Precision Determination of Corotation Radii in Galaxy Disks: Tremaine–Weinberg versus Font–Beckman for NGC 3433

Determination of the corotation radius in a galaxy disk, a key component of density wave theory (Lindblad 1938; Lin & Shu 1964) has been relevant to cosmology since the prediction by Weinberg (1985), Hernquist & Weinberg (1992), and Tremaine & Ostriker (1999) that the bars in barred galaxies should have been dynamically braked by their dark matter halos on timescales significantly shorter than the Hubble time. This prediction was focused by Debattista & Sellwood (2000) and by Sellwood & Debattista (2006) into a quantitative prediction of this effect, on the ratio of the corotation radius to the length of the bar. The difficulty of determining corotation radii has been gradually overcome during the last decade. A known method (TW), first proposed by Tremaine & Weinberg (1984), and first used on the S0 galaxy NGC 936 (Merrifield & Kuijken 1995), combines the surface density distribution of stellar light from photometric imaging with line-of-sight velocity distribution using long-slit spectra. The technique has been usefully applied to a few tens of objects, notably by Aguerri et al. (2003), Corsini et al. (2007), and Aguerri et al. (2015), with uncertainties that lie in the range of 10%–40%. An interesting generalization of the method for galaxies with multiple pattern speeds and corotation radii was presented in Meidt et al. (2008). The Tremaine–Weinberg method has been considered standard, but has been limited in applicability due to long observation times with conventional long-slit spectra. Integrated field spectra have recently improved the situation, but over relatively restricted fields. For example, the PPAK IFU with which the CALIFA survey was conducted (Sanchez et al. 2012) has "an extremely wide field" of only 1.3 arcmin². Debattista & Williams (2004) showed that a Fabry–Perot spectrometer could be used for 2D absorption line spectroscopy of a stellar population, obtaining very precise TW results for NGC 7079, but their field was 1.4 arcmin² compared to the field of the GHαFaS Fabry–Perot spectrometer used in the present work, of 12 arcmin².

More recently two of the present authors presented the Font–Beckman method, (FB; Font et al. 2011, 2014a, 2014b), based on the change in phase, at corotation, of the radial component of the velocity in the disk, from inward to outward, or vice versa, first brought out explicitly by Kalnajs (1978). This was used to determine corotation radii in over 100 galaxies, with uncertainties typically in the range of 5%–10%. It was applied to 2D velocity fields obtained in Hα emission at good spatial (∼1 arcsec) and spectral (∼8 km s⁻¹) resolution, i.e., using the ionized gas as a tracer. Although FB is considerably faster, and can in practice reach larger disk radii, than TW, it is most useful for star-forming i.e., later-type galaxies where there is sufficiently extended Hα emission, and is complementary to TW. Indeed, TW is intended for use with the stellar components of galaxies, as it ideally requires continuity, while FB is designed for use with the ISM, and does not entail this. The aim of the work presented here was to find a galaxy that could serve as a test bed, where the data allowed us to apply both techniques, so that we could compare them directly, helping observers to judge where best each could be applied. We also hoped that by using both methods on a given object we might optimize the precision with which corotation could be measured, again giving hints for wider applications. In Section 2 we describe the observations, in Section 3 we give a brief summary of the two methods being compared, in Section 4 we show the results of the application of both methods, and in Section 5 we discuss the results and draw conclusions.

NGC 3433, an SAB(s)b galaxy (Buta et al. 2015) was chosen because its declination, at ∼+10°, makes it observable from the latitude of the Very Large Telescope (VLT) at Paranal, and also from the William Herschel Telescope (WHT) on La Palma. This means that we could use and compare 2D spectral data from the Multi Unit Spectrograph Explorer (MUSE; Bacon et al. 2014) on the VLT and from our own instrument GHαFaS (Hernandez et al. 2008) on the 4.2 m WHT. A further criterion is that the full optical disk of NGC 3433 fits a single GHαFaS field of 3.4 × 3.4 arcmin. For comparison, in Figure 1 we show the field of MUSE, (red square) compared with the field of GHαFaS (yellow square). We can see one advantage of GHαFaS: its field is over 10 times the area of the MUSE field. However, the internal field of NGC 3433, including the bar region, is covered by the field of MUSE, so the GHαFaS field is needed for the interstellar emission in the outer spiral arms.

