© The Authors 2023

1 Introduction

Semi-regular variable (SRV) stars are intriguing objects. They are evolved, pulsating, red giant stars that reflect the future of our own Sun. SRVs are distinguished from Mira-type stars by their smaller brightness amplitude, which is less than 2.5 mag(V). In particular, type “a” SRV stars (SRa) show an identifiable and sustainable primary pulsation period, and in many cases secondary pulsation periods.

Semi-regular variable stars, which are bright and so have been observed for centuries, are recognized as important standard candles that also follow a number of period-luminosity relations (Trabucchi et al. 2017; Lebzelter et al. 2019). This correspondence contributes to our understanding of their evolution and makes SRVs valuable to extragalactic and cosmological studies. Even after so many observations and studies, SRVs continue to bring up a number of questions, including ones related to the role of convection, stellar and circumstellar magnetic fields, atmospheric dynamics, winds, and how these stars interact with their circumstellar medium (Kerschbaum & Hron 1992; Kudashkina 2019).

One of the more surprising questions arises from linear polarization measurements of some SRV stars. Kruszewski et al. (1968) and Shawl (1975a,b) presented early polarization measurements for V CVn (HD 115898). This M4e-M6eIIIa late asymptotic giant branch (AGB) star has a highly variable polarization and is located at a distance of 501 pc (Gaia Collaboration 2020). Linear polarization measurements of unresolved sources can measure deviations from circular symmetry as projected onto the sky. Thus, the detection of linear polarization from a star may indicate an asymmetry due to the presence of convection, wind bow shocks, or disks (Shrestha et al. 2021).

Kruszewski et al. (1968) presented time-domain polarization measurements of V CVn showing that the polarization fraction changes over the approximate 194 day primary pulsation period, while the position angle remains constant. What is more curious is that the polarization fraction appears to be anticorrelated to the brightness of the star; when the star is brightest, the polarization is at a minimum, and when the star is dimmest, the polarization is at a maximum. This surprising behavior was confirmed to be a function of wavelength by Magalhaes et al. (1986a,b). While still unexplained, this phenomenon is not unique to V CVn.

Kruszewski et al. (1968) identified similar polarization variability in the star L₂ Puppis. This result added to the challenge of understanding these two stars, but resolved polarimetric observations provided information about the circumstellar structure of the L₂ Pup system. Kervella et al. (2015) used VLT SPHERE/ZIMPol polarimetric observations to discover the presence of a circumstellar disk. This disk is remarkable as its asymmetry projected on the sky explains the constant position angle of the unresolved linear polarization, but not the variation of the polarization itself.

Neilson et al. (2014) reviewed time-domain polarization measurements for V CVn and confirmed the periodic variability of the polarization while the position angle stays approximately constant. They also confirmed that the maximum value of the polarization appeared to correspond to minimum brightness, and the minimum polarization to maximum brightness. They suggested that the polarization and position-angle time-domain observations could be explained by a pulsation-driven, spherically symmetric dusty wind that interacts with an asymmetric stellar wind bow shock. The presence of a constant bow shock could explain the constant position angle. When the star approaches maximum brightness, it also approaches maximum radius and minimum effective temperature. Thus, dust will form in the spherically symmetric outflow, and that dust will be more likely to interact with scattering photons from the star. This will cause the polarization to decrease to a minimum. Conversely, as the shell expands, the density decreases and the shell will interact with the bow shock, creating a highly asymmetric system and resulting in a maximum polarization value, as shown by Shrestha et al. (2021).

An alternative model was proposed by Safonov et al. (2019) based on differential speckle polarimetric observations of V CVn. They introduced the idea of scattering arcs or reflection nebulae in the circumstellar envelope of the star; hereafter, we refer to these simply as “blobs”. In their model, some blobs are in the background, and some are in the foreground of the system. In this geometry, the authors argued that the polarization variability could be explained if the star undergoes a form of nonradial variability of its brightness, which could be either dipolar pulsation or rotational variation, such that when the star appears brightest, an observer behind the star would see V CVn at maximum brightness, and vice versa. Because one blob appears to be behind the star, when we see V CVn at minimum light, the distant blob will see intercepted photons from the star at maximum brightness on the far side, and thus there will be more backscattered photons, and hence the polarization will be greatest.

