Spectroscopic Comparison of Metal-rich RRab Stars of the Galactic Field with their Metal-poor Counterparts

For et al. (2011b) describe a scattered light problem peculiar to the du Pont echelle spectrograph, and they outline a procedure used to remove scattered light from their observations. This special technique is required because the projected length of the entrance aperture of the du Pont echelle, constrained by prismatic order separation, is too small longward of 5000 Å to permit subtraction of scattered light by use of pixels adjacent to the projected stellar image. In this paper we use a greatly simplified procedure, namely, subtraction of a constant fraction of the stellar flux at each pixel along each echelle order. This model attributes all scattered light to small-angle scattering into regions immediately adjacent to each spectral order. It ignores large-angle scattering that could, for example, flood high (blue–violet) orders with red light from distant low (red) spectral orders in the echelle format. Large-angle scattering into blue–violet orders can be important in the case of cool, red stars, but RR Lyrae are not very red, so our simplified correction works well. We demonstrate this with measurements of equivalent widths (EWs) in the spectrum of one of our radial velocity standard stars, the metal-poor subgiant HD 140283. This star is somewhat redder (B − V = 0.50) than the RRab stars, for which colors lie in the range 0.1 < B − V < 0.5 during their pulsation cycles.

We added 38 du Pont echelle spectra of HD 140283, flattened them, subtracted 0.1 from the normalized continua in all orders, and renormalized. The signal-to-noise ratio (S/N) in this spectrum increases from 250 at 4000 Å to 500 at 5500 Å. For this test we used the IRAF/splot⁶ package to measure EWs of the Fe i and Fe ii lines that were measured in a Suburu spectrum of HD 140283 by Gallagher et al. (2010). We correlate these two EW sets in Figure 2, plotting the Suburu values (labeled EWS) versus those from du Pont (EWD) in panel (a), and line-by-line differences EWS − EWD as functions of line strength EWS in panel (b) and wavelength in panel (c). The agreement is good: $\langle \mathrm{EWS}-\mathrm{EWD}\rangle =-0.48$ mÅ, with standard deviation σ = 1.01 mÅ from 83 lines in common.

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The EW differences in panel (c) of the figure allow us to investigate whether large angle scattering from the brighter low (red) orders in the du Pont spectrograph produces wavelength-dependent residuals. The lack of significant variation with wavelength leads us to conclude that our scattered light corrections are acceptable, i.e., they play an insignificant role in the error budget of our abundance analyses.

We used the co-added spectra to first derive their model atmospheric parameters. We also decided to re-derive models and abundances for the 10 stars originally discussed by For et al. (2011b), in order to ensure that their parameters were on the same system that we employed for the rest of our sample. We used spectra of one co-added phase bin at ϕ = 0.5 for their stars. Here we highlight the main analytical points and note any substantive differences between the present and previous studies.

We adopted the line list of For et al. (2011b), updating the transition probabilities with new laboratory values for Ti i (Lawler et al. 2013), Ti ii (Wood et al. 2013), and Fe i (Den Hartog et al. 2014), combining these with values from O'Brian et al. (1991) for lines with excitation energies χ < 2.3 eV. For these lines we measured EWs using an Interactive Data Language (IDL)⁷ code developed by Roederer et al. (2010) and Brugamyer et al. (2011). These line lists and interpolated ATLAS (Kurucz (2011 and references therein) model atmospheres were used as inputs to the current version of the LTE line analysis code MOOG (Sneden 1973).⁸ In the analyses we eliminated those lines that are too weak for reliable measurement.

The results of these calculations were line-by-line abundances for all species. The parameters of the models were altered and the line analyses repeated to derive: (a) T_eff, so that there would be no average trend of abundances with excitation energy derived from Fe i lines; (b) ξ_t, so that there would be no average trend of abundances from Fe i with EW; (c) log g, so that mean abundances derived from Fe i and Fe ii agreed well, and secondarily the mean abundance derived from Ti i would not be in severe disagreement with that from Ti ii; and (d) [M/H], so that the assumed model metallicity was close to the derived [Fe/H] value. This procedure took typically three or fewer iterations. The rapidity of convergence was a result of two RRab properties. First, within the confines of traditional analyses, the RRab stars occupy a relatively small region of model parameter space, typically 6000 < T_eff < 7000 K, 1.7 < log g < 2.7, and 2.5 < ξ_t < 3.5 km s⁻¹. Second, these stars are warm enough that the free electrons needed for the dominant H⁻ continuous opacity are largely donated from H itself instead of the easily ionized metals, and so the widely ranging metallicities (−2.5 < [M/H] < 0.0) do not severely affect the model atmosphere structures.

Final model atmosphere parameters for the program stars in all co-added phase bins are listed in Table 2. In considering these values the reader should keep in mind that they result from the application of static, plane-parallel model atmospheres and LTE line analysis techniques to stellar atmospheres that are highly dynamic. In less than a day the RRab stars cycle through correlated variations of 1000 K in Teff and 1 dex in log g and ξ_t ( e.g., For et al. 2011b). The derived spectroscopic gravities have static and dynamic components. Moreover, the derived microturbulent velocities reflect both the standard small-scale turbulent velocities that are applied in analyses of non-variable stellar atmospheres and radial velocity gradients within the line-forming regions of RR Lyrae pulsators.

We also derived several abundance ratios [X/Fe] for each program star at each pulsational phase. However, abundance ratios are of secondary interest in this paper, so given the S/N limitations of our spectra we have chosen to report mean abundance ratios [X/Fe] for each star only for species that were available for analysis at most phases of most stars: Ca i, Sc ii, Ti i and ii, Cr i and ii, Mn i, and Ba ii, along with the mean [Fe/H] metallicities. All abundance ratios are referenced to the solar photospheric values recommended in the Asplund et al. (2009) review. Those values are in many cases based on analytical assumptions (e.g., non-LTE, multi-stream modeling) that are beyond the scope of our standard abundance analysis techniques. Recently, a series of papers (Grevesse et al. 2015; Scott et al. 2015a, 2015b) has updated the solar abundances, computing them in a variety of ways. The solar abundance differences they derive are typically not large, usually less than 0.05 dex for the species of interest here. These are small uncertainties compared to the other error sources in our work.