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In Table 1 we give the basic parameters of NGC 3433. These include the position and the morphological type from the standard literature, and also the morphological parameters we have inferred during the course of the work reported in this article. These include the position angle, the inclination angle, and the length of the bar. Our kinematic analysis, explained below, carried out using a sequence of concentric tilted rings, was the basis of most of the values in Table 1, as shown, but we used the Spitzer 3.6 μm image to measure the bar length (this was also used by Herrera-Endoqui et al. 2015), and also to derive a purely morphological estimate of the disk position angle. We note here that the measurement of the inclination angle from three distinct data sets yields very consistent values. The most precise of these, using the full disc Hα kinematics from GHαFaS, has an intrinsic error of only 08 and lies well within the error limits of the two values obtained from the inner disk from the MUSE data. The position angle shows a significant variation of some 9° between the MUSE values and the GHαFaS value. We attribute this to the presence of the bar, which must affect both the isophotes and the isovels in the inner part of the galaxy, the part observed with MUSE. This inference is supported by the measured angular separation between the bar and the disk major axes, which is close to 10°. In our TW derivation we have used the inner position angle for the MUSE data and the GHαFaS inner disk data, and the outer position angle for the GHαFaS outer disk data. We should make it clear that this discrepancy does not represent a variational uncertainty in the determinations. Where we show these determinations, below, we will deal further with the measurement uncertainties.

Table 1.  Parameters of NGC 3433

Parameter	Value	Reference
Morphological type	SAB(s)b	Buta et al. (2015)
R.A. (J2000)	10h52m03870
Decl. (J2000)	+10d08m5392
Inclination Angle	342 ± 14a	This work
Position Angle	248 ± 11a	This work
	235 ± 37b	This work
Bar Length	15.5 arcsec	Herrera-Endoqui et al. (2015)
	15.4 ± 1.2 arcseca	This work

Notes.

^aThese are the average values obtained from fitting tilted ring models to the three velocity maps obtained from each of the velocity fields shown in Figure 2. ^bThis value is obtained by ellipse fitting to the 3.6 μm Spitzer model.

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2.1. Observations with GHαFaS

GHαFaS is a Fabry–Perot interferometer at the Nasmyth focus of the WHT. It has a free spectral range of 8.5 A, corresponding to 390 km s⁻¹ at the rest wavelength of Hα, divided into 60 independent channels yielding a formal spectral resolution of 6.5 km s⁻¹. The observations were performed on 2015 March 27th, with an average value for the seeing of 1.2 arcsec. The angular resolution was seeing-limited, with a pixel scale of ∼0.2 arcsec pixel⁻¹. The observing time on NGC 3433 was 3 hr; the reduced observations (see Hernandez et al. 2008; Font et al. 2011) yield a wavelength-calibrated data cube. We apply principal component analysis (PCA, Steiner et al. 2009) to this cube. PCA conserves flux and yields a cleaned cube, in which the spectra are smoothed and the noise is minimized, making it easier to measure the parameters of the emission line spectra (line center, FWHM, and integrated flux) derived from the cube. After applying PCA, a 2D Gaussian smoothing, with a FWHM of 5 pixels, was applied, and the moment maps (corresponding to integrated surface brightness, peak velocity, and velocity dispersion) were derived. We need only the peak velocity map for our purposes here, which is displayed in panel (a) of Figure 2.

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2.2. The MUSE Observations

We used archive observations from the MUSE, on the VLT of the European Southern Observatory (ESO). The wavelength coverage is from 4650 to 9300 Å over a field of view of 1 × 1 arcmin². NGC 3433 was observed with MUSE in 2014 February 20th (Program ID: A-9100), using the wide field mode, with an exposure time of 3900 s divided among 13 exposures. The atmospheric seeing, which limited the angular resolution, ranged from 0.6 to 0.8 arcsec. The data were published on 2014 July 28th and were reduced to a single data cube using MUSE data reduction software (pipeline version v1.0.1) in the Reflex environment. This combines the calibration files to correct the science frames for bias, dark, flat, geometric calibration, and sky emission, while performing the wavelength and photometric calibrations. The final data product is a cube of 315 × 316 × 3682 spaxels, with an effective angular size 0.2 × 0.2 arcsec and a formal spectral resolution of 1.25 Å, corresponding to some 50 km s⁻¹. With these data, as with the GHαFaS data described above, significantly higher effective spectral resolution can be achieved given a sharp enough line or set of lines, and a high enough signal-to-noise ratio. Note that the PCA technique is also applied to the MUSE data.

