The GaiaESO Survey: Empirical determination of the precision of stellar radial velocities and projected rotation velocities^{⋆,}^{⋆⋆}
^{1}
Astrophysics Group, Keele University,
Keele,
Staffordshire
ST5 5BG,
UK
email:
r.j.jackson@keele.ac.uk
^{2}
Institute of Astronomy, University of Cambridge,
Madingley Road, Cambridge,
CB3 0HA,
UK
^{3}
Moscow MV Lomonosov State University, Sternberg Astronomical
Institute, 119992
Moscow,
Russia
^{4}
INAF–Osservatorio Astrofisico di Arcetri, Largo E. Fermi
5, 50125
Florence,
Italy
^{5}
Research School of Astronomy & Astrophysics, Australian
National University, Cotter
Road, Weston Creek,
ACT
2611,
Australia
^{6}
Rudolf Peierls Centre for Theoretical Physics, Keble
Road, Oxford,
OX1 3NP,
UK
^{7}
GEPI, Observatoire de Paris, CNRS, Université Paris
Diderot, 5 place Jules
Janssen, 92190
Meudon,
France
^{8}
Centre for Astrophysics Research, STRI, University of
Hertfordshire, College Lane
Campus, Hatfield
AL10 9AB,
UK
^{9}
Lund Observatory, Department of Astronomy and Theoretical
Physics, Box 43,
221 00
Lund,
Sweden
^{10}
Institute of Astronomy, University of Edinburgh,
Blackford Hill, Edinburgh
EH9 3HJ,
UK
^{11}
INAF–Osservatorio Astronomico di Palermo, Piazza del Parlamento
1, 90134
Palermo,
Italy
^{12}
Departamento de Física, Ingeniería de Sistemas y Teoría de la
Señal,
Universidad de Alicante, Apdo.
99, 03080
Alicante,
Spain
^{13}
ESA, ESTEC, Keplerlaan 1, Po Box
299
2200 AG,
Noordwijk, The
Netherlands
^{14}
MaxPlanck Institut für Astronomie, Königstuhl 17, 69117
Heidelberg,
Germany
^{15}
INAF–Padova Observatory, Vicolo dell’Osservatorio 5, 35122
Padova,
Italy
^{16}
Instituto de Astrofísica de AndalucíaCSIC,
Apdo. 3004, 18080
Granada,
Spain
^{17}
Instituto de Astrofísica de Canarias, 38205, La Laguna, Tenerife, Spain
^{18}
Universidad de La Laguna, Dept. Astrofísica, 38206,
La Laguna, Tenerife,
Spain
^{19}
Royal Observatory of Belgium, Ringlaan 3, 1180
Brussels,
Belgium
^{20}
INAF–Osservatorio Astronomico di Bologna, via Ranzani
1, 40127
Bologna,
Italy
^{21}
Department of Physics and Astronomy, Uppsala
University, Box
516, 751 20
Uppsala,
Sweden
^{22}
Dipartimento di Fisica e Astronomia, Sezione Astrofisica,
Università di Catania, via S. Sofia
78, 95123
Catania,
Italy
^{23}
ASI Science Data Center, Via del Politecnico SNC,
00133
Roma,
Italy
^{24}
Laboratoire Lagrange (UMR 7293), Université de NiceSophia
Antipolis, CNRS, Observatoire de la Côte d’Azur, CS 34229, 06304
Nice Cedex 4,
France
^{25}
Department for Astrophysics, Nicolaus Copernicus Astronomical Center, ul. Rabiańska
8, 87–100
Toruń,
Poland
^{26}
Institut d’Astronomie et d’Astrophysique, Université libre de
Brussels, Boulevard du
Triomphe, 1050
Brussels,
Belgium
^{27}
Instituto de Física y Astronomiía, Universidad de
Valparaiíso, Chile
^{28}
European Southern Observatory, Alonso de Cordova 3107 Vitacura, Santiago de Chile,
Chile
^{29}
INAF–Osservatorio Astrofisico di Catania, via S. Sofia
78, 95123
Catania,
Italy
^{30}
Astrophysics Research Institute, Liverpool John Moores
University, 146 Brownlow
Hill, Liverpool
L3 5RF,
UK
^{31}
Departamento de Ciencias Físicas, Universidad Andrés
Bello, República 220,
8370134
Santiago,
Chile
^{32}
Millennium Institute of Astrophysics, Av. Vicuña Mackenna 4860, 7820436 Macul,
Santiago,
Chile
^{33}
Pontificia Universidad Católica de Chile,
Av. Vicuña Mackenna 4860, 7820436
Macul, Santiago,
Chile
^{34}
Instituto de Astrofísica e Ciências do Espaço, Universidade do
Porto, CAUP, Rua das
Estrelas, 4150762
Porto,
Portugal
Received: 2 April 2015
Accepted: 25 May 2015
Context. The GaiaESO Survey (GES) is a large public spectroscopic survey at the European Southern Observatory Very Large Telescope.
Aims. A key aim is to provide precise radial velocities (RVs) and projected equatorial velocities (vsini) for representative samples of Galactic stars, which will complement information obtained by the Gaia astrometry satellite.
Methods. We present an analysis to empirically quantify the size and distribution of uncertainties in RV and vsini using spectra from repeated exposures of the same stars.
Results. We show that the uncertainties vary as simple scaling functions of signaltonoise ratio (S/N) and vsini, that the uncertainties become larger with increasing photospheric temperature, but that the dependence on stellar gravity, metallicity and age is weak. The underlying uncertainty distributions have extended tails that are better represented by Student’s tdistributions than by normal distributions.
Conclusions. Parametrised results are provided, which enable estimates of the RV precision for almost all GES measurements, and estimates of the vsini precision for stars in young clusters, as a function of S/N, vsini and stellar temperature. The precision of individual high S/N GES RV measurements is 0.22–0.26 km s^{1}, dependent on instrumental configuration.
Key words: stars: kinematics and dynamics / open clusters and associations: general
Based on observations collected with the FLAMES spectrograph at VLT/UT2 telescope (Paranal Observatory, ESO, Chile), for the Gaia ESO Large Public Survey (188.B3002).
Full Table 2 is only available at the CDS via anonymous ftp to cdsarc.ustrasbg.fr (130.79.128.5) or via http://cdsarc.ustrasbg.fr/vizbin/qcat?J/A+A/580/A75
© ESO, 2015
1. Introduction
The GaiaESO survey (GES) is a large public survey programme carried out at the ESO Very Large Telescope (UT2 Kueyen) with the FLAMES multiobject instrument (Gilmore et al. 2012; Randich & Gilmore 2013). The survey will obtain high and intermediateresolution spectroscopy of ~10^{5} stars, the majority obtained at resolving powers of R ~ 17 000 with the GIRAFFE spectrograph (Pasquini et al. 2002). The primary objectives are to cover representative samples of all Galactic stellar populations, including thin and thick disc, bulge, halo, and stars in clusters at a range of ages and Galactocentric radii. The spectra contain both chemical and dynamical information for stars as faint as V ~ 19 and, when combined with complementary information from the Gaia satellite, will provide full threedimensional velocities and chemistry for a large and representative sample of stars. The GES began on 31 December 2011 and will continue for approximately 5 years. There are periodic internal and external data releases, and at the time of writing, data from the first 18 months of survey operations have been analysed and released to the survey consortium for scientific exploitation – the “second internal data release”, known as iDR2. Part of the same data have also been released to ESO through the second GaiaESO phase 3 and will soon be available to the general community.
The GES data products include stellar radial velocities (RV) and projected rotation velocities (vsini). A thorough understanding of the uncertainties in RV and vsini is an essential component of many aspects of the GES programme. For instance, the GES data are capable of resolving the kinematics of clusters and star forming regions, but because the RV uncertainties are not negligible compared with the observed kinematic dispersion, an accurate deconvolution to establish intrinsic cluster velocity profiles, massdependent kinematic signatures, net rotation, etc., relies on a detailed knowledge of the RV uncertainties (e.g. Cottaar et al. 2012; Jeffries et al. 2014; Lardo et al. 2015; Sacco et al. 2015). Searching for binary members of clusters and looking for outliers in RV space also requires an understanding of the uncertainty distribution in order to optimise search criteria and minimise falsepositives. Similarly, inverting the projected rotation velocity distribution to a true rotation velocity distribution (e.g. Chandrasekhar & Münch 1950; Dufton et al. 2006) or comparison of the rotation velocity distributions of different samples requires an understanding of how uncertainties in vsini broaden the observed distribution and impose a lower limit to the rotation that can be resolved (Frasca et al. 2015).
These examples illustrate that not only does one wish to know the level of uncertainty in RV and vsini as a function of stellar spectral type, the spectrum signaltonoise ratio (S/N), the rotation rate and possibly other variables, but it is also important to understand whether the uncertainties are normally distributed or perhaps have extended tails that might be better represented in some other way (e.g. Cottaar et al. 2014). The procedures for reducing and analysing the GES spectra will be fully detailed in forthcoming data release papers, but ultimately the RVs and vsini are estimated with a detailed chisquared fitting procedure (Koposov et al., in prep. and Sect. 2.3). Fitting uncertainties can of course be computed, but these are often minor contributors to the overall repeatability of the measurements and therefore underestimate the total uncertainty. In this paper we empirically determine the uncertainties and their probability distribution based upon repeated measurements of the same stars in GES. Our analysis is limited to the >90 per cent of spectra measured with the GIRAFFE spectrograph and deals only with the precision of the measurements, rather than their absolute accuracy.
In Sect. 2 we describe the GES data and the database of repeat measurements for RV and vsini that is available for characterising their uncertainties. In Sect. 3 we show how the differences in RV and vsini measured between repeated observations can be used to determine the underlying distribution of measurement uncertainty, represented by simple scaling functions that depend on S/N and vsini. In Sect. 4 we investigate how these scaling functions alter with stellar properties. Section 5 considers how the measurement uncertainties change for different observational configurations within GES. In Sect. 6, we conclude and provide parametric formulae and coefficients that allow an estimation of the RV and vsini precision of GES measurements.
2. Repeat measurements of radial velocities and projected rotation velocities
2.1. GES observations
The GES employs the FLAMES fibrefed, multiobject instrument, feeding both the UVES highresolution (R ~ 45 000) and GIRAFFE intermediate resolution (R ~ 17 000) spectrographs. More than 90 per cent of the spectra are obtained with GIRAFFE and we deal only with these data here. The Medusa fibre system allows the simultaneous recording of spectra from ≃100 stars in each pointing. The stars in a single pointing are usually related by scientific interest (e.g. a cluster or a bulge field) and cover a limited range of brightness (usually less than a 4 mag spread). A further ≃15 fibres are normally allocated to patches of blank sky.
The GIRAFFE spectrograph permits the recording of a limited spectral range and this is selected through the use of ordersorting filters. Eight of these have been used in the GES (HR3, HR5A, HR6, HR10, HR11, HR14A, HR15N, HR21), each of which records a spectrum over a fixed wavelength range, although just three filters (HR10, HR15N, HR21) are used for the large majority of observations:

