Core mass

Figure 6. Central density as a function of stellar core mass (in solar mass units) for the critical states, from Bisnovatyi-Kogan (2002).

PO4 core masses is transformed to optical displacement coordinates ... 

Deep searches for LPVs in the LMC (Wood, Bessell and Paltoglou 1985 Reid, Glass and Catchpole 1987) show that the LPV population is dominated by objects of age a few times 10 years which, at the time of their exit from the AGB, have envelope mass -0.35 M0, core mass -0.65 M0 and Mbol = -5. These objects are presumably the immediate precursors of most of the planetary nebulae in the Magellanic Clouds, and their properties will be compared below with those of the planetary nebulae. 

From the observations of 99Tc in Mira type AGB stars and from assumed period-luminosity relations [21] for such stars, it is concluded that 99Tc may be observed in AGB stars as dim as Mbol -4.0 (Moore 0.6 MG). As current AGB evolution models show that the 22Ne neutron source is only active in AGB stars with Mbol < -6.0 (Mcote > 0.8 MG), one concludes that the 13C source must be the nucleosynthesis source in the low-core mass objects. 

Malaney [13] finds that the 13C source and the high-core mass 22Ne source always produce large amounts of 95Zr, and only a low-core mass 22Ne source (Mcore =0.8 MG, Nn - 10B n/cm3) is able to match the low 9BZr abundance. 

Figure 2 Comparison between observations and evolutionary tracks applicable for sdB, sdOB and classical sdO stars. HB is the horizontal branch with core mass Mcore = 0.475 MQ, on which some values of q = Mcore/M 0j.aj are indicated. CC denotes the track of Caloi and Castellani (1985), the tracks G1 and G2 are from Gingold (1976). Post-AGB tracks applicable for CSPN are from Schonberner (1983 SI M = 0.546 MQ, S2 M = 0.565 Mq). The hatched strip marks the area in which Groth et al. (1985) have found photospheric convection. Stellar symbols are the same as in Fig. 1. 

Both points disagree with observations The observed main sequence width requires only a moderate core mass increase (cf. Mermilliod and Maeder, 1986), the LBVs exist, and very massive WNE and WC stars are not observed (cf. references in Langer, 1987 Doom, 1987). Evolutionary calculations without overshooting avoid both discrepancies. We conclude that convective overshooting is not very efficient in massive H-burning stars, but that the Schwarzschild-criterion may be a fair approximation in order to determine the size of the convective core. 

The luminosity of SK -69 202 is consistent with a helium core mass of about 6 solar masses, and this corresponds to a main sequence mass of order 15 solar masses. An important question that will affect the future behavior of the supernova is how much of the 9 solar masses not in the core was present on the surface of the star when it exploded. If mass loss was large, then the mass on the star could have been small, although we know that the surface layers were hydrogen-rich from the optical spectra. 

Models of the supernova progenitor have indicated that SN 1987A possessed a helium-core mass of M(je = 6 M , with a total mass at the time of outburst in the range of 10-20 Mg, and a carbon-core mass of 4 M0 (Woosley et al. 1987). If the s-process enhancements produced in the carbon core are approximately those computed by Lamb et al. (1977) for this mass regime, and if these are subsequently mixed via... 

Another way of parameterizing the problem of the progenitor structure of SN 1987A is to specify the luminosity, the total mass, and the core mass, and then to examine the behavior of the core radius, Rhb. as a function of Teff (or R). Figure 2 shows the result for models with the standard envelope composition, M = 15 M and MHe = 4 M . For fixed L, M, Mne. and RHe. if there is a solution with large Teff then there are, in fact, three solutions to the stellar structure equations, with different values of Te and R. A similar pattern exists for models with cores of 6 M . The result is not particularly sensitive to the specific parameters assumed. 

Figure 2. The radius of the core in units of 1010 cm is given as a function of effective temperature for a total mass of 15 M and a helium core mass of 4 M . The curves are labeled with the luminosity in units of 1000 L . Note that for fixed total mass, core mass, core radius, and luminosity, there are three solutions with different effective temperature. These three solutions have different density structures. 

Figure 3. The radius is given as a function of Lagrangian mass for a series of models with total mass 15 M , and helium core mass 4 M , for