# HYDROGEN AND HELIUM SHOCK PHENOMENA DURING RISING LIGHT IN RR LYRAE
FUNDAMENTAL MODE PULSATORS

George W. Preston  Carnegie Observatories, 813 Santa Barbara Street, Pasadena,
CA 91101, USA<EMAIL_ADDRESS>Christopher Sneden
Department of Astronomy and McDonald Observatory, The University of Texas,
Austin, TX 78712, USA<EMAIL_ADDRESS>Merieme Chadid  University
of Côte d’Azur, Nice Sophia–Antipolis University, CNRS–UMR 7250, CS 34229,
06304 NICE Cedex 4, France<EMAIL_ADDRESS>
###### Abstract

We present measurements of H and He emission and absorption lines produced in
RRab fundamental mode pulsators during primary light rise. The lines define
universal progressions of rise and decay in metal-poor RRab stars. Such a
progression cannot be constructed for He in metal-rich RRab (those with [Fe/H]
$>$ $-$0.8) because weak He i emission is detected in only two of the six
metal-rich RRab in our survey. Great variety exists in the phase variations of
the blue- and red-shifted absorption components of the 5876 Å line during pre-
and post-emission phases. Detection of measurable He ii 4686 Å emission in
eight RRab, three of them Blazhko variables, provides an additional constraint
on ionization of Helium.

††facilities: Du Pont (echelle spectrograph)††software: IRAF (Tody 1986, Tody
1993), SPECTRE (Fitzpatrick & Sneden 1987)

##  1 INTRODUCTION

Two weakly-visible H$\alpha$ emission episodes attributed to shock phenomena
occur in RRab stars during declining light. The first, dubbed the “third
apparition” (Preston, 2011), is identified by a weak red-shifted H$\alpha$
emission wing visible during about one third of the pulsation cycle following
maximum light. Ad hoc explanations of this apparition have been advanced by
Chadid & Preston (2013) and Gillet et al. (2017, 2019). A second shock,
identified by violet-shifted H$\alpha$ emission during the “bump” preceding
minimum light (Gillet & Crowe, 1988), is discussed at length in the foregoing
references. The existence of this shock was anticipated theoretically by the
calculations of Hill (1972). Both emissions are accompanied by absorption
line-doubling produced in outflowing and infalling atmospheric layers.

Chadid et al. (2014) detected multi-shocks propagating through the atmosphere
of RR Lyrae stars. They identified new light curve properties111 nicknamed
$jump$, $lump$, $rump$, $bump$, and $hump$, as introduced by Christy 1966 and
expanded by Chadid et al. 2014 induced by five shock waves, with different
amplitudes and origins. Recently Prudil et al. (2020) have used broad band
photometry to identify (hump) shock phenomena in a large sample of field RRab.
Metallic-line absorption phenomena, particularly shock-induced increases in
macroturbulence during rising in stable RRab, were reviewed by Preston (2009),
and studied in great detail more recently by Preston et al. (2019).

This paper deals with strong shock phenomena (“first apparition”, the hump)
during primary light-rise, discovered spectroscopically by Sanford (1949) and
interpreted soon thereafter by Schwarzschild (1952). All previous discussions
of this primary shock in RR Lyrae stars have been based solely on the
behaviors of Balmer emission and absorption lines in the context of the
Schwarzschild model. Wallerstein (1959) explored shock conditions in the Pop
II Cepheid W Virginis. He calculated, by taking into account the energy of
ionization in application of the Rankine-Hugoniot relations, that a shock
velocity higher than 80 km s-1 is able to ionize Hydrogen and neutral Helium
and, additionally, a small fraction of singly-ionized Helium. This condition
is fulfilled by typical RRab stars (to be discussed in §3.1.3), so it came as
no great surprise that Preston (2009, 2011) detected He i emission and
absorption lines during light-rise in 10 RRab stars and very weak He ii
4685.68 Å emission in 3 RRab stars. These H and He emissions, produced in the
wakes of shock waves, can provide quantitative insight into the stratification
of the outflowing and infalling atmospheric layers.

In this paper we present new information about the time evolution of the
pulsation velocities, equivalent widths, and line-of-sight broadening of the
emission and absorption of the most prominent lines of H, He i and He ii in
our spectra. The format of the paper is as follows. In §2 we describe the data
set; in §3 we present Hydrogen and Helium emission and absorption line
measurements; and in §4 we discuss issues raised by the behavior of the He i
$\lambda$5876 line. §5 contains our concluding remarks.

##  2 THE SPECTROSCOPIC DATA SET

Figure 1:  Variations of H$\alpha$, He i 5875.62 Å, and He i 6678.15 Å, in the
rising light shock phases of RV Oct. Dotted lines indicate the rest
wavelengths of these lines as defined by the metallic lines. The phases are
written in the middle panel.  Figure 2:  Mean line profiles of H$\alpha$, four
He i lines and a He ii line at the phase of maximum He emission in program
stars RV Oct and X Ari.

###  2.1 Observations and Reductions

For this study we mine the database of several thousand echelle spectra of RR
Lyrae stars that were obtained with the du Pont 2.5 m telescope at Las
Campanas Observatory between 2006 and 2014. The resolving power of these
spectra is $R\equiv\ \lambda/\delta\lambda$ = 27,000 at 5000 Å. We concentrate
on a subset of stars analyzed by Chadid et al. (2017) that have many spectra
in the phase range $\phi$ = 0.8$-$1.1. In Table 1 we list the observed
photometric pulsational properties and metallicities of the RR Lyraes
considered in the present work. The pulsational data are taken from Chadid et
al. Table 1. The adopted metallicities are means of the [Fe i/H] and [Fe ii/H]
values (with assumed log $\epsilon$(Fe)⊙ = 7.50) listed in Table 2 of that
paper.

RR Lyrae stars pulsate with periods mostly confined to the range 0.25 d $<$ P
$<$ 0.75 d. Target exposure times, limited to small fractions of these
periods, never exceeded 600 s ($\sim$0.01P), which resulted in relatively poor
signal-to-noise ratios in the blue spectral region, S/N $\sim$ 15$-$20 for
most of the stars in our sample. This created challenges for derivation of
photospheric quantities ($T_{\rm eff}$, log g, $V_{mic}$, [Fe/H] metallicity,
[X/Fe] abundance ratios) from these spectra, because the great majority of
detectable metallic lines in RR Lyrae stars occur at wavelengths less than
5000 Å. On the other hand, the H and He transitions used in this investigation
lie in the yellow and red region, $\lambda$ $>$ 5000 Å, where our spectra have
much higher $S/N$ $\sim$ 50$-$80.

The du Pont echelle observations were reduced to final spectra as described in
detail by For et al. (2011a). These spectra have been used in multiple papers
since that time. Therefore the reduction procedures will not be repeated here.
The individual du Pont multi-order wavelength calibrated spectra of all stars
that we obtained in the 2006-2014 time period, whether or not they have been
used in the present paper, can be obtained at the Zenodo data depository222
https://zenodo.org/record/5794389#.YcD3XBPMIo8

###  2.2 The Transitions

In Figure 1 we extend Figure 1 of Preston (2009) by showing the variation with
phase near maximum light of RV Oct for H$\alpha$ and the two He i lines with
strongest emission, 5875.62 Å and 6678.15 Å. The rapid appearance and fadeout
of He i emission lines are illustrated in the sequence of RV Oct spectra in
the phase range 0.66 $\lesssim$ $\phi$ $\lesssim$ 1.00. Discussion of detailed
H$\alpha$ phase variations has been included in previous papers in this series
for RRab stars (Chadid et al., 2017) and for RRc (Sneden et al., 2017). For RV
Oct the He i emission occurs only in the narrow phase range $\phi$ =
0.92$-$0.94, and we will show in §3.1.3 that a similarly narrow phase range
for He emission occur in all of our RRab stars.