Most observations of targets in clusters and star forming regions are made using ordersorting filter HR15N. The wavelength range of this filter (6444–6816 Å) includes both the Hα and lithium lines and can provide useful information on the effective temperature (T_{eff}), gravity (log g), age and magnetic activity of the target stars (Lanzafame et al. 2015).

Most targets in the halo, bulge and disc fields are observed using both filters HR10 and HR21. The main goals here are to provide accurate stellar parameters and chemical abundances.
GES fields are usually observed in observation blocks (OBs) comprising two science exposures of equal duration. In addition, for filters HR10 and HR15N a short “simcal” exposure is interleaved between the science exposures. The “simcal” observation illuminates five dedicated fibres with a thoriumargon (ThAr) lamp, providing a means of monitoring the wavelength calibration. In the HR21 observations, this role was fulfilled by emission lines in the sky spectra and no “simcal” exposures were performed.
2.2. Data reduction
Full details of the GES GIRAFFE data reduction will be given in a forthcoming paper (Lewis et al., in prep.). In brief, the raw data frames are corrected for a bias level using zero exposure bias frames and the resulting images are divided by normalised daytime tungsten lamp exposures to remove pixeltopixel sensitivity variations. The multiple spectra in each CCD frame are traced using the tungsten lamp exposures and then extracted using the optimal algorithm described by Horne (1986). Given the readout noise and gain of the CCD, this algorithm also yields an estimated S/N in the extracted spectral pixels, and it is this estimate that is propagated through subsequent analysis steps leading to the final reported S/N of the spectra. Extracted daytime tungsten lamp spectra are used to correct the overall shape of the spectrum and calibrate the individual transmission efficiencies of each fibre. The wavelength calibration proceeded in two stages. Deep exposures of a daytime ThAr lamp are used to define a polynomial relationship between extracted spectral pixel and wavelength. Then, for observations using filters HR10 or HR15N the wavelength calibration is modified by an offset determined from the positions of prominent arc lines in the nighttime “simcal” exposures. For observations using filter HR21 the offset applied to the wavelength calibration is determined from the position of prominent emission lines in the sky spectra. Spectra are rebinned into 0.05 Å pixels using this wavelength solution and sky is subtracted using a median of the sky spectra corrected for the differing responses of each fibre.
2.3. Radial velocity and projected rotation velocity estimates
The resulting survey spectra are processed and analysed by working groups organised in a workflow described by Gilmore et al. (2012). The RV and vsini estimates used in this report are determined using a pipeline developed by the Cambridge Astronomical Survey Unit (CASU) which follows the general method described by Koposov et al. (2011). Details of the pipeline used to analyse the GES data will be described in a forthcoming paper (Koposov et al., in prep.). A first pass used a standard crosscorrelation method with a grid of synthetic template spectra at a range of temperatures, metallicities and gravities (Munari et al. 2005) to give an initial RV estimate. The second pass used a direct modelling approach that fits each spectrum with a loworder polynomial multiplied by a template spectrum, with the RV, vsini, T_{eff}, log g, metallicity and polynomial coefficients as free parameters. The best fit parameter set is found by chisquared minimisation with emission lines excluded from the fitting process. The fitting process is then repeated using a finer grid to determine optimum values of RV and vsini with the other parameters held constant at their previously determined values.
The chisquared minimisation yields an estimate of the uncertainty in the best fit parameters. However, in the case of GES data, this underestimates the measurement uncertainty, in part due to the analysis step where spectra are rebinned but chiefly due to systematic uncertainties in wavelength calibration (Jeffries et al. 2014). For this reason an empirical determination of the measurement precision is preferred; the measurement uncertainty is estimated by comparing repeated measurements of RV and vsini for the same star.
2.4. Selected data
To empirically characterise the RV and vsini uncertainties and how they depend on stellar parameters requires a database containing a large number of repeat observations of the same stars and a broad range of stellar types and rotational broadening. For these reasons, and especially to ensure a range of vsini, we initially focused on GES data for eight open clusters that were observed using the HR15N filter. These clusters have ages in the range 1 to 600 Myr (see Table 1), covering both premain sequence and main sequence objects. Only a fraction of the targets in each pointing will be actual cluster members, but we expect that cluster members will dominate any subsample of lowmass stars with high vsini, since older field stars are not expected to rotate quickly. To provide a sample with older ages and lower gravities, a field consisting mainly of red giants, observed on repeated occasions as part of the GESCoRoT collaboration, was included.
Numbers of short and long term repeat GIRAFFE observations of RV and vsini used for open clusters with ordersorting filter HR15N.
The data were restricted to observations made with two equal length exposures per OB. Since this is the usual mode of GIRAFFE observations this leads to no significant loss of data. Using this standard arrangement simplifies the analysis and allows two distinct classes of measurement uncertainty to be identified:

