3.1.7 Transformations of astrometric data and error propagation

Author(s): Alexey Butkevich, Lennart Lindegren

The Epoch transformation and transformation to galactic coordinates is covered here.

Epoch propagation of prior information

This section contains the description of general procedure for the transformation of the source parameters from one the initial epoch T0 to the arbitrary epoch T. The rigorous treatment of the epoch propagation including the effects of light-travel time was developed by Butkevich and Lindegren (2014). However, for the propagation of the prior information to the Gaia reference epoch, it is sufficient to use the simplified treatment, which was employed in the reduction procedures used to construct the Hipparcos and Tycho catalogues, since the light-time effects are negligible at milli-arcsecond accuracy (ESA 1997, Vol. 1, Sect. 1.5.5).

The epoch propagation is based on the standard astrometric model assuming the uniform rectilinear motion with respect to the solar-system barycentre. In the framework of this model, the barycentric position of a source at the epoch T is

𝒃(T)=𝒃(T0)+(T-T0)𝒗. (3.7)

where 𝒃(T0) is the barycentric position at the initial epoch T0 and 𝒗 the constant space velocity. To simplify the expressions, we use subscript 0 to denote quantities at T0 and the corresponding un-subscripted variables when they refer to the epoch T. Furthermore, the epoch difference t=T-T0 is used as the time argument:

𝒃=𝒃0+t𝒗. (3.8)

The expression for the space velocity in terms of the source parameters reads

𝒗=AVϖ0(𝝁0+𝒓0μr0)=AVϖ0(𝒑0μα*0+𝒒0μδ0+𝒓0μr0), (3.9)

where 𝝁0 is the proper motion at the initial epoch, three unit vector constitute the normal triad [𝒑0,𝒒0,𝒓0] and AV=4.740 470 446 equals the astronomical unit expressed in kmyrs-1. This relation implies that the parallax and proper motions are expressed in compatible units, for instance, mas and mas yr-1, respectively.

Propagation of the source parameters

The propagation of the barycentric direction 𝒖=𝒃/b is given by Equation 3.8. Squaring both sides of Equation 3.8 and making use of obvious relations 𝒃0𝒗0=b02μr0 and v02=b02(μr02+μ02), we find

b2=b02(1+2μr0t+(μ02+μr02)t2), (3.10)

where μ02=μα*02+μδ02. Introducing the distance factor

f=b0/b=[1+2μr0t+(μ02+μr02)t2]-1/2, (3.11)

the propagation of the barycentric direction is

𝒖=[𝒓0(1+μr0t)+𝝁0t]f (3.12)

and the propagation of the parallax becomes

ϖ=ϖ0f. (3.13)

The celestial coordinates (α, δ) at epoch T are obtained from 𝒖 in the usual manner and the normal triad associated with the propagated direction is

[𝒑,𝒒,𝒓]=[-sinα-sinδcosαcosδcosαcosα-sinδsinαcosδsinα0cosδsinδ]. (3.14)

Direct differentiation of Equation 3.12 gives the propagated proper motion vector:

𝝁=d𝒖dt=[𝝁0(1+μr0t)-𝒓0μ02t]f3, (3.15)

and the propagated radial proper motion is found to be

μr=dbdtϖA=[μr0+(μ02+μr02)t]f2. (3.16)

To obtain the proper motion components (μα*, μδ) from vector 𝝁 it is necessary to resolve the latter along the tangential vectors 𝒑 and 𝒒 at the propagate direction:

μα*=𝒑𝝁,μδ=𝒒𝝁. (3.17)

The tangential vectors are defined in terms of the propagated 𝒖 or (α, δ) at the epoch T according to Equation 3.14.

The above formulae describe the complete transformation of (α0, δ0, ϖ0, μα*0, μδ0, μr0) at epoch T0 into (α, δ, ϖ, μα*, μδ, μr) at the arbitrary epoch T=T0+t. The transformation is rigorously reversible: a second transformation from T to T0 recovers the original six parameters.

