CHARA Near-infrared Imaging of the Yellow Hypergiant Star ρ Cassiopeiae: Convection Cells and Circumstellar Envelope

Yellow hypergiants (YHGs) represent a brief yet pivotal evolutionary phase for massive stars as they transition from or to red supergiant phases (de Jager 1998). These YHGs typically have initial masses of 20−40M_⊙ (Ekström et al. 2012; Kourniotis et al. 2022) and exhibit high luminosity ranging from $5.4\leqslant \mathrm{log}(L/{L}_{\odot })\leqslant 5.8$ (Neugent et al. 2012; Dorn-Wallenstein et al. 2023; Humphreys et al. 2023). Their effective temperatures fall approximately within the range of T_eff ∼ 4000−7500 K (excluding eruptive episodes; Neugent et al. 2012). One of their defining characteristics is their remarkable variability in spectral type in a short period of time. This spectral variability is linked to another unique feature: their inherent instability around a temperature of T_eff ∼ 7000−12,000 K. In this temperature region, YHGs experience an instability zone known as the “Yellow Void” (Nieuwenhuijzen & de Jager 1995; de Jager 1998; Nieuwenhuijzen & de Jager 2000; Stothers & Chin 2001; Glatzel & Kraus 2024). In this yellow void region, the effective gravity force becomes zero, leading to episodic mass loss in these stars. As the star loses mass, its effective temperature decreases, causing it to move redward on the H-R diagram and leave the yellow void. Subsequently, the temperature starts to increase, moving the star move blueward again on the H-R diagram until it enters the yellow void once more. The star may oscillate between hot and cool evolutionary phases, undergoing multiple “blue loops” (de Jager 1998; Stothers 2012). For instance, ρ Cas lost matter at a rate of up to 5.4 × 10⁻² M_⊙ yr⁻¹ during the mass-loss episode it experienced in the years 2000–2001 (Lobel et al. 2003). These mass-loss episodes often manifest as visual dimming events followed by subsequent brightening in the star’s light curve. The dimming occurs because newly ejected material condenses into dust around the star, temporarily obscuring it from our view. This phenomenon is similar to the dimming events observed in other massive stars like Betelgeuse (Montargès et al. 2021) and RW Cep (Anugu et al. 2023). Spectroscopic observations during these dimming events reveal a rapid drop in temperature and the presence of expanding circumstellar shells. These shells are detected by analyzing specific spectral lines, such as CO band emissions and the wings of the Hα line (Lobel et al. 2003).

The significant mass loss experienced during the YHG phase profoundly influences the star’s fate, ultimately determining its position on the H-R diagram at the end of its life (de Jager 1998). This mass loss can lead to the transformation of the YHG stars into hot supergiants such as luminous blue variables, B[e] supergiants, or Wolf–Rayet stars, as well as influence the type of supernova and resulting remnant. Furthermore, a few potential YHG supernova progenitors have been identified, but controversy surrounds them because they do not fully meet the mass and luminosity criteria typically associated with YHGs (Maund et al. 2011; Bersten et al. 2012; Georgy 2012; Groh 2014; Mauerhan et al. 2015; Klochkova 2019; Kilpatrick et al. 2021).

