Stellar evolution along the AGB as revealed by the shape of Miras’ visual light curves

We start by measuring as accurately as possible the time and magnitude of each maximum and each minimum of the curve. For sample A, this can be done only approximately because of two difficulties that are met: one is the presence of flat maxima, with a very small enhancement usually located at its later edge; defining the time of the maximum as the time of the centre of the plateau or as the time of the enhancement implies differences reaching up to 8% of the period. The same difficulty is met in the analysis of Cepheid light curves, as remarked and clearly illustrated in Figure 2 of the study presented by Antonello (1993). This happens when a hump on the ascending branch reaches the maximum of the curve and eventually gets confused with it. The second difficulty relates to the evaluation of the time at minimum light and is mostly caused by the frequent lack of observations in this low apparent brightness region of the light curve.

For sample B, we collect the observations stored in the AAVSO database in time bins spanning typically 2% of the period. Each bin contains the number of observations, the mean magnitude and its rms value with respect to the mean. To locate the maxima, we produce a parabolic fit to these data in a time interval of 7 bins centred on the date where the maximum is expected. In the rare cases where the fit gives a date deviating by more than a bin from the centre of the interval, we may increase the size of the interval and/or iterate with an interval shifted accordingly. This procedure defines maximum dates to better than 1% of the period and maximum apparent luminosities to better than 0.1 units of magnitude. The main source of systematic uncertainty attached to the measurement of the dates is the occasional occurrence of very flat maxima.

Locating the minima proceeds similarly, except that we use broader intervals, typically of 17 bins, in order to cope with the much smaller number of observations, several bins being empty. We start with an interval centred at the phase corresponding to the mean asymmetry between ascending and descending branches and iterate the fit when necessary. We estimate that this procedure defines minimum dates to better than 2% of the period and minimum apparent brightness to better than 0.1 units of magnitude.

Having obtained for each maximum $i$ of the light curve a time $T_{\rm max,i}$ and a magnitude $M_{\rm max,i}$ and for its following minimum a time $T_{\rm min,i}$ and a magnitude $M_{\rm min,i}$, we define eleven parameters as follows:

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  the period $P$, mean value of the time separating two successive maxima, $P$=$<$$T_{\rm max,i+1}-T_{\rm max,i}$$>$, and $\Delta P$, the rms value with respect to $P$, both measured in units of days.

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  the mean phase of the minimum, $\varphi_{\rm min}$=$<$$(T_{\rm min,i}-T_{\rm max,i})$/$(T_{\rm max,i+1}-T_{\rm max,i})$$>$. It is the complement to unity of the parameter $f$ introduced by Vardya (1988) and is related to the asymmetry parameter $\alpha=1-2\varphi_{\rm min}$. $\Delta\varphi_{\rm min}$ is the rms value of this quantity with respect to $\varphi_{\rm min}$. In sample A, a direct evaluation of $\varphi_{\rm min}$ is often not possible, because of the insufficient phase coverage near luminosity minimum. We provide instead an approximate evaluation of the asymmetry parameter $\alpha$, obtained from a comparison of the slopes of the ascending and descending branches.

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  the mean brightness at maximum light, $M_{\rm max}$=$<$$M_{\rm max,i}$$>$, and its rms value with respect to the mean, $\Delta M_{\rm max}$. $M_{\rm max}$ depends on the distance of the star to the Sun, but $\Delta M_{\rm max}$ provides a first measure of the global irregularity of the curve. Another, independent measure is given by the mean of the absolute value of the difference between the magnitudes of two successive maxima, $\Delta^{\prime}M_{\rm max}$=$|M_{\rm max,i+i}-M_{\rm max,i}|$. For random magnitude fluctuations having a Gaussian distribution, $\Delta M_{\rm max,i+i}$=1.08$\times$$\Delta M_{\rm max}$. A difference between the two parameters reveals the nature of the source of the observed magnitude fluctuations: when fluctuations between successive oscillations are small but when the magnitude of the maxima displays strong dependence on time, one expects $\Delta M_{\rm max}$$>$$\Delta^{\prime}M_{\rm max}$; if instead the magnitude of the maxima stays constant on average, but fluctuates strongly from one oscillation to the next, $\Delta M_{\rm max}$$<$$\Delta^{\prime}M_{\rm max}$. There are examples of both cases. In practice, rather than using $\Delta M_{\rm max}$ and $\Delta^{\prime}M_{\rm max}$ separately, we may prefer to use their mean value, $\Delta M_{0}$=($\Delta M_{\rm max}$+$\Delta^{\prime}M_{\rm max}$)/2 and their ratio, $R_{\rm\Delta M}$$=$$\Delta^{\prime}M_{\rm max}/\Delta M_{\rm max}$.

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  The mean amplitude of the oscillations, measured in units of magnitude, $A$=$<$$M_{\rm min,i}-1/2(M_{\rm max,i}+M_{\rm max,i+1})$$>$; it is sufficient to average the amplitudes of the descending and ascending branches as their means must be equal when the number of oscillations is large enough. $\Delta A$ is the rms value of this quantity with respect to $A$; it provides a third measure of the global irregularity of the curve.

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  A parameter $K_{\rm MB}$, referred to as the Merchán Benítez index; it makes use of the finding of Uttenthaler et al. (2019) reported in Figure 13 of the Merchán Benítez et al. (2023) article, according to which a clear separation splits the curves in two distinct families in the K$-$[22] vs $P$ plane. Above a line of equation K–[22]$\sim$0.011($P$–125), the curves are mostly associated with spectral type M and are smooth, at strong variance with the population below the curve. We define accordingly $K_{\rm MB}$$=$$0.011(P-125)-(K-[22])$.

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  Finally, we noted that some light curves display significantly larger dispersions than average between the magnitudes reported by different observations at a same time. At first sight, one would think that this simply reflects instrumental errors and is of no relevance to the present study. However, the effect is observed to depend on the nature of the star, being stronger for some carbon stars, which excludes an interpretation in pure terms of instrumental uncertainties. In order to study it, we select a time interval of $\sim$6% of a period, centred on each maximum and record the number of observations that it contains, $N_{\rm dis,i}$, and the rms value of the measured magnitudes with respect to their mean, $\Delta M_{\rm dis,i}$, and retain their mean values as two additional global parameters, $N_{\rm dis}$=$<$$N_{\rm dis,i}$$>$ and $\Delta M_{\rm dis}$=$<$$\Delta M_{\rm dis,i}$$>$.

