4.1.4 Standard model of stellar motion

Author(s): Lennart Lindegren

In the astrometric processing described in this chapter, the motions in the BCRS of all sources beyond the solar-system (i.e., stars and extragalactic objects) are modelled using the ‘standard model of stellar motion’. This model was also used in the construction of the Hipparcos Catalogue (ESA 1997, Volume 1, Section 1.2.8). The model assumes that the source is moving with uniform velocity relative to the Solar-System Barycentre (SSB), and its barycentric position $\boldsymbol{b}(t)$ is thus described by the linear model

 $\boldsymbol{b}(t)=\boldsymbol{b}_{\text{ep}}+(t-t_{\text{ep}})\boldsymbol{v}$ (4.1)

with six parameters, namely the components in BCRS of $\boldsymbol{b}_{\text{ep}}$ (unit: m) and $\boldsymbol{v}$ (unit: m s${}^{-1}$). $t_{\text{ep}}$ is the reference epoch of the catalogue. The barycentric coordinate direction (unit vector) to the source at time $t$ (in TCB) is then

 $\boldsymbol{\bar{u}}_{\text{B}}(t)=\left\langle\boldsymbol{b}_{\text{ep}}+(t-t% _{\text{ep}})\boldsymbol{v}\right\rangle\,,$ (4.2)

where the angular brackets signify vector normalisation: $\langle\boldsymbol{a}\rangle=\boldsymbol{a}|\boldsymbol{a}|^{-1}$. Equation 4.2 ignores the finite speed of light. In principle, the barycentric coordinate direction measured at time $t$ corresponds to the barycentric position of the source at the time $t-|\boldsymbol{b}|c^{-1}$, several (or many) years earlier. However, it would be highly impractical to take this into account because the distance $|\boldsymbol{b}|$ is rarely known to sufficient accuracy. The standard model is therefore parametrised by quantities representing the position and motion of the source as they appear from the SSB at a given time. Thus, the time argument in $\boldsymbol{\bar{u}}_{\text{B}}(t)$ must always be interpreted as the time of light arrival at the SSB, not as the time of light emission from the source.

For similar reasons it would be highly impractical to use the rectangular components of $\boldsymbol{b}_{\text{ep}}$ and $\boldsymbol{v}$ as the parameters of the standard model. By convention, the six parameters of the standard model are instead defined as follows (Klioner 2003a; Lindegren et al. 2012):

• The barycentric right ascension $\alpha$ and declination $\delta$ at the reference epoch are defined in terms of the barycentric coordinate direction at the reference epoch, expressed in the BCRS as

 $\boldsymbol{\bar{u}}_{\text{B}}(t_{\text{ep}})=\begin{bmatrix}\cos\alpha\cos% \delta\\ \sin\alpha\cos\delta\\ \sin\delta\end{bmatrix}\,.$ (4.3)
• The components of the proper motion in right ascension $\mu_{\alpha*}$ and in declination $\mu_{\delta}$ at the reference epoch are defined in terms of the derivatives of the barycentric coordinate direction at the reference epoch, expressed in the BCRS as

 $\left.\frac{\text{d}\boldsymbol{\bar{u}}_{\text{B}}}{\text{d}t}\right|_{t=t_{% \text{ep}}}=\begin{bmatrix}-\sin\alpha\\ \cos\alpha\\ 0\end{bmatrix}\mu_{\alpha*}+\begin{bmatrix}-\cos\alpha\sin\delta\\ -\sin\alpha\sin\delta\\ \cos\delta\end{bmatrix}\mu_{\delta}\,.$ (4.4)

(The notations $\boldsymbol{r}$, $\boldsymbol{p}$, and $\boldsymbol{q}$ are later introduced for the three unit vectors appearing in Equation 4.3 and Equation 4.4.)

• The parallax $\varpi$ is related to the barycentric distance to the source according to

 $\varpi=|\boldsymbol{b}|\,A_{\text{u}}^{-1}\,,$ (4.5)

where $A_{\text{u}}=149\,597\,870\,700$ m is the astronomical unit (Capitaine 2012).

• The sixth astrometric parameter quantifies the radial motion of the source relative to the SSB, and may be taken to be the radial velocity $v_{r}$ (more precisely the astrometric radial velocity; see Lindegren and Dravins 2003), or alternatively the ‘radial proper motion’

 $\mu_{r}=v_{r}\varpi A_{\text{u}}^{-1}\,.$ (4.6)

As can be seen by comparing Equation 4.5 and Equation 4.6 the radial proper motion is the relative change in distance per unit time, and can be expressed in the same unit as the transverse proper motion components $\mu_{\alpha*}$ and $\mu_{\delta}$.

Although there are thus six astrometric parameters that could be fitted to the observations according to the standard model, the radial velocity is always taken from spectroscopic measurements (ground-based catalogues or Gaia’s Radial Velocity Spectrometer). Therefore, only $\alpha$, $\delta$, $\varpi$, $\mu_{\alpha*}$, and $\mu_{\delta}$ are in practise fitted to the data; they are commonly referred to as the ‘five astrometric parameters’. Nevertheless, for applying the model, and propagating the parameters between epochs, all six astrometric parameters are needed. $\mu_{r}=0$ is assumed if the radial velocity is not known.

Referring to $v_{r}$ or $\mu_{r}$ as the sixth astrometric parameter is potentially confusing in connection with the so-called six-parameter solutions discussed in Section 4.3.4, where the pseudocolour $\hat{\nu}_{\text{eff}}$ is estimated as an additional (sixth) parameter after the standard five.

However, while the pseudocolour may be called a source parameter, it is not an astrometric parameter.

The conventional unit for $\alpha$ and $\delta$ is degrees or radians, for $\varpi$ it is mas, and for $\mu_{\alpha*}$, $\mu_{\delta}$, and $\mu_{r}$ it is mas per Julian year of 31 557 600 s TCB, abbreviated mas yr${}^{-1}$. Differential quantities in $\alpha$ and $\delta$ ($\Delta\alpha*\equiv\Delta\alpha\cos\delta$ and $\Delta\delta$), including uncertainties, are expressed in mas.

That the standard model of stellar motions neglects light-time effects has some non-trivial consequences (Stumpff 1985; Butkevich and Lindegren 2014). Apart from the obvious fact that the currently measured directions to the stars represent their actual positions in space at a much earlier epoch, we need to consider the following.

1. 1.

For the modelling of the apparent motions of stars in our Galaxy, the standard model is always adequate at the precision of Gaia, in the sense that the neglected light-time effects at most produce prediction errors in position of $\sim$10 $\mu$as after a ten-year interval, and much smaller errors for most stars over much longer intervals of time.

2. 2.

For the interpretation of the astrometric parameters in terms of the physical motion of stars in the Galaxy, it may be necessary to take light-time effects into account for a wider range of objects. The dominant effect is that the Doppler factor, equal to $(1-v_{r}/c)^{-1}$, where $c$ is the speed of light, needs to be included when calculating the (true) space velocity of a star from its (apparent) astrometric parameters (cf. Klioner 2003a).

These two effects should not be confused with the far more important perspective acceleration (e.g., van de Kamp 1981), which is a purely geometrical effect caused by the changing distance to the source and changing angle between the velocity vector and the line of sight. It is an observationally well-established effect that needs to be taken into account in the astrometric solutions for all high-velocity, nearby stars (cf. de Bruijne and Eilers 2012). The perspective acceleration is fully accounted for in the standard model, provided that the radial velocity is known and used in the model.

While the standard model is routinely fitted to all non-solar-system sources observed by Gaia, it will give a bad fit for a substantial fraction of the sources that have manifestly non-uniform space motions or other complications. These include astrometric binaries and exoplanetary systems, resolved or partially resolved non-single stars with significant orbital motion, variability-induced movers (VIMs), stars with surface structure, and various kinds of extended objects. The Gaia observations of such sources are analysed in the special DPAC ‘Object Processing’, using a range of dedicated procedures. It is however important that the standard model is fitted also to these sources, as it provides in most cases a meaningful approximation to the astrometric parameters, and since the goodness-of-fit can be used to select sources for the object processing.

The standard model is also used for quasars and other sufficiently point-like extragalactic objects. Their parallaxes and proper motions are fitted exactly as for stars, even though it is known a priori that they are very small. This is important in order not to force (i.e. bias) the solution and because the classification of the source could be wrong; but also because the fitted parameters are needed for aligning the reference frame of Gaia (Section 4.3.2) and in the quality assessment (Section 4.5.10).