YHGs serve as invaluable laboratories for investigating the mechanisms behind mass-loss events and for testing theories of stellar evolution, given the active and pivotal role of the YHG phase in post-red-supergiant evolution. However, their rarity presents a significant challenge. They are relatively scarce because of their short-lived phase, typically lasting less than 10⁴ yr. Additionally, their substantial mass loss can lead to self-obscuration by dust formed from the cooled ejecta, which makes them difficult to detect directly. Currently, we are aware of approximately 35 objects classified as YHGs or YHG candidates in our the Milky Way (de Jager 1998) and neighboring galaxies of the Local Group, such as the Small Magellanic Cloud (Neugent et al. 2010; Kourniotis et al. 2022), the Large Magellanic Cloud (Neugent et al. 2012; Dorn-Wallenstein et al. 2023; Humphreys et al. 2023), and M33 (Humphreys et al. 2006; Drout et al. 2012). Consequently, only a handful have been extensively studied, with IRC+10420 and ρ Cas being two of the most prominent examples. Both of these stars have asymmetric circumstellar shells and have undergone episodic, high-mass-loss events spanning more than a hundred years. IRC+10420, with T_eff = 7930 ± 140 K (Nieuwenhuijzen & de Jager 2000) has been seen to be evolving in the H-R diagram from the red to the blue and is believed to be crossing the Yellow Void to evolve as a low-luminosity Luminous Blue Variable. This dramatic shift is evidenced by a change in spectral type from F8Ia to early A5Ia in 30 yr, indicating a rapid temperature increase of ∼120 K per year (Oudmaijer et al. 1996; Klochkova et al. 1997; Oudmaijer 1998; Klochkova et al. 2002). Additionally, it experienced a mass-loss rate of 2 × 10⁻³ M_⊙ yr⁻¹ until about 2000 yr ago, and since then the rate has decreased (Shenoy 2016). This expelled material has sculpted a complex circumstellar environment, filled with knots, arcs, shells, and even jet-like structures (Tiffany et al. 2010).

In this study, we focus on ρ Cas, an archetype YHG. Table 1 summarizes the properties of ρ Cas, confirming its status as a YHG. With a mass of ∼40M_⊙, luminosity range $\mathrm{log}(L/{L}_{\odot })\,\sim 5.48-5.72$, and effective temperature averaging ∼7000 K, ρ Cas aligns perfectly with the established parameters for YHGs. Additionally, ρ Cas is a well-known for its historical outburst episodes of dimming and rebrightening, with a visual brightness change of approximately 1.5−2 mag (see Figure 1). It experienced four documented outbursts in the last century that happened in 1945–1947 (Beardsley 1961), 1985-1986 (Boyarchuk et al. 1988; Leiker et al. 1989), 2000–2001 (the “Millennium outburst”; Lobel et al. 2003), and 2013–2014 (Kraus et al. 2019). Additionally, ρ Cas is known for a dramatic decrease in the effective temperature by 3000 K due to the ejection of a hydrogen shell (Lobel et al. 2003). These outbursts in ρ Cas resulted in the formation of circumstellar material and multiple detached shells, which were observed using spectroscopic Hα wings, Ca ii, and CO band emissions (Sargent 1961; Lambert et al. 1981; Lobel et al. 1998, 2003; Kraus et al. 2019). A high-spectral-resolution analysis by Lobel et al. (2003) reported that, during the 2000–2001 outburst, the mass loss in this star reached a peak of approximately 5.4 × 10⁻² M_⊙ yr⁻¹, resulting in the formation of an optically thick circumstellar gas shell of 3 × 10⁻² M_⊙ during 200 days after outburst. The analysis of the visual light curve reveals that the outburst episodes are not only becoming more frequent, but also exhibiting a reduction in their duration (Maravelias & Kraus 2022). This observed pattern hints at the possibility that ρ Cas may be transitioning into the next phase of its evolutionary cycle traversing the Yellow Void (Kraus et al. 2019, references within). Analysis of infrared photometry data suggests dust formation occurred between 1973 and 1983, likely from material ejected during the powerful 1946 outburst (Jura & Kleinmann 1990). However recent mid-infrared observations find no evidence of subsequent dust formation after 1946 (Shenoy 2016). This suggests that ρ Cas might be in an earlier evolutionary stage compared to other hypergiants (such as IRC+10420 and HD 179821) that show a significant infrared excess from dust shells (Castro-Carrizo et al. 2007; Quintana-Lacaci et al. 2008).

