The Pulsation Modes of Mira Variables

Introduction

A convincing identification of the pulsation mode of the Mira variables has yet to be made. This is somewhat surprising since the ratio of fundamental-mode period to first-overtone period in a Mira-like red giant variable is $\raisebox{-0.6ex}{$\,\stackrel {\raisebox{-.2ex}{$\textstyle >$}}{\sim}\,$}$2 (e.g. Fox and Wood 1982). Any reasonable estimate of the radius and mass of a Mira variable, coupled with a theoretical period-mass-radius relationship for each of the first few modes of pulsation, should determine the pulsation mode. This type of analysis has been performed many times in the past (e.g. Wood 1975a, 1990b; Whitelock 1986; Tuthill et al. 1994; Haniff, Scholz and Tuthill 1995; Feast 1996), using either direct measurements of Mira radii or a temperature scale based on radius measurements and model atmospheres. A mass of ∼1M_$\odot$ is usually assumed, based on kinematic studies of Mira variables in the Galaxy which indicate that typical Miras with periods of a few hundred days are old disk stars (Feast 1963; Jura and Kleinmann 1992). Generally, these studies strongly favour the first overtone as the mode of Mira pulsation, although adoption of a warmer temperature scale appropriate for non-variable red giants can produce models consistent with fundamental-mode pulsation in Miras.

However, there is a major difficulty with the assumption that the Miras pulsate in the first-overtone mode. This is that, because the radii of first-overtone pulsators are very large, the gravities are small and the pulsation velocities achieved in the stellar atmosphere are much smaller than those observed (Hill and Willson 1979; Willson 1982; Bowen 1988; Wood 1990a; Bessell, Scholz and Wood 1996). Once again, the situation is not totally clear cut since high mass (M $\raisebox{-0.6ex}{$\,\stackrel {\raisebox{-.2ex}{$\textstyle >$}}{\sim}\,$}$2M_$\odot$) or short period (P $\raisebox{-0.6ex}{$ \stackrel {\raisebox{-.2ex}{$\textstyle <$}}{\sim} $}$250 days) theoretical overtone models of Mira variables can be made with pulsation velocity amplitudes close to those that are observed (Wood 1974; Tuchman 1991). Basically, the cause of the uncertainties listed above is the difficulty of determining the radii of Miras, which are very extended and gradually fade away into the interstellar medium rather than having sharp edges like the Sun.

Theoretical models of red giant pulsation show that as a star evolves up the AGB, it develops unstable overtone modes, the most unstable of which becomes of lower order as luminosity increases: towards the end of the AGB the fundamental is the most unstable mode (Fox and Wood 1982). If the Mira variables are indeed fundamental-mode pulsators, then we might expect to see two distinct (K,logP) relations among the LPVs in the LMC, one corresponding to fundamental-mode pulsation (the Miras) and one corresponding to first-overtone pulsation (lower amplitude LPVs). Most importantly, there should be a distinct separation between these sequences (at a given luminosity) of ΔlogP∼0.34, corresponding to the typical fundamental to first-overtone period ratio P₀/P₁∼2.2. On the other hand, if the Miras are first-overtone pulsators, then one might expect another sequence of LPVs in the (K,logP) plane corresponding to 2nd and higher overtone pulsation, but this sequence should be separated from the Mira sequence by ΔlogP of only ∼0.11, corresponding to P₁/P₂∼1.3.