As an initial step in the kinematic analysis we need to derive the rotation curve of the galaxy from the velocity map, using the ROTCUR task of the GIPSY⁴ package. This technique yields a circular velocity for each of the tilted rings employed in the method, and these points can then be plotted against the galactocentric radii of the rings. We used the method on each of the velocity maps shown in Figure 1, for stars and gas in the MUSE data and for gas alone in the GHαFaS data. We first plotted separate rotation curves for each of the three data sets using the best-fit polynomials with the expression

$$\begin{eqnarray*}&&{V}_{\mathrm{rot}}(r)=a\cdot \displaystyle \frac{r/b}{{[1/3+2/3{(r/b)}^{1/2}]}^{3}},\end{eqnarray*}$$

where a is a velocity with a value that turns out to be close to the maximum velocity of the curve, and b is a radius whose value is close to that at which this velocity is attained. As a second step we plotted all the points derived from the three velocity fields in a single graph, and performed a single polynomial fit to the combined data set. This is shown in Figure 3, together with its associated band of uncertainty. We used the curve in Figure 3 as a starting point for the FB derivation of the resonance radii, as described below.

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The two methods we used to derive the corotation radii for NGC 3433 have been described in detail in the literature, so here we will give a very brief summary of each. An important point, is that the methods measure two different parameters. The TW method measures the pattern speed, (in the original formulation, of the bar in a barred galaxy), from which the corotation radius can be derived using the rotation curve. The FB method determines a corotation radius directly, and its corresponding pattern speed can then be determined, again from the rotation curve. Finding agreement between the two measured values would strengthen the validity of both methods. To find the pattern speed using TW (see Merrifield & Kuijken 1995) we need to measure two quantities: the line-of-sight velocity of the stellar population, measured along a set of long spectral slits parallel to the major axis of the disk, and the continuum emission from that population, measured over the same field, using the expression

$$\begin{eqnarray*}&&{{\rm{\Omega }}}_{P}=\displaystyle \frac{1}{\sin (i)}\cdot \int \displaystyle \frac{l(x)\cdot [{v}_{\mathrm{los}}(x)-{v}_{0}]}{l(x)\cdot [x-{x}_{0}]}{dx},\end{eqnarray*}$$

where the integrals are performed along the slit, v_los(x) is the mean line-of-sight velocity of the stars at position x on the slit, v_o is the systemic velocity of the galaxy, the square brackets indicate the average value along the slit weighted by l(x) the luminosity of the stars at each point, and i is the angle of inclination of the galaxy disk to the line of sight. To improve the signal-to-noise ratio the process can be repeated along as many slits as can be observed, and an effective mean can be derived. In principle, this method assumes a continuity equation for the population whose velocity is measured, which would imply that only an older stellar population should be used. We will see that this condition may be relaxed and TW can be applied to the velocity field of the interstellar gas. However, it has mainly been used for earlier types of disk galaxies because dust obscuration must affect the photometric continuity of the stellar continuum signal.

To find a corotation radius using FB we use the change in phase of the non-circular (radial) velocity vector across corotation. Using a 2D map of the line-of-sight velocity, we first derive a mean rotation curve, and use this to form a 2D circular velocity model. This model is subtracted off from the initial field leaving a residual field of (projected) non-circular velocity. The third step is to identify the positions in the map where the residual velocity is zero, and (to avoid problems of noise and projection effects) to eliminate the sub-set of these pixels where the change in radial velocity, from inflow to outflow or vice versa, has an amplitude smaller than twice the rms uncertainty in the velocity measurements. The fourth step is to plot a histogram of the numbers of the remaining pixels against galactocentric radius. This histogram shows very well defined peaks, which we identify as corotation radii.

