Author(s): Lennart Lindegren

The astrometric model is a recipe for calculating the proper direction ${\bm{u}}_{i}(t)$ to a source ($i$) at an arbitrary time of observation ($t$) in terms of its astrometric parameters ${\bm{s}}_{i}$ and various auxiliary data, assumed to be known with sufficient accuracy. The auxiliary data include an accurate barycentric ephemeris of the Gaia satellite, ${\bm{b}}_{\text{G}}(t)$, including its coordinate velocity $\text{d}{\bm{b}}_{\text{G}}/\text{d}t$, and ephemerides of all other relevant solar-system bodies. The details of the model have been outlined in Section 3.2 of Lindegren et al. (2012) or Section 3.1.4 and only a short introduction is given here.

As explained in Section 3.1.3, the astrometric parameters refer to the ICRS and the time coordinate used is TCB. The reference epoch ${t}_{\text{ep}}$ is preferably chosen to be near the mid-time of the mission in order to minimize statistical correlations between the position and proper motion parameters.

The transformation between the kinematic and the astrometric parameters is non-trivial (Klioner 2003), mainly as a consequence of the practical need to neglect most of the light-propagation time $t-{t}_{*}$ between the emission of the light at the source (${t}_{*}$) and its reception at Gaia ($t$). This interval is typically many years and its value, and rate of change (which depends on the radial velocity of the source), will in general not be known with sufficient accuracy to allow modelling of the motion of the source directly in terms of its kinematic parameters according to Equation 3.1. The proper motion components ${\mu}_{\alpha *i}$, ${\mu}_{\delta i}$ and radial velocity ${v}_{ri}$ correspond to the ‘apparent’ quantities discussed by in Sect. 8 of Klioner (2003).

The coordinate direction to the source at time $t$ is calculated with the same ‘standard model’ as was used for the reduction of the Hipparcos observations (ESA (1997), Vol. 1, Sect. 1.2.8), namely

$${\overline{\bm{u}}}_{i}(t)=\u27e8{\bm{r}}_{i}+({t}_{\text{B}}-{t}_{\text{ep}})({\bm{p}}_{i}{\mu}_{\alpha *i}+{\bm{q}}_{i}{\mu}_{\delta i}+{\bm{r}}_{i}{\mu}_{ri})-{\varpi}_{i}{\bm{b}}_{\text{G}}(t)/{A}_{\text{u}}\u27e9$$ | (3.99) |

where the angular brackets signify vector length normalization, and $[{\bm{p}}_{i}{\bm{q}}_{i}{\bm{r}}_{i}]$ is the ‘normal triad’ of the source with respect to the ICRS (Murray 1983). In this triad, ${\bm{r}}_{i}$ is the barycentric coordinate direction to the source at time ${t}_{\text{ep}}$, ${\bm{p}}_{i}=\u27e8\bm{Z}\times {\bm{r}}_{i}\u27e9$, and ${\bm{q}}_{i}={\bm{r}}_{i}\times {\bm{p}}_{i}$. The components of these unit vectors in the ICRS are given by the columns of the matrix

$${\U0001d5a2}^{\prime}[{\bm{p}}_{i}{\bm{q}}_{i}{\bm{r}}_{i}]=\left[\begin{array}{ccc}\hfill -\mathrm{sin}{\alpha}_{i}\hfill & \hfill -\mathrm{sin}{\delta}_{i}\mathrm{cos}{\alpha}_{i}\hfill & \hfill \mathrm{cos}{\delta}_{i}\mathrm{cos}{\alpha}_{i}\hfill \\ \hfill \mathrm{cos}{\alpha}_{i}\hfill & \hfill -\mathrm{sin}{\delta}_{i}\mathrm{sin}{\alpha}_{i}\hfill & \hfill \mathrm{cos}{\delta}_{i}\mathrm{sin}{\alpha}_{i}\hfill \\ \hfill 0\hfill & \hfill \mathrm{cos}{\delta}_{i}\hfill & \hfill \mathrm{sin}{\delta}_{i}\hfill \end{array}\right].$$ | (3.100) |

${\bm{b}}_{\text{G}}(t)$ is the barycentric position of Gaia at the time of observation, and ${A}_{\text{u}}$ the astronomical unit. ${t}_{\text{B}}$ is the barycentric time obtained by correcting the time of observation for the Römer delay; to sufficient accuracy it is given by

$${t}_{\text{B}}=t+{\bm{r}}_{i}^{\prime}{\bm{b}}_{\text{G}}(t)/c,$$ | (3.101) |

where $c$ is the speed of light. See Section 3.2 of Lindegren et al. (2012) for further details.

The iterative updating of the sources is described in Section 5.1 of Lindegren et al. (2012). There the method of identifying potential outliers and estimating the excess source noise ${\u03f5}_{i}$ is also outlined together with the calculation of the partial derivatives of the coordinate direction with respect to the source parameters.

The perspective acceleration modelled by ${\mu}_{r}={v}_{r}\varpi /{A}_{\text{u}}$ in Equation 3.99 is totally negligible for the vast majority of sources, and ${\mu}_{r}=0$ (at the reference epoch) is therefore normally assumed, equivalent ${v}_{r}=0$. For Gaia DR2 the effect was however taking into account by assuming non-zero values for the 53 nearby, high-velocity Hipparcos stars listed in Table 3.3. The accumulated effect over a time interval $T$ is $\mathrm{\Delta}=|{v}_{r}|\mu \varpi {T}^{2}/{A}_{\text{u}}$, where $\mu ={({\mu}_{\alpha *}^{2}+{\mu}_{\delta})}^{1/2}$ is the total proper motion. Table 3.3 contains the sources for which the predicted $\mathrm{\Delta}$, calculated for $T=1.75$ yr using Hipparcos astrometry (van Leeuwen 2007), exceeds 0.023 mas.

