First Detection of Shock Waves in RRc Stars

The cross-correlation radial velocity of four RRc stars is measured by the cross-correlation method (M. Chadid et al. 2017). Figure 2 shows the cross-correlation radial velocity for AS123811, Y Crv, AS132448, AU Vir, AS211933, YZ Cap, and AS190212, MT Tel, respectively. The radial velocity of the correlation profile shows an asymmetric curve with a descending branch lasting shorter than the ascent one, and representing ∼30% of the pulsation period of the star, with a subtle elbow appearing just before the peak of the light curve, around pulsation phase ϕ = 0.5.

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4.2.1. Metallic Radial Velocity

We measured the heliocentric radial velocity (HRV) for the metallic ionized line Fe II λλ4923.921 Å. The Fe II HRV curves for AS123811, Y Crv, AS132448, AU Vir, AS211933, YZ Cap, and AS190212, MT Tel, respectively, are shown in Figure 2. From the HRV V_r(t), we calculate the pulsation velocity in the stellar rest frame using the formula

where p is the correction factor for geometrical projection and limb darkening and V_* is the center-of-mass velocity. Since the metallic absorption line Fe II λλ4923.921 Å is formed near the photosphere, is close to the photospheric motion, and V_* is assumed to be equal to the so-called γ-velocity, in which we use the average of the HRV curve over one whole pulsation period. Here, a value of 1.36 is adopted for p as discussed by G. Burki et al. (1982).

4.2.2. Acceleration and Radius Variation

The acceleration of the stellar surface is given from the derivative of , while the radius variation is given from the integration of :

where n is an integer, R₀ is the mean stellar radius, which we estimate using the method from M. Marconi et al. (2005). The empirical radius used for RRc stars is given by Equation (3):

where the range of the metal content (heavy-element abundance) Z for RR Lyrae stars is from 10⁻⁴ to 10⁻², corresponding to a [Fe/H] from −2.5 to −0.0 (H. A. Smith 2004), and we assume that is linear to [Fe/H] by the reference and data from J. Fernley (1993) and M. D. Criscienzo et al. (2004). Thu, we use linear interpolation to get the relation between them:

Figure 3 shows the pulsation velocity, acceleration, and radius variation (ΔR/R) for AS123811, AS132448, AS190212, and AS211933, respectively.

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The most striking feature is that the acceleration curves show a primary acceleration which occurs during the peak in the light curve around phase 0.894, 0.891, 0.879, and 0.879 for AS123811, AS132448, AS190212, and AS211933, respectively, when the radius is minimum, and a secondary acceleration, during the subtle elbow in the radial velocity curve, around phase 0.416, 0.412, 0.466 and 0.421, respectively. The amplitude of the pulsation at the photospheric level is about 17.70%, 16.61%, 15.92%, and 12.58% R_⊙ for AS123811, AS132448, AS190212, and AS211933, respectively. Y Crv has the largest pulsation amplitude and, consequently, the most extended stellar photosphere. There is substantial star-to-star variation in acceleration maxima, with AS211933 reaching 7.0 m s⁻² while AS190212 exhibits little variation in its acceleration throughout its pulsational phases, while the secondary acceleration is negative and negligible for the four RRc stars. This explains the deceleration of the photosphere during the subtle radial velocity elbow.

Table 2 gives the average of the HRV curve over one pulsation period for the metallic absorption line Fe II λλ4923.921 Å primary acceleration, radius variation, and mean radius.

Table 2. Main Dynamic Parameters of Four RRc Stars for Metallic Absorption Lines Fe II λλ4923.921 Å

Star Name	γ-Velocity	Primary Acceleration	Second Acceleration	ΔR/R⊙	R0
	(km s−1)	(m s−2)	(m s−2)		(R⊙)
AS123811	121.14	6.3024	−0.9339	0.1770	4.748
AS132448	86.06	6.0118	−0.6957	0.1661	5.100
AS190212	67.54	5.1788	−1.2715	0.1592	5.039
AS211933	−90.47	6.9428	−0.9519	0.1258	4.265

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Figure 2 shows the hydrogen, Hαλλ6563 Å , radial velocity curve for AS123811, AS132448, AS190212, and AS211933, respectively. The hydrogen radial velocity curve shows an asymmetric curve with a descent branch lasting shorter than the ascent one, and representing ∼20% of the pulsation period of the star, with an elbow appearing just before the peak of the light curve. All RRc stars of this study possess prominent Hα radial velocity maxima and stationary values near phase ϕ = 0.50. This phenomenon, while observed in the metallic lines as negligible, is significantly more pronounced in the Hα cores, originating from the upper atmospheric layers.

