© The Authors 2025

1. Introduction

Yellow hypergiants (YHGs) are very massive late-type stars with temperatures roughly between 4000 K and 8000 K (Nieuwenhuijzen & de Jager 2000). They have very extended atmospheres and high mass-loss rates, and they are subject to continuous ordinary quasi-periodic pulsations. Once in one or more decades, they release a massive outburst. The radii of YHGs vary between 350 R_⊙ and 750 R_⊙, also depending on the adopted distance (Lobel et al. 2003; the present paper). As a result, they are very rewarding objects (since the late 18th century) for visual, photographic, and spectroscopic observers and since the 1950s also for photometric observers around the world. The latter group of observers are presently well-equipped with multi-colour photo-electric photometers.

The quality of spectroscopic and theoretical studies is rapidly increasing, and models of YHGs and their evolution have considerably improved in recent years. Dedicated YHG research studies after the 1980s are offered in such works as Maeder & Meynet (1987), de Jager (1980), Lobel et al. (1994), Nieuwenhuijzen & de Jager (1995), de Jager & Nieuwenhuijzen (1997), Stothers & Chin (2001), Lobel (1997), Lobel et al. (1998) and Nieuwenhuijzen et al. (2012). Nevertheless, the precise to-and-fro tracks of pulsations and outbursts in the Hertzsprung–Russell diagram (HRD; log T_eff versus log L/L_⊙) have as yet not been investigated, and thus it is the focus of this paper. Another pivotal goal is to elucidate why, when, and under what physical circumstances the onset of massive YHG outbursts can be expected.

During their numerous to-and-fro tracks, the YHGs gradually approach a theoretical boundary around T_eff ≃ 8200 K after an episode of large mass loss. This was predicted by Nieuwenhuijzen & de Jager (2000) and so too was the location of the YHGs with respect to an area of the HRD called the ‘yellow void’ (YEV). This region is depicted in their Fig. 1. The gradual approach to this area due to the fast evolution of YHGs is called ‘bouncing against the void’ (de Jager 1998). Only after even more excessive mass loss may YHGs cross the 8200 K boundary and evolve on a blueward track into this area, later baptised by Nieuwenhuijzen & de Jager (1995) as the ‘yellow evolutionary void’ (YEV). Nieuwenhuijzen & de Jager (2000) checked the reality of the YEV and showed that inside it the extended atmospheres of bluewards evolving YHGs become highly unstable (Nieuwenhuijzen et al. 2012).

Fig. 1. LTV of ρ Cas between 1962 and 2020. Top panel: Median (B − V)obs versus the reduced JD. The dark blue dots define the dark-blue unsteady median. Red dots represent the photometric observations with an available Teff value (green numbers). Bottom panel: δ (B − V) = (B − V)0 – (B − V)dJN versus the reduced JD. Red dots and green numbers are similar to those in the top panel (see discussion in Sect. 2.2).

An in-depth analysis of multi-colour photometric observations, preferably supported by reliable temporal T_eff-determinations of YHGs, has been lacking for over a century. For this reason, we embarked on the present photometric study (Paper II) with the goal of clarifying a large number of not yet understood YHG properties. This work is a continuation of Paper I (van Genderen et al. 2019). An important result of Paper I was that the temperatures based on a careful analysis of the spectrum (which we indicate with T_eff) are always larger than temperatures derived from (B − V)_obs observations (which we label with T_PHOT). The B magnitudes obviously suffer more from the selective continuum opacity than the V band magnitudes. Hence, V and B magnitudes are not consistent with respect to each other, and this is especially important when the mass-loss rates are high, causing larger gas densities and hence higher selective continuum opacities. It is of note that in the caption of Fig. 1 in Paper I, the YEV is incorrectly assigned to the right-hand side of the 8200 K boundary when the YEV is actually to the left of it. In other words, the YEV is located approximately between 8200 K and 12 000 K.

We discuss in Sect. 2.6 of Paper I that the ratios Ampl V/Ampl (B − V) of individual pulsations signal the presence of enhanced mass-loss episodes, which also indicate increased selective continuum opacities. A discussion on the sources for the variable transparency of YHG atmospheres is presented in Lobel (1997, Chap. 1.) and Lobel et al. (1992). Lobel (2001) presented non-LTE calculations in cool supergiant atmosphere models that become dynamically unstable because of the decrease of the stability-integral (⟨Γ₁⟩) over a major fraction of the extended atmosphere as a result of the partial thermal and photo-ionisation of hydrogen. The global atmospheric instability of YHGs at the cool YEV boundary results from very extended partial hydrogen-ionisation zones where this volumetric pressure-weighted heat-capacity integral decreases close to the radiative value of 4/3, resulting in strong overall atmospheric compressibility for a small bulk modulus (Lobel et al. 1992; Lobel 2001). The effects of the T_eff variations due to the ordinary atmospheric pulsations alter the local kinetic gas temperature structure as well as the temperature of the ambient stellar radiation field. This changes the thermal conditions in the extended partial hydrogen H-ionisation zone, which modifies the selective continuum opacities in B and V. The spectroscopic, radial velocity, and photometric studies of the outburst of ρ Cas in the year 2000 by Lobel et al. (2003) revealed new and fundamental properties of the YHGs.

Another important result described in Paper I is that the long-term variations (LTVs) in the median curve sketched through all YHG ordinary pulsations and the outbursts, especially of the colour index (B − V)_obs, are important. After a number of calculations and assumptions of the distance and interstellar reddening and extinction, the computed luminosity and T_eff values revealed cyclic to-and-fro tracks in the HRD. The timescales can sometimes amount to only hundreds of days. The tracks by outbursts are also responsible for the T_eff alternating between roughly 4000 K and 8000 K. Their zig-zag movements in the HRD are required for losing large amounts of mass and subsequently for short permanent evolutionary shifts to the blue in the HRD. It is of note that we use ‘track’ as well as ‘loop’ to describe the movements in, for example, the HRD and in the LTV colour index curves. Whether the movement is truly straight, slightly curved, or irregular is never precisely measurable.

Thanks to the pioneering work of de Jager & Nieuwenhuijzen (1995), the current availability of many new T_eff values in Kraus et al. (2019), and the T_eff sets offered in Lobel et al. (2003) and in Klochkova et al. (2014), we are able to construct a number of T_eff calibration relations using simultaneous BV photometry. This is our main goal and means of making progress in this study. In the present paper, we unveil new properties and relationships between a number of variable astrophysical parameters of ρ Cas, and we draw interesting new conclusions based on these temperatures.

The ordinary atmospheric oscillations are one of the primary characteristics of the YHGs. They are important since they can be used to probe internal physical processes. The YHGs show continuous quasi-periodic ordinary pulsations that can be characterised as ‘weakly chaotic’, that is, each pulsation cycle is unique according to the models for ‘semi-regular’ variables of Wisse (1979) and of Icke et al. (1992; see also Paper I, Sect. 2.2). A concise overview of information of all observations used and discussed in this paper is provided in Table 1.

The body of this paper and part of the Appendix focus exclusively on the evolution of ρ Cas = HD 224014 during the time interval from 1963 to 2023. This is thanks to the availability of a long-term sequence of mainly BV photometry data (see Table 1), which were simultaneously observed with a sufficient amount of available T_eff values. The latter values combine the spectroscopic datasets of ρ Cas of Lobel et al. (2003), Klochkova et al. (2014), and (the largest) Kraus et al. (2019). The three ordinary datasets are labelled Lo, Kl, and Kr, respectively, and they are referenced often in this paper.

As the spectroscopic temperatures are provided by three different methods, we cannot exclude systematic shifts between ordinary datasets. Considering the estimated uncertainties by the respective authors, we assumed that it would not be an underestimation that a T(Sp), and thus also in T_eff, may be in error by less than ±100 K. (Sect. 3.2). The above is apart from the earlier photographic (PG) and visual (VIS) observations of 1905–1963 and 1905–1986 in Appendix G and Appendix H, respectively.

This paper mainly consists of eight sections and seven diagrams. The seventh diagram is the highlight of this paper. To avoid lengthy descriptions, a relatively large number of in-depth discussions concerning some particular features and properties of ρ Cas, HR 8752, HR 5171A, and HD 17982 have been moved to a number of Appendices that accord with the section of the paper they belong to.

The Appendix offers new and useful information about the last three objects because of their recent evolution into a second region of potential instability in the HRD. This is also in view of the pulsation pattern recently observed in HR 5171A and because of the peculiar UBV variability observed in HD 179821 between 1993 and 1999, which has prompted a debate on whether it is a true YHG.

The new formulae required for our calculations are provided in Appendix A. From Sect. 1 it is clear that YHGs show various types of complex connections between physical processes. It is therefore inevitable that our explanations of some features can often only be provided using arguments that are based on previous knowledge and understanding, which we also discuss in the next sections.

In Sect. 2, we offer an in-depth photometric analysis of the unsteady LTV of the median of (B − V)_obs in the time interval 1962–2020. Its unsteady behaviour is complicated and requires a complex explanation. To advance our understanding of the behaviour of ρ Cas, we developed four temperature (T_eff) calibration relations. They are important for studying brightness variations in the HRD caused by ordinary pulsations and the outbursts in log L/L_⊙ and log T_eff and for finding correlations between the various physical properties and the onset of ρ Cas’ outbursts.

We reached our goals with Fig. 7 by constructing a convincing correlation diagram between a number of physical properties, which includes conditions necessary for the onset of the rare outbursts. We emphasize that all figures in the main sections and Appendices are solely based on T_eff measurements, although in several figures (e.g. Figs. 4, 5, 6) we first made use of the four temperature calibration relations combined with simultaneous BV observations in Fig. 2, mentioned above, of which the average is called the 4T method, within an uncertainty of a few hundred degrees. However, this method is based on the spectroscopic temperatures by Lo, Kl, and Kr, which are considered to be T_eff within an uncertainty of about 100 K.

Fig. 2. Four temperature calibration relations versus photometric parameters for determining an average Teff for relatively reliable photometric observations (see Sect. 3).

We also investigated HR 8752, HR 5171A, and HD 17982 because of interesting properties and the current status of their evolution. The YHG HR 8752 = V509 Cas started to climb out of the first instability region after the last outburst in 1973, revealing decreasing pulsation activity. It is currently on a blue loop (Nieuwenhuijzen et al. 2012) and became a stable star somewhere between 1996 and 2017 (see Table 1). We also show the current pulsation activity of YHG HR 5171A = V766 Cen in Fig. E.6. Finally, we show a very short-term BV brightness variation running in opposite directions in HD 179821 = V1427 Aql.

2. The LTV of the median of (B − V)_obs between 1962 and 2020

To ease the discussion on ‘ordinary’ continuous quasi-periodic pulsations they are numbered with respect to an outburst, we introduce a special nomenclature for the ordinary pulsation maxima and minima. The maximum before an outburst is called max1, the outburst minimum is min1, and the maximum after ascending from the outburst, is called max2. These three points, plus the median (often abbreviated med.) between the descending and ascending branches of an outburst, are called the four characteristic points of an outburst. The numbering of ordinary pulsation maxima preceding and following the outburst of 2013 is shown in Fig. A.1 and names and epochs for outbursts of 1986, 2000, and 2013 are given in Table E.1.

For example, when starting with four ordinary pulsations before max1, and ending with three after max2, the naming convention is as follows: min-4, max-3, min-3, max-2, min-2, max-1, min-1, max0, min0, max1, min1, max2, min2, max3, min3, max4, min4, max5, min5, etc.

We note that in Figs. A.1 and A.2, showing the ordinary pulsation sequences of ρ Cas, we use this type of numbering for the maxima only. max1 and max2 are recognizable in Fig. A.1 as numbers 1 and 2 representing the outburst in 2013.

The next subsection is devoted to two panels of Fig. 1. Both are discussed separately, although it is unavoidable to mention and refer to both of them in the same subsection, notwithstanding that their content is not equal.