Inspection of the He i lines for the three earliest phases ($\phi$ = 0.856,
0.893, 0.912) in the right-had panels of Figure 1 reveals apparent He
absorption in the 6678 Å line, but neither emission nor absorption in the 5876
Å line. The absorption seen at 6678 Å throughout most of the pulsation cycle
is, in fact, due to Fe i not He i. Most lines of Fe i in the red spectral
region of Fe i are very weak, but multiplet 268 (Moore, 1972) is the exception
with three strong lines: 6546.25, 6592.92, and 6677.99 Å. These lines have
similar transition probabilities (Kramida et al., 2019)333
https://physics.nist.gov/PhysRefData/ASD/lines_form.html and have substantial
absorptions in the solar spectrum (Moore et al., 1966). For $\lambda$6546 log
$gf$ = $-$1.54 and EW⊙ = 103 mÅ, somewhat blended with a weak Ti i line; for
$\lambda$6592 log $gf$ = $-$1.47 and EW⊙ = 123 mÅ; and for $\lambda$6678 log
$gf$ = $-$1.42 and EW⊙ = 122 mÅ. In the RV Oct phases shown in Figure 1 we
always detected the $\lambda$6592 line and tentatively identified
$\lambda$6546 above the continuum noise. However, the maximum-light phases
under study here also have the highest photospheric temperatures. For RV Oct
and other RRab stars, $\langle\mbox{$T_{\rm eff}$}\rangle$ $\sim$ 6150 K at
$\phi$ $\sim$ 0.85 but rises sharply in about an hour to $\langle\mbox{$T_{\rm
eff}$}\rangle$ $\sim$ 7150 K at $\phi$ $\sim$ 0.95 (For et al., 2011b).
Neutral-species transitions such as those of Fe i weaken with rising
temperature. If we examine the RV Oct spectrum at earlier phases, all three
lines are easily detected. For our spectrum at $\phi$ = 0.370 ($T_{\rm eff}$
$\sim$ 6100 K, For et al. 2011b), the equivalent widths are $EW$ = 27 mÅ for
$\lambda$6546, $EW$ = 35 mÅ for $\lambda$6592, and $EW$ = 44 mÅ for
$\lambda$6678\. All of these lines weaken to $EW$ $\lesssim$ 5 mÅ at $\phi$
$\sim$ 0.9, but our conclusion holds: the $\lambda$6678 absorption is due to
Fe i, not He i.

Preston (2009) showed spectra of the He i $\lambda$5876 in emission during the
rising-light phases of RV Oct, but also argued that four other He emission
lines could be seen in these phases. In Figure 2 we show the presence of
multiple He i lines in RV Oct and the very metal-poor X Ari, and one He ii in
both stars as well. To assist in these identifications, we co-added individual
observations with maximum $\lambda$5876 emission, five of them for RV Oct and
eight for X Ari, thus substantially increasing their $S/N$ values. The spectra
that were combined were obtained essentially at a single phase for each star.
Qualitatively the emission strengths are approximately the same in both stars.

Figure 3:  A partial Grotrian diagram for He i. Transitions detected by
Preston (2009) are in red, with the wavelengths labeled. No permitted dipole
transitions connect the triplet and the ground states.

The detected He i lines in RR Lyraes arise from very high excitation states of
two multiplet systems. In Figure 3 we show a partial Grotrian diagram for this
species, modeled after Moore & Merrill (1968), using data from the NIST atomic
line database (Kramida et al., 2019)444
https://physics.nist.gov/PhysRefData/ASD/lines. The very high excitation
energies of the 21P and 23P He i transitions ensure that these lines detected
in RR Lyrae stars cannot be formed in the same atmospheric layers that give
rise to the metallic lines with typical excitation energies $\sim 1-3$ eV.

We omitted the 7281.48 Å He i from Figure 2. This line is extremely weak in RV
Oct and not obviously present in X Ari. Additionally, the $\lambda$7280
spectral region is contaminated with many telluric H2O features that were not
cancelled in our reduction procedures, so we drop it from further
consideration here. However, we should note that Zhang et al. (2005) found
this transition most useful in constructing temperature diagnostics for
planetary nebulae. We suggest that measures of He i $\lambda$7281 in spectra
with S/N higher than ours may prove useful in analysis of RRab shocks as well.

Figure 4:  Two representative H$\alpha$ lines near emission maximum. Top
panel: the H$\alpha$ profile in X Ari at phase $\phi$ = 0.909 in one of its
pulsational cycles (black line). A Gaussian profile is fitted to the violet
wing of the observed emission (blue curve) ignoring the broad violet and red
wings present in most stars near emission maximum. A second fit to the broad
wings is plotted as a red curve. Bottom panel: the same Gaussian fit (blue
curve) is made to the observed H$\alpha$ profile for the more typical case at
a later phase, $\phi$ = 0.928, when there is no red emission component to
constrain the Gaussian velocity center. Because of the general absence of this
constraint, we adopted the photospheric velocity defined by metal lines to
locate the centroids of all Gaussian fits.

##  3 MEASUREMENTS OF EMISSION AND ABSORPTION LINES

###  3.1 The Emission Lines

The onset of observable shock waves in RRab stars always begins with the
sudden, simultaneous appearance of H and He emission lines. These rise to
maximum strength on time scales of minutes and decline on similar but slightly
different timescales. They are followed by the appearance of violet- and red-
displaced absorption components.

####  3.1.1 H$\alpha$ Emission Equivalent Widths

We began with equivalent width (EW) measurements of the emission components of
Balmer H$\alpha$ (6562.81 Å). Measurements of EWs and wavelengths of the
Balmer lines are complicated by the fact that the emission profiles are
overlain (thus partially masked) by strong red-shifted absorptions of gas
infalling from the previous pulsation cycle.

We measured the H$\alpha$ emission EWs in two ways: first by direct
integration of emission parts of the observed features, and second by modeling
the Gaussian pure emission profiles that best match the blue wings of the
observed profiles. Our procedures are illustrated in Figure 4 for program star
X Ari at two phases near maximum emission.

Figure 5:  The correlation of H$\alpha$ emission equivalent widths measured by
direct integration and a Gaussian model approximation, as defined in the text.
Data between dotted lines show that Gaussian model EWs exceed measured values
by a factor of 2.

To effect direct integrations, we measured EWs of the violet emission fluxes
in pixel ranges chosen by use of the IRAF (Tody, 1986, 1993)555
http://ast.noao.edu/data/software $splot$ e-cursor option. In Figure 4 for
example, these ranges are $\sim$6557.0$-$6562.5 Å (top panel) and
6560.0$-$6562.5 Å (bottom panel).