From the HRV V_r(t), we calculate the pulsation velocity in the stellar rest frame . The acceleration of the stellar surface is given from the derivative of , while the radius variation is given from the integration of following the formula described in Section 4.2.2. Figure 4 shows the acceleration, radius variation, and pulsation velocity for AS123811, AS132448, AS190212, and AS211933, respectively. The most striking feature is that the acceleration curves show a primary acceleration which occurs during the peak in the light curve around phase 0.950, 0.910, 0.901, and 0.918 for AS123811, AS132448, AS190212, and AS211933, respectively, when the radius is minimum, and a secondary acceleration, during the elbow in the radial velocity curve, around phase 0.597, 0.640, 0.545, and 0.568, respectively. The average of the HRV curve over one pulsation period for primary acceleration, radius variation, and mean radius of four RRc stars is given in Table 3. The amplitude of the pulsation at the high atmospheric level is large for the four RRc stars and reaches a value around 33.10%, 34.88%, 19.80%, and 23.54% R_⊙, respectively. AU Vir has the largest pulsation amplitude and, consequently, the most extended stellar high atmosphere. There is substantial star-to-star variation in primary acceleration maxima, with AS123811 reaching ∼12 m s⁻² while AS132448 exhibits a small variation in its acceleration throughout its pulsational phases. The secondary acceleration is significant for the high atmosphere of four RRc stars. The AU Vir star has the largest secondary acceleration, reaching a value of 1.07 m s⁻², indicating a more pronounced dynamic effect on the high atmosphere compared to the subtle secondary accelerations observed in the photosphere of four RRc stars. The radius variation curves ΔR(t) also reveal a distinct asymmetry around the mean radius. The expansion phase is steeper and shorter in duration than the contraction, which is more gradual. This asymmetry is consistent with nonlinear pulsation dynamics; the rapid expansion corresponds to the passage of a primary shock, while the slower contraction reflects radiative damping and delayed atmospheric response. Such asymmetry is a well-known signature of shock propagation in pulsating stars and further supports the shock interpretation of our observations.

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5.3.1. Hydrogen Line Broadening Phenomena

Figures 5, 6, 7, and 9 show the evolution of Hαλλ6563 Å over the pulsation period in Y Crv, AU Vir, MT Tel, and YZ Cap, respectively. The Hαλλ6563 Å profile shows a significant broadening phenomenon during the pulsation period. Figures 8, 10, 11, and 12 demonstrate such a broadening phenomenon. The FWHM and the equivalent width (EQW) are derived from a single Gaussian fit. The FWHM curve is strongly variable during the pulsation period. Such variation is produced by the expansion itself ( C. De Jager & G. De Jonge 1978), axial rotation vsini (G. Kovacs & J. R. Buchler 1990), turbulence (G. Burki et al. 1982), and the velocity gradients (M. Chadid & D. Gillet 1996a). The FWHM curve shows a significant peak centered at ϕ = 0.90, hereafter the primary peak, occurring at the primary acceleration. A second peak occurs at ϕ = 0.55, hereafter the secondary peak, lower than the primary peak, occurring during the secondary acceleration. The EQW curve shows a very strong increase as well at the pulsation amplitude maximum, and a strong peak at the same phases centered at phase 0.90. The most striking feature in comparison with the FWHM and EQW of metallic lines is that the FWHM and EQW curves of metallic lines show no significant variation during the pulsation period for the four RRc stars.