2.1. Upper panel of Figure 1

This panel discusses the LTV of the median (B − V)_obs versus the reduced JD of ρ Cas between 1962 and 2020. We note that as the early ordinary photographic and visual observations starting in 1885 are less accurate, we discuss them separately in Appendix G and Appendix H (see Figs. G.1: 1905–1963 and H.1: 1905–1986, Table 1).

The first panel of Fig. 1 contains ∼40 ordinary pulsations and the three outbursts of 1986, 2000, and 2013. It is drawn by the dark blue unsteady curve connecting the selected dark-blue dots. These dots are selected by will.

The median has a maximum amplitude of 0$\stackrel{\text{m}}{.}$4. A detailed version of the LTV of the median (B − V)_obs is shown in Fig. 12 of Paper I. In fact, this curve represents the movements in the HRD.

It is important to emphasize that we did not draw all ∼40 ordinary pulsations along the entire median (B − V)_obs-curve. If done so, it would show a saw-tooth having small amplitudes of about ±0$\stackrel{\text{m}}{.}$1. We pay special attention to their presence by discussing the bottom panel of Fig. 1.

The small red bar drawn in the left-hand corner of the top panel represents the first spectral type determination made in the early 1960s (F8) by Humphreys (1978). The colour index appeared to be relatively blue in accordance with its position in Fig. 1 (see Sect. 3.3 of Paper I; de Jager et al. 1988; Arellano Ferro & Mendoza 1993). Our goal is to find the cause of the unsteady character.

The red dots in the bottom and top panels show (B − V)_obs-observations having an available T_eff-value (green numbers) in the three spectroscopic datasets of Lo, Kl, and Kr. Kraus et al. (2019) warned for unreliable temperatures due to saturated absorption lines, or possibly filled/polluted by emission. Only part of the outbursts of 2000 (with max1) and 2013 (alas no max1) are schematically sketched with dotted curves.

The ordinary data points and temperatures indicated in both panels sometimes represent an average of a small number of ordinary data points, falling in almost the same area. Since the distribution of the plotted ordinary data points and temperatures are slightly different in both panels, no precise match between the plotted red dots and the marked temperatures can be guaranteed. The data of Fig. 1 (top and bottom panels) are listed in Appendix, Tables A.1 and A.2 (see Table 1).

The fact that a number of individual data points of Lo and Kr show high and low temperatures (especially in outbursts) that are significantly above and below the LTV of (B − V)_obs, indicates that fluctuations of the LTV in (B − V)_obs are at least partly due to temperature variations. This also concerns the second panel.

Since we know from Paper I that B suffers more from the selective continuum opacity than V, it follows that (B − V) is not a reliable measure of T_eff and always shows variable T_PHOT-values due to the ordinary pulsation cycles. We conclude that the selective continuum opacity is also variable. Thus, T_eff and the selective continuum opacity variations are responsible for the unsteady median of (B − V)_obs. Our next goal is to study both contributions. It is therefore appropriate to refer to the LTVs of the medians observed in V, B and (B − V) in Figs. 11 and 12 of Paper I. They reveal that the unsteadiness is minimal in V, and largest in B and (B − V), which provides evidence the above assertion is correct.

It is also appropriate to point out that we can classify the 1986 and 2000 outbursts as ‘normal’, meaning they first show a gradual reddening of (B − V)_obs until the outburst, followed by a gradual blueing. This behaviour can be traced from the one of 1946 to 1986 (see Fig. H.1). Besides, their minimum brightness (min1) are almost equal. The 2013 outburst is a one-off event showing quite a different morphology. Its pulsation pattern is since fading away (see Figs. A.1 and A.2).

The pulsations sequences of the 1986 and 2000 environments are relatively red compared to those of 2013, being significantly bluer. This means that the ordinary pulsations of 2013 are likely hotter and that the continuum opacity is lower. Indeed, this suspicion appears to be correct thanks to the use of four temperature calibration relations (T_eff) versus simultaneously observed photometric BV-parameters. They are constructed in Sect. 3, and called the ‘4T method’ using the Lo, Kl, and Kr spectroscopic datasets. It appears that the T_eff-values for the pulsations in the 2013 environment are indeed even larger by ∼1000 K.

We discuss below in the panels of Fig. 1 that the continuum opacity during the 1986 and 2000 outbursts is indeed larger. Obviously, T_eff and the selective continuum opacity vary opposite to each other.

The majority of the T_eff-values from the three sets Lo, Kl, and Kr were obtained sufficiently isolated from each other, apart from a very short overlap between the Kl and Kr sets. This is supported by the jumps in (B − V)_obs: red in 2000 during the Lo set, bluest during the Kl set, and again significantly redder during the Kr set. It shows that each set was obtained when the stellar atmospheric conditions with respect to temperature and selective continuum opacity were different.

Since we are mainly interested in how long the special outburst conditions last, we embarked upon the analysis shown in the bottom panel of Fig. 1. However, we first point out some interesting features and properties of ρ Cas that need more attention. They are presented in Appendix A. It appears that prior, during, and after the 2013 outburst the photometric and physical properties differ from those of both outbursts in 1986 and 2000 (see Appendix A.3.1 and Appendix A.3.2).

Another peculiarity is that max1 of the 2000 outburst in the top panel should be much bluer considering its large temperature of 7600 K compared to the observed value of (B − V)_obs (top panel). A possible explanation is offered in Appendix A.3.3. This is related to the subject above about peculiar photometric variations observed in (UBV) of HD 179821. It shows an LTV of the medians more or less similar to that of YHGs. Arkhipova et al. (2009) also earlier reported peculiar and fast photometric variations of its continuum formation level (see Figs. A.1 and A.2 and Table 1).

Conclusions:

1. A careful analysis, supported by our 4T method (Sect. 3) and applied to many ordinary pulsation cycles before and after the three outbursts in 1986, 2000, and 2013, reveals that T_eff-values of the 2013 environment are larger by ∼1000 K with lower continuum opacity, and consequently bluer (B − V)_obs. The opposite is the case for the 1986 and 2000 outbursts.

2. This is also the case during the eruption of 1905 (see the note at the beginning of this subsection, when (B − V)_obs is red).

3. The maximum change of the latter parameter amounts to 0$\stackrel{\text{m}}{.}$4. The effects of the variable T_eff and continuum opacity on (B − V)_obs have opposite signs.

4. The outbursts of 1986 and 2000 are classified as ‘normal’ based on their similarity to the previous ones in 1905 and 1946: these four outbursts are preceded by a gradual reddening in (B − V)_obs until the outburst, and followed by a gradual bluing.

5. The 1905 and 1946 outbursts are as far as that is concerned ‘normal’ as well. Conversely, the one of 2013 is a one-off event: the very last outburst in view of its ordinary pulsations are currently fading away.

6. A general survey of the properties of the atmosphere of YHGs is presented in Appendix A.

2.2. Bottom panel of Figure 1: Introduction of a new parameter

The same red dots in the upper panel with a known T_eff are plotted in the bottom panel of Fig. 1 but as a function of δ (B − V). We introduce a new parameter for measuring the variation of selective continuum opacity. Because T_eff and the selective continuum opacity have opposite signs, we search for the time-scale of stable T_eff/continuum opacity regimes, in case they exist at all. The new parameter is defined as the observed (B − V)_obs corrected for the interstellar reddening (0$\stackrel{\text{m}}{.}$45) (see Fig. A.2). Labelled with (B − V)₀, this parameter is subject to T_eff/continuum opacity variations. One subtracts it from the ‘statistical’ parameter provided by de Jager & Nieuwenhuijzen (1987) (private communication, and in their Appendix), (labelled by us as (B − V)dJN). It serves as the ‘zero selective continuum opacity scale’ without interstellar reddening (ISR), although we believe it may contain a minor amount of the selective continuum opacity:

$$\begin{array}{c}\hfill \delta (B-V)={(B-V)}_0-(B-V)\mathrm{dJN}.\end{array}$$

The latter assertion requires an explanation for which we refer to de Jager & Nieuwenhuijzen (1987) and Nieuwenhuijzen & de Jager (2000). We note that at the time of these publications the existence of the selected continuum opacity typical for the YHGs was unknown. When they started in 1987 to determine statistical relations between some physical parameters for all types of stars (in total about 200 stars), they included 10 unspecified hypergiants, probably in slightly different evolutionary stages. Later in the paper of 2000 these authors checked and analysed the area in the HRD using a much larger collection of very luminous stars including supergiants, YHGs, and LBVs (see their Fig. 1). They outlined them in the YEV and in the adjoining blue instability region together with many other luminous stars. Obviously, many of them are in very different evolutionary states, presumably having different physical properties. We therefore think that any deteriorating contribution of selective continuum opacity to their ‘zero-continuum opacity scale’ remained small.

In addition we are aware that YHGs crossing the ∼8200 K YEV boundary signal a permanent blue loop evolution, such as HR 8752 has shown between 2005 and 2017 (Table 1; Sect. 7.1), and also ρ Cas after 2013 (Sect. 7), as well as the two other objects mentioned in the previous paragraph. As a result, ρ Cas became physically different with evolution, including its atmosphere.

We use the (B − V)dJN-scale as a continuum opacity free parameter in the above equation. We assume that δ (B − V) represents any variation in the continuum opacity/temperature regime (see first paragraph) along the unsteady median in the top panel of Fig. 1. We still require the parameter s for deriving the (B − V)dJN value by applying the following sequence:

T_eff ↣ s ↣ (B − V)dJN, then subtract it from (B − V)₀.

We assume that as long (B − V)_obs stays reasonably stable (hence the atmosphere as well), this will also be the case for the T_eff-value and the continuum opacity. The above sequence of computations is also applicable to outbursts, but sometimes shows larger deviations as they are considered a different type of stellar pulsation. For example, on their way to the deep minimum, min1, δ (B − V) becomes even equal to zero when the temperature declines to ∼4400 K (e.g. in the 2000 outburst). This is normal because T_eff and the selective continuum opacity decline in tandem.

In Table 2 we put the use of the δ (B − V) into practice. The Lo dataset is an ideal case to start our testing because the (B − V)_obs is, apart from max1 and min1 of the outburst, very stable from the 7250 K data point. This is done by collecting the necessary photometric properties of all data points having a known T_eff-value (see the δ (B − V)-values). In case of stability for the individual data points, successive δ (B − V)-values should be almost equal, apart from the extreme temperatures belonging to the outbursts. The last column of Table 2 confirms this.

Table 2 also lists three ordinary data points of only the Kr set, just as a matter of interest, including two near min1 and one near max2. The last column shows the results. Most of the Lo ordinary data points are concentrated along the horizontal line, apart from min1 as expected. The concentration along the horizontal line indicates that the temperature and continuum opacity regime at the time lasted 3000 d. After all, (B − V)_obs stayed almost constant (in view of the straight line) between 7250 K on the left, to 7600 K on the right. Similar stable regimes exist at later dates with a very blue ${(B-V)}_{\mathrm{obs}}$. The Kl ordinary dataset partly overlaps the Kr set, and is bluer than the next Kr set (including the three ordinary data points in Table 2) because the (B − V)_obs is redder (see bottom panel of Fig. 1). Both sets last 1000 d and 2000 d, respectively (see, however, the second paragraph below).

A surprising property of the δ (B − V)-values is the very small scatter and homogeneity along the three horizontally running line pieces. This supports our claim that an unsteady (B − V)_obs is caused by T_eff- and selective continuum opacity variations shown by this formula.

However, we have to be cautious to claim absolute stability of the atmosphere over such a long period of time. The reason is that one should not forget about the presence and inevitable contributions of the 40 ordinary pulsations between 1964 and 2020 to the temperature- and continuum opacity variations. Their (B − V)_obs variations are about four times smaller (∼0$\stackrel{\text{m}}{.}$1) than the one from the LTV. We therefore assume that ‘total stability’ lasting for a few thousand days should not be taken too literally.