The polarization variability of V CVn can thus be explained by both of these scenarios, but they each depend on the circumstellar medium (CSM) structures of V CVn and L₂ Pup. Understanding the CSMs of these stars offers important insights into how evolved semi-regular variable stars lose mass and potentially explains their future evolution into planetary nebulae. In a collaboration between professional and amateur astronomers, we considered new time-domain polarization observations of V CVn (led by co-authors Steenken and Simpson) that offer new tests of the two models mentioned. This is an especially valuable collaboration because V CVn is too bright for most large telescopes. Therefore it was advantageous to perform this investigation in cooperation with amateur astronomers. In Sect. 2, we discuss the tools and methods used to obtain new observations of V CVn. In Sect. 3, we compare the polarimetric observations with the visual brightness of the star. In Sect. 4, we present a review of potential analogs to V CVn and L₂ Pup, and we suggest that these stars are actually the prototypes of a class of polarimetric semi-regular variable stars (“V CVn type” stars). We discuss the various CSM and polarization models for V CVn to discern if and how these new observations might provide new insights into this system. Finally, in Sect. 5 we present our conclusions and suggestions for future observations.

2 Observations and data reduction

Optical polarization observations and photometry of V CVn were carried out independently by two of the co-authors (Steenken and Simpson; hereafter “observers”) operating Schmidt-Cassegrain-Telescopes (SCT). The observers were located in Munich, Germany (Steenken, Obi) and in Parton, Scotland (Simpson, Ob2). Both telescopes have dual-beam polarimeters with V-band filters and cooled CCD cameras mounted at the Cassegrain focus. Customary SCTs were used, with OB1 having a focal length of 2800 mm and aperture of 280 mm, and OB2 using a two different SCTs with focal lengths of 1500 and 2000 mm and apertures of 150 and 200 mm (Ob2) on a German equatorial mount. Object centering and autoguiding was achieved with separate guide scopes and guiding cameras.

Both polarimeters were built and calibrated in 2018–2019 to measure the linear optical polarization (P) of stars brighter than 9 mag, with an error of +0.1% polarization and ±5 degrees of polarization position angle (PA) in the relevant polarization range of V CVn in the past, that is P = 0.3–6.9%. An achromatic half-wave plate (HWP) was manually rotated into the positions 0, 22.5, 45, and 67.5 degrees to rotate the polarization plane by 0, 45, 90, and 135 degrees. The 0 degree position was aligned with the celestial north direction. The instrumental 0 degrees position was been nominally aligned with the celestial NS reference and offsets from this were checked during each observing session, using measurements obtained either by plate solving a stacked measurement image and/or by disabling tracking so as to generate an EW trace on measured images and therefore an NS reference. These offsets were removed during data processing. Calibration, which was performed later, uses high polarization standards to correct instrumental position angle offsets arising from any half-wave plate or Wollaston prism misalignments. A Wollaston prism splits of the star into a parallel and a perpendicular beam. The beams were passed through a V-band filter and focused onto a CCD detector of 1392 × 1040 pixels for a cooled 16-bit Astro camera. The optical design is shown in Fig. 1.

The images were captured and analyzed with standard image capturing and photometry software (for Ob1: AstroArt 6¹ and for Ob2: AstroImageJ²). The exposure times vary from 2 to 15 s depending on the star brightness, weather conditions, and telescope. For one observation, 40–100 images in each rotation position of the HWP were exposed. Batch processing was used for the dark field correction, stellar alignment, and photometry of the parallel and perpendicular stellar images. Aperture radii selection, based on the image’s full width at half maximum [FWHM] – typically 1.5–2 times FWHM for the signal and 3–5 times FWHM respectively for inner or outer annuli for sky background evaluation – provided a flux measurement that is robust against seeing variations, giving confidence in the method even when the object images were slightly distorted. Signal-to-noise ratios (S/N) of over 400 of each image taken as calculated by the photometry software were achieved, subject to atmospheric conditions. Figure 2 shows an example of an image made by Ob 1.