The visual inspection performed in the preliminary phase of the study had shown the difficulty to classify the light curves in terms of the lack of smoothness of the ascending branch. At first sight, many light curves appear to display regular featureless oscillations, while others display well-pronounced humps on most of the oscillations. However, a detailed inspection reveals continuity between these two groups: many curves display humps on only a small fraction of the oscillations, and the amplitude of such humps is often very small. We therefore produced for each curve of sample B a mean normalised oscillation profile, together with the distribution of the rms deviation of the individual oscillation profiles with respect to this mean profile. Normalised profiles cover from maximum $i$ to maximum $i+1$ and are defined as follows: the abscissa $x$ measures the phase in 50 bins of 0.02 width, defined as 0 at $T_{\rm max,i}$ and 1 at $T_{\rm max,i+1}$; on the descending branch, $x<\varphi_{\rm min,i}$, $y$=$(M-M_{\rm min,i})/(M_{\rm max,i}-M_{\rm min,i})$ and on the ascending branch, $x>\varphi_{\rm min,i}$, $y$=$(M-M_{\rm min,i})/(M_{\rm max,i+1}-M_{\rm min,i})$, $M$ being the mean magnitude in bin $x$. Like $x$, $y$ varies from 0 to 1, being equal to 0 at minimal apparent brightness and equal to 1 at maximal apparent brightness. In some cases, rather than normalising the oscillation profile over a whole period, we restrict it to a single branch, descending or ascending, with $x$ spanning from 0 to 1 for $\varphi$ spanning from 0 to $\varphi_{\rm min}$ or from $\varphi_{\rm min}$ to 1, respectively, each time using 25 bins.

In the analysis of sample A curves, hoping to progress in the search for criteria efficient at discriminating between different light curves, we attempted at grouping them in classes associated with a specific shape of the oscillation profile. To this end, we defined five typical profile types illustrated in Figure 2. Types and display no hump on the ascending branch, while types to do. The difference between types and is the width of the oscillation, larger in the latter case. The hump on the rising branch is clear for type , typically between phases of 5/8 and 7/8, closer to minimum for type and closer to maximum for type . The exercise proved to be difficult in many cases, for which the choice between two successive profiles of a pair was not obvious. This gave evidence for continuity between the sequence of profiles depicted in Figure 2, a result that may suggest that stars on the AGB would typically evolve from profile to profile according to such a sequence. Part of the difficulty was that a same curve often displays a broader variety of oscillation profiles than the difference between the typical profiles of the pair of candidate types. In spite of such difficulty, evidence for significant correlations between the mean oscillation profile of curve and its spectral type was obtained. In particular, the Mno spectral type was found to be present in profile types and exclusively, while profile type is the exclusive domain of spectral type C.

For the light curves of sample A that are also part of sample B, we performed a 6th degree polynomial fit of the normalised profile of each oscillation of the light curves; the result is always of very good quality and we used it to define humpy curves from their behaviour in the central phases of the ascending branch, excluding the minimum and maximum regions. The match with the assignment of profile types was in general excellent.

In the case of these 24 sample B curves, with the aim to obtain a simpler and more rigorous definition of humpy and hump-free curves than obtained from the assignment of profile types to , we tried to distinguish them on the only basis of the mean normalised profile of the ascending branch. To the normalised profile of each branch, we fit a curve depending on a single parameter $\lambda$: namely, writing $\xi=2x-1$, $\eta=2y-1$, $\eta$=Sgn$\times\frac{1}{2}\xi(\xi^{2}-3)+\lambda[1-\xi^{2}(2-\xi^{2})]$ with Sgn=+1 on the descending branch and $-$1 on the ascending branch; $\lambda$ is the value of $\eta$ at $\xi$=0 ($x=\frac{1}{2}$). They are the simplest possible polynomial forms having maxima and minima at the proper place and distorted by a single parameter, $\lambda$. Table 2 lists the best fit values of $\lambda$ and $p$, the rms deviation between fit and data, for each branch separately and Figure 3 displays the results in the $p$ vs $\lambda$ plane.

They show a good match between the set of curves considered as hump-free from the assignment of profile types and the set having $p<0.1$ on the ascending branch, with the exception of three cases: S Her, R Cam and S UMa. These display clear humps near maximum light but their mean profiles are well described by a hump-free form having a large value of $\lambda$. This is better illustrated in Figure 4, which compares profiles of the four stars having $\lambda>0.3$ on the ascending branch (Figure 3 left), R Cam, S UMa, T Cam and S Cam. All four have a type profile, the first three are of spectral type S and S Cam is of spectral type C. This illustrates the difficulty to reveal the presence of a hump when it approaches maximal luminosity, as had been remarked earlier.

A similar difficulty is met for humps near minimum luminosity. Figure 5 displays the profiles of the three curves having $p>0.1$ on the descending branch (Figure 3 centre), R Cas, U Her and RU Her, to which a type profile had been assigned. They display clear humps on the lower part of the ascending branch but their descending branches are perfectly well-behaved. Indeed all 24 curves display a hump-free descending branch.

