# The Axial Rotation and Variable Macroturbulence of RR Lyrae and Red Horizontal Branch Stars

George Preston, Christopher Sneden, Merieme Chadid, Ian Thompson, Stephen Slectman

► To cite this version:

George Preston, Christopher Sneden, Merieme Chadid, Ian Thompson, Stephen Slectman. The Axial Rotation and Variable Macroturbulence of RR Lyrae and Red Horizontal Branch Stars. The Astronomical Journal, 2019, 157 (4), pp.153. 10.3847/1538-3881/ab0ae1. hal-03655200

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# The Axial Rotation and Variable Macroturbulence of RR Lyrae and Red Horizontal Branch Stars

George W. Preston<fnref n="1" />, Christopher Sneden<fnref n="2" />, Merieme Chadid<fnref n="3" />, Ian B. Thompson<fnref n="1" />, and Stephen A. Shectman<fnref n="1" />

## Abstract

We have derived relations between FWHMs and equivalent widths of metallic absorption lines in the spectra of RR Lyrae stars to estimate new upper limits on the axial equatorial rotational velocities of RR Lyrae and metal-poor red horizontal branch (RHB) stars. We have also derived the variations of RR Lyrae macroturbulent velocities during the pulsation cycles. In RRab cycles, the line widths are dominated by phase-dependent convolutions of axial rotation and macroturbulence, which we designate as $V_{\text{macrot}}$. The behavior of $V_{\text{macrot}}$ is remarkably uniform among the RRab stars, but the behavior of $V_{\text{macrot}}$ among RRc stars varies strongly from star to star. The RRab stars exhibit an upper limit on $V_{\text{macrot}}$ of $5 \pm 1$ km s$^{-1}$ with weak evidence of an anticorrelation with $T_{\text{eff}}$. The RRc minima range from $2$ to $12$ km s$^{-1}$. The abrupt decline in large rotations with decreasing $T_{\text{eff}}$ at the blue boundary of the instability strip and the apparently smooth continuous variation among the RRab and RHB stars suggests that HB stars gain/lose surface angular momentum on timescales that are short compared to HB lifetimes. The $V_{\text{macrot}}$ values for our metal-poor RHB stars agree well with those derived by Fourier analysis of an independent but less metal-poor sample of Carney et al.; they conform qualitatively to the expectations of Tanner et al. A general conclusion of our investigation is that surface angular momentum as measured by $V_{\text{rot}} \sin(i)$ is not a reliable indicator of total stellar angular momentum anywhere along the HB.

**Key words:** methods: observational – stars: atmospheres – stars: oscillations – stars: variables: RR Lyr – techniques: spectroscopic

**Supporting material:** machine-readable table

## 1. Introduction

Variations of turbulence during the pulsation cycles of classical Cepheids (Breitfellner & Gillet 1993; Stift & Gillet 1994; Bersier & Burki 1996) and RR Lyrae stars (Chadid & Gillet 1996; Fokin et al. 1999) have been well documented. Individual studies have different definitions of turbulence and procedures for its extraction from stellar line profiles. The disentanglement of stellar line broadening due to axial rotation from turbulence, however defined, is difficult in all of these investigations, which mainly have been concerned with the generation of turbulence and its effect on pulsating atmospheres.

This problem of disentanglement is illustrated in Figure 1, which shows the observed and synthetic spectra of X Ari, a very metal-poor (MP; [Fe/H] = -2.6) RRab star, at phase $\phi = 0.37$. In the top panel, we show a computed spectrum (magenta) of a small spectral region containing the Ca I 4425.44 Å and Fe I 4427.32 Å lines. This spectrum was generated with the synthetic spectrum software and model atmosphere parameters described by Chadid et al. (2017), and transition probabilities were adjusted to match the equivalent widths (EWs) of the two main lines. This computed spectrum only has line broadening that results from thermal and microturbulent velocity and clearly does not match the observed spectrum (black dots). The green spectrum shows this synthesis with added Gaussian smoothing to account for the du Pont echelle spectrograph slit function ($R \equiv \lambda/\Delta\lambda \simeq 27,000$, or FWHM $\simeq 0.16$ Å at 4426 Å). This accounts for most but not all of the observed line profiles. The bottom panel shows two convolutions with identical FWHM

values but very different Maxwellian macroturbulent and rotational parameters (blue, $V_{\text{mac}} = 7.5$ km s$^{-1}$, $V_{\text{rot}} \sin(i) = 0.0$ km s$^{-1}$; red, $V_{\text{mac}} = 4.0$ km s$^{-1}$, $V_{\text{rot}} \sin(i) = 6.0$ km s$^{-1}$). These profiles appear to be nearly identical in the wings. Because rotation remains virtually constant during a pulsation cycle, any variation of FWHM with phase must be due to thermal and turbulent motions of gas elements within RRab atmospheres.

For 27 RRab stars, Peterson et al. (1996) found a minimum FWHM value near pulsation phase $\phi = 0.35$, the approximate phase of maximum radius, derived from cross-correlation of metal lines for radial velocity during the stellar pulsation cycles. They argued that the variation of FWHM during pulsation cycles is due to variable turbulence and the minimum value is an unresolvable combination of turbulence and axial rotation. They derived an upper limit of $V_{\text{rot}} \sin(i) < 10$ km s$^{-1}$ for the RR Lyrae stars, similar to that reported for classical Cepheids by Bersier & Burki (1996).

On the other hand, many blue horizontal branch (BHB) stars with $T_{\text{eff}} < 11,500$ K<fnref n="4" /> are rapid rotators, some attaining $V_{\text{rot}} \sin(i) \sim 40$ km s$^{-1}$ (Peterson et al. 1995; Kinman et al. 2000; Behr 2003a, 2003b; Recio-Blanco et al. 2004). Standard HB theory tells us that if these BHB rapid rotators evolve with constant luminosity and angular momentum, then their rotation rates will go as $V_{\text{rot}} \sin(i) \propto T_{\text{eff}}^2$. In this case, many of them

<footnote n="1">¹Carnegie Observatories, 813 Santa Barbara Street, Pasadena, CA 91101, USA; gwp@obs.carnegiescience.edu, ian@obs.carnegiescience.edu, shec@obs.carnegiescience.edu</footnote>

<footnote n="2">²Department of Astronomy and McDonald Observatory, The University of Texas, Austin, TX 78712, USA; chris@verdi.as.utexas.edu</footnote>

<footnote n="3">³University of Côte d'Azur, Nice Sophia-Antipolis University, ARTEMIS-UMR 7250, CS 34229, F-06304 Nice, France; Merieme.Chadid@unice.fr

*Received 2018 July 16; revised 2019 February 24; accepted 2019 February 25; published 2019 March 25*</footnote>

<footnote n="4">⁴ As first noted by Grundahl et al. (1999), HB stars with $T_{\text{eff}} \lesssim 11,500$ K have normal (approximately solar) [X/Fe] values, while HB stars hotter than this value display abundance anomalies commonly associated with radiative levitation and gravitational settling (e.g., Michaud et al. 1983; Quievy et al. 2009; Théado et al. 2012). We adopt 11,500 K as the upper temperature boundary for BHB stars to be included in our discussion.</footnote>
<figure><img src="image_3.png" /></figure>

**Figure 1.** Top panel: observed and synthetic profiles of Ca I 4425.44 Å and Fe I 4427.32 Å, two unsaturated lines in the spectrum of RRab X Ari. The black dots represent the observed spectrum. The magenta line is the raw synthetic spectrum, broadened only by thermal and microturbulent velocities assumed in the spectrum computations. The green line shows the raw spectrum broadened by the spectrograph slit function only. Bottom panel: same observed spectrum compared to two synthetic spectra. Each of these has the same thermal, microturbulent, and spectrograph slit broadening, but the red line has $V_{\text{mac}} = 4.0 \text{ km s}^{-1}$ and $V_{\text{rot}} \sin(i) = 6.0 \text{ km s}^{-1}$, and the blue line has $V_{\text{mac}} = 7.5 \text{ km s}^{-1}$ but no rotational broadening. A small vertical shift has been added to the blue line for plotting clarity.

will pass through the instability strip rotating more rapidly than $V_{\text{rot}} \sin(i) \sim 10 \text{ km s}^{-1}$. None of them do so; hence, the “Peterson Conundrum.”

Here we address this puzzle by the study of additional samples of RRab and RRc RR Lyrae variables and red horizontal branch (RHB) stars. In Section 2, we describe the spectroscopic data sets and our extraction of the observed broadening of metallic lines for each star in individual pulsation phases. In Sections 3 and 4, for RRab and RRc stars, we derive $V_{\text{macrot}}$ quantities, which are combinations of macroturbulent and rotational broadening effects. We compare the $V_{\text{macrot}}$ values to values of microturbulent velocities $V_{\text{mic}}$. In Section 5, we derive values of $V_{\text{macrot}}$ for very MP RHB stars and compare them to values for RHB samples used in previous studies. Finally, we consider the implications of the complex behavior of $V_{\text{macrot}}$ along the HB in Section 6.

## 2. Observations and Procedures

### 2.1. The Observational Database

The main observational data for this study were provided by several thousand echelle spectra of 34 RRab stars and 15 RRc stars acquired with the du Pont 2.5 m telescope of the Las Campanas Observatory in 2006–2014.<fnref n="5" /> The typical spectral resolving power is, as introduced above, $R \equiv \lambda/\Delta\lambda \simeq 27,000$ at 5000 Å. Exposure times are limited to small fractions of the stellar pulsation periods, never exceeding 600 s ($\simeq 0.01\text{P}$), and the typical signal-to-noise ratio (S/N) is $S/N \sim 15–20$. The S/N was improved by factors of two or more at all phases by coaddition of spectra in small phase intervals. Reduction