Fluxes of parallel and perpendicular beams were measured for every image i with the photometry software. Polarization (P), including correction for bias, and PA were calculated as described in Clarke (2010). For the error analysis, the study by Patat & Romaniello (2006) was used.

To calculate the measured Stokes Q_m and U_m parameters, parallel and perpendicular flux measurements are combined in the following way for angle positions θ:

(1)

(2)

(3)

where N_θ is the fractional photon count between parallel and perpendicular beams at the specified angel of the half-wave plate.

Any optical design of a polarimeter has to deal with instrumental polarization caused by the telescope or the optical components in the polarimeter, including the filter. To remove the instrumental polarization effects, the Stokes parameters of unpolarized stars in Table 1 were measured as zero-point offsets and calibrated Stokes parameters calculated using

(4)

(5)

Calibration standard high polarization stars such as ϕ Cas and 2H Cam were measured in agreement with the published values of Hsu & Breger (1982), as shown in Table 2. Both the instrumental set-ups and zero-point offsets were kept unchanged during the observing seasons from January until September to ensure comparable results. Calibration stars were observed each season to determine the stability and the reliability of the polarization position angle.

V-band photometry of V CVn was obtained directly before or after the polarimetric measurements with nearby comparison stars. Ob1 selected nearby star HD 116957, with mag(V) = 5.87, as a comparison. Ob2 used an ensemble of the four AASVO comparison stars³. The measured polarization is composed of the intrinsic polarization of V CVn and the interstellar polarization.

The star BD+46 1863 is located only about 2 arcmin southwest of V CVn at a distance of about 1670 pc and was used estimate the interstellar polarization. The measurements performed on the nearby stars TYC 3460-1345-1 and TYC 3460-2045-1 gave values of polarization between 0 and 0.5%. However, due to the high uncertainties, these measurements were not used. The estimate is as shown in Table 3 and is the basis for the calculation of the intrinsic polarization of V CVn.

Intrinsic Stokes Q_intr and U_intr parameters were calculated according to

(6)

(7)

The intrinsic polarization P_intr and intrinsic polarization angle PA_intr were calculated according to

(8)

(9)

The standard error of the mean values of Stokes Q, U were calculated according to Clarke (2010) using the student-t distribution for 95% confidence limits. Errors of P and PA and were calculated as follows:

(10)

(11)

In the data reduction process, polarization was de-biased using the Wardle & Kronberg (1974) formulation:

(12)

A total of 227 measurements were taken on 51 nights in 2020, 89 nights in 2021, and 87 nights in 2022, with a table of measurements given in Appendix B⁴. During this time span, V CVn went through five pulsation cycles, as shown in Fig. 3. The polarization measurements covered the first pulsation cycle in 2020, 2021, and 2022 (hereafter 1/2020, 1/2021, and 1/2022).

Fig. 1 Optical design of dual-beam polarimeter, with components along the optical path identified and labeled.

Fig. 2 Example of one image of V CVn split into parallel (bright star at top) and perpendicular beams (bright star at bottom) from 3 June 2022 at a 0 degree rotation position. The star below V CVn is BD+46 1863.

Fig. 3 V-band light curve of V CVn from 2020-2022. The red triangles are the photometry measurements provided by Ob1 and Ob2. The gray dots are for AASVO data (CCD measurements only). The three pulsation cycles covered by polarization measurements in this study in red, namely at the left pulsation cycle 1/2020 in the middle 1/2021 and at the right cycle 1/2022. It can be seen that the data from AASVO agrees well with the data measured by Ob1 and Ob2. Pulsation cycle 1/2021 showed the lowest brightness of all minima considered.