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In this paper, our objective is to study the fundamental parameters of ρ Cas to gain insight into the mechanisms driving its mass loss. The main mechanisms that contribute to mass loss are pulsationally driven and radiation pressure line driven stellar winds (Smith 2014). The study of the Great Dimming episode of Betelgeuse suggests that the trigger for mass loss may be related to the combined effects of giant convective cells on the stellar surface, pulsations, and the magnetic field of the star (Dupree et al. 2022; Humphreys & Jones 2022). However, distinguishing between these mechanisms remains challenging. Our goal is to enhance our understanding of these underlying mechanisms by imaging the stellar surface and circumstellar environment of ρ Cas.

Here, we present the first high-angular-resolution images of ρ Cas photosphere, captured using the Georgia State University Center for High Angular Resolution Astronomy (CHARA) Array situated at the Mount Wilson Observatory (ten Brummelaar et al. 2005; Schaefer et al. 2020). The CHARA images reveal unprecedented details of the stellar surface, including giant convective cells. These cells are detected as hot and cool spots on the stellar surface, indicative of stellar activity, which may have triggered the episodic mass-loss events observed in the past. We describe the interferometric observations and the reduction of the data from the CHARA Array in Section 2. As a first step in the data analysis, we derived diameter measurements using uniform disk (UD) and limb-darkened disk (LDD) model fitting (Section 3). We utilized three different image reconstruction techniques to extract images from the interferometric observations. The image reconstructions of the continuum and carbon monoxide (CO) spectral line bands are provided in Section 4 and Section 5, respectively. Finally, we compare ρ Cas with other hypergiants and discuss the implications of these observations for models of dimming and mass ejection in Section 6.

We observed ρ Cas at four epochs using the CHARA Array between 2023 October and November (see Table 2). The CHARA Array is the world’s largest optical and near-infrared interferometer, having baseline lengths spanning from B = 34 to 331 m. In the H-band (λ = 1.50−1.74 μm) and K-band (2.00−2.38 μm) wavelength range, it provides angular resolutions of approximately λ/2B ∼ 0.5 and 0.6 mas, respectively. We used the H-band MIRC-X (Anugu et al. 2020) and K-band MYSTIC (Setterholm et al. 2022) beam combiners in a cophasing observing mode. In this mode, we collected data simultaneously from both MIRC-X and MYSTIC instruments, with MYSTIC serving as the primary fringe tracker, controlling the CHARA main delay lines, while MIRC-X functioned as the secondary science beam combiner, managing internal differential delay lines. MIRC-X and MYSTIC are six telescope beam combiners and deliver 15 squared visibilities (V²) and 10 independent closure phases (ϕ) in each operating wavelength band. The observations were carried out with a spectral resolving power of R = 190 for MIRC-X and R = 278 for MYSTIC. The observations cover 30 and 52 spectral channels across the near-infrared H and K-bands.

To ensure better calibration of our observations, we employed multiple calibrators (see Table 2). We chose calibrators by cross-checking with the Gaia archive to avoid known binaries. These calibrators are used to correct any visibility loss in the target observations caused by time-variable factors, including fluctuations in atmospheric coherence time, vibrations, differential dispersion, and birefringence in the beam path.

The raw data sets are available on the CHARA data archive portal, ⁹ and these raw data sets are reduced using the standard mircx pipeline (Anugu et al. 2020; Le Bouquin et al. 2024). The pipeline provides uncalibrated oifits files of the science target comprising measurements of the spectral distributions of flux, squared visibility, and closure phases. To assess whether the calibrator stars displayed any binary signatures, we calibrated the data from these calibrator stars against each other. To detect binary signatures in these data, we fit models using a grid-search script from the Python-based Parametric Modeling of Optical InteRferomEtric Data software, known as PMOIRED (Mérand 2022; Mérand et al. 2024). We found that the calibrator stars did not exhibit any binary signatures. The calibration was achieved with the IDL mircx_cal.script (Monnier et al. 2012; Anugu et al. 2020).