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5.3.2. Hydrogen Emission Line and Line-doubling Phenomena

Hα blueshifted emission and line doubling are detected in AU Vir, Y Crv, and YZ Cap between ϕ ≈ 0.75–0.97, when primary acceleration (ϕ ≈ 0.90) occurs near minimum radius. This is the strongest acceleration phase, causing the expansion of the stellar atmosphere. The Hα absorption lines show doubling of their cores during the primary acceleration. According to the classical Schwarzchild mechanism (M. Schwarzschild 1954), we explain the doubling phenomenon by the propagation of a strong shock wave in the atmosphere of RRc stars. The brightest Hα emissions observed during the primary acceleration are directly produced in the radiative wake of the shock waves. This main shock is created by the κ and γ mechanisms, respectively. The intensity of the main shock is correlated with the strength of the κ and γ mechanisms. Henceforth, we refer to Sh_He+H.

The maximum of the FWHM and EQW of the Hα occurs when the two blue and red doubling components have the same intensity. Strong FWHM and EQW peak centered at ϕ = 0.90, mainly due to the line-doubling phenomenon caused by the Sh_He+H.

The intensity of the Sh_He+H shock is not significant in the most metal-poor star, MT Tel. MT Tel exhibits neither Hα emission nor line doubling, indicating weaker Sh_He+H shock propagation due to its low metallicity. In contrast, the less metal-poor stars Y Crv and YZ Cap display the strongest Sh_He+H main shock, accompanied by the highest dynamical acceleration (Tables 2 and 3), yet the shock is weaker in the most metal-poor ones. This suggests that even within a population of metal-poor stars, small variations in metallicity influence shock strength. The more metal-poor a star is, the closer it lies to the cooler fundamental red edge, leading to a deeper convective envelope and thicker compression zones. Consequently, the Sh_He+H shock crosses the metal-poor photosphere at a lower optical depth and with a smaller Mach number. As the shock interacts with the density inhomogeneities typical of convective envelopes, it generates turbulent velocities that contribute to shock dissipation.

We attribute the weaker Sh_He+H shock in the most metal-poor stars to their larger convective envelopes, which enhance turbulence and promote shock dissipation. This behavior is rooted in structural differences between metal-poor and metal-rich stars. Lower metallicity reduces the overall opacity of stellar material, resulting in more extended and deeper convective envelopes (H. A. Smith 2004; M. Chadid et al. 2017). This in turn influences the atmospheric response to pulsation and the development of radiative shocks. Early theoretical insights into this opacity–convection relationship were already noted by S. A. Zhevakin (1953).

Table 2 demonstrates a correlation between metallicity and shock intensity: the most metal-poor stars, such as MT Tel, exhibit low dynamical acceleration, indicating a weaker Sh_He+H shock. This trend can be explained by the structure of convective envelopes; the more metal-poor a star is, the deeper its convective envelope, which amplifies turbulence and facilitates shock dissipation. Conversely, stars with slightly higher metallicity have less developed convective envelopes, leading to reduced dissipation and stronger Sh_He+H shock. The Sh_He+H shock develops further as it propagates outward, following the atmospheric extension of the star. In more extended atmospheres, the Sh_He+H shock continues to travel outward into the upper layers, where its amplitude increases as the gas density decreases. As a result, the shock reaches higher Mach numbers in the upper atmosphere. Table 3 highlights how extended atmosphere influences Sh_He+H shock propagation in different stars. Y Crv has the most extended atmosphere (18% R_⊙), allowing the shock to develop further and reach higher Mach numbers in the upper layers (highest dynamical acceleration 11.4 m s⁻¹). In contrast, MT Tel has the least extended atmosphere (15% R_⊙), limiting shock amplification (lowest dynamical acceleration 8.5 m s⁻¹ ). These variations indicate that stars with more extended atmospheres provide more space for the shock to strengthen as it propagates outward, while those with less extended atmospheres experience weaker shock evolution.