3 Results

The relationship between intrinsic polarization and brightness over the course of all three pulsation cycles is shown in Fig. 4. The polarization in the V band is highly variable, and is generally anticorrelated with brightness. The larger polarizations were measured around brightness minima and are consistent with earlier studies by Serkowski & Shawl (2001) and Davidson et al. (2014).

Thanks to the high cadence of measurements covering three pulsation cycles, the temporal relationship between the star’s light curve and polarization can be analyzed in more detail. Pulsation cycle 1/2021 showed the strongest obscuration associated with the highest measured intrinsic polarization of all three observed pulsation cycles. Of particular interest is the observation that the maximum polarization does not occur exactly at the time of the minimum brightness of V CVn. We observe that in the three cycles the polarization maximum can precede the brightness minimum by several weeks or follow the brightness minimum by weeks. Especially in cycle 1/2021, polarization led by about 16 days, and in cycle 1/2022 polarization lagged again by about 20 days. This lead/lag phenomenon was observed around brightness minima and is indicated by a gray arrow in Fig. 4. The pulsation cycle 1/2020 is shown in the left column of the figures. Polarization maximum of 1.47% was reached at JD 2459024. The brightness at 7.68 mag(V) minimum occurred 11 days (JD 2459013) before. Polarization was lagging the light curve. In the pulsation cycle 1/2021 (middle column), polarization was leading the light curve. The maximum of intrinsic polarization of 2.70% was measured at JD 2459338 16 days before the first minimum of the light curve, which occured at JD 2459354 and reached 7.99 mag(V). In this pulsation cycle, the star darkened significantly more and reached much higher polarization values than in the other two cycles. In the pulsation cycle 1/2022 (right column), polarization maximum followed the light curve minimum again. The second light minimum occurred on JD 2459743 at 7.79 mag(V), which was followed by the polarization maximum of 2.11% on JD 2459763 20 days later. Due to the irregular course of the light curve and the daily variations in brightness and polarization, an exact determination of the shift is not possible. Polarization showed strong variations while going through the maximum. There are indications that phase shifts between the curves may occur when the polarization changes significantly within a few weeks. During periods of little change in brightness, there appears to be little correlation between the curves. The position angle PA generally stays within a range of 80–120 degrees, except when the intrinsic polarization drops below 0.7%.

Fig. 4 Photopolarimetry of observed pulsation cycles. Intrinsic polarization curves in red are shown in the upper row. The light curves in blue are shown in the middle row. The intrinsic position angle (PA) is shown in the lower row in black. In all three pulsation cycles’ polarization and photometry are generally anticorrelated, and this is especially true around times of brightness maxima and minima. However, the curves are not exactly anti-correlated as indicated by the gray arrows. A phase shift between polarization and total light is most clearly visible around the brightness minimum of pulsation cycle 1/2021, but also seen in pulsation cycles 1/2020 and 1/2022. Comparing between the 3 cycles reveals that the polarization curve can either lead or lag the light curve. The intrinsic PA generally ranges from 80 to 120 degrees, except when the star is brighter than 7 mag(V) and/or the polarization drops significantly below 0.7%. At such times, the PA can drop below 50 degrees.

4 Discussion

4.1 A new class of V CVn-type stars

The Combined General Catalog of Variable Stars (Samus et al. 2004) lists 348 SRa and 1201 SRb stars, yet polarimetric measurements have only been published for a few of them. As a consequence, it is unknown whether significant variation of the polarization during a pulsation cycle of more than 1% is typical or exceptional for semi-regular variables.