On the other hand, the existence of unstable atmospheric conditions over a relatively long period of time do exist. They often include sequences of ordinary pulsation cycles we investigated in Sect. 2.6 of Paper I. HR 8752 and ρ Cas, and occasionally also HD 17981, show gas layers having enhanced selective continuum opacity. They survive the successive ordinary pulsations and show abnormal amplitude-ratios Ampl V/Ampl (B − V). Instead of the normal ratio of 1.7 ± 0$\stackrel{\text{m}}{.}$5, they can amount to six or seven, sometimes accompanied by Hα emission lines.

The average δ (B − V)-values for the three lines Lo, Kl+Kr, and Kr amount to 0$\stackrel{\text{m}}{.}$73, 0$\stackrel{\text{m}}{.}$46 and 0$\stackrel{\text{m}}{.}$60, respectively, and their durations of 3000 d, 1000 d and 2000 d, respectively, represent the amount of the continuum opacity in the (B − V)_obs-magnitude scale.

The three horizontal line pieces can be used as a ‘correction’ or ‘elimination’ (in magnitudes), by shifting the fluctuations of (B − V)_obs upwards in order to obtain a horizontal median without any fluctuations. Because we need these corrections in other paper sections, such as Sect. 4, we dub them ‘the three corrections’.

The data of Fig. 1 are offered in Tables A.1 and A.2, while Fig. A.3 is mentioned in Sect. 2 (bottom panel). The latter section shows the diagram δ (B − V) versus the T_eff. The small scatter for each set is remarkable. Appendix A also presents discussions about a number of particular properties of ρ Cas, including the 2013 outburst (Table 1; see also Appendix A.3.1).

Conclusions:

1. We introduce a new photometric parameter δ (B − V). It is a measure of the amount of selective continuum opacity (correlated with the T_eff), responsible for the reddening of the 1962–2020 median (B − V)_obs. It represents the difference between two types of (B − V) indices. The first one is observed and corrected for interstellar reddening with selective continuum opacity. The other one is considered to be almost free of continuum opacity. Using the new parameter we find that during the spectroscopic sets Lo, Kl, and Kr, ρ Cas passed through three different, but apparently stable, T_eff/continuum opacity regimes. They last 3000 d to 1000 d, being proportional to the durations of these ordinary datasets. The stability lasts remarkably long, after all, they are well isolated from each other and in significantly different epochs. However, the stability should not be taken too literally because of the presence of the more modest contributions of ∼40 ordinary pulsations. They also modify the temperature and the selective continuum opacity of the atmosphere, although to a lesser degree. This is substantiated by the detection of enhanced mass loss episodes during a number of sequences of ordinary pulsations confirmed by the presence of an Hα emission line profile, not only in ρ Cas (de Jager et al. 1997) and HR 8752, but also in HD 179821 (see Paper I, Chap. 2.6.), and in this paper Appendix A.3.4 (see Table 1).

2. The frequently short-lasting fluctuations in (B − V)_obs between 2006 and 2015 are presumably due to hot convection cells and shell ejections causing short fluctuations of the LTV in the (B − V)_obs-curve. The opacity is almost zero for min1 of the ‘normal’ 2000 outburst, and 0$\stackrel{\text{m}}{.}$15 for min1 of the ‘abnormal’ 2013 outburst. Its T_eff ∼ 4900 K is a few hundred degrees higher as a consequence of the shallow depth (Kraus et al. 2019).

3. For the Lo, Kl, and Kr datasets (1993–2016) the average of the new continuum opacity parameter δ (B − V) amounts to 0$\stackrel{\text{m}}{.}$73, 0$\stackrel{\text{m}}{.}$46, and 0$\stackrel{\text{m}}{.}$60, respectively. These values can be used to shift their respective intervals of (B − V)_obs upwards, transforming the unsteady median (B − V)_obs into a straight median. It represents stability of ρ Cas that can be corrected for the interstellar reddening (0$\stackrel{\text{m}}{.}$45). However, because the exact contributions of the amounts of selective continuum opacity affecting the V- and B-magnitudes remain unknown, we are also in the dark about the precise total amount of the selective continuum opacity.

3. The construction of four temperature calibration relations

Section 2 started with describing the construction of three temperature calibration relations between known T_eff-values and simultaneously observed photometric BV-parameters. These relations are necessary to determine from any observed photometric magnitude V_obs, B_obs, and (B − V)_obs the T_eff (hence including interstellar extinction and reddening). Next, physical parameters such as the stellar luminosity and radius can be calculated if the distance is known for some sequences of interest, and then plotted in the HRD. For example, we investigate the tracks of the outbursts and of a number of ordinary pulsation cycles that precede and succeed the outbursts.

The use of temperature calibration relations is similar to those of Lobel et al. (2003). They found a linear relationship between their small T_eff dataset and the visual brightness V during the time of the 2000 outburst of ρ Cas.

In this paper we can dispose of sufficient amount of V-, B-, and (B − V)-magnitude and T_eff-values in order to construct three calibration relations well-separated from each other because of the different epochs shown in Fig. 1. The T_eff-values are from Lo (epoch 1993–2000), Kl (epoch 2008–2010), and Kr (epoch 2008–2016) (see Sects. 2 and 3). The provided accuracies amount from 40 K to 160 K. A small overlap exists between the Kl and Kr datasets, two temperatures at the start of the Kr set coincide with the Kl set.

Figure 2 shows four temperature calibration relations. As they don’t appear linear, we yet sketched them as such, being the simplest approximation. We omit the one at the bottom right for the moment, but it is discussed later. The other three diagrams show T_eff versus the three observed magnitudes V, B, and (B − V). We note that we offer more V than B magnitudes. Notice also the scale differences of the three magnitude axes. Average lines have been sketched by eye, although we cannot guarantee they should be linear and/or running parallel from 8000 K down to 4000 K. The reason for these uncertainties is the limited number of available temperatures and their relatively large scatter, in particular for the data points in the 2000 and 2013 outbursts. The reason is that outbursts represent a different type of stellar pulsation compared to the continuous ordinary pulsations, while the number of available T_eff-values is still too low.

The temperature calibration diagrams show a relatively large intrinsic scatter (see below for a clarification) amounting to ∼500 K. It is therefore not surprising that some overlap exists between a relatively large number of ordinary data points in neighbouring datasets (see also their distribution in the top panel of Fig. 1). Obviously, if one wants to calculate the T_eff-value of a photometric data point between two relations it is necessary to obtain an interpolated temperature.

The above mentioned ‘intrinsic scatter’ does not need further clarification because already discussed in Sect. 2. As a result, the morphology of the observed V, B light, and (B − V) colour curves of each pulsation cycle is unique, not to mention the very diverse morphologies of the light and colour curves of the outbursts (see e.g. Figs. E.1-E.3, and Table E.2 of the Appendix).

The uniqueness of the pulsations of long-period variables is in accordance with the light curve models of Icke et al. (1992) and of Wisse (1979) (see Sect. 2.2 of Paper I). Hence, any fixed V-magnitude has a different B-magnitude and a different (B − V) colour index. Therefore, we can apply these three parameters as almost independent from each other. We have only selected reliable ordinary photometric datasets, and in case of two sources an average magnitude value is used. The photometric BV data are partly from the Lo, Kl, and Kr sets, while most of the photometric data were collected by G.W.H. (in Paper I: see the note to the title of Appendix B and M), and in this paper (see Table 1: AAVSO Database, and Figs. A.1 and A.2 of Appendix A).

Conclusion:

We first constructed two temperature calibration diagrams for ρ Cas based on three relatively isolated spectroscopic datasets having T_eff-values shown in Fig. 1, besides simultaneously obtained V- and B-magnitudes. These temperatures have been obtained in three epochs: 1993–2000, 2008–2010, and 2008–2016. The amount of data points is for the time being acceptable, although the scatter in the derived temperatures of ±500 K is relatively large. It includes all kinds of intrinsic and coincidental errors. We therefore embarked upon constructing a third diagram for (B − V)_obs. This appeared to improve the reliability of the average T_eff-values. For the same reason we have below constructed a fourth calibration relation (for the data used in Fig. 2 see Tables B.1 and B.2).

3.1. The fourth calibration relation

To further increase the weight of the mean temperature determinations discussed above, a fourth temperature calibration diagram was constructed which also requires the interstellar reddening (ISR). The latter reddening amounts to E(B − V) = 0.45 (Lobel et al. 2003). The two axes of the fourth graph of Fig. 2 represent two versions of the colour index. The first one is (B − V)dJN on the horizontal axis (see Sect. 2). This parameter likely contains low statistical opacity value. It has been used in Sect. 2 for the second panel of Fig. 1. The second version of the colour index, but which suffers from the usual amount of opacity, is (B − V)₀ = ((B − V)_obs – 0$\stackrel{\text{m}}{.}$45) on the vertical axis. The scale difference between the two axes is useful for our purpose. For clarity reasons a few values of T_eff are represented by the vertical dotted lines. The sequence of steps for constructing this calibration relation is as follows: first determine the parameter s from each available T_eff belonging to the Lo, Kr, Kl datasets and having a known colour index (B − V)_obs. Next, use s for finding (B − V) dJN by consulting the tables of de Jager & Nieuwenhuijzen (1987; private communication and their Appendix).

In the next step we determine (B − V)₀ by correcting (B − V)_obs for the ISR of 0$\stackrel{\text{m}}{.}$45. The ordinary data points are plotted in Fig. 2 (bottom right) for depicting the fourth temperature calibration diagram. It is looking normal due to the three oblique and parallel running lines to the left. This is in contrast to the dashed lines that run with a sloping angle until horizontal, signaling zero (or near-zero) opacity. They are based solely on the data points of the 2000 (Lo) and 2013 (Kr) outburst minima min1, or ∼min1 (meaning close to min1). For clarity reasons a few T_eff-values based on the temperature scale of de Jager & Nieuwenhuijzen (1987) are marked with dashed vertical lines, where 7600 K is the derived T_eff-value for the 2000 outburst max1 (Lobel et al. 2003). Table B.1 offers for the Lo, Kl, and Kr datasets the observed magnitudes and colour indexes (B − V)_obs, besides the (B − V)dJN-values necessary in the fourth relation.

In the next section we test the reliability of the four calibration diagrams of Fig. 2 provided in Table B.2. While we described above the sequence of steps for calculating (B − V)dJN when T_eff is available, we explain below the steps for finding from (B − V)_obs the fourth relation.

The following procedure has to be used for any observation. The sequence of steps to find the individual T_eff-values for any observation is as follows: correct (B − V)_obs to (B − V)₀, and next with the aid of the fourth calibration relation of Fig. 2 find the (B − V)dJN-value, and subsequently consult the dJN tables for the s parameter (probably slightly deteriorated by continuum opacity) for calculating the T_eff-value.

Conclusions:

1. We have constructed a fourth temperature calibration relation between the IS reddening-free colour index (B − V)₀ and the (B − V)dJN having a low, statically derived, opacity contribution (see de Jager & Nieuwenhuijzen 1987, private communication and in their Appendix).

2. The sequence of steps for obtaining the fourth temperature calibration diagram completes our ‘4T method’. When these steps are used for all individual photometric observations the T_eff uncertainty decreases to ∼300 K.

3.2. Testing the reliability of the 4T method

To investigate the reliability of our 4T method we arbitrarily selected a set of 13 data points having reliable V- and B-magnitudes according to three conditions. They are well distributed from faint to bright magnitudes, are reliable magnitudes, and should include observations from the 2013 outburst among which 11 are from Kr and two from Kl. We note that a small overlap exists between both sets. The purpose is to determine the T_eff-values with the aid of the 4T method. Next we determine the averages, the averages of individual differences, as well as the standard deviations (st.dev.). The latter are listed in Table 3.

The observations we select are collected in Table 3 based on these results. We conclude that the use of our 4T method provides satisfactory results. A single T_eff 4T determination has an average accuracy of ±264 K, with rare maxima of 400 K or 500 K (see also Table B.2 of Appendix B).

Conclusions:

1. First we have constructed three temperature calibration diagrams for ρ Cas: T_eff versus simultaneous observations of V_obs, B_obs, and of (B − V)_obs. The three T_eff–datasets are obtained for the three different epochs of Lo, Kl, and Kr. The three calibration relations run parallel and lie at some distance of each other. This is because the T_eff-values are observed in different isolated epochs for their own temperature and opacity regimes (see the first and second panels of Fig. 1).