The abundance ratios were computed for the same ionization-state abundances in numerator and denominator, e.g., [Ca/Fe] from Ca i and Fe i, and [Sc/Fe] from Sc ii and Fe ii. These ratios have been computed from those derived at each phase for a star. At the bottom of this table we also list the mean line-to-line abundance scatters for each species $\langle \sigma \rangle$ and the mean number of lines employed in the calculations $\langle n\rangle$ in individual star/phase abundance derivations. These numbers can vary from star to star and phase to phase, but suggest that the σ values are never less than ≃0.1. Additionally, some species are represented by three or fewer transitions. Such abundance results should not be over-interpreted.

Table 3.  Mean Abundance Ratios

Star	[Fe/H]	[Fe/H]	[Ca/Fe]	[Sc/Fe]	[Ti/Fe]	[Ti/Fe]	[Cr/Fe]	[Cr/Fe]	[Mn/Fe]	[Ba/Fe]
	i	ii	i	ii	i	ii	i	ii	i	ii
WY Ant	−1.82	−1.77	0.32	0.05	0.39	0.29	0.07	0.03	−0.50	...
BS Aps	−1.51	−1.44	0.28	−0.05	0.27	0.18	0.00	0.07	−0.34	−0.07
XZ Aps	−1.65	−1.65	0.30	0.05	0.32	0.35	0.02	0.10	−0.31	−0.07
DN Aqr	−1.77	−1.74	0.36	0.01	0.34	0.27	0.03	0.11	−0.46	−0.09
SW Aqr	−1.39	−1.36	0.35	−0.17	0.29	0.13	0.02	0.04	−0.35	−0.14
X Ari	−2.60	−2.60	0.35	0.25	0.45	0.44	−0.11	0.21	−0.75	−0.68
RR Cet	−1.57	−1.56	0.31	0.10	0.28	0.24	−0.02	0.07	−0.33	−0.01
W Crt	−0.76	−0.74	0.30	−0.24	0.04	0.03	−0.06	−0.01	−0.26	−0.55
DX Del	−0.52	−0.58	0.07	−0.43	−0.28	−0.11	−0.30	−0.05	0.01	−0.14
SX For	−1.80	−1.79	0.39	−0.02	0.29	0.24	−0.01	0.07	−0.46	−0.13
DT Hya	−1.43	−1.35	0.29	−0.03	0.29	0.27	0.21	0.14	−0.33	...
V Ind	−1.62	−1.59	0.34	−0.02	0.27	0.21	0.05	0.07	−0.43	−0.13
SSLeo	−2.36	−2.32	0.41	0.03	0.48	0.42	0.04	0.29	−0.75	0.05
ST Leo	−1.32	−1.29	0.32	−0.10	0.28	0.23	0.18	0.16	−0.44	−0.03
Z Mic	−1.46	−1.46	0.27	−0.09	0.27	0.25	0.07	0.08	−0.40	−0.06
RV Oct	−1.47	−1.44	0.30	−0.06	0.29	0.21	0.09	0.03	−0.52	−0.32
UV Oct	−1.69	−1.69	0.29	−0.03	0.25	0.23	0.06	0.10	−0.48	−0.23
V0445 Oph	−0.05	−0.19	−0.10	−0.58	−0.51	−0.20	−0.15	0.07	0.11	−0.23
AV Peg	−0.14	−0.20	0.12	−0.43	−0.38	−0.24	−0.23	−0.02	−0.11	−0.30
HH Pup	−0.95	−0.90	0.06	−0.71	−0.16	−0.27	−0.08	0.02	−0.08	−0.33
AN Ser	0.06	0.03	0.01	−0.51	−0.46	−0.36	−0.26	−0.01	0.03	−0.32
VY Ser	−1.88	−1.87	0.35	0.09	0.30	0.28	−0.01	0.08	−0.51	0.02
v1645 Sgr	−1.81	−1.79	0.31	−0.03	0.31	0.22	−0.06	0.17	−0.34	−0.11
W Tuc	−1.77	−1.74	0.35	0.02	0.31	0.25	−0.05	0.15	...	0.19
CD Vel	−1.71	−1.73	0.29	−0.03	0.31	0.25	0.19	0.03	−0.50	−0.09
AS Vir	−1.59	−1.54	0.31	−0.13	0.26	0.15	0.18	0.10	−0.42	−0.22
ST Vir	−0.85	−0.86	0.20	−0.24	0.03	0.06	−0.08	0.01	−0.17	−0.31
UU VIr	−0.93	−0.92	0.23	−0.59	0.10	0.10	0.00	0.09	−0.08	−0.29
$\langle \sigma \rangle$	0.14	0.16	0.15	0.11	0.11	0.11	0.11	0.13	0.10	0.09
$\langle \#\rangle$	97	22	8	3	5	7	2	3	2	2

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The derived absolute quantities should be viewed with caution. Relative values from star to star among our RRab sample should be much more reliable. All spectral features studied in this paper are due to "photospheric" absorptions by α- and Fe-group elements in a restricted strength (EW) range. Thus their lines are formed in similar atmospheric layers, and their abundance ratios should be less sensitive to modeling uncertainties than are the individual abundances.