Shortterm repeats are where empirical estimates of uncertainties are obtained by comparing RV and vsini values for individual targets derived from spectra measured in each of the individual exposures within an OB. The targets are observed using the same GIRAFFE fibre in the same configuration and are calibrated using the same wavelength solution. In this case the uncertainty is expected to be caused primarily by noise in the target spectra and inherent uncertainties in the reduction and analysis processes. Any drift in wavelength calibration over time, perhaps due to temperature or pressure changes, is expected to be small since the time delay between exposures is always <3000 s and normally <1500 s; there should also be no movement of the fibres and any effects due to imperfect scrambling in the fibre or changing hour angle (see Sect. 6) should also be small. The assumption is also made that any significant velocity shifts due to binary motion on such short timescales will be rare enough to be neglected.

Longterm repeats are where uncertainties are estimated by comparing the mean values of RV and vsini measured in one OB with those measured for the same target in a second OB, where the fibre allocation and configuration on the plate is changed between OBs. In this case the empirical uncertainties are due to the combined effects of noise in the spectra, the analysis techniques plus any external uncertainties in the wavelength calibration or possibly differences due to the particular fibre used for a target or the hour angle of the observation. Binary motion may also contribute to any observed velocity shifts. A subset of these longterm repeat observations were observations of the same star taken on the same night but in a different fibre configuration. These are invaluable in assessing the relative importance of binaries to the velocity shifts. The data used in comparing RV measurements were selected to have S/N> 5 (for the combined spectra in an OB) and those data used to compare vsini have S/N> 5 and vsini> 5 km s^{1}. Table 1 shows the number of short and long term comparisons of RV and vsini available for each cluster. Table 2 shows the time, date, field centre coordinates, exposure times and numbers of targets for each of the GIRAFFE OBs used in this paper. Values of RV, vsini, S/N and stellar properties are taken from the iDR2 iteration of analysis of the GES data, first released by the Cambridge Astronomical Unit to the GES working groups in May 2014 and subsequently placed in the GES archive at the Wide Field Astronomy Unit at Edinburgh University^{1}.
Table 2Log of VLT/Flames observations used in the analysis of RV and vsini measurement precision.
3. Normalised distributions of measurement uncertainty
Figures 1 and 2 show the general characteristics of the observed RV precision, which is defined by the distribution of , the change in RV between shortterm repeat pairs of observations for individual targets divided by . Figure 1 shows  E_{RV}  for ~8500 short term repeats. There is a strong dependence on S/N and vsini such that the measurement precision cannot be represented by a distribution dependent on just one of these parameters. Figure 2 compares the distributions of E_{RV} for short and longterm repeats. The peak height is reduced and the full width half maximum (FWHM) is increased for longterm repeats. There is thus an apparent increase in measurement uncertainty for targets with high S/N when compared to the precision assessed using shortterm repeats of the same stars.
Our general approach is to divide E_{RV} (and the corresponding E_{vsini}) by some function of the target, signal and spectrograph properties, in order to identify the underlying normalised distributions of measurement precision. If the underlying distributions are Gaussian then these normalising functions, S_{RV} and S_{vsini}, would correspond to the standard deviations of E_{RV} and E_{vsini} as a function of S/N, vsini and stellar properties. S_{RV} and S_{vsini}, are used here in a more general sense in order to normalise the E_{RV} and E_{vsini} distributions to an as yet unknown underlying distribution which could be nonGaussian.
Initially, we make the simplifying assumption that the normalising functions depend only on the S/N and vsini of the target star and on the spectrum resolution and pixel size, which are set by the GIRAFFE ordersorting filter.
Fig. 1
The empirical uncertainty in RV precision () estimated from the change in RV between shortterm repeat observations of cluster targets (see Tables 1 and 2) using ordersorting filter HR15N. The size of the symbol indicates the measured value of vsini. 

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Fig. 2
Comparison of the probability density of E_{RV} for short and longterm repeats (see Tables 1 and 2) using ordersorting filter HR15N. The black line shows results for shortterm repeats (i.e. pairs of observations within the same OB). The red histogram shows results for longterm repeats (i.e. spectra of the same targets but taken from different OBs where individual targets are allocated to different fibres). 

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3.1. Normalising functions
RV and vsini are estimated by matching the wavelength offset and line width of a rotationally broadened template spectrum to the measured spectrum. To assess the dependency of uncertainty in RV on S/N and vsini it can be shown (see Appendix A) that the distribution of E_{RV} values measured from short term repeats scales approximately according to W^{3/2}/ (S/N) where W is the FWHM of individual lines in a template spectrum, rotationally broadened to match the line width of the measured spectrum. In this case (also see Appendix A), the RV precision for short term repeats should scale as (1)where C ≈ 0.895c/R_{λ}, R_{λ} is the resolving power of the spectrograph, c is the speed of light and B is an empirically determined parameter that will depend on the type of star being observed. This is consistent with the variation of uncertainty in RV with S/N predicted by Butler et al. (1996) for photon limited errors.
In the case of longterm repeats there is an additional contribution to the measurement uncertainty due to variations in wavelength calibration. This is independent of S/N and vsini and therefore adds a fixed component A in quadrature to the short term uncertainty such that the distribution of E_{RV} for longterm repeats scales as (2)where A will be an empirically determined constant and B and C are as defined in Eq. (1).
The relative precision of vsini used in this paper is defined as (i.e. a fractional precision), where Δvsini is the change between repeat observations and ⟨ vsini ⟩ is their mean value. To find the normalising function for the E_{vsini} distribution we make the assumption that W increases as a function of vsini according to the rotational broadening function given by Gray (1984) and that the uncertainty in W varies as W^{3/2}/ (S/N). In this case the uncertainty for shortterm repeats (see Appendix A) scales as
(3)Again, a constant term is added in quadrature to account for additional sources of uncertainty present in the case of longterm repeats, such that the distribution of E_{vsini} scales as (4)where α and β will be empirically determined constants and C is the same function of spectral resolution featured in Eq. (1).
Fig. 3
Analysis of the empirical uncertainty for shortterm repeat observations of RV using filter HR15N. The solid line in plot a) shows the variation of E_{RV} × (S/N)/(1 + ( [ vsini ] /C)^{2})^{3/4} with S/N. The horizontal line indicates the value of parameter B in Eq. (1) fitted to the full dataset. Blue crosses show the estimated values B as a function of S/N corrected for the measured variation of B with T_{eff} (see Sect. 4.1 and Table 3). Plot b) shows the variation of E_{RV} × S/N with vsini. The solid line show the relationship predicted using the theoretical value of C and the value of B from plot a). The dashed line shows a curve of similar functional form using parameters B and C fitted to the binned data. In plots a) and b) the yaxis shows an estimate of the standard deviation based on the MAD divided by 0.72 (see Sect. 3.2). Plot c) shows the cumulative probability distribution (CDF) of the normalised uncertainty in RV for shortterm repeats. The red solid line shows results for measured data, the dashed line shows the cumulative distribution of a Gaussian with unit dispersion, and the diamond symbols show the cumulative distribution function for a Student’s tdistribution with ν = 6. 

