3 The pulsation masses of carbon LPVs

3.1 The method

To estimate pulsation masses, we made use of the PMR relations of Wood (1990) for the fundamental mode and of those of Wood et al. (1983) for the first overtone. Calculated for oxygen-rich LPVs, they were already applied by Groenewegen & de Jong (1994) to LPVs in the LMC, either oxygen-rich or carbon-rich. For the fundamental, we thus adopted the period

$$P_{0} = 0.00851~\left(R_{\rm {p}}/R_{\odot}\right)^{1.94}~\left(M/M_{\odot}\right)^{-0.90}$$

$${\rm for}\:\:\: M/M_{\odot}\le 1.5$$ (6)

and

$$P_{0} = 0.00363~\left(R_{\rm {p}}/R_{\odot}\right)^{2.09}~\left(M/M_{\odot}\right)^{-0.77}$$

$${\rm for}\:\:\: M/M_{\odot}\ge 2.5$$ (7)

and an interpolation formula in the intermediate mass-range. Concerning the first overtone, the period is

$$% P_{1} = Q~\left(R_{\rm {p}}/R_{\odot}\right)^{1.50}~\left(M/M_{\odot}\right)^{-0.50}$$ (8)

where

$$Q = 0.038+5.5 \times 10^{-5}\left( P_{1} -100\right)$$

$${\rm for}\:\:\: M/M_{\odot}\le 0.85\:\:\:{\rm and}\:\:\:P_{1}\ge 100$$

$$Q = 0.038+4.5 \times 10^{-5}\left( P_{1} -150\right)$$

$${\rm for}\:\:\: 0.85\le M/M_{\odot}\le 1.5\:\:\:{\rm and}\:\:\:P_{1}\ge 150$$

$$Q = 0.038+2.5 \times 10^{-5}\left( P_{1} -300\right)$$

$${\rm for}\:\:\: 1.50\le M/M_{\odot}\le 2.5\:\:\:{\rm and}\:\:\:P_{1}\ge 300$$ (9)

and Q=0.038 in any other case. The accuracy of the radii is not sufficient to allow individual derivation of pulsation masses, and the true parallaxes used were intended essentially for statistical use. The expressions of pulsation masses obtained from (6), (7) and (8) are not proportional to true parallaxes. Averaging individual masses is thus subject to the bias described by Smith & Eichhorn (1996). A method consists of averaging quantities linear in $\varpi \propto \left( R/R_{\odot}\right)^{-1},$ i.e. $\left( M_{{\rm p}}/M_{\odot} \right)^{-\gamma}$ with $\gamma \simeq 0.46,$ 0.37 and 0.33 respectively for Eqs. (6), (7) and (8). The appropriate version of Eq. (9) has to be selected by trial and error. This is much like averaging inverse radii. Since we are missing information about possible biases on periods, we suppose that their influence can be neglected here. A second method is to consider a fictitious star with inverse radius equal to the average inverse radius of the studied sample and period equal to the average period. Those quantities are then introduced in Eqs. (6) to (9). Differences between both methods are most often small and never exceed 15%. We finally adopted the second method, which we consider as slightly more favorable. If the distribution for a given sample was strongly non-Gaussian, a bias toward lower masses might intervene. This is not the case for the HC- and CV-samples studied in Sect. 3.2.

 

 

Table 1: Mean pulsation masses $\left <M/M_{\odot }\right >$ estimated for the photometric groups from Eqs. (6) to (9) and the second method described in Sect. 3.1. Both fundamental (F) and first-overtone (O) modes were used with similar results. First overtone pulsation was assumed for CV6-CV7 stars with very large radii (O$\arcmin$). Adopted mean masses $\left (M_{\rm {ad}}\right )$are also quoted together with dispersions, for documented groups (both modes): HC5, CV1, CV2, CV3, CV4, CV5, and CV6, illustrating the increase of the average mass along the photometric sequence. Values quoted for HC3, HC4, CV7 and SCV are only indicative. The case of some underluminous CV5-stars is also investigated (see text for details). The spanned range is nearly 0.5-4.2 $~M_{\odot},$ in good agreement with results of Sect. 7.5.2 in Paper III. Also quoted the mean density referred to the solar value $\rho ' =10^{8}~\langle \rho \rangle /\langle \rho_{\odot} \rangle,$ and the mean surface gravity $g'=10^{3}~\langle g \rangle$ in SI units, i.e. 10 times its value in CGS units.