The derived temperatures of our stars vary with phase in a manner broadly consistent with the results of For et al. (2011a). Immediately after maximum light the temperatures decline, rapidly at first and then more slowly from phase ∼0.3 to phase 0.6, the upper phase limit of our binned spectra. The T_eff values for individual stars, divided as defined in Section 1 into metal-rich and metal-poor regimes at [Fe/H] = −1, populate two distinct regimes in correlations with phase, as shown in Figure 3. We have added curves to the figure to represent quadratic regression lines for the two metallicity groups. Model T_eff of metal-rich stars are approximately 250 K hotter than those of metal-poor stars at all phases in the range 0.2 < ϕ < 0.6. Thus we confirm Skarka (2014) conclusion that metal-rich RRab stars are hotter than metal-poor RRab stars at a given B − V color.⁹

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The temperatures and gravities are correlated with metallicities, as illustrated in Figure 4. Solid black lines denote linear regressions of the quantities using the entire metallicity range. The horizontal dashed lines denote mean values of the metal-poor and metal-rich stars. Standard deviations of these regressions for T_eff and for log g, ∼100 K and ∼0.2 dex, respectively are indistinguishable, i.e., simple statistical analysis offers no guidance about choice of regressions. The physical bases for the natures of the correlations are uncertain. We defer further discussion of these two different statistical representations to Section 5.

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The most metal-poor star in our sample, X Ari ([Fe/H] = −2.6), lies closer to the horizontal regressions for T_eff and log g. Data for additional very metal-poor stars collected and analyzed in a uniform way may help to resolve this uncertainty in choice of regressions. We are aware of three such stars: NR Lyr, [Fe/H] = −2.54 (Nemec et al. 2013); CS 22881-029, [Fe/H] = −2.75; and CS 30317-056, [Fe/H] = −2.85 (Hansen et al. 2011).

Spectroscopic metallicities have been derived for RR Lyr stars beginning with the ΔS parameter introduced by Preston (1959) to relate the strengths of the Ca ii K line to the H i Balmer lines appearing on low-resolution spectra. Many observations and calibrations of ΔS as a function of [Fe/H] have been published over the last several decades. The largest single source of K-line metallicities is the Lay94 the 302-star RRab survey. Previous RR Lyr large-sample high-resolution spectroscopic metallicity and abundance ratio surveys include: Clementini et al. (1995, 10 stars); Lambert et al. (1996, 18 stars); Fernley & Barnes (1996, 9 stars); For et al. (2011, 11 stars); Liu et al. (2013, 23 stars); Nemec et al. (2013, 41 stars); and Pancino et al. (2015, 18 stars). We relate the metallicites reported in those studies to the [Fe/H] values derived here, using Layden's [Fe/H] values as the common baseline.

In Figure 5 we plot the differences between [Fe/H] values of various high-resolution studies (including our work) and those of Lay94 as a function of the [Fe/H]_Lay94. We have added simple linear regression lines to illustrate the mean trends, which have positive slopes in five of the seven cases. Inspection of this figure suggests that the most significant differences among most high-resolution studies are metallicity scale offsets.

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To attempt to merge together all of the high-resolution studies, we first repeated the analyses of Clementini et al. (1995) and Pancino et al. (2015), using only lines in common with our work. We adopted their published EWs and our transition probabilities, and derived model parameters in the manner described in Section 3.1. In general our results were not significantly different than theirs.

Then we shifted the [Fe/H] values of other studies onto our scale by using the regression lines shown in Figure 5, and updated regression fits to reflect our new analyses of Clementini et al. (1995) and Pancino et al. (2015) metallicities to calculate the offset between the other studies and ours at a metallicity [Fe/H]_Lay94 = −1.25. This metallicity point is of course arbitrary but is approximately in the middle of the observed RR Lyr metallicity distribution. In Figure 6 we redraw Figure 5 with the re-analyses and applied [Fe/H] offsets. Then all high-resolution analyses are merged in Figure 7, and we add an additional point to include the [Fe/H] value for the eponymous star RR Lyr from the analysis by Kolenberg et al. (2010). There is general agreement on the relationship between high-resolution spectroscopic metallicities and the Lay94 values, and reasonably small scatter in this relationship.. We stress that this exercise is not intended to be advocacy for our own RR Lyr metallicity scale. It simply suggests that with small scale shifts the [Fe/H] values of major RR Lyr high-resolution studies can be brought into good agreement.

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Two stars deserve special comment because they help anchor the high and low ends of the RR Lyr metallicity scale. As such they have been well represented in past and present analyses. At the very metal-poor limit there is X Ari, with [Fe/H]_Lay94 =−2.40. There is some discord among the high-resolution results: [Fe/H] = −2.50 (Clementini et al. 1995, or −2.51 in our reanalysis); −2.47 (Lambert et al. 1996); −2.74 (Nemec et al. 2013); −2.14 (Pancino et al. 2015, or −2.20 in our re-analysis); and −2.60, this study. From all five studies, $\langle [\mathrm{Fe}/{\rm{H}}]\rangle =-2.49$ with σ = 0.22. Application of the scale offsets described above does not significantly shrink the total envelope of values, as can be seen in the lower panel of Figure 7. The revised high-resolution mean value is $\langle [\mathrm{Fe}/{\rm{H}}]\rangle =-2.61$ with σ = 0.25. The shift from [Fe/H]_Lay94 for X Ari is in agreement with the general trend line of our analysis, but renewed attention to this star would be welcome.

SW And is one of the most metal-rich RR Lyr stars with [Fe/H]_Lay94 = −0.38. Most high-resolution studies report higher metallicities for this star, but with scatter: [Fe/H] = −0.06 (Clementini et al. 1995, or −0.24 in our re-analysis); −0.26 (Fernley & Barnes 1996); −0.32 (Lambert et al. 1996); −0.07 (Liu et al. 2013); and +0.20 (Nemec et al. 2013). The means before and affer re-analyses and offsets are $\langle [\mathrm{Fe}/{\rm{H}}]\rangle =-0.10$, σ = 0.20, and $\langle [\mathrm{Fe}/{\rm{H}}]\rangle =-0.27$, σ = 0.14; for this star the high-resolution analyses come into better accord with one another.

With the scale shifts described above, a common relationship emerges between the high-resolution multi-star metallicity surveys considered here and the Lay94 K-line metallicities. Our own data indicate that [Fe/H]_highres = 1.100 × [Fe/H]${}_{\mathrm{Lay}94}\ +0.055$. Future high-resolution studies can and should revisit this relationship, which provides an important calibration for K-line surveys that now can be efficiently conducted for large samples of Galactic and extragalactic stellar systems.