Source identifier | HIP | ${v}_{r}$ | $\mathrm{\Delta}$ | Name |

[km s${}^{-1}$] | [mas] | |||

Gaia DR2 4472832130942575872 | 87937 | $-$110.51 | 1.975 | Barnard’s star |

Gaia DR2 4810594479417465600 | 24186 | 245.19 | 1.694 | Kapteyn’s star |

Gaia DR2 2552928187080872832 | 3829 | 263.00 | 0.573 | Van Maanen 2 |

Gaia DR2 1872046574983507456 | 104214 | $-$65.74 | 0.313 | 61 Cyg A |

Gaia DR2 1872046574983497216 | 104217 | $-$64.07 | 0.297 | 61 Cyg B |

Gaia DR2 4034171629042489088 | 57939 | $-$98.35 | 0.239 | Groombridge 1820 |

Gaia DR2 5853498713160606720 | 70890 | $-$22.40 | 0.208 | $\alpha $ Cen C (Proxima) |

Gaia DR2 6412595290592307840 | 108870 | $-$40.00 | 0.163 | $\u03f5$ Ind |

Gaia DR2 3340477717172813568 | 26857 | 105.83 | 0.144 | Ross 47 |

Gaia DR2 4847957293277762560 | 15510 | 87.40 | 0.141 | e Eri |

Gaia DR2 6307365499463905536 | 74234 | 310.77 | 0.126 | |

Gaia DR2 6307374845312759552 | 74235 | 310.12 | 0.124 | |

Gaia DR2 2306965202564506752 | 439 | 25.38 | 0.112 | |

Gaia DR2 3195919528988725120 | 19849 | $-$42.32 | 0.109 | |

Gaia DR2 5918660719981686144 | 86990 | $-$115.00 | 0.109 | |

Gaia DR2 6697578465310949376 | 99461 | $-$126.90 | 0.109 | |

Gaia DR2 1057318835428596096 | 55360 | 60.40 | 0.063 | |

Gaia DR2 6553614253923452800 | 114046 | 8.81 | 0.058 | |

Gaia DR2 1129149723913123456 | 57544 | $-$111.65 | 0.058 | |

Gaia DR2 6254033894120917760 | 76901 | 84.92 | 0.058 | |

Gaia DR2 6583272171335359360 | 105090 | 20.11 | 0.055 | |

Gaia DR2 2739689239311660672 | 117473 | $-$71.32 | 0.052 | |

Gaia DR2 1057879895596316416 | 56936 | $-$118.00 | 0.050 | |

Gaia DR2 880495478530513920 | 38541 | $-$234.45 | 0.050 | |

Gaia DR2 3519785523672576384 | 60559 | 51.17 | 0.046 | |

Gaia DR2 5955305209191546112 | 86214 | $-$60.00 | 0.044 | |

Gaia DR2 6228695270697905280 | 72511 | $-$39.60 | 0.043 | |

Gaia DR2 3796072592206250624 | 57548 | $-$31.09 | 0.040 | |

Gaia DR2 470826482635704064 | 21088 | 27.90 | 0.038 | |

Gaia DR2 5588607120530408832 | 37853 | 106.16 | 0.038 | |

Gaia DR2 6228695236338166528 | 72509 | $-$38.80 | 0.038 | |

Gaia DR2 4278722497040124032 | 91668 | 196.00 | 0.037 | |

Gaia DR2 4937000898855759104 | 10138 | 57.00 | 0.036 | |

Gaia DR2 2358524597030794112 | 5643 | 28.09 | 0.033 | |

Gaia DR2 6885776098199761024 | 104432 | $-$58.27 | 0.032 | |

Gaia DR2 266846631637524864 | 23518 | 64.54 | 0.030 | |

Gaia DR2 5425628298649940608 | 47425 | 142.00 | 0.030 | |

Gaia DR2 385334230892516480 | 1475 | 11.62 | 0.030 | |

Gaia DR2 6232511606838403968 | 73184 | 26.79 | 0.029 | |

Gaia DR2 4293318823182081408 | 94761 | 35.88 | 0.028 | |

Gaia DR2 2452378776434276992 | 8102 | $-$16.68 | 0.028 | |

Gaia DR2 2261614264930275072 | 96100 | 26.78 | 0.027 | |

Gaia DR2 19316224572460416 | 12114 | 25.85 | 0.026 | |

Gaia DR2 2007876324466455424 | 110893 | $-$33.94 | 0.026 | |

Gaia DR2 5690980582306104448 | 48336 | 61.60 | 0.026 | |

Gaia DR2 1637645127018395776 | 86162 | $-$28.58 | 0.026 | |

Gaia DR2 6894054664842632448 | 106255 | $-$57.70 | 0.025 | |

Gaia DR2 6378584028690858496 | 113229 | 46.60 | 0.025 | |

Gaia DR2 6847167606385195648 | 99825 | $-$55.10 | 0.024 | |

Gaia DR2 4268226078065241600 | 94349 | $-$42.40 | 0.024 | |

Gaia DR2 823773494718931968 | 49908 | $-$25.73 | 0.024 | |

Gaia DR2 5339892367683264384 | 55042 | $-$35.00 | 0.024 | |

Gaia DR2 5902750168276592256 | 75181 | $-$66.50 | 0.023 |