On the night of 2023 October 28, there was a relatively higher systematic visibility bias attributed to atmospheric seeing conditions. During our investigation, we performed imaging analyses both with and without incorporating the data from this particular night. The results did not exhibit substantial differences that might impact the overall outcomes of our study. Therefore, we opted to exclude the 2023 October 28 data set in this paper.

Figure 2 illustrates the (u, v) plane coverage of the combined observations, which provides a good uniform sampling and yields high-quality image reconstructions. The effective resolution of interferometric synthetic images is fundamentally determined by the maximum extent of the (u, v) coverage. The sparseness or incompleteness in the filling of the (u, v) plane tends to create artifacts in these images. Figure 3 shows the visibilities V² and closure phases (ϕ) of ρ Cas in the K band. The closure phases exhibit large, nonzero values that vary significantly with wavelength and show markedly different behaviors among the various triplets of telescopes. This indicates that the target has a strongly asymmetric spatial flux distribution.

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We initially focused on determining the angular diameter of the star from the continuum data set. To achieve this, we employed two distinct stellar disk models. The first model is the uniform disk model (UD), while the second is a power-law limb-darkened disk model (LDD), I(μ)/I₀ = μ^α , where μ is the cosine of the angle between the surface normal and line of sight and it is 0 ≤ μ ≤ 1. The power-law coefficient is α, which varies α ≥ 0. We combined all the epochs into a single data set and used PMOIRED software for this model fitting. The best-fit parameters are summarized in Table 3. Applying the uniform disk model, we calculated the star diameter to be 2.0 ± 0.1 mas. On the other hand, using the LDD model, we obtained a slightly different diameter of ${2.1}_{-0.02}^{+0.01}$ mas, with the limb-darkening coefficient α_H = 0.47 ± 0.04 at H band and α_K = 0.38 ± 0.08 at K band. It is important to note that both models yielded relatively high fitting errors, as indicated by the reduced chi-squared χ² > 7 (see Figure 4). We also fit LDD for the individual epochs and found that the diameter and α changed by ±0.1. The reason for the error scatter occurring between epochs is that this simple two-parameter model is inadequate, as the star exhibits asymmetry in its shape (see Appendix, Appendix B). This limb-darkening fit overestimates α in the presence of surface features. See the estimations from the image reconstructions in Section 4.

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Figure 3 illustrates that the V² in the spectral channels covering the CO bandhead emission line, i.e., between wavelengths in the range of λ = 2.3−2.38 μm is lower than the continuum, suggesting that the target is more resolved as compared to the continuum. We measured the diameter of the molecular features of the CO first-overtone bands, and it is 2.91 ± 0.19 mas. The extended structure observed above the photosphere (with a diameter of ${2.08}_{-0.02}^{+0.01}$ mas) is interpreted as a circumstellar CO shell environment. It is also confirmed in the image reconstruction (see Section 5). The presence of a circumstellar CO shell is a common phenomenon in massive evolved stars (e.g., Perrin et al. 2005; Ohnaka et al. 2011; Montargès et al. 2014; Anugu et al. 2023). The limb-darkened angular diameter measurements are summarized in Table 3. We expect that the star appears larger at wavelengths where the opacity is larger, and variations in the reported diameters from the visible to near-IR to CO-band appear to be approximately consistent with opacity variations.

We also investigated whether ρ Cas might be a compact binary system considering its high $v\sin i=25$ km s⁻¹ (Table 1), such as found for the yellow hypergiant HR 5171 A (HD 119796) that has been identified as a contact binary through interferometric observations (Chesneau et al. 2014). To explore this possibility, we conducted a search for a companion using the PMOIRED grid-search method within a field of view up to 50 mas. However, no signals indicative of the presence of a companion were detected. Long-term spectroscopic monitoring (8.5 yr, Lobel et al. 2003) and photometric observations (Maravelias & Kraus 2022) also support the conclusion that there is no companion associated with ρ Cas.