In Section 3.1 we described the derivation of relative abundances of several elements that can be studied even with moderate S/N spectra of metal-poor RR Lyr stars; see Table 3. Here we compare our [X/Fe] values to those reported by other abundance studies of RR Lyr stars and by several surveys of non-variables over a wide metallicty range. In Figure 8 we plot [X/Fe] ratios as a function of metallicity from: (b) this study (black dots); from other RR Lyr abundance studies (Clementini et al. 1995, Fernley & Barnes 1996, Lambert et al. 1996, Liu et al. 2013, and Pancino et al. 2015; red dots); and (c) several large-sample studies of Galactic thin and thick disk main sequence and subgiant stars (Reddy et al. 2003, Reddy et al. 2006, Bensby et al. 2014 green dots), and of Galactic halo low metallicity stars of various evolutionary states (Roederer et al. 2014; also green dots). Relative abundance ratios are to first order insensitive to absolute metallicity scale, so we go through the data in Figure 8, species by species.

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In considering the data of Figure 8, note first that abundance ratios determined in our work are consistent with those reported in other RR Lyr studies. Second, for most elements this agreement extends also to comparisons between the RR Lyr and non-variable stars. Third, the RR Lyr metal-poor abundance ratios are always in accord with those of non-variable stars. However, there are a couple of discordant cases to be noted among the abundances of metal-rich stars, and so here we will go through the figure on species-by-species basis.

• 1.  
  Ca, Figure 8 panel (a): all reported abundances are based on Ca i lines, which are plentiful (our values come from eight lines on average; Table 3). The [Ca/Fe] ratios of RR Lyr stars agree very well with those from other stellar sample over the entire metallicity range.
• 2.   
  Sc, Figure 8 panel (b): all reported abundances are based on a handful of Sc ii lines. For halo stars the Sc abundances of RR Lyr stars agree well with their non-variable counterparts, but for disk stars the [Sc/Fe] values for RR Lyr stars are 0.3−0.4 dex smaller than those of disk main sequence and subgiant stars. One expects [Sc/Fe] ≃ 0 at [Fe/H] ≃ 0, so the RR Lyr abundances surely are wrong here. Resolution of this problem, shared much more mildly by Ti ii and Ba ii (see below), is beyond the scope of this paper. Note that the first ionization potential of Sc is low (IP = 6.56 eV), ensuring that ${N}_{\mathrm{Sc}{\rm{II}}}$ ≫ ${N}_{\mathrm{Sc}{\rm{I}}}$, as is the second ionization potential (12.80 eV, lower than even that of hydrogen, 13.60 eV). It is entirely possible that limitations of standard abundance analyses applied to the variable RR Lyr stars are appearing in the LTE calculations of Sc ionization.
• 3.   
  Ti, Figure 8 panel (c): we have chosen to plot only abundances from Ti ii here, since in general this species has stronger transitions than does Ti i. There is a small Ti abundance offset, ≃0.2 dex, between RR Lyr and non-variable disk stars.
• 4.  
  Cr, Figure 8 panel (d): we were only able to measure a few lines of each Cr species, and so we chose to average their abundance ratios for this figure. Within the uncertainties of RR Lyr analyses, there is agreement with the non-variables at all metallicities.
• 5.   
  Mn, Figure 8 panel (e): the well-known sharply declining [Mn/Fe] values with decreasing metallicity are reproduced well in RR Lyr abundance surveys. There are too few points for stars with [Fe/H] < −2 to draw firm conclusions on a possible negative offset relative to non-variable stars in this metallicity regime.
• 6 .  
  Ba, Figure 8 panel (f): the agreement between RR Lyr abundance ratios and those of non-variables is satisfactory, given the large star-to-star scatter among all samples.

In summary, the abundance ratios derived for our RR Lyr sample are mostly consistent with those seen in other stellar populations; further investigation of Sc and Ba would be welcome.

Thirteen orders of each RR Lyrae spectrum spanning the wavelength interval λλ 4000–4600 Å were flattened and stitched together by use of an IRAF script prepared by I. Thompson (2016, private communication). This is the same spectral interval used in the radial velocity investigations of Preston & Sneden (2000) and Chadid & Preston (2013). Radial velocities of metal lines in these spectra were obtained by use of the IRAF/fxcor package. Masks centered on Hδ and Hγ were employed to avoid deleterious broadening of the cross-correlation function by these wide, strong lines. The reference spectrum for metal-lines is that of CS 22874–009 for which we adopted RV = −36.6 km s⁻¹, the value used by Preston & Sneden (2000). The reference spectrum for Hα is that of the metal-poor giant CS 22892–052 for which we adopted RV = +14.2 km s⁻¹. This particular value contributed to an appreciable systematic correction noted in the following paragraph.¹⁰ Outputs of the fxcor package include VHELIO, the radial velocity corrected for the Earth's orbital and diurnal motions, and HJD, the Heliocentric Julian Date needed to calculate pulsation phases.

Numerous observations of HD140283, our primary radial velocity standard, for which RV = −170.9 km s⁻¹ (Kollmeier et al. 2013) were used to apply a systematic correction of −2.9 km s⁻¹ to measured Hα radial velocities. Spectra of du Pont echelle order 53, which contains Hα near its center, were prepared for measurement as follows. For each stellar spectrum we made an average spectrum of adjacent orders 52 and 54. We divided order 53 by a spline fit to the continuum of this average spectrum, thus obtaining a flattened normalized spectrum of order 53 in which fluxes at Hα can be measured in units of the continuum flux at 6560 Å. These spectra were also used to measure the radial velocity of Hα.