For measurements of the Gaussian profiles we modeled the emission with
Gaussians that best match the observed H$\alpha$ emission wings. Following
Chadid & Preston (2013) we adopted the photospheric velocity defined by metal
lines for the centers of our Gaussian fits. Then, as illustrated in Figure 4,
we were presented with options for the best Gaussian approximations for the
emission wings. Inspection of the observed H$\alpha$ in the top panel reveals
that a single Gaussian cannot be fitted to the entire emission part of the
total profile. In this case we fit a Gaussian to the violet component of X Ari
at phase $\phi$ = 0.909, ignoring the broad violet and red extensions, and a
second fit to the broad extensions. The extended emission wings are a common,
perhaps ubiquitous, characteristic of the H$\alpha$ emission profiles of RRab
stars near emission maximum. They disappear at later phases, as illustrated in
the Figure 4 bottom panel for X Ari at $\phi$ = 0.928 . The blue Gaussian
emission wing had to suffice in the absence of a red emission component.

Figure 6:  Top panel: H$\alpha$ emission EWs measured via direct integration
plotted versus phases. For each star $\phi$ = 0.00 was set to be the phase of
minimum radial velocity. The red points for RR Cet and the green points for RV
Oct have been connected with solid lines of their colors. Bottom panel: the
emission EWs scaled to a common flux maximum and shifted to a common emission
peak phase; see text for descriptions of these H$\alpha$ emission
manipulations. Here the blue line represents the mean trend of scaled emission
versus shifted phase. The symbols and colors of points are identified in this
panel’s legend and are consistent with those of Figure 5.

We compare our emission EWs from direct measurement with those derived from
Gaussian model profiles in Figure5. The two sets are clearly correlated, but
there are systematic offsets between measures of different stars. Compare for
example those of WY Ant (cyan-filled circles) and X Ari (black solid circles)
in this figure. However, for the bulk of strong emissions (equivalent widths
$EW$ $>$ 1 Å) the Gaussian model profile values exceed direct integration ones
by a factor of $\sim$2, as shown by the solid line drawn through the data. The
correlation is strongest for the strongest H$\alpha$ emission features, i.e.,
those contained between the dotted-lines in Figure 5 (data for WY Ant
excepted). Weaker emissions are more difficult to measure primarily because of
uncertainty in proper continuum location. The systematic offsets and larger
scatters in the regression for smaller EWs is evident in the figure.

The linear correlation between emission EWs derived from direct integration
and from Gaussian model fits yields confidence that H$\alpha$ relative
emission strengths can be derived from the observed line profiles. For the
remainder of this paper we will employ the EWs from direct integration only,
as these measurements involve fewer assumptions than those from Gaussian
integration.

####  3.1.2 Phase variation of H$\alpha$ emission

The emission appears abruptly, rises to maximum on a timescale of a few
minutes, and declines on a similar timescale, as illustrated in Figure 6 for
seven RRab stars with the strongest emission in our sample. At first sight the
data in the top panel seem to form a hopeless jumble. But inspection quickly
reveals that the jumble is caused largely by systematic star-to-star phase
shifts. This is illustrated by lines connecting the data for RR Cet (red) and
RV Oct (green). The points for these two stars can be superposed in phase by
applying mean shifts of $+$0.0060 and $-$0.0095 to the RR Cet and RV Oct,
respectively.

Figure 7:  Measured EWs (top panels) and scaled emission EWs shifted to phase
= 0.92 (bottom panels) for He i 5876 Å and 6678 Å in the manner employed for
H$\alpha$ in Figure 6. The symbols and colors of points are identified in
panel (d) and are consistent with those of Figure 5

By trial and error we found phases shifts, $\Delta\phi$, that align all
emission maxima at an arbitrarily adopted phase $\phi$ = 0.92. These phase
shifts are listed in Table 2. Additionally, we used the absolute fluxes of
Kurucz (2011, 2018)666 Available at http://kurucz.harvard.edu/grids.html model
atmospheres to convert measured fluxes in the du Pont echelle orders to fluxes
relative to the continuum flux at H$\alpha$, all scaled to the average value
of these measurements in the phase interval 0.915 $<$ $\phi\ +\Delta\phi$ $<$
0.925. We applied these flux normalizations and small phase shifts to the
emission phase data for the seven stars in the top panel of Figure 6 to
produce the much more coherent flux versus phase variation in the bottom
panel. A blue-colored line in this panel represents the mean trend of these
data. The small scatter of individual points from this line, $\sim\pm$0.002 in
shifted phase, $\sim\pm$0.1 in scaled EW, strongly suggests that this is a
standard feature in H$\alpha$ behavior near maximum light in metal-poor RR
Lyraes.

####  3.1.3 Helium i Emission

Figure 8:  Phases that produce alignment of emission maxima for He i 5876 Å
are plotted versus those for H$\alpha$. The two shifts are strongly correlated

Identifications of five emission lines of He i were reported by Preston (2009)
in du Pont echelle spectra of RV Oct during rising light. We restrict
attention to the leading lines of the series (3P0 $-$ ${}^{3}D$), 5876 Å, and
(1P0 $-$ 1D), 6678 Å, because they are the only lines measurable in
substantial numbers of the spectra obtained in our survey. The behaviors of
these two lines are much simpler than those of Balmer lines. Because of their
much larger excitation energies ($>$20 eV), the He i series do not produce
detectable absorption by infalling gas during the initial emission line
phases. This simplicity is clearly illustrated in the middle panels of Figure
1 (phases $\phi$ = 0.927, 0.928) above. Therefore, flux and velocity
measurements can be made without regard to the complications encountered above
for H$\alpha$.

Emission EW measurements of 5876 Å and 6678 Å were made in the same manner as
described in §3.1.2 for H$\alpha$. Peak EWs for these He i lines are smaller
than that of H$\alpha$ by factors of 6 and 12, respectively, so measurements
were limited to a smaller range of phases, and measurement accuracies were
reduced accordingly.

In Table 2 we list the estimated phase shifts $\Delta\phi$ for He i 5876 Å
needed to construct Figure 7 (b). We emphasize that the phase shifts
$\Delta\phi$ for He i 5876 Å emissions were determined independently from
those derived for H$\alpha$ described in §3.1.2. Although both lines arise in
the recombination wake of the shock, the two line forming regions cannot be
identical because of the large difference in the ionization potentials
($\sim$11 eV) of H i and He i. Figure 8 shows that the two shifts are strongly
correlated, but the slope of the regression is less than unity.

####  3.1.4 Helium II $\lambda$4686 Emission

Figure 9:  Spectra of the He ii 4686 Å region of eight RRab stars arranged
bottom to top in order of increasing [Fe/H]. The relative flux values for RV
Oct are correct, and the spectra for the other stars are shifted vertically
for display purposes. The average spectrum shown in blue at the top of the
plot was formed by a straight mean of the indivdual spectra. Dotted vertical
lines are located at the rest wavelengths of the He ii line and the
photospheric Mg i line.