Propagation of errors (covariances)

The uncertainties in the source parameters α, δ, ϖ, μα*, μδ, μr and correlations between them are quantified by means of the 6×6 covariance matrix 𝑪 in which the rows and columns correspond to the parameters taken in the order given above. The general principle of (linearised) error propagation is well known and briefly summarized below. The covariance matrix 𝑪0 of the initial parameters and the matrix 𝑪 of the propagated parameters are related as

𝑪=𝑱𝑪0𝑱, (3.18)

where 𝑱 is the Jacobian matrix of the source parameter transformation:

𝑱=(α,δ,ϖ,μα*,μδ,μr)(α0,δ0,ϖ0,μα*0,μδ0,μr0) (3.19)

Thus, the propagation of the covariances requires the calculation of all 36 partial derivatives constituting the Jacobian 𝑱.

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 μr0. If the initial radial velocity vr0 has the standard error σvr0 and is assumed to be statistically independent of the astrometric parameters in the catalogue, then the required additional elements in 𝑪0 are

[C0]i6=[C0]6i=[C0]i3(vr0/A),i=15, (3.20)

(Michalik et al. 2014). If the radial velocity is not known, it is recommended that vr0=0 is used, together with an appropriately large value of σvr0 (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.

The elements of the Jacobian matrix are given hereafter:

J11 =α*α*0=𝒑𝒑0(1+μr0t)f-𝒑𝒓0μα*0tf (3.21)
J12 =α*δ0=𝒑𝒒0(1+μr0t)f-𝒑𝒓0μδ0tf (3.22)
J13 =α*ϖ0=0 (3.23)
J14 =α*μα*0=𝒑𝒑0tf (3.24)
J15 =α*μδ0=𝒑𝒒0tf (3.25)
J16 =α*μr0=-μα*t2 (3.26)
J21 =δα*0=𝒒𝒑0(1+μr0t)f-𝒒𝒓0μα*0tf (3.27)
J22 =δδ0=𝒒𝒒0(1+μr0t)f-𝒒𝒓0μδ0tf (3.28)
J23 =δϖ0=0 (3.29)
J24 =δμα*0=𝒒𝒑0tf (3.30)
J25 =δμδ0=𝒒𝒒0tf (3.31)
J26 =δμr0=-μδt2 (3.32)
J31 =ϖα*0=0 (3.33)
J32 =ϖδ0=0 (3.34)
J33 =ϖϖ0=f (3.35)
J34 =ϖμα*0=-ϖμα*0t2f2 (3.36)
J35 =ϖμδ0=-ϖμδ0t2f2 (3.37)
J36 =ϖμr0=-ϖ(1+μr0t)tf2 (3.38)
J41 =μα*α*0=-𝒑𝒑0μ02tf3-𝒑𝒓0μα*0(1+μr0t)f3 (3.39)
J42 =μα*δ0=-𝒑𝒒0μ02tf3-𝒑𝒓0μδ0(1+μr0t)f3 (3.40)
J43 =μα*ϖ0=0 (3.41)
J44 =μα*μα*0=𝒑𝒑0(1+μr0t)f3-2𝒑𝒓0μα*0tf3-3μα*μα*0t2f2 (3.42)
J45 =μα*μδ0=𝒑𝒒0(1+μr0t)f3-2𝒑𝒓0μδ0tf3-3μα*μδ0t2f2 (3.43)
J46 =μα*μr0=𝒑[𝝁0f-3𝝁(1+μr0t)]tf2 (3.44)
J51 =μδα*0=-𝒒𝒑0μ02tf3-𝒒𝒓0μα*0(1+μr0t)f3 (3.45)
J52 =μδδ0=-𝒒𝒒0μ02tf3-𝒒𝒓0μδ0(1+μr0t)f3 (3.46)
J53 =μδϖ0=0 (3.47)
J54 =μδμα*0=𝒒𝒑0(1+μr0t)f3-2𝒒𝒓0μα*0tf3-3μδμα*0t2f2 (3.48)
J55 =μδμδ0=𝒒𝒒0(1+μr0t)f3-2𝒒𝒓0μδ0tf3-3μδμδ0t2f2 (3.49)
J56 =μδμr0=𝒒[𝝁0f-3𝝁(1+μr0t)]tf2 (3.50)
J61 =μrα*0=0 (3.51)
J62 =μrδ0=0 (3.52)
J63 =μrϖ0=0 (3.53)
J64 =μrμα*0=2μα*0(1+μr0t)tf4 (3.54)
J65 =μrμδ0=2μδ0(1+μr0t)tf4 (3.55)
J66 =μrμr0=[(1+μr0t)2-μ02t2]f4 (3.56)