In RRab stars, the ballistic shock, Sh_PM3, described in M. Chadid et al. (2014) follows the main Sh_He+H shock. The Sh_PM3 ballistic shock is produced by compression heating and induces the secondary acceleration. In RRab stars, this shock (M. Chadid et al. 2014) is strong enough to interact with the extended atmosphere, producing a detectable Hα emission and also the Hα line-doubling phenomenon, as detected for the first time by M. Chadid et al. (2017). In their study, M. Chadid et al. (2017) show that Hα line doubling is detected in the metal-poor RRab stars, but not in the metal-rich ones during Sh_PM3. They explain that the greater the dynamical gravity, the larger the Sh_He+H shock intensity that causes a higher level of dynamical imbalance, i.e., the nonsynchronization of motions in the upper and lower photospheric layers. Stronger collision induces higher supersonic motion, Sh_PM3. Thus, secondary acceleration becomes more important.

M. Chadid et al. (2017) attribute these detections to the fact that the metal-poor RRab atmospheres are more extended, and the amplitudes of the ballistic motions of the metal-poor RRab atmospheres are greater than those of metal-rich RRab. In fact, the ballistic photospheric motion has its greater amplitude in metal-poor relative to metal-rich RRab stars. The greater extensions of the metal-poor envelopes complicate their atmospheric motions, and the nonsynchronization effect greatly changes atmospheric motion structure and creates chaotic behavior, mainly in the upper atmospheres. M. Chadid et al. (2017) also explain that the region of formation of spectral lines is greatly enlarged with concomitant effects on line profiles. This enlarged region of line formation in metal-poor stars is the origin of the greater duration of Hα doubling and the peak Hα emission flux. Hα emission is generally weaker in metal-rich RRab stars.

In this study, we detect that a secondary acceleration appears around pulsation phase 0.55–0.60 in AU Vir, Y Crv, YZ Cap, and MT Tel, a pronounced elbow in the radial velocity curve, coinciding with the timing of Sh_PM3. We detect that the stronger the dynamical gravity, the more important secondary acceleration (Tables 2 and 3), as reported by M. Chadid et al. (2017) in RRab atmospheres. However, in all four RRc stars, no Hα emission nor Hα line doubling is detected during the Sh_PM3 ballistic shock. This confirms that the Sh_PM3 temperature is insufficient to trigger hydrogen emission. This means that the Sh_He+H shock intensity in RRc induces a lower level of dynamical imbalance, i.e., the nonsynchronization of motions in the upper and lower photospheric layers to RRab. On the other hand, RRc stars are less extended than RRab stars, leading to rapid dissipation. Henceforth, we refer to Sh_Ball in RRc stars.

We refer to the gravitational collapse shock as Sh_Gravity, a newly proposed shock phenomenon in RR Lyrae stars. This return shock occurs at the onset of atmospheric contraction, emerging when the outward-moving atmospheric layers fall back toward the stellar surface under gravity, typically around pulsation phase ϕ = 0.30. In Eulerian coordinates, this is a receding shock, while in the Lagrangian frame, it is advancing. As the infalling gas collides with denser layers, it undergoes rapid recompression, producing a localized shock wave. This recompression excites hydrogen atoms, resulting in redshifted Hα emission.

Despite generating detectable spectral signatures, Sh_Gravity has little impact on radial velocity or light curves. The absence of a pronounced velocity profile discontinuity, such as the classical elbow, indicates that the shock is confined to outer atmospheric layers, without significantly disturbing the global dynamics of the star. Moreover, the associated Hα emission is too weak to noticeably affect the overall luminosity.

Sh_Gravity coincides with the Lump observed in RRab light curves, first identified by M. Chadid & G. Preston (2013). It bears similarities to the Sh_PM shock, which also produces redshifted Hα emission at ϕ ≈ 0.35, indicating inward-moving shocked gas. In RRab stars, Sh_PM is a strong, global shock, accompanied by a discontinuity in the radial velocity curve (the Bulge) and a luminosity feature (the Lump).

We propose that Sh_Gravity results from the same fundamental hydrodynamical process as Sh_PM, but develops within the thinner and more compressible atmosphere of RRc stars. This likely explains its localized, transient character and its different energy dissipation profile.

Our spectroscopic data reveal redshifted Hα emission in RRc stars over a broader phase interval than typically observed in RRab stars (see Table 4).

