# 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 $T_{0}$ 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

 $\boldsymbol{b}(T)=\boldsymbol{b}(T_{0})+(T-T_{0})\,\boldsymbol{v}\,.$ (3.7)

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

 $\boldsymbol{b}=\boldsymbol{b}_{0}+t\,\boldsymbol{v}\,.$ (3.8)

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

 $\boldsymbol{v}=\frac{A_{\mathrm{V}}}{\varpi_{0}}\left(\boldsymbol{\mu}_{0}+% \boldsymbol{r}_{0}\mu_{r0}\right)=\frac{A_{\mathrm{V}}}{\varpi_{0}}\left(% \boldsymbol{p}_{0}\mu_{\alpha*0}+\boldsymbol{q}_{0}\mu_{\delta 0}+\boldsymbol{% r}_{0}\mu_{r0}\right)\,,$ (3.9)

where $\boldsymbol{\mu}_{0}$ is the proper motion at the initial epoch, three unit vector constitute the normal triad $\left[\boldsymbol{p}_{0},\boldsymbol{q}_{0},\boldsymbol{r}_{0}\right]$ and $A_{\mathrm{V}}=4.740\,470\,446$ equals the astronomical unit expressed in $\mathrm{km}\ \mathrm{yr}\ \mathrm{s}^{-1}$. 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 $\boldsymbol{u}=\boldsymbol{b}/b$ is given by Equation 3.8. Squaring both sides of Equation 3.8 and making use of obvious relations $\boldsymbol{b}^{\prime}_{0}\boldsymbol{v}_{0}=b_{0}^{2}\mu_{r0}$ and $v_{0}^{2}=b_{0}^{2}\left(\mu_{r0}^{2}+\mu_{0}^{2}\right)$, we find

 $b^{2}=b_{0}^{2}\left(1+2\mu_{r0}t+\left(\mu_{0}^{2}+\mu_{r0}^{2}\right)t^{2}% \right)\,,$ (3.10)

where $\mu_{0}^{2}=\mu_{\alpha*0}^{2}+\mu_{\delta 0}^{2}$. Introducing the distance factor

 $f=b_{0}/b=\left[1+2\mu_{r0}t+\left(\mu_{0}^{2}+\mu_{r0}^{2}\right)t^{2}\right]% ^{-1/2},$ (3.11)

the propagation of the barycentric direction is

 $\boldsymbol{u}=\left[\boldsymbol{r}_{0}\left(1+\mu_{r0}t\right)+\boldsymbol{% \mu}_{0}t\right]f$ (3.12)

and the propagation of the parallax becomes

 $\varpi=\varpi_{0}f\,.$ (3.13)

The celestial coordinates ($\alpha$, $\delta$) at epoch $T$ are obtained from $\boldsymbol{u}$ in the usual manner and the normal triad associated with the propagated direction is

 $\left[\boldsymbol{p},\boldsymbol{q},\boldsymbol{r}\right]=\begin{bmatrix}-\sin% \alpha&-\sin\delta\cos\alpha&\cos\delta\cos\alpha\\ \cos\alpha&-\sin\delta\sin\alpha&\cos\delta\sin\alpha\\ 0&\cos\delta&\sin\delta\end{bmatrix}\,.$ (3.14)

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

 $\boldsymbol{\mu}=\frac{\mathrm{d}\boldsymbol{u}}{\mathrm{d}t}=\left[% \boldsymbol{\mu}_{0}\left(1+\mu_{r0}t\right)-\boldsymbol{r}_{0}\mu_{0}^{2}t% \right]f^{3}\,,$ (3.15)

and the propagated radial proper motion is found to be

 $\mu_{r}=\frac{\mathrm{d}b}{\mathrm{d}t}\frac{\varpi}{A}=\left[\mu_{r0}+\left(% \mu_{0}^{2}+\mu_{r0}^{2}\right)t\right]f^{2}\,.$ (3.16)

To obtain the proper motion components ($\mu_{\alpha*}$, $\mu_{\delta}$) from vector $\boldsymbol{\mu}$ it is necessary to resolve the latter along the tangential vectors $\boldsymbol{p}$ and $\boldsymbol{q}$ at the propagate direction:

 $\mu_{\alpha*}=\boldsymbol{p}^{\prime}\boldsymbol{\mu}\,,\quad\mu_{\delta}=% \boldsymbol{q}^{\prime}\boldsymbol{\mu}\,.$ (3.17)

The tangential vectors are defined in terms of the propagated $\boldsymbol{u}$ or ($\alpha$, $\delta$) at the epoch $T$ according to Equation 3.14.