We used the Gaussian fitting function of IRAF/fxcor package to locate the centers of Hα absorption features. In the course of our measurements we discovered that the "Doppler" core of the Hα line is noticeably asymmetric during substantial fractions of pulsation cycles, the slope of one wing (sometimes red, sometimes violet) being greater than the other. In all such cases we fit a Gaussian to the flux minimum and the steepest wing of the profile, arbitrarily attributing the other broader wing to unspecified asymmetry in line-of-sight gas motion, perhaps due to weak shocks at some pulsation phases. We do not know why the profiles are asymmetric, but we needed a rule in order to measure them in a consistent manner. For a description of the shock complexity that accompanies RR Lyrae pulsation see the observational evidence presented by Chadid & Preston (2013). The magnitude of effects resulting from use/non-use of our measurement procedure is almost always less than 2 km s⁻¹, and usually less than 1 km s⁻¹, metrics that characterizes the inherent uncertainty of Hα radial velocity estimates made with the du Pont echelle spectrograph.

Emission flux measurements were made with the IRAF/splot package as sums of pixel counts in the Hα profile above a reference continuum. When the emission profile lies entirely above the continuum, the measurement is unambiguous. However, when the emission is flanked by violet and red absorption components, an inescapable ambiguity arises about the portions of the flux between the absorption components that are due to overlapping absorption wings and those due to bona fide line emission. In instances when the flux maximum between absorption components lies below the continuum level, skeptics may deny the existence of any emission. Justification for emission measurements in such cases rests on an appeal to phase continuity; namely, it would be unreasonable to suppose that emission ceased exactly at the moment when the apparent emission peak fell below continuum level. On the other hand, deciding exactly when to discontinue emission measurements and to begin measurement of violet-shifted absorption velocities requires a subjective judgment that we base on apparent radial velocity. When a violet absorption minimum first appears, its apparent velocity is strongly negative relative to the velocity measured minutes later when the violet and red minima have equal depths. Examples of rapidly increasing sequences of such "apparent velocities" near phase 0.9 can be seen in left panels of a number of stars in Figure 9. We deliberately included such (spurious) measures in the RV plots to illustrate how we decided when to treat the apparent violet minima as valid indicators of mass motion in the atmosphere. In our view the proper moment occurs when a cusp in the expanding velocities occurs near mid-rising light. Prior to that moment we regard the violet-displaced flux minimum in the profile merely as an apparent absorption minimum located between the violet Stark absorption wing and the violet wing of approximately undisplaced central emission.

During primary light rise Balmer line emission (different in each star and overlain by red-displaced absorption) appears briefly in many RRab spectra, as illustrated in Figure 6 of Preston & Paczynski (1964). We use the emission flux in the violet wing of the Hα profile as an admittedly imperfect proxy for the total (un-measurable) flux. Measurements of this Hα flux were made as follows. A reference continuum was chosen at the violet-displaced flux minimum in the Hα profile, as illustrated by the horizontal red line in Figure 10. Uncertainties of unknown magnitude associated with this choice of reference level undoubtedly occur from exposure to exposure during primary light rise because of rapidly changing contributions from two absorption profiles to the total flux. Flux counts in the emission profile above the reference level (the red horizontal line in Figure 10), are expressed in units of the local continuum fluxes at the phases of observation. To obtain true relative fluxes it would be necessary to multiply the plotted fluxes by the phase-dependent continuum fluxes at 6560 Å, which increase rapidly during the Hα emission phase sequences.

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Table 4 contains a summary of quantities derived from the radial velocity and flux measurements. For five stars we list two metal velocity amplitudes that differ by as much as 4 km s⁻¹ in different cycles. Our survey data do not provide enough information to determine whether these velocity differences are due to low-amplitude Blazhko modulation or to "irregularity" of the sort reported by Chadid (2000). For consistency we use the largest of these values in all graphical displays that follow.

From the du Pont echelle heliocentric radial velocity V_r(t), we derive a pulsational velocity $\dot{R}(t)$ in the stellar rest frame using:

$$\begin{eqnarray}&&\dot{R}(t)=-p({V}_{r}(t)-{V}_{* })\end{eqnarray} \tag{ 1 }$$

where p is the value of the correction factor for geometrical projection and limb darkening and V_* the center-of-mass velocity. Because we derive radial velocities from metallic lines formed near the photosphere, the pulsational velocity $\dot{R}(t)$ is approximately that of the photosphere. We adopt in this study the projection facto p = 1.36. Use of this approximation is discussed by Chadid & Preston (2013). Using polynomial fitting, we calculate the γ-velocity as the average value of the heliocentric radial velocity curve over one pulsation period. The derivative of $\dot{R}(t)$ gives the dynamical acceleration $\ddot{R}(t)$ of the stellar surface layer in which metallic absorption lines are formed:

$$\begin{eqnarray}&&\ddot{R}(t)=d\dot{R}(t)/{dt}\end{eqnarray} \tag{ 2 }$$

Performing the integration of $\dot{R}(t)$, we derive the photospheric radius variation ΔR(t):

$$\begin{eqnarray}&&{\rm{\Delta }}R(t)=R(t)-{R}_{{ph}}={\int }_{0}^{t}\dot{R}(t){dt}\end{eqnarray} \tag{ 3 }$$

where R_ph is the mean photospheric radius of the star.

Radius and acceleration curves are respectively computed from integration and derivative of radial velocity curve. During the pulsation period (0 < ϕ < 1) the gaps in the measurement data have been interpolated linearly and, in order to minimize the noise, the raw radial velocity curves have been convoluted by a sliding window of Δϕ = 0.1 width.

Table 5 contains a summary of photospheric motion parameters derived from the photospheric heliocentric radial velocity of the metal-poor and metal-rich RR ab stars. The acceleration and the radius variation curves for individual metal-poor and metal-rich RR ab stars, as well as the means for the two groups, are shown in Figure 11. The acceleration curves exhibit two significant peaks. An acceleration peak (hereafter primary acceleration) occurs near phase ϕ = 0.90 ± 0.02; it corresponds to the phase of minimum radius. A much smaller secondary acceleration peak occurs near phase ϕ = 0.72 ± 0.02. These secondary accelerations are clearly visible for several of the stable RRab stars in Figure 5 of Chadid & Preston (2013). Any acceleration associated with the lump,¹¹ rump, and jump in the RV curve (Chadid et al. 2014) that may be present around ϕ = 0.30, ϕ = 0.10 and ϕ = 0.70 respectively are at or below the level of detection.