As noted in §2.2, He ii 4686 Å 32D$-$42Fo is analogous to H i Paschen
$\alpha$. Preston (2009) reported weak He ii 4686 Å emission in three Blazhko
variables (AS Vir, UV Oct, and V1645 Sgr) observed during large amplitude
cycles when their metallic-line radial velocity amplitudes were 65, 70, and 64
km s-1, respectively.777 Details of the radial velocity measurement procedure
may be found in Preston & Sneden (2000) and Chadid et al. (2017). These
emission lines together with those of five stable RRab with similar RV
amplitudes are displayed in Figure 9. The average spectrum formed from these
spectra at the top of the figure shows unambiguous detection of the He ii
line. The largest measured equivalent widths in Figure 9, i.e., those for RV
Oct and X Ari, are $\sim$30 mÅ.

By converting average Blazhko RV amplitude for AS Vir, UV Oct, and V1645 Sgr,
66.3km s-1, to pulsation amplitude, dR/dt888 We derive the pulsational
velocity dR/dt in the stellar rest frame using formula (1) in Chadid et al.
(2017), as described in their §4.3 . with projection factor, $p$ = 1.3, we
obtain an estimate of a critical pulsation velocity amplitude, 82 km s-1,
above which observable $\lambda$4686 He ii emission occurs in our spectra.

We chose projection factor $p$ = 1.3 for calculation of dR/dt after review of
the $p$-values, derived by a variety of methods, that are summarized in Figure
5 of Neilson et al. 2012). These $p$-values, all sensitive to assumptions
about limb darkening and atmospheric stratification, range from 1.2 to 1.4. We
adopt a conservative one decimal-place value for $p$ that reflects our
uncertainties about RRab atmospheric parameters, particularly during rising
light.

Finally, strong absorption due to Mg i at $\lambda$4703 illustrates the
seemingly paradoxical occurrence of lines of He ii (IP = 54.4 eV) and Mg i (IP
= 7.6 eV) in the same spectra. This circumstance requires pronounced
temperature stratification in the shock wake, to be discussed in a work in
progress.

###  3.2 Absorption Lines

Immediately after the phase of maximum emission, red-shifted absorption lines
of H and He begin to appear in most metal-poor RRab spectra. This section is
devoted to a description of their behavior.

####  3.2.1 H$\alpha$ Absorption

Figure 10:  The variations of H$\alpha$ pulsation velocity dR/dt (km s-1),
equivalent width $EW$ (mÅ), and Maxwell line-of-sight velocity dispersion
$\sigma_{Max}$ (km s-1) with phase during rising light for six RRab stars. Red
and blue symbols are measures of lines produced by infalling and out-flowing
atmospheric layers, respectively.

We measured three quantities useful for discussion of atmospheric structure
during rising light: radial velocities (RV km s-1), equivalent widths (EW Å),
and Doppler widths ($\sigma_{Max}$ km s-1). To accomplish this task we used
IRAF/splot with the “d” cursor option. $\sigma_{Max}$ is the Maxwellian line-
of-sight velocity dispersion defined by equation 1 of Unsöld & Struve (1949)
We display the phase variations of these quantities for six RRab stars in the
montage of Figure 10. Values of the pulsation velocities dR/dt and velocity
dispersions $\sigma_{Max}$ of the H$\alpha$ absorptions at phase $\sim$0.97,
when they first become measurable, are given in the top half of Table 3.

The average H$\alpha$ velocity difference between infall and outflow in column
4 of Table 3 is about twice the range of the continuous velocity variation of
the metal lines. Radial velocity is a location marker in the atmosphere. Metal
lines are formed in the region where continuum optical depth is $\tau$ $\sim$
0.1$-$0.2, near the photosphere ($\tau$ $\sim$ $\frac{2}{3}$), while the
heavily saturated cores of H$\alpha$ produced in the outflowing and infalling
gas must be formed high in the atmosphere at very low continuum optical
depths. The large persistent differences between H$\alpha$ and metal
velocities when both are measured during rising light signal the existence of
velocity gradients in the atmosphere detected by Chadid & Gillet (1996) and
discussed by Bono et al. (1994) and Fokin & Gillet (1997). These gradients
produce line asymmetries that will be the subject of a paper in preparation.

The regression coefficients used to construct the phase variations in Figure
10 for dR/dt and $\sigma_{Max}$ are given in Table 4. We chose linear
equations for all regressions with one exception: the large $S/N$ values of
the $\sigma_{Max}$ data for X Ari permitted use of a quadratic fit. We see a
tendency for maxima in $\sigma_{Max}$ near phase 0.97 in the outflows of the
other five stars, so we surmise that this quadratic behavior may well be a
general characteristic of the MP stars that is best studied with higher $S/N$
than that available in our spectra.

Two H$\alpha$ absorption components appear near phase 0.96 and remain
measurable after maximum light for durations that vary greatly from star to
star. The average difference between infall and outflow velocities,
characteristically $\sim$165 km s-1 near maximum light, differs little from
star to star in our sample. This similarity is due at least in part to a
selection effect created by our choice of RRab stars with strong H$\alpha$
emission.

The EWs of the outflow (B) and infall (R) components are approximately equal
($\sim$1.5 Å) at phase 0.97 for all six stars discussed here. The outflow
values continue to increase for some 0.02 cycles past RV minimum. Thereafter,
they remain constant or decline slightly. The infall EWs decline continually
after their first appearance, more slowly after phase 0.02, and they generally
persist beyond phase 0.10. Remarkably, infall absorption for RV Oct can be
measured to phase 0.2.

The line-of-sight velocity dispersions for H$\alpha$ are residual values after
removal of instrumental broadening by Gaussian deconvolution, and much smaller
effects due to thermal broadening, microturbulence and the Stark effect. We
discuss the (lack of) a significant contribution by the Stark effect to our
measured line widths in §3.2.3.

Outflow dispersions at phase 0.97 are typically $\sim$90 km s-1, then they
generally increase slightly until phase of RV minimum, and remain constant
thereafter until the lines disappear. Infall dispersions at phase 0.97 are
smaller ($\sim$62.0 $\pm$ 5.7 km s-1), decline slowly and remain approximately
constant at $\sim$50 km s-1 until they disappear. W Tuc is an apparent
exception. Our measures of its infall dispersion yield a constant value of
72.0 $\pm$ 6.9 km s-1 in the phase interval 0.97 $<$ $\phi$ $<$ 1.10.

####  3.2.2 He i Absorption

Figure 11:  (left) The variations of He i pulsation velocity dR/dt (km s-1),
(middle) equivalent width $EW$ (mÅ), and (right) velocity dispersion
$\sigma_{Max}$ (km s-1) with phase during rising light for the same six RRab
stars shown in Figure 10. Red and blue symbols are measures of lines produced
by infalling and outflowing atmospheric layers, respectively. Unfilled dark
green symbols denote emission features.  Figure 12:  H$\alpha$ and He i
pulsation velocities from the left-hand windows of Figures 10 and 11 combined
in one panel each for the six stars. The symbol colors are labeled in the
panel for RR Cet. Squares are used for the H$alpha$ data; open squares denote
those measurements made with IRAF $fxcor$, and filled squares for those made
with IRAF $splot$. Data points for He 5876 Å in this Figure represent more
data than do those in Figures 10 and 11 because of co-addition.  Figure 13:
The distance D traversed by the primary shock from the metallic line-forming
region just above photosphere to the layer that produces out-flowing H$\alpha$
absorption. In the left panel D is expressed in units of km, and in the right
panel D is shown in units of stellar radii. Data for six stars are taken from
Table 13. Black lines show the mean trend of the data from a linear
regression. Red lines show radii for Oo I and II groups taken from Table 1 of
Marconi et al. (2005).