The above formulae describe the complete transformation of ($\alpha_{0}$, $\delta_{0}$, $\varpi_{0}$, $\mu_{\alpha*0}$, $\mu_{\delta 0}$, $\mu_{r0}$) at epoch $T_{0}$ into ($\alpha$, $\delta$, $\varpi$, $\mu_{\alpha*}$, $\mu_{\delta}$, $\mu_{r}$) at the arbitrary epoch $T=T_{0}+t$. The transformation is rigorously reversible: a second transformation from $T$ to $T_{0}$ recovers the original six parameters.

### Propagation of errors (covariances)

The uncertainties in the source parameters $\alpha$, $\delta$, $\varpi$, $\mu_{\alpha*}$, $\mu_{\delta}$, $\mu_{r}$ and correlations between them are quantified by means of the $6\times 6$ covariance matrix $\boldsymbol{C}$ 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 $\boldsymbol{C}_{0}$ of the initial parameters and the matrix $\boldsymbol{C}$ of the propagated parameters are related as

 $\boldsymbol{C}=\boldsymbol{J}\boldsymbol{C}_{0}\boldsymbol{J}^{\prime}\,,$ (3.18)

where $\boldsymbol{J}$ is the Jacobian matrix of the source parameter transformation:

 $\boldsymbol{J}=\frac{\partial\left(\alpha,\delta,\varpi,\mu_{\alpha*},\mu_{% \delta},\mu_{r}\right)}{\partial\left(\alpha_{0},\delta_{0},\varpi_{0},\mu_{% \alpha*0},\mu_{\delta 0},\mu_{r0}\right)}$ (3.19)

Thus, the propagation of the covariances requires the calculation of all 36 partial derivatives constituting the Jacobian $\boldsymbol{J}$.

### Initialization of $\boldsymbol{C}_{0}$:

The initial covariance matrix $\boldsymbol{C}_{0}$ must be specified in order to calculate the covariance matrix of the propagated astrometric parameters $\boldsymbol{C}$. 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 $\boldsymbol{C}_{0}$ 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 $\boldsymbol{C}_{0}$, this matrix must therefore be augmented with a sixth row and column related to the initial radial proper motion $\mu_{r0}$. If the initial radial velocity $v_{r0}$ has the standard error $\sigma_{v_{r0}}$ and is assumed to be statistically independent of the astrometric parameters in the catalogue, then the required additional elements in $\boldsymbol{C}_{0}$ are

 $\displaystyle\left[C_{0}\right]_{i6}=\left[C_{0}\right]_{6i}=\left[C_{0}\right% ]_{i3}\left(v_{r0}/A\right),\quad i=1\dots 5\,,$ (3.20) $\displaystyle\left[C_{0}\right]_{66}=\left[C_{0}\right]_{33}\left(v_{r0}^{2}+% \sigma_{v_{r0}}^{2}\right)/A^{2}+\left(\varpi_{0}\sigma_{v_{r0}}/A\right)^{2}$

(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 $\sigma_{v_{r0}}$ (set to, for example, the expected velocity dispersion of the stellar type in question), in which case $\left[C_{0}\right]_{66}$ 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 $T_{0}$), 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:

 $\displaystyle J_{11}$ $\displaystyle=\frac{\partial\alpha*}{\partial{\alpha*}_{0}}=\boldsymbol{p}^{% \prime}\boldsymbol{p}_{0}\left(1+\mu_{r0}t\right)f-\boldsymbol{p}^{\prime}% \boldsymbol{r}_{0}\mu_{\alpha*0}tf$ (3.21) $\displaystyle J_{12}$ $\displaystyle=\frac{\partial\alpha*}{\partial\delta_{0}}=\boldsymbol{p}^{% \prime}\boldsymbol{q}_{0}\left(1+\mu_{r0}t\right)f-\boldsymbol{p}^{\prime}% \boldsymbol{r}_{0}\mu_{\delta 0}tf$ (3.22) $\displaystyle J_{13}$ $\displaystyle=\frac{\partial\alpha*}{\partial\varpi_{0}}=0$ (3.23) $\displaystyle J_{14}$ $\displaystyle=\frac{\partial\alpha*}{\partial\mu_{\alpha*0}}=\boldsymbol{p}^{% \prime}\boldsymbol{p}_{0}tf$ (3.24) $\displaystyle J_{15}$ $\displaystyle=\frac{\partial\alpha*}{\partial\mu_{\delta 0}}=\boldsymbol{p}^{% \prime}\boldsymbol{q}_{0}tf$ (3.25) $\displaystyle J_{16}$ $\displaystyle=\frac{\partial\alpha*}{\partial\mu_{r0}}=-\mu_{\alpha*}t^{2}$ (3.26) $\displaystyle J_{21}$ $\displaystyle=\frac{\partial\delta}{\partial{\alpha*}_{0}}=\boldsymbol{q}^{% \prime}\boldsymbol{p}_{0}\left(1+\mu_{r0}t\right)f-\boldsymbol{q}^{\prime}% \boldsymbol{r}_{0}\mu_{\alpha*0}tf$ (3.27) $\displaystyle J_{22}$ $\displaystyle=\frac{\partial\delta}{\partial\delta_{0}}=\boldsymbol{q}^{\prime% }\boldsymbol{q}_{0}\left(1+\mu_{r0}t\right)f-\boldsymbol{q}^{\prime}% \boldsymbol{r}_{0}\mu_{\delta 0}tf$ (3.28) $\displaystyle J_{23}$ $\displaystyle=\frac{\partial\delta}{\partial\varpi_{0}}=0$ (3.29) $\displaystyle J_{24}$ $\displaystyle=\frac{\partial\delta}{\partial\mu_{\alpha*0}}=\boldsymbol{q}^{% \prime}\boldsymbol{p}_{0}tf$ (3.30) $\displaystyle J_{25}$ $\displaystyle=\frac{\partial\delta}{\partial\mu_{\delta 0}}=\boldsymbol{q}^{% \prime}\boldsymbol{q}_{0}tf$ (3.31) $\displaystyle J_{26}$ $\displaystyle=\frac{\partial\delta}{\partial\mu_{r0}}=-\mu_{\delta}t^{2}$ (3.32) $\displaystyle J_{31}$ $\displaystyle=\frac{\partial\varpi}{\partial{\alpha*}_{0}}=0$ (3.33) $\displaystyle J_{32}$ $\displaystyle=\frac{\partial\varpi}{\partial\delta_{0}}=0$ (3.34) $\displaystyle J_{33}$ $\displaystyle=\frac{\partial\varpi}{\partial\varpi_{0}}=f$ (3.35) $\displaystyle J_{34}$ $\displaystyle=\frac{\partial\varpi}{\partial\mu_{\alpha*0}}=-\varpi\mu_{\alpha% *0}t^{2}f^{2}$ (3.36) $\displaystyle J_{35}$ $\displaystyle=\frac{\partial\varpi}{\partial\mu_{\delta 0}}=-\varpi\mu_{\delta 0% }t^{2}f^{2}$ (3.37) $\displaystyle J_{36}$ $\displaystyle=\frac{\partial\varpi}{\partial\mu_{r0}}=-\varpi\left(1+\mu_{r0}t% \right)tf^{2}$ (3.38) $\displaystyle J_{41}$ $\displaystyle=\frac{\partial\mu_{\alpha*}}{\partial{\alpha*}_{0}}=-\boldsymbol% {p}^{\prime}\boldsymbol{p}_{0}\mu_{0}^{2}tf^{3}-\boldsymbol{p}^{\prime}% \boldsymbol{r}_{0}\mu_{\alpha*0}\left(1+\mu_{r0}t\right)f^{3}$ (3.39) $\displaystyle J_{42}$ $\displaystyle=\frac{\partial\mu_{\alpha*}}{\partial\delta_{0}}=-\boldsymbol{p}% ^{\prime}\boldsymbol{q}_{0}\mu_{0}^{2}tf^{3}-\boldsymbol{p}^{\prime}% \boldsymbol{r}_{0}\mu_{\delta 0}\left(1+\mu_{r0}t\right)f^{3}$ (3.40) $\displaystyle J_{43}$ $\displaystyle=\frac{\partial\mu_{\alpha*}}{\partial\varpi_{0}}=0$ (3.41) $\displaystyle J_{44}$ $\displaystyle=\frac{\partial\mu_{\alpha*}}{\partial\mu_{\alpha*0}}=\boldsymbol% {p}^{\prime}\boldsymbol{p}_{0}\left(1+\mu_{r0}t\right)f^{3}-2\boldsymbol{p}^{% \prime}\boldsymbol{r}_{0}\mu_{\alpha*0}tf^{3}-3\mu_{\alpha*}\mu_{\alpha*0}t^{2% }f^{2}$ (3.42)