While V CVn is not the only semi-regular evolved star with significant variable polarization, it by far has the greatest number of polarization measurements obtained since the 1960s. Table A.1 lists the basic stellar parameters of five other semi-regular stars that show polarimetric behavior similar to V CVn itself. The following points summarize their observed properties. UZ Ari (IRC +20052) appears similar to V CVn in many respects. It is situated at high galactic latitude (−31 deg) at a similar distance (539 pc), and has a tangential velocity of 59 km s⁻¹. The period of UZ Ari is 163 days, about 30 days shorter than that of V CVn. UZ Ari has a spectral class of M8, with an apparent brightness that is significantly lower than V CVn, and it varies between 11.8 and 12.6 mag(V). Baug et al. (2014) were able to measure the diameter of the star in the K-band during three lunar occultations. They found values of 4.5–6.0 ± 0.5 mas without a correlation to phase. The few published polarization measurements in the V band vary between 2 and 3.5%, with a decade-long stable PA around 130 degrees (Kruszewski & Coyne 1976; Baug et al. 2014). If future polarization measurements of UZ Ari confirm the stable range of PA, the physical origin of the asymmetry could be the same as in V CVn.

AK Peg has a very similar period of variability and spectral type to V CVn. At its distance of 1329 pc, the star ranges in brightness from 8.6–10.2 mag(V). The few polarization measurements published by Serkowski & Shawl (2001) indicated polarization values between 1.5 and 3.4%, which is a fairly large intrinsic level of polarimetric variability similar to UZ Ari.

RX Boo has variability that is classified as an SRb, with a distance of 156 pc. Using the Infrared Optical Array Imaging Interferometer (IOTA), Ragland et al. (2006) measured a stellar diameter of 17.5 mas in the H band that showed no asymmetry. Unfortunately, only a few older polarization measurements are available for this star (Vartanyan 1968), with values ranging from 0.4 to 1.9%. The change in polarization is comparable to UZ Ari and AK Peg.

Z UMa is an SRb star at a distance of 296 pc. Dyck & Jennings (1971) published polarization measurements ranging between 0.2 and 1.8%. Measurements at five epochs by Ob1 in 2021 (Fig. 5) confirmed variable polarization from 0.13 to 1.0%, showing an anticorrelation with brightness. Again, the amplitude of variable polarization is comparable to the previous three objects.

L₂ Pup is an SRb star on the southern hemisphere. The polarization measurements of Magalhaes et al. (1986b) reveal a high level of polarization variabilty between 0 and 8%, accompanied by a steady PA of 165–180 degrees. If the few historical polarization V-band measurements are compared with photometric data from the AASVO, the polarization appears anticorrelated with the brightness level. This level of variable polarization is considerably larger than the four preceding stars, but is comparable to V CVn.

Due to a distance of only 56 pc, and the superior imaging performance of VLT SPHERE/ZIMPol, it has been possible to detect a dust disk responsible for the asymmetry of L₂ Pup (Kervella et al. 2015). For the other stars listed above, the reasons for the asymmetries remain unclear. None of these stars are located in the galactic plane, with its higher interstellar density, but some of them show high tangential or radial velocities.

In Fig. 6, we see hints of a relationship between the tangential velocity and the observed maximum polarization. Stars with significant tangential velocities and slow winds, such as those from cool, evolved stars, would be expected to host a bow shock along the plane of the sky. That bow shock, if unresolved, would create a significant asymmetry on the plane of the sky with an associated polarization signature (Shrestha et al. 2021). The amount of polarization would depend on both the shape and density of the bow shock, along with the properties of the stellar wind. Shrestha et al. (2021) showed that polarization is a function of the type of dust in the bow shock, the density of the local interstellar medium, and the wind properties of the stars in question, so the apparent correlation of Fig. 6 could be consistent with the model proposed by Neilson et al. (2014). The polarization would be variable if, during the phase where V CVn is brightest, the star ejects a dusty shell whose density is sufficient to wash out the bow shock signal and make the system appear more symmetric (Shrestha et al. 2021).

Safonov et al. (2019) argued that V CVn cannot have a bow shock, due to a low ISM density expected from its high galactic latitude. However, the star is located only a few arcmin away from a molecular cloud discovered by Reach et al. (1994). The molecular cloud G107.4 +70.9 is one of the densest clouds in that region, and lies roughly at the same distance as V CVn, according to a study from Gladders et al. (1999). It is plausible that the interstellar density near the fast-moving V CVn could be large enough to allow the formation of a bow shock.