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Chadid & Preston (2013) call primary acceleration the dynamical gravity of the star. We derive the average of the gravitational acceleration,

$$\begin{eqnarray}&&g\,=\,G\displaystyle \frac{M}{{R}^{2}}\end{eqnarray} \tag{ 4 }$$

where G is the gravitational constant (G = 6.67 × 10⁻¹¹), and M and R are respectively the mass and radius of the star.

Figure 11 shows that almost all metal-poor RRab stars have radius variations larger and primary accelerations smaller than those of metal-rich RRab stars. Mean radii of metal-poor and metal-rich RRab stars occur at very nearly the same phases, ϕ = 0.65 ± 0.02 during infall and ϕ = 0.12 ± 0.02 during outflow. The amplitude of radius variation is larger by 50% in the metal-poor as it is in the metal-rich RRab stars. The radius variations indicate that RRab stars are relatively extended stars for about half of their pulsation cycles; the minimum radius occurs near ϕ = 0.92 ± 0.02 and the maximum value at ϕ = 0.38 ± 0.02.

de Boer & Maintz (2010) claim to have detected a rapid collapse phenomenon in the atmosphere of RR ab stars by the use of multichannel Strömgren photometry. They suggest that at phase ϕ = 0.90 the atmosphere begins to collapse, quickly reaching a high gas density; then within a small phase interval Δϕ = 0.10, the atmosphere expands again. No rapid collapse phenomena associated with the radius variation as suggested by de Boer & Maintz (2010) are detected in our spectroscopic data.

The physical origin of the primary acceleration, the dynamical gravity of the star, is due to the ${{Sh}}_{{\rm{H}}+\mathrm{He}}$, the main shock produced by the κ and γ mechanisms, processes based on the effects of radiative opacity in He⁺ and He⁺⁺ ionization zones that drive the pulsational instability in RR ab stars (Cox 1980). The secondary acceleration is caused by the shock Sh_PM3, described in Chadid et al. (2014), that produces compression heating. We suggest that primary acceleration is a better proxy for the intensity of the ${{Sh}}_{{\rm{H}}+\mathrm{He}}$ main shock. Accordingly, we use rise time, the very short time interval of the descending branch of the radial velocity curve, to characterize the shock intensity.

Nearly all RR ab stars of this study possess more or less prominent secondary radial velocity maxima, the elbow near phase ϕ = 0.72 ± 0.02. The phase and visibility of this maximum vary from star to star. They occur late for WY Ant, and are indistinct for DT Hya and Z Mic. The secondary acceleration is largest for the metal-rich star HH Pup, which also exhibits the largest dynamical gravity. The metal-poor star VY Ser has the lowest dynamical gravity in our sample.

We collect all of the derived motion parameters in Figure 12, using the primary acceleration (dynamical gravity) as the common variable on the figure's horizontal axis. The various correlations seen in Figure 12 will be discussed in subsequent sections. Inspection of this figure shows that sevaral motion parameters have fundamentally different relationships with primary acceleration, such as radius variation (panel (b)), duration of Hα doubling Δt (panel (d), discussed in Section 7.1), and Hα velocity amplitude ΔRV_Hα (panel (e)). In contrast, the other motion parameters show a single trend but different (albeit overlapping) regions occupied by the metal-poor and metal-rich RR Lyr stars.

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We suspect that the violet emission edge of the Hα that first appears near phase ϕ = 0.62 ± 0.02 in the top panel of Figure 13 is responsible for the small, apparent increase in radial velocity seen in the bottom panel of Figure 13 at this phase. This emission pushes measured radial velocities to spuriously large values, enhancing the brief secondary velocity maximum. The apparent deceleration, the elbow,¹² that follows is due to heretofore unresolved line-doubling during the Sh_PM3 shock. A line component of longer wavelength produced by ballistic infalling gas also appears suddenly at phase 0.63 and strengthens as the original line weakens and gradually disappears. Because IRAF/fxcor does not clearly resolve this line-doubling in most of our spectra, we usually do not detect the velocity discontinuity depicted schematically by the blue lines in the bottom panel of Figure 13. Note that both line components are formed by infalling gas at the interface of the Sh_PM3 ballistic shock. Such doubled profiles are not present in any of our spectra of metal-rich RRab stars.

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The Sh_PM3 ballistic shock (Chadid et al. 2014) crosses the inward moving atmosphere during the phase interval phase Δϕ [0.63, 0.85]. Hα line emission is produced in the radiative wake of the ballistic shock. Figure 14 illustrates schematically the sequence of Hα profiles during the phase interval 0.62–0.76. The Hα profile shows first a redshifted absorption line (hereafter original line). At phase ϕ = 0.62 a blueshifted emission appears while the velocity of the original line increases and its residual intensity decreases. A new second redshifted absorption line appears at phase ϕ = 0.63 with a velocity greater and a residual intensity smaller than the original line. At phase ϕ = 0.66, the residual intensity of the second redshifted line increases to be equal to that of the original line. At phase ϕ = 0.73, the second redshifted line strengthens as the original line weakens and gradually disappears.

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We interpret these variations in the Hα profile during the bump in the light curve as follows. We assume that the Sh_PM3 ballistic shock crossing the infalling Hα line forming zone has an inward motion in the Eulerian rest frame although is moving outward in the mass Lagrangian coordinate system, and that its existence is strongly dependent on the amplitude of the ballistic motion. Initially the velocity of Sh_PM3 is not large enough to produce resolved doubling. The shock, moving outward through mass shells, is accelerated during its propagation; consequently a second redshifted absorption component is observed with a higher velocity than the original line. Its residual intensity increases as the original line weakens at phase ϕ = 0.66. At this time the Hα doubling is clearly visible and the shock amplitude is near 50 km s⁻¹. The speed of sound is almost entirely dependent on its temperature, being roughly equal to $0.11\times \surd {T}_{\mathrm{eff}}$ in km s⁻¹ units. Adopting T ∼ T_eff = 6050 K for RV Oct (Table 2), the Mach number of the the shock Sh_PM3 is approximately 8.