 $\displaystyle J_{45}$ $\displaystyle=\frac{\partial\mu_{\alpha*}}{\partial\mu_{\delta 0}}=\boldsymbol% {p}^{\prime}\boldsymbol{q}_{0}\left(1+\mu_{r0}t\right)f^{3}-2\boldsymbol{p}^{% \prime}\boldsymbol{r}_{0}\mu_{\delta 0}tf^{3}-3\mu_{\alpha*}\mu_{\delta 0}t^{2% }f^{2}$ (3.43) $\displaystyle J_{46}$ $\displaystyle=\frac{\partial\mu_{\alpha*}}{\partial\mu_{r0}}=\boldsymbol{p}^{% \prime}\left[\boldsymbol{\mu}_{0}f-3\boldsymbol{\mu}\left(1+\mu_{r0}t\right)% \right]tf^{2}$ (3.44) $\displaystyle J_{51}$ $\displaystyle=\frac{\partial\mu_{\delta}}{\partial{\alpha*}_{0}}=-\boldsymbol{% q}^{\prime}\boldsymbol{p}_{0}\mu_{0}^{2}tf^{3}-\boldsymbol{q}^{\prime}% \boldsymbol{r}_{0}\mu_{\alpha*0}\left(1+\mu_{r0}t\right)f^{3}$ (3.45) $\displaystyle J_{52}$ $\displaystyle=\frac{\partial\mu_{\delta}}{\partial\delta_{0}}=-\boldsymbol{q}^% {\prime}\boldsymbol{q}_{0}\mu_{0}^{2}tf^{3}-\boldsymbol{q}^{\prime}\boldsymbol% {r}_{0}\mu_{\delta 0}\left(1+\mu_{r0}t\right)f^{3}$ (3.46) $\displaystyle J_{53}$ $\displaystyle=\frac{\partial\mu_{\delta}}{\partial\varpi_{0}}=0$ (3.47) $\displaystyle J_{54}$ $\displaystyle=\frac{\partial\mu_{\delta}}{\partial\mu_{\alpha*0}}=\boldsymbol{% q}^{\prime}\boldsymbol{p}_{0}\left(1+\mu_{r0}t\right)f^{3}-2\boldsymbol{q}^{% \prime}\boldsymbol{r}_{0}\mu_{\alpha*0}tf^{3}-3\mu_{\delta}\mu_{\alpha*0}t^{2}% f^{2}$ (3.48) $\displaystyle J_{55}$ $\displaystyle=\frac{\partial\mu_{\delta}}{\partial\mu_{\delta 0}}=\boldsymbol{% q}^{\prime}\boldsymbol{q}_{0}\left(1+\mu_{r0}t\right)f^{3}-2\boldsymbol{q}^{% \prime}\boldsymbol{r}_{0}\mu_{\delta 0}tf^{3}-3\mu_{\delta}\mu_{\delta 0}t^{2}% f^{2}$ (3.49) $\displaystyle J_{56}$ $\displaystyle=\frac{\partial\mu_{\delta}}{\partial\mu_{r0}}=\boldsymbol{q}^{% \prime}\left[\boldsymbol{\mu}_{0}f-3\boldsymbol{\mu}\left(1+\mu_{r0}t\right)% \right]tf^{2}$ (3.50) $\displaystyle J_{61}$ $\displaystyle=\frac{\partial\mu_{r}}{\partial{\alpha*}_{0}}=0$ (3.51) $\displaystyle J_{62}$ $\displaystyle=\frac{\partial\mu_{r}}{\partial\delta_{0}}=0$ (3.52) $\displaystyle J_{63}$ $\displaystyle=\frac{\partial\mu_{r}}{\partial\varpi_{0}}=0$ (3.53) $\displaystyle J_{64}$ $\displaystyle=\frac{\partial\mu_{r}}{\partial\mu_{\alpha*0}}=2\mu_{\alpha*0}% \left(1+\mu_{r0}t\right)tf^{4}$ (3.54) $\displaystyle J_{65}$ $\displaystyle=\frac{\partial\mu_{r}}{\partial\mu_{\delta 0}}=2\mu_{\delta 0}% \left(1+\mu_{r0}t\right)tf^{4}$ (3.55) $\displaystyle J_{66}$ $\displaystyle=\frac{\partial\mu_{r}}{\partial\mu_{r0}}=\left[\left(1+\mu_{r0}t% \right)^{2}-\mu_{0}^{2}t^{2}\right]f^{4}$ (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 $(\alpha_{\text{G}},\,\delta_{\text{G}})=(192\hbox{.\!\!^{\circ}}85948,\,+27% \hbox{.\!\!^{\circ}}12825)$ and the galactic longitude of the first intersection of the galactic plane with the equator is $l_{\Omega}=32\hbox{.\!\!^{\circ}}93192$.