We computed the same absorption quantities dR/dt (km s-1), $EW$ (mÅ), and
$\sigma_{Max}$ that were derived for H$\alpha$. These are displayed in Figure
11. Because He i 5876 Å is much weaker than H$\alpha$ we co-added files into
phase bins of 0.01P for this figure. Emission features are included in the
regression for dR/dt to illustrate the continuity with outflowing absorption
that is expected, because the emission and blue absorption features must arise
in the wake of the shock.

The small values of dR/dt for He $\lambda$5876, similar to those for the metal
lines in the early expansion phases (0.90 $<$ $\phi$ $<$ 0.95), surprise us
because the thermal conditions for the production of metal lines and He
$\lambda$5876 are so different. Likewise, the velocity dispersion at these
phases are also similar to those of the metals at these phases reported in
considerable detail by Preston et al. (2019). Note also the reappearance of He
$\lambda$5876 emission in the post maximum spectra of RV Oct, WY Ant, and X
Ari, first reported for RV Oct in Preston (2009). This reappearance is
discussed in §4 below.

We interpret the He $\lambda$5876 data in a preliminary way by superposing it
on the H$\alpha$ data for six stars in Figure 12. Integration of the He
$\lambda$5876 regression from the inception of emission until it reaches the
outflow velocity of H$\alpha$ produces an estimate of the distance traversed
by the shock from the phase of emergence from the photosphere to its arrival
at the level of formation of the H$\alpha$ absorption. The results of such
integration for our six stars are shown in Figure 13.

Although not important in the context of our investigation, we call attention
to the systematic differences between H$\alpha$ velocities measured with
IRAF/$splot$-$d$ and IRAF/$fxcor$ evident in Figure 12. These differences are
largest in the weak infalling (R) post-maximum components. $splot$-$d$ employs
a deblending procedure that attempts to recover the (Gaussian) central
wavelengths, EWs, and FWHMs of the B and R components by a best fit algorithm.
The $fxcor$ measurements rely on the measurer’s choice of pixels to include in
the $fxcor$ sampling of the blended B and R components. This choice is
difficult to make for the R component when it appears as a weak feature in the
red wing of a strong B component. Accordingly, we use the $splot$-$d$ results
in our discussion, including the $fxcor$ data in Figure 12 only as a warning
to those who might contemplate similar measurements.

####  3.2.3 Stark Broadening

After correcting measured FWHMs for instrumental, thermal and microturbulent
broadening by use of Gaussian subtraction we converted FWHM to Maxwellian
line-of-sight velocity dispersion as described in Preston et al. (2019) and
presented in Table 3 and in the right-most panels of Figures 10 and 11. Before
discussing these results we consider whether or not Stark broadening, a common
effect in stellar lines of H and He, contributes to the widths of the Doppler
cores of H$\alpha$ and He i $\lambda$5876 in RRab atmospheres, here taken to
have characteristic values of $T_{\rm eff}$ = 7,000 K and Ne = 1014 cm-3.

For H$\alpha$ Kowalski et al. (2017) calculate FWHM = 0.14 Å ($\sigma_{Max}$ =
5 km s-1) for the parameters ($T_{\rm eff}$ = 10,000 K and Ne = 1014 cm-3)
that they deem appropriate for solar and stellar flares. This value agrees
well with a three order-of-magnitude downward extrapolation of the regression
of Griem (1983) for T = 20,000 K used by Kielkopf & Allard (2014) as a fit to
laboratory data collected in the range 1017 $<$ Ne $<$ 1021. We conclude that
Stark broadening is an insignificant contributor to our $\sigma_{Max}$ values
for H$\alpha$ in MP RRab stars.

For He i $\lambda$5876 Å we refer to the regression used by Gigosos et al.
(2014) in their Figure 24b to fit data obtained in a number of laboratory
experiments. A small justifiable extrapolation of their regression to stellar
Ne values indicates that the Stark effect produces a negligible broadening of
this line at T $<$ 10,000 K, Ne = 1014 cm-3.

Accordingly, in the absence of any additional broadening processes, we
conclude that the $\sigma_{Max}$ values in Table 3 and Figures 10 and 11 are
reliable measures of line-of-sight velocity dispersions that occur in MP RRab
atmospheres during primary light rise.

##  4 THE NATURE, ORIGIN AND PERSISTENCE OF HE LINES IN RRab

Figure 14:  Line profiles of He i 5876 Å and H$\alpha$ for six metal-poor RRab
at four pulsation phases.  Figure 15:  Line profiles of He i 5876 Å and
H$\alpha$ for six metal-rich RRab at four pulsation phases.

Our discussion centers on Figures 14 and 15. In these figures dashed vertical
lines are plotted at the rest wavelengths of metal lines, which are formed in
the near-photosphere gas layers. We derived these rest wavelengths by cross-
correlation of metallic-line spectra in the wavelength region 4000$-$4600 Å,
using IRAF/fxcor as discussed by Preston & Sneden (2000). Masks removed Balmer
H$\delta$ and H$\gamma$, and the great majority of metallic absorption lines
that create the cross-correlation signal are those of Fe i with excitation
potentials between 1 eV and 3 eV. Dashed verticals at rest wavelengths closely
bisect the He i 5876 Å emission features in all stars of both figures, but
they lie slightly red-ward of the H$\alpha$ emission maxima as expected from
the considerations in §3.2.1. The weak He i lines in metal-rich stars conform
to expectations based on H$\alpha$ in Chadid et al. (2017). However, time-
evolution of the line profiles of He i 5876 Å and H$\alpha$ in these figures
pose several problems.

In previous investigations we divided our stars into metal-poor and metal-rich
abundance groups at [Fe//H] = $-$1, rounding to the nearest integer the
abundance boundary, $-$0.8, used by Zinn (1985) to separate disk globular
clusters from those of the halo. Here we revert to Zinn’s division point,
primarily because it places UU Vir among the metal-poor RRab, where it belongs
in Figures 14 and 15 and in the various correlations presented in Chadid et
al. (2017). That such precision in choice of the division point affects our
discussion strongly suggests, that the boundary is, in fact, blurred, i.e.,
the abundance distributions of halo and disk overlap. We reject the notion
that UU Virginis is a class sui generis.

Following expectations of Schwarzchild’s (1952) model, we locate the H and He
emissions in outward-flowing gas following its passage through a strong shock
(Wallerstein, 1959). The different appearances of the initial emission line
profiles of He and H arise as a circumstance imposed by infalling gas. H is so
abundant that it produces absorption in atmospheres of all stars, types O
through M. Infalling gas above the shocks will have temperatures that are
consistent with those implied by broad-band colors and metallic lines in the
spectra (T $\sim$ 6500, e.g., For et al. 2011b). This gas is responsible for
the strong red-ward displaced H$\alpha$ absorption. Because He i lines, all
with excitation potentials$>$20 eV (Figure 3), occur only in OB atmospheres
with $T_{\rm eff}$ $>$ 10,000K, He emission flux produced in the wake of the
shock is not distorted initially by absorption of infalling gas. Therefore
$\lambda$5876 is an ideal diagnostic for investigating the time-evolution of
shocks in RRab atmospheres.