<footnote n="5">⁵ Spectra used in this investigation can be obtained at 10.5281/zenodo.2575102.</footnote>
<table><caption>**Table 1**
The Stellar Sample</caption><thead><tr><th>Star Name</th><th>[Fe/H]</th><th>P<br>(days)</th><th>HJD<sub>0</sub><br>(days)</th><th>V<br>Max. Light</th><th>log(T<sub>eff</sub>)<br>(K)</th><th>⟨FWHM<sub>upsat</sub>⟩<br>(km s<sup>-1</sup>)</th><th>σ(FWHM<sub>upsat</sub>)<br>(km s<sup>-1</sup>)</th><th>⟨V<sub>macrot</sub>⟩<br>(km s<sup>-1</sup>)</th><th>σ(V<sub>macrot</sub>)<br>(km s<sup>-1</sup>)</th></tr></thead><tbody><tr><td colspan="10"><strong>MP RRab Calibration</strong></td></tr><tr><td>WY Ant</td><td>-1.66</td><td>0.574344</td><td>3835.592</td><td>10.4</td><td>3.810</td><td>10.2</td><td>0.3</td><td>5.3</td><td>0.2</td></tr><tr><td>XZ Aps</td><td>-1.57</td><td>0.587266</td><td>3836.276</td><td>11.9</td><td>3.809</td><td>10.5</td><td>0.2</td><td>5.5</td><td>0.2</td></tr><tr><td>SW Aqr</td><td>-1.24</td><td>0.459303</td><td>1876.138</td><td>10.6</td><td>3.829</td><td>10.5</td><td>...</td><td>5.5</td><td>...</td></tr><tr><td>X Ari</td><td>-2.50</td><td>0.651170</td><td>1890.064</td><td>9.2</td><td>3.803</td><td>11.7</td><td>0.2</td><td>6.3</td><td>0.1</td></tr><tr><td>RR Cet</td><td>-1.52</td><td>0.553029</td><td>1900.312</td><td>9.3</td><td>3.813</td><td>11.2</td><td>...</td><td>5.9</td><td>...</td></tr><tr><td>SX For</td><td>-1.62</td><td>0.605342</td><td>1870.406</td><td>10.9</td><td>3.805</td><td>10.7</td><td>1.2</td><td>5.6</td><td>0.8</td></tr><tr><td>VX Her</td><td>-1.60</td><td>0.455359</td><td>2699.996</td><td>10.1</td><td>3.828</td><td>8.6</td><td>0.6</td><td>4.1</td><td>0.5</td></tr><tr><td>DT Hya</td><td>-1.22</td><td>0.567978</td><td>3835.608</td><td>12.5</td><td>3.814</td><td>10.7</td><td>0.9</td><td>5.6</td><td>0.6</td></tr><tr><td>SS Leo</td><td>-1.83</td><td>0.626335</td><td>1873.056</td><td>10.5</td><td>3.809</td><td>9.5</td><td>0.6</td><td>4.8</td><td>0.4</td></tr><tr><td>ST Leo</td><td>-1.29</td><td>0.477984</td><td>5322.597</td><td>11.0</td><td>3.827</td><td>10.6</td><td>0.4</td><td>5.5</td><td>0.3</td></tr><tr><td>Z Mic</td><td>-1.28</td><td>0.586928</td><td>3930.509</td><td>11.3</td><td>3.808</td><td>13.1</td><td>0.1</td><td>7.2</td><td>0.1</td></tr><tr><td>RV Oct</td><td>-1.34</td><td>0.571170</td><td>3835.895</td><td>10.5</td><td>3.816</td><td>10.6</td><td>0.4</td><td>5.5</td><td>0.3</td></tr><tr><td>VY Ser</td><td>-1.82</td><td>0.714105</td><td>5325.340</td><td>9.8</td><td>3.796</td><td>11.3</td><td>...</td><td>6.0</td><td>...</td></tr><tr><td>AT Ser</td><td>-2.05</td><td>0.746568</td><td>5326.814</td><td>11.0</td><td>...</td><td>12.1</td><td>1.5</td><td>6.0</td><td>1.0</td></tr><tr><td>W Tuc</td><td>-1.64</td><td>0.642243</td><td>5454.570</td><td>10.9</td><td>3.810</td><td>11.2</td><td>...</td><td>6.0</td><td>...</td></tr><tr><td colspan="10"><strong>MP RRab Blazhko<sup>b</sup></strong></td></tr><tr><td>BS Aps</td><td>-1.51</td><td>0.582560</td><td>3836.378</td><td>11.9</td><td>3.808</td><td>11.0</td><td>0.7</td><td>5.8</td><td>0.4</td></tr><tr><td>DN Aqr</td><td>-1.77</td><td>0.633766</td><td>5364.300</td><td>10.8</td><td>3.785</td><td>11.5</td><td>1.6</td><td>6.1</td><td>1.1</td></tr><tr><td>S Ara<sup>c</sup></td><td>...</td><td>0.451883</td><td>2137.338</td><td>10.3</td><td>3.827</td><td>8.2</td><td>0.8</td><td>3.7</td><td>0.6</td></tr><tr><td>RV Cap</td><td>-1.72</td><td>0.447748</td><td>5328.400</td><td>10.6</td><td>3.825</td><td>10.6</td><td>...</td><td>5.6</td><td>...</td></tr><tr><td>RV Cet</td><td>-1.32</td><td>0.623421</td><td>5455.460</td><td>10.6</td><td>3.804</td><td>11.2</td><td>0.6</td><td>6.0</td><td>0.4</td></tr><tr><td>UV Oct</td><td>-1.69</td><td>0.542573</td><td>3836.795</td><td>9.2</td><td>3.815</td><td>9.9</td><td>1.3</td><td>5.0</td><td>0.9</td></tr><tr><td>V Ind</td><td>-1.62</td><td>0.479602</td><td>1873.466</td><td>9.6</td><td>3.825</td><td>8.6</td><td>0.4</td><td>4.1</td><td>0.3</td></tr><tr><td>V1645 Sgr</td><td>-1.81</td><td>0.552910</td><td>3932.480</td><td>11.0</td><td>3.810</td><td>11.1</td><td>0.6</td><td>5.9</td><td>0.4</td></tr><tr><td>CD Vel</td><td>-1.71</td><td>0.573509</td><td>3835.914</td><td>11.7</td><td>3.811</td><td>10.7</td><td>0.4</td><td>5.6</td><td>0.3</td></tr><tr><td>AM Vir</td><td>-1.45</td><td>0.615085</td><td>1888.395</td><td>11.2</td><td>3.806</td><td>10.8</td><td>0.8</td><td>5.7</td><td>0.5</td></tr><tr><td>AS Vir</td><td>-1.59</td><td>0.553400</td><td>3836.325</td><td>11.7</td><td>3.812</td><td>10.5</td><td>0.8</td><td>5.5</td><td>0.5</td></tr><tr><td colspan="10"><strong>MR RRab Stable</strong></td></tr><tr><td>W Crt</td><td>-0.76</td><td>0.412013</td><td>1871.640</td><td>10.9</td><td>3.836</td><td>9.6</td><td>...</td><td>4.8</td><td>...</td></tr><tr><td>DX Del</td><td>-0.52</td><td>0.472619</td><td>5323.417</td><td>9.8</td><td>3.822</td><td>11.2</td><td>0.6</td><td>5.9</td><td>0.4</td></tr><tr><td>V445 Oph</td><td>-0.05</td><td>0.397024</td><td>1939.030</td><td>10.5</td><td>3.831</td><td>9.3</td><td>0.8</td><td>4.6</td><td>0.6</td></tr><tr><td>AV Peg</td><td>-0.14</td><td>0.390382</td><td>5360.677</td><td>10.0</td><td>3.836</td><td>8.8</td><td>1.0</td><td>4.1</td><td>0.7</td></tr><tr><td>HH Pup</td><td>-0.95</td><td>0.390745</td><td>1869.661</td><td>10.6</td><td>3.837</td><td>8.5</td><td>0.5</td><td>4.0</td><td>0.1</td></tr><tr><td>AN Ser</td><td>-0.06</td><td>0.522072</td><td>2701.153</td><td>10.5</td><td>3.811</td><td>7.6</td><td>...</td><td>3.3</td><td>...</td></tr><tr><td>ST Vir</td><td>-0.85</td><td>0.410806</td><td>5323.225</td><td>11.0</td><td>3.806</td><td>10.0</td><td>...</td><td>5.1</td><td>...</td></tr><tr><td>UU Vir</td><td>-0.93</td><td>0.475609</td><td>1886.488</td><td>10.1</td><td>3.826</td><td>9.8</td><td>0.2</td><td>5.0</td><td>0.1</td></tr><tr><td colspan="10"><strong>MP RRc Stable</strong></td></tr><tr><td>as014500-3003</td><td>-2.28</td><td>0.377380</td><td>1869.252</td><td>11.1</td><td>3.851</td><td>13.1</td><td>0.10</td><td>7.4</td><td>0.06</td></tr><tr><td>as023706-4257</td><td>-1.88</td><td>0.311326</td><td>1869.352</td><td>8.8</td><td>3.851</td><td>11.4</td><td>0.38</td><td>6.1</td><td>0.26</td></tr><tr><td>as094541-0644</td><td>-2.10</td><td>0.350225</td><td>1869.729</td><td>10.6</td><td>3.851</td><td>17.1</td><td>0.19</td><td>9.8</td><td>0.12</td></tr><tr><td>as095328+0203</td><td>-1.76</td><td>0.324697</td><td>1869.701</td><td>9.9</td><td>3.851</td><td>11.4</td><td>0.03</td><td>6.1</td><td>0.02</td></tr><tr><td>as101332-0702</td><td>-1.73</td><td>0.316000</td><td>1869.668</td><td>10.8</td><td>3.851</td><td>15.8</td><td>0.77</td><td>9.2</td><td>0.48</td></tr><tr><td>as123811-1500</td><td>-1.39</td><td>0.329045</td><td>1884.454</td><td>11.4</td><td>3.851</td><td>13.2</td><td>0.18</td><td>7.3</td><td>0.12</td></tr><tr><td>as132225-2042</td><td>-0.96</td><td>0.235934</td><td>1888.578</td><td>10.7</td><td>3.851</td><td>12.8</td><td>...</td><td>7.0</td><td>...</td></tr><tr><td>as132448-0658</td><td>-2.04</td><td>0.343237</td><td>1900.300</td><td>11.4</td><td>3.851</td><td>16.3</td><td>0.12</td><td>9.3</td><td>0.07</td></tr><tr><td>as143322-0418</td><td>-1.48</td><td>0.249632</td><td>1907.348</td><td>10.6</td><td>3.851</td><td>4.8</td><td>0.25</td><td>1.3</td><td>0.32</td></tr><tr><td>as190212-4639</td><td>-2.58</td><td>0.316900</td><td>1954.795</td><td>8.8</td><td>3.851</td><td>7.7</td><td>0.07</td><td>3.4</td><td>0.06</td></tr><tr><td>as203145-2158</td><td>-1.17</td><td>0.310712</td><td>1873.230</td><td>11.3</td><td>3.851</td><td>12.5</td><td>0.02</td><td>6.8</td><td>0.02</td></tr><tr><td>as211932-1507</td><td>-1.50</td><td>0.273458</td><td>1873.307</td><td>11.1</td><td>3.851</td><td>9.9</td><td>...</td><td>5.0</td><td>...</td></tr><tr><td colspan="10"><strong>MP RRc Blazhko<sup>d</sup></strong></td></tr><tr><td>as110522-2641</td><td>-1.60</td><td>0.294457</td><td>4900.170</td><td>11.7</td><td>3.851</td><td>15.6</td><td>...</td><td>8.8</td><td>...</td></tr><tr><td>as162158+0244</td><td>-1.83</td><td>0.323698</td><td>4900.435</td><td>12.6</td><td>3.851</td><td>25.1</td><td>...</td><td>12.1</td><td>...</td></tr><tr><td>as200431-5352</td><td>-2.66</td><td>0.300241</td><td>4900.470</td><td>11.0</td><td>3.851</td><td>15.4</td><td>2.00</td><td>8.7</td><td>1.28</td></tr><tr><td colspan="10"><strong>MP RHB</strong></td></tr><tr><td>CS22186-005</td><td>-2.72</td><td>...</td><td>...</td><td>13.0</td><td>3.796</td><td>15.14</td><td>...</td><td>8.44</td><td>...</td></tr><tr><td>CS22190-007</td><td>-2.57</td><td>...</td><td>...</td><td>14.2</td><td>3.756</td><td>11.73</td><td>...</td><td>6.64</td><td>...</td></tr><tr><td>CS22191-029</td><td>-2.70</td><td>...</td><td>...</td><td>14.1</td><td>3.778</td><td>14.99</td><td>...</td><td>8.42</td><td>...</td></tr><tr><td>CS22875-029</td><td>-2.65</td><td>...</td><td>...</td><td>13.7</td><td>3.778</td><td>15.31</td><td>...</td><td>8.57</td><td>...</td></tr><tr><td>CS22878-121</td><td>-2.37</td><td>...</td><td>...</td><td>14.0</td><td>3.736</td><td>10.32</td><td>...</td><td>5.71</td><td>...</td></tr><tr><td>CS22879-097</td><td>-2.45</td><td>...</td><td>...</td><td>14.2</td><td>3.763</td><td>13.16</td><td>...</td><td>7.46</td><td>...</td></tr><tr><td>CS22879-103</td><td>-2.10</td><td>...</td><td>...</td><td>14.3</td><td>3.762</td><td>14.80</td><td>...</td><td>8.27</td><td>...</td></tr><tr><td>CS22881-039</td><td>-2.68</td><td>...</td><td>...</td><td>15.1</td><td>3.785</td><td>13.62</td><td>...</td><td>7.59</td><td>...</td></tr></tbody></table>
<table><caption>**Table 1**
(Continued)</caption><thead><tr><th>Star Name</th><th>[Fe/H]</th><th>P<br>(days)</th><th>HJD<sub>0</sub><br>(days)</th><th>V<br>Max. Light</th><th>log(T<sub>eff</sub>)<sup>a</sup><br>(K)</th><th>⟨FWHM<sub>unsat</sub>⟩<br>(km s<sup>-1</sup>)</th><th>σ(FWHM<sub>unsat</sub>)<br>(km s<sup>-1</sup>)</th><th>⟨V<sub>macrot</sub>⟩<br>(km s<sup>-1</sup>)</th><th>σ(V<sub>macrot</sub>)<br>(km s<sup>-1</sup>)</th></tr></thead><tbody><tr><td>CS22882-001</td><td>-2.48</td><td>...</td><td>...</td><td>14.8</td><td>3.775</td><td>15.21</td><td>...</td><td>8.45</td><td>...</td></tr><tr><td>CS22883-037</td><td>-1.89</td><td>...</td><td>...</td><td>14.7</td><td>3.771</td><td>15.72</td><td>...</td><td>8.93</td><td>...</td></tr><tr><td>CS22886-043</td><td>-2.15</td><td>...</td><td>...</td><td>14.7</td><td>3.778</td><td>15.24</td><td>...</td><td>8.47</td><td>...</td></tr><tr><td>CS22888-047</td><td>-2.34</td><td>...</td><td>...</td><td>14.6</td><td>3.778</td><td>14.93</td><td>...</td><td>8.34</td><td>...</td></tr><tr><td>CS22891-184</td><td>-2.55</td><td>...</td><td>...</td><td>13.8</td><td>3.748</td><td>12.20</td><td>...</td><td>6.90</td><td>...</td></tr><tr><td>CS22896-110</td><td>-2.66</td><td>...</td><td>...</td><td>13.6</td><td>3.740</td><td>10.88</td><td>...</td><td>6.02</td><td>...</td></tr><tr><td>CS22898-043</td><td>-2.84</td><td>...</td><td>...</td><td>14.1</td><td>3.782</td><td>15.31</td><td>...</td><td>8.59</td><td>...</td></tr><tr><td>CS22937-072</td><td>-2.73</td><td>...</td><td>...</td><td>14.0</td><td>3.736</td><td>10.13</td><td>...</td><td>5.62</td><td>...</td></tr><tr><td>CS22940-077</td><td>-2.92</td><td>...</td><td>...</td><td>14.1</td><td>3.732</td><td>10.65</td><td>...</td><td>5.95</td><td>...</td></tr><tr><td>CS22940-121</td><td>-2.89</td><td>...</td><td>...</td><td>14.2</td><td>3.732</td><td>10.34</td><td>...</td><td>5.69</td><td>...</td></tr><tr><td>CS22945-056</td><td>-2.90</td><td>...</td><td>...</td><td>14.1</td><td>3.763</td><td>14.93</td><td>...</td><td>8.40</td><td>...</td></tr><tr><td>CS22948-006</td><td>-2.62</td><td>...</td><td>...</td><td>15.1</td><td>3.744</td><td>11.60</td><td>...</td><td>6.51</td><td>...</td></tr><tr><td>CS22951-077</td><td>-2.42</td><td>...</td><td>...</td><td>13.6</td><td>3.728</td><td>10.27</td><td>...</td><td>5.66</td><td>...</td></tr><tr><td>CS22955-174</td><td>-2.88</td><td>...</td><td>...</td><td>14.4</td><td>3.752</td><td>14.15</td><td>...</td><td>8.08</td><td>...</td></tr></tbody></table>