The risk of confusion of a hump with the minimum of luminosity is a warning that such a humpy curve may have been assigned an profile type, in the same way as the S Cam curve might have been assigned an profile type. In order to check on this, we display in Figure 6 the normalised profiles of the ascending branches of the nine curves retained in the hump-free family. While none of these gives evidence for the presence of a clear hump, it is difficult to reliably ascertain their absence. The right panel of Figure 3 displays the dependence of $\lambda$ (ascending branch) on the phase $\varphi_{\rm min}$ at minimal luminosity. In the hump-free family, for which the fit is good and $\lambda$ accordingly meaningful, three curves of spectral type Mno, R Aql, R Boo and R Tri, have $\lambda$ near zero and $\varphi_{\rm min}$ near 0.5, implying that they are close to a sine wave. But the other six curves are slightly asymmetric, with $\lambda$ between $-$0.24 and $-$0.33 and $\varphi_{\rm min}$ between 0.56 and 0.62. The three curves of the humpy family having $\varphi_{\rm min}$$>$0.57 are R Cas, U Her and W And, all having profiles showing a hump on the lower part of the ascending branch. While this confirms the validity of the assignment of the curves having $\varphi_{\rm min}$$>$0.56 to the humpy and hump-free families, respectively, it also illustrates how marginal is the distinction between these and invites prudence.

In strong contrast with the other curves of sample A, which all display technetium in their spectrum when it has been searched for, the Mno spectral type family is expected to include stars that have recently entered the TP-AGB or, possibly, that are still in the E-AGB and will soon do so. As remarked by Uttenthaler et al. (2019), it may include TP-AGB stars for which the absence of technetium in their spectrum would be caused by the thermal pulses being too weak to be followed by significant third-dredge up events: only the stronger thermal pulses on the upper AGB are expected to be followed by such third-dredge-up events. It is therefore interesting to learn whether an inspection of the shapes of the light curves may help with differentiating between stars on the E-AGB and stars on the TP-AGB. Also in strong contrast with the other curves of sample A, the Mno curves are exclusively made of and profile types. Yet, while having such strong identity, the Mno curves of sample A seem to host two different families of curves, as illustrated in Figure 7, which displays their distribution in the $\Delta M_{0}$ vs $\alpha$ and colour index, $C_{0}$=$\frac{1}{2}$(K+[3.4])$-$[22], vs period planes.

A first nearly symmetric family, defined as having $\alpha$$>$$-$15% ($<$$\alpha$$>$=$-10\pm 3$%), is relatively regular ($<$$\Delta M_{\rm max}$$>$=0.34$\pm$0.04, $<$$\Delta^{\prime}M_{\rm max}$$>$=0.41$\pm$0.04), has small colour indices ($<$K$-$[22]$>$=1.8$\pm$0.4, $<$[3.4]$-$[22]$>$=1.7$\pm$0.3) and relatively short periods ($<$$P$$>$=252$\pm$42 days). The complementary family is of course asymmetric ($<$$\alpha$$>$=$-21\pm 2$%), less regular ($<$$\Delta M_{\rm max}$$>$=0.53$\pm$13, $<$$\Delta^{\prime}M_{\rm max}$$>$=0.65$\pm$20) and has larger colour indices ($<$K$-$[22]$>$=2.6$\pm$0.3, $<$[3.4]$-$[22]$>$=2.3$\pm$0.2) and larger periods ($<$$P$$>$=349$\pm$55 days). The first family contains eight curves, generally of profile type . The only exception is R Leo, of profile type , with the largest period of this first family (312 days) and accordingly a positive value of $K_{\rm MB}$, in contrast with the other curves of the first family, all having negative $K_{\rm MB}$ values. The second family contains five curves, of which three are of profile type and two, W Dra and T UMa, are of profile type . While the cases of R Leo and W Dra may be considered as marginal, a higher density of observations being necessary for stating with strong confidence the validity of the curve parameters, the case of T UMa deserves special attention. Its curve displays high quality and density of observations, making it belong to sample B. It has a low value of $\alpha$ ($-$20%), or equivalently a high value of $\varphi_{\rm min}$ (0.58), characteristic of the second family. But it has a low value of the period (256 days) characteristic of the first family. As all other parameters take values intermediate between those of the first and second family, it suggests that T UMa is in a state intermediate between the first and second family. If Mira variables would typically evolve from the first to second family, T UMa would then be currently doing so. Among the five other Mno-type curves selected in sample B, three are in the first family, R Aql, R Boo and R Tri, with $K_{\rm MB}$ values of $-$0.6, $-$0.5 and $-$0.2, respectively; the other two, R Cas and U Her, are in the second family with $K_{\rm MB}$ values of +1.0 and +0.7, respectively.

It is important at this point to remark that there is no reason to assume that stars evolve from one family to the other. At an early stage of the TP-AGB, or late stage of the E-AGB, we may expect the selected stars to be essentially characterised by their initial mass and their age on the AGB: being in general hosted in the solar neighbourhood, they are likely to have metallicity close to solar, $\sim$0.015. Such stars, if having a mass similar to or lower than the solar mass, are expected to spend a very long time in the Mno family and never reach the Myes, S and C spectral types (Straniero et al., 2023; Miller Bertolami, 2016). Indeed, R Aql, a clear member of the first Mno family, is often quoted in the published literature as having recently experienced a thermal pulse (Wood & Zarro, 1981). The Mno family must therefore host both such stars and stars that will later evolve to the Myes, S and C spectral types. A major question that needs therefore to be answered is whether, and if yes how, these two types of stars are related to the two types of Mno families that we have identified. Moreover, all stars are expected to transition from the E-AGB to the TP-AGB when they have exhausted the helium-burning layer around the inert core. Another major question that needs therefore to be answered is whether, and if yes how, the distinction between E-AGB and TP-AGB is related to the two types of Mno families that we have identified. It is essential to keep both questions in mind when discussing the results obtained in the present work.

As clearly illustrated in the pair of leftmost panels of Figure 1, the main difference between “M stars with no bump” and “M stars with bump”, as defined by Merchán Benítez et al. (2023), is a difference of period. For stars evolving across the boundary, one needs to understand what causes the significant period increase associated with this transition. Our results, while broadly confirming those obtained by Merchán Benítez et al. (2023) offer a more detailed picture and imply that the line of equation K$-$[22]$\sim$0.011($P-125$) shown in Figure 1 is in fact related to the distinction between the first family of Mno-type curves, having $K_{\rm MB}$ negative, and the other curves, the observed continuity between the two suggesting that stars may evolve from one family to the other.