In RRc stars, redshifted emission begins significantly earlier than in RRab stars, where it typically appears near ϕ ≈ 0.3, suggesting differences in shock dynamics. For instance, the presence of redshifted emission as early as ϕ ≈ 0.00 in MT Tel points toward the jump (Sh_PM1) shock described by M. Chadid et al. (2014), which marks the initial stage of atmospheric compression. Conversely, the persistence of emission up to ϕ ≈ 0.3 in AU Vir, Y Crv, and YZ Cap aligns with the Lump (Sh_PM) shock seen in RRab stars M. Chadid et al. (2014).

These observations suggest that in RRc stars, the Sh_PM1 and Sh_PM shocks may not be distinct events but instead represent a continuous, blended shock signature. This behavior results from the compact and compressible nature of RRc atmospheres, which enables faster transitions between different shock regimes.

Moreover, the longer duration and greater intensity of redshifted Hα emission in RRc stars point toward a relatively stronger Sh_Gravity compared to RRab stars. In RRab stars, more extended atmospheres act as buffers; they absorb and dissipate the energy of the infall more gradually, reducing the likelihood of hydrogen excitation and emission.

Since Sh_Gravity is powered by gravitational infall, its intensity is proportional to the kinetic energy of the collapsing layers:

At first glance, this relation suggests that denser atmospheres (higher ρ) should produce stronger shocks. However, the observational signatures, such as Hα emission, depend not only on the energy injected but also on how that energy is transported and dissipated within the atmosphere. In denser envelopes, the shock front can be more strongly damped through radiative diffusion and thermal conduction, which limits local heating and delays hydrogen excitation.

By contrast, in the lower-density atmospheres of RRc stars, the infall velocity v_infall tends to be higher, and the same or even lower total energy can lead to more effective local heating. The lower density reduces radiative damping, allowing the shock to heat the gas more efficiently and trigger observable emission. These stars also have lower atmospheric mass and higher compressibility, which leads to faster energy dissipation and shorter shock durations.

This compressibility is reflected in the atmospheric pressure scale height:

where g is the surface gravity. In RRc stars, higher g values lead to smaller H, meaning more compact atmospheres that respond more quickly to dynamic perturbations.

In contrast, RRab stars, with denser and more extended envelopes, retain shock energy longer, resulting in more sustained but often less intense emission features.

Finally, metallicity also modulates shock visibility. Since the opacity κ scales with the metal content Z,

metal-poor atmospheres (typical of RRc stars) exhibit lower opacity, which facilitates shock propagation while limiting radiative losses, thereby contributing to sharper, more transient emission signatures.

To sum up, it is the overall atmospheric structure, particularly density, compressibility, and metallicity, that governs the nature and observability of shock phenomena in RR Lyrae stars. RRc stars favor rapid, intense, but short-lived shocks, while RRab stars support slower, more buffered shock evolution.

In radiative shocks, the postshock gas velocity (V_post) and the shock velocity (V_s) are related through mass conservation and the compression ratio r = ρ_post/ρ_pre, where ρ denotes the gas density. For an ideal monoatomic gas with adiabatic index γ = 5/3, the Rankine–Hugoniot conditions yield a theoretical compression ratio of r = 4 for strong adiabatic shocks. In the presence of radiative cooling, r can increase to values ≳7 depending on shock strength and cooling efficiency (Y. B. Zeldovich & Y. P. Raizer 1966; B. T. Draine 2011).

For RRc stars, which exhibit relatively modest radiative energy losses and partially ionized hydrogen layers, a compression ratio of r = 2.5 is considered physically realistic. This choice reflects moderate shock compression and avoids overestimating the shock strength that would result from assuming r = 3 or higher, which are more typical of RRab stars. We adopt r = 2.5 as a physically realistic compression ratio for partially radiative, nonadiabatic shocks, following the classical treatments of Y. B. Zeldovich & Y. P. Raizer (1966) and B. T. Draine (2011). This value represents an intermediate case between adiabatic conditions (r ≈ 4) and fully radiative shocks (r > 4). To assess sensitivity, we tested variations in r between 2.0 and 3.0. These produce changes in the derived shock velocity V_s of less than 15%, leaving the relative shock ranking across the stars unaffected.

