© ESO, 2015

1. Introduction

The Gaia-ESO survey (GES) is a large public survey programme carried out at the ESO Very Large Telescope (UT-2 Kueyen) with the FLAMES multi-object instrument (Gilmore et al. 2012; Randich & Gilmore 2013). The survey will obtain high- and intermediate-resolution spectroscopy of ~10⁵ 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 three-dimensional 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 Gaia-ESO 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, mass-dependent 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 false-positives. 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 signal-to-noise 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 chi-squared 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 fibre-fed, multi-object instrument, feeding both the UVES high-resolution (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 order-sorting 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 order-sorting 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 thorium-argon (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 pixel-to-pixel 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 day-time 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 night-time “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 cross-correlation 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 low-order 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 chi-squared 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 chi-squared minimisation yields an estimate of the uncertainty in the best fit parameters. However, in the case of GES data, this under-estimates the measurement uncertainty, in part due to the analysis step where spectra are re-binned 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 pre-main 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 low-mass 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 GES-CoRoT collaboration, was included.

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:

• Short-term 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.

• Long-term 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 long-term 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 co-ordinates, 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 University¹.

  Table 2

  Log 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 short-term 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 long-term repeats. The peak height is reduced and the full width half maximum (FWHM) is increased for long-term repeats. There is thus an apparent increase in measurement uncertainty for targets with high S/N when compared to the precision assessed using short-term 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 non-Gaussian.

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 order-sorting filter.

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 long-term 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 long-term 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 short-term 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 long-term 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).

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 non-Gaussian 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 t-distributions 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 over-estimate the increase in measurement uncertainty with vsini (see Eq. (1)). Figure 3c shows the cumulative distribution function (CDF) of E_RV for short-term 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 t-distribution 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 short-term 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 t-distribution 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 long-term repeats, E_RV/S_RV, matches the normalised distribution of uncertainty from short-term 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 “long-term” repeats where the repeat observations were taken on the same observing night is indistinguishable from that of E_RV/S_RV,0 for short-term 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.

Figure 3a shows an increase in the estimated value of B for S/N> 100. This does not significantly affect the estimate of parameters A, B and C described above but does reflect the variation of B with stellar properties. Lower S/N bins contain a mix of stars such that variations of B with stellar properties average out. However the smaller samples in the high S/N bins contain a higher fraction of stars with larger T_eff and, as we show in Sect. 4.1, B increases with T_eff. The blue crosses in Fig. 3a show T_eff-corrected values of B as a function of S/N, using the values discussed in Sect. 4.1 and reported in Table 3. These show a more uniform variation of B with 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 semi-empirical 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 short-term repeats. This shows a more pronounced tail than the normalised distribution of E_RV precision (Fig. 3c) such that a broader Student’s t-distribution 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 long-term repeats with the equivalent distribution for the short-term 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.

Figure 4a shows an increase in the estimated value of β for S/N> 100 due to the higher proportion of hotter stars in this bin. This variation is reduced when the estimated value of β is corrected for the measured variation of β with T_eff discussed in Sect. 4.4 and reported in Table 3.

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.

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 best-fit 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 temperature-dependent 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 order-sorting 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 pre-main 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 long-term 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 short-term repeats, normalised with S_RV,0 (Eq. (1)), with (i) all the long-term repeats, with uncertainties normalised to S_RV (Eq. (2)); (ii) a separate distribution of E_RV/S_RV for just those long-term repeats where the repeat observations were on the same night (nullifying the effects of all but the rarest, short-period binaries). By design, the three CDFs are very close in the interquartile range; but whilst the CDF for long-term repeats has a more pronounced tail, better described by a Student’s t-distribution with ν = 3, the long-term repeats within a night are indistinguishable (with a Kolmogorov-Smirnov test) from the short-term repeats, following a Student’s t-distribution 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 t-distribution multiplied by S_RV as given by Eq. (2).

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 best-fitting “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 long-term repeats. There is much less difference between these CDFs than that found between the short- and long-term repeat estimates of RV precision. This is not unexpected since measurements of vsini should be much less effected by binarity. A Student’s t-distribution with ν = 2 fits either the short- or long-term 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 order-sorting 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 long-term 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.

Only constant B has to be found, and this can be done using the distribution of E_RV found from short-term repeats. This allows the measurement precision of RV for a given filter to be estimated even when there are no long-term repeat measurements to make an independent empirical analysis. In the case of filters HR10 and HR21 it turns out that there are enough long-term repeat measurements, albeit over a restricted range of vsini values, to test this hypothesis.

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 order-sorting 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 long-term 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 long-term 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 short-term repeats) or 2 (for long-term repeats). Also shown is the CDF of a Student’s t-distribution with ν = 6 which, as for the HR15N data, appears to be an excellent representation of the distribution due to short-term repeats. The data are sparse for long-term 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 long-term repeats within a night are similar to those for short-term repeats, but we assume that, like the HR15N data, this will the case for data taken with HR10 and HR21.

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 t-distribution with ν = 6, whilst for vsini the uncertainty distribution can be approximated by Eq. (4) multiplied by a normalised Student’s t-distribution 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 order-sorting 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 mis-centering 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 short-term repeats (i.e. without the inclusion of the A (α) term), or () for long-term repeats.

• 4.

  For accurate modelling of RV data one should use S_RV (S_vsini) multiplied by a Student’s t-distribution 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 t-distribution 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 t-distributions. 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.

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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.B-3002. 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 Gaia-ESO 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 RPG-2012-541. 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 Gaia-ESO workshops and conferences supported by the ESF (European Science Foundation) through the GREAT Research Network Programme.

References

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 short-term 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

All Figures