However, issues arise with the evolution of He 5876 line profiles presented in
the top three spectra for each star in Figure 14. Stars in this Figure are
arranged in order of decreasing maximum emission flux (from DT Hya upper left
to UU Vir lower right). He absorption in the infalling gas appears in all six
stars at or soon after phase 0.92. It strengthens initially as emission
declines and it persists long after the observable emission disappears. This
behavior is particularly noticeable in the panels for X Ari, RV Oct, SS Leo,
and UU Vir.

The origin of infalling He i 5876 Å absorption presents a puzzle. The p3Po
level of He $\lambda$5876 is metastable: radiative excitation from the 1S
ground state to this state is forbidden, and in any event the lifetime of 23Po
is a mere 5.63 ms (Łach & Pachucki 2001; Hodgman et al. 2009). In contrast,
prominent $\lambda$5876 absorption (EW $\sim$ 50 mÅ) is present in infalling
gas throughout the phase interval 0.92 $<$ $\phi$ $<$ 1.00, persisting for
$\sim$1 hour after the phase of emission maximum as illustrated in Figures 11
and 14.

Collisional excitation cannot populate the 23Po level of infalling He because
this gas has not yet reached the shock. So, how is this level populated? The
only remaining possibility is recombination following ionization by a source
of “hot” radiation. However, the broad-band colors of these stars during
rising light of by Monson et al. (2017) provide no convincing evidence for
such a long-lived source of hot radiation that would have to be present during
all of the rising branch of the pulsation cycle and beyond.

Next, we call attention to an unexpected change in the He i 5876 Å profiles
that occurs during a few minutes near phase 0.97, intermediate between the
phases of the bottom two spectra in each panel of Figure 14. Such variations
of this line profile, first noted in spectra of RV Oct (Preston, 2009), are
reminiscent of the profile variations of Fe ii $\lambda$4923 encountered in
the Blazhko variable S Arae by Chadid et al. (2008), and attributed by them to
a hypersonic shock that develops high in the atmosphere. Our He i phenomenon
is seen most clearly by comparing the bottom two spectra of panels for X Ari,
RV Oct, SS Leo, and UU Vir of Figure 14. The symptoms of this phenomenon,
present in varying degrees in all RRab stars in our survey, are these:

  1. 1.

The profile suddenly broadens dramatically. The average radial velocity of the
red-shifted absorption for the stars in Figure 14 (DT Hya excepted) increases
from 60.3 $\pm$ 3.7 km s-1 to 105.0 $\pm$ 1.9 km s-1 while that for the
violet-shifted component decreases from $-$10.5 $\pm$ 8.0 km s-1 to $-$23.4
$\pm$ 6.5 km s-1. The extent to which the measured absorption velocities are
shifted from their true positions by putative emission wings is unknown.

  2. 2.

The velocity of the violet-shifted component is difficult to quantify
unambiguously because of its asymmetry; the violet wing RV Oct extends
approximately to an escape velocity, $\sim$200 km s-1, calculated for
canonical values of the mass (0.65 M⊙) and radius (5.5 R⊙).

  3. 3.

Weak ($\sim$2%) emission reappears between the absorption components. Chi-
square analysis confirms its reality in X Ari, RV Oct, and SS Leo. It is a
reality to be reckoned with.

Finally, we consider the incongruent properties of two stars lying together at
top-left in Figure 15.

  1. 1.

HH Pup, contrary to expectations due to its weak H and He emissions, has the
largest photometric (1.35 mag) and radial velocity (69.7 km s-1) amplitudes
and the largest primary acceleration in our entire spectroscopic sample
(Chadid et al., 2017). It also has the shortest pulsation period (0.39 d),
which it shares with AV Peg (also in the mid-right panels of Figure 15). It is
an archetypal example of the low latitude, short period RRab population first
identified by Kukarkin (1949) and subsequently as metal-rich by Preston
(1959), Layden (1994), and Layden et al. (1996).

  2. 2.

AN Ser, the most metal-rich star in our sample, [Fe/H] = +0.04, has by far the
longest photometric period (0.52 d) of the metal-rich RRab. It belongs neither
to Oosterhoff’s type I metal-poor cluster population, nor to Kukarkin’s short-
period low-latitude metal-rich population. Note that DX Del (P = 0.47 d,
[Fe/H] = $-$0.44) in the right column of Figure 15 is a cousin of HH Pup.

Location of HH Pup, AN Ser, and DX Del in a proper evolutionary sequence of
stellar populations is a challenge for the future. We continue to seek
explanations of results presented above. We defer our exploration of the
systematically weaker H and He emission in metal-rich RRab to a paper in
preparation.

##  5 SUMMARY

We report quantitative data about spectral features of H$\alpha$, He i, and He
ii measurable with spectra obtained during the course of an 8-year survey of
some three dozen RRab stars with representative periods and metallic
abundances. We present graphical and tabular descriptions of the time
evolutions of the emission line equivalent widths of H$\alpha$, triplet He i
$\lambda$5876, and singlet He i $\lambda$6678 of RRab stars with strongest He
i emission. We also show time evolutions of the radial velocities, equivalent
widths, and turbulent velocities of H$\alpha$ and He i $\lambda$5876
absorptions measured for outflow and infall gas during the shock phases.

A principal conclusion of our work is that radial velocities of He i
$\lambda$5876 define outward traveling waves in metal-poor RRab that we use to
estimate the extent of RRab atmospheres traversed by the primary shock waves.
We call attention to the problem of populating the 23Po level of He i in
infalling gas during primary shock phases. We identify several RRab that do
not fit in extant theory of Galactic stellar evolution. Finally we provide
well-documented evidence for the occurrence of He ii $\lambda$4686 emission in
eight RRab stars during primary light rise.

As always we thank the support staff for their efficient, cheerful assistance
in our work at LCO. We also thank Giuseppe Bono and Harriet Dinerstein for
helpful comments that improved our presentation. This work has been supported
by NSF grant AST-1616040 (C.S.).