From our previous studies using FB (Font et al. 2011, 2014a, 2014b) we have shown that there is usually more than one corotation in a galaxy, and we have always been able, using the morphology, to assign each corotation to a feature: a main bar, a nuclear bar, or an annular section of the spiral arms. This has produced results akin to those of Meidt et al. (2008), who modified the simplest form of TW to assign several corotation radii to identified morphological features of the galaxies over concentric annuli of galactocentric radius. Because FB normally uses the emission from the interstellar medium, it is most suited to later-type galaxies, and is complementary to TW. However, for the galaxy under study we could use both methods, giving us the opportunity to compare them.

4.1. Using TW

The MUSE spectra give a wide range of stellar absorption lines, but also interstellar Hα emission, so we could use the 2D spectral map to derive pattern speeds from both types of velocity fields, and then go on to infer the corotation radii. Without trying to segment the data radially, TW gave us a clear pattern speed for the main bar (which is quite oval) using both stars and gas, with the inferred corotation radius at 18.3 ±4.1 arcsec using the Ca near-IR triplet in absorption for the stellar component, (following the technique of using multiple lines to enhance the effective velocity precision for TW described in García-Lorenzo et al. 2015), and at 18.2 ±3.5 arcsec using interstellar Hα emission. It was easier to make radial segments with the GHαFaS observations, and we could measure a wider set of corotations using TW and GHαFaS. These yielded a corotation at 17.9 ± 4.5 arcsec, when the GHαFaS data are confined to the same region as the MUSE observations, which is in good agreement with the MUSE value using TW, and a corotation at 58.9 ± 8.2 arcsec using the complete GHαFaS map.

In Figure 4 we show the TW plots using both instruments. In the left panel we show detailed results for the MUSE and GHαFaS observations. Each point represents the value for a pseudo-slit of the ratio of the integrated value of the line-of-sight velocity and the integrated distance, along the slit, both weighted by the luminosity. The pattern speed is derived from the slope of the graph. The left panel shows plots using stars and gas from MUSE, and the central zone, which is dominated by the bar, from the GHαFaS data. The right panel shows the full disk fit for the GHαFaS data. Luminosity weighting for the Hα observations used the nearby stellar continuum.

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We note here that the quoted uncertainties in the values of the pattern speeds obtained using TW combine the measured uncertainty in the inclination angle of the disk, the uncertainty in the position angle of the inner zone for the TW plots in the left panel of Figure 4, the uncertainty in the position angle of the outer disk for the TW plots in the right panel of Figure 4, and the uncertainties in the slopes of each of the TW plots presented in both panels of Figure 4. When converting a pattern speed to the corresponding value of the corotation radius, the uncertainty in the frequency curve was also taken into account, but as will be seen in Figure 5, the resulting estimated uncertainties are clearly conservative.

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4.2. Using FB

Using the MUSE map to derive corotation radii directly with FB, and first taking the interstellar Hα emission line map, we found three values: 8.9 ± 0.5 arcsec, 18.5 ± 1.4 arcsec, which is in good agreement with the TW value above, and 31.5 ±3.0 arcsec. If we use FB with the stellar absorption spectral map from MUSE, we find only one corotation, at 18.3 ± 2.7 arcsec, in excellent agreement with the three values obtained with TW above. Turning to the GHαFaS map and deriving corotations with FB we find one at 8.2 ± 0.6 arcsec, another at 17.8 ±1.9 arcsec, a third at 31.9 ± 2.4 arcsec, and a fourth at 59.5 ±2.5 arcsec. The radius of the innermost corotation is in excellent agreement with the values found using FB with MUSE. The second corotation radius is in excellent agreement with the values found using TW with both gas and stars, from the MUSE map, as well as with the values found using FB with both stars and gas in MUSE. The third corotation radius in excellent agreement with the value found using FB with gas, from the MUSE data. The fourth corotation radius derived using FB and gas, with GHαFaS cannot be compared with any other value, because this radius is well outside the MUSE field, and GHαFaS does not obtain a stellar velocity map. All the observed values of the four resonances are shown in Table 2, where we can see the agreement between the two methods in all cases, well within the uncertainty limits, and also the agreement between the values obtained using the kinematics from the stellar spectra and those using the kinematics from the interstellar spectra.