Figure 8 displays the distribution in the irregularity $\Delta M_{0}$ vs asymmetry $\alpha$ and colour index $C_{0}$ vs period $P$ planes of the curves of sample A having an Myes or S spectral type. All have positive $K_{\rm MB}$ values, in agreement with the assignments made by Merchán Benítez et al. (2023). Clear correlations are observed between the curve parameters. The more symmetric curves are also more regular and have smaller colour indices than the more asymmetric curves, which are irregular and have larger colour indices. Moreover, the more symmetric curves are exclusively of and profile types, while the more asymmetric are exclusively of the and profile types.

This encourages having a global look at all sample A curves of oxygen-rich star, including the second family of Mno spectral types but excluding the first family. This is done in Figure 9, the left panel of which displays the dependence of the regularity parameters, $\Delta M_{\rm max}$ and $\Delta^{\prime}M_{\rm max}$, on the colour parameters, K$-$[22] and [3.4]$-$[22]; the right panel displays the dependence of the asymmetry parameter $\alpha$ on the mean colour parameter. They give clear evidence for significant correlations: more symmetric curves have smaller colour indices, probably meaning smaller dust mass-loss rate, and are more regular; more asymmetric curves, with an ascending branch faster than the descending branch, have larger colour indices and are more irregular. However, while Mno-type curves, associated with stars that just entered the TP-AGB, are clustered in the latter group, Myes- and S-type curves populate the whole range of parameters. Moreover, the Mno-type curves are of profile types or , while Myes- and S-type curves have profile types spanning from to . When having an or profile type, they tend to be on the asymmetric and irregular side, when having a or profile type, they tend to be on the symmetric and regular side. For stars evolving along the AGB from Mno to Myes and S types, this suggests that some of their light curves evolve all the way to a state of high symmetry and regularity with a low dust mass-loss rate while others span only a smaller path along such an evolution. In order to better understand this evolution, we study in the next sub-section how the hump, when present, evolves along the ascending branch.

The evolution of type profiles for oxygen-rich stars of sample A is well illustrated by the mean values taken, for each of them separately, by the asymmetry parameter $\alpha$, the mean colour index $C_{0}$, and the mean regularity parameter, $\Delta M_{0}$. They are, respectively, for the profile type, ($-$20%, 2.1, 0.58), for the profile type, ($-$19%, 2.5, 0.59), for the profile type, ($-$3%, 1.4, 0.45) and for the profile type, ($-$4%, 1.2, 0.30): the and profiles are more asymmetric, more dusty and more irregular; the and profiles are more symmetric, less dusty and more regular. In order to obtain a more precise evaluation of this evolution, we use the curves of sample A that have been selected in sample B to study the evolution of the hump on the ascending branch.

We select a sample of light curves displaying a clear hump in many of their oscillations and, for each selected oscillation, fit a sixth-degree polynomial to the normalised profile of the ascending branch. The hump interval, [$x_{\rm hump1}$, $x_{\rm hump2}$], is then defined as giving the best linear fit in it, the normalised magnitude spanning accordingly from $y_{\rm hump1}$ to $y_{\rm hump2}$. The left panel of Figure 10 illustrates the procedure. The result is summarised in Table 3 and illustrated in the central panel of Figure 10. Defining $\Delta x_{\rm hump}$ as $|x_{\rm hump2}-x_{\rm hump1}|$ and $x_{\rm hump}$ as $\frac{1}{2}$($x_{\rm hump1}+x_{\rm hump2}$) and, similarly, $\Delta y_{\rm hump}$ as $|y_{\rm hump2}-y_{\rm hump1}|$ and $y_{\rm hump}$ as $\frac{1}{2}$($y_{\rm hump1}+y_{\rm hump2}$), we find that the mean values of $x_{\rm hump}$ are poor discriminants between different curves, the rms deviations of $x_{\rm hump}$ with respect to the mean taking values as large as the differences between the mean values of different curves. In contrast, the central panel of Figure 10 shows that $y_{\rm hump}$ has rms dispersions around its mean significantly smaller than the differences between the mean values of different curves and its mean value is seen to span a broad range along the ascending branch. Typically, the rms dispersion of $y_{\rm hump}$ is of the order of a tenth, and $\Delta y_{\rm hump}$ is of the order of 14% of the oscillation amplitude. The right panel of Figure 10 displays the dependence on $y_{\rm hump}$ of three parameters: the mean colour index, $C_{0}$, the phase at minimal light, $\varphi_{\rm min}$ and the mean irregularity, $\Delta M_{0}$.

One may wonder whether the presence of a hump, during which the rate of increase of the luminosity is smaller than outside it, implies a lengthening of the period, the resulting slowing-down not being compensated by speeding-up either before or after it. To answer this question, we select 8 curves displaying a clear hump and plot, for each oscillation, the relative period, $P$/$<$$P$$>$, as a function of the delay $\Delta_{\rm slow}$ caused by the hump; this delay is the difference between $\Delta x_{\rm hump}$ and the time spent to climb $\Delta y_{\rm hump}$ in the absence of hump, namely $k(1–\varphi_{\rm min})\Delta y_{\rm hump}$; here, $k$ accounts for the faster rate of luminosity increase at the time of the hump compared with a linear increase ($k$=1). We evaluated $k$ separately for each of the 8 selected curves and obtained $k$=0.88$\pm$0.04. We checked carefully the validity of the procedure. The result is illustrated in the left panel of Figure 11 and shows the absence of significant correlation: [d$P$/$<$$P$$>$]/d$\Delta_{\rm slow}$$\sim$0.0$\pm$0.2 while a value of 1 is expected in the absence of compensation. This implies that the hump delay is somehow compensated by speeding-up elsewhere. In order to understand whether this speeding-up occurs on the ascending or descending branch, we show in the central panel of the figure the dependence of $\varphi_{\rm min}$$-$$<$$\varphi_{\rm min}$$>$ on $\Delta_{\rm slow}$. The absence of correlation shows clearly that the compensation occurs on the ascending branch: the presence of a hump implies a change of shape, but not of duration of the ascending branch.