G	m	n	$\langle P_{0,1} \rangle$	$\langle \frac{R}{R_{\odot}} \rangle_{0,1}$	$\langle \frac{M}{M_{\odot}}\rangle$	$\langle M_{\rm {bol}} \rangle$	$\langle T_{\rm {eff}} \rangle$	$\rho '$	g'	$\frac{M_{{\rm ad}}}{M_{\odot}}$	Comments
HC3	F	1	125	75	0.26:	-3.15	3945	61:	13:	0.3:	C3938=V CrA
HC4	F	1	110	93	0.47:	-3.65	3865	59:	15:	0.5:	C3319=TT CVn; CH
HC5	F	5	$290\pm70$	$161\pm83$	0.53	-4.18	3400	13	5.6
	O	2	$74\pm5:$	$137\pm34$	0.68	-3.76	3460	26:	10:	$0.6\pm0.2$
CV1	F	6	$299\pm98$	$153\pm55$	0.46	-3.60	3300	13	5.3
	O	8	$99 \pm42$	$159\pm42$	0.59	-3.76	3290	15	6.4	$0.55\pm0.15$
CV2	F	13	$339\pm71$	$241\pm113$	1.05	-4.42	3050	7.5	5.0
	O	16	$147\pm46$	$231\pm90$	0.94	-4.34	3020	7.6	4.8	$1.0\pm0.2$
CV3	F	10	$355\pm73$	$285\pm81$	1.52	-4.58	2880	6.6	5.1
	O	12	$159\pm54$	$302\pm84$	1.62	-4.73	2910	5.9	4.9	$1.6\pm0.3$
CV4	F	14	$353\pm55$	$299\pm129$	1.74	-4.77	2770	6.5	5.4
	O	8	$178\pm53$	$358\pm157$	2.16	-4.96	2780	4.7	4.6	$1.9\pm0.35$
CV5	F	13	$393\pm51$	$311\pm133$	1.68	-4.40	2670	5.6	4.8
	F	10	$384\pm30$	$387\pm115$	3.15	-4.91	2670	5.4	6.0		without 3 underluminous stars
	F	3	423:	188:	0.48	-3.28	2645	7.3	3.7	0.5:	underlum.: RU Pup, T Lyn, RZ Peg
	O	15	$201\pm26$	$376\pm114$	2.14	-4.76	2660	4.0	4.1
	O	14	$200\pm24$	$396\pm100$	2.25	-4.87	2660	3.6	3.9	$2.7\pm0.5$	without 1 underluminous star
	O	1	217:	220:	0.45	-3.61	2675	4.2	2.5	0.5:	underlum.: AC Pup
CV6	F	13	$442\pm49$	$479\pm110$	4.67	-4.86	2460	4.3	5.6
	O	4	$212\pm23$	$422\pm119$	2.78	-4.58	2465	3.7	4.3
	O$\arcmin$	7	$458\pm67$	$779\pm97$	3.97	-5.88	2250	0.84	1.8	$4.2\pm0.8$	O$\arcmin$ = supposed 1st overtone
CV7	O$\arcmin$	7	$444\pm107$	$1040\pm370$	8.24	-5.72	1970	0.73	2.1	8.2:	O$\arcmin$ = supposed 1st overtone
SCV	F	1	365:	406:	3.83:	-5.49	2880	5.7:	6.4:		V346 Aur
	O$\arcmin$	3	227:	533:	5.04:	-5.83	2995	3.3:	4.9:	4.7:	O$\arcmin$ = supposed 1st overtone

3.2 The results

The available data is practically limited to the HC5, SCV and CV1-CV7 photometric groups, some of them being poorly documented. The adopted period (either fundamental mode or first overtone) is the mean of individual values in the sample. The mean radius was obtained from the values of the $C_{\rm {R}}$-coefficient as described in Sects. 3.1 and 4 of Paper III. The results were given in Table  $\ref{mass}$ of Paper III. The mean pulsation mass increases along the sequence of groups from $\simeq$ $0.6~{M}_{\odot}$ at HC5-CV1 to $\simeq$ $4~M_{\odot}$ at CV6-CV7.