Galactic coordinates

The positions and proper motions of non-solar system objects derived from Gaia observations are expressed in the International Celestial Reference System (ICRS). This is an inertial (non-rotating) reference system, which since 1998 replaces the various earlier celestial reference frames (referred to by names such as FK5, FK4, J2000, B1950, equinox and equator of 1950.0, etc.).

For galactic research it is often desirable to use galactic coordinates instead of ICRS. Unfortunately there is no accurate transformation from ICRS to galactic coordinate sanctioned by the IAU. (The existing IAU resolution from 1958 defines the galactic axes with reference to the equatorial B1950 system, which cannot be accurately transformed to the ICRS; see Murray 1989.) We therefore adopt the same definition as was used in the Hipparcos Catalogue (Vol. 1, Sect. 1.5.3 of ESA 1997). According to this, the ICRS coordinates of the north galactic pole are (αG,δG)=(192.85948,+27.12825) and the galactic longitude of the first intersection of the galactic plane with the equator is lΩ=32.93192.

Transformation of position

Transformation of astronomical spherical coordinates (α, δ in ICRS; l and b in the galactic system) and of the corresponding proper motions (μα*, μδ and μl*, μb, respectively) is best done by using vectors and matrix algebra (see Ch. 4 in van Altena 2012). A given point on the celestial sphere is then represented by a unit vector, whose components in the two systems are

𝒓ICRS=[XICRSYICRSZICRS]=[cosαcosδsinαcosδsinδ] (3.57)


𝒓Gal=[XGalYGalZGal]=[coslcosbsinlcosbsinb]. (3.58)

In terms of these column matrices the transformation from ICRS to the galactic system is obtained through the matrix multiplication

𝒓Gal=𝑨G𝒓ICRS, (3.59)


𝑨G =𝑹z(-lΩ)𝑹x(90-δG)𝑹z(αG+90) (3.60)
=[-0.0548755604162154-0.8734370902348850-0.4838350155487132+0.4941094278755837-0.4448296299600112+0.7469822444972189-0.8676661490190047-0.1980763734312015+0.4559837761750669] (3.61)

is a fixed orthogonal matrix (the transpose of the matrix 𝑨G defined in Vol. 1, Eq. 1.5.11 of ESA 1997). 𝑹i(θ) is the 3×3 matrix representing a rotation of the coordinate frame by the angle θ about axis i. Since 𝑨G is orthogonal, the inverse transformation to Equation 3.59 is

𝒓ICRS=𝑨G𝒓Gal. (3.62)

Given (α,δ), application of Equation 3.57 and Equation 3.59 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(YGal,XGal),b=atan2(ZGal,XGal2+YGal2). (3.63)

Note that Equation 3.63 works also for vectors that are not of unit length.