**Notes.**

<sup>a</sup> For the RRab stars, the log T<sub>eff</sub> values are the means over the pulsational cycles, as given by Skarka (2014). For the RRc stars, we have adopted a constant for log T<sub>eff</sub>.

<sup>b</sup> Nominal periods; see Table 3 and text.

<sup>c</sup> We gathered du Pont echelle spectra for this star but did not include it in our model atmosphere analyses in previous papers.

<sup>d</sup> Blazhko variation reported by Szczygiel & Fabrycky (2007).

<table><caption>**Table 2**
Atomic Lines</caption><thead><tr><th>λ<br>(Å)</th><th>Species</th><th>EW<sub>☉</sub><sup>a</sup><br>(mÅ)</th></tr></thead><tbody><tr><td>4045.82</td><td>Fe I</td><td>1174</td></tr><tr><td>4062.45</td><td>Fe I</td><td>98</td></tr><tr><td>4063.60</td><td>Fe I</td><td>787</td></tr><tr><td>4067.99</td><td>Fe I</td><td>133</td></tr><tr><td>4071.75</td><td>Fe I</td><td>723</td></tr><tr><td>4076.64</td><td>Fe I</td><td>127</td></tr><tr><td>4077.72</td><td>Sr II</td><td>428</td></tr><tr><td>4121.32</td><td>Co I</td><td>125</td></tr><tr><td>4134.68</td><td>Fe I</td><td>129</td></tr><tr><td>4143.42</td><td>Fe I</td><td>134</td></tr></tbody></table>

**Note.**

<sup>a</sup> EWs from the Moore et al. (1966) Solar Atlas.

(This table is available in its entirety in machine-readable form.)

procedures that resulted in wavelength-calibrated extractions of the du Pont echelle spectra are described in detail by For et al. (2011a) and need not be repeated here. Basic data for the program stars are given in Table 1, where stars are listed in groups discussed in Sections 3–5 below. For this paper, we split the stellar sample into stars that are MP ([Fe/H] < −1) and those that are metal-rich (MR; [Fe/H] ≥ −1), consistent with the definition adopted by Chadid et al. (2017) and Sneden et al. (2017, 2018). The mean temperatures for the RRab stars in this table are taken from Skarka (2014), and those for the RRc stars (which have only small T<sub>eff</sub> variations) are an assumed constant value of 7100 K. Standard deviations accompany average values when more than one estimate is available.

We also obtained many du Pont echelle spectra of the well-known very MP ([Fe/H] < −2) subgiant HD 140283, which we used as a radial velocity standard. This star has been

analyzed repeatedly with high spectral resolution, and the mean of entries in the PASTEL database (Soubiran et al. 2016) suggests T<sub>eff</sub> = 5700 K, log g = 3.6, and [Fe/H] = −2.4. It has much narrower absorption lines than RR Lyrae and other HB stars.

Observations of 24 very MP RHB field stars were obtained with the *Magellan* MIKE spectrograph (Bernstein et al. 2003) in a mode that delivered R ≃ 40,000. The combination of *Magellan’s* larger aperture and longer exposure times yielded S/N > 100 at 5000 Å in all cases. Reductions and analyses of the *Magellan* / MIKE spectra are discussed in Preston et al. (2006).

## 2.2. Procedure

We first derived estimates of line-of-sight velocity dispersion caused by the combination of macroturbulence and axial rotation after removal of all other measurable line-broadening sources. In this section, we discuss how we avoid line-damping problems and correct for instrumental broadening. Then, in Section 3, we consider the effects of microturbulent and thermal broadening.

Cross-correlation procedures developed to study galactic and stellar rotation (Simkin 1974; Tonry & Davis 1979, 1981) are not useful for RR Lyrae stars. During typical RR Lyrae pulsation cycles, temperatures vary by as much as 1000 K, effective gravities vary by more than an order of magnitude, and turbulent velocities vary by amounts greater than their axial rotations. No stars in the color–magnitude diagrams (CMDs) of old MP populations can serve as suitable templates for variable stars with these characteristics. Accordingly, we developed a variant of the method adopted by Hosford et al. (2009), described in Section 2.2.1 below.