One may also wonder whether the presence of a hump is related to the presence of a shock wave, in which case it might be enhanced by the strength of the preceding oscillation; we check on this by plotting the relative amplitude $A$/$<$$A$$>$ of the descending branch of the preceding oscillation as a function of $\Delta_{\rm slow}$. The result, displayed in the right panel of Figure 11, shows indeed a small, but barely significant correlation of the expected sign, [d$A$/$<$$A$$>$]/d$\Delta_{\rm slow}$$\sim$0.3$\pm$0.1.

In the selection of sample A light curves, M-type curves for which technetium was either not searched for or, when searched for, was found neither clearly present nor clearly absent, were deliberately ignored. Having now obtained a clearer view of the general properties displayed by the analysed light curves, we can recover such curves from the Merchán Benitez sample of 548 Mira stars, requiring that they allow for the rigorous parameterisation of the sample B selection. We select this way eight additional M-type curves: three for which technetium had not been searched for (R UMa, V Cas and U UMi), two for which the presence of technetium had been considered doubtful (R Dra and RT Cyg), two for which it had been considered possible (S CrB and R CVn) and one for which it had been considered probable (T Cas). The curve parameters are listed in Table 1 and illustrated in Figure A1. Figure 12 below compares the 8 new M-type curves to the M-type curves belonging to both sample A and sample B. It suggests the following assignments for the new curves: V Cas, RT Cyg and R Dra, with profiles of type , to the first family of Mno spectral type; S CrB and R UMa, with profiles of type , to the second family of Mno spectral type; U UMi, T Cas and R CVn, with profile types and , to the Myes spectral type.

It would, of course, be important to extend the studies presented here to bandpasses other than visible, but this is a major effort beyond the scope of the present article. Unfortunately, light curves at wavelengths outside the visible bandpass are scarce and include much smaller sets of observations. Figure 13 displays the better examples of such observations for the curves of sample B. It shows, in addition to the visible, B, V and R, the Johnson infrared bandpass having a mean and full width at half maximum of 806 nm and 149 nm, respectively. The main observation is that when a hump is present (absent) on the visible curve, it is also present (absent) on the infrared curve but its detailed shape may be different. This observation is consistent with the generally accepted belief that the presence of a hump is related to global properties of the star pulsation but its detailed shape is affected by the nature of the outer atmosphere. Over 50 years ago, Lockwood & Wing (1971) studied the near infrared ($\lambda$=1.04 $\mu$m) light curves of 24 Mira variables, 15 of which are part of our sample A, and measured the ratio $R_{\rm I/V}$ of their oscillation amplitude to that in the visible bandpass. Its mean value is 20%. It is remarkable that the Mno curves of sample B are split in two families as clearly in terms of $R_{\rm I/V}$ as they are in terms of $\varphi_{\rm min}$ and $K_{\rm MB}$: R Cas and U Her have ($R_{\rm I/V}$, $\varphi_{\rm min}$, $K_{\rm MB}$)=(29%, 0.61, 1.0) and (27%, 0.58, 0.7), respectively, while R Aql and R Boo have ($R_{\rm I/V}$, $\varphi_{\rm min}$, $K_{\rm MB}$)= (17%, 0.53, $-$0.6) and (15%, 0.52, $-$0.5), respectively.

The analysis of sample A curves of spectral type Mno presented in Section 4.1 has given clear evidence for the presence in this group of two distinct families. Figure 21 illustrates this result for sample B. The first family is more symmetric with $\varphi_{\rm min}$ between 0.50 and 0.55, compared with between 0.57 and 0.65 for the second family. It has smaller periods, below 300 days, is more regular and has smaller colour indices than the second family. It is exclusively made of type profiles, while the second family contains also type profiles. It sits on the left-hand side of the colour vs period diagrams first produced by Uttenthaler et al. (2019), with $K_{\rm MB}$ negative, while the second family occupies also the right-hand side. When observed in infrared (Section 4.5), the ratio $R_{\rm I/V}$ of the infrared oscillation amplitude to that in the visible bandpass is smaller than 20% while it exceeds 25% for the second family. It includes the curves of R Aql, R Boo, R Tri, V Cas, RT Cyg, and R Dra. The second family includes the curves of R Cas, U Her, T UMa, S CrB and R UMa.

All these curves have large oscillation amplitudes, on average 4.9 mag for the first family and 5.4 mag for the second family: they are clearly identified as Mira variables. When analysed in terms of Fourier transforms, a very powerful tool abundantly used in the published literature, they are seen to pulsate in the fundamental mode. However, for what concerns us here, Fourier transforms are not helpful; AGB stars are not organ pipes and what we are after is to understand the shape of a single oscillation: expressing the presence of a hump in terms of overtone contributions would not help with revealing the underlying physics. Yet, it is important to understand at which stage of their evolution along the AGB, stars can be expected to pulsate as Mira variables. For a star of solar metallicity having an initial mass between 1 and 3 solar masses, the time spent on the E-AGB is an order of magnitude larger than the time spent on the TP-AGB as oxygen-rich. Precisely, the ratio takes values of 17, 9, 7 and 10 for initial masses of 1, 1.5, 2 and 3 solar masses, respectively (Miller Bertolami, 2016). While, on most of the E-AGB, stars are not expected to pulsate as Mira variables but rather as low-amplitude semi-regulars (Yu et al., 2020), it is conceivable that they do at the end of the E-AGB, say in the last 10% or so. By then, their radius and luminosity have considerably increased.

We recall that at the end of the E-AGB, the star has a small degenerate core of carbon and oxygen in its centre, surrounded by a thin helium-rich shell, itself surrounded by a thin hydrogen-rich shell, both shells being separated by a thin inter-shell layer (Herwig, 2005). During the whole TP-AGB, the star is in a state fully governed by what happens in this pair of shells. When running out of helium fuel at the end of the E-AGB, the star starts burning hydrogen around the depleted helium shell: when this will have produced enough helium ashes in the inter-shell layer, the star will experience its first thermal pulse, which will be followed by several others at intervals typically two orders of magnitude shorter than the time spent on the E-AGB. What happens in the convective envelope surrounding them and in the stellar atmosphere further out is, of course, essential to decide how the star looks like when observed from far away and to understand the mass loss mechanism that defines its lifetime on the AGB; but it obscures the basic dynamics causing the evolution of the star and considerably complicates the interpretation of what is observed.