Transformation of proper motion

The transformation of the proper motion components (μα*,μδ) to (μl*,μb) (where μl*=μlcosb) requires the use of the four auxiliary column matrices

𝒑ICRS=[-sinαcosα0],𝒒ICRS=[-cosαsinδ-sinαsinδcosδ] (3.64)


𝒑Gal=[-sinlcosl0],𝒒Gal=[-coslsinb-sinlsinbcosb]. (3.65)

Geometrically, 𝒑ICRS and 𝒒ICRS represent unit vectors in the directions of increasing α and δ, respectively, expressed by their Cartesian components in ICRS. Similarly, 𝒑Gal and 𝒒Gal 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

𝝁ICRS=𝒑ICRSμα*+𝒒ICRSμδ, (3.66)

and in the galactic system as

𝝁Gal=𝒑Galμl*+𝒒Galμb. (3.67)

These column matrices transform exactly as any other Cartesian vector, namely

𝝁Gal=𝑨G𝝁ICRS (3.68)


𝝁ICRS=𝑨G𝝁Gal. (3.69)

Applying Equation 3.64, Equation 3.66, and Equation 3.68 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 3.67, using the orthogonality of 𝒑Gal and 𝒒Gal:

μl*=𝒑Gal𝝁Gal,μb=𝒒Gal𝝁Gal. (3.70)

For completeness we give also the corresponding calculation in ICRS:

μα*=𝒑ICRS𝝁ICRS,μδ=𝒒ICRS𝝁ICRS. (3.71)

Error propagation

The statistical errors associated with the astrometric parameters α, δ, ϖ, μα*, and μδ are given by the standard uncertainties σα*, σδ, σϖ, σμα*, and σμδ together with the correlation coefficients ρ(α,δ), ρ(α,ϖ), etc. For notational convenience we may number the five parameters 0, 1, , 4; thus σ0=σα*, ρ01=ρ(α,δ), etc. Let

𝒆[e0e1e2e3e4]=[Δα*ΔδΔϖΔμα*Δμδ] (3.72)

be a vector containing the errors, that is the differences between the measured and true astrometric parameters, expressed in mas or mas yr-1 (with Δα*=Δαcosδ). The measurements are assumed to be unbiased, so the expectation of the error vector is

E[𝒆]=𝟎 (3.73)

and its covariance is the symmetric positive definite matrix

𝑪=E[𝒆𝒆]=[E[e0e0]E[e0e1]E[e0e4]E[e1e0]E[e1e1]E[e1e4]E[e4e0]E[e4e1]E[e4e4]] (3.74)

with diagonal elements Cii=σi2 and off-diagonal elements Vij=σiσjρij (with ρji=ρij).

The transformation from ICRF to galactic coordinates is a strongly non-linear function. However, the errors ei are very small and the error vector in galactic coordinates

𝒈[g0g1g2g3g4]=[Δl*ΔbΔϖΔμl*Δμb] (3.75)

is therefore obtained by the linear transformation

𝒈=𝑱𝒆, (3.76)


𝑱=(l*,b,ϖ,μl*,μb)(α*,δ,ϖ,μα*,μδ)=[l*α*l*δl*μδbα*bδbμδμbα*μbδμbμδ] (3.77)

is the Jacobian of the transformation. (The cosb or cosδ factor implied by the asterisk is never differentiated; thus l*/α*=(l/α)cosb/cosδ, etc.) Clearly E[𝒈]=𝑱E[𝒆]=𝟎, so the galactic parameters are also unbiased, with covariance matrix

𝑪Gal=E[𝒈𝒈]=𝑱𝑪𝑱. (3.78)

It remains to determine 𝑱. Since ϖ is unchanged by the transformation we have J22=1 and Ji2=J2i=0 for i2. 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 3.66–Equation 3.70 we find

𝑱=[𝑮0001000𝑮] (3.79)


𝑮=[𝒑Gal𝒒Gal]𝑨G[𝒑ICRS𝒒ICRS] (3.80)

is an orthogonal 2×2 matrix. Geometrically, 𝑮 describes the local rotation from ICRS to galactic coordinates in the tangent plane of the celestial sphere at the position of the source.