In the second phase of our data analysis, we concentrated on creating model-independent aperture synthesis images of the continuum wavelength spectrum data. The incomplete (u,v) coverage from the interferometry presents an ill-posed inverse problem, hindering the ability to derive a unique image from the measured set of visibilities. The reconstruction of images involves the minimization of a cost function comprising both data and regularization terms. The data consist of squared visibility V² and closure phase ϕ and the regularization term incorporates a set of a priori assumptions about the target, such as positivity and smoothness. We merged all MIRC-X and MYSTIC epochs into respective combined data sets to perform image reconstructions on the consolidated (u, v)-coverage data sets. Appendix A shows the image reconstructions from the best (u, v)-coverage individual epochs.

4.1. Strategy of the Image Reconstructions

We employed a diverse range of imaging techniques for these image reconstructions for the purpose of enhancing the reliability of the reconstructed images: (i) OITOOLS (Martinez et al. 2021; Norris et al. 2021), ¹⁰ (ii) ROTational Image Reconstruction (ROTIR; Martinez et al. 2021), ¹¹ and (iii) SURFace imagING (SURFING; Roettenbacher et al. 2016, 2017; Martinez et al. 2021; Anugu et al. 2023). OITOOLS and ROTIR are written in Julia, while SURFING is written in the IDL programming language.

OITOOLS utilizes a quasi-Newton method that incorporates rapid gradient minimization through Variable Metric Limited Memory with Bounds (VMLMB). This image tool is specifically designed for two-dimensional imaging applications.

On the other hand, ROTIR and SURFING are specialized in imaging three-dimensional surfaces of stars. These techniques assign specific intensity values to each element on the three-dimensional surface of a star, allowing for the creation of detailed stellar surfaces. Parameters such as angular diameter, equatorial to polar axis ratio, inclination, position angle of the rotation pole, limb-darkening coefficient, rotation period, epoch, and visibility at a spatial frequency of 0 are considered in the fitting.

The initial step in image reconstruction involved determining the best-fit power-law limb-darkened angular diameter, which sets the outer boundary for the assigned flux. The best-fit diameters, denoted θ = θ_LDD, are presented in Table 3.

In OITOOLS, we applied positivity constraints and edge-preserving regularization ℓ₂ − ℓ₁. The regularizer weight (hyperparameter) was optimized using the L-curve method. To mitigate the effects of local minima, we initiated 1000 reconstructions from random initial generated images. We also searched a range of pixel scales (0.05–0.2 mas) and fields of view (6.4–12 mas) and eventually settled on pixel scale 0.05 mas, with a final field of view of 6.4 × 6.4 mas. Despite varying different regularizers, a consistent large-scale asymmetry was observed in the images (top row in Figure 5). For additional verification, we reconstructed images from data averaged over 10 minutes integrations, assessing their robustness against those from shorter intervals (150 s). Both integration sets gave the nearly the same images.

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In ROTIR and SURFING, the star is assumed to be a sphere, so the ratio of the equatorial to polar axes is set to 1. The rotational distortion of the star is probably small. We computed the Roche figure for the mass, polar radius, and projected equatorial velocity given in Table 1, and we found that the difference between the projected major and minor axes is less than 3%. We fix the stellar inclination and the position angle of the axis of rotation in the global model because the time between the epochs of observation is significantly less than the approximately 3 yr rotation period expected for ρ Cas. We arbitrarily set the epoch to the first night of our observation.

For ROTIR spheroid image reconstruction, we fitted several stellar parameters with initial uniform prior distributions within a wide range of values taken from Table 3: the angular size of the radius at the pole (1.05 mas), surface temperature, and limb-darkening power-law coefficient (α_H ∼ 0.47 and α_K ∼ 0.38, Figure 4). We used a total variation regularizer with weight of 0.05. The middle row of Figure 5 shows the ROTIR images.

The SURFING software fits the stellar diameter as part of the image reconstruction process, which resulted in diameters of 2.04 and 2.10 mas for the H-band and K-band wavelengths, respectively. We used fixed limb-darkening coefficients (α_H ∼ 0.47 and α_K ∼ 0.38) derived from Section 3. The bottom row of Figure 5 shows the SURFING images.