From the conservation of mass flux across the shock in the stellar rest frame, and assuming V_pre = 0:

Solving for V_post yields

Thus, for r = 2.5:

However, the observed Hα emission arises not from the immediate postshock interface, but from a radiative cooling zone located downstream, where the gas has been compressed and then decelerated. To account for this, we introduce a deceleration factor β defined by

Combining both relations:

Assuming β = 0.6 and r = 2.5 gives

This approximation is used throughout the paper to estimate shock velocities from Doppler measurements of the Hα emission, while accounting for both physical compression and postshock deceleration effects.

7.1.1. Doppler Measurement of the Hα Emission

The Hα emission arises from the radiative cooling region immediately behind the shock front, where recombination of hydrogen produces Balmer radiation. The emitting gas, compressed and decelerated postshock, exhibits a Doppler shift that reflects a velocity lower than the instantaneous postshock flow.

The observed velocity is determined via

This formulation improves upon empirical estimates by incorporating a physically motivated compression ratio (r = 2.5) and a deceleration factor β suited to the cooling zone properties of RRc stars. The resulting scaling,

7.1.2. Temperature Effects and Method Validity

7.1.3. Mach Number Estimation

The shock strength is quantified by the Mach number:

with the adiabatic sound speed:

(B. W. Carroll & D. A. Ostlie 2017), assuming γ = 5/3, T = 7250 K, and μ = 1.3.

Table 5 presents the shock velocities and corresponding Mach numbers for r = 2.5 and β = 0.6 in all three RRc stars analyzed.

Table 5. Shock Velocity and Mach Number for Outward and Rebound Shocks Assuming r = 2.5 and β = 0.6

Star	Vs (He+H)		Vs (Gravity)
	(km s−1)		(km s−1)
AS123811	323.6	30.2	231.2	21.6
AS132448	231.2	21.6	166.5	15.6
AS211933	166.5	15.6	138.7	13.0

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Table 6. Classification of Shock Regimes in RRc Stars

Shock Regime	Mach Number	Observable Signatures	Example Star
Transonic	1–2	No emission, calm profiles	MT Tel (infall)
Supersonic	2–5	Minor asymmetries, no Hα emission	⋯
Low hypersonic	5–8	Hα line doubling, weak emission	YZ Cap
Mild hypersonic	8–15	Strong Hα broadening, emission peak	AU Vir
Strong hypersonic	15–25	Intense Doppler shift, no He lines	⋯
Ultrahypersonic	>25	Sharp Hα transients, extreme velocities	Y Crv

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These results confirm that both Sh_He+H (main outward shock) and Sh_Gravity (gravitational return shock) are strongly supersonic in all three stars.

The strongest shock is observed in AS123811, with a peak Mach number of 30.2 for the main Sh_He+H event. This suggests a highly dynamic atmospheric expansion, consistent with an extended atmosphere and efficient energy transfer by the κ and γ mechanisms. The redshifted component, associated with Sh_Gravity, also reaches a high Mach number of 21.6, indicating a vigorous infall and efficient recompression of the upper layers.

AS132448 shows intermediate shock intensities with Mach numbers of 21.6 (outward) and 15.6 (inward), while AS211933 presents the weakest shocks among the three stars, with Mach numbers of 15.6 and 13.0, respectively. These differences are likely influenced by variations in metallicity and atmospheric extension.

A possible explanation lies in the relative atmospheric extension, as shown in Table 3, AS123811 (Y Crv) has the most extended atmosphere (18 R_⊙), allowing shocks to develop more fully as they propagate. In contrast, AS211933 (YZ Cap) has a more compact atmosphere, where the limited spatial development suppresses the amplification of shock velocities.

Furthermore, the trend in Mach number correlates with the strength of Hα features. AS123811 exhibits the highest FWHM and EQW at ϕ ≈ 0.90, coinciding with the phase of maximum dynamical acceleration and the strongest Sh_He+H. The detection of significant redshifted emission in this star also confirms a strong Sh_Gravity contribution.