Fig. 5 Polarization measurements (red triangles) and photometry (blue dots) of Z UMa in 2021 (Ob1). Five epochs of photopolarimetric measurements were obtained. The data give indications of an anticorrelation between the measured polarization and brightness.

Fig. 6 Maximum polarization versus tangential velocity of the stars in Table A.1, indicating a higher maximum polarization with increased tangential velocity.

4.2 Possible models of the polarimetric variability

The polarimetric variability of V CVn is surprising and counterintuitive. When the star reaches maximum light, the polarization approaches a minimum, and vice versa. One might expect that if the source of the polarization were constant – which would yield a constant position angle – then at maximum light, the number of polarized photons would also be at a maximum. However, since the polarization is a normalized quantity, one would find that the polarization is also constant. Satisfying the observations requires either of the following scenarios: (1) the source of the polarization interacts with a different number of photons relatively to what the observer sees (Safonov et al. 2019); or (2) the source of the polarization changes, yet it maintains the same deviation of symmetry on the sky (Neilson et al. 2014).

The results of this work impose an additional requirement that a viable model should explain observed time differences between the polarization maximum and the light minimum. If we consider the first option, Safonov et al. (2019) suggested that one of the dusty “blobs” is behind the star relative to the observer (see Sect. 1). They suggested that if the light variation is not due to radial pulsation, and instead is some type of dipolar variability or rotational variability, then when the observer measures minimum light, the other side of the star appears at maximum light; hence, the blob “sees” the greatest number of polarizing photons that get backscattered to the observer. This model is intriguing, but requires the stellar variability to be asymmetric and non-radial. The pulsation amplitude of V CVn is about 2 mag(V), and there are no known stars that pulsate with this amplitude non-radially. If this were a rotational phenomenon, combining a period of about ≈194 d with an assumed stellar radius on the order of 100 R_⊙ would imply the rotation rate is about 25 km s⁻¹ and is the same order as the critical rotation rate for the star.

An additional challenge for this model is the apparent lead and lag time between polarization and flux. If the issue were only that the polarization maximum lagged behind the luminosity, one might explain it as a light-time delay of t_d = 22d_b/c, where d_b is the distance between the star and blob. Safonov et al. (2019) reported a radius of the blobs around V CVn of 35 ± 1 mas. Using the latest distance data from Gaia Collaboration (2020) of 501 pc, this results in a distance of the bipolar cloud of 17.5 AU. The time delay caused by the light travelling between the assumed bipolar cloud and the star is then 2.5 h, and thus by far not enough to explain the observed time difference in our observational data between the light and polarization curves of up to two weeks.

Neilson et al. (2014) suggested that the variable polarization is caused by a pulsation-driven dusty wind shell (see also Willson 2000; Höfner & Freytag 2022) that collides with a stellar wind bow shock, which is expected since V CVn is a runaway star. They argue that the shell is driven outwards from the star near maximum light, and because the shell is symmetric, the polarization is then at a minimum. However, the position angle will remain constant because of the constant presence of the asymmetric bow shock around the shell. As the shell expands and its density decreases, the polarization is set more by the near-constant density bow shock. This result is consistent with the apparent connection between the polarization amplitude and the tangential velocity observed for V CVn and its potential analogs, as described in Sect. 4.1. However, this model does not explain the apparent lead/lag time.

Additionally, instead of an asymmetry due to the presence of bow shocks, the wind may simply be asymmetric and variable. Simulations performed by Aronson et al. (2017) using nonspherical circumstellar shells that contain clumped material predicted significant net polarization signals. If the maximum polarization occurs near the minimum light, then the wind at that phase could be most asymmetric, and at maximum light and minimum polarization the wind would be most symmetric. It is worth noting that this argument is independent of wind density, since the polarization observed for V CVn is normalized and is thus only concerned with changes in the symmetry of the wind as projected against the sky.