2. We next construct the fourth calibration diagram based on the (B − V)dJN-scale and the observed colour index corrected for interstellar reddening (B − V)₀. This diagram is introduced for increasing the weight of the calculated average in the 4T method.

3. In addition we always interpolate between two neighbouring calibration relations in case an ordinary data point falls between two of them.

4. The use of these four temperature calibration relations for calculating a mean T_eff for any photometric observation of ρ Cas is called ‘the 4T method’. It turns out that this average has a standard deviation amounting to ±264 K with incidental peaks to 400 K (Table 3). Klochkova et al. (2014) and Kraus et al. (2019) assert that the errors estimated for their T_eff-values amount to 40–160 K. However, as the three spectroscopic sets are based on different methods, we expect some additional uncertainty of the order of ±100 K. We conclude that all temperatures used in diagrams like the Figs. 3, 4, 5 and 6 can be considered as T_eff (Sect. 1.3).

Fig. 3. Illustration showing how the observed unsteady median (B − V)obs in Fig. 1 can be transformed into a linear median (dashed drawn line) with the aid of the three vertical arrows discussed in Sect. 2.

Fig. 4. HRD for ρ Cas in 1962–2020. The encircled area in the HRD, having a central Teff = 6700 K ± 700 K and a luminosity of 5.42, locates the blue dots showing zig-zaging changes in the observed median (B − V)obs of Fig. 1.

Fig. 5. Same as Fig. 4 but for the interval 2008–2017 (in Appendix Table D.1). Due to the different scales used for the luminosity relative to the one for the temperature, the circle in Fig. 4 is an ellipse. The diagram shows the ‘to-and-fro’ tracks, consisting of ordinary pulsations before and after the 2013 outburst.

Fig. 6. Three ρ Cas outburst tracks of 1986, 2000, and 2013 in the HRD.

Fig. 7. Diagram showing the data points of the four median curves in 1962–2020 of ρ Cas, as well as the four characteristic points of the outbursts. The time-scale is reduced JDs. The time-scale of the wave pattern in the four panels is ∼15 y. From bottom to top the panels show Teff, (B − V)obs, stellar radius R/R⊙, and the average quasi-periods P(d) (number of pulsations used are marked with brackets). Ordinary data points are dark-blue while the four characteristic points of the outbursts are red. The latter are plotted in panels 1–3 representing max1, the median between the descending and ascending branch, min1, and max2 (see Sect. 8.2).

4. Location of ρ Cas in the HRD

The goal of this section is to determine the intrinsic location of ρ Cas in the HRD. In Fig. 3. we apply the three δ (B − V)-values in the bottom panel of Fig. 1 and shift them upwards with the aid of ‘the three corrections’ of the median. They clearly show (within an uncertainty of ±0$\stackrel{\text{m}}{.}$1) a linear median of (B − V)_obs = 0$\stackrel{\text{m}}{.}$7 (first dashed line). It is labelled ‘constant (B − V)_obs’ and LTV ‘eliminated’, assuming it stays linear towards the very left-hand side.

The second step corrects for the colour excess E(B − V) = 0$\stackrel{\text{m}}{.}$45 (Lobel et al. 2003) (second dashed line) for a (B − V)_obs = 0$\stackrel{\text{m}}{.}$25, corresponding to T_eff = 6200 K in the de Jager and Nieuwenhuijzen (1987) temperature scale.

It appears to us that the temperatures and luminosity used for ρ Cas, and shown in various H-R diagrams of the papers by, for example, Nieuwenhuijzen & de Jager (2000; Fig.), employ rather different values indicating there was no consensus at the time. These temperature values range between 6000 K and 8000 K, while the log of the luminosity ranges from 5.5 to 5.7.

We therefore prefer to apply a more accurate method for determining T_eff. We calculate the average of 41 T_eff-values we select from Lo, Kr, and Kl, obtained for at least one of the following simultaneously observed photometric parameters: V_obs, B_obs, and (B − V)_obs. These photometric parameters are based on the photometry by G.W.H. listed in Table 1 (see )Fig. A.1, and on Kr (Kraus et al. 2019) who show V-magnitudes provided by observer Wolfgang Vollmann and by the Bright Star Monitor programme, and also on Maravelias & Kraus (2022) who derived V-magnitudes using both visual and digital observations. We omit any temperatures of outbursts below 6000 K because they belong to the fainter parts of these outbursts (see Figs. 4, 5, and 6) and their properties appear rather different from normal pulsations.

It is of note that all the temperatures we used are based on analyses of spectral lines. Hence, they are almost independent of the selective continuum opacity.

The first result for these 41 temperatures is a mean T_eff-value of 6700 K ± 700 K st.dev. We can make two comments. First, the st.dev. is large because we are dealing with a variable star. Second, the new average temperature is 500 K larger than the previous one (upper dotted line). The cause may well be that ‘the three corrections’ should be increased by 0$\stackrel{\text{m}}{.}$07 as this is the difference in (B − V)_obs between the upper dashed line and the dotted one. In that case both temperatures become equal. It may also result from a mix of uncertainties in the colour excess, as well as being due to the presence of still some continuum opacity contained in the (B − V)dJN data.

The next step is to calculate the mean (B − V)dJN for the above list of 41 temperatures and BV-parameters of which we could use 39 data points. This is accomplished with the following sequence of steps, including the s-parameter:

T_eff–V_obs–B_obs–(B − V)_obs–s from T_eff–(B − V)dJN from s.

By averaging the (B − V)dJN-values we calculate 0.18 ± 0.07 st.dev. Next, we also compute the mean T_eff using the average of the 41 T_eff-values of the datasets of Lo (but excluding extremely low temperatures in Lo and Kr), and the sets of Kl and Kr. It is of note that all these spectral temperatures are independent of IS extinction. Our result is a mean T_eff ∼ 6700 K ± 700 K shown with the dotted line in Fig. 3.

We calculate the luminosity with the s-parameter and the Bolometric Correction-value (see tables in de Jager & Nieuwenhuijzen (1987; private communication and in their Appendix). It requires the V-magnitude, the colour excess E(B − V) = 0.45, and the shorter distance we consider more reliable (see Appendix A) of 2.5 kpc ± 0.3 kpc (see also Sect. 2.9 of Paper I). We also need M_V, M_bol (see formula in Appendix A). Hence, we calculated log L/L_⊙ = 5.42 ± 0.3 and log T_eff = 3.83 ± 0.04. The location of ρ Cas in the HRD is close to the YEV as defined in Nieuwenhuijzen & de Jager (2000; Fig. 1). While they (accidentally) use the same T_eff, their luminosity is larger than 5.7 because they used a larger distance to ρ Cas.

Conclusion:

We present a reliable method for finding the intrinsic location of ρ Cas in the HRD by using 41 spectroscopically derived temperatures T_eff together with simultaneously observed and variable photometric BV-parameters between 2000 and 2015. The result is 6700 K ± 700 st.dev. corresponding to a (B − V)_obsdJN = 0.18 ± 0.07. We find that log L/L_⊙ = 5.42 ± 0.3 using a shorter distance of 2.5 ± 0.03 kpc compared to the classic value. The location of ρ Cas in the HRD is very close to the YEV as defined, but very uncertain (see next section), by Nieuwenhuijzen & de Jager (2000; Fig. 1), although we calculated a slightly lower luminosity.

5. The domain of the LTVs of the (B − V)₀ from 1962 to 2020 and their zig-zag tracks in the HRD

Before discussing the Figs 4, 5 and 6, we have to comment on the fact that the dashed line marks not precisely the border of the Yellow Void, but lying significantly below 8200 K. We understood from de Jager and Nieuwenhuijzen (private communication) that an accurate model of the Yellow Void does not exist yet (e.g. Fig. 1 in Nieuwenhuijzen et al. (2012) and in their earlier papers, and our Paper I: Fig. 1.). Therefore, their model locations, temperatures, tracks, boundaries, adopted in our figures, should be considered as only temporary.

We show in Fig. 4 a portion of the HRD over a short distance to the right of the ∼8200 K boundary of the YEV. The circle marks the intrinsic domain of the (B − V)_obs-data points in Fig. 1. The temperatures and luminosity of each blue dot are calculated in Sect. 4, and listed in Table C.1 of the Appendix. We note that some dots coincide and are therefore not identifiable.

The medians of the three outbursts, at about halfway the brightness descending and ascending to min1 (large green dots for 1986, black dots for 2000, and red dots for 2013), are preceded and succeeded by two ordinary data points for the pulsations (lines drawn with an arrow). They show the direction of the zig-zagging just before and after the outburst. We consider the plotted medians during the outbursts least reliable. The height differences of max1 and max2 are the main problem. As a result we often have to use approximate values for the temperature and brightness. Nevertheless, they are sufficiently reliable to conclude that the outburst tracks move relatively far away from the circle, in contrast to those of the few pulsations shown in this plot. After all, both are different types of pulsations. We note that the two tracks max1 towards the medians of 1986 and 2000 are running parallel. Some peculiarities of the 2013 outburst are discussed in Appendix A.3.1.

We find that the median of 2013 is hotter than those of 1986 and 2000, and is around 6400 K. In Sect. 2 we argued that at the time all the pulsation sequences of the 2013 outburst are hotter by ∼1000 K compared to those of the 1986 and 2000 outbursts. Such a large difference can be expected. The analysis in Sect. 2 reveals that the increase in temperature to 2013 appears to be dominant with respect to the continuum opacity. For the 1986 and 2000 outbursts this dominance is the opposite.

The tracks of the 1986 and 2000 outbursts until their medians are restricted to the same narrow strip in log L/L_⊙ versus log T_eff, running steeply down towards the right-hand side. In this case we can better describe these tracks as ‘to-and-fro’ tracks, rather than ‘zig-zag’ tracks.

Conclusions:

1. Figure 4 depicts for the first time an HRD showing the detailed ‘to-and-from’ pattern of a YHG (ρ Cas).

2. The circular domain in Fig. 4, near the boundary of the YEV, contains the individual (B − V)_obs observations of the unsteady median of ρ Cas (see Fig. 1). The central temperature and luminosity (marked with the plus-sign) is 6700 K ± 700 K st.dev., and log L/L_⊙ = 5.42 ± 0.03, respectively.

3. The tracks from max1 to the medians of the 1986 and 2000 outbursts appear to be restricted to a narrow strip running steeply down to the right. The much weaker, but hotter, outburst in 2013 stays behind because its amplitude is about half of the other two outbursts. Consequently, its median is also hotter (∼6400 K) than the ones of the 1986 and 2000 outbursts (∼6000 K and ∼5900 K, respectively). Its luminosity was higher as well by ∼15%.

6. The 2008–2017 time series of ρ Cas in the HRD

Figure 5 shows a more extended HRD with a larger portion of the YEV left of the 8200 K boundary for the 2008–2017 interval. It includes the 2013 outburst. The region to the right shows the ∼10 000 K boundary, roughly separating the active from the quiescent Luminous Blue Variables or LBVs, also called S Dor variables (van Genderen 2001; de Jager et al. 2001). The structure of both HRD-regions is investigated and discussed in various papers, showing some differences between the location of various LBV types in such works as Wolf (1989), van Genderen (2001), Smith et al. (2004), and Clark et al. (2005).

For the construction of Fig. 5 we use the following spectroscopic and photometric datasets; the spectroscopic sets of Kl and Kr, the V-magnitude based on visual AAVSO data transformed to the Johnson system (Maravelias & Kraus 2022), and the BV photometry (Table 1). The photometric data based on two different sources and made in the same night are averaged. For the photometric BV-data of special interest, but without a temperature such as max1, the median and max2 of the 2013 outburst, and max3 and max5, we use the 4T method. The results of calculated log T_eff- and log L/L_⊙-values are listed in Table D.1 of the Appendix D.