CCD observations of star clusters in the LMC have revealed many LPVs with amplitudes in V of a few tenths of a magnitude to several magnitudes. Pre-CCD searches for LPVs in the Magellanic Clouds (see references in Wood, Moore and Hughes 1991) have revealed ∼1000 LPVs. However, these objects are of relatively large amplitude. The largest survey, by Hughes (1989), revealed LPVs of amplitude ΔI$\raisebox{-0.6ex}{$\,\stackrel {\raisebox{-.2ex}{$\textstyle >$}}{\sim}\,$}$0.5. Hughes (1989) categorised those objects with amplitude ΔI <0.9 as semiregular (SRa) variables by analogy with the amplitude definition for Galactic semiregular variables (a Galactic LPV with visual light amplitude <2.5 mag. and a reasonably regular light curve is classified as a semiregular, while LPVs with larger amplitudes are classified as Mira variables). The larger-amplitude LPVs in the LMC were classified by Hughes as Miras. The SRa and Mira classes in the Galaxy and the LMC seem to merge smoothly into each other in properties. In particular, Hughes and Wood (1990) showed that the LMC Miras and SRa variables fell on the same (K,logP) relation to within the errors of single-phase observations. This result shows that the Hughes (1989) SRas can not be assigned to first- overtone mode pulsation while the Miras are assigned to fundamental- mode pulsation since the period ratio of ∼2.2 between these two modes would lead to well separated (K,logP) relations if two separate pulsation modes were involved. However, a situation where the Miras were first-overtone pulsators and the SRas were second-overtone pulsators could not be ruled out (the ratio P₁/P₂ is relatively small and typically ∼1.3).

The first aim of this chapter is to look for secondary (K,logP) sequences among the newly discovered LPVs in the LMC. Since many of these LPVs have amplitudes lower than those of previously known LPVs in the LMC, there is a reasonable chance that these objects could be pulsating in modes higher than that of the Miras. It should be noted that Wood (1975b) and Barthes and Tuchman (1994) have previously tried to find multiple modes in the light curves of individual local Miras in order to identify the dominant pulsation mode but the signal-to-noise of the possible secondary modes in these large amplitude stars is very low and the results are very uncertain.

In a second approach, theoretical models are fitted to giant branches at luminosities below those at which large-amplitude variability starts, and then extend those giant branches into the Mira regime where the models are compared with observations of LMC Miras in order to identify the Mira pulsation mode. A brief discussion of some of the results presented here is given in Wood (1995) along with a general discussion of the problem of determining the mode of pulsation of the LPVs.

Observational data

The LPVs used in this study were discovered around the LMC cluster NGC 1850 at the northern end of the LMC bar (Chapter 3), and the pair of clusters NGC 2058 and NGC 2065 at the southern end of the LMC bar (Chapter 4). Infrared J and K magnitudes were obtained on two occasions for each object using the 2.3m telescope at Siding Spring Observatory. The infrared camera CASPIR was used on the dates 29 May 1994 and 14 May 1995, while the single channel infrared photometer was used on 19 December 1994 with a 10 ^′′ aperture. The infrared observations are given in Table 5.1. Table 5.2 shows mean K magnitudes and J-K colours, along with the periods, mean I magnitudes and mean V-I colours from Chapters 3 and 4. The J and K means are averages of the data in Table 5.1, while the V and I means are magnitude means obtained from least-squares Fourier fits to the time series of V and I magnitudes.

Table 5.1: Infrared observations of LPVs

Field	Variable	Date	K	J-K
			mag.	mag.
NGC 1850	347	940529	12.41	0.72
		950514	12.10	0.77
	587	940529	12.06	1.11
		950514	12.17	1.33
	670	940529	10.49	1.85
		950514	10.50	1.81
	1038	940529	9.23	1.40
		950514	8.73	1.45
	1906	940529	11.15	1.73
		950514	11.14	1.77
	2093	940529	11.42	1.32
		950514	11.53	1.30
	2268	940529	10.86	1.27
		950514	10.84	1.34
NGC 2058-65	V403	941219	11.40	1.28
		950514	11.49	1.25
	V446	941219	10.96	1.86
		950514	10.77	1.51
	V523	941219	10.48	1.37
		950514	10.11	1.37
	V569	941219	10.93	1.72
		950514	10.87	1.57
	V607	941219	11.95	1.16
		950514	11.87	1.18
	V627	941219	11.86	1.37
		950514	11.93	1.09
	V741	941219	10.74	1.84
		950514	10.65	1.90
	V1111	941219	10.92	1.72
		950514	11.22	2.26
	V1243	941219	11.49	1.41
		950514	11.49	1.30
	V1402	941219	11.01	1.37
		950514	11.03	1.18
	V2056	941219	11.02	1.44
		950514	11.01	1.30
	V3094	941219	11.05	1.25
		950514	11.20	1.28

Note: Date is in the form year-month-day.