In summary, RRc stars exhibit distinct yet complementary shock phenomena, Sh_He+H and Sh_Gravity, each with supersonic velocities and measurable spectral signatures. The variability in Mach number across stars provides insights into the influence of atmospheric structure, metallicity, and pulsation strength on the dynamics of shock propagation.

Our direct measurements of shock velocities in RRc stars reveal Mach numbers that significantly exceed the traditional classification thresholds discussed in the literature. In particular, maximum Mach values reach up to . This estimate assumes a typical sound speed of c_s ∼ 10.7 km s⁻¹ in RRc photospheric layers (for T ∼ 7250 K, μ ∼ 1.3), and Doppler-inferred shock velocities reaching v_shock ∼ 324 km s⁻¹ (as in AS123811), well beyond the “hypersonic” regime typically defined by .

The classification scheme proposed by (M. Chadid et al. 2017), based on spectroscopic diagnostics in RRab stars, defines four regimes: transonic, supersonic, hypersonic (He I emission), and ultrahypersonic (He II emission). However, it is primarily grounded in emission features and indirect temperature diagnostics, rather than direct shock velocity measurements.

In contrast, our method deriving V_s from Doppler shifts of the Hα line and converting it into Mach numbers using a sound speed of c_s ≈ 10.7 km s⁻¹ (for T ≈ 7250 K) offers a more direct kinematic quantification of the shock strength. As a result, we find that RRc stars exhibit hypersonic shock velocities, yet without He I or He II emission lines.

This apparent discrepancy can be reconciled by considering the compact and low-opacity nature of RRc atmospheres. The shocks, while extremely fast, are short lived and may not sustain the high postshock temperatures required to excite helium lines before dissipating. This implies that RRc stars occupy a distinct shock regime, one characterized by extremely high Mach numbers, but lacking extended radiative wakes typically associated with helium ionization.

We therefore propose an extension to the standard shock classification framework, specific to RRc stars, rather than a revision of the original scheme. This approach complements the work of M. Chadid et al. (2017) by introducing a parallel classification adapted to a different subclass of RR Lyrae variables.

This updated classification is summarized below:

• 1.  
  Ultrahypersonic. extremely fast shocks in compact RRc atmospheres, with no He I/II emission due to short radiative timescales.
• 2.  
  Strong hypersonic. sharp Hα transients, large Doppler shifts, but still no helium lines.
• 3.  
  Mild hypersonic. clear Hα doubling, strong line broadening, occasionally weak emission.
• 4.  
  Low hypersonic. incipient line doubling, small FWHM increases.
• 5.  
  Supersonic. minor velocity structure, no emission.
• 6.  
  Transonic. no Doppler effects, steady absorption.

This classification reflects the specific atmospheric dynamics of RRc stars, where extremely high shock velocities can occur within compact, radiatively efficient envelopes. The absence of He I/II lines at high Mach numbers suggests rapid shock dissipation before helium excitation, contrasting with the longer-lived, helium-emitting shocks in RRab stars.

These findings not only highlight the presence of hypersonic shocks in RRc stars but also emphasize the need to reassess how we interpret stellar shock phenomena across different subclasses of RR Lyrae variables.

While traditional spectroscopic diagnostics such as the presence of He I and He II emission lines remain effective in extended RRab atmospheres, they may be insufficient for characterizing the more compact and rapidly evolving shock environments of RRc stars. The absence of helium lines, despite extremely high shock velocities, underscores the limitations of emission-based classification schemes when applied universally.

By combining direct Doppler velocity measurements with shock compression physics, our approach offers a complementary pathway to probe the dynamical structure of RRc atmospheres. The proposed classification provides a framework to interpret shock signatures in compact stars that may otherwise appear spectroscopically quiet.

This work also opens the door to a broader investigation of dynamical diversity among RR Lyrae stars, suggesting that the interplay between metallicity, atmospheric structure, and pulsation amplitude may create a richer variety of shock regimes than previously recognized.