### Transformation of position

Transformation of astronomical spherical coordinates ($\alpha$, $\delta$ in ICRS; $l$ and $b$ in the galactic system) and of the corresponding proper motions ($\mu_{\alpha*}$, $\mu_{\delta}$ and $\mu_{l*}$, $\mu_{b}$, 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

 $\boldsymbol{r}_{\text{ICRS}}=\begin{bmatrix}X_{\text{ICRS}}\\ Y_{\text{ICRS}}\\ Z_{\text{ICRS}}\end{bmatrix}=\begin{bmatrix}\cos\alpha\cos\delta\\ \sin\alpha\cos\delta\\ \sin\delta\end{bmatrix}$ (3.57)

and

 $\boldsymbol{r}_{\text{Gal}}=\begin{bmatrix}X_{\text{Gal}}\\ Y_{\text{Gal}}\\ Z_{\text{Gal}}\end{bmatrix}=\begin{bmatrix}\cos l\cos b\\ \sin l\cos b\\ \sin b\end{bmatrix}\,.$ (3.58)

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

 $\boldsymbol{r}_{\text{Gal}}=\boldsymbol{A}_{\text{G}}^{\prime}\boldsymbol{r}_{% \text{ICRS}}\,,$ (3.59)

where

 $\displaystyle\boldsymbol{A}_{\text{G}}^{\prime}$ $\displaystyle=\boldsymbol{R}_{z}(-l_{\Omega})\boldsymbol{R}_{x}(90\,^{\circ}-% \delta_{\text{G}})\boldsymbol{R}_{z}(\alpha_{\text{G}}+90\,^{\circ})$ (3.60) $\displaystyle=\begin{bmatrix}-0.0548755604162154&-0.8734370902348850&-0.483835% 0155487132\\ +0.4941094278755837&-0.4448296299600112&+0.7469822444972189\\ -0.8676661490190047&-0.1980763734312015&+0.4559837761750669\end{bmatrix}$ (3.61)

is a fixed orthogonal matrix (the transpose of the matrix $\boldsymbol{A}_{\text{G}}$ defined in Vol. 1, Eq. 1.5.11 of ESA 1997). $\boldsymbol{R}_{i}(\theta)$ is the $3\times 3$ matrix representing a rotation of the coordinate frame by the angle $\theta$ about axis $i$. Since $\boldsymbol{A}_{\text{G}}^{\prime}$ is orthogonal, the inverse transformation to Equation 3.59 is

 $\boldsymbol{r}_{\text{ICRS}}=\boldsymbol{A}_{\text{G}}\boldsymbol{r}_{\text{% Gal}}\,.$ (3.62)

Given $(\alpha,\,\delta)$, 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:

 $l=\text{atan2}(Y_{\text{Gal}},\,X_{\text{Gal}})\,,\quad b=\text{atan2}\left(Z_% {\text{Gal}},\,\sqrt{X_{\text{Gal}}^{2}+Y_{\text{Gal}}^{2}}\right)\,.$ (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 $(\mu_{\alpha*},\,\mu_{\delta})$ to $(\mu_{l*},\,\mu_{b})$ (where $\mu_{l*}=\mu_{l}\cos b$) requires the use of the four auxiliary column matrices