Table 5.2: Mean magnitudes and colours of LPVs

	Variable		P(days)	<K >	<J >- <K >	<V >	<V >- <I >
			(days)	mag.	mag.	mag.	mag.
	NGC 1850
	347		109.5	12.26	0.74	16.01	1.80
	587		145.6	12.11	1.22	17.80	3.10
	670		330.0	10.49	1.83	17.01	2.89
	1038		597.0	8.98	1.43	17.70	4.60
	1906		230.0	11.15	1.75	17.13	2.73
	2093		208.0	11.48	1.31	17.81	3.10
	2268		134.5	10.85	1.31	17.46	3.60
	NGC 2058-65
	403		47.0	11.44	1.27	16.34	2.27
	446		131.0	10.86	1.68	16.38	2.54
	523		410.0	10.30	1.37	16.91	3.36
	569		238.0	10.90	1.65	16.64	2.46
	607		65.0	11.91	1.17	16.60	2.20
	741		283.0	10.69	1.87	16.83	2.74
	1111		293.0	11.07	1.99	17.58	2.86
	1243		88.0	11.49	1.35	17.06	2.75
	1402		135.0	11.02	1.27	17.32	3.37
	2056		154.0	11.02	1.37	17.65	2.73
	3094		103.0	11.12	1.26	17.94	3.69

The observed (K,logP) relation

The LPVs are plotted in the (K₀,logP) plane in Figure 5.1. Reddenings of E(B-V)=0.15 and 0.12 for the fields of NGC 1850 and NGC 2058-65, respectively have been assumed , along with absorption A(K)=0.4E(B-V). The line is the (K₀,logP) relation for Mira variables of M and C spectral type in the LMC from Feast et al. (1989).

Figure 5.1: The LPVs in the (K₀,logP) plane. Circles are LPVs near NGC 1850 and triangles are LPVs near NGC 2058-65. The line is the LMC Mira relation of Feast et al. (1989).

It is immediately apparent from Figure 5.1 that only about half the LPVs in this sample lie on the Mira sequence in the (K₀,logP) plane. We note that Feast et al. (1989) find that the LMC Miras scatter about the mean (K₀,logP) relation by only 0.15 magnitudes. Even single phase observations of Miras in the LMC produce a scatter about the mean (K₀,logP) relation of only 0.26 mag. (Hughes and Wood 1990). In Figure 5.1, there is clearly a second sequence of LPVs parallel to the Mira sequence but separated from it by ΔlogP ∼ -0.35, corresponding to periods at a given luminosity that are shorter by a factor of ∼2.2. Recalling the discussion in the Introduction, we note that this ratio is remarkably close to the period ratio of fundamental mode to first overtone. Therefore, the suggestion is that the Mira variables are fundamental-mode pulsators while the second, shorter-period sequence of LPVs in Figure 5.1 consists of first-overtone pulsators.

Amplitude distributions

As a general consideration, overtone pulsators are expected to have amplitudes which are smaller than those of fundamental-mode pulsators. If the two LPV sequences found above correspond to fundamental and overtone pulsation, then different amplitude distributions might be expected for the two sequences. Figure 5.2 shows the distribution of I amplitude (ΔI) for stars on the two sequences. Amplitudes of stars on the Mira sequence cover a wide range from 0.1 $\raisebox{-0.6ex}{$ \stackrel {\raisebox{-.2ex}{$\textstyle <$}}{\sim} $}$ΔI$\raisebox{-0.6ex}{$ \stackrel {\raisebox{-.2ex}{$\textstyle <$}}{\sim} $}$2.0, with most stars have ΔI$\raisebox{-0.6ex}{$\,\stackrel {\raisebox{-.2ex}{$\textstyle >$}}{\sim}\,$}$0.5, while the overtone pulsators are confined to ΔI$\raisebox{-0.6ex}{$ \stackrel {\raisebox{-.2ex}{$\textstyle <$}}{\sim} $}$0.5. A large number of LMC LPVs with amplitudes ΔI$\raisebox{-0.6ex}{$\,\stackrel {\raisebox{-.2ex}{$\textstyle >$}}{\sim}\,$}$0.5 magnitudes were studied by Hughes and Wood (1990). All but perhaps 8 out of 300 LPVs in this sample lie on the Mira sequence in the (K,logP) plane, confirming that larger-amplitude LPVs pulsate in the same mode, which have been identified above with the fundamental. (Since there are some uncertainties in period determination for the stars in Hughes and Wood (1990), and given that Hughes and Wood had only single-phase K magnitudes rather than mean K magnitudes, it is quite plausible that the small number (∼8) of stars in Figure 10 of Hughes and Wood that appear to lie on the overtone sequence are misplaced.)