Table 2.  Corotation Radii and Pattern Speeds for NGC 3433

Resonance	Method	Corotation Radius		Pattern Speed
		(arcsec)	(kpc)	(km s−1 kpc−1)
1	FBg(γ)	8.2 ± 0.6	1.3 ± 0.1	85.1 ± 3.9
1	FBg(μ)	8.9 ± 0.5	1.4 ± 0.1	81.0 ± 3.8
2	FBg(γ)	17.8 ± 1.9	3.6 ± 0.4	49.2 ± 3.7
2	TWg(γ)	19.1 ± 2.1	3.9 ± 0.5	46.7 ± 3.7
2	FBg(μ)	18.5 ± 1.4	3.7 ± 0.3	47.8 ± 3.0
2	TWg(μ)	20.0 ± 2.7	4.0 ± 0.6	45.0 ± 4.2
2	FBs(μ)	18.3 ± 2.7	3.7 ± 0.5	48.2 ± 5.2
2	TWs(μ)	18.1 ± 3.0	3.7 ± 0.6	46.7 ± 4.9
3	FBg(γ)	31.9 ± 2.4	6.4 ± 0.5	30.9 ± 2.0
3	FBg(μ)	31.5 ± 3.0	6.4 ± 0.6	31.3 ± 2.6
4	FBg(γ)	59.5 ± 2.4	12.1 ± 0.5	17.6 ± 0.7
4	TWg(γ)	60.6 ± 7.2	12.3 ± 1.4	17.3 ± 1.8

Note. Second column gives the method used: TW_s refers to the Tremaine–Weinberg method using the stellar component; TW_g is the Tremaine–Weinberg method using the ionized IS gas; FB_g is the Font–Beckman Method using the ionized IS gas; and FB_s is the Font–Beckman method using the stellar component. The instrument used is indicated in brackets; γ means GHαFaS and μ means MUSE. Numbers in boldface are the primary parameters derived by the method used (pattern speed for Tremaine–Weinberg, and corotation radius for Font–Beckman).

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It is instructive to show how the values of those corotations where we could use both TW and FB compare, taking into account the uncertainties in their derivations. In each panel of Figure 5 we have plotted the frequency curve (Ω against galactocentric radius) obtained using the relevant fitted rotation curve from Figure 3; the uncertainties associated with this curve are shown as shaded regions in green, which are calculated taking into account the uncertainties of the rotation curve and the standard deviation of the fitting parameters to the rotation curve. For a given corotation we have plotted the radius, derived directly using FB, as a vertical dashed line, with the uncertainty as a blue shaded surrounding bar. For the comparison we have plotted the pattern speed derived using TW as a horizontal dashed line, with the associated uncertainty as a pink shaded surrounding bar. In the three panels we have shown these measurements for, respectively, the GHαFaS Hα field, the MUSE Hα field, and the MUSE stellar field. We can see that for all the corotations the crossing points of the vertical bars with the Ω curves are in agreement with the crossing points of the horizontal bars with the curves, and the two crossing points are in fact indistinguishable in all four cases shown. This implies that the uncertainties that we have computed as described above are in fact conservative. It also implies that the two methods give identical results, which we take as strong support for FB given that TW is an established technique.

In Figure 6 we have summarized the information in Table 2 for ease of visualization. In the left panel we have shown the corotation radii against the method used to obtain each value. We show the results of the FB determinations in red, as these are the direct measurements, and the results of the TW determinations in black, as these are derived by combining the measured pattern speed with the angular velocity of the rotation curve. In the right panel we have shown the pattern speeds. In this case the direct determinations using TW are plotted in red and the values using FB, which are obtained by combining the measured pattern speeds with the angular velocity of the rotation curve, are plotted in black. It should be clear that the larger corotation radii correspond to the smaller values of pattern speed, so the plots are in this sense inversely related.

