Author(s): Lennart Lindegren

Gaia measures the positions of star images in the focal plane, from which the astrometric solution reconstructs the celestial positions, proper motions, and parallaxes of the sources together with the attitude and calibration parameters. Because the measurements are essentially relative within the field of view, the resulting coordinate system of positions and proper motions is also relative in the sense that its orientation and spin with respect to the celestial reference system (ICRS) are not precisely defined from the measurements themselves. The source and attitude parameters obtained in AGIS are thus expressed in a reference system that is internally consistent and well-defined, but slightly different from ICRS.

The purpose of the frame alignment process is to correct the source and attitude parameters so that they are expressed in a reference system that coincides with ICRS as closely as possible. Gaia observations of extragalactic sources (quasars) are used for this. Quasars are sufficiently distant that their peculiar motions can be neglected; therefore, they define a kinematically non-rotating reference system. Moreover, a subset of the quasars, known as the ICRF, have accurate positions in ICRS determined by radio–interferometric (VLBI) observations. Gaia observations of their optical counterparts allow the origins of $\alpha $ and $\delta $ to be aligned with the ICRS.

Section 6.1 in Lindegren et al. (2012) gives the rigorous definition of the rotation correction, then derives a linear approximation applicable to the small corrections that exist in practise. Only the small-angle approximation is described here, as it is sufficient in (nearly) all practical cases and much easier to explain. A similar process was used to align the Hipparcos Catalogue with the extragalactic reference frame (Lindegren et al. 1992).

The ICRS may be represented by the vector triad $\U0001d5a2=[\bm{X}\bm{Y}\bm{Z}]$, where $\bm{X}$, $\bm{Y}$, and $\bm{Z}$ are unit vectors pointing towards $(\alpha ,\delta )=(0,\mathrm{\hspace{0.17em}0})$, (90 ${}^{\circ}$, 0), and (0, 90 ${}^{\circ}$), respectively. Since ICRS is non-rotating relative to distant quasars, the directions of these vectors are fixed. The reference system of the source and attitude parameters resulting from the astrometric solution can similarly be represented by a vector triad $\stackrel{~}{\U0001d5a2}=[\stackrel{\mathbf{~}}{\bm{X}}\stackrel{\mathbf{~}}{\bm{Y}}\stackrel{\mathbf{~}}{\bm{Z}}]$ which deviates slightly from $\U0001d5a2$. Moreover, $\stackrel{~}{\U0001d5a2}$ may rotate slowly with respect to $\U0001d5a2$. At any particular time $t$ the difference between the two systems can be represented by a rotation vector $\bm{\epsilon}$ such that

$$\U0001d5a2=\stackrel{~}{\U0001d5a2}+\bm{\epsilon}\times \stackrel{~}{\U0001d5a2}+O({\epsilon}^{2}).$$ | (3.91) |

The last term indicates that we are in the regime of the small-angle approximation,
meaning that terms of the order of ${|\bm{\epsilon}|}^{2}$ can be neglected. If
$\mu $as precision is aimed at, that $|\bm{\epsilon}|$ must consequently be
less than about 0.5 ${}^{\mathrm{\prime \prime}}$. Note that $\bm{\epsilon}$ is time-dependent,
due to the rotation of $\stackrel{~}{\U0001d5a2}$, and that it is defined in the sense of a
*correction* to the orientation of $\stackrel{~}{\U0001d5a2}$.

The astrometric source model (Section 3.3.3) constrains the sources to move uniformly through space, which implies that the time-dependence of $\bm{\epsilon}$ must be linear. As long as the correction remains small, it can therefore be written

$$\bm{\epsilon}(t)={\bm{\epsilon}}_{0}+(t-{t}_{\text{ep}})\bm{\omega},$$ | (3.92) |

where ${\bm{\epsilon}}_{0}$ is the orientation correction at $t={t}_{\text{ep}}$ (the reference epoch of the catalogue), and $\bm{\omega}$ is the spin correction.

The celestial position $(\alpha ,\delta )$ and proper motion $({\mu}_{\alpha *},{\mu}_{\delta})$ of a source in the ICRS (relative to $\U0001d5a2$) are defined by means of Equation 3.3 and Equation 3.4, where the components of the vectors ${\overline{\bm{u}}}_{\text{B}}({t}_{\text{ep}})$ and ${\text{d}{\overline{\bm{u}}}_{\text{B}}/\text{d}t|}_{t={t}_{\text{ep}}}$ are actually the projections of these vectors on $\U0001d5a2$. The corresponding source parameters obtained in the astrometric solution, denoted $(\stackrel{~}{\alpha},\stackrel{~}{\delta})$ and $({\stackrel{~}{\mu}}_{\alpha *},{\stackrel{~}{\mu}}_{\delta})$, are similarly defined by the projections of the same vectors on $\stackrel{~}{\U0001d5a2}$. It is readily shown that