### 2.2.1. Elimination of Line-damping Effects

For each spectrum, we measured the FWHM and EW of metal lines chosen from a list of some 200 unblended lines
<figure><img src="image_4.png" /><figcaption>**Figure 2.** The FWHM<sub>meas</sub> vs. EW for lines measured in the du Pont echelle spectrum of HD 140283. These measurements were made on a spectrum formed by the merger of 12 separate observations. There are three EW regions: the discarded lines, EW < 40 mÅ; the plateau lines, 40 mÅ ≤ EW ≤ 80 mÅ; and the damping lines, EW > 80 mÅ. The division of EW regions is discussed in the text.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Figure 3.** Plots of FWHM vs. wavelength for plateau lines in MP RRab stars, arranged in order of increasing [Fe/H] from top left to bottom right.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>**Figure 4.** (Top) The histogram of the difference FWHM (damping regression at EW = 80 mÅ) minus FWHM (plateau) is strongly peaked near 0.5 km s⁻¹ and (middle and bottom) varies little with pulsation phase and FWHM.</figcaption></figure>

(mostly due to Fe I) in the wavelength range 4100–5300 Å that we judged to be unblended by inspection of the solar line identifications of Moore et al. (1966). These lines are listed in Table 2. Less than half of them proved to be measurable in any particular spectrum. We expect that these lines are unblended in our spectra of RR Lyrae and RHB stars and in the spectrum of HD 140283. Measurements of these lines are labeled with the subscript “meas.” Then, using two methods described below, we culled from these measurements subsets of unsaturated “plateau” lines for each of the observed pulsation phases of each RR Lyrae star. The measurements of the lines surviving this cut are labeled with the subscript “plat.”

(1) Our spectra with characteristic S/N ~ 20–30 and resolution of R ~ 27,000 are inadequate for direct detection of rotation profiles or analysis by Fourier methods. Hence, our recourse was to use a variant of the procedure of Hosford et al. (2009), who empirically determined the average minimum widths of weak metal lines as part of their study of lithium in MP main-sequence turnoff stars ($T_{\text{eff}}$ ~ 6200 K, log g ~ 3.8).

They demonstrated that in a plot of EW versus FWHM, the lines of their stars with EW < 90 mÅ lie on a “plateau” of constant FWHM free of detectable damping wings; see their Figure 1. We found this procedure to be adequate for our purposes. We followed Hosford et al. (2009) in Figure 2 with a plot of $FWHM_{\text{obs}}$ versus EW for the lines of Fe I and Fe II in HD 140283.

In somewhat hotter RR Lyrae stars, more MP by one to two orders of magnitude, depression of the continuum by hosts of weak unresolved lines is of greatly diminished importance. The effect of line opacity on the energy distributions of solar-type stars was treated in detail long ago by Melbourne (1960) and Wildey et al. (1962). Such stars display a marked decline in atomic line blanketing with increasing wavelength from 4000 to 5500 Å. This is illustrated in Figure 1 of Wildey et al., which also shows the relative unimportance of line blanketing in the MP ([Fe/H] ~ -2.0) subdwarf HD 19445.

All of our measures of EW and FWHM were made by using the Gaussian profile fitting routine of IRAF/splot. In this
<figure><img src="image_7.png" /><figcaption>**Figure 5.** Montage showing how plateau FWHM<sub>obs</sub> (40 mÅ ≪ EW ≪ 80 mÅ) varies during the pulsation cycle of star WY Ant. Pulsation phases (relative to φ = 0.0 at maximum light) are shown in the bottom right corner of each panel.</figcaption></figure>

procedure, the computer operator chooses the local continuum
level at each stellar line by interaction with a graphic display of
the spectrum. If unrecognized metallic lines were preferentially
depressing the adopted continua at shorter wavelengths, we
should see this effect as a positive correlation of FWHM with
wavelength, but there are no such correlations in Figure 3
displayed for plateau lines at phases near V<sub>macrot</sub> minimum
(phase φ ~ 0.38) in the wavelength range 4000–5300 Å. The
stars in this montage are displayed from top left to bottom right
in order of increasing [Fe/H]. It is instructive to note that the
plateau value FWHM<sub>plat</sub> ~ 15 km s<sup>-1</sup> is very nearly the same
for all stars plotted in the montage except for Z Mic, a
remarkable result that will be further documented in
Section 3.1.1 by the small dispersion of V<sub>macrot</sub> minima near
phase 0.38. We conclude that continuum placement is not a
problem at phases near V<sub>macrot</sub> minimum, when the lines are
narrowest and of greatest interest to us.

We used an upper bound to the plateau of 80 mÅ to identify
the onset of increases of FWHM<sub>meas</sub> with increasing EW due to
damping in our RR Lyrae spectra. Because the Hosford et al.
(2009) spectra had nearly double the resolving power
(R = 47,000) and many times higher S/N (⟨S/N⟩ ~ 100) than

those of our RR Lyrae spectra, they were able to measure metal
lines as weak as 5 mÅ. We did not trust our lines with
EW < 40 mÅ to yield reliable FWHM because of a “personal
equation” effect that we found in our measurements of weak
lines, i.e., those for which the central depths are comparable to
the continuum noise of the spectra. Accidental negative noise
in the wings of a line tends to produce broad absorption
features that we frequently regarded as unmeasurable. How-
ever, accidental positive noise in the wings tends to produce
spuriously narrow but easily measurable features. These effects
become more important with decreasing spectral resolution and
S/N, becoming particularly noticeable near minimum light
(pulsation phase φ ~ 0.85), when the lines become broad.
Decisions about measurability will vary from one person to
another and for the same person at different times, hence
“personal equation.” The systematic effect arising from these
biases, derived from measurements of hundreds of lines in
eight RRab stars at all pulsation phases, is small but detectable
(1.07 ± 0.65 km s⁻¹). The effect varies with pulsation phase,
going through a minimum near phase φ ≃ 0.35 when the lines
are intrinsically narrowest. To summarize, we calculated the
<figure><img src="image_8.png" /><figcaption>**Figure 6.** Variations with pulsation phase $\phi$ of $T_{\text{eff}}$ (panel (a)), log g (panel (b)), $V_{\text{mic}}$ (panel (c)), and $v_{\text{th}}$ (panel (d)) for the RRab stars studied by For et al. (2011b; dots) and Chadid et al. (2017; open circles). Blue symbols denote MP stars, and red symbols denote MR stars.</figcaption></figure>

average FWHM<sub>plat</sub> in the more restrictive domain 40 mÅ < EW < 80 mÅ.

(2) The damping regression for EW > 80 mÅ produces a second, independent estimate of FWHM<sub>plat</sub>. This regression considers stronger, more accurately measured lines than those that lie on the plateau. Its numerical value, derived by the method of least squares for each spectrum, depends only slightly on the choice of the plateau EW upper limit. The utility of this regression evaluated at EW = 80 mÅ as an estimator of the plateau value is simply an empirical fact, supported by the statistical comparisons of the regression and plateau estimates in Figure 4. The top left panel of this figure is a histogram of 370 measured differences that we define as FWHM(R - P) ≡ FWHM (regression at EW = 80 mÅ) minus FWHM(plateau) for RRab stars binned at intervals of 0.5 km s<sup>-1</sup>. The average difference, &lang;FWHM(R - P)&rang; = 0.35 ± 0.05 km s<sup>-1</sup>, is smaller than the measurement errors. The bottom two panels of Figure 4 show that there is little variation of the difference with either pulsation phase or FWHM. Which of the two estimates is more reliable depends in a complicated way on S/N and metallicity. The regression values are valuable for observations

made during rising light and for MR stars, situations in which we frequently are unable to measure weak lines.

We adopt the average of these two estimates as our final value of FWHM<sub>plat</sub> when both are available.

### 2.2.2. Correction for Instrumental Line Broadening

Instrumental broadening is comparable to the intrinsic widths of metal lines in our RR Lyrae spectra, so careful removal of it is important for the success of our analysis. We used the profiles of emission lines produced by a thorium-argon (ThAr) hollow cathode tube to remove the effects of instrumental broadening. The ThAr exposures were obtained with the equipment and procedures described below.

The optical paths of starlight and ThAr light for the du Pont echelle are nearly identical except for the central obscuration of some stellar light by the Cassegrain secondary mirror. A lens with the focal ratio of the telescope projects an image of the ThAr hollow cathode onto the entrance aperture of the spectrograph by a motor-operated mirror inserted into the telescope beam. The ThAr spectra were obtained before and after observations at each position of the telescope. When
<figure><img src="image_9.png" /><figcaption>Figure 7. Variations of $V_{\text{macrot}}$ with phase for MP (top panel) and MR (bottom panel) RRab stars. Solid curves are mean variations as described in the text.</figcaption></figure>

we followed a star for lengthy intervals of time, we recorded the ThAr spectrum at intervals of 20–60 minutes; time intervals between ThAr exposures varied as required by various combinations of airmass, decl., and hour angle. We coadded successive pairs of ThAr exposures to make wavelength-calibrated extractions of stellar spectra obtained between successive pairs.

The temperature dependence of the focal adjustment of the echelle spectrograph is well calibrated. This calibration was employed to alter the focus of the spectrograph on those rare occasions when significant temperature variation during the night required its use.

Following the suggestion of an anonymous referee, we also measured the FWHM of the [O I] 5577 Å airglow line, which lies some 200 Å longward of the upper limit of the spectral region used in our analyses, on all of the stellar spectra obtained on a dozen nights in 2006–2012. We compare these airglow measurements with measurements made of ThAr spectra obtained on the same nights in the Appendix.

We approximated corrections for all line-broadening processes by using the additive property, $\sigma_a = (\sigma_b^2 + \sigma_c^2)^{1/2}$, of the Maxwell

velocity distribution, in which $a, b$, and $c$ are parameters of the Gaussian error function. We assume that for RR Lyrae stars, the turbulent velocities on all length scales are isotropic throughout the metallic line-forming regions. To begin, we remove the effect of instrumental broadening, characterized by FWHM$_{\text{inst}}$, from the average values of FWHM$_{\text{plat}}$ to obtain FWHM$_{\text{unsat}}$, according to

$$ \text{FWHM}_{\text{unsat}} = (\langle \text{FWHM}_{\text{plat}} \rangle^2 - \langle \text{FWHM}_{\text{inst}} \rangle^2)^{1/2}, \quad (1) $$

in which $\langle \text{FWHM}_{\text{inst}} \rangle$ is an average value described below. We then convert FWHM$_{\text{unsat}}$ to a Maxwellian velocity dispersion by use of $\sigma_{\text{unsat}} = (2\sqrt{\ln 2})^{-1} \text{FWHM}_{\text{unsat}}$. All further corrections are made in units of line-of-sight velocity dispersion $\sigma$.

Removal of instrumental broadening was made to the average value $\langle \text{FWHM}_{\text{plat}} \rangle$ rather than to the values of individual lines, because some measured widths of unsaturated lines may be narrower than $\langle \text{FWHM}_{\text{inst}} \rangle$. These narrowest lines belong in the average of a normal distribution, but they cannot be corrected for instrumental broadening by Equation (1).

