Astron. Astrophys. 323, 449-460 (1997)

4. Selfconsistent determination of the distance to U Ant

The two dust shells around U Ant can be related directly to two consecutive thermal pulses in the recent AGB evolution of U Ant. We develop a selfconsistent method to determine the distance to U Ant assuming that this is the case. We assume solar-like initial elemental abundances for this star since an N-type carbon star like U Ant in the solar neighborhood should be younger than the Sun.

There is a tight relation between the core mass and the luminosity of a star in the H-shell burning phase (Paczyski 1970). We obtained the core mass - luminosity relation for the solar abundances by interpolating the compilation of Groenewegen & de Jong (1993) based on the results by Boothroyd & Sackman (1988a), Becker & Iben (1979), and Iben & Truran (1978).

For the solar abundances mean molecular weight µ = 0.6031, heavy element abundance Z = 0.017, and total metallicity of carbon, nitrogen, and oxygen = 0.791Z, we find for the first thermal pulse stage

and for the full amplitude pulse stage,

Here and M denote the core mass, luminosity, and initial mass, respectively, of the star in units of solar values. The symbol "(1)" indicates that the quantities are of the first thermal pulse. There is also a well-know relation between the core mass and the interpulse period for thermally pulsating AGB stars (Paczyski 1975). The relation for stars of solar-like abundances is derived as below in the same way as in Groenewegen & de Jong (1993) based on the results by Boothroyd & Sackman (1988b),

where denotes the interpulse period in years.

Groenewegen et al. (1992) determined the distance to U Ant as 280pc from the observed bolometric flux and the adopted carbon star luminosity of 7050 , which gives a relation between the distance and the true luminosity of this star,

where D denotes the distance to the star in units of parsec. The obtained shell separation in the projected distance is related to the interpulse period through

where and are the shell separation in arcseconds and average expansion velocity of the shells in km s^-1, respectively. The superscripts in and out indicate that the quantities are of the inner and outer shells, respectively. Solving the four equations (1) or (2) (or (3) or (4)), (5), (6), and (7) for the obtained and adopted of 21 km s^-1, we have determined selfconsistently that D=436pc, L=1.7x10⁴ , =1.4x10⁴ years, and =0.77 for the full amplitude pulse case of a star with initial mass of 4 , and 324pc, 9.4 , 1.0x10⁴ years, and 0.80 for the first thermal pulse case (Fig. 9). A change in the initial mass of 1 influences the stellar luminosity by 5%, which corresponds to a change of less than 15pc in distance. Revising the distance from 280pc to 436pc affects the shell thickness, the inner radii of the shells, and the total mass in the shells for the best fitted model. For the distance of 436pc, the shell thickness becomes 3.1 cm, and we obtain dust shell parameters shown in Table 5.

Fig. 9. The luminosity of U Ant as a function of its distance. Thick lines show the relation obtained by combining the core mass-interpulse period and the core mass-luminosity relations. The thick solid lines are for the full amplitude thermal pulse stage and the broken lines for the first thermal pulse stage. Top, middle, and bottom lines for each stage indicate the cases of the average outer shell expansion velocity of 31, 21, and 11 km s-1, respectively. The bends of the broken lines are due to the interpolation of equations (1) and (2) in Sect.4 for 0.8 Mc 0.85 case. For the full amplitude stage the initial mass of 4 is assumed. The thin solid line shows the luminosity obtained with the observed bolometric flux

Table 5. Best fit dust shell model for the 60µm image (D=436pc)

We note that this method of distance determination has been proposed earlier by Paczysky (1975). He attempted to apply it to FG Sge, but the analysis was not very conclusive.

The core mass-luminosity relation may break down in the presence of hot bottom envelope burning if the star in question was massive when it was on the main sequence (Blöcker & Schönberner 1991). Although it is very difficult to estimate the initial mass of U Ant, the interpulse period determined from the above procedure suggests that the initial mass would be between 3 and 5 (cf. Vassiliadis & Wood 1993). The relation probably holds for this case and the uncertainty in the obtained distance arising from the core mass-luminosity relation is likely less than 10% here (see figure 12 in Vassiliadis & Wood 1993).

One of the main uncertainties in the above discussion resides in the average expansion velocity of the outer dust shell, while the uncertainty of the inner shell's average expansion velocity does not affect very much the estimate of the time lag between the two shells. For the inner shell it is probably a good approximation to use the CO gas expansion velocity.

However, the average expansion velocity of the outer shell may differ significantly from the adopted value of 21 km s^-1 due to the deceleration by the presence of interstellar medium (Young et al. 1993b). The present-day dust expansion velocity in the outer shell may be as small as 10 km s^-1 due to the deceleration, if the outer shell had an initial expansion velocity similar to that of the inner one. Detached CO gas envelopes around S Sct and TT Cyg show expansion velocites of 17.3 and 13.5 km s^-1, while their linear sizes are estimated to be 5.5 cm and 5.3 cm for the distances of 540pc and 1000pc, respectively (Olofsson et al. 1993), which are not very different from the size of the outer shell of U Ant. Their distances from the galactic plane are 32pc and 85pc, which are comparable to that of U Ant (124pc and 93pc for the full amplitude and first thermal pulse cases, respectively).

This supports the idea that the present-day gas expansion velocity of U Ant in the outer shell which is the lower limit of the present-day dust expansion velocity would be somewhat but not very much smaller than 21 km s^-1, and the average dust expansion velocity should be less affected. The gas/dust drift also makes the estimate of the average dust expansion velocity somewhat uncertain (Gilman 1972; Goldreich & Scoville 1976; Kwan & Hill 1977). The drift velocity can be as large as 9 km s^-1 or 5 km s^-1 for a mass loss rate of 5  yr^-1, a gas expansion velocity of 21 km s^-1, and a luminosity of 1.7 when the momentum transfer efficiency factor is 0.05 (Sopka et al. 1985) or 0.015 (Huggins, Olofsson, & Johansson 1988), respectively. Taking account of the two uncertainty factors we adopt 21 10 km s^-1 as a reasonable range for the average dust expansion velocity of the outer shell and we indicate the influence of changes in the expansion velocity in Fig. 9 (see the legend).

The core mass-luminosity and core mass-interpulse period relations are rather robust after close examination by many researchers. We find a distance which satisfies both the observed bolometric flux and the theoretically discovered relations (Fig. 9). The interpulse period suggests that the star should be relatively massive (3.5 M 5.0 ) initially (cf. Vassiliadis & Wood 1993). This initial mass range and the obtained luminosity matches those for carbon stars with a large envelope expansion velocity (greater than 17.5 km s^-1) by Barnbaum et al. (1991). They derived a scale height of 107 pc, which suggests an initial mass range of 2.5-5  , and typical luminosities of 1.5 for the carbon stars. The mass loss behavior predicted for such a relatively massive star in Vassiliadis & Wood does not show a significant change in the mass loss rate. The present results together with the CO observations by Olofsson et al. (1993) show, however, that the contrast between the high mass loss rates ( 10 ) in the shells and the present-day low mass loss rate ( 10 ) reaches two orders of magnitude. In present stellar evolution calculations, the treatment of mass loss behavior is often arbitrary. Our results suggest that massive carbon stars show a significant change in the mass loss rate over a certain fraction of each interpulse period along the AGB evolution. Further studies are needed to reveal the dependence of mass loss behavior of AGB stars on their initial mass as well as on their evolutionary status. The evolution of mass loss in low- and intermediate-mass carbon stars will be further investigated by our observations of their extended dust shells with the Infrared Space Observatory (ISO).

© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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