Figure 5 shows the to-and-fro tracks of ∼30 selected ordinary data points preceding and succeeding the outburst max1 (marked with red star symbol). As we discussed in Sect. 2, the temperatures at that time were often high, close to, or even higher than the ∼8200 K YEV boundary. Some examples are max1 of the 2013 outburst, and a number of ordinary pulsation maxima before and after the outburst, such as max0, max-1, max3 and max5 (the nomenclature is explained in Sect. 2). For clarity we omit too many neighbouring data points and tracks.

It is obvious that the 2013-max1 have just crossed the ∼8200 K boundary, as well as a number of ordinary pulsations before and after 2013. All with almost the same high temperatures, such as max0, max3, and even max5 (see Figs. A.1 and A.2 of Appendix A). In more detail max2, the median, and min1 are plotted with large red dots. The small drawn blue dots in Fig. 5, connected by blue lines, represent the selected data points and tracks, respectively, of ordinary data points preceding the 2013 outburst (red star symbol). The maxima and minima of the pulsations are shown with small red dots (see max-1, ∼max0 (meaning close to max0), to ∼min0 to max1 (red star)). The blue tracks marked with blue dots eventually become red when approaching the outburst max1. Thereafter, the red track further continues to the median and min1 (large red dot), and next returning back to the median value of max2 (large red dot). The red track then continues to the finish via two ordinary data points after max2. The two isolated small red dots, viz. max3 and max5, are located close to max1.

The sudden switch to the right from the 6000 K median to 4900 K and 5850 K (from the Kr dataset) into the direction of min1 is most surprising. The radius is expected to increase to ∼700–800 R_⊙. This part of the outburst track remains at almost constant luminosity. To complete the gap in the AAVSO V-lightcurve of Kraus et al. (2019), in Fig. 3 at JD 24 55650-55900, we used some AAVSO visual observations (Maravelias & Kraus 2022), together with a few photoelectric V- and B-magnitudes from G.W.H. This is why the visual lightcurve shows less gaps and is therefore more useful.

Figure 5 demonstrates that Maravelias & Kraus (2022) are correct. In 2013 ρ Cas bounced against the ∼8200 K instability boundary of the YEV. We think that ρ Cas also crossed it. Surprisingly, this is also the case for some ordinary pulsations before and after 2013 shown in Fig. 5 (see in the time-series of Table I in Appendix A, and Figs. A.1 and A.2). The temperatures vary between 7000 K and 8000 K, or more, for example, max-1, max0, max3, and even max5 (and possibly also max6 in 2017 at ∼JD 24 57900).

In Paper I we expressed the assumption that the outbursts prefer ‘red circumstances’, that is to say they tend to occur when the LTV of (B − V) are red (similar to our experience with the 1986 and 2000 outbursts discussed in Sect. 2). This appears to be also true for the 1973 outburst of HR 8752 (see Nieuwenhuijzen et al. (2012) and Fig. 7 of Paper I). We can add the 1975 outburst of HR 5171A, and also the Otero outburst in 2000 (see Figs. 13 and 14 of Paper I).

We therefore consider an outburst in ‘blue circumstances’ as a rare phenomenon. Indeed, ρ Cas has commenced to enter into the next evolutionary phase. It was obviously dynamically very unstable before and after 2013, but also hotter than usual and subject to shell ejections and, among others, to hot convection cells as well. As a result, ρ Cas showed complicated light and colour curves at that time (see Sect. 2 where this is analysed and discussed at the end of Appendix D).

HR 8752 became a stable star somewhere between 1996 and 2017. Its photometry shows a stable star in Fig. E.5. We assume that stability started in 2005 or slightly later, 32 y after the last outburst in 1973. The next evolutionary phase for ρ Cas is approaching, which is supported by our observations mentioned in Table 1, and presented in Figs. A.1 and A.2. Their (BV) photometry shows a sequence of decreasing pulsation amplitudes and quasi-periods since around 2019, following pulsation the max7 until 2023. Adopting the HR 8752 time-scale, the stability of ρ Cas may be expected in ∼32 yr after the last outburst of 2013, or possibly around the year 2045.

Conclusions including the analysis in Appendix D:

1. Section 6 is devoted to the ‘to-and-fro’ pattern of the 2008-2017 time-series, including the 2013 outburst in the HRD. It is based on the T_eff-dataset of Kr, slightly overlapping the one of Kl, we discuss in Sects. 2, 3, 4, and in the present one. The temperatures we use for the selected photometric observations in Fig. 5 mainly consist of genuine T_eff-values that are partly based on our 4T method.

2. The to-and-fro tracks of the ordinary pulsations, and of the 2013 outburst, run within a less than ∼700 K wide, oblique, linear path from ∼8000 K to the median of T_eff∼6000 K. Its most striking feature is the sudden switch after the median running horizontally (at almost constant luminosity) to min1, having a temperature slightly below 5000 K. At last the track returns to max2, but likely not precisely to all the way there.

3. The 2013 outburst occurred for ‘blue circumstances’, in contrast to the 1986 and 2000 outbursts (see Fig. 1).

4. Based on our analysis presented at the end of Appendix D we conclude that Lobel et al. (2003) are correct. Pulsation No. 7 (Fig. 2 of Paper I) was the very first pulsation (marked in red with 8 in Appendix D) just after the 2000 outburst, obviously being abnormal with too deep a brightness minimum. We show that the subsequent pulsations until the pulsations following the 2013 outburst are abnormal as well. In the 2013-domain ρ Cas bounced against the 8200 K YEV boundary, even crossed it. It provides proof that it shed sufficient mass for entering into the YEV-region.

5. The next evolutionary stage is to move further on a blue loop, entering the next instability region in the HRD at temperatures above 10 000 K (e.g. the S Dor region). A sequence of decreasing pulsation periods and smaller amplitudes started in 2019 according to the BV photometry we present in Table 1 and in Figs. A.1 and A.2. It indicates that ρ Cas will follow HR 8752 towards dynamical stability, possibly around 2045.

7. The 1986, 2000, and 2013 ρ Cas outburst tracks on the HR diagram

Figure 6 depicts the to-and-fro tracks of the three outbursts in 1986, 2000 and 2013. About half of the temperatures for part of the data points are based on T_eff measurements, while the remainder on our 4T method. They are represented by the four characteristic points as usual. The max0 and min0 of the preceding pulsation of the 2000 and 2013 outbursts (shown in Fig. 5) are also plotted (see Table E.1 of the Appendix). As a matter of interest we added in Fig. 6 the durations △ D in days, of the three minima min1, measured 0$\stackrel{\text{m}}{.}$1 above the faintest point of the V-lightcurves. According to de Jager et al. (2001) the next variable-star phase could be an S Dor variable. The approximate S Dor-location in the HRD is from, for example, Smith et al. (2004) and Clark et al. (2005).

The location of the three max1 of the 1986, 2000, and 2013 outbursts gradually approaches the 8200 K YEV boundary in steps of ∼500 K. The one of 2013 also crossed it, we discuss in Sects. 6 and 1.1. Each new outburst cycle from max1 (at T_eff ∼ 7000 K to 8000 K) to the deep minimum min1 (at T_eff ∼ 4000 K to 4500 K), and back to max2 (at high T_eff) is required for the YHG to shed a sufficient amount of mass.

The three tracks, including their medians, almost coincide, while thereafter the 1986 and 2000 tracks abruptly switch to the right-hand side as during the 2013 outburst (see Appendix A.3), however at a lower luminosity (see the light and colour curves of the three outbursts in, for example, Fig. 13 of Lobel et al. 2003, and in this paper Figs. E.1 to E.3 of the Appendix).

The width of the path (shown in Figs. 5 and 6) of the to-and-fro tracks, including a few ordinary quasi-periodic pulsations, is surprisingly small by ∼500 K. Inaccuracies can contribute to this width, in particular for the temperature and luminosity, as well as the variable continuum opacity. The relationship between luminosity and temperature we find for this narrow path is:

$$\begin{array}{c}\hfill logL/{\mathrm{L}}_{\odot }=-2.23+2log{T}_{\mathrm{eff}}.\end{array}$$

The results are tabulated in Table E.1 of the Appendix. The too red (B − V)₀-values of the three maxima max1, viz. ∼0$\stackrel{\text{m}}{.}$80–1$\stackrel{\text{m}}{.}$0, are discussed in Appendix A.3.3. One can differentiate between opacity and temperature variations because the latter T_eff-values are known, which is helping to improve our insight into the wavelength dependencies on the opacity.

At the start of an atmospheric outburst the star is compressed and T_eff has increased to 7500 K – 8000 K. Its opacity is high due to the large mass-loss rate. Subsequently, the star rapidly expands to about twice its normal size of ∼800 R_⊙ (with errors within 100 R_⊙) according to the model of Lobel et al. (2003) and based on V_rad observations. Within ∼200 d it reaches the minimum T_eff of ∼4400 K (min1). It this point the hydrogen gas in the expanded atmosphere has almost fully recombined and the selective continuum opacity is about zero.

After some time the star shrinks back to a more normal size at max2 for a total duration of about one year. Lobel et al. (2003) hydrostatic calculations in spherical geometry predict the correct time-scales for the outburst velocities.

Similar mechanical processes, more or less analogous to this fast atmospheric expansion, can be thought of resembling the release of elastic potential energy stored in the springs of a trampoline after its compression. The fast global atmospheric expansion results from the release of hydrogen (H⁺) ionisation-recombination energy stored in its voluminous and extended partial hydrogen ionisation zone (Lobel et al. 2003).

Another analogy can be found in geology for a small part in the crust of Earth-like planets such as the Earth and Moon. When a massive meteorite hits one of both, the crust in the central part at the crater bottom becomes compressed and hotter, sometimes over many kilometers deep, but which immediately rebounds, albeit rather slowly (see Carruthers 2019, pg. 57–59; Reimold & Gibson 2005).

Another arresting feature which is apparent to us, is that all detailed light and colour curves of the outburst minima show a sort of twist, or bump, just after the start to recovery from the deep brightness minimum. Even the 1945-minimum in the photographic lightcurve of Gaposhkin (1949), also shown in Fig. 13 of Lobel et al. (2003), reveals a bump (the visual AAVSO curve is not sufficiently accurate for this), shown in Figs. E.1., E.2, and E.3 of Appendix E. This feature often appears at the start of the ascending branch. It supports our suspicion that both branches evolve quite differently. The 1986 LTV median (B − V)_obs-curve (see Fig. E.1) shows the feature between the numbers marked 2 and 3. This is however not a genuine minimum because it coincides with the rising branch of V, while the genuine min1 in 1986 (no observations are available) occurred presumably around ∼JD 24 46550 and should be significantly redder (based on the lightcurve shown in Fig. E.1). The 1986-lightcurve is also shown in Fig. 13 of Lobel et al. (2003).

Figure E.1 shows the 2013-min1 V-brightens after about 40 d, while (B − V) remains red for at least two months, and while the B-magnitude stays obviously behind. The explanation for both features can be illustrated by drawing the individual 1986 B- and V-measurements at min1, by hand on graphic paper. The result is surprising. Indeed, we find that the first part of the ascending lightcurve, from location 2 to 3, appears to be due to a much faster brightness increase in V than in B. The latter magnitude even tends to be lagging behind for a while.

A possible scenario for this is a modest increase of selective continuum opacity which causes a temporal retardation of the brightening in B, but not in V. Then, as T_eff increases faster, V overtakes the B-brightness, lasting for about two months. This explains the short reddening track from 2 to 3. Thereafter, following a further temperature increase, the B-brightening overtakes the one in V (see the track from 3 to b and c). The light and colour curve of the 2013 min1 (right-hand side of Fig. E.1) in V and (B − V) are also a good example of such a scenario, although the total dimming was only half of the 1986 outburst. As a result, the temperature only decreased to 4850 K (Kraus et al. 2019), although some selective continuum opacity appears to be present.

It is also striking that a few months prior to the outbursts of 1986, 2000, and 2013, a strong P Cygni Hα profile appears in the spectra, but which disappears in the deep brightness minima min1. It signals that fast contraction phases of the atmosphere precede these outbursts. The strong Hα profile for example, preceding the 2013 outburst, becomes visible in Nov 2012 between max0 and max1 when the photospheric absorption lines show red-shifted Doppler displacements. A new blue-shifted absorption component appeared near the centre of Hα absorption core. Thereafter, it disappeared or became blended by the centre of the Hα-line.