The FWHM$_{\text{inst}}$ was evaluated as follows. We measured 591 FWHM values of some four dozen lines in each of 14 ThAr spectra chosen at random from spectra gathered in 2006–2012.
<figure><img src="image_10.png" /><figcaption>**Figure 8.** Average variations of $V_{\text{macrot}}$ with phase for MP (blue) and MR (red) RRab stars. The minimum value of $V_{\text{macrot}}$ for MR stars is significantly lower than that for MP stars, and the rising branch is shifted to a later phase.</figcaption></figure>

Details of this investigation are given in the Appendix. We used the average value of all of these measurements, $\langle \text{FWHM}_{\text{inst}} \rangle = 11.36 \pm 0.52 \text{ km s}^{-1}$, in Equation (1) to calculate the FWHM$_{\text{unsat}}$ for each RR Lyrae spectrum. This velocity corresponds to resolving power $R = 26,700 \pm 200$, very close to the value of $R \simeq 27,000$ usually adopted for the du Pont echelle spectrograph.

### 3. $V_{\text{macrot}}$ from Measurements of EW and FWHM$_{\text{unsat}}$ for RRAB Stars

For the calibration stars in Table 1, we have enough spectra to construct FWHM$_{\text{meas}}$ versus EW relations throughout their pulsation cycles. For each star, we combined observations made over several years into small phase intervals ($\sim 0.05$P) to increase S/N. The average numbers of phase bins for MP and MR stars are 22 and 18, respectively. For each spectrum, we used IRAF/splot<fnref n="6" /> to measure EW and FWHM$_{\text{meas}}$ for all usable lines in our list. Results from these measurements are listed in the sixth through ninth columns of Table 1.

From the FWHM$_{\text{meas}}$ versus EW diagrams, direct evidence of the variation of macroturbulence with pulsation phase is easily seen. In Figure 5, we illustrate this variation for WY Ant with diagrams for eight phases during its pulsation cycle. The FWHM$_{\text{meas}}$ plateau value is high ($\sim 23 \text{ km s}^{-1}$) immediately after maximum light ($\phi = 0.065$); see the top left panel of Figure 5. It reaches a minimum ($\sim 15 \text{ km s}^{-1}$) in the bottom left panel and increases steadily during the remainder of the declining light shown in the right panels.

The FWHM$_{\text{plat}}$ subsets of these measurements were converted to FWHM$_{\text{unsat}}$ by Equation (1) and from these to $V_{\text{macrot}}$. The FWHM$_{\text{plat}}$ value is similar to the damping

regression value at all phases, as shown in the middle panels of Figure 4. In Table 1, we provide estimates of the minimum value of $V_{\text{macrot}}$ for each star, calculated as the average $V_{\text{macrot}}$ for observations that lie in the phase interval $0.30 < \phi < 0.45$.

#### 3.1. The Behaviors of Macroturbulence and Microturbulence in RRab Stars

##### 3.1.1. The Stable RRab Stars

Microturbulence, produced by motions on length scales smaller than the photon mean free path, broadens all spectral features, but for weak lines on the linear portion of the curve of growth (COG), the EWs are unchanged. Microturbulence will desaturate lines on the damping portion of the COG, thus increasing their EWs. Fortunately, $V_{\text{mic}}$ can be derived from spectrum analysis (Gray 2008) with standard codes (e.g., MOOG;<fnref n="7" /> Sneden 1973). We used the microturbulence values of For et al. (2011b) in our calculations. Macroturbulence, produced by motions on length scales that are large compared to the photon mean free path, acts to broaden all absorption lines. Macroturbulent velocities exceed the microturbulent velocities of Richardson's (1920) eddies using the Siedentopf model of turbulent convection (Woolley & Stibbs 1953; Ruediger 1989), to which we subscribe.

After computing $\sigma_{\text{unsat}} = (2\sqrt{\ln 2})^{-1}$ FWHM$_{\text{unsat}}$ for each spectrum of each star, we removed microturbulent and thermal velocity dispersions. These quantities were taken directly or computed from the atmospheric parameters for RRab stars derived by For et al. (2011b) and Chadid et al. (2017). In Figure 6, we show the variations in $T_{\text{eff}}$, log g, $V_{\text{mic}}$, and $V_{\text{th}}$ as functions of pulsation phase for these stars. The gravities (panel (b)) are based on ionization equilibrium constraints only, as discussed in For et al. and Chadid et al. Their values are shown

<footnote n="6">⁶ IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under cooperative agreement with the National Science Foundation.</footnote>

<footnote n="7">⁷ http://www.as.utexas.edu/~chris/moog.html</footnote>
<figure><img src="image_11.png" /><figcaption>**Figure 9.** Radial velocity (top panel) and FWHM<sub>cc</sub> of the cross-correlation function (bottom panel) vs. phase for the RRab Blazhko star UV Oct. Data were obtained by cross-correlation with a template spectrum of CS 22874–009 (Preston & Sneden 2000). The colors indicate different observing seasons, denoted in the figure legend as yymm, where yy are the last digits of the year and mm are the digits of the month; e.g., 0604 is 2006 April.</figcaption></figure>

<table><caption>**Table 3**
Ephemerides for Blazhko Stars Used in Figure 10</caption><thead><tr><th>Star</th><th>HJD<sub>0</sub></th><th>P</th><th>δφ</th></tr></thead><tbody><tr><td>AM Vir</td><td>1888.395</td><td>0.615085</td><td>-0.010</td></tr><tr><td>AS Vir</td><td>3836.325</td><td>0.553400</td><td>0.000</td></tr><tr><td>BS Aps</td><td>3836.370</td><td>0.582560</td><td>0.015</td></tr><tr><td>RV Cap</td><td>5328.400</td><td>0.447748</td><td>-0.010</td></tr><tr><td>RV Cet</td><td>5455.460</td><td>0.623421</td><td>0.025</td></tr><tr><td>S Ara</td><td>2137.338</td><td>0.451883</td><td>-0.030</td></tr><tr><td>UV Oct</td><td>3836.795</td><td>0.542573</td><td>0.010</td></tr><tr><td>V1645 Sgr</td><td>3932.480</td><td>0.552910</td><td>-0.010</td></tr><tr><td>V Ind</td><td>1873.466</td><td>0.479602</td><td>-0.030</td></tr><tr><td>V494 Sco</td><td>5364.435</td><td>0.427266</td><td>-0.045</td></tr></tbody></table>

here only for completeness, as we will not use them in our subsequent calculations. Additionally, all quantities show significant scatter in the phase domains of rising and maximum light (0.8 < φ < 1.1); these parts of the pulsational cycles will not be used here.

We computed thermal velocities from $V_{th} = (2kT/m)^{1/2}$, adopting $m = 54$ amu,<fnref n="8" /> and these values are shown in panel (d) of Figure 6. These velocities are nearly constant, ranging only over $1.38 \lesssim V_{th} \lesssim 1.53$, due to the modest RRab temperature variations (panel (a)) that are muted by the square-root function. Over most of the pulsational cycles, $V_{th} \approx 1.4$ km s⁻¹. Additionally, the microturbulent velocities shown in panel (c) are consistently larger, $V_{mic} \approx 2V_{th}$ at all phases, so we have assumed a single microturbulent velocity of 2.8 km s⁻¹ in all subsequent calculations. We use these data to plot $V_{macrot} = (\sigma_{unsat}^2 - \sigma_{mic}^2 - \sigma_{th}^2)^{1/2}$ versus phase in Figure 7. The top and bottom panels contain, respectively, data for MP and MR stars.

We also calculated average variations of $V_{macrot}$ with phase (solid curves in Figure 7) as follows. We sorted all of the data by phase and then calculated averages $\langle\phi\rangle$ and $\langle V_{macrot}\rangle$ for data binned in successive groups of 10 measurements. We superpose

<footnote n="8">⁸ The mean value of the atomic masses of the lines participating in the velocity calculations is 54 amu.</footnote>
<figure><img src="image_12.png" /><figcaption>**Figure 10.** The $V_{\text{macrot}}$ vs. phase for nine Blazhko RRab stars. The solid blue curve denotes the mean points calculated for data in successive 0.05 phase intervals.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>**Figure 11.** Variation with phase of $V_{\text{macrot}}/V_{\text{mic}}$, an empirical spectrum of turbulence, for the MP and MR RRab stars. The blue curve represents the mean trend for the MP RRab. Its values were computed in phase intervals of $\Delta\phi = 0.05$; see Table 4.</figcaption></figure>

the mean MP and MR curves without the individual data points in Figure 8. Minimum $V_{\text{macrot}}$ values for all of the RR Lyrae stars in this investigation and the MP RHB stars to be discussed in Section 5 are presented in Table 1. The minimum averages, computed from data for each RRab in the phase interval $0.30 < \phi < 0.45$, are $5.31 \pm 0.13$ and $4.40 \pm 0.22 \text{ km s}^{-1}$, respectively. These averages differ by an amount three times larger than the sum of their mean probable errors. Furthermore, the persistent offset of the two variations after phase $\phi = 0.5$ in

Figure 8 indicates a small but real difference between the behaviors of $V_{\text{macrot}}$ in MP and MR stars.

Although we can only claim that the $V_{\text{macrot}}$ minima in Figure 7 provide upper limits on $V_{\text{rot}} \sin(i)$ for the MP and MR RRab, the small scatter of individual stars near these minima strongly suggests that macroturbulence, not rotation, dominates the minimum values. Random inclination of rotation axes alone should create dispersion in the minimum values, and we are uncomfortable with the notion that all RRab stars rotate at the
<figure><img src="image_14.png" /><figcaption>**Figure 12.** Top panel: $V_{\text{macrot}}$ vs. phase for 12 RRc stars. Bottom panel: same as the top panel but with the $V_{\text{macrot}}$ values in each star arbitrarily shifted to coincide with the data point for AS 1433 at phase 0.25. The solid curve connects the data points phase-averaged in groups of five.</figcaption></figure>

same speed; thus, the actual rotations of RRab stars may lie well below our calculated upper limits.

### 3.1.2. The Blazhko RRab stars

Complex variations of the metallic radial velocities and line widths of Blazhko RRab stars near their light maxima (radial velocity minima) defy summary description. One example, UV Oct, is shown in Figure 9. The incomplete sampling of Blazhko phases for UV Oct is typical for the Blazhko stars in our data set. Because phases near maximum light contribute nothing to our exploration of axial rotation, we restricted our attention to observations made during declining light ($0.20 < \phi < 0.55$), where the line widths of stable RRab stars achieve their minimum values. The ephemerides in Table 3 produce phases of radial velocity minima near zero for our Blazhko variables (Chadid & Preston 2013; Preston, unpublished). Values of $V_{\text{macrot}}$ plotted versus these phases are displayed in Figure 10. Modest phase corrections, adopted by

eye inspection and listed in the fourth column of Table 3, were added to phases calculated with these ephemerides to reduce horizontal scatter in this figure. Blue line segments connect the mean points of observation calculated at intervals of $0.05P$. The lowest mean point, $V_{\text{macrot}} = 4.77 \pm 0.32 \text{ km s}^{-1}$, occurring at $\phi = 0.38$, is slightly smaller than the mean value for stable RRab stars, $\langle V_{\text{macrot}} \rangle = 5.31 \pm 0.13 \text{ km s}^{-1}$. Stable and Blazhko RRab stars yield the same upper limit on axial rotation of $5 \pm 1 \text{ km s}^{-1}$.

