3 The "barycentric radial-velocity measure''

In order to eliminate ambiguities of classical radial-velocity concepts, Lindegren et al. (1999) proposed a stringent definition which was later adopted as Resolution C1 at the IAU General Assembly in Manchester (Rickman 2002). This recommends that accurate spectroscopic radial velocities should be given as the barycentric radial-velocity measure  $cz_{\rm B}$, which is the measured (absolute) line shift corrected for gravitational effects of the solar-system bodies and effects of the observer's displacement and motion relative to the solar-system barycentre. Thus,  $cz_{\rm B}$ does not include corrections e.g. for the gravitational redshift of the star or convective motions in the stellar atmosphere, and therefore cannot directly be interpreted as a radial motion of the star.

From its definition it is clear that the radial-velocity measure is not a unique quantity for a given star (at a certain time), but refers to particular spectral features observed under specific conditions (resolution, etc.). Lest the radial-velocity measure should become meaningless at the highest level of accuracy, these features and conditions should be clearly specified along with the results (cf. Sect. 7.2).

Let $z_{\rm obs}=(\lambda_{\rm obs}-\lambda_{\rm lab})/\lambda_{\rm lab}$be the observed shift of a certain spectral feature, or (more usually) the mean shift resulting from many such features in a single spectrum. In order to compute $cz_{\rm B}$ we need to eliminate the effects of the barycentric motion of the observer and the fact that the observation is made from within the gravitational field of the Sun. Lindegren & Dravins (2002) give the following formula, which for present purposes is accurate to better than 1 m s^-1:

 

$$1+z_{\rm B} = (1+z_{\rm obs}) \left(1 - \frac{\textstyle\Phi_{\rm... ...^2}\right)^{-1} \left(1+\frac{\textstyle v_{\rm LOS}} {\textstyle c}\right) ~ .$$ (1)

Here c=299 792 458 m s^-1 is the speed of light, $\Phi_{\rm obs}$ the (positive) Newtonian potential at the observer, $\vec{v}_{\rm obs}$ the barycentric velocity of the observer and $v_{\rm LOS}=\vec{k}{'}\vec{v}_{\rm obs}$ the line-of-sight velocity of the observer. ($\vec{k}$ is the unit vector from the observer towards the star, taking into account stellar proper motion and parallax, but not aberration and refraction.) The required radial-velocity measure is the barycentric line shift $z_{\rm B}$multiplied by the speed of light; it is thus expressed in velocity units although it cannot readily be interpreted as a velocity at the sub-km s^-1 accuracy level.

The main correction in Eq. (1) is the last factor, caused by the observer's barycentric motion along the line of sight. $v_{\rm LOS}$is normally provided to sufficient accuracy by standard reduction softwares (Sect. 5.4). The other correction factor is caused by gravitational time dilation and transverse Doppler effect at the observer, and amounts to

$$\left(1 - \frac{\textstyle\Phi_{\rm obs}}{\textstyle c^2} - ... ...e 2c^2}\right)^{-1} \simeq 1+(1.48 \pm 0.03)\times 10^{-8} ~ ,$$ (2)

where $\pm 0.03\times10^{-8}$ is the amplitude of periodic variations caused by the ellipticity of the Earth's orbit around the Sun. This amplitude corresponds to a maximum error of 0.1 m s^-1 in the radial-velocity measure. For the present purpose this factor can therefore be regarded as constant. Equations (1)-(2) define the required transformation from observed line shifts to the barycentric radial-velocity measure.

Because the Doppler effect as well as the barycentric correction is multiplicative in the wavelength $\lambda$, the analysis of lineshifts is best made in the logarithmic domain. We introduce $\Lambda=\ln\lambda$, and use the dimensionless u to designate a small shift in $\Lambda$. The shift can also be expressed in velocity units as d=cu. This is related to the usual spectral shift z through $1+z=\exp(d/c)$, which by expansion gives $cz=d+d^2/(2c)+d^3/(6c^2)+\cdots$. cz and d are therefore alternative ways of expressing spectral shifts in velocity units, differing in the second-order terms, but none of them strictly representing physical velocity v. d has the advantage over czthat the various effects (or corrections) are additive in this variable. The distinction between them disappears, to the nearest m s^-1, for shifts <17 km s^-1. Second-order relations are adequate to the same precision for shifts <600 km s^-1.

The radial-velocity measure for a star should be referred to the mean epoch of observation, expressed as the barycentric time of arrival ($t_{\rm B}$), i.e. the time of observation ( $t_{\rm obs}$) corrected for the Rømer delay associated with the observer's motion around the solar-system barycentre. Neglecting terms due to relativity and wavefront curvature, which together are less than 1 ms for stellar objects, this correction can be computed as

$$t_{\rm B} = t_{\rm obs} + \vec{k}'\vec{r}_{\rm obs}/c$$ (3)

(Lindegren & Dravins 2002), where $\vec{r}_{\rm obs}$ is the barycentric position of the observer at the time of observation and $\vec{k}$ the previously defined unit vector towards the star.

Figure 1: This flowchart outlines the reduction process described in the text. The grey box contains processes that are included in the ELODIE software package TACOS. All spectra of stars and the Moon are piped through the upper branch of processes, and thus receive identical treatment. The lunar spectra are also passed through the lower branch in order to determine the long-term drift correction which effectively defines the zero point of the final radial-velocity measures.