The mean pulsation periods are quoted for each mode, identified as described in Sect. 2. The adopted mode is doubtful for a few percent of the data that might prove to be misclassified later on, but the mean values should not be seriously affected. The mean values of photospheric radii are given for variables pulsating in either mode. They can differ from values quoted in Table 3 of Paper III which includes all the carbon giants of our initial sample, even if they are not LPVs (e.g. even for irregular variables). It was shown in Table 4 of Paper III that, on average, the SR- and Mira-variables are slightly brighter than the Lb-ones. We have also operated a detailed comparison, group per group, with the conclusion: there is no significant difference in terms of $M_{\rm {bol}}$ between the irregular Lb-stars and the whole sample. The difference noted in Table 4 of Paper III is actually an artifact of the concentration of irregular variables in earlier (thus less luminous) photometric groups (say HC5 to CV3) than SRs and Miras (see Fig. 8 of Paper II).

Small mean pulsation masses are found in the groups HC5 and CV1. One is tempted to question the applicability of the PMR relations as written in Sect. 3.1. They were established for oxygen-rich variables, and the effect of opacities specific for carbon-rich atmospheres on such models remains to be investigated. We however note that

• the values 0.5-0.6 $~M_{\odot}$ are physically possible, allowing only tiny envelopes depleted by mass loss,
• the $\left(0.6\pm0.2\right)~M_{\odot}$-mean of HC5-stars is compatible with what is expected from CH stars among HC-stars (represented in HC5, e.g. C327=V Ari). It has been argued that they all are members of double stars (McClure & Woodsworth 1990) whose companion is a white dwarf of nearly $0.6~M_{\odot}.$ The statistical study of 8 systems yielded $\left<M\right>\simeq \left(0.8\pm 0.1\right)~M_{\odot}.$

Incidentally, we obtained the same value ( $0.6~{M}_{\odot}$) for the RCB-variable C4098=V CrA (F9g) whose period is 75 days. The other end of the distribution ( $4~M_{\odot}$) is not well-established. The existence of a few massive CV6-CV7 ($\ga$ $4~M_{\odot}$), with strong mass loss and optically thick circumstellar shells, seems indicated on the grounds of the available data. The HIPPARCOS sample contains unfortunately only Miras of intermediate optical thickness like R Lep. Some SCV-stars intermediate between the bright (intrinsic) S stars and the CV-stars may also be more massive than the above $4~M_{\odot}$ upper limit.

Figure 4: A diagram of mean absolute bolometric magnitudes vs. mean pulsation masses in solar unit on a log scale (Table  $\ref{mass}$). For the sake of clarity, only typical dispersions in both coordinates, are shown as arrows. The TP-AGB zone is delineated by theoretical lines (onset and tip; Z=0.008) and a carbon star formation line (CSFL) is shown (see text for details).

An interesting question comes from the underluminous carbon giants already found when studying the period-luminosity diagrams of the present Galactic carbon stars (Sample 3 of Bergeat et al. 1998), and those of their analogues in the LMC. Those CV5-CV6 stars are found at $\left<M_{\rm {bol}}\right>\simeq -3.3$ that is about 1.4 mag below the locus of standard CV5-stars (say $\left<M_{\rm {bol}}\right>\simeq -4.7$). Samples without underluminous stars have been also considered, resulting in increased estimates of mean masses. The mean mass deduced for 4 underluminous CV5-stars, namely RU Pup, T Lyn, RZ Peg and AC Pup, amounts to $\left(0.5\pm0.15\right)~{M_{\odot}},$ a value again compatible with a practically stripped core. Such underluminous giants are also observed in the LMC (Bergeat et al. 1998). Spurious results from HIPPARCOS may accidentally contribute but their existence seems beyond doubt. The above mean mass we found suggests that they experienced strong mass loss, evolving from standard CV-stars. We note that their mean masses, luminosities and radii are similar to those of the CV1- or HC5-group, the main difference lying in mean effective temperatures about 650 to 800 K higher in the latter groups. More speculatively, they could be low-luminosity "interpulse'' objects or post-TP AGB stars evolving toward HC5-CV1 at nearly constant luminosity. Such a possibility is suggested by theoretical tracks (e.g. Lattanzio 1987; Sackman et al. 1993). Many RCB variables and HdC stars with similar low masses are considered as born-again objects (Sect. 7.2 of Paper III). The results in Table  $\ref{mass}$ can be summarized as follows; (1) a marked increase of mean masses along the HC5-CV7 sequence, to be discussed later, (2) the absence of large systematic differences between mean values deduced from both modes, (3) the consistency of extreme mean values (0.5- $4~M_{\odot}$) with prediction from evolutionary tracks and mass loss operating, (4) the mean stellar density steadily decreases along the sequence as expected, and (5) the mean surface gravity is practically constant, close to 0.005 SI that is 0.5 CGS $\left(\log g\simeq-0.3\right).$