### 3.2. An Empirical "Spectrum of Turbulence" in RRab Stars

For stable RRab stars, we combine the variations of macroturbulence (Figure 7) and microturbulence (Figure 6) by forming their ratio. The microturbulent velocities for these stars were derived by For et al. (2011b) and Chadid et al. (2017) from phase-combined spectra, so there are fewer of these values than of $V_{\text{macrot}}$. This is especially true of the Chadid et al. results, which were usually based on only three or
<figure><img src="image_15.png" /><figcaption>**Figure 13.** Correlations of $\Delta V_{\text{macrot}}$ amplitude (the difference between maximum and minimum $V_{\text{macrot}}$ values) with (top panel) radial velocity amplitude and (bottom panel) visual light amplitude during pulsation cycles.</figcaption></figure>

four combined phase points. We matched the phases of the $V_{\text{macrot}}$ and $V_{\text{mic}}$ data for as many points per star as possible, interpolating the $V_{\text{mic}}$ values where possible and not attempting any extrapolations into phase intervals not covered by both data sets. The results of this exercise are displayed in Figure 11, where MP and MR stars are distinguished by different symbols and colors. The solid line segments connect the mean values of the data points calculated in phase intervals of 0.05$P$. These mean points are listed in Table 4. This variation of the $V_{\text{macrot}}/V_{\text{mic}}$ ratio with phase is in accord with the minimal requirements of Richardson (1920), Kolmogorov (1941), and the Siedentopf model (Woolley & Stibbs 1953) at all phases, namely, that $V_{\text{macrot}}$ always exceeds $V_{\text{mic}}$. We offer this ratio as a primitive empirical description of a “spectrum of turbulence,” defined by the two length scales associated with macro- and microturbulence, in RRab atmospheres during their pulsation cycles.

## 4. $V_{\text{macrot}}$ from EWs and FWHMs of RRc Stars

The measurements of EW and FWHM for the RRc stars and the conversion of these measurements to estimates of

$V_{\text{macrot}}$ proceeded in the same manner followed for RRab stars in Section 3. We chose lines for measurement from the same line list. Because the RRc stars are systematically hotter than the RRab stars, the metallic absorption lines are systematically weaker. However, the agreement between the plateau and regression estimates shown in Figure 4 is hardly impaired: $\langle \text{FWHM}(R-P) \rangle = 0.46 \pm 0.06 \text{ km s}^{-1}$ for RRc stars is only 0.11 km s$^{-1}$ larger than the value derived for RRab stars, $0.35 \pm 0.05 \text{ km s}^{-1}$.

The variations of $V_{\text{macrot}}$ with phase for RRc stars, presented in the top panel of Figure 12, differ markedly from the uniform behavior of the RRab stars in Figure 7. Minimum values of $V_{\text{macrot}}$ occur at earlier phases, near phase $\phi = 0.3$, and range from 2 to 12 km s$^{-1}$. Maximum values occur near maximum light. We create a characteristic variation of $V_{\text{macrot}}$ with phase in the bottom panel of Figure 12 by using additive constants that superpose data for all stars near phase $\phi \simeq 0.25$ at $V_{\text{macrot}} = 6.62 \text{ km s}^{-1}$, the mean value for the sample. This exercise is an instructive, albeit mathematically incorrect, way to combine the data. The amplitude of the solid blue curve that connects the mean points of the individual observations is
<figure><img src="image_16.png" /><figcaption>**Figure 14.** The $V_{\text{macrot}}$ values for RHB stars of this study and the $V \sin i$ values of Behr (2003b) and Carney et al. (2008) are plotted vs. $\log T_{\text{eff}}$.</figcaption></figure>

<table><caption>**Table 4**
Mean Values of $V_{\text{macrot}}/V_{\text{mic}}$ for MP RRab Stars</caption><thead><tr><th>φ</th><th>V<sub>macrot</sub>/V<sub>mic</sub></th></tr></thead><tbody><tr><td>0.025</td><td>4.37</td></tr><tr><td>0.075</td><td>3.71</td></tr><tr><td>0.125</td><td>3.11</td></tr><tr><td>0.175</td><td>2.93</td></tr><tr><td>0.225</td><td>2.55</td></tr><tr><td>0.275</td><td>2.27</td></tr><tr><td>0.325</td><td>1.91</td></tr><tr><td>0.375</td><td>1.81</td></tr><tr><td>0.425</td><td>2.08</td></tr><tr><td>0.475</td><td>2.40</td></tr><tr><td>0.525</td><td>2.69</td></tr><tr><td>0.575</td><td>2.98</td></tr><tr><td>0.625</td><td>3.28</td></tr><tr><td>0.675</td><td>3.41</td></tr><tr><td>0.725</td><td>3.23</td></tr><tr><td>0.775</td><td>3.31</td></tr><tr><td>0.825</td><td>3.41</td></tr><tr><td>0.875</td><td>3.78</td></tr><tr><td>0.925</td><td>4.05</td></tr><tr><td>0.975</td><td>4.32</td></tr><tr><td>1.025</td><td>4.37</td></tr></tbody></table>

~4 km s⁻¹, much smaller than the amplitude for RRab stars (~10 km s⁻¹).

We can understand the behavior of the RRc stars in the top panel of Figure 12 by noting that macroturbulent velocity amplitude, the difference between maximum and minimum values of $V_{\text{macrot}}$ achieved in a pulsation cycle, scales with pulsation amplitude for all RR Lyraes, as shown by the regressions in Figure 13. Accordingly, we imagine that less macroturbulence in RRc stars with their smaller pulsation amplitudes permits detection of smaller axial rotations in RRc stars than those encountered among the RRab. Outstanding

<table><caption>**Table 5**
$T_{\text{eff}}$ and $V_{\text{rot}} \sin(i)$ Values for RHB Stars in Two Previous Studies</caption><thead><tr><th>Source</th><th>Star</th><th>T<sub>eff</sub></th><th>log T<sub>eff</sub></th><th>V<sub>rot</sub> sin (i)</th></tr></thead><tbody><tr><td>Behr (2003b)</td><td>HD 25532</td><td>5553</td><td>3.745</td><td>7.7</td></tr><tr><td>Carney et al. (2008)</td><td>HD 25532</td><td>5320</td><td>3.726</td><td>4.8</td></tr><tr><td>Difference</td><td>...</td><td>233</td><td>0.019</td><td>2.9</td></tr><tr><td>Behr (2003b)</td><td>HD 184266</td><td>5760</td><td>3.760</td><td>9.3</td></tr><tr><td>Carney et al. (2008)</td><td>HD 184266</td><td>5490</td><td>3.740</td><td>5.0</td></tr><tr><td>Difference</td><td>...</td><td>233</td><td>0.021</td><td>4.3</td></tr><tr><td>Mean difference</td><td>...</td><td>252</td><td>0.020</td><td>3.6</td></tr></tbody></table>

examples are AS 190212-4639 (4 km s⁻¹) and AS 143322-0418 (~1 km s⁻¹). The large upper limit of $V_{\text{macrot}}$ (~12 km s⁻¹) for RRc suggests that BHB rapid rotators adjacent to the instability strip lose their surface angular momentum during evolution through the RRc portion of the instability strip, a possibility worthy of further exploration with better data.

## 5. $V_{\text{macrot}}$ from EWs and FWHMs of MP RHB Stars

We used *Magellan* MIKE echelle spectra (outlined in Section 2) to estimate $V_{\text{macrot}}$ for 22 MP RHB stars. We adopted FWHM<sub>inst</sub> = 7.20 km s⁻¹ derived from ThAr lines to remove instrumental broadening, and we borrowed $V_{\text{th}}$ and $V_{\text{mic}}$ data from Preston et al. (2006) to calculate values of $V_{\text{macrot}}$.

These $V_{\text{macrot}}$ values are systematically 1–2 km s⁻¹ larger than those of the RRab stars, and they decline smoothly with decreasing temperature along the RHB portion of the horizontal branch. This decline is shared by the values of $V_{\text{rot}} \sin(i)$ derived by Behr (2003b) and Carney et al. (2008), as illustrated in Figure 14. Behr (2003b) removed macroturbulent broadening from his measured line widths by the conservative assumption that macroturbulent velocities equal the microturbulent velocities derived from his spectrum analyses. The
<figure><img src="image_17.png" /><figcaption>**Figure 15.** Histograms of the [Fe/H] distributions of three MP RHB samples. Strong selection effects are evident.</figcaption></figure>

mean value and standard deviation of these values for the Behr sample presented here are 2.11 ± 0.44 km s⁻¹. Carney et al. (2008) derived much larger macroturbulent velocities (6–12 km s⁻¹) from Fourier transform analyses of line profiles provided by high-resolution, high-S/N spectra of six RHB stars. Carney et al. remarked that “rotation and macroturbulence play comparable roles in the line broadening of the observed RHB stars,” whereas Behr, by virtue of his assumed small macroturbulent velocity, found that rotation dominates in his sample. The data assembled in Figure 3 of Grassitelli et al. (2015) support the conclusions of Carney et al. A comparison of the only two stars, HD 25532 and HD 184266, common to the Behr and Carney et al. samples is presented in Table 5. We suspect that the larger $V_{\text{rot}} \sin(i)$ values of Behr in this table arise, at least in part, from his assumed low macroturbulent velocities. Because our $V_{\text{macrot}}$ values lie comfortably between/along the regression lines of Carney et al. and Behr in Figure 14, we conclude that our $V_{\text{macrot}}$ values are very nearly $V_{\text{rot}} \sin(i)$ values; i.e., the macroturbulent velocities of our RHB stars lie below our limit of detectability.

We omitted HD 195636 ($V_{\text{rot}} \sin(i) = 20.6$ km s⁻¹, $T_{\text{eff}} = 5399$ K, [Fe/H] = −2.74; Behr 2003b) from Figure 14

<table><caption>**Table 6**
Metallicity Data for Three RHB Samples</caption><thead><tr><th>Source</th><th>⟨[Fe/H]⟩</th><th>σ</th><th>f([Fe/H] > −1.7)</th></tr></thead><tbody><tr><td>This study</td><td>−2.59</td><td>0.27</td><td>0.00</td></tr><tr><td>Carney et al. (2008)</td><td>−1.95</td><td>0.35</td><td>0.17</td></tr><tr><td>Behr (2003b)</td><td>−1.53</td><td>0.36</td><td>0.80</td></tr></tbody></table>

because it lies so far from the locus of all other MP RHB stars. The star is an indubitable MP, giant, rapid rotator (see discussion by Preston 1997).