 $\boldsymbol{p}_{\text{ICRS}}=\begin{bmatrix}-\sin\alpha\\ \cos\alpha\\ 0\end{bmatrix}\,,\quad\boldsymbol{q}_{\text{ICRS}}=\begin{bmatrix}-\cos\alpha% \sin\delta\\ -\sin\alpha\sin\delta\\ \cos\delta\end{bmatrix}$ (3.64)

and

 $\boldsymbol{p}_{\text{Gal}}=\begin{bmatrix}-\sin l\\ \cos l\\ 0\end{bmatrix}\,,\quad\boldsymbol{q}_{\text{Gal}}=\begin{bmatrix}-\cos l\sin b% \\ -\sin l\sin b\\ \cos b\end{bmatrix}\,.$ (3.65)

Geometrically, $\boldsymbol{p}_{\text{ICRS}}$ and $\boldsymbol{q}_{\text{ICRS}}$ represent unit vectors in the directions of increasing $\alpha$ and $\delta$, respectively, expressed by their Cartesian components in ICRS. Similarly, $\boldsymbol{p}_{\text{Gal}}$ and $\boldsymbol{q}_{\text{Gal}}$ 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

 $\boldsymbol{\mu}_{\text{ICRS}}=\boldsymbol{p}_{\text{ICRS}}\mu_{\alpha*}+% \boldsymbol{q}_{\text{ICRS}}\mu_{\delta}\,,$ (3.66)

and in the galactic system as

 $\boldsymbol{\mu}_{\text{Gal}}=\boldsymbol{p}_{\text{Gal}}\mu_{l*}+\boldsymbol{% q}_{\text{Gal}}\mu_{b}\,.$ (3.67)

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

 $\boldsymbol{\mu}_{\text{Gal}}=\boldsymbol{A}_{\text{G}}^{\prime}\boldsymbol{% \mu}_{\text{ICRS}}$ (3.68)

and

 $\boldsymbol{\mu}_{\text{ICRS}}=\boldsymbol{A}_{\text{G}}\boldsymbol{\mu}_{% \text{Gal}}\,.$ (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 $\boldsymbol{p}_{\text{Gal}}$ and $\boldsymbol{q}_{\text{Gal}}$:

 $\mu_{l*}=\boldsymbol{p}_{\text{Gal}}^{\prime}\boldsymbol{\mu}_{\text{Gal}}\,,% \quad\mu_{b}=\boldsymbol{q}_{\text{Gal}}^{\prime}\boldsymbol{\mu}_{\text{Gal}}\,.$ (3.70)

For completeness we give also the corresponding calculation in ICRS:

 $\mu_{\alpha*}=\boldsymbol{p}_{\text{ICRS}}^{\prime}\boldsymbol{\mu}_{\text{% ICRS}}\,,\quad\mu_{\delta}=\boldsymbol{q}_{\text{ICRS}}^{\prime}\boldsymbol{% \mu}_{\text{ICRS}}\,.$ (3.71)

### Error propagation

The statistical errors associated with the astrometric parameters $\alpha$, $\delta$, $\varpi$, $\mu_{\alpha*}$, and $\mu_{\delta}$ are given by the standard uncertainties $\sigma_{\alpha*}$, $\sigma_{\delta}$, $\sigma_{\varpi}$, $\sigma_{\mu\alpha*}$, and $\sigma_{\mu\delta}$ together with the correlation coefficients $\rho(\alpha,\delta)$, $\rho(\alpha,\varpi)$, etc. For notational convenience we may number the five parameters 0, 1, $\dots$, 4; thus $\sigma_{0}=\sigma_{\alpha*}$, $\rho_{01}=\rho(\alpha,\delta)$, etc. Let

 $\boldsymbol{e}\equiv\begin{bmatrix}e_{0}\\ e_{1}\\ e_{2}\\ e_{3}\\ e_{4}\end{bmatrix}=\begin{bmatrix}\Delta\alpha*\\ \Delta\delta\\ \Delta\varpi\\ \Delta\mu_{\alpha*}\\ \Delta\mu_{\delta}\end{bmatrix}$ (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 $\Delta\alpha*=\Delta\alpha\cos\delta$). The measurements are assumed to be unbiased, so the expectation of the error vector is

 $\text{E}\left[\boldsymbol{e}\right]=\boldsymbol{0}$ (3.73)

and its covariance is the symmetric positive definite matrix

 $\boldsymbol{C}=\text{E}\left[\boldsymbol{e}\boldsymbol{e}^{\prime}\right]=% \begin{bmatrix}\text{E}\left[e_{0}e_{0}\right]&\text{E}\left[e_{0}e_{1}\right]% &\cdots&\text{E}\left[e_{0}e_{4}\right]\\ \text{E}\left[e_{1}e_{0}\right]&\text{E}\left[e_{1}e_{1}\right]&\cdots&\text{E% }\left[e_{1}e_{4}\right]\\ \vdots&\vdots&\ddots&\vdots\\ \text{E}\left[e_{4}e_{0}\right]&\text{E}\left[e_{4}e_{1}\right]&\cdots&\text{E% }\left[e_{4}e_{4}\right]\end{bmatrix}$ (3.74)

with diagonal elements $C_{ii}=\sigma_{i}^{2}$ and off-diagonal elements $V_{ij}=\sigma_{i}\sigma_{j}\rho_{ij}$ (with $\rho_{ji}=\rho_{ij}$).

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

 $\boldsymbol{g}\equiv\begin{bmatrix}g_{0}\\ g_{1}\\ g_{2}\\ g_{3}\\ g_{4}\end{bmatrix}=\begin{bmatrix}\Delta l*\\ \Delta b\\ \Delta\varpi\\ \Delta\mu_{l*}\\ \Delta\mu_{b}\end{bmatrix}$ (3.75)

is therefore obtained by the linear transformation

 $\boldsymbol{g}=\boldsymbol{J}\boldsymbol{e}\,,$ (3.76)

where

 $\boldsymbol{J}=\frac{\partial(l*,\,b,\,\varpi,\,\mu_{l*},\,\mu_{b})}{\partial(% \alpha*,\,\delta,\,\varpi,\,\mu_{\alpha*},\,\mu_{\delta})}=\begin{bmatrix}% \frac{\partial l*}{\partial\alpha*}&\frac{\partial l*}{\partial\delta}&\cdots&% \frac{\partial l*}{\partial\mu_{\delta}}\\ \frac{\partial b}{\partial\alpha*}&\frac{\partial b}{\partial\delta}&\cdots&% \frac{\partial b}{\partial\mu_{\delta}}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{\partial\mu_{b}}{\partial\alpha*}&\frac{\partial\mu_{b}}{\partial\delta}% &\cdots&\frac{\partial\mu_{b}}{\partial\mu_{\delta}}\end{bmatrix}$ (3.77)

is the Jacobian of the transformation. (The $\cos b$ or $\cos\delta$ factor implied by the asterisk is never differentiated; thus $\partial{l*}/\partial\alpha*=(\partial l/\partial\alpha)\cos b/{\cos\delta}$, etc.) Clearly $\text{E}\left[\boldsymbol{g}\right]=\boldsymbol{J}\text{E}\left[\boldsymbol{e}% \right]=\boldsymbol{0}$, so the galactic parameters are also unbiased, with covariance matrix

 $\boldsymbol{C}_{\text{Gal}}=\text{E}\left[\boldsymbol{g}\boldsymbol{g}^{\prime% }\right]=\boldsymbol{J}\boldsymbol{C}\boldsymbol{J}^{\prime}\,.$ (3.78)

It remains to determine $\boldsymbol{J}$. Since $\varpi$ is unchanged by the transformation we have $J_{22}=1$ and $J_{i2}=J_{2i}=0$ for $i\neq 2$. It is also readily seen that the proper motion errors transform in the same way as the positional errors (if we regard $\boldsymbol{p}$ and $\boldsymbol{q}$ as fixed and not subject to errors); then from Equation 3.66–Equation 3.70 we find

 $\boldsymbol{J}=\begin{bmatrix}\boldsymbol{G}&0&0\\ 0&1&0\\ 0&0&\boldsymbol{G}\end{bmatrix}$ (3.79)

where

 $\boldsymbol{G}=\left[\boldsymbol{p}_{\text{Gal}}\quad\boldsymbol{q}_{\text{Gal% }}\right]^{\prime}\boldsymbol{A}_{\text{G}}^{\prime}\left[\boldsymbol{p}_{% \text{ICRS}}\quad\boldsymbol{q}_{\text{ICRS}}\right]$ (3.80)

is an orthogonal $2\times 2$ matrix. Geometrically, $\boldsymbol{G}$ describes the local rotation from ICRS to galactic coordinates in the tangent plane of the celestial sphere at the position of the source.