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3.2. Parameters for normalising the RV measurement precision
Parameters A, B and C defining the normalising function S_{RV} are fitted to match the measured distribution of E_{RV} using a dataset of 8,429 repeat observations, with S/N> 5, taken using filter HR15N. Since we expect (and it turns out) that the distributions of these quantities are not Gaussians and have significant nonGaussian tails, we choose to use the median absolute deviation (MAD) to characterise the observed distribution, rather than the square root of the mean variance which could be heavily biased by outliers. An estimate for the standard deviation then follows by noting that the MAD of a unit Gaussian distribution is 0.674, such that MAD/0.674 gives an estimate of the standard deviation. As we shall see, the distributions more closely follow Student’s tdistributions with ν degrees of freedom, for which we determine (by Monte Carlo simulation) the corresponding corrections of 0.82 for ν = 2, 0.77 for ν = 3 and 0.72 for ν = 6. Uncertainties in the standard deviations (68 per cent confidence intervals) as a function of sample size are also estimated using the same Monte Carlo simulations.
Defining A, B and C is then done in three steps.

1.
B is found by finding the MAD of (E_{RV} × (S/N)/(1 + ( [ vsini ] /C)^{2})^{3/4}, using the theoretical value of C determined in Appendix A (C = 15.8 km s^{1} for filter HR15N, and see step (2) below). Figure 3a shows values of B estimated from data in equal bins of S/N. For S/N< 100 the average values per bin are close to B = 5.0 km s^{1} for the full dataset. There is more scatter for S/N> 100 but the variation is not excessive considering the larger uncertainties due to the smaller numbers of data per bin. This indicates that the functional form of the normalising function derived in Appendix A is applicable to the GIRAFFE RV data.

2.
C is then checked by comparing the curve of S_{RV,0} × (S/N), calculated using “empirical” values of B and C fitted to the measured values of E_{RV} × (S/N) as a function of vsini, with the curve predicted using B and C based on the theoretical value of C determined in Appendix A. Figure 3b shows that these two curves are very similar for the two methods, indicating that the theoretical value of C can be used to predict the scaling of measurement uncertainty in RV with vsini. In fact the uncertainty on the fitted slope is largely due to the relatively small proportion of fast rotating stars. For this reason, having confirmed that the data are consistent with the theory in Appendix A, we prefer to use the theoretical value of C rather than an uncertain empirical value. The theoretical value for parameter C is a minimum that assumes any broadening of the spectral lines beyond the spectral resolution is due to rotation. This is reasonable for most types of star in the GES, given the modest resolution of the GIRAFFE spectra, but if C were underestimated then we would overestimate the increase in measurement uncertainty with vsini (see Eq. (1)). Figure 3c shows the cumulative distribution function (CDF) of E_{RV} for shortterm repeats normalised with S_{RV,0}, together with the CDF of a unit Gaussian distribution. The distribution of measurement uncertainties follows the Gaussian distribution over the central region (− 1 ≤ E_{RV}/S_{RV,0} ≤ 1), but larger uncertainties are more frequent than predicted by the Gaussian. The measured distribution of E_{RV}/S_{RV,0} is better represented by a Student’s tdistribution with ν = 6 degrees of freedom, ν. This value of ν represents the integer value that provides the best fit to the normalised uncertainty of shortterm repeats at the 5th and 95th percentiles (see Fig. 3). Having determined this, steps (1) and (2) are iterated, dividing the MADs by the appropriate factor of 0.72 (for a Student’s tdistribution with ν = 6) to estimate a true standard deviation and produce the final results.

3.
The value of A that is added in quadrature to S_{RV,0} is set to A = 0.25 ± 0.02 km s^{1}. This value is chosen so that the normalised CDF of observed E_{RV} found from longterm repeats, E_{RV}/S_{RV}, matches the normalised distribution of uncertainty from shortterm repeats (E_{RV}/S_{RV,0}), but only between the upper and lower quartiles. We choose only to match this range because the tails of the distribution are expected to be different owing to the likely presence of binaries. We show in Sect. 4.3 that this assumption is justified because the distribution of E_{RV}/S_{RV} for those “longterm” repeats where the repeat observations were taken on the same observing night is indistinguishable from that of E_{RV}/S_{RV,0} for shortterm repeats both in the core and the tails of the distribution. The value of A defines the minimum level of uncertainty that can be achieved for GES spectra with high S/N.
3.3. Parameters for normalising the v sin i precision
Constants α, β and C that define the normalising function S_{vsini} are fitted to match the measured distribution of E_{vsini} for a subset of the data comprising 2004 observations with vsini> 5 km s^{1}. Again, parameters are evaluated in three steps with the MAD being used to estimate the true standard deviations and the analysis being iterated once the true distribution of E_{vsini}/S_{vsini} is known.

1.
First β is found by determining the MAD of E_{vsini}(S/N)( [ vsini ] /C)^{2}/(1 + ( [ vsini ] /C)^{2})^{5/4}, using the theoretical value of C determined in Appendix A. The variation of the uncertainty with S/N shows some scatter (see Fig. 4a) and consequently there is an ±8 per cent uncertainty in the estimated value of β.

2.
C is then checked by comparing the measured values of E_{vsini} × (S/N) as a function of vsini with the curve predicted using β and C based on the theoretical value of C determined in Appendix A. Figure 4b shows reasonable agreement between the semiempirical curve and the measured data indicating that a scaling function of the form S_{vsini} using the theoretical value of C can be used to predict the variation of measurement uncertainty with S/N and vsini. Figure 4c shows the CDF of E_{vsini}/S_{vsini,0} for shortterm repeats. This shows a more pronounced tail than the normalised distribution of E_{RV} precision (Fig. 3c) such that a broader Student’s tdistribution with ν = 2 is a better fit to the CDF between the 5th and 95th percentiles.