We seek an explanation for the small values of macroturbulent velocities in our RHB sample in the work of Tanner et al. (2013) and Grassitelli et al. (2016), who reported that at a given effective temperature, the optically thin superadiabatic layers of model atmospheres with higher metallicity possess larger convective velocities. This is an abundance effect: higher atmospheric opacities produce optically thin layers in regions of lower gas density. Their calculations induced us to construct the metallicity distributions of the three RHB samples, which we display as histograms in Figure 15. It is evident at a glance that major selection effects are present. The RHB stars of the present study were discovered in the course of high-resolution
<figure><img src="image_18.png" /><figcaption>**Figure 16.** Variation of $V_{\text{rot}} \sin(i)$ (BHB stars) and $V_{\text{macrot}}$ (RR Lyrae and RHB stars) with $\log(T_{\text{eff}})$ along the MP HB. The black vertical line denotes the Grundahl limit at 11,500 K. For the instability strip, the vertical lines show the fundamental blue edge (FBE, blue), the first-overtoned red edge (FOR, green), and the fundamental red edge (FRE, red). Black parabolic trajectories define the regions of the instability strip traversed by BHB rapid rotators that evolve with constant luminosity and angular momentum ($V_{\text{rot}} \propto T_{\text{eff}}^2$).</figcaption></figure>

spectroscopic investigations (Preston et al. 2006; Roederer et al. 2014) of the most MP stars, [Fe/H] $\le -2$, identified in the survey of Beers et al. (1992), while the samples of Behr (2003b) and Carney et al. (2008) were chosen by exploration of more abundance-inclusive, photometrically defined samples. The content of the histograms is summarized in Table 6.

The mean [Fe/H] values of this study and that of Behr (2003b) differ by one full order of magnitude. The sample fractions with [Fe/H] $> -1.7$ in these three studies range from 0.0 to 0.8. This comparison of our very MP sample with that of Carney et al. (2008) supports the conclusion of Tanner et al. (2013): macroturbulent velocities in the optically thin layers that produce metal absorption lines increase with increasing metallicity. Perhaps this presentation will stimulate renewed interest in abundance-dependent macroturbulence on the HB.

## 6. Discussion

In this paper, we have combined high-resolution spectra of RR Lyrae and RHB stars collected at Las Campanas Observatory over three decades with literature studies of RHB and BHB stars to derive rotation rates or meaningful upper limits for stars along the HB. In Figure 16, we present a summary plot showing $V_{\text{macrot}}$ or $V_{\text{rot}} \sin(i)$ as a function of effective temperature along the HB. The astonishing variations of measured rotational velocity with $T_{\text{eff}}$ clearly indicate that surface angular momentum cannot be a proper measure of the total angular momentum of stars along the HB.

To begin discussion of Figure 16, we recall that the large values of $V_{\text{rot}} \sin(i)$ for many MP BHB stars first encountered by Peterson et al. (1995) at $T_{\text{eff}} < 11,500$ K were totally unexpected in view of the small projected rotational velocities
<figure><img src="image_19.png" /><figcaption>**Figure 17.** The $V_{\text{rot}} \sin(i)$ distributions of globular cluster (top panel) and field (bottom panel) BHB stars in the compilation of Cortés et al. (2009).</figcaption></figure>

($V_{\text{rot}} \sin(i) < 5 \text{ km s}^{-1}$) of their evolutionary antecedents in globular clusters (Lucatello & Gratton 2003). Then, the abrupt decline in rotation at the blue edge of the instability strip (Peterson et al. 1996) was an additional surprise, appropriately labeled a “conundrum” by Preston (2011) and Preston & Chadid (2013). This investigation reduces the upper limit of RRab rotation from 10 to $5 \text{ km s}^{-1}$ and, additionally, suggests that rapid BHB rotation disappears during RRc evolution. Should the arguments of Tanner et al. (2013) apply to RR Lyrae stars, the upper limit on RR Lyrae rotation would be further reduced.

Extant $V_{\text{rot}} \sin(i)$ distributions differ from cluster to cluster in the Galactic halo. However, whether field and globular cluster HB star samples are drawn from the same parent population is an open question. The $V_{\text{rot}} \sin(i)$ distributions of the Cortés et al. (2009) compilations of field and cluster HB stars are similar, as noted previously by Cortés et al. and shown in Figure 17. This similarity encouraged us to include the cluster HB stars in Figure 16. Only a handful of the ~150 known Galactic globular clusters have been investigated for rotation, and the field sample is limited to HB stars near the solar circle, so it is premature to draw firm conclusions about this issue.

The use of “during” in the description of RRc evolution deserves comment because of the complicated nature of HB

evolution in and near the instability strip. This is best appreciated by inspection of the evolutionary tracks of Lee & Demarque (1990) in Figure 18 plotted for $Y = 0.23$ with masses and abundances that bracket the models adopted for the MP globular clusters M3, M15, and M92 by VandenBerg et al. (2016). For nearly all of the models plotted in Figure 18, approximately horizontal evolution at near-constant luminosity toward higher effective temperatures occupies ~2/3 of their HB lifetimes (see Figure 1 of de Santis & Cassisi 1999).

Second-parameter issues play no role in our study. Evolution from the zero-age horizontal branch (ZAHB) to higher temperature, the “blue loops” in Figure 18 are the result of a hydrogen-burning phenomenon common to all HB evolutionary sequences (Sweigart 1987; Lee & Demarque 1990; Castellani et al. 1991; Dorman 1992; Dotter et al. 2007) for reasons discussed in some detail by Sweigart, by Dorman, and by Dotter et al. Early on, Rood (1973) realized that these loops could not provide the range in HB color distribution required of a second parameter. However, the blue loops would act to blur the pattern of $V_{\text{macrot}}$ values of RHB and RR Lyrae stars seen in Figure 16, if ZAHB stars were to retain their initial surface angular momenta during migration from the RHB and/or RR Lyrae domains toward the BHB. For $[\text{Fe/H}] < -1.6$, evolution proceeds steadily from the BHB through the
<figure><img src="image_20.png" /><figcaption>**Figure 18.** Evolutionary tracks for HB stars with masses and abundances that bracket the models adopted by VandenBerg et al. (2016) for the globular clusters M5, M15, and M92. Blue, green, and red vertical lines mark the fundamental blue edge, first-overtone red edge, and fundamental red edge of the instability strip.</figcaption></figure>

instability strip to the RHB and on to the AGB. However, for stars with [Fe/H] > −1.6, evolution can be quite complicated. As a consequence of stochastic mass loss, an RRc star-to-be may arrive on the ZAHB variously as an RHB, RRab, RRc, or BHB star, all of these evolving to higher temperatures for most of their HB lifetimes. From the metallicity distributions of our RR Lyrae samples, shown in Figure 19, we see that roughly half of them, those with [Fe/H] < −1.6, evolve monotonically from the BHB through the instability strip to the AGB. At higher metallicity, a star can arrive on the ZAHB as an RHB star, evolve through the RRab and RRc states to the BHB, and then go back across the instability strip to the AGB. However, Figure 19 indicates that the stars retain no memory of their initial ZAHB locations.

We observe a decline in mean $V_{\text{rot}} \sin(i)$ during the transition from BHB to RRc, then a gentle rise in the transition from RRab to RHB, followed by a decline during subsequent RHB evolution toward the AGB. Absent a mechanism for altering the total angular momentum of HB stars, we attribute these regularities in behavior of $V_{\text{rot}} \sin(i)$ among the RR Lyrae and

RHB stars to indications that angular momentum is continuously being redistributed within these stars on timescales that are short, ~$10^7$ yr, compared to HB lifetimes. Sills & Pinsonneault (2000) discussed schemes for such redistribution.

As regards removal of surface angular momentum that might accompany mass loss during RR Lyrae evolution, the only credible investigation of which we are aware is the 3D numerical simulation of Stellingwerf (2013), which produced a hot corona with outflow. Stellingwerf did not estimate the mass-loss rate, and we are loathe to speculate in this matter. Whether or not these notions are correct, we believe that they are legitimate food for thought worthy of presentation to theoreticians.

Finally, we note the inclusion in Figure 16 of HD 195636 with $\log T_{\text{eff}} = 3.732$, $V_{\text{rot}} \sin(i) = 20.6 \text{ km s}^{-1}$, and [Fe/H] = −2.7 (Behr 2003b) lying far above all other RHB stars. We deliberately excluded this star from the discussion of Section 5 simply because it is such an egregious outrider. A long history of investigations, summarized by Preston (1997), confirms that it is a very MP RHB or AGB star with lines abnormally broad for its presumed
<figure><img src="image_21.png" /><figcaption>**Figure 19.** Abundance distributions of the RR Lyrae stars studied in this paper.</figcaption></figure>

evolutionary state. Further speculation about the history of HD 195636 is beyond the scope of this paper.

We thank all of the Las Campanas Observatory support personnel for their help during the course of our endeavor, with particular regards to several du Pont telescope operators for their efforts in assisting with the observations required to produce this paper. We also thank Michel Breger, Marcio Catelan, Bill Cochran, and Phillip MacQueen for helpful comments that improved our presentation. This work has been supported in part by NSF grant AST1616040 to C.S. and the University of Texas Rex G. Baker, Jr., Centennial Research Endowment.

# Appendix

## Estimates of FWHM<sub>inst</sub> of the du Pont Echelle Spectrograph

We explore two ways to estimate the instrumental width, FWHM<sub>inst</sub>, of the du Pont echelle spectrograph.

(1) Using procedures described in Section 2.2.2, we made 591 measurements of EW (arbitrary flux units) and FWHM (km s<sup>-1</sup>) in six echelle orders of 14 ThAr spectra chosen at random from observations made during observing runs in 2006, 2007, 2009, 2011, and 2012. The results are plotted in Figure 20 as a plot of

FWHM versus EW, where the horizontal line represents the average value, 11.36 ± 0.52 km s<sup>-1</sup>, of all of the measurements. The average and standard deviation for the 151 strongest lines is reduced only slightly to 11.25 ± 0.45 km s<sup>-1</sup>. Inspection of Figure 20 reveals an obvious skew toward larger FWHM values. This asymmetry arises from degradation of the echelle instrumental profile with increasing angular distance from the optical axis of the spectrograph. This degradation is due to optical aberrations and an inevitable slight tilt of the CCD chip with respect to the focal surface of the spectrograph camera.