Figure 5.2: The distribution of total pulsation amplitude in I for LPVs on the Mira sequence (solid curve) and for LPVs on the second, shorter period sequence (dotted curve).

Theoretical Asymptotic Giant Branch sequences

We now make theoretical models for the LMC LPVs in order to confirm the mode identifications suggested above. A theoretical examination of period ratios P₀/P₁ and P₁/P₂ in LPVs shows that these ratios can vary significantly depending on the model used for the LPV (see Figure 3 of Fox and Wood 1982). In order to compare the results in Figure 5.1 directly with theory, we construct models appropriate for LPVs in the LMC. Our starting point is the observational (I,V-I) diagram (Figure 5.3) for stars in the field around NGC 1850. The data for this figure come from Chapter 3, and we have assumed a reddening E(B-V)=0.15, E(V-I)=1.3 E(B-V), A_V=3.1E(B-V) and an LMC distance modulus of 18.5. The LPVs around both NGC 1850 and NGC 2058-65 are plotted on Figure 5.3, with a reddening of E(B-V)=0.12 assumed for the NGC 2058-65 LPVs. The old giant branch, and presumably the LPV population, for the field around NGC 2058 and NGC 2065 is similar to that around NGC 1850 but with a much more patchy reddening.

Figure 5.3: Stars in the 10^′×10^′ field around the cluster NGC 1850 plotted in the (I,V-I) plane (small dots). The circles and triangles are mean I and V-I values for the LPVs around NGC 1850 and NGC 2058-65, respectively. Theoretical AGB tracks for stars of mass 0.8, 1.0 and 1.5M_$\odot$ are also shown.

Asymptotic giant branches (AGBs) have been constructed theoretically to fit the observational data in Figure 5.3. A series of models was made for masses of 0.8, 1.0 and 1.5M_$\odot$. This range of masses was chosen to fall within the range defined at the lower end by the core mass for AGB stars of ∼0.6M_$\odot$ and at the upper end by a mass corresponding to that of stars formed in the burst of star formation which began in the LMC ∼3×10⁹ years ago (Butcher 1977; Bertelli et al. 1992). For each luminosity and mass, an envelope model was constructed by integrating from surface to core in the manner described in Fox and Wood (1982), with the core mass appropriate for the model luminosity (Wood and Zarro 1981). The mixing length was adjusted so that the AGB sequences had the correct temperature at relatively low luminosities (M _bol∼ -2, M_I∼ -2.5) where the conversion from T_eff to V-I is well calibrated. A value of 2.5 pressure scale heights was found to fit the observed giant branch T_eff. The sequences were extended to higher luminosities assuming a constant mixing length. The V-I colours and bolometric corrections needed to plot the theoretical models in Figure 5.3 were derived from the model atmospheres of Kurucz (1993) for T_eff ≥4000K. At lower temperatures the (V-I,logT_eff) calibration of Bessell (1979) was used along with the bolometric correction BC_I to I from Bessell and Wood (1984). By this procedure we eliminate the need to adopt a temperature scale for the LPVs themselves. The temperatures and radii of the LPVs are a direct outcome of the assumption that the mixing length is constant up the AGB. Such an assumption generally reproduces giant branches (e.g. in globular clusters) quite well. In constructing the AGB sequences, and in the linear pulsation models described below, the OPAL opacities (Iglesias and Rogers 1993) were used, supplemented by molecular opacities added as described in Chiosi, Wood and Capitanio (1993). Abundances Y=0.30 and Z=0.008 were adopted.