Asymmetric stellar winds are rare, but they can be caused by rapid rotation as suggested for Be supergiants (e.g., Granada et al. 2010; Georgy et al. 2011), magnetic fields as seen in hot, massive stars with strong magnetic fields (e.g., Erba et al. 2021; Subramanian et al. 2022), or perhaps convection as suggested for massive evolved red supergiant stars (e.g., Kamiński 2019; López Ariste et al. 2022). There is no evidence that V CVn is a rapidly rotating star. However, given its evolved stage of evolution, its critical rotational velocity is small. Rotation might be important, and it could be consistent with a small population of variable-polarization evolved stars. Furthermore, one would expect the rotation to be slow; however, it is possible there were dynamic events between the progenitors of these stars, and a companion spun up the progenitor (e.g., Staritsin 2022), while a supernova explosion or another dynamic event ejected the companion (Dorigo Jones et al. 2020). In that case, the progenitor would evolve into a runaway red giant star with rapid rotation. While this scenario may be possible, it is not obvious how to test this idea, and the proposed scenario should be a rare occurrence at best.

Strong magnetic fields (>1kG) are not common among evolved stars (Grunhut et al. 2010); however, strong magnetic fields in hot stars are known to shape their stellar winds and make them asymmetric (e.g., ud-Doula & Owocki 2002). In hot (O- and B-type) stars, the magnetic field tends to be nearly dipolar, while in evolved stars the magnetic field is more tangled and similar to that of the Sun. It is not clear that magnetic fields in this case can impact the wind structure in a meaningful manner, but the existence of a maser (Wolak et al. 2012) around V CVn implies that follow-up measurements would be a valuable test, and could be related to the observed polarization lead/lag.

Convection is unlikely to be the main source of polarizing photons in these red giant stars. Convection is not constant, and will create spots of some lifetime over different parts of a star that is not periodic. As such, the variable polarization should not be correlated with the brightness variation.

Although convection cannot be the main source of polarization, it is possible that the polarization lead/lag is caused by convection. It is known that, for pulsating Cepheids, convection can perturb their regular light curves to vary cycle-to-cycle (Derekas et al. 2012, 2017; Neilson & Ignace 2014), and this is a potential cause of pulsation instability in variable red giant and supergiant stars. For V CVn, it is likely that convection and granulation make the times of maximum and minimum light unstable and variable in the same way. However, if the stellar wind is due to a dust shell being formed in the photosphere, then the wind is correlated more with the variability of the stellar radius and not the variability of the stellar luminosity. In Fig. 7, we sketch the relationship between the pulsating star and the bow shock. We assume that with each pulsation cycle a different pattern of convection cells is created. In a pulsation cycle -where the dust is most dense first along the line of sight and only delayed in the direction of the bow shock- the brightness minimum occurs before the polarization maximum, and the polarization curve lags behind the brightness curve, as observed in pulsation cycle 1/2020, and vice versa in cycle 1/2021.

Fig. 7 Schematic representation of the interaction between the pulsating star and the bow shock. Top: at maximum brightness and minimum polarization, the dust is accelerated from the photosphere in a dense shell (circle) and the mass-loss rate is greatest. The star moves relatively to the interstellar medium in the direction of the arrow. The wind shell is denser than the bow shock (light blue) and dominates the fraction of scattered light. Bottom: around minimum light there is less radiative acceleration, and dust forms in the photosphere, while the mass-loss rate is much smaller. The shell formed at the previous flux maximum has expanded asymmetrically. The cloud-like structure of the dust envelope is supposed to show turbulence. A bow shock (blue) dominates the observed polarization and is incorporated into the smaller density CSM. A lead/lag of the polarization and the light-curve can occur if convection causes the dust density in the line of sight and in the bow shock to develop differently.