We tested the hypothesis that skew is introduced by position-dependent aberrations by using the (x, y) pixel coordinates of the ThAr lines in the CCD image frames to calculate the radial distance r(x, y) of each line from the central coordinates of the CCD chip (x = 1024, y = 1024). The results of this exercise are shown in Figure 21. We found that ⟨FWHM⟩ increases with r(x, y) by ~0.4 km s<sup>-1</sup> over the usable range of r(x, y), in qualitative agreement with expectations.

(2) Following the suggestion of an anonymous referee, we measured the FWHM of the airglow line [O I] 5577 Å in 792 spectra of RR Lyrae stars observed in 2006–2012. This single telluric line is not ideal for our purposes because it lies at a single location on the CCD chip some 1000 Å (20 echelle orders) longward of the bulk of our ThAr and stellar line measurements.
<figure><img src="image_22.png" /><figcaption>Figure 20. The FWHM of ThAr emission lines plotted vs. the relative integrated line fluxes.</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>Figure 21. The FWHM of ThAr lines plotted vs. radial pixel distance from the center of the CCD chip. The red line is a linear regression derived from the plotted data points.</figcaption></figure>

Our measurements are displayed in Figure 22 as a plot of FWHM versus EW of the airglow emission in units of normalized stellar continua. The [O I] emission feature is characteristically weak in our spectra because of our short (<600 s) exposure times. The red horizontal line in this figure represents the average of all measurements, $10.86 \pm 0.62 \text{ km s}^{-1}$. The scatter of individual FWHM about this mean value is free of skew because it is located at the same near-central location on the CCD chip in all spectra. The mean FWHM of the [O I] distribution is $0.50 \text{ km s}^{-1}$ smaller than that of the ThAr lines. No reasonable assumptions about thermal broadening of [O I] 5577 Å in the Earth's atmosphere or Th lines in hollow cathode lamps can produce this difference, the cause of which remains unknown to us.

We consider how our conclusions about $V_{\text{macrot}}$ depend on our choice of $\text{FWHM}_{\text{inst}}$ by calculating the difference in $V_{\text{macrot}}$ (5577) – $V_{\text{macrot}}$ (ThAr) as a function of phase. In Figure 23, we display these differences for the 12 RRab stars identified in

the figure legend. In this exercise, we used $10.86 \text{ km s}^{-1}$, the average for all measurements of [O I] 5577 Å, and $11.36 \text{ km s}^{-1}$, the average for all measurements of ThAr lines. The smaller $\text{FWHM}_{\text{inst}}$ derived from [O I] 5577 Å produces larger $V_{\text{macrot}}$ values. These differences in $V_{\text{macrot}}$, which vary with pulsation phase, are generally smaller than $0.4 \text{ km s}^{-1}$ near phase $\phi = 0.38$, the phase we use to set the upper limit on $V_{\text{rot}} \sin(i)$. This difference, $0.4 \text{ km s}^{-1}$, provides an independent estimate of the uncertainty in our upper bound of $5 \text{ km s}^{-1}$ for the $V_{\text{rot}} \sin(i)$ of MP RRab stars. We prefer the larger FWHM derived from the ThAr lines, because these lines are distributed more nearly like those of the metallic absorption lines that occur at random locations in all spectral orders.

(3) Finally, consider the narrow-lined HD 140283, our radial velocity standard. It has a measured $\text{FWHM}_{\text{meas}} = 12.15 \text{ km s}^{-1}$ that narrowly exceeds our adopted $\text{FWHM}_{\text{inst}} = 11.36 \text{ km s}^{-1}$. We calculate $V_{\text{macrot}} = 1.6 \text{ km s}^{-1}$ for HD 140283 by use of
<figure><img src="image_24.png" /><figcaption>**Figure 22.** The FWHM<sub>meas</sub> of the night sky [O I] 5577 Å emission line plotted vs. its EW in units of normalized stellar fluxes.</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>**Figure 23.** Increases in $V_{\text{macrot}}$ if FWHM<sub>inst</sub> is reduced from 11.36 to 10.86 km s<sup>-1</sup>, plotted as a function of pulsation phase for 12 RRab stars.</figcaption></figure>

these FWHM values, $T_{\text{eff}} = 5750$ K, and $V_{\text{mic}} = 1.50$ km s<sup>-1</sup>. Collet et al. (2009) used their 3D hydrodynamic code to derive $V_{\text{rot}} \sin(i) = 2.58$ km s<sup>-1</sup> for HD 140283. The difference between their value and ours, 1.0 km s<sup>-1</sup>, is consistent with our estimate of the uncertainty in our calculations, and it is unclear which of these two values is to be preferred.

## ORCID iDs

Christopher Sneden https://orcid.org/0000-0002-3456-5929

## References

*   Beers, T. C., Preston, G. W., & Shectman, S. A. 1992, AJ, 103, 1987
*   Behr, B. B. 2003a, ApJS, 149, 67
*   Behr, B. B. 2003b, ApJS, 149, 101

*   Bernstein, R., Shectman, S. A., Gunnels, S. M., Mochnacki, S., & Athey, A. E. 2003, Proc. SPIE, 4841, 1694

*   Bersier, D., & Burki, G. 1996, A&A, 306, 417

*   Breitfellner, M. G., & Gillet, D. 1993, A&A, 277, 553

*   Carney, B. W., Gray, D. F., Yong, D., et al. 2008, AJ, 135, 892

*   Castellani, V., Chieffi, A., & Pulone, L. 1991, ApJS, 76, 911

*   Chadid, M., & Gillet, D. 1996, A&A, 315, 475

*   Chadid, M., & Preston, G. W. 2013, MNRAS, 434, 552

*   Chadid, M., Sneden, C., & Preston, G. W. 2017, ApJ, 835, 187

*   Collet, R., Asplund, M., & Nissen, P. E. 2009, PASA, 26, 330

*   Cortés, C., Silva, J. R. P., Recio-Blanco, A., et al. 2009, ApJ, 704, 750 de Santis, R., & Cassisi, S. 1999, MNRAS, 308, 97

*   Dorman, B. 1992, ApJS, 81, 221

*   Dotter, A., Chaboyer, B., Jevremović, D., et al. 2007, AJ, 134, 376

*   Fokin, A. B., Gillet, D., & Chadid, M. 1999, A&A, 344, 930

*   For, B.-Q., Preston, G. W., & Sneden, C. 2011a, ApJS, 194, 38

*   For, B.-Q., Sneden, C., & Preston, G. W. 2011b, ApJS, 197, 29

*   Grassitelli, L., Fossati, L., Langer, N., et al. 2015, A&A, 584, L2

*   Grassitelli, L., Fossati, L., Langer, N., et al. 2016, A&A, 593, A14
Gray, D. F. 2008, The Observation and Analysis of Stellar Photospheres (Cambridge: Cambridge Univ. Press)

Grundahl, F., Catelan, M., Landsman, W. B., Stetson, P. B., & Andersen, M. I. 1999, ApJ, 524, 242

Hosford, A., Ryan, S. G., García Pérez, A. E., Norris, J. E., & Olive, K. A. 2009, A&A, 493, 601

Kinman, T., Castelli, F., Cacciari, C., et al. 2000, A&A, 364, 102

Kolmogorov, A. 1941, DoSSR, 30, 301

Lee, Y.-W., & Demarque, P. 1990, ApJS, 73, 709

Lucatello, S., & Gratton, R. G. 2003, A&A, 406, 691

Melbourne, W. G. 1960, ApJ, 132, 101

Michaud, G., Tarasick, D., Charland, Y., & Pelletier, C. 1983, ApJ, 269, 239

Moore, C. E., Minnaert, M. G. J., & Houtgast, J. 1966, The Solar Spectrum 2935 A to 8770 A, National Bureau of Standards Monograph (Washington, DC: US GPO)

Peterson, R. C., Carney, B. W., & Latham, D. W. 1996, ApJL, 465, L47

Peterson, R. C., Rood, R. T., & Crocker, D. A. 1995, ApJ, 453, 214

Preston, G. W. 1997, AJ, 113, 1860

Preston, G. W. 2011, AJ, 141, 6

Preston, G. W., & Chadid, M. 2013, EAS, 63, 35

Preston, G. W., & Sneden, C. 2000, AJ, 120, 1014

Preston, G. W., Sneden, C., Thompson, I. B., Shectman, S. A., & Burley, G. S. 2006, AJ, 132, 85

Quievy, D., Charbonneau, P., Michaud, G., & Richer, J. 2009, A&A, 500, 1163

Recio-Blanco, A., Piotto, G., Aparicio, A., & Renzini, A. 2004, A&A, 417, 597

Richardson, L. F. 1920, RSPSA, 97, 354

Roederer, I. U., Preston, G. W., Thompson, I. B., et al. 2014, AJ, 147, 136

Rood, R. T. 1973, ApJ, 184, 815

Ruediger, G. 1989, Differential Rotation and Stellar Convection. Sun and the Solar Stars (Berlin: Akademie Verlag)

Sills, A., & Pinsonneault, M. H. 2000, ApJ, 540, 489

Simkin, S. M. 1974, A&A, 31, 129

Skarka, M. 2014, MNRAS, 445, 1584

Sneden, C. 1973, ApJ, 184, 839

Sneden, C., Preston, G. W., Chadid, M., & Adamów, M. 2017, ApJ, 848, 68

Sneden, C., Preston, G. W., Kollmeier, J. A., et al. 2018, AJ, 155, 45

Soubiran, C., Le Campion, J.-F., Brouillet, N., & Chemin, L. 2016, A&A, 591, A118

Stellingwerf, R. F. 2013, arXiv:1310.0535

Stift, M. J., & Gillet, D. 1994, in ASP Conf. Ser. 64, Cool Stars, Stellar Systems, and the Sun, ed. J.-P. Caillault (San Francisco, CA: ASP), 735

Sweigart, A. V. 1987, ApJS, 65, 95

Szczygiel, D. M., & Fabrycky, D. C. 2007, MNRAS, 377, 1263

Tanner, J. D., Basu, S., & Demarque, P. 2013, ApJ, 778, 117

Théado, S., Alecian, G., LeBlanc, F., & Vauclair, S. 2012, A&A, 546, A100

Tonry, J., & Davis, M. 1979, AJ, 84, 1511

Tonry, J. L., & Davis, M. 1981, ApJ, 246, 666

VandenBerg, D. A., Denissenkov, P. A., & Catelan, M. 2016, ApJ, 827, 2

Wildey, R. L., Burbidge, E. M., Sandage, A. R., & Burbidge, G. R. 1962, ApJ, 135, 94

Woolley, R., & Stibbs, D. W. N. 1953, The Outer Layers of a Star (Oxford: Clarendon)