Thus a shock Mach number around 8 is necessary to produce the Hα line-doubling phenomenon. Later at phases 0.76–0.85, the original line gradually disappears. At phase ϕ = 0.85, the shock leaves the Hα forming zone as the star approaches the phase of the minimum radius. Soon thereafter, the shock ${{Sh}}_{{\rm{H}}+\mathrm{He}}$, associated to the primary acceleration, emerges from the photosphere, and reverses the motion of infalling atmospheric layers.

The weak Hα emission, associated with the inward shock Sh_PM3, appears blueshifted, which is inconsistent with what we expect to see, namely redshifted emission. This is puzzling. We conclude that more than one mechanism must be at work to produce the observed profile. We suggest that an atomic mechanism, some combination of Stark and Doppler broadening, and a geometric mechanism—some unanticipated motions of gas layers—may be responsible. We simply do not know how these mechanisms combine to produce what is observed.

In closing this section we call attention to the visibility of the elbow in the metal–RV curves of Figure 9. The elbow is well-marked, or at least identifiable in most metal-poor RRab stars. However, it is not present in metal-rich ST Vir, DX Del, V445 Oph, AV Peg, and AN Ser, and it is only marginally visible in large-amplitude HH Pup and W Crt. We shall see in Section 7 that this distinction is related to the different atmospheric extents of metal-poor and metal-rich atmospheres.

The strong, albeit imperfect, correlation of V-light amplitude with metallic line RV amplitude in Figure 15 shows, first of all, that these two amplitudes are equally acceptable indicators of RRab pulsation amplitude. Residuals of individual stars in these regressions can be understood as a combination of errors in the amplitude estimates and small (non-Blazhko) irregularities in pulsation amplitudes in some stars (Chadid 2000). In particular, the rms scatters of ASAS V magnitudes that produce the V-light amplitudes in Table 4 are strongly correlated with apparent magnitude because signal-to-noise decreases with decreasing luminosity in the ASAS program. The well-known period versus light amplitude relations among RRab stars (Clement & Shelton 1999) and their associated period versus velocity amplitude relations undoubtedly contribute to the scatter as well. Investigations of the latter are beyond the scope of this paper. Slopes of the regressions calculated separately for metal-rich (red) and metal-poor (blue) in this diagram differ by less than the sum of their standard errors, regardless of the choice of independent variable in the least-squares solutions. Thus metal-poor and metal-rich RRab stars share a common relation between luminosity amplitude and photospheric velocity amplitude.

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However, the Hα lines of metal-poor and metal-rich stars which are formed at very low optical depths (Section 5) behave very differently in the two groups of stars, as illustrated in Figure 16. The Hα velocity amplitudes of the metal-rich stars are systematically smaller by ∼20 km s⁻¹ than those of metal-poor stars at a given metal velocity amplitude. The velocity structures and geometrical extents of the outer atmospheres of the two groups must be very different.

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The durations of double Hα lines in the two abundance groups shown in Figure 17 provide an additional indication of different atmospheric structures. We define duration as the time interval Δt during which IRAF/splot can identify two absorption minima in the Hα profile. The durations are markedly longer in stars with large Hα emission fluxes. We note that three of the four stars with no observed Hα emission flux (AV Peg, V445 Oph, and W Crt) have the very short periods that attracted Kukarkin's (1949) attention long ago. The fourth, DX Del (P = 0.473 days), has the lowest light and RV amplitudes in our metal-rich sample.

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The ranges in appearance and duration of line-doubling is illustrated in the montages of Hα profiles for metal-rich AV Peg and metal-poor RV Oct in Figure 18. The profiles of AV Peg are typical of the four stars mentioned above, i.e., they are devoid of Hα emission in our du Pont echelle spectra. A violet absorption wing turns into a resolved absorption feature that replaces the red component on the short timescales plotted in Figure 17. In AV Peg line doubling can be seen only in the brief interval from ϕ = 0.93 to ϕ = 0.96, Δt = 0.03 P = 1000 s, corresponding to 3% of pulsation period. The same datum for RV Oct is Δt = 0.17 P = 8400 s, corresponding to 17% of pulsation period; the durations differ by a factor ∼8. These durations are measures of the times that outflowing gas sweeps up and reverses the flow of infalling gas. They define the lifetime of the ${{Sh}}_{{\rm{H}}+\mathrm{He}}$ main shock in the upper atmosphere. The lifetimes of the main shocks in metal-poor and metal-rich stars differ by a factor ∼8. For fluxes less than 1 continuum unit, the metal-poor durations exceed metal-rich durations by a factor of two or less. Note, however, that durations as large as 0.12 days ∼10000 s occur among the metal-poor stars plotted in Figure 17.

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The panel (d) of Figure 12 tells us that large photospheric dynamical gravities (i.e., strong shocks) in the photospheres of metal-rich stars are of short duration, while weaker shocks in metal-poor stars are of longer duration. We make a dynamical gravity division between the metal-poor and metal-rich families near to $\ddot{R}$ = 30 km s⁻². To this, we add the well-marked correlation between Hα flux and duration of the weaker shocks of metal-poor stars shown in Figure 17. Finally, recall that the photospheric radius variation is larger for the metal-poor RRab stars than for the metal-rich stars (bottom panel of Figure 11); the average difference in the panel (b) of Figure 12 is approximately ΔR = 0.17 R_⊙.

All the regressions in Figure 12 indicate that dynamical gravity, the primary acceleration, is the empirical, unifying, causal parameter of the various observed envelope phenomena. Used as independent variable it produces simple linear regressions for secondary acceleration, radius variation, rise-time, duration of Hα doubling, and hydrogen and metallic radial velocity amplitudes. The slopes and intercepts of the metal-rich regressions (red-straight lines) are different from those of their metal-poor counterparts (blue-straight lines). The differences between the metal-rich linear regressions and those of their metal-poor counterparts are more pronounced in the upper-atmospheric parameters (duration of Hα doubling, Hα radial velocity amplitude and Hα emission) than those of the photosphere (secondary acceleration, metal radial velocity amplitude and rise-time). Finally, the radius variation is larger by a factor ∼1.8 for metal-poor stars than that of their metal-rich counterparts.