In order to suggest possible interpretations of the presence of two different families in the curves of Mno spectral type, we need to identify a physical mechanism that distinguishes between two different groups among Mira variables that have not yet experienced a significant third-dredge-up event. Two examples of such a possible distinction, suggesting two different scenarios, come immediately to mind: between stars at the end of the E-AGB and stars on the TP-AGB, and between low mass stars that will not evolve to the Myes spectral type and higher mass stars that will.

In the first scenario, one of the Mno families would include E-AGB stars, the other TP-AGB stars. All E-AGB stars are expected to transition to the TP-AGB phase when they run out of helium fuel around the inert core. The E-AGB group includes therefore stars of all possible masses; in contrast, the TP-AGB group includes all low mass stars that do not experience strong enough third-dredge-up events to allow their evolution to the Myes spectral type but excludes the higher mass stars that have already evolved to the Myes spectral type. The E-AGB group is therefore more homogeneous than the TP-AGB group and must be identified with the first Mno family. In favour of such a scenario, is its good qualitative match with what we observe: the E-AGB stars are expected to be characterised by only a few parameters, essentially their mass and their age on the AGB, and can therefore be expected to experience relatively regular and simple oscillations; for the same reason, their transition to the TP-AGB can be expected to take place at a well-defined stage of their evolution, suggesting a clear distinction between the two families in a two-parameter space such as the K$-$[22] vs $P$ plane. In contrast, the second family would be made of stars that experience thermal pulses, which implies contributions from complex convection mechanisms, and would include low mass stars that are experiencing mass-loss, another complex mechanism, up to the end of their AGB life. The second Mno family would therefore be expected to cover larger domains than the first Mno family in any parameter space that characterizes the shape of the light curve and/or the evolutionary state of the star. Such an interpretation would then imply that when transiting from E-AGB to TP-AGB, namely from helium- to hydrogen-burning around the inert core, the star experiences significant changes in the dynamics that govern its interior, resulting, in particular, in an increase of the pulsation period. The transition would also cause a significant change in the shape of the light curve, taking the form of a slowing-down episode near minimal light, that would progressively evolve to higher phases and become a hump in the cases of the stars evolving to the Myes spectral type. The simultaneity between the appearance of such a slowing-down episode and the increase of the period might have suggested that the two events are related, the slowing-down not being compensated by speeding-up during the remaining part of the period, which would then increase accordingly. However, the study of the delay caused by the hump that was presented in Section 4.3 invalidates such a suggestion.

In contrast, the second scenario would imply that the transition from E-AGB to TP-AGB does not result in major changes of the pulsation properties of the star or, if it does, that the transition occurs before the star having evolved to a state that enables its pulsating as a Mira variable; namely, in this latter case, the whole sample of stars studied in the present article would be made of TP-AGB stars. Then, the second Mno family would probably be made of stars that will soon experience sufficient third-dredge-up events to enter the Myes spectral type and the first Mno family of low mass stars that will never enter the Myes spectral type but spend their entire lifetime on the AGB as Mno stars. At variance with the first scenario, no star would transition between the two Mno families: those of the first Mno family would stay there until the end of their AGB life, those of the second Mno family would then transition to the Myes spectral type. The difference of periods between the two families would then be the simple result of the different star masses and the enhanced irregularity of the second Mno family would be the effect of stronger thermal pulses causing more active convection.

These two extreme scenarios form the basis on which more elaborate scenarios could be constructed. Both may allow some continuity between the two families: the first scenario with curves of stars in the process of transiting between the E- and TP-AGB; the second scenario as a result of the continuity of the star mass distributions, implying a corresponding continuity of the strength of the convective mixing following thermal pulses and eventually leading to third-dredge-up events.

In Section 4.2, to each curve of oxygen-rich stars selected in sample A, we have assigned a profile type, ranking from to , the three latter displaying a hump on the ascending branch at increasing phases. We noted that the curves having oscillations of and profile types are more asymmetric, more dusty and more irregular; those having oscillations of and profile types are more symmetric, less dusty and more regular. In Section 4.3 we studied curves of sample B displaying a clear hump and we evaluated its magnitude, normalised to the oscillation amplitude, $y_{\rm hump}$. We found that its rms dispersion with respect to its mean is of the order of a tenth, and that the width of the hump is of the order of 14% of the oscillation amplitude. On average (Figure 10 right), we have shown that the parameters $\varphi_{\rm min}$, $C_{0}$ and $\Delta M_{0}$ are decreasing functions of $y_{\rm hump}$. We also established that the slowing-down caused by the hump is essentially compensated by speeding-up elsewhere on the ascending branch and does not cause an increase of the pulsation period. A weak, barely significant positive correlation between the strength of the hump and the amplitude of the descending branch of the preceding oscillation allowed for a possible contribution of a shock wave.

This suggests that when entering the TP-AGB, the curves develop a hump, first confused with the luminosity minimum and then progressively climbing the ascending branch when and if the star transitions to the Myes spectral type. However, while the hump reaches up to maximal light in the case of some carbon stars, such as S Cam, it does not always do so, far from it. Instead, each spectral type is seen to cover the whole corresponding $y_{\rm hump}$ range rather than clustering at its end. This is illustrated in Table 4, which lists, for each spectral type and each profile type separately, the number of curves selected in samples A and B. We recall that the assignment of type profiles is less reliable for sample A than for sample B, which, however, includes only half as many curves. But the assignments made in sample B can be trusted as reliable and their diversity was clearly illustrated in Figure 15 in the case of carbon stars. It looks like if, when progressing along the TP-AGB from M to C spectral types, the star experiences something that causes an apparent slowing-down episode of the pulsation, accompanied by an episode of compensating speeding-up, when it is on the ascending branch of the light curve. The phase of the hump would start at minimal light and would increase together with the C/O ratio that measures the progression of the star on the TP-AGB. However, this phase would stop increasing at some point, the star continuing its evolution along the TP-AGB while the hump stays at a fixed phase.