Conclusions:

1. The brightness- and colour curves of the outbursts of ρ Cas are unique, showing many morphological differences, sharing this characteristic with the quasi-periodic pulsations.

2. Figure 6 shows the gradual approach of the three max1 of the 1986, 2000, and 2013 outbursts to the cool YEV boundary, while the one of 2013 even entered into it. The difference in T_eff amounts to ∼500 K. The luminosity is lowest for the 1986 outburst. The cyclic tracks or loops between 8000 K and 4000 K are necessary to shed sufficient mass as was theoretically predicted by Nieuwenhuijzen & de Jager (2000).

3. It appears that all detailed light, and especially the colour curves of the deep outburst minima (including 1946 in Fig. 13 of Lobel et al. 2003) show a twist or bump just after the start of recovery from min1. All pass-bands of the Johnson filter system are sensitive to it, but more so at longer wavelengths than at shorter ones. It is not surprising that all outburst minima show this twist or bump, just after the start of the ascending branch, lasting for some months. In summary, we find that shortly after the brightness increase from min1 to max2, a very early but slow increase of the selective continuum opacity temporarily delayed the brightening of the short-wavelength magnitude B (or I) compared to the brightening in the long-wavelength magnitude V (or R). A possible cause might be that immediately after the start of the increase to max2 sufficient amounts of H⁻-anions can increase the selective continuum opacity.

4. The cyclic to-and-fro tracks of the three outbursts in the HRD almost coincide. Only the moment of a sudden switch to the right-hand side, following the median to the continuum opacity-free location (min1), occurs at quite a different luminosity, in agreement with the T_eff-differences of min1, being largest in 2013 and smallest in 2000. However, the 1986-min1 is unreliable as it is not precisely known (Fig. E.1). In Appendix E we discuss a number of subjects entitled: Light and colour curves of outbursts show morphological differences. The transformation of (V − R)₀ to (B − V)₀. Peculiar complicated spectral features in the spectra of YHGs. Interfering pulsations in outbursts? More information on HR5171A and Fig. E.6.

7.1. The past decades of HR 8752 as a YHG in transition after its 1973 outburst

Based on an early comparative study of the detailed spectroscopic variability of the YHGs HR 8752 and ρ Cas, both could then still be called spectroscopic twins (Gorlova et al. 2006). Observations made between 1991 and 1995 revealed that they still shared many properties, but not enough to be called spectral twins. This was also the case when compared to HR 5171A (Israelian et al. 1999; Lobel et al. 2015; Klochkova 2019). One of the reasons is that HR 8752 has lost already some typical YHG characteristics between 1991 and 1995. The most complete LTVs of the UBV-parameters of HR 8752 to date of 1941–1994, a time span of 53 y, have been depicted in Figs. 6 and 7 of Paper I. It also includes the location of the last 1973 outburst with a spectroscopically observed min1 by Luck (1975) of T(Sp) = 4000 K ± 300 K, and (B − V)_obs = 1.75.

A comprehensive overview of the gradual physical changes in HR 8752, almost a dozen years before the analysis of fundamental changes in HR 8752 by Nieuwenhuijzen et al. (2012), was presented by de Jager et al. (2001). The latter authors assumed that the temperature of HR 8752 started to increase in 1985 following a number of episodes with large mass-loss. According to our LTV of HR 8752, however, the temperature already started to increase in 1977.5 (see above), when suddenly only the V-brightness declined. They hypothesised that HR 8752 commenced to traverse the YEV and next entering a region of dynamical instability, to become a P Cyg-like star, or an S Dor variable. The study by Nieuwenhuijzen et al. (2012) offers much spectroscopic and photometric data shown in many graphs. They investigate changes of photometric and physical parameters, such as V and (B − V), T_eff, the surface gravity g, the radius R, and the luminosity L in their Figs. 5 to 13. A large part also discusses theory, generally based on evolutionary models of Maeder et al. (1989), Maeder & Meynet (1987, 1988), and Meynet et al. (1994). Their study delivered much new insights into the physical state during its recent start of blue-loop evolution after the 1973 outburst. One of the fundamental differences of HR 8752 with respect to e.g. ρ Cas is its totally different brightness history. According to early records too faint: V ∼ 6 around 1840, and at present ∼5.1 (see LTV of V between 1946 and 1990 (Paper I, Fig. 6)). HR 8752 was almost invisible before that time and probably even up to medieval times (Zsoldos 1986a; Nieuwenhuijzen et al. 2012, Sect. 3.5, Fig. 13 and Sect. 5.6).

For gaining some insight in the photometric evolution of HR 8752 after the 1973 outburst, we demonstrate by means of simple chronology, by mentioning the date, the observed median brightness and the colour index V and (B − V), respectively. 1973: 5.0-1.75; 1974: 4.95-1.73; 1977.5: 4.8-1.6; 1995/1996: 5.15-0.85; 2000-2005: no photometry; 2005-2017: no photometry; 2017-2023: 5.3-0.85.

In Appendix E.2 we discuss the variation of some physical parameters, such as the linear shortening of the pulsation periods from 600 d in 1974 until 100 d in 1994 (unknown to Nieuwenhuijzen et al. 2012). We show in Fig. E.4 the gradual decrease of amplitudes in V, B, and (B − V), varying between 0$\stackrel{\text{m}}{.}$1-0$\stackrel{\text{m}}{.}$3, 0$\stackrel{\text{m}}{.}$2-0$\stackrel{\text{m}}{.}$4, and 0$\stackrel{\text{m}}{.}$05-0$\stackrel{\text{m}}{.}$2, respectively. We comment on the T, g, R, and L variations by comparing with their figures. The moment of total absence of ordinary pulsations should have happened between 2005 and our first BV photometry (see Table 1) in 2017 (Fig. E.5).

Obviously, the brightness in V did not vary much, contrary to (B − V) which became appreciably bluer. T_eff increased slowly between 1973 and 1977.5, and subsequently faster to 1995/1996, but hardly changed in 2017 through 2023. Our photometry of V (5^m.3) and (B − V) (0^m.9) fitted after some extrapolation very well to their Figs. 13 and 11, respectively. The most radical time span was between 1974 and 1995/1996 when the pulsation periods linearly decreased from ∼600 d to 100 d (see Fig. E.4). The temperature increased from ∼5000 K to 7200 K in 1995/1996 according to Nieuwenhuijzen et al. (2012), while the surface gravity rose as well, in contrast to the luminosity and radius which decreased (see Nieuwenhuijzen et al. 2012: their Figs. 29 and 20, resp.). Hence, the star became smaller and hotter.

This time span was followed by the second radical one between 1995/1996 and 2000/2005 when the temperature increased to 7800 K. The third radical time span happened somewhere between 2005 and 2017 (date of our first BV photometry mentioned above) when HR 8752 was likely a stable star of ∼8000 K. Perhaps, it will gradually become an S Dor variable with a high mass-loss rate, and thereafter, possibly more speculative, a B[e] or a WR-star. One expects that this star will end in an SN explosion.

Conclusions:

1. Based on our study of HR 8752 (and by Nieuwenhuijzen et al. 2012) we discuss the evolution of its V-brightness and the observed colour index (B − V), together with a number of physical properties. We find the disappearance of its quasi-periodic pulsations with the increase in temperature and surface gravity, and a decline of luminosity and stellar radius. The most radical times over which most physical parameters changed appreciably occurred between 1973/1974 (T_eff = 5000 K) and 1995/1996 (T_eff = 7200 K), and subsequently between 1995/1996 and 2000/2005 (T_eff = 7800 K).

2. To cite Nieuwenhuijzen et al. (2012) at the end of their Abstract about the future of HR 8752: ‘it is on its way to stability, but in the course of its future evolution it still has to go through a second potential unstable region’. Like de Jager et al. (2001), we speculate that this could well be the S Dor phase. We started to monitor the star photometrically in 2017 (Table 1) when it turned out that HR 8752 was already stable, shown in Fig. E.5 based on our BV-data.

3. The total time span between the outburst in 1973 and 2000-2005, when HR 8752 had crossed the YEV, lasted ∼30 y, although the precise date for complete stability is unknown. It appears that AAVSO V magnitudes exists between these years, showing no significant light variations.

4. Applying the evolutionary time-scale of HR 8752 to ρ Cas, which started in 2019 to decrease its pulsation periods (see Sect. 6), we expect that ρ Cas will need another ∼30 y for also becoming a stable star, or somewhere between 2045 and 2050.

7.2. Report on the present pulsation activity of HR 5171A

In view of the behaviour of HR 8752 described above, we consider it advisable to also inspect the present pulsation activity of HR 5171A. In Sect. 3.4.3 of Paper I we reported on the suddenly smaller pulsation amplitudes in HR 5171A moving on a blue loop. In Fig. E.6 of Appendix E (see Table 1), we plot and discuss the AAVSO visual lightcurve of HR 5171A = V766 Cen between 2015 and 2022. The graph is prepared by E.B (see Table 1) who also contributed visual observations, while R.T. was responsible for the grey one-sigma curves. Figure 15 of Paper I depicts the time span of 2015-2018. The reason to show it at the time is because of its decreasing pulsation amplitudes which starts rather abruptly in 2015. According to the longer time-series until 2022, shown in Fig. E.6, this is not the case.

Conclusion:

The abrupt decline of the pulsation activity of HR 5171A in 2015 was obviously a short-lasting event. From Fig. E.6 we find that a few interesting publications exist about H-band images of HR 5171A by Wittkovski et al. (2017a,b,c).

8. In pursuit of the solution

Figure 7 presents the highlight of this paper. It consists of four panels. We speculate it can serve as a prototype for other YHGs. The data in Fig. 7 are offered in Table F.1 of Appendix F. All blue and red dashed lines/curves represent blue and red tracks, respectively.

8.1. The first panel

Because of the close connection between the four panels, we cannot always avoid to refer to some other one. The first panel shows the median LTV T_eff-curve using 34 blue points in Fig. 1 of Sect. 2, and four characteristic dots of the outbursts in red colour. We note that some data points coincide. The temperatures are based on the 4T method using Fig. 2 in Sect. 3. The results are independent of the interstellar extinction. The scatter amounts to about ±400 K, in agreement with Sect. 3.

The total amplitude of the LTV of T_eff amounts to ∼1000 K. The curve hovers between 6000 K and 7000 K. The red dots on top of each of the two branches, max1 (to the left) and max2 (to the right) are between 7000 K and 8000 K. The medians of the outbursts are situated between the two branches (also at red dot). They hover between 6000 K and 6500 K. The min1 are situated below the two branches, hovering between 4500 K (1986 and 2000) and 4850 K (2013). We wondered which percentage of the T_eff variations in the first panel is responsible for the (B − V)_obs variations in the second panel, and how many percent for the continuum opacity variations. This has been accomplished by transforming small steps in T_eff variations (black lines shown at the bottom of the first panel) into matching steps in the (B − V)dJN-scale. The percentages of T_eff are indicated above the short black arrows along the reduced JD-axis. The remainder must be due to continuum opacity variations. We only calculate averages in relatively large steps (from right to left), tabulated in Table 4.

With regards to the accuracy of these percentages we find that the uncertainty of the curves due to the scatter of the data points is unknown since part of it can be intrinsic. We therefore think that the reliability of these percentages should not be overestimated. The same applies to the marked outbursts. Part of the differences are due to morphological differences between the light and colour curves.

8.2. The second panel

The same 34 blue dots we mention above, together with the four characteristic points of the outbursts, are plotted in the second panel of Fig. 7 versus their median (B − V)_obs-values. We note that for all data points (42 in total) the log T_eff- and log L/L_⊙-values are used to illustrate the zig-zag behaviour in their domain of the HRD in Figs. 4 and 5.