3.
Finally, the value of α that represents the effect of wavelength uncertainty for long term repeats is found by matching the normalised E_{vsini}/S_{vsini} distribution from 463 longterm repeats with the equivalent distribution for the shortterm repeats between the upper and lower quartiles, giving α = 0.047 ± 0.003. This corresponds to the minimum fractional uncertainty in vsini that can be obtained from GES spectra with high S/N and large vsini. This optimum result is most readily achieved in spectra with vsini = 2C (i.e. 31 km s^{1}). Figure 4b shows that, for a given (S/N), fractional uncertainties rise at both higher and lower values of vsini, and rise drastically for vsini< 10 km s^{1} due to the limited spectral resolution.
Constants describing the scaling function of measurement precision in RV and vsini as a function of S/N and vsini (see Eqs. (2) and (4)).
4. The effect of stellar properties
In Sect. 3 the constants defining the normalising functions S_{RV} and S_{vsini} were estimated by fitting data from an inhomogeneous set of stars. The values obtained represent average values. In this section we determine how these “constants” vary with stellar properties, in particular T_{eff}, gravity, metallicity, and age. We make the simplifying assumption that uncertainties in RV and vsini scale with S/N and vsini as described in the last section and that only the parameters B in Eq. (1) and β in Eq. (3) depend on stellar properties. This follows because parameters A and α represent uncertainties due to changes in wavelength calibration with time and fibre configuration, and parameter C should depend only on the spectral resolution (Eq. (A.4)).
4.1. Variation of S_{RV} with effective temperature
Values of T_{eff} determined from an analysis of the iDR2 spectra are available in the GES archive for 75 per cent of HR15N targets considered in this paper. It is labelled T_{eff} in the archive. The E_{RV} values are divided between 5 evenly spaced bins of temperature between 3000 K and 7000 K and analysed as described in Sect. 3.2. The results in Fig. 5 show a slow increase of B with temperature for T_{eff}< 5200 K such that B is within ±10 per cent of the mean value in Fig. 3b. However, above 5200 K, B increases rapidly with temperature to twice its mean value at T_{eff} ~ 7000 K.
Fig. 4
Analysis of the empirical uncertainty for shortterm repeat observations of vsini using ordersorting filter HR15N. Plot a) shows the variation of E_{vsini} × (S/N)( [ vsini ] /C)^{2}/ (1 + ( [ vsini ] /C)^{2})^{5/4} with S/N . The horizontal line indicates the value of parameter β in Eq. (3) fitted to the full dataset. Blue crosses show the estimated values β as a function of S/N corrected for the measured variation of β with T_{eff} (see Sect. 4.4 and Table 3). Plot b) shows the variation of E_{vsini} × S/N with vsini. The solid line show the relationship predicted using the theoretical value of C and the value of β from plot a). In plots a) and b) the yaxis shows an estimate of the standard deviation based on the MAD divided by 0.82 (for ν = 2, see Sect. 3.3). Plot c) shows the cumulative probability distribution (CDF) of the normalised uncertainty in vsini for shortterm repeats. The red solid line shows results for measured data, the dashed line shows the cumulative distribution of a Gaussian with unit dispersion, and the diamond symbols show the cumulative distribution function for a Student’s tdistribution with ν = 2. 

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The dashed lines in Fig. 5 also show results plotted as a function of the “template” temperature (known in the GES archive as log T_{eff}). This is the logarithm of the temperature of the bestfit synthetic spectrum that was used to determine RV and vsini in the pipeline. This is likely to be less accurate than the T_{eff} derived from a full spectral analysis, but a key advantage is that it is available for all iDR2 targets with a RV and vsini. In fact, the B values estimated using the “template” temperature have a very similar trend with T_{eff} and so may be used directly to estimate temperaturedependent values of B and S_{RV} where these are required.
4.2. Variation of S_{RV} with gravity, metallicity and age
Values of log g and [Fe/H] (labelled as log g and FeH in the GES archive) obtained from a detailed spectral analysis by the GES working groups are presently available for about 75 per cent of the targets observed with ordersorting filter HR15N. Analysing these data in bins of log g (see Fig. 6a) shows only a ~25 per cent change in the estimated value of parameter B over a 2 dex range in log g. Analysis in bins of [Fe/H] (see Fig. 6b) shows a similarly small change in B with metallicity over the range of metallicities −1 < [ Fe/H ] < 1. Below this, the estimated value of parameter B appears to increase sharply with decreasing [Fe/H] but in truth there are too few data points for filter HR15N with [ Fe/H ] < − 1 to estimate parameter B with any degree of accuracy. We confirmed that any variation seen in Fig. 6 is not due to differences in temperature – the median values of log T_{eff} are very similar in all binned subsamples.
Although the fundamental cause of any variation of RV precision with age would likely be due to the evolution of log g in premain sequence stars, it is nevertheless important to confirm that the prescription for calculating RV precision is valid at all ages, since studying the dynamics of young clusters is a key GES objective. Figure 7 shows the variation of B with stellar age. The adopted ages for cluster stars are those given in Table 1. For this plot, we attempted to separate genuine cluster members from field objects by selecting according to RV. For most cluster datasets there was a clear RV peak corresponding to the cluster, so cluster members were selected from a range ±5 km s^{1} either side of this peak with little contamination. However, no selection by RV was made for the COROT sample or for the cluster NGC 6633, since neither showed a clear peak in their RV distributions. We assume these datasets contain mostly older (>1 Gyr) field stars. Figure 7 shows in any case that there is a weak dependence of B on age. However, it can be seen that this small variation is directly linked to the decreasing median temperatures of the cluster samples at younger ages.
4.3. Variation of S_{RV} with time between observations
In our model of RV uncertainty we assume that A represents some additional uncertainty arising from random changes in wavelength calibration with time and the effects of changes in fibre allocation. We fitted A, using the interquartile range of the uncertainty distribution in longterm repeats, in an effort to avoid modelling tails that might be due to binary motion. This simplifying assumption can be tested by plotting values of A determined for samples with increasing time differences between observations. Figure 8a shows that the value of A depends only weakly on the time between observations, increasing from 0.23 ± 0.02 km s^{1} for measurements made in different configurations on the same night to 0.26 ± 0.02 km s^{1} for intervals of up to 100 days between observations. This confirms that the A value is not unduly influenced by any binaries in the sample.
The effect of binaries is far more apparent in the tails of the distributions. Figure 8b compares CDFs of the normalised distribution of measurement uncertainty derived from the change in RV between shortterm repeats, normalised with S_{RV,0} (Eq. (1)), with (i) all the longterm repeats, with uncertainties normalised to S_{RV} (Eq. (2)); (ii) a separate distribution of E_{RV}/S_{RV} for just those longterm repeats where the repeat observations were on the same night (nullifying the effects of all but the rarest, shortperiod binaries). By design, the three CDFs are very close in the interquartile range; but whilst the CDF for longterm repeats has a more pronounced tail, better described by a Student’s tdistribution with ν = 3, the longterm repeats within a night are indistinguishable (with a KolmogorovSmirnov test) from the shortterm repeats, following a Student’s tdistribution with ν = 6. This is consistent with a fraction of the sample being binary stars that show genuine RV changes between observations on timescales longer than a day. It also justifies an assumption that the true uncertainties in a single RV measurent are best represented by the ν = 6 Student’s tdistribution multiplied by S_{RV} as given by Eq. (2).
Fig. 5
Variation of parameter B of the scaling function for uncertainty in RV (S_{RV}) with effective temperature. The solid line shows results for filter HR15N as a function of T_{eff} (see Sect. 4.1). Dashed lines show results for filters HR10, HR15N and HR21 as a function of the temperature of the template spectrum fitted in the CASU pipeline (see Sect. 2.3). Numbers equal the sample size per bin. 

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Fig. 6
Variation of parameter B of the scaling function for uncertainty in RV (S_{RV}) with log g and metallicity for ordersorting filters, HR10, HR15N and HR21. a) Shows the value of B determined for data in three equal bins of log g. Numbers indicate the sample size. b) Shows the value of B determined for data in three equal bins of [Fe/H]. For filter HR15N only the values shown in the upper two bins are reliable due to the low number of targets with [ Fe/H ] < − 1. 