5 Conclusions

The new high-cadence photopolarimetry of V CVn presented here raises new questions about the nature of the polarization variability both of this star and other semi-regular variable stars. In particular, we find that the polarization maximum does not occur at the same time as light minimum, and that time will be a lead or a lag. This phenomenon is not naturally explained by the models proposed by either Neilson et al. (2014) or Safonov et al. (2019). Our investigation supports the hypothesis that the variable polarization is caused by a bow shock and that the observed phase shifts between polarization and light curves are caused by convective processes in the stellar photosphere. As such, we find that more high-cadence polarization measurements and simulations of V CVn-type stars would be important if we wish to understand the relationship between their stellar properties, their motions, and their circumstellar media.

Acknowledgements

The authors gratefully thank the Referee for the constructive comments and recommendations which helped to improve the paper. We acknowledge with thanks the variable star observations from the AAVSO International Database contributed by observers worldwide and used in this research. We thank https://zenodo.org/ for hosting our data (https://zenodo.org/record/7997101). This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. M.S. is supported by an STFC consolidated grant number (ST/R000484/1) to LJMU. H.N. acknowledges funding from NSERC.

Appendix A The sample of variable-polarization semi-regular variable stars

Appendix B The recorded photometric measurements of V CVn

References

All Tables

All Figures

	Fig. 1 Optical design of dual-beam polarimeter, with components along the optical path identified and labeled.
In the text

	Fig. 2 Example of one image of V CVn split into parallel (bright star at top) and perpendicular beams (bright star at bottom) from 3 June 2022 at a 0 degree rotation position. The star below V CVn is BD+46 1863.
In the text

Fig. 3 V-band light curve of V CVn from 2020-2022. The red triangles are the photometry measurements provided by Ob1 and Ob2. The gray dots are for AASVO data (CCD measurements only). The three pulsation cycles covered by polarization measurements in this study in red, namely at the left pulsation cycle 1/2020 in the middle 1/2021 and at the right cycle 1/2022. It can be seen that the data from AASVO agrees well with the data measured by Ob1 and Ob2. Pulsation cycle 1/2021 showed the lowest brightness of all minima considered.

In the text

Fig. 4 Photopolarimetry of observed pulsation cycles. Intrinsic polarization curves in red are shown in the upper row. The light curves in blue are shown in the middle row. The intrinsic position angle (PA) is shown in the lower row in black. In all three pulsation cycles’ polarization and photometry are generally anticorrelated, and this is especially true around times of brightness maxima and minima. However, the curves are not exactly anti-correlated as indicated by the gray arrows. A phase shift between polarization and total light is most clearly visible around the brightness minimum of pulsation cycle 1/2021, but also seen in pulsation cycles 1/2020 and 1/2022. Comparing between the 3 cycles reveals that the polarization curve can either lead or lag the light curve. The intrinsic PA generally ranges from 80 to 120 degrees, except when the star is brighter than 7 mag(V) and/or the polarization drops significantly below 0.7%. At such times, the PA can drop below 50 degrees.

In the text

	Fig. 5 Polarization measurements (red triangles) and photometry (blue dots) of Z UMa in 2021 (Ob1). Five epochs of photopolarimetric measurements were obtained. The data give indications of an anticorrelation between the measured polarization and brightness.
In the text

	Fig. 6 Maximum polarization versus tangential velocity of the stars in Table A.1, indicating a higher maximum polarization with increased tangential velocity.
In the text

Fig. 7 Schematic representation of the interaction between the pulsating star and the bow shock. Top: at maximum brightness and minimum polarization, the dust is accelerated from the photosphere in a dense shell (circle) and the mass-loss rate is greatest. The star moves relatively to the interstellar medium in the direction of the arrow. The wind shell is denser than the bow shock (light blue) and dominates the fraction of scattered light. Bottom: around minimum light there is less radiative acceleration, and dust forms in the photosphere, while the mass-loss rate is much smaller. The shell formed at the previous flux maximum has expanded asymmetrically. The cloud-like structure of the dust envelope is supposed to show turbulence. A bow shock (blue) dominates the observed polarization and is incorporated into the smaller density CSM. A lead/lag of the polarization and the light-curve can occur if convection causes the dust density in the line of sight and in the bow shock to develop differently.

In the text