We note that the domain before and after the 2013 outburst shows an irregular behaviour in all panels. It is a consequence of its unique evolutionary state we discussed in the previous sections.

The two rows with green numbers 1–24 along the reduced JD axes at the top and bottom of the third panel indicate the time intervals, likely due to non-radial pulsations (marked green bars), or a mixture with radial pulsations (marked dotted bars). More details are discussed in the next subsection. Empty gaps indicate time-intervals with mainly radial pulsations, especially the ones close to the outburst.

The green ellipse to the right of the fourth panel shows the 2013 outburst domain with the time differences between all short lasting atypical bumps, peaks, and shoulders with respect to the maxima of ordinary pulsations. Its isolated location in relation to the median curve of P of the quasi-periods (but also showing fluctuations) guarantees that we are not dealing with normal pulsations (see Fig. 1 and Fig. A.1 by G.H.W. in the Appendix, and the visual light curve in Fig. 3 of Kraus et al. 2019). This time-interval contains all atypical features and lasted 5-6 y, including the 2013 outburst. The data points in Fig. 7 are listed in Table F.1.

Connecting the max1 of 2013, 2000 and 1986, representing the outbursts of ρ Cas in the first panel of Fig. 7, with the max1 of 1946: blue circle on the left of the 1946 outburst in Fig. H.1, it appears that the temperature shows a declining trend from 2013 to 1946, (no max1 known for 1905). An approximate range for max1 is from about 8000 K to 6000 K.

A similar pattern is offered by min1 of 2013, 2000 and 1986 in the second panel of Fig. 7, up to the three min1 in Fig. H.1 of 1986 (again), 1946 and 1905. Their (B − V)_obs values become redder from 2013 to 1905. As the selective continuum opacity is now zero, (B − V)_obs has become a more reliable T_eff indicator. However the 1986 outburst, is unreliable by the absence of a precise minimum brightness (Fig. 6 and Fig. E.1), and bracketed in Fig. 7 second panel. An approximate range of the min1 temperatures amounts to between 4900 K (2013) and about 4000 K (1905 and 1946, but both are not very accurate). Both are based on a rough approximation of the T_eff dependency of the (B − V)_obs of the min1 in Fig. 7 first panel. The most recent outburst (2013) is called ‘old’, with respect to the previous ones like 1905, which is called ‘young’.

We conclude that a sequence of outbursts, with a gradually rising T_eff is necessary to loose mass until the total stellar mass decreased sufficiently for crossing the ∼8200 K YEV boundary. Thereafter, the YHG evolves on a blue track through the YEV from the beginning to the next potentially unstable region in the HRD, presumably becoming an S Dor variable.

8.3. The third and fourth panels

Because of the close connection between the third and fourth panel, we combine their discussion. The third panel represents the median curve of the radius variations of all ordinary data points shown in the first and fourth panel. We used the formulae for M_bol and T_eff (see Appendix A) with a distance of 2.5 kpc (see Sect. 2.9 of Paper I). The time-scale of the wavy pattern amounts to ∼15 y with an amplitude of ∼100 R_⊙. It runs in anti-phase with the temperature- and colour index-curves. The stellar radius is smaller at high temperatures and vice versa. The maximum radius variation of the median curve amounts to ∼100 R_⊙. The median values hover between 350 R_⊙ and 450 R_⊙. The radii of the two characteristic points of the three outbursts, max1 and min1, are of the same order of 750 R_⊙ and 350 R_⊙. The radii are within the estimated error, similar to those based on radial velocity observations that are independent of distance (see Lobel et al. 2003 with 800 R_⊙ and 400 R_⊙, and an uncertainty of ±100 R_⊙). These results support the reliability of our 4T calibration relations.

The fourth panel shows the trend of the mean period, represented by a dashed curve, of a number (bracketed) of ordinary pulsations hovering between 200 d and 500 d. We investigated in total ∼50 pulsation cycles. Additionally, the outbursts are represented by a single red dot for which the duration between max1 and max2 is used as a pulsation period, although in the case of 1986 and 2013 longer than any ordinary pulsation. We don’t know whether that has any physical significance.

The reason that the dashed curve does not intersect with any dot in the neighbourhood of the 2013 eruption is on purpose. ρ Cas showed strong dynamical instability, and ordinary pulsations behaved abnormal (see e.g. Appendix D). They all represent peculiar light and colour curve variations, with secondary peaks and shoulders, before 2013 (but after the 2000-eruption), as well as during and after the 2013 eruption. As a consequence most time scales do not represent a proper ordinary pulsations. Therefore the dashed curves must be considered as arbitrary.

It is not always easy to decide whether a pulsation cycle is radial or non-radial. Large amplitude pulsations and long variability periods in YHGs are typically caused by radial pulsations. This is especially in view of the radial velocity (V_rad) observations. The phase difference between V_rad and V amounts to a quarter of P, which points to atmospheric pulsations. The three pulsations just before the 2000 outburst show considerably larger V_rad pulsation amplitudes (Lobel et al. 2003). They signal large movements in the gas layers of the atmosphere. The amplitudes of V also increased, although they are thought to relate more to the variations in temperature. This occurs even up to the weaker 2013 outburst. It is reasonable to assume that (and supported by the relations in the first, third, and fourth panels) whenever the pulsation amplitudes increase prior to an outburst on a red track, the photospheric radius and temperature increase as well. As a result, the coherency in the radial pulsation movements will also increase, which occurs shortly before the outbursts over a short blue track (see Fig. 7).

We think that the pulsations of ρ Cas are becoming gradually more radially organised over a sufficient amount of pulsation cycles in order to drive an outburst event. A regular succession of almost pure radial pulsations is capable of lifting a sufficient amount of atmospheric gas mass into space. Only global radial oscillations are able to do so because of the large total gas momentum (mass times velocity) they can impart to the upper layers of the extended atmosphere. This appears impossible during the regular non-radial pulsation cycles. With the globally strong radial pulsations the recombination of hydrogen (H⁺) can release a large amount of ionisation-recombination energy that drives the fast atmospheric expansion into an outburst (Lobel et al. 2003). We observe that the pulsations periods of the V light curves tend to lengthen while the V-amplitudes increase.

Almost pure non-radial pulsations are found in points 1 to 4, marked with the solid green bar in the fourth panel. This is nicely confirmed with the non-radial pulsation in 1970 investigated by Lobel et al. (1994), and shown by the short green bar above the longer one in the fourth panel. It coincides with the minimum of P. Thereafter follows a period (without marked green bar) coinciding with increasing pulsation periods along a red track, that also signal gradually more radial pulsations prior to the 1986 outburst. The dotted bars, for example marked with green numbers below the reduced JD-axis in the fourth panel (between Nos. 10 and 15), represent a mix of radial and non-radial pulsations preceding the 2000 outburst.

The results in Fig. 6 of the previous section are most relevant for this chapter. Nieuwenhuijzen & de Jager (2000) predicted that each cycle from max1 (T_eff = 7000 K–8000 K) via min1 T_eff = 4000 K–4500 K, and back to max2 (with large T_eff) is necessary for a YHG to lose sufficient amounts of mass in order to evolve across a blue loop into the YEV, and still further into the next instability region. We find that ρ Cas has reached this stage, as well as HR 8752 (see references in Table 1).

The gradual increase of the max1 temperatures in Fig. 7 (first panel) from 1986 via 2000 to 2013(by ∼500 K) explains their increasing luminosity shown in Fig. 6. The green numbers 1–24 at the bottom of the third and fourth panel of Fig. 7, represent small time intervals, which are useful for the discussion on the sizes of R/R_⊙, P (both read from the smooth curves) and the type of pulsations: radial or non-radial. This numbering is also used for the three characteristic points of the three outbursts: 7(1), 7(2), 7(3), 15(1), 15(2), etc. We omitted the medians.

Conclusions on the third and fourth panel:

1. The fourth panel of Fig. 7 shows a continuous cycle of average quasi-periods (blue dots with the number of used pulsations in brackets) having a time-scale of ∼15 y. Between max1 and max2, as a special type of pulsation (red dots) at the top of the cycle, is the duration of the outburst which counts most. We find the latter duration to be longest, but we don’t know if this has any significant physical meaning. The fourth panel shows a sort of ‘symbiosis’ between the explosive state with very large mass-loss rate, and the onset of radial pulsations prior to the outbursts. To the left, after a short-lasting red track, follows a short-lasting blue track. The entire 2013 time interval shows many atypical lightcurve features. Receding blue tracks to the right, on a blue track, show decreasing periods. From the above we assume that the outbursts are predictable five years in advance.

2. The normal outbursts of 1986 and 2000 happened in a red (B − V)_obs-minimum, including the 2013 outburst, albeit a small local one (see second panel). As expected, the entire 2010-2015 domain showed strong fluctuations in (B − V)_obs.

3. The relations in the third (R/R_⊙) and fourth (P(d)) panels are running in phase. The longest periods (500 d) also show the largest radii (450 R_⊙), and vice versa. Both relations run in anti-phase with the first (T_eff) and the second ((B − V)_obs) panels. Hence, they show the largest temperatures and bluest (B − V)_obs when the stellar radii are smallest and the quasi-periods shortest.

4. Only a regular succession of almost strict radial pulsations can cause an outburst in YHGs. Under the condition of global coherence can radial atmospheric movements lift sufficient amounts of mass into space due to the large total momentum radial pulsations can impart. In a YHG outburst the global pulsation-decompression releases large amounts of hydrogen (H⁺) ionisation-recombination energy which drives the fast atmospheric expansion and subsequent cooling, thereby further enhancing this hydrogen-recombination thereby sustaining the global atmospheric decompression and cooling down. We present in Appendix F: Summary and a general image of the information offered by the four panels.

9. Conclusions

1. The highlight of this paper is provided in Fig. 7 (thanks to the construction of the T_eff calibration relations, mentioned in Conclusion 2 below). It depicts four cyclic curves from between 1962 and 2020 running in phase or in anti-phase and showing a close connection between a number of prominent physical properties and the onset of the three massive atmospheric outbursts in 1986, 2000, and 2013. In the following paragraph, we discuss the four panels of Fig. 7. The first panel (bottom) concerns the T_eff/selective continuum opacity (for which H⁻-anions are responsible). Both are variable but opposite to each other. The second panel (second from bottom) shows the median (B − V)_obs-curve of all ordinary pulsations and outbursts between 1962 and 2020. The third panel shows the trend of the stellar radius R/R_⊙. The fourth panel (top) shows the trend of the quasi-periods for about 40 ordinary pulsations. The onset of the three outburst events are represented by red dots (for which H⁺ atoms are responsible through the release of an enormous amount of recombination energy). Their ‘periods’ represent the time interval between max1 and max2. Each outburst marks an evolutionary cycle, between ∼4500 K and ∼7500 K. It is obvious that the 2013 outburst already crossed the 8200 K yellow evolutionary void (YEV) boundary. It appears that a YHG at high temperature, such as the 2013 outburst, shows more chaos in the ordinary pulsation cycles, with many secondary short-lasting bumps and peaks in the pulsation light curves (time differences between these short cycles are located in the green ellipse in the fourth panel). The 1946 outburst fits the cyclic behaviour (see Fig. H.1 showing the cycle from 1905 to 1986).

2. We constructed four new temperature calibration relations based on reliable T_eff observations of three spectroscopic datasets (named Lo, Kl, and Kr) together with simultaneously observed V_obs, B_obs, and (B − V)_obs photometry by these authors and also by G.W.H. (see Table 1). The average uncertainty based on our ‘4T method’ amounts to ±264 K, with incidental peaks from 400 K to 500 K. The new relations are independent of IS reddening and extinction. The fourth relation is based on the T_eff-s parameter scale set by de Jager & Nieuwenhuijzen (1987), which we call the ‘dJN temperature scale’. The relations are of crucial importance for successful analyses in various parts of our YHG investigation. Based on our new relations, precise evolutionary tracks for the outbursts and ordinary pulsations can be depicted in detail in the HRD for the very first time.