Theoretical pulsation models for the LPVs

For the theoretical AGB models shown in Figure 5.3, linear, radial, non-adiabatic pulsation periods were computed with the code described in Fox and Wood (1982), updated to include the new opacities as described above. The results of these calculations have been compared to the observations of LPVs in the (K,logP) plane in two ways. In the first comparison, the period ratios obtained from the models have been used, but the periods themselves are not required to be very accurate. This method avoids the possibility that the mixing length may vary up the giant branch, giving incorrect values of T_eff, and hence period, at a given luminosity or K magnitude (period ratios are relatively insensitive to changes in stellar parameters).

To test if the observations are consistent with fundamental-mode pulsation in the Mira variables, it is assumed that the model fundamental-mode periods corresponded to the observed Mira periods. K magnitudes are then derived for the models from the observed Mira (K₀,logP) relation of Feast et al. (1989). Then all models automatically lie on the Mira (K₀,logP) relation, while the period ratios prescribe the theoretical positions of the sequences corresponding to first, second and third-overtone (Figure 5.4). A similar test of the first-overtone hypothesis for Mira pulsation was made by forcing the first-overtone period to coincide with the observed Mira (K₀,logP) relation (Figure 5.5). Note that the period ratios shown in Figures 5.4 and 5.5 do not vary greatly with stellar mass or luminosity or mixing length. Therefore, uncertainties in the parameters of the models should not lead to errors in mode identification while only period ratios are being considered.

Figure 5.4: The observations of LPVs compared with theoretical models in the (K,logP) plane. Points are as in Figure 5.1. The theoretical fundamental-mode period has been forced to fit the observed Mira relation of Feast et al. (1989) and period ratios in the models are then used to position the curves corresponding to 1st, 2nd and 3rd overtone.

Figure 5.5: As for Figure 5.4 except that the first-overtone period has been forced to fit the observed Mira (K,logP) relation.

The results shown in Figures 5.4 and 5.5 confirm the tentative conclusions reached in Section 5.3. If the Miras in the LMC are assumed to be pulsating in the fundamental mode, then the second sequence of LPVs corresponds well with pulsation in the first or second overtone. In particular, there should be an unpopulated gap between the fundamental and first-overtone sequences, as is indeed observed. On the other hand, if the Miras are pulsating in the first- overtone mode, then the second and third-overtone pulsators fall in the gap where no LPVs are observed. The second observed sequence of LPVs would have to correspond to ∼5th overtone pulsation with no unstable modes between the 5th and 1st overtone. This is a very unlikely situation since the theoretical pulsation models constructed here and by Fox and Wood (1982) are mostly unstable in the 2nd and 3rd overtones, although use of the mixing length theory of convection for energy transport means that the theoretical growth rates are suspect. Finally, if it assumed that the Miras are first- overtone pulsators, then the predicted fundamental-mode sequence is unpopulated, in spite of the fact that models exhibit strong instability for the fundamental mode. All these considerations strongly suggest that the Miras are fundamental-mode pulsators.

In the second comparison between theory and observation, the AGB models are used directly to predict the (M_bol,logP) relation expected for fundamental and overtone pulsation and then compared the predictions with the observed Mira relation of Feast et al. (1989). Again, it should be noted that estimates of Mira T_eff values were not explicitly required here. The procedure used was to adjust the mixing length to get the correct giant branch temperature at relatively low luminosities and to then rely on the assumption that the mixing length does not vary along the AGB to get the T_eff values (and radii) of the model AGB stars. Although this assumption may be questioned, we note that evolutionary tracks constructed with constant mixing length in general reproduce red giant sequences well.