## References

  * Bono et al. (1994) Bono, G., Caputo, F., & Stellingwerf, R. F. 1994, ApJL, 432, L51, doi: 10.1086/187509
  * Chadid & Gillet (1996) Chadid, M., & Gillet, D. 1996, A&A, 315, 475
  * Chadid & Preston (2013) Chadid, M., & Preston, G. W. 2013, MNRAS, 434, 552, doi: 10.1093/mnras/stt1040
  * Chadid et al. (2017) Chadid, M., Sneden, C., & Preston, G. W. 2017, ApJ, 835, 187, doi: 10.3847/1538-4357/835/2/187
  * Chadid et al. (2008) Chadid, M., Vernin, J., & Gillet, D. 2008, A&A, 491, 537, doi: 10.1051/0004-6361:200810270
  * Chadid et al. (2014) Chadid, M., Vernin, J., Preston, G., et al. 2014, AJ, 148, 88, doi: 10.1088/0004-6256/148/5/88
  * Christy (1966) Christy, R. F. 1966, ApJ, 144, 108, doi: 10.1086/148593
  * Fokin & Gillet (1997) Fokin, A. B., & Gillet, D. 1997, A&A, 325, 1013
  * For et al. (2011a) For, B.-Q., Preston, G. W., & Sneden, C. 2011a, ApJS, 194, 38, doi: 10.1088/0067-0049/194/2/38
  * For et al. (2011b) For, B.-Q., Sneden, C., & Preston, G. W. 2011b, ApJS, 197, 29, doi: 10.1088/0067-0049/197/2/29
  * Gigosos et al. (2014) Gigosos, M. A., Djurović, S., Savić, I., et al. 2014, A&A, 561, A135, doi: 10.1051/0004-6361/201322866
  * Gillet & Crowe (1988) Gillet, D., & Crowe, R. A. 1988, A&A, 199, 242
  * Gillet et al. (2019) Gillet, D., Mauclaire, B., Lemoult, T., et al. 2019, A&A, 623, A109, doi: 10.1051/0004-6361/201833869
  * Gillet et al. (2017) Gillet, D., Mauclaire, B., Garrel, T., et al. 2017, A&A, 607, A51, doi: 10.1051/0004-6361/201629897
  * Griem (1983) Griem, H. R. 1983, Phys. Rev. A, 28, 1596, doi: 10.1103/PhysRevA.28.1596
  * Hill (1972) Hill, S. J. 1972, ApJ, 178, 793, doi: 10.1086/151835
  * Hodgman et al. (2009) Hodgman, S. S., Dall, R. G., Byron, L. J., et al. 2009, Phys. Rev. Lett., 103, 053002, doi: 10.1103/PhysRevLett.103.053002
  * Kielkopf & Allard (2014) Kielkopf, J. F., & Allard, N. F. 2014, Journal of Physics B Atomic Molecular Physics, 47, 155701, doi: 10.1088/0953-4075/47/15/155701
  * Kowalski et al. (2017) Kowalski, A. F., Allred, J. C., Uitenbroek, H., et al. 2017, ApJ, 837, 125, doi: 10.3847/1538-4357/aa603e
  * Kramida et al. (2019) Kramida, A., Yu. Ralchenko, Reader, J., & and NIST ASD Team. 2019, NIST Atomic Spectra Database (version 5.7.1), [Online]. Available: https://physics.nist.gov/asd [Aug 11 2020] National Institute of Standards and Technology, Gaithersburg, MD.
  * Kukarkin (1949) Kukarkin, B. V. 1949 (Moscow: Government Printing Office), 182
  * Kurucz (2011) Kurucz, R. L. 2011, Canadian Journal of Physics, 89, 417, doi: 10.1139/p10-104
  * Kurucz (2018) Kurucz, R. L. 2018, in Astronomical Society of the Pacific Conference Series, Vol. 515, Workshop on Astrophysical Opacities, ed. C. Mendoza, S. Turck-Chiéze, & J. Colgan, 47
  * Łach & Pachucki (2001) Łach, G., & Pachucki, K. 2001, Phys. Rev. A, 64, 042510, doi: 10.1103/PhysRevA.64.042510
  * Layden (1994) Layden, A. C. 1994, AJ, 108, 1016, doi: 10.1086/117132
  * Layden et al. (1996) Layden, A. C., Hanson, R. B., Hawley, S. L., Klemola, A. R., & Hanley, C. J. 1996, AJ, 112, 2110, doi: 10.1086/118167
  * Marconi et al. (2005) Marconi, M., Nordgren, T., Bono, G., Schnider, G., & Caputo, F. 2005, ApJ, 623, L133, doi: 10.1086/430273
  * Monson et al. (2017) Monson, A. J., Beaton, R. L., Scowcroft, V., et al. 2017, AJ, 153, 96, doi: 10.3847/1538-3881/153/3/96
  * Moore (1972) Moore, C. E. 1972, A multiplet table of astrophysical interest - Pt.1: Table of multiplets - Pt.2: Finding list of all lines in the table of multiplets, NSRDS-NBS 40 (Washington: Nat. Bur. Stds.)
  * Moore & Merrill (1968) Moore, C. E., & Merrill, P. W. 1968, Partial Grotrian Diagrams of Astrophysical Interest (National Bureau of Standards Monograph, Washington: US Government Printing Office (USGPO))
  * Moore et al. (1966) Moore, C. E., Minnaert, M. G. J., & Houtgast, J. 1966, The solar spectrum 2935 A to 8770 A (National Bureau of Standards Monograph, Washington: US Government Printing Office (USGPO))
  * Neilson et al. (2012) Neilson, H. R., Nardetto, N., Ngeow, C.-C., Fouqué, P., & Storm, J. 2012, A&A, 541, A134, doi: 10.1051/0004-6361/201118550
  * Preston (1959) Preston, G. W. 1959, ApJ, 130, 507, doi: 10.1086/146743
  * Preston (2009) —. 2009, A&A, 507, 1621, doi: 10.1051/0004-6361/200912984
  * Preston (2011) —. 2011, AJ, 141, 6, doi: 10.1088/0004-6256/141/1/6
  * Preston & Sneden (2000) Preston, G. W., & Sneden, C. 2000, AJ, 120, 1014, doi: 10.1086/301472
  * Preston et al. (2019) Preston, G. W., Sneden, C., Chadid, M., Thompson, I. B., & Shectman, S. A. 2019, AJ, 157, 153, doi: 10.3847/1538-3881/ab0ae1
  * Prudil et al. (2020) Prudil, Z., Dékány, I., Smolec, R., et al. 2020, A&A, 635, A66, doi: 10.1051/0004-6361/201936704
  * Sanford (1949) Sanford, R. F. 1949, ApJ, 109, 208, doi: 10.1086/145124
  * Schwarzschild (1952) Schwarzschild, M. 1952, in Transactions IAU, Vol. VIII, , 811
  * Sneden et al. (2017) Sneden, C., Preston, G. W., Chadid, M., & Adamów, M. 2017, ApJ, 848, 68, doi: 10.3847/1538-4357/aa8b10
  * Tody (1986) Tody, D. 1986, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 627, Instrumentation in astronomy VI, ed. D. L. Crawford, 733, doi: 10.1117/12.968154
  * Tody (1993) Tody, D. 1993, in Astronomical Society of the Pacific Conference Series, Vol. 52, Astronomical Data Analysis Software and Systems II, ed. R. J. Hanisch, R. J. V. Brissenden, & J. Barnes, 173
  * Unsöld & Struve (1949) Unsöld, A., & Struve, O. 1949, ApJ, 110, 455, doi: 10.1086/145221
  * Wallerstein (1959) Wallerstein, G. 1959, ApJ, 130, 560, doi: 10.1086/146745
  * Zhang et al. (2005) Zhang, Y., Liu, X. W., Liu, Y., & Rubin, R. H. 2005, MNRAS, 358, 457, doi: 10.1111/j.1365-2966.2005.08810.x
  * Zinn (1985) Zinn, R. 1985, ApJ, 293, 424, doi: 10.1086/163249