3. For the first time (also thanks to Conclusion 2.) it is possible to show the evolutionary cyclic loops and nearby ordinary pulsations in detail in the HRD, or log T_eff versus log L/L_⊙ in Figs. 5, 6, and 7. According to the theoretical prediction by de Jager and Nieuwenhuijzen (see Sect. 7), each cycle is necessary in order to lose a sufficient amount of mass until the total stellar mass has sufficiently decreased to cross the 8200 K YEV boundary and to subsequently move on a blue track through the YEV, thus evolving to the next potentially unstable region in the HRD (presumably that will be the one for the S Dor variables). Indeed, we find that the distances between the max1 of each outburst from 1986, 2000, and 2013 in Fig. 6 shift after each cycle by ∼500 K towards the blue, causing the 2013 cycle to cross the 8200 K boundary. A precise analysis becomes even more compelling by connecting the outburst maxima (max1) of 2013 to 2000 and 1986 to 1946 (in Sects. 8 to 8.3, Fig. 7 and Fig. H.1). The approximate decline of the temperatures range from 8000 K to 6000 K. The approximate decline for the outburst minima (min1) of 2013 to 2000, 1986, 1946 to 1905, ranges from 4900 K to 4000 K. As in the minima (min1), the selective continuum opacity is zero, and the (B − V)_obs becomes a reliable temperature indicator.

4. We find that the ordinary pulsation periods in the V light curves become longer and the amplitudes become larger prior to an outburst. This also concerns the ordinary pulsations approaching the 1946 outburst shown in the original PG light curve of Gaposhkin (1949). Strong radial pulsations appear to be necessary for the onset of the outbursts. It is our opinion that the pulsations gradually become more radially organised over a sufficient number of pulsation cycles. Only a regular succession of almost global radial pulsations is capable of lifting sufficient amounts of mass into space due to the large total gas momentum they can impart for driving the fast atmospheric expansion of an outburst.

5. The location of ρ Cas that we calculated in the HRD (log T_eff versus log L/L_⊙) is 6700 K ±700 K st.dev. The large standard deviation results in a variable star, where log T_eff = 3.83 and log L/L_⊙ = 5.42 ±0.3. Its location is close to the YEV defined by Nieuwenhuijzen & de Jager (2000) in their Fig. 1. They report the same T_eff but a higher luminosity of 5.7 because they adopted a larger distance (see Sect. 4).

6. HR 8752 entered the YEV and evolved on a blue track after 1996. It appeared to be a stable YHG between 2017 and 2023 (Fig. E.5, Table 1). This has been discovered by R.J.v.B. and M.S. based on BV photometry. The observed photometric parameters during this epoch are as follows: V = 5.30, B = 6.15, and (B − V) = 0.85. These data are not much different from the extrapolations of the medians plotted in Figs. 6 and 7 of Paper I. Between 1996 and 2017, the star seemed to be observed in V by many observers, but the variations are not significant at all. Thus, HR 8752 was already constant in 1996. This is based on consulting the AAVSO International Database Data Usage Guidelines¹.

7. The same evolutionary transformation is currently occurring in ρ Cas. It started just after the 2000 outburst with pulsations until the most recent 2013 outburst (see the discussion on the analysis of the pulsations at the start of Fig. E.1). The pulsation cycles until 2019 (at pulsation No. 6) became more chaotic, and the amplitudes have decreased systematically since then (see Figs. A.1 and A.2 and Table 1, where both light and colour curves are relative to the same comparison star). Figure A.1 shows the BV photometry by G.W.R. from 2003 to 2020 and Fig. A.2 by E.J.v.B and M.S. between 2015 and 2023. The mean observed photometric parameters in 2023 are as follows: V = 4.41, B = 5.66, and (B − V) = 1.25. They are not much different from the extrapolations of the medians plotted in Figs. 11 and 12 of Paper I. Using HR 8752 as the prototype of YHGs, ρ Cas should become stable around 2045.

8. HR 5171A appeared to resume its normal quasi-periodic pulsation pattern in early 2018 after the gradual brightness decline and decreasing amplitudes at the end of 2016 (see Fig. 15 in Sect. 3.4.3 of Paper I). Because of the interesting experience with HR 8752 and ρ Cas, E.B. checked the present stability of HR 5171A by collecting AAVSO visual observations. The resulting light curve of HR 5171A until 2023 is shown in Fig. E.6 (see Appendix E.5 and Table 1). Two other items of interest are Appendix E.3 and Appendix E.4.

9. The variable star type of HD 179821 is currently uncertain. Arkhipova et al. (2009) assumed that the very vast U, BV, and V brightness variations, which sometimes vary in opposite directions (see Fig. 9 of Paper I), are not typical of YHGs. They are due to variations of the continuum formation level (optical depth variations) in variable mass-loss episodes. On the other hand, the mean amplitude ratio Ampl V/Ampl (B − V) for 14 selected pulsation cycles amount to 1.7 ± 0$\stackrel{\text{m}}{.}$5 (st.dev.), similar to ρ Cas and HR 8752. Ikonnikova et al. (2018) have suggested that HD 179821 is very similar to a YHG. The absence of any outburst so far as well as the presence of peculiar photometric variations, such as in the light curves of our YHGs, cause us to speculate that HD 179821 could be a late-type predecessor of YHGs (see Appendix A.3.4; Table 1 of Appendix A).

10. The original PG and VIS light curves in 1905-1946 and 1905-1962, respectively Fig. G.1; Table 1), including AAVSO visual observations in 1941–1962, have been transformed by us to be more surveyable. The AAVSO dataset was brightened by 0$\stackrel{\text{m}}{.}$3 in order to correct them to the photoelectric Johnson V-scale. All of these datasets are more supplementary to the medians of our LTVs of 1963–2020, presented in Paper I and in this paper. Our photometric analysis shows that the PG and VIS and corrected AAVSO magnitude scales are almost equal the Johnson B- and V-scales, within an uncertainty of 0$\stackrel{\text{m}}{.}$1 to 0$\stackrel{\text{m}}{.}$2. The survey of the complete PG dataset of 1885 yielded two new outbursts: one in 1895 and one in 1905. The analysis of the VIS observations was much more complicated due to a mysterious and sudden brightness decline of the VIS data in 1919 to 1920, amounting to 0$\stackrel{\text{m}}{.}$4 (while leaving the PG magnitudes unaffected), and it remained constant until somewhere between 1934 and 1941. It appeared to be displaced by a brightness excess of ∼0$\stackrel{\text{m}}{.}$3. The total increase, therefore, amounts to 0$\stackrel{\text{m}}{.}$4 + 0$\stackrel{\text{m}}{.}$3 = 0$\stackrel{\text{m}}{.}$7. The original brightness decline of 0$\stackrel{\text{m}}{.}$4 in 1919 was restored between 1934 and 1941. The VIS brightness excess gradually faded away until about 1957. It is of note that the VIS brightness decline of 1919–1920 was considered by Hassenstein (1934) to be real. Its cause may be related to some shell expansion event, to which only the long-wavelengths in the spectrum are sensitive. When the shell fell back between 1934 and 1941, the atmospheric density temporarily increased, possibly explaining the temporal VIS brightness excess, but this increased density faded away between 1946 and 1957. Figure H.1 shows the LTV of the median (B − V)_obs in 1905–1946 and extended to the 1986 outburst versus the reduced JD. The figure depicts a few red and blue evolutionary tracks. The track marked with the blue plus signs represents the V brightness decline of 1919–1920. The blue dots in 1941–1946 represent part of the gradual decline in the excess until 1947 (thereafter no B magnitudes are available, and there are hence no (B − V)_obs values). (See Sect. 8.2. for an interesting discussion on young and old outbursts: The young ones (1905, 1946) and the old ones (1986–2000–2013) become hotter in max1 and min1). Our investigation improves the understanding of extreme stars, such as ρ Cas and it recurring outbursts, and also contributes to the broader knowledge of the YHGs, their variability, and importance for stellar evolution. In particular, continuous high-resolution spectroscopic monitoring of the outburst events is urgently needed for further advancing YHG investigations.

10. Data availability

The appendices corresponding to this article are available at Zenodo.

Acknowledgments

A.L. acknowledges in part funding from the European Union’s Framework Programme for Research and Innovation Horizon 2020 (2014-2020) under the Marie Skłodowska-Curie grant Agreement No. 823734: Physics of Extreme Massive Stars (EC-POEMS). A.L. acknowledges in part funding from the ESA/PRODEX Belgian Federal Science Policy Office (BELSPO) related to the Gaia Data Processing and Analysis Consortium. RT is grateful for support from the UKRI Future Leaders Fellowship (grant MR/T042842/1). This work was supported by the STFC [grants ST/T000244/1, ST/V002406/1]. We regretfully have to inform the reader that two co-authors of Paper I (by van Genderen et al. 2019), and partly of the present paper passed away: Prof. Dr. C. de Jager on April 27, 2020 at the age of 100, and Dr. H. Nieuwenhuijzen on June 25, 2021 at the age of 85. We shall badly miss their professional skill as well as their warm collegiality. We are extremely grateful to the Leiden Observatory Computer Group for their inexhaustible patience to help and advice in case of computer and LaTeX problems: Dr. E. Deul (head), Mr. A. Vos, Dr. D. Jansen, Dr. L.Lenoci, Dr. H. Intema, Dr. B.P. Venemans, Mr. E. van der Kraan and Mr. R.G. Kuijvenhoven. Their incredible well organised support during the corona years, by day and by night, during weekends and holidays, was incredible! All people of the Leiden Observatory owe them a lot. A.M.v.G. is most grateful to his beloved wife Constance, for her understanding and her indispensable support for so many years! We would like to thank the following colleagues for their stimulating and fruitful correspondence on various subjects addressed in this paper: Prof.dr. Christiaan Sterken, Prof.dr. H.J.G.L.M. Lamers for advices concerning papers on the stellar atmospheric opacity. And at last, but not at least: we are very grateful for the acuteness of the anonymous referee. We benefitted much of the invaluable comments and suggestions, improving the presentation of this paper.

References

All Tables

All Figures

Fig. 1. LTV of ρ Cas between 1962 and 2020. Top panel: Median (B − V)obs versus the reduced JD. The dark blue dots define the dark-blue unsteady median. Red dots represent the photometric observations with an available Teff value (green numbers). Bottom panel: δ (B − V) = (B − V)0 – (B − V)dJN versus the reduced JD. Red dots and green numbers are similar to those in the top panel (see discussion in Sect. 2.2).

In the text

	Fig. 2. Four temperature calibration relations versus photometric parameters for determining an average Teff for relatively reliable photometric observations (see Sect. 3).
In the text

	Fig. 3. Illustration showing how the observed unsteady median (B − V)obs in Fig. 1 can be transformed into a linear median (dashed drawn line) with the aid of the three vertical arrows discussed in Sect. 2.
In the text

	Fig. 4. HRD for ρ Cas in 1962–2020. The encircled area in the HRD, having a central Teff = 6700 K ± 700 K and a luminosity of 5.42, locates the blue dots showing zig-zaging changes in the observed median (B − V)obs of Fig. 1.
In the text

	Fig. 5. Same as Fig. 4 but for the interval 2008–2017 (in Appendix Table D.1). Due to the different scales used for the luminosity relative to the one for the temperature, the circle in Fig. 4 is an ellipse. The diagram shows the ‘to-and-fro’ tracks, consisting of ordinary pulsations before and after the 2013 outburst.
In the text

	Fig. 6. Three ρ Cas outburst tracks of 1986, 2000, and 2013 in the HRD.
In the text

Fig. 7. Diagram showing the data points of the four median curves in 1962–2020 of ρ Cas, as well as the four characteristic points of the outbursts. The time-scale is reduced JDs. The time-scale of the wave pattern in the four panels is ∼15 y. From bottom to top the panels show Teff, (B − V)obs, stellar radius R/R⊙, and the average quasi-periods P(d) (number of pulsations used are marked with brackets). Ordinary data points are dark-blue while the four characteristic points of the outbursts are red. The latter are plotted in panels 1–3 representing max1, the median between the descending and ascending branch, min1, and max2 (see Sect. 8.2).

In the text