Such humps are also observed on the light curves of other pulsating stars. In the case of RR Lyrae (Prudil et al., 2020, and references therein) they track shock waves caused by a rapid compression of the star atmosphere: they are the result of a sudden stop of the infalling photosphere and immediate outward expansion. Kudashkina & Rudnitskij (1994) have remarked that a similar mechanism could be at stake in the case of Mira variables. Using a simple model of the propagation of shock waves in the star atmosphere, heating the gas to temperatures sufficient for ionisation, they show that the duration of the resulting hump is of the order of a tenth of a period, in good agreement with observations. In the case of Cepheids (Bono et al., 2000; Marconi et al., 2024, and references therein) the hump is on the ascending branch for periods larger than  11 days and on the descending branch otherwise (the so-called Hertzsprung progression Hertzsprung, 1926). State-of-the-art non-linear convective pulsation models, accounting for the movement of pressure waves in the star interior including reflections and resonances, provide descriptions of the Hertzsprung progression in agreement with observations (Marconi et al., 2024).

However, in the case of Mira light curves, and in spite of numerous and detailed descriptions that have been available for decades (Schorr, 2016; Templeton et al., 2005; Marsakova & Andronov, 2007; Lebzelter, 2011; Merchán Benítez et al., 2023), we lack a reliable interpretation of what is causing them. The suggestion (e.g. Lebzelter, 2011) that they may simply result from a resonance between the fundamental and the first overtone modes is not satisfactory, the light curves of Mira stars having large oscillation amplitudes implying that the stars are in conditions allowing them to comfortably pulsate in the fundamental mode. Merchán Benítez et al. (2023) suggest instead that dust being formed inefficiently in the wake of an outwardly propagating shock wave would fall back toward the star instead of being accelerated outwards and would hit the outwardly moving shock wave of the next pulsation cycle (Hinkle et al., 1982), thereby causing humps or secondary maxima in the light curve.

Figure 9 has given evidence for significant correlations between the parameters describing the sample A light curves of oxygen-rich stars, spectral-types M to S. The more symmetric curves, exclusively made of and profile types, were observed to be more regular and have smaller colour indices than the more asymmetric curves, exclusively made of and profile types, which are irregular and have larger colour indices. Figure 14 has shown that a similar trend is followed by the curves of carbon stars, which, however, for a same colour index, are more regular and more symmetric than the curves of oxygen-rich stars. Figure 22 displays the dependence on colour index, $C_{0}$, of the regularity parameters, $\Delta M_{\rm max}$ and $\Delta^{\prime}M_{\rm max}$ (left) and of the phase at minimal light, $\varphi_{\rm min}$ (right) for the curves of sample B. It qualitatively confirms the general trend found for sample A: the average slopes for sample A (Figure 9) were 0.19 mag of $\Delta M_{0}$ and 0.045 of $\varphi_{\rm min}$ per unit of magnitude of the mean colour index $C_{0}$. They are now, for sample B, 0.23 mag of $\Delta M_{0}$ and 0.054 of $\varphi_{\rm min}$.

While reliable and robust, the above result is only valid on average and the curve parameters shown in Figure 22 display major dispersion with respect to the mean. This can be understood in part as an effect of the thermal pulses that occur at a typical rate of one every 10⁵ years (Miller Bertolami, 2016; Marigo et al., 2013). While very short, only a few years, these pulses are so violent, with temperatures up to a few 10⁸ K, that they can be expected to disturb the interior and the atmosphere of the star for a significant fraction of the inter-pulse phase. In particular, this argument has been used by Merchán Benítez et al. (2023) to explain the occasional observation of period changes, the more drastic the closer to the thermal pulse. Some parameters, such as the C/O ratio, the amount of technetium in the spectrum, the ¹²C/¹³C ratio, increase progressively after each third dredge-up episode and are not expected to display significant scatter. But others, such as temperature and turbulence, can be expected to be deeply affected, causing major temporary changes in the pulsation regime. However, such an interpretation seems unlikely to apply to the near totality of the curves illustrated in Figure 22. Such major scatter must be seen in the context of other puzzling features that Mira stars light curves display over the TP-AGB phase, such as major changes that affect their shape from one cycle to the next, as discussed in Section 6; these include, in particular, the value of $y_{\rm hump}$ and the intensity with which a hump may occur, but also, to a lesser extent, the maximal luminosity and the period. Another puzzling feature worth mentioning in this context is the absence of S- and C-types meandering period changes (Merchán Benítez et al., 2023). Also of relevance is the possibility for the curve of a given star to stop being a Mira variable, either momentarily or definitively. This may happen on the occasion of thermal pulses but may also happen in other occasions. In particular, we noted that many of the sample A curves of carbon stars are closer to the semi-regular type of pulsation than to the Mira type.

What makes it difficult to devise sensible interpretations of observations that must necessarily be made from outside the star, is that its appearance is the consequence of a long chain of causally connected events: it starts from nuclear burning around the inert core, continues with the properties of the convective zone and ends with the inner and outer atmospheres of the star. Ideally, one would like to understand nuclear burning and convection before having to deal with the complexity of the nascent wind, dust formation and radiation pressure. In this context, the latter, not to mention the impact of a possible companion, seem unfortunate complications that prevent a clear identification and description of the more basic mechanism at stake inside the star. In particular, quantities such as the mass loss rate (Uttenthaler et al., 2024, and references therein) or the ¹²C/¹³C isotopic ratio are known to be both episodic and anisotropic: their properties are very complex and their evaluation requires a detailed understanding of the radiative transfer taking place in the atmosphere, of the shock chemistry of relevance and of the dust formation process (e.g. Darriulat et al., 2024; Hoai et al., 2024). As a result, very large uncertainties are attached to such quantities: apart from general trends, well documented in the published literature, they seem unlikely to help significantly with the understanding of the features addressed in the present article. Yet, while strongly affected by the properties of the star atmosphere, light curves are also apt at revealing features taking place inside the star, which makes a detailed study of their shapes particularly valuable.