Figure 5 displays the images from the OITOOLS (top row), ROTIR (middle row), and SURFING (bottom row) software, respectively. A visual inspection of the images reveals that there are common characteristics and some differences between them. We were concerned whether the ring-shaped bright spots observed in the images (Figure 5) are real or result from the incomplete (u, v) coverage. To explore how the reconstructed images were affected by various imaging and optimization choices, we generated synthetic data sets. The synthetic OIFITS data sets were generated numerically using a limb-darkened disk model based on the power-law coefficient and with the angular diameter θ_LDD = 2.1 mas measured from the observed data (from Table 3) and also mimicking our actual observations, including the coverage (u, v) and the signal-to-noise ratios. The image reconstructions were generated from these data sets and these did not show same image patterns (Figure 6), indicating that the light and dark patterns are not an artifact of the incomplete coverage (u, v). Appendix A displays individual epoch image reconstructions that exhibit excellent agreement with the image reconstructions derived from the combined data.

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Figures 7 and 8 show the fitting residuals for the ROTIR image reconstruction. The fitting residuals for ROTIR and SURFING are also qualitatively similar in appearance. The reduced χ² values were 1.3 for MIRC-X data fits and 1.2 for MYSTIC data fits. We did not perform image reconstructions with the full K-band wavelengths, including the CO lines. However, we isolated the part of the spectrum recording the CO lines (λ = 2.3−2.38 μm) and conducted image reconstructions that we discuss in Section 5.

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4.2. Comparison between Imaging Results

We isolated the CO lines (λ = 2.3−2.38 μm) from the K-band data and performed image reconstructions using OITOOLS and SURFING, employing the same methods outlined in the previous section. Figure 9 shows the images, revealing two components: a central star photosphere and an outer circumstellar envelope. The OITOOLS image reveals an asymmetric shape of the circumstellar envelope that appears smaller along the northeast to southwest direction. The OITOOLS and SURFING software measure the outer diameter of the circumstellar envelope as 2.91 ± 0.19 mas and 2.96 mas, respectively. We have higher confidence in the OITOOLS image in this case compared to the SURFING software, because the circumstellar envelope does not necessarily need to be a spherical shape. ROTIR was not implemented for these particular data, as it also enforces a spherical shape, which may not be suitable for this asymmetric shape of the circumstellar envelope.

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6.1. Physical Size of Photosphere and Circumstellar Envelope

6.2. Convection Cells and Mass Loss

6.3. The Occurrence of Secondary Long Periods Driven by Giant Convection Cells?

6.4. Limb Brightening?

In this paper, we present high-angular-resolution (∼0.5 mas) observations of the yellow hypergiant ρ Cas that we made using the CHARA Array, employing the MIRC-X and MYSTIC instruments in H- and K-band wavelengths. Fundamental parameters of ρ Cas were determined using two model-fitting techniques and three different image reconstruction techniques analyzing the data. Our model fitting constrains the stellar diameter to 2.09 ± 0.02 mas. Assuming a distance range of 2.5–3.1 kpc, ρ Cas has a radius of 564−700 R_⊙ in the near-infrared.

The primary result of our study is the first surface imaging of ρ Cas, showcasing unprecedented details of its photosphere. These images reveal four to five bright spots, which we interpret as manifestations of convection cells. We suggest that these convection cells might have played a pivotal role in ejecting gas clouds from the stellar surface, thereby contributing to the episodic mass-loss events recorded previously.

Our analysis also shows that the V² and ϕ data appear different in the CO bandhead emission lines (λ = 2.31−2.38 μm), leading to the detection of an extended circumstellar envelope of size with a diameter of 2.91 ± 0.19 mas. Such envelopes are characteristic of cool and luminous stars.

