3.3.3 Astrometric source model

Author(s): Lennart Lindegren

The astrometric model is a recipe for calculating the proper direction 𝒖i(t) to a source (i) at an arbitrary time of observation (t) in terms of its astrometric parameters 𝒔i and various auxiliary data, assumed to be known with sufficient accuracy. The auxiliary data include an accurate barycentric ephemeris of the Gaia satellite, 𝒃G(t), including its coordinate velocity d𝒃G/dt, 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 tep 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 μα*i, μδi and radial velocity vri 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

𝒖¯i(t)=𝒓i+(tB-tep)(𝒑iμα*i+𝒒iμδi+𝒓iμri)-ϖi𝒃G(t)/Au (3.99)

where the angular brackets signify vector length normalization, and [𝒑i𝒒i𝒓i] is the ‘normal triad’ of the source with respect to the ICRS (Murray 1983). In this triad, 𝒓i is the barycentric coordinate direction to the source at time tep, 𝒑i=𝒁×𝒓i, and 𝒒i=𝒓i×𝒑i. The components of these unit vectors in the ICRS are given by the columns of the matrix

𝖢[𝒑i𝒒i𝒓i]=[-sinαi-sinδicosαicosδicosαicosαi-sinδisinαicosδisinαi0cosδisinδi]. (3.100)

𝒃G(t) is the barycentric position of Gaia at the time of observation, and Au the astronomical unit. tB is the barycentric time obtained by correcting the time of observation for the Römer delay; to sufficient accuracy it is given by

tB=t+𝒓i𝒃G(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 ϵi 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 μr=vrϖ/Au in Equation 3.99 is totally negligible for the vast majority of sources, and μr=0 (at the reference epoch) is therefore normally assumed, equivalent vr=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 Δ=|vr|μϖT2/Au, where μ=(μα*2+μδ)1/2 is the total proper motion. Table 3.3 contains the sources for which the predicted Δ, calculated for T=1.75 yr using Hipparcos astrometry (van Leeuwen 2007), exceeds 0.023 mas.

Table 3.3: Sources for which the astrometric solution was corrected for perspective acceleration using vr0.
Source identifier HIP vr Δ 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 α Cen C (Proxima)
Gaia DR2 6412595290592307840 108870 -40.00 0.163 ϵ 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
Notes. vr is the assumed radial velocity taken from SIMBAD (Wenger et al. 2000). Δ is the predicted size of the effect calculated as described in the text. Common names are only given for the first ten entries.