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Fig. 7
Variation of parameter B of the scaling function for uncertainty in RV (S_{RV}) with target age for observations with ordersorting filter HR15N. Square symbols show the value of B for targets identified as possible cluster members from their RV versus the nominal age of the cluster (see Table 1). No selection by RV is made for NGC 6633 or COROT and we assume the stars have mean ages > 1 Gyr. The dotted line indicates the median value of T_{eff} (right hand axis values) for members identified in each cluster. 

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Fig. 8
Dependence of the scaling function for uncertainty in RV (S_{RV}) as a function of time between observations. Plot a) shows how scaling parameter A varies with time between repeat observations. Numbers indicate size of the sample used to determine A. Plot b) shows the CDFs of the normalised distribution of RV precision for shortterm repeats (black line), longterm repeats (red line) and long term repeats where the observations were taken on the same observing night (blue dashed line). The CDFs for the shortterm repeats and the longterm repeats within a night are indistinguishable using a twotailed KolmogorovSmirnov test. 

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Fig. 9
Variation of the scaling function for vsini with temperature and time between observations. Plot a) shows how the β parameter in Eq. (3) varies with T_{eff}. The solid line shows results using T_{eff} from a detailed spectral analysis; the dashed line shows the results using the “template” temperature (see Sect. 4.4). Labels indicate the sample size per bin. Plot b) shows the CDF of the normalised distributions of vsini uncertainty for short and long term repeats. Also shown (as small diamonds) is a Student’s tdistribution with ν = 2. 

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4.4. Variation of S_{v sin i} with temperature and time between observations
In Fig. 9a we show how β, the parameter in the scaling function governing vsini precision (see Eq. (3)), depends on stellar temperature and the time between observations . The data were divided into 5 equal bins of temperature. Results are shown using the temperature derived from detailed spectral analysis (T_{eff}) and the bestfitting “template” temperature (log T_{eff}). There appears to be little variation with T_{eff} below 6000 K using either temperature estimate, but like the parameter B governing RV precision, there is a rapid growth in β for hotter stars – by about a factor of 2 at T_{eff} ≃ 7000 K.
Figure 9b compares the CDF of the normalised measurement precision in vsini for short and longterm repeats. There is much less difference between these CDFs than that found between the short and longterm repeat estimates of RV precision. This is not unexpected since measurements of vsini should be much less effected by binarity. A Student’s tdistribution with ν = 2 fits either the short or longterm repeat CDFs equally well.
There are too few stars in our sample with vsini> 5 km s^{1} for a detailed investigation of how β might vary with age, log g or metallicity subsets.
5. Measurement uncertainties using different instrumental configurations
So far the analyses have been restricted to observations with the HR15N ordersorting filter. In this section we consider how these results can be extended to the other GES observational setups. We used all of the “GES_MW” (GES Milky Way Programme) fields, consisting of more than 20 000 RV measurements from individual spectra taken with the HR10 and HR21 filters. Unfortunately there are too few measurements through these filters with vsini> 5 km s^{1} to constrain the vsini precision in the same way that was done for HR15n observations.
These precision of the HR10 and HR21 RV measurements were compared with those predicted using the simple model described in Appendix A. For this comparison it is assumed that:

The uncertainty in RV precision scales asS_{RV} (See Sect. 3.1)

Parameter C characterising the dependence of RV uncertainty on vsini depends on the spectral resolution as 0.895c/R_{λ} (see Eq. (A.4)).

Parameter A that determines the difference in RV precision for short and longterm repeats corresponds to a displacement of the spectrum on the detector in the dispersion direction, measured in pixels, rather than a fixed velocity difference. The number of physical CCD pixels contributing each spectrum along the dispersion direction is 4096, so where Δλ is the wavelength range of the filter (see Table 3). In the case of filter HR15, A = 0.25 km s^{1} corresponds to δ_{pix} = 0.061 pixels, which we will assume is the same for the other filters.
Figure 10 shows an analysis for all field stars that were observed in the GES_MW fields. Figures 10a and d show the variation of the standard deviation of E_{RV} × S/N with vsini. There are few data points with large vsini values; therefore the error bars become large with increasing vsini. Even so, the curve corresponding to the value of B evaluated for the full dataset using the value of C predicted from Eq. (A.4) is consistent with the empirically measured uncertainties.
Figures 10b and e show how the standard deviation of (E_{RV})(S/N)/(1 + ( [ vsini ] /C)^{2})^{3/4} (estimated using the MAD) varies with S/N. For S/N< 100 both plots show reasonable agreement (within 10 per cent) between the measured data and the line showing a single value of B evaluated for the full dataset using theroretical C. Agreement is less good for data with S/N> 100. However, any inaccuracy here will have little effect on the estimated uncertainty in RV for the majority of stars which are slow rotators since, at high values of S/N, the uncertainty of these stars is dominated by the constant term, A in the expresssion for S_{RV} (see Eqs. (1) and (2)).
The mean values of B are 2.3 km s^{1} for filter HR10 and 7.1 km s^{1} for filter HR21, compared with 5.0 km s^{1} for HR15N i.e. for spectra with similar vsini and S/N RVs estimated from spectra taken with HR10 are more precise. The temperature dependence, illustrated in Fig. 5, is also different in detail. Figures 6a and b show the variation of B with log g and [Fe/H]. The trends are similar to the variation found for ordersorting filter HR15N i.e. B is almost independent of gravity and changes only slowly with metallicity.
Parameter A was determined in two ways. First, it was determined using the measured data for the relatively small sample of longterm repeats as described in Sect. 3.1. This gave values of A = 0.18 ± 0.02 km s^{1} for filter HR10 and 0.28 ± 0.02 km s^{1} for filter HR21. These compare with the predicted values of 0.22 km s^{1} and 0.26 km s^{1} inferred by scaling the δ_{pix} value determined for filter HR15N by the ratio of their pixel sizes in km s^{1}.
Figures 10c and f shows the CDFs of the normalised uncertainty for short and longterm repeats using filters HR10 and HR21 respectively. In each case S_{RV} is evaluated using the appropriate theoretical values of A and C and the mean empirical value of B determined for each filter, and these are inserted into Eq. (1) (for shortterm repeats) or 2 (for longterm repeats). Also shown is the CDF of a Student’s tdistribution with ν = 6 which, as for the HR15N data, appears to be an excellent representation of the distribution due to shortterm repeats. The data are sparse for longterm repeats, but the distributions appear to have more extended tails, consistent with the idea that they contain RV shifts due to binary systems. We do not have sufficient data to test whether the uncertainty CDFs for longterm repeats within a night are similar to those for shortterm repeats, but we assume that, like the HR15N data, this will the case for data taken with HR10 and HR21.
Fig. 10
Analysis of the empirical uncertainty for short term repeat observations of RV using filter HR10 and HR21. Plots a) and d) show the variation of E_{RV} with vsini. Red lines show the curves predicted using the model value of C (see Eq. (A.4)). The dashed lines shows the curves predicted using values of B and C fitted to the binned data. Plots b) and e) show the variation in the estimated value of parameter B in Eq. (1) with S/N. Red lines show values of B fitted to the full dataset for each filter. Blue crosses show the estimated values B as a function of S/N corrected for the measured variation of B with T_{eff} (see Table 3). Plots c and f show the normalised uncertainty for short term repeats and long term repeats. The black curve shows the CDF for shortterm repeats. The red line shows the CDF for longterm repeats. Blue diamonds show a Student’s tdistribution with ν = 6, which matches the distribution for shortterm repeats well. 