Table 1: Program Star Data Name | $P$ | $T_{0}$ | $V$ | $V_{amp}$ | [Fe/H]
---|---|---|---|---|---
| (d) | (d) | (mag) | (mag) |
WY Ant | 0.574340 | 1870.76 | 10.37 | 0.85 |  $-$1.80
X Ari | 0.651172 | 1890.10 | 9.24 | 0.94 |  $-$2.60
RR Cet | 0.553030 | 2143.63 | 9.26 | 0.82 |  $-$1.56
W Crt | 0.412011 | 1871.64 | 10.90 | 1.10 |  $-$0.59
DX Del | 0.472650 | 2415.85 | 9.81 | 0.70 |  $-$0.44
DT Hya | 0.567970 | 1872.11 | 12.53 | 0.98 |  $-$1.39
SS Leo | 0.626330 | 1873.06 | 10.47 | 1.00 |  $-$1.88
ST Leo | 0.477983 | 2384.94 | 10.99 | 1.19 |  $-$1.30
RV Oct | 0.571184 | 1891.66 | 10.53 | 1.13 |  $-$1.46
V445 Oph | 0.397030 | 1939.03 | 10.54 | 0.81 |  $-$0.10
AV Peg | 0.390385 | 2758.11 | 9.95 | 0.96 |  $-$0.13
HH Pup | 0.390745 | 1869.65 | 10.57 | 1.24 |  $-$0.74
AN Ser | 0.522080 | 2701.17 | 10.46 | 1.01 |  $+$0.04
W Tuc | 0.642260 | 1869.52 | 10.90 | 1.11 |  $-$1.76
UU Vir | 0.475597 | 1886.54 | 10.06 | 1.08 |  $-$0.92
Table 2: Phase Shifts and Flux Scale factors Quantity | RR Cet | RV Oct | SS Leo | ST Leo | W Tuc | WY Ant | X Ari
---|---|---|---|---|---|---|---
flux scale factors
H$\alpha$ | 1.136 | 1.085 | 0.910 | 1.450 | 1.104 | 1.040 | 0.770
He i 5876 Å | 1.135 | 0.850 | 0.780 | 1.200 | 1.640 | 1.020 | 1.000
phase shifts $\Delta\phi$
H$\alpha$ | 0.0060 |  $-$0.0095 |  $-$0.0035 |  $-$0.0220 |  $-$0.0035 | 0.0090 | 0.0061
He i 5876 Å | 0.0080 |  $-$0.0050 |  $-$0.0005 |  $-$0.0180 | 0.0005 | 0.0085 | 0.0053
Table 3: Kinematic Properties of RRab Stars During Primary Light Rise aaQuantities measured at phase $\phi$ $\simeq$ 0.97 Star | dR/Dt | dR/dt | dR/dt | $\sigma_{Max}$ | $\sigma_{Max}$ | $\sigma_{Max}$
---|---|---|---|---|---|---
| H$\alpha$(B) | H$\alpha$(R) | H$\alpha$(B-R) | H$\alpha$(B) | H$\alpha$(R) | H$\alpha$(B-R)
RR Cet | 86.4 |  $-$71.6 | 158.0 | 85.8 | 57.8 | 28.0
RV Oct | 93.6 |  $-$72.1 | 165.7 | 81.7 | 57.8 | 23.9
SS Leo | 87.5 |  $-$76.2 | 163.7 | 83.2 | 59.1 | 24.1
W Tuc | 109.9 |  $-$71.9 | 181.8 | 94.0 | 72.3 | 21.7
WY Ant | 87.9 |  $-$73.3 | 161.2 | 87.3 | 64.8 | 22.5
X Ari | 88.4 |  $-$74.2 | 162.6 | 87.8 | 60.1 | 27.7
mean | 92.3 |  $-$73.2 | 165.5 | 86.6 | 62.0 | 24.7
$\sigma$ | 8.2 | 1.6 | 7.7 | 3.9 | 5.2 | 2.4
| He i | He i | He i | He i | He i | He i
RR Cet | 57.8 |  $-$87.7 | 145.5 | 43.8 | 28.6 | 15.2
RV Oct | 58.0 |  $-$73.8 | 131.8 | 39.8 | 23.3 | 16.5
SS Leo | 53.0 |  $-$73.5 | 126.5 | 40.6 | 26.5 | 14.1
W Tuc | 52.2 |  $-$69.6 | 121.8 | 45.2 | 36.1 | 9.1
WY Ant | 48.2 |  $-$74.6 | 122.8 | 39.6 | 39.3 | 0.3
X Ari | 53.1 |  $-$62.9 | 116.0 | 51.6 | 43.3 | 8.3
mean | 53.7 |  $-$73.7 | 127.4 | 43.4 | 32.9 | 10.6
$\sigma$ | 3.4 | 7.4 | 9.4 | 4.2 | 7.2 | 5.5
Table 4: Regression constants for dR/dt and $\sigma_{Max}$ Star | feature | dR/Dt(B) | dR/dt(B) | dR/dt(R) | dR/dt(R) |  $\sigma_{Max}$(B) |  $\sigma_{Max}$(B) |  $\sigma_{Max}$(R) |  $\sigma_{Max}$(R)
---|---|---|---|---|---|---|---|---|---
|  | a0 | a1 | a0 | a1 | a0 | a1 | a0 | a1
RR Cet | H$\alpha$ | 186.554 |  $-$103.250 | 523.387 |  $-$613.388 |  $-$204.589 | 299.338 | 258.439 |  $-$206.717
RV Oct | H$\alpha$ |  $-$34.041 | 131.576 | 249.885 |  $-$331.951 |  $-$72.621 | 159.074 | 201.159 |  $-$147.791
SS Leo | H$\alpha$ |  $-$216.277 | 313.160 | 614.894 |  $-$709.663 |  $-$249.830 | 343.312 | 272.169 |  $-$219.620
W Tuc | H$\alpha$ | 12.285 | 100.665 | 134.548 |  $-$212.862 | 56.774 | 38.419 | 93.955 |  $-$22.313
WY Ant | H$\alpha$ | 59.292 | 31.410 | 283.985 |  $-$368.308 |  $-$76.043 | 168.379 | 258.183 |  $-$199.415
X Ari | H$\alpha$ |  $-$146.453 | 242.098 | 137.306 |  $-$218.010 | 35.781 | 53.628 | 326.617 |  $-$274.785
RR Cet | He i |  $-$850.430 | 936.342 | 572.532 |  $-$680.664 |  $-$78.794 | 126.436 |  $-$217.945 | 254.132
RV Oct | He i |  $-$1020.236 | 1111.608 | 313.749 |  $-$399.558 | 84.620 |  $-$46.234 |  $-$32.516 | 57.567
SS Leo | He i |  $-$865.432 | 946.876 | 614.894 |  $-$709.663 |  $-$124.863 | 170.547 |  $-$6.733 | 35.940
W Tuc | He i |  $-$809.648 | 888.546 | 360.180 |  $-$443.044 |  $-$148.486 | 199.692 |  $-$96.769 | 136.933
WY Ant | He i |  $-$689.407 | 760.464 | 464.721 |  $-$556.043 |  $-$107.779 | 151.970 |  $-$202.714 | 249.459
X Ari | He i |  $-$781.674 | 860.612 | 437.561 |  $-$515.920 | aaQuadratic fit: $\sigma_{Max}$ = $-$8531 + 17733$\phi$ $-$9160$\phi^{2}$ | aaQuadratic fit: $\sigma_{Max}$ = $-$8531 + 17733$\phi$ $-$9160$\phi^{2}$ | bbQuadratic fit: $\sigma_{Max}$ = $-$10910 + 22426$\phi$ $-$11478$\phi^{2}$ | bbQuadratic fit: $\sigma_{Max}$ = $-$10910 + 22426$\phi$ $-$11478$\phi^{2}$