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Figure 7 shows the histograms which we used to derive the corotation radii, using data from both instruments and the velocity fields of both gas and stars. On the left is the histogram from the GHαFaS velocity field. We detect four clear peaks, whose radii are defined by the Gaussian fits to the histogram. In the center panel we have the equivalent histogram from the MUSE Hα emission velocity field. The resonance peaks agree very well with those from GHαFaS, although, given the restricted angular size of the MUSE field, we find only the inner three resonances. On the right we show the MUSE histogram produced using the absorption lines of the stellar population. We can see that there is only one peak, at the corotation corresponding to the bar, which we would expect to be the strongest resonant feature. This peak is much wider than those in the previous figure, and we attribute this to the fact that the stellar orbits are "more three-dimensional" than the gas orbits. The stars have a greater scale height above the disk plane than the gas, and a significant component of velocity perpendicular to the plane. Our technique of finding zeros in the radial velocity is thus partly compromised by false zeros due to components perpendicular to the plane. This has not, however, affected the value of the corotation radius obtained using FB with the stellar component, determined by the peak of the distribution, as listed in Table 2. Taking the mean value of the bar corotation radius and our measured bar length, we find that the ratio of the two has the value 1.2 ± 0.1.

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In Figure 7, we also show how the resonances are coupled in the sense we reported in Font et al. (2014a). This means that given two pattern speeds, for example, Ω₁ = 85.5 ± 4.1 km s⁻¹ kpc⁻¹ and Ω₂ = 46.8 ± 3.3 km s⁻¹ kpc⁻¹, with corotations at 8.9 ± 0.5 arcsec and 18.5 ± 1.4 arcsec, respectively (see Table 2), it turns out that the radius of the OLR of Ω₁ (OLR₁) is consistent with the corotation radius of Ω₂ (CR₂), within uncertainties, and also the inner 4:1 resonance of Ω₂ (I4:1₂) occurs at the same radial position as the corotation of Ω₁ (CR₁). This interlocking pattern is displayed in panel (b) of Figure 7, in which the radial positions of each resonance are colored and numbered according to their associated corotation. This figure also shows that we have identified two interlocking patterns between the three resonances found: between the first and the second resonances, and between the second and the third resonances. Finally, using the outermost resonance found only with GHαFaS in Hα, we also detect the same characteristic coupling pattern between the third resonance and the fourth, outermost resonance.

Even though density wave theory has deep historical roots (Lindblad 1938; Lin & Shu 1964) and innumerable subsequent research contributions, there are several aspects of the theory that could still benefit from rigorous observational support. Foyle et al. (2011), for example, questioned the long-term stability of spiral arms, based on measured sequences of star formation indicators across the arms, but the predictions tested were for density wave models with a single corotation radius, while Font et al. (2014b) showed that virtually all the star-forming disk galaxies they observed showed multiple corotations. In this context, as well as to underpin attempts to use the rotation of bars in tests of cosmological models, it is valuable to quantify the parameters of a density wave system by comparing two independent methods for determining its basic parameters. We have done this here and give the following conclusions:

• 1.  
  It is possible to determine the pattern speeds and corotation radii of spiral galaxies quite accurately. The values of the formal uncertainty for the corotation radii obtained for NGC 3433 are of the order of 3%.
• 2.  
  (a) We used two very different instrumental and telescope setups, with results that are the same for both.
• (b)  
  We used two methods that employ different observational input parameters and derive different output parameters. TW yields values for the pattern speed and FB yields values for the corotation radii. Using the rotation curve, it is then easy to compute the other output parameter for each method. The results are in complete agreement.
• (c)  
  We used the velocity fields of the stellar component and of the ionized gas component for both TW and FB, with results that give full agreement.

Using the ratio of the corotation radius to bar length to test the hypothesis that bars in galaxies should be braked by dynamical friction, Font et al. (2017) showed that the uncertainty in the measured length of the bar, rather than that in the radius of corotation, is the limiting factor in deriving this ratio. Here, we have gone further, showing that corotation radii in disk galaxies can be among the most precisely measurable parameters in disks, and that the general principles of density wave theory are strongly supported by these measurements.

We thank the anonymous referee for a series of perceptive comments that caused us to make significant improvements to the article. The research was carried out with funding from project P/308603 of the Instituto de Astrofísica de Canarias. The William Herschel Telescope is operated on the island of La Palma by the Isaac Newton Group in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias. This work is based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere under ESO programme A-9100. B.G-L acknowledges support from the Spanish Ministerio de Economia y Competitividad (MINECO) by the grant AYA2015-68217-P.