The periods of the fundamental mode and 1st, 2nd and 3rd overtones for 0.8, 1.0 and 1.5M_$\odot$ AGB stars obtained in this manner are plotted in the (M_bol,logP) plane in Figure 5.6. In this rather crowded diagram, there are four lines for each mass, corresponding to the fundamental and 1st, 2nd and third-overtone modes, respectively. The fundamental mode has the longest period at any luminosity and mass, the 1st overtone is next longest, and so on. Also shown in Figure 5.6 is the mean (M_bol,logP) relation for LMC Mira variables, plotted assuming an LMC distance modulus of 18.5. The LMC LPVs studied in this chapter are also shown, where M_bol has been derived from <K > using the bolometric correction BC_K to K given in Wood, Bessell and Fox (1983) and a distance modulus of 18.5. Stars with (J - K)₀ > 1.5 are assumed to be carbon stars and are shown by solid symbols. The most luminous (M_bol < -4.5) Mira variables in this study, which are carbon stars, do not closely fit the mean (M_bol,logP) relation possibly because only two observations were obtained per star and the amplitudes are relatively large.

Figure 5.6: Periods of the fundamental, 1st, 2nd and 3rd overtone modes of model AGB stars of mass 0.8, 1.0 and 1.5M_$\odot$ (thin lines). The thick line is the mean (M_bol,logP) relation for Mira variables in the LMC from Feast et al. (1989). Circles are LPVs near NGC 1850 and triangles are LPVs near NGC 2058-65, where carbon stars (defined as objects with (J-K)₀ > 1.5) are shown as solid symbols and M stars are shown as open symbols.

It is clear from Figure 5.6 that the observed Mira (M_bol,logP) sequence in the LMC is readily explained in terms of fundamental-mode pulsation in stars of increasing mass as the luminosity increases. We note that kinematic studies of Mira variables in the Galaxy show just such an increase in mass with pulsation period (Feast 1963; Jura and Kleinmann 1992). The LPVs observed here, and lying on the short period sequence parallel to the Mira sequence, are well-explained as overtone pulsators. On the other hand, the predicted first-overtone periods are much too short to be consistent with LMC Mira pulsation.

Summary

Infrared observations have been obtained of a set of small and large- amplitude LPVs in the LMC and shown that the LPVs fall on two distinct sequences in the (K,logP) plane. One sequence is the well-known Mira sequence while the second sequence runs parallel to the Mira sequence but has periods smaller by a factor of ∼2.2. Theoretical models have been made for AGB stars on the prominent LMC old giant branch and pulsation periods have been derived for the fundamental, 1st, 2nd and 3rd overtone modes. Using period ratios only, we showed that if the Miras are fundamental-mode pulsators then the second sequence of stars can be readily explained by first or second-overtone pulsation. However, if the Miras were first-overtone pulsators, then the smaller-amplitude LPVs would need to be pulsating in a high (∼5th) overtone and we would not expect to see the distinct gap which is observed between the two sequences unless all modes between the 1st and 5th overtone were stable. Similarly, the absence of a sequence of LPVs corresponding to fundamental-mode pulsation would require this mode to be stable, in disagreement with theoretical models. Observations presented here and by Hughes and Wood (1990) show that LPVs of relatively large amplitude ( ΔI$\raisebox{-0.6ex}{$\,\stackrel {\raisebox{-.2ex}{$\textstyle >$}}{\sim}\,$}$0.5) all lie on the Mira sequence while all LPVs on the second sequence have small amplitudes ( ΔI$\raisebox{-0.6ex}{$ \stackrel {\raisebox{-.2ex}{$\textstyle <$}}{\sim} $}$ 0.5). This result is consistent with expectations if the LPVs on the Mira sequence are fundamental-mode pulsators and the LPVs on the second sequence are overtone pulsators. Finally, comparison of the actual model periods with the observed Mira periods shows good agreement between fundamental-mode periods and observed Mira periods. Taken together, these results present strong evidence in favour of the Miras being fundamental-mode pulsators. Although the current sample of stars is relatively small, a study of a larger sample of stars using the MACHO database has begun in order to provide a better sample for comparison with the theory.

If the conclusions above are accepted, then some explanation is required for the large radii that have been derived for Miras from angular diameter or T_eff measurements. Possible explanations include overestimation of Mira distances, and underestimation of the atmospheric opacity and/or extension caused by pulsation in the models of Bessell et al. (1996), which were used by Tuthill et al. and Haniff et al. to interpret their observations. A better understanding of the large observed radii in Miras is clearly needed.