Future follow-up observations of ρ Cas with the CHARA Array will provide a comprehensive monitoring of the dynamics of the convection cells on the star’s photosphere. Furthermore, linking these observational data with future episodes of stellar dimming will be crucial to advance our understanding of the mass-loss phenomena and their driving mechanism in massive stars. We are planning to continue the CHARA Array observations to explore the correlation between the evolving convection cells in the images and the associated photometric variations.

Yellow hypergiants are extremely rare; currently, only about 10 objects have been identified in our the Milky Way. In particular, only five of these objects (ρ Cas, HR 8752, IRC +10420, HD 179821, RW Cep) are accessible for observation in the northern sky. Among them, ρ Cas and RW Cep are only large enough for stellar surface imaging, making them crucial for follow-up imaging studies. We aim to extend CHARA observations to the remaining yellow hypergiants to measure their physical diameters and gain a detailed understanding of these objects.

We express our sincere gratitude to the anonymous reviewer for the valuable and constructive comments, which contributed to the improvement of the paper. We thank Stephen Ridgway for his insightful comments, which contributed to the improvement of the paper. This work is based upon observations obtained with the Georgia State University Center for High Angular Resolution Astronomy Array at Mount Wilson Observatory. The CHARA Array is supported by the National Science Foundation under grant No. AST-1636624, AST-1908026, and AST-2034336. Institutional support has been provided from the GSU College of Arts and Sciences and the GSU Office of the Vice President for Research and Economic Development. F.B. acknowledges funding from the National Science Foundation under grant No. AST-1814777. J.D.M. acknowledges funding for the development of MIRC-X (NASA-XRP NNX16AD43G, NSF AST-1909165) and MYSTIC (NSF ATI-1506540, NSF AST-1909165). R.M.R. acknowledges support from the Heising-Simons Foundation 51 Pegasi b Fellowship. S.K. acknowledges support from the European Research Council through a Starting Grant (Grant Agreement No. 639889) and Consolidator Grant (Grant Agreement ID 101003096). M.M. acknowledges funding from the Programme Paris Region fellowship supported by the Région Ile-de-France. This project has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant agreement No. 945298. We acknowledge with thanks the variable star observations from the AAVSO International Database contributed by observers worldwide and used in this research. This research has made use of the Jean-Marie Mariotti Center Aspro and SearchCal services. We acknowledge funding from NSF-ATI grant 1909858 for the development of the Silmaril instrument.

Facility: CHARA - Center for High Angular Resolution Astronomy interferometer.

Software: PMOIRED (Mérand et al. 2024), OITOOLS (Norris et al. 2021), ROTIR (Martinez et al. 2021), SURFING (Roettenbacher et al. 2016), mircx_pipeline (Le Bouquin et al. 2024).

In order to evaluate the robustness of image reconstructions, we reconstructed individual epoch images for two nights that have good (u, v)-coverage data and higher signal-to-noise ratio: 2023 October 25 and 2023 November 14. The images from ROTIR and OITOOLS are depicted in Figures 10 and 11. The individual epoch images exhibit good agreement with the combined images, reinforcing the reliability of the reconstruction process.

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In Figure 12, the intensity values along the X- and Y-axes are plotted based on the H-band OITOOLS image. We selected the region on the image where no bright spots are present. The intensity values at the center are initially low, gradually increase, and subsequently decrease toward the edges. The intensity does not exhibit a smooth transition from the center to the edge, creating challenges in accurately fitting the limb-darkening model to the observed data (Section 3).

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In Figure 12, we overplotted the limb darkening with power-law coefficients α = 0.47 and 0.38, measured in Section 3, for comparison. Based on the differences between the observations and theoretical expectations, it may be that ρ Cas is experiencing limb brightening.

Figure 13 compares the photometry across K-band wavelengths observed with the MYSTIC instrument. The ρ Cas flux measured from six telescopes is averaged, and then it is divided by the flux of the calibrator stars for the spectral calibration. Since MYSTIC does not measure the absolute flux, we have normalized it by taking the mean flux observed at λ = 2.0 μm.

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