To conclude, the motions of the upper-atmospheric and near-photospheric regions exhibit very different behaviors, and these differences are more pronounced in metal-poor RRab stars than in their metal-rich counterparts.

Panel (a) of Figure 12 shows the regression between the secondary and primary accelerations for metal-poor and metal-rich RRab stars: the stronger the dynamical gravity, the more important is the secondary acceleration. Both groups share a similar physical process of the amplification of the secondary acceleration with the primary acceleration which indicates dynamical coupling interplay between the ${{Sh}}_{{\rm{H}}+\mathrm{He}}$ main shock and the Sh_PM3 ballistic shock. The greater the dynamical gravity, the larger the Mach number ${{Sh}}_{{\rm{H}}+\mathrm{He}}$ that causes a higher level of dynamical imbalance, i.e., the non-synchronization of motions in the upper and lower photospheric layers. Stronger collision induces higher supersonic motion, Sh_PM3. Thus secondary acceleration becomes more important. Two distinct correlations between metal-line (near-photospheric) radial velocity amplitude and primary acceleration exist for both metal-poor and metal-rich RRab stars (panel (f) of Figure 12). The amplitude of the photospheric ballistic motion is larger when the photosphere is crossed by a stronger shock, i.e., the more important the primary acceleration, the larger the metal radial velocity amplitude. The comparison of primary accelerations in the top panel of Figure 11 demonstrates that the intensity of the ${{Sh}}_{{\rm{H}}+\mathrm{He}}$ main shock is more important in the photospheres of metal-rich stars than those of metal-poor stars.

The most striking characteristic of the ballistic photospheric motion is its greater magnitude in metal-poor relative to metal-rich RRab stars (Figure 12). Longer period implies longer ballistic motion. The greater extensions of the metal-poor envelopes complicates their atmospheric motions; the non-synchronization effect greatly changes atmospheric motion structure and creates chaotic behavior, mainly in the upper atmospheres. Moreover, the region of formation of spectral lines is greatly enlarged with concomitant effects on line profiles. This enlarged region of line formation in metal-poor stars is the origin of the greater duration of Hα doubling (Figure 12, panel (d)), Hα radial velocity amplitude (panel (e) of Figure 12) and the peak Hα emission flux.

Hα emission is generally weaker in metal-rich RRab stars (Figure 19). No emission could be detected in four of the eight metal-rich RRab stars in our sample (AV Peg, V445 Oph, ST Vir, and DX Del), and the strongest emission in the remaining four occurred in the lowest-metallicity star of the metal-rich group (UU Vir, [Fe/H] = −0, 93). We attribute this metallicity effect to the fact that the metal-poor RRab atmospheres are more extended, and the amplitudes of the ballistic motions of the metal-poor RRab atmospheres are greater than those of metal-rich RRab.

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The bottom panel of Figure 19 shows that strongest Hα emission in RRab stars is restricted to those with the largest hydrogen radial velocity amplitudes. A threshold effect seems to be present: a steep gradient of Hα emission occurs near Hα radial velocity amplitude ∼115 km s⁻¹, when the duration of Hα line-doubling is ∼10% of the pulsation period. The emission flux increases abruptly when the dynamical gravity exceeds $\ddot{R}\sim 20$ km s⁻², independent of the metallicity, pulsation period and radius variation. We call this phenomenon the hydrodynamic gradient. This mechanism applies to metal-poor and perhaps to metal-rich RRab stars as well. Only one metal-rich RRab star in our sample (UU Vir, [Fe/H] = −0, 93) shows this phenomenon, and it is the most metal-poor of the group. The hydrodynamic gadient mechanism is strongly dependent of the temperature and the gravitational acceleration of the star, and affects the stars with lower temperatures (<6100 K) and slighter gravities ($\mathrm{log}g\lt 1.9)$ (Figure 4). Evidently the upper atmosphere reaches a high-hypersonic regime when the dynamical gravity exceeds $\ddot{R}$ ∼ 20 km s⁻². Accordingly, we divide the metal-poor RRab stars into two types, type 1 and type 2, and we define four main shock regimes in the upper atmospheres of RR Lyrae stars:

• 1.  
  Trans-sonic regime. No hydrogen emission or line doubling is observed. This regime could characterize RRc and RRd.
• 2.  
  Supersonic regime. Hydrogen line doubling and only a very small or even no hydrogene emission is observed. This regime occurs mostly in the metal-rich RRab.
• 3.  
  Hypersonic regime. Additional He I emission line is observed. The temperature must be higher than 10 000 K, with a threshold Mach number Mach_{He i}. The hypersonic regime occurs in metal-poor RRab type 1 and a few metal-rich RRab with higher dynamical gravity.
• 4.  
  High-hypersonic regime. He II emission line is observed. The temperature and energy respectively higher than 30 000 K and 24.59 eV are able to ionize a neutral helium atom with a threshold Mach number Mach_{He II}. The atmosphere is very extended and a circumstellar envelope around the star may occur. This should characterize the metal-poor RRab type 2 that could be cooler and more convective. This regime has been already identified in two metal-poor RR ab stars: AS Vir and V1645 Sgr (Preston 2011).

Finally, we consider briefly the Hα emission of RRab stars in the context of other post-main sequence pulsating stars. We note the absence of H emission in most classical Cepheids with periods less than 10 days, i.e., those Cepheids with lowest luminosity and least extended atmospheres. Among the remainder, the Hα emission of the metal-poor RRab stars is weaker than those seen in W Virginis, RV Tauri, and Mira stars. The last of these families comprises a sequence of increasing luminosity and atmospheric extent. Wider atmospheric extension in these cooler pulsators is accompanied by larger Mach numbers of Sh${}_{{\rm{H}}+\mathrm{He}}$. and stronger shocks.