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6. Discussion and summary
We have shown that the normalisation functions given in Eqs. (1)–(4) are reasonable descriptions of how uncertainties in RV and vsini scale with S/N and vsini. The recommended average parameters of A, B, C defining the scaling function for RV are given in Table 3 for observations performed with the three main instrumental configurations used for GIRAFFE observations in the GES. Average values of α, β, and C that define the scaling function for vsini are also given for filter HR15N. The uncertainties given by Eqs. (2) and (4) are not normally distributed; they have more extended tails. The uncertainty distribution for a given observation of RV is better represented by the value of Eq. (2) multiplied by a normalised Student’s tdistribution with ν = 6, whilst for vsini the uncertainty distribution can be approximated by Eq. (4) multiplied by a normalised Student’s tdistribution with ν = 2.
Equations (2) and (4) decouple the influences of spectral type and the spectrograph; A, C and α are properties of the instrumental setup, whilst B and β depend on the type of star observed. The dependence on gravity, age and metallicity, over the range −1 < [ Fe/H ] < 1, is weak; but the temperature dependence becomes strong for T_{eff}> 5200 K, such that B and β increase with T_{eff} and the precision worsens. This is presumably the result of a decreasing number of strong, narrow lines in the spectra of hotter stars. The temperature dependent B and β values are listed in Table 3 and should be used in conjunction with the mean values of A, C and α. There are insufficient observations of stars with vsini> 5 km s^{1} using ordersorting filters HR10 and HR21, so we cannot estimate β for such observations. It should also be noted that for reasons of sample size, the calibration of B and β is limited to 3200 ≤ T_{eff} ≤ 7200 K.
Parameter A is between 0.22 and 0.26 km s^{1}, dependent on instrumental setup, and represents the best precision with which RV can be obtained from an individual GES spectrum with low rotational broadening and large S/N. The origin of this term is unclear; it partly arises from uncertainties in the wavelength calibration and the application of calibration offsets from the “simcal” fibres or sky emission lines. However, various tests have shown these cannot be entirely responsible and we suspect there are additional contributions that may be associated with movement of the fibres at the spectrograph slit assembly or target miscentering in the fibres combined with imperfect signal scrambling.
The analyses we present were derived from results in the iDR2 GES data release and the coefficients in Table 3 are applicable to those data and also to the more recent iDR3 update that used the same pipeline analysis. The exact use of these results depends on the purpose of any particular investigation. To estimate what approximates to a particular confidence interval on a RV (vsini) value, the following procedure is recommended:

1.
Use the instrumental setup and an estimated stellar temperature (preferably from the GES analysis) to choose the appropriate values of B (β) and C from Table 3; calculate S_{RV,0} (S_{vsini,0}) from Eq. (1) (Eq. (3)) using the measured vsini and S/N.

2.
Choose the A (α) value appropriate for the instrumental setup from Table 3 and calculate S_{RV} (S_{vsini}) from Eq. (2) (Eq. (4)).

3.
If combining results from repeated observations, these should be weighted using ( for shortterm repeats (i.e. without the inclusion of the A (α) term), or () for longterm repeats.

4.
For accurate modelling of RV data one should use S_{RV} (S_{vsini}) multiplied by a Student’s tdistribution with ν = 6 (ν = 2) as a probability distribution for the uncertainty. More crudely, a confidence interval can be estimated by multiplying S_{RV} (S_{vsini}) by the appropriate percentile point of a Student’s tdistribution with ν = 6 (ν = 2). For example, to estimate a 68.3 per cent error bar, multiply by 1.09 (1.32), or for a 95.4 per cent error bar multiply by 2.51 (4.50).
Note that whilst the 68.3 per cent confidence intervals are quite close to the value expected for a normal distribution with a standard deviation of S_{RV} (S_{Vsini}), the 95.4 per cent confidence intervals are significantly larger due to the broader tails of the Student’s tdistributions. We do not recommend extrapolating these estimates to even larger confidence intervals since we have insufficient data to reliably constrain the distribution at these values. It seems likely that at the conclusion of GES there will be sufficient data (roughly 5 times as much) to significantly improve this situation. A larger dataset will also allow us to study how vsini precision varies with T_{eff} and log g and between differing observational setups.
Acknowledgments
R.J.J. wishes to thank the UK Science and Technology Facilities Council for financial support. Based on data products from observations made with ESO Telescopes at the La Silla Paranal Observatory under programme ID 188.B3002. These data products have been processed by the Cambridge Astronomy Survey Unit (CASU) at the Institute of Astronomy, University of Cambridge, and by the FLAMES/UVES reduction team at INAF/Osservatorio Astrofisico di Arcetri. These data have been obtained from the GaiaESO Survey Data Archive, prepared and hosted by the Wide Field Astronomy Unit, Institute for Astronomy, University of Edinburgh, which is funded by the UK Science and Technology Facilities Council. This work was partly supported by the European Union FP7 programme through ERC grant number 320360 and by the Leverhulme Trust through grant RPG2012541. We acknowledge the support from INAF and Ministero dell’ Istruzione, dell’ Università’ e della Ricerca (MIUR) in the form of the grant “Premiale VLT 2012”. The results presented here benefit from discussions held during the GaiaESO workshops and conferences supported by the ESF (European Science Foundation) through the GREAT Research Network Programme.
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Appendix A: Variation of measurement precision with radial and projected rotation velocities
We consider below how the measurement precision of RV and vsini scale with S/N and vsini for shortterm repeats where there are no changes in setup or wavelength calibration between observations. We make the simplifying assumption that the precision in RV scales as E_{RV} ∝ W^{3 / 2}/ (S/N) where W, is the FWHM of a Gaussian profile representing the characteristic absorption line profile of the measured spectrum. This approximate relation can be deduced from the results of Landman et al. (1982). These authors showed that, for the ideal case of a Gaussian line profile of amplitude a, mean value m, and standard deviation s, sampled using binned data with a uniform Gaussian noise of rms amplitude ϵ per bin, the statistical uncertainties in the estimated values of m and s are given by; (A.1)where Δ_{x} is the uniform bin width and Δ_{x} ≪ s.
In the present case a varies with equivalent width, (EW), of the characteristic absorption line as , where h is the amplitude of the continuum. If the depth of the absortion line, a, is small compared to the continuum, h then measurement uncertainty is ϵ ≈ h/ (S/N). Substituting these values in Eq. (A.1) using the relation gives; (A.2)
Appendix A.1: Effect of v sin i on FWHM of the absorption line
For a slowly rotating star, assuming that any sources of broadening other than rotation are much smaller than the intrinsic spectrograph resolution, the FWHM of an individual absorption line is , where is the mean wavelength and R_{λ} the resolving power of the spectrograph. For fast rotating stars the width of the spectral lines is increased by rotational broadening. Gray (1984) gives the rotational broadening kernel as (A.3)where , λ is wavelength (over the range −Λ <λ< Λ) and u is the limb darkening coefficient.
Convolving a spectrum with this kernel increases the FWHM of individual lines approximately as where λ_{rms} is the rms of the broadening kernel (). Evaluating γ_{rms} from Eq. (A.3) gives; (A.4)where
Appendix A.2: Scaling of uncertainty in RV and v sin i
To determine how the uncertainty in radial velocity, E_{RV} scales with S/N and vsini we assume E_{RV} ∝ σ_{m}. For a given spectra Δx and EW are independent of W and S/N so that (from Eqs. (A.2) and (A.4)) E_{RV} scales with vsini and S/N as, (A.5)where B is an empirically determined constant and C depends on R_{λ} and u. A value of u = 0.6 is used in this paper (Claret et al. 1995) giving C = 0.895c/R_{λ}.
The uncertainty in the estimated value of vsini is determined from the uncertainty in the estimated absorption line width, σ_{W} (Eqs. (A.2) and (A.4)) as; (A.6)Using this expession the uncertainty in the normalised value of vsini, (∝ σ_{vsini}/ [ vsini ]) scales with vsini and S/N as; (A.7)where β is an empirically determined constant.
All Tables
Numbers of short and long term repeat GIRAFFE observations of RV and vsini used for open clusters with ordersorting filter HR15N.
Log of VLT/Flames observations used in the analysis of RV and vsini measurement precision.
Constants describing the scaling function of measurement precision in RV and vsini as a function of S/N and vsini (see Eqs. (2) and (4)).
All Figures
Fig. 1
The empirical uncertainty in RV precision () estimated from the change in RV between shortterm repeat observations of cluster targets (see Tables 1 and 2) using ordersorting filter HR15N. The size of the symbol indicates the measured value of vsini. 