$$\begin{array}{cc}\hfill (\stackrel{~}{\alpha}-\alpha )\mathrm{cos}\delta & ={({\bm{\epsilon}}_{0}\times \bm{r})}^{\prime}\bm{p}={\bm{q}}^{\prime}{\bm{\epsilon}}_{0}\hfill \\ \hfill \stackrel{~}{\delta}-\delta & ={({\bm{\epsilon}}_{0}\times \bm{r})}^{\prime}\bm{q}=-{\bm{p}}^{\prime}{\bm{\epsilon}}_{0}\hfill \end{array}\mathit{\hspace{1em}}\}$$ | (3.93) |

and

$$\begin{array}{cc}\hfill {\stackrel{~}{\mu}}_{\alpha *}-{\mu}_{\alpha *}& ={(\bm{\omega}\times \bm{r})}^{\prime}\bm{p}={\bm{q}}^{\prime}\bm{\omega}\hfill \\ \hfill {\stackrel{~}{\mu}}_{\delta}-{\mu}_{\delta}& ={(\bm{\omega}\times \bm{r})}^{\prime}\bm{q}=-{\bm{p}}^{\prime}\bm{\omega}\hfill \end{array}\mathit{\hspace{1em}}\}$$ | (3.94) |

where $\bm{p}$, $\bm{q}$, $\bm{r}$ are the unit vectors introduced in Section 3.3.3. With ${\epsilon}_{0X}$, ${\epsilon}_{0Y}$, ${\epsilon}_{0Z}$, ${\omega}_{X}$, ${\omega}_{Y}$, ${\omega}_{Z}$ denoting the components of ${\bm{\epsilon}}_{0}$ and $\bm{\omega}$ in $\U0001d5a2$, these relations can be written in matrix form as

$$\left[\begin{array}{c}\hfill (\stackrel{~}{\alpha}-\alpha )\mathrm{cos}\delta \hfill \\ \hfill \stackrel{~}{\delta}-\delta \hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill -\mathrm{sin}\delta \mathrm{cos}\alpha \hfill & \hfill -\mathrm{sin}\delta \mathrm{sin}\alpha \hfill & \hfill \mathrm{cos}\delta \hfill \\ \hfill \mathrm{sin}\alpha \hfill & \hfill -\mathrm{cos}\alpha \hfill & \hfill 0\hfill \end{array}\right]\left[\begin{array}{c}\hfill {\epsilon}_{0X}\hfill \\ \hfill {\epsilon}_{0Y}\hfill \\ \hfill {\epsilon}_{0Z}\hfill \end{array}\right]$$ | (3.95) |

and

$$\left[\begin{array}{c}\hfill {\stackrel{~}{\mu}}_{\alpha *}-{\mu}_{\alpha}\hfill \\ \hfill {\stackrel{~}{\mu}}_{\delta}-{\mu}_{\delta}\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill -\mathrm{sin}\delta \mathrm{cos}\alpha \hfill & \hfill -\mathrm{sin}\delta \mathrm{sin}\alpha \hfill & \hfill \mathrm{cos}\delta \hfill \\ \hfill \mathrm{sin}\alpha \hfill & \hfill -\mathrm{cos}\alpha \hfill & \hfill 0\hfill \end{array}\right]\left[\begin{array}{c}\hfill {\omega}_{X}\hfill \\ \hfill {\omega}_{Y}\hfill \\ \hfill {\omega}_{Z}\hfill \end{array}\right].$$ | (3.96) |

Given the position and proper motion differences for the same sources expressed in $\stackrel{~}{\U0001d5a2}$ and $\U0001d5a2$, these equations can be used to obtain a least-squares estimate of ${\bm{\epsilon}}_{0}$ and $\bm{\omega}$. The same equations can then be used to correct the source parameters so that they refer to $\U0001d5a2$ instead of $\stackrel{~}{\U0001d5a2}$.

The astrometric solution that produced source parameters relative to $\stackrel{~}{\U0001d5a2}$ also produced an attitude estimate in the same reference system. This attitude should also be aligned with $\U0001d5a2$ if it will be used for the further analysis (e.g., the secondary source update). This is done by applying a time-dependent transformation of the attitude parameters, as described in Section 6.1.3 of Lindegren et al. (2012).

In practise ${\bm{\epsilon}}_{0}$ is obtained by comparing the radio positions of ICRF sources ($\alpha $ and $\delta $ in the above equations) with the positions of their optical counterparts as obtained in the astrometric solution ($\stackrel{~}{\alpha}$, $\stackrel{~}{\delta}$). The second version of the ICRF, ICRF2 (Fey et al. 2015) contains at least some 2000 sources that are optically bright enough to be measured by Gaia. For Gaia DR2 a prototype version of ICRF3 was used (see below).

