas yr −1 (with ∆α∗ = ∆α cos δ). The measurements are assumed to be unbiased, so the expectation of the error vector is E [e] = 0 (3.23) and its covariance is the symmetric positive definite matrix C = E ee = E [e 0 e 0 ] E [e 0 e 1 ] · · · E [e 0 e 4 ] E [e 1 e 0 ] E [e 1 e 1 ] · · · E [e 1 e 4 ] .. . .. . ... .. . E [e 4 e 0 ] E [e 4 e 1 ] · · · E [e 4 e 4 ] (3.24) with diagonal elements C ii = σ 2 i and o ff-diagonal elements V i j = σ i σ j ρ i j (with ρ ji = ρ i j ). The transformation from ICRF to galactic coordinates is a strongly non-linear function. However, the errors e i are very small and the error vector in galactic coordinates g ≡ g 0 g 1 g 2 g 3 g 4 = ∆l∗ ∆b ∆ ∆µ l∗ ∆µ b (3.25) is therefore obtained by the linear transformation g = Je , (3.26) where J = ∂(l∗, b, , µ l∗ , µ b ) ∂(α∗, δ, , µ α∗ , µ δ ) = ∂l∗ ∂α∗ ∂l∗ ∂δ · · · ∂l∗ ∂µ δ ∂b ∂α∗ ∂b ∂δ · · · ∂b ∂µ δ .. . .. . ... ... ∂µ b ∂α∗ ∂µ b ∂δ · · · ∂µ b ∂µ δ (3.27) is the Jacobian of the transformation. (The cos b or cos δ factor implied by the asterisk is never di fferentiated; thus ∂l∗/∂α∗ = (∂l/∂α) cos b/cos δ, etc.) Clearly E g = JE [e] = 0, so the galactic parameters are also unbiased, with covariance matrix C Gal = E gg = JCJ . (3.28) It remains to determine J. Since is unchanged by the transformation we have J 22 = 1 and J i 2 = J 2i = 0 for i 2. It is also readily seen that the proper motion errors transform in the same way as the positional errors (if we regard p and q as fixed and not subject to errors); then from Equation 3.16–Equation 3.20 we find J = G 0 0 0 1 0 0 0 G (3.29) 160 Yüklə 5,01 Kb. Dostları ilə paylaş:
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