In Section 3.3, we briefly commented that all curves of sample B display a hump-free and smooth normalised profile of the descending branch. This is important information, worth additional comments. Even if we know of a few, extremely rare cases of oscillations that seem to display a small hump on the descending branch, we can consider this as a robust and reliable result. The left pair of panels of Figure 23 compares the superimposed normalised profiles of all oscillations of the ascending branches of the sample B curves with those of the descending branches. Curves of the first Mno family are shown separately from the other 24 curves. The figure offers a spectacular illustration of the difference between the ascending branches of these 24 curves and either their descending branches or both the ascending and descending branches of the first Mno family. In order to better quantify such a result, we fit each normalised profile of the ascending and descending branches observed in the oscillations of the 32 light curves of sample B with the form defined in Section 3.3; it uses a single parameter $\lambda$ that measures, for each branch separately, the deviation from mean magnitude ($y=\frac{1}{2}$) at mean phase ($x=\frac{1}{2}$). The distributions of the number of profiles as a function of $\lambda$ are displayed in the right panels of Figure 23. The (mean, rms) values of $\lambda$ are ($-$0.01, 0.16) for the descending branches of the first Mno family, ($-$0.07, 0.17) for its ascending branches and ($-$0.01, 0.17) for the descending branches of the 24 other curves. Of course, the similarity revealed by comparing these three distributions does not mean that all curves share a same branch profile. They span different fractions of the period, measured by the different values taken by $\varphi_{\rm min}$, and different fractions of the oscillation amplitude. But it means that no special event seems to take place when the luminosity of the star decreases, nor when it increases in the case of the first Mno family. This is at strong variance with the clear evidence for something specific happening, taking the form of an apparent slowing-down episode, when the luminosity increases, in the 24 other curves. It provides another illustration of the results presented in sub-sections 7.1 and 7.2.

The results obtained in the present article deliberately ignored peculiarities that might have distracted us from trying to reveal general features. Yet several such peculiarities have been met and, while it is beyond the scope of the present article to address them, we know that science often progresses by detailed scrutiny of anomalous cases and we briefly list the main ones in the present sub-section.

X Oph is a star of spectral type Mno with a curve displaying an profile type but its very small oscillation amplitude ($A$=1.7 mag compared with 5.1 mag) and its very broad minima (and correspondingly narrow maxima, $W_{\rm\frac{1}{2}}$=0.36 compared with 0.44) make it very different from the Mno curves which have been studied in the present article. The star has a bright K-type giant companion (Fernie, 1959).

R Cen and R Nor are also stars of spectral type Mno, with respective periods of 572 and 498 days, but they display a double-peaked oscillation profile. It has been recurrently argued in the literature that the periods are in fact half these values and that oscillations of higher and lower apparent luminosity alternate (Templeton & Willson, 2004). Hawkins et al. (2001) argue that R Cen is probably experiencing a helium-shell flash and that the most likely cause of double-peaking is a resonance between two modes, but García-Hernández et al. (2013) show instead that R Cen is a massive hot-bottom-burning AGB star. One often reads in the published literature that “R Cen used to have an unusual double-peaked light curve, but by 2001 this had reverted to an almost normal single-peaked curve”, but this statement is incorrect. From a study of lithium in the spectra of R Cen and R Nor, Uttenthaler et al. (2012) suggest that both are undergoing the hot-bottom-burning phase.

R Aql was found in Section 6 (Figure 20 right) to show evidence for a systematic variability of its period, which is indeed known to have declined from 320 days in 1915 to 264 days in 2010 (Zhao-Geisler et al., 2012). This is surprising for a star of spectral type M that displays no technetium in its spectrum, has a low ¹²C/¹³C ratio of only 8, and has been identified in the present work as a clear candidate of the first Mno family. Wood & Zarro (1981) have shown that such a period change could very well be interpreted as the consequence of a recent thermal pulse, in which case the curve should belong to the second, rather than first, Mno family if the first scenario is retained: the case of R Aql seems therefore to favour the second scenario.

R Gem, a star of spectral type S with a curve displaying a clear a-type oscillation profile, stands out as being quite regular ($\Delta M_{0}$=0.27 mag, $R_{\rm\Delta M}$=1.3) together with a very large asymmetry ($\alpha$$\sim$$-$30%), two properties that usually do not go together. It has a $K_{\rm MB}$ value of 0.8. As it has a period very close to a year (370 days) it suffers from being always observed at the same phase at a given time of the year, but this does not explain its peculiarity.

R Cyg, another star of spectral type S, has a light curve displaying spectacular episodes of alternating maxima of luminosity, that were discussed in Section 6 together with other extreme cases of irregularity including R Lep, R Cas and W Aql. Extreme values of the period include: below 200 days, T Her (165 days) and above 500 days, R Cen (572 days), BH Cru (529 days) and S Cas (613 days). R Lep is a carbon rich star which has been the target of many observations (Rau et al., 2017; Asaki et al., 2023, and references therein). W Aql is an S-type star, which has also been abundantly observed (Danilovich et al., 2014; Brunner et al., 2018); De Beck & Olofsson (2020), from a study of the chemistry of the gaseous component of its CSE, have concluded that it appears considerably closer to a C-type star than to an S-type star. Mayer et al. (2013) have detected a companion at a separation of 160 au. BH Cru is known for the high variability of its period (Loidl et al., 2001). Zijlstra et al. (2004), from a study of its evolution, argued that it is unlikely to be related to an ongoing thermal pulse. Walker (2009) studied changes in the light curve and Uttenthaler et al. (2011) discussed the period changes and the transition from spectral type S to C. S Cas has been observed to have a high mass loss rate, of $\sim$1.8 10^-6 solar masses per year, by McDonald et al. (2018). The pulsation period of R Hya has been observed to decline and Uttenthaler & Lebzelter (2010), from a study of technetium and lithium in its spectrum, have argued that, while this decline may be related to a recent thermal pulse, such pulses would not be strong enough to generate a third dredge-up. The CSE has been a target for numerous observations (Homan et al., 2021; Joyce et al., 2024, and references therein).