The determination of $\bm{\omega}$ can use a much larger set of quasars, because Equation 3.96 does not require their precise positions in ICRS to be known, only their proper motions in ICRS (${\mu}_{\alpha *}$, ${\mu}_{\delta}$). The latter are expected to be very small, but not zero. Apart from various random effects, including the peculiar motions of quasars and optical variability causing centroid displacements (Bachchan et al. 2016), the most important systematic proper motion pattern for quasars is expected to be produced by the secular variation of stellar aberration due to the galactocentric acceleration of the solar-system barycentre (Kopeikin and Makarov 2006). The galactocentric acceleration vector $\bm{g}$ should have a magnitude of $g\simeq 2\times {10}^{-10}$ m s${}^{-2}$, pointing more or less towards the galactic centre. The effect on the quasars is an apparent streaming motion towards the galactic centre, described by

$$\begin{array}{cc}\hfill {\mu}_{\alpha *}& ={\bm{p}}^{\prime}\bm{a}\hfill \\ \hfill {\mu}_{\delta}& ={\bm{q}}^{\prime}\bm{a}\hfill \end{array}\mathit{\hspace{1em}}\}$$ | (3.97) |

where $\bm{a}=\bm{g}/c$ and $c$ is the speed of light. The magnitude of the effect is thus $a=g/c\simeq 4.3$ $\mu $as. It is not wise to assume that this effect is precisely known; instead the components of $\bm{a}$ in the ICRS should be introduced as additional unknowns when solving for $\bm{\omega}$. This leads to an augmented version of Equation 3.96 with six unknowns,

$$\left[\begin{array}{c}\hfill {\stackrel{~}{\mu}}_{\alpha *}\hfill \\ \hfill {\stackrel{~}{\mu}}_{\delta}\hfill \end{array}\right]=\left[\begin{array}{cccccc}\hfill -\mathrm{sin}\delta \mathrm{cos}\alpha \hfill & \hfill -\mathrm{sin}\delta \mathrm{sin}\alpha \hfill & \hfill \mathrm{cos}\delta \hfill & \hfill -\mathrm{sin}\alpha \hfill & \hfill \mathrm{cos}\alpha \hfill & \hfill 0\hfill \\ \hfill \mathrm{sin}\alpha \hfill & \hfill -\mathrm{cos}\alpha \hfill & \hfill 0\hfill & \hfill -\mathrm{sin}\delta \mathrm{cos}\alpha \hfill & \hfill -\mathrm{sin}\delta \mathrm{sin}\alpha \hfill & \hfill \mathrm{cos}\delta \hfill \end{array}\right]\left[\begin{array}{c}\hfill {\omega}_{X}\hfill \\ \hfill {\omega}_{Y}\hfill \\ \hfill {\omega}_{Z}\hfill \\ \hfill {a}_{X}\hfill \\ \hfill {a}_{Y}\hfill \\ \hfill {a}_{Z}\hfill \end{array}\right].$$ | (3.98) |

If the quasars are reasonably distributed over the celestial sphere, the correlation between the least-squares estimates of $\bm{\omega}$ and $\bm{a}$ will be small (Bachchan et al. 2016). The additional unknowns will therefore not weaken the solution for $\bm{\omega}$, but are essential to eliminate possible biases caused by the acceleration terms.

The orientation, spin, and acceleration parameters represent the lowest-order terms of a general expansion of position and proper motion differences in terms of vector spherical harmonics (Mignard and Klioner 2012). Other terms are expected to be negligible and can be used to check the internal consistency of the Gaia reference frame.

For Gaia DR2, Equation 3.95 was used to align the Gaia reference frame
to ICRF at the reference epoch J2015.5, using the optical counterparts of VLBI sources
in a prototype version of ICRF3 made available to DPAC by the IAU Working Group
*Third Realisation of International Celestial Reference Frame* (P. Charlot, chair).
The prototype ICRF3 lists 4262 sources, of which 2844 were used for the alignment.
Equation 3.96 was used to make the Gaia reference frame non-rotating,
based on 492 009 sources presumed to be quasars. These included, in addition to the 2844
ICRF3 sources, 489 165 sources found by cross-matching a provision version of the
Gaia catalogue with the AllWISE AGN catalogue
(Secrest et al. 2015, 2016). The acceleration parameters
in Equation 3.98 were not used as the proper motions in Gaia DR2
still have too large systematics for a meaningful determination of them. More details
are given in Mignard et al. (2018).