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In the text 
Fig. 2
Comparison of the probability density of E_{RV} for short and longterm repeats (see Tables 1 and 2) using ordersorting filter HR15N. The black line shows results for shortterm repeats (i.e. pairs of observations within the same OB). The red histogram shows results for longterm repeats (i.e. spectra of the same targets but taken from different OBs where individual targets are allocated to different fibres). 

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In the text 
Fig. 3
Analysis of the empirical uncertainty for shortterm repeat observations of RV using filter HR15N. The solid line in plot a) shows the variation of E_{RV} × (S/N)/(1 + ( [ vsini ] /C)^{2})^{3/4} with S/N. The horizontal line indicates the value of parameter B in Eq. (1) fitted to the full dataset. Blue crosses show the estimated values B as a function of S/N corrected for the measured variation of B with T_{eff} (see Sect. 4.1 and Table 3). Plot b) shows the variation of E_{RV} × S/N with vsini. The solid line show the relationship predicted using the theoretical value of C and the value of B from plot a). The dashed line shows a curve of similar functional form using parameters B and C fitted to the binned data. In plots a) and b) the yaxis shows an estimate of the standard deviation based on the MAD divided by 0.72 (see Sect. 3.2). Plot c) shows the cumulative probability distribution (CDF) of the normalised uncertainty in RV for shortterm repeats. The red solid line shows results for measured data, the dashed line shows the cumulative distribution of a Gaussian with unit dispersion, and the diamond symbols show the cumulative distribution function for a Student’s tdistribution with ν = 6. 

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In the text 
Fig. 4
Analysis of the empirical uncertainty for shortterm repeat observations of vsini using ordersorting filter HR15N. Plot a) shows the variation of E_{vsini} × (S/N)( [ vsini ] /C)^{2}/ (1 + ( [ vsini ] /C)^{2})^{5/4} with S/N . The horizontal line indicates the value of parameter β in Eq. (3) fitted to the full dataset. Blue crosses show the estimated values β as a function of S/N corrected for the measured variation of β with T_{eff} (see Sect. 4.4 and Table 3). Plot b) shows the variation of E_{vsini} × S/N with vsini. The solid line show the relationship predicted using the theoretical value of C and the value of β from plot a). In plots a) and b) the yaxis shows an estimate of the standard deviation based on the MAD divided by 0.82 (for ν = 2, see Sect. 3.3). Plot c) shows the cumulative probability distribution (CDF) of the normalised uncertainty in vsini for shortterm repeats. The red solid line shows results for measured data, the dashed line shows the cumulative distribution of a Gaussian with unit dispersion, and the diamond symbols show the cumulative distribution function for a Student’s tdistribution with ν = 2. 

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In the text 
Fig. 5
Variation of parameter B of the scaling function for uncertainty in RV (S_{RV}) with effective temperature. The solid line shows results for filter HR15N as a function of T_{eff} (see Sect. 4.1). Dashed lines show results for filters HR10, HR15N and HR21 as a function of the temperature of the template spectrum fitted in the CASU pipeline (see Sect. 2.3). Numbers equal the sample size per bin. 

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In the text 
Fig. 6
Variation of parameter B of the scaling function for uncertainty in RV (S_{RV}) with log g and metallicity for ordersorting filters, HR10, HR15N and HR21. a) Shows the value of B determined for data in three equal bins of log g. Numbers indicate the sample size. b) Shows the value of B determined for data in three equal bins of [Fe/H]. For filter HR15N only the values shown in the upper two bins are reliable due to the low number of targets with [ Fe/H ] < − 1. 

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In the text 
Fig. 7
Variation of parameter B of the scaling function for uncertainty in RV (S_{RV}) with target age for observations with ordersorting filter HR15N. Square symbols show the value of B for targets identified as possible cluster members from their RV versus the nominal age of the cluster (see Table 1). No selection by RV is made for NGC 6633 or COROT and we assume the stars have mean ages > 1 Gyr. The dotted line indicates the median value of T_{eff} (right hand axis values) for members identified in each cluster. 

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In the text 
Fig. 8
Dependence of the scaling function for uncertainty in RV (S_{RV}) as a function of time between observations. Plot a) shows how scaling parameter A varies with time between repeat observations. Numbers indicate size of the sample used to determine A. Plot b) shows the CDFs of the normalised distribution of RV precision for shortterm repeats (black line), longterm repeats (red line) and long term repeats where the observations were taken on the same observing night (blue dashed line). The CDFs for the shortterm repeats and the longterm repeats within a night are indistinguishable using a twotailed KolmogorovSmirnov test. 

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In the text 
Fig. 9
Variation of the scaling function for vsini with temperature and time between observations. Plot a) shows how the β parameter in Eq. (3) varies with T_{eff}. The solid line shows results using T_{eff} from a detailed spectral analysis; the dashed line shows the results using the “template” temperature (see Sect. 4.4). Labels indicate the sample size per bin. Plot b) shows the CDF of the normalised distributions of vsini uncertainty for short and long term repeats. Also shown (as small diamonds) is a Student’s tdistribution with ν = 2. 

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In the text 
Fig. 10
Analysis of the empirical uncertainty for short term repeat observations of RV using filter HR10 and HR21. Plots a) and d) show the variation of E_{RV} with vsini. Red lines show the curves predicted using the model value of C (see Eq. (A.4)). The dashed lines shows the curves predicted using values of B and C fitted to the binned data. Plots b) and e) show the variation in the estimated value of parameter B in Eq. (1) with S/N. Red lines show values of B fitted to the full dataset for each filter. Blue crosses show the estimated values B as a function of S/N corrected for the measured variation of B with T_{eff} (see Table 3). Plots c and f show the normalised uncertainty for short term repeats and long term repeats. The black curve shows the CDF for shortterm repeats. The red line shows the CDF for longterm repeats. Blue diamonds show a Student’s tdistribution with ν = 6, which matches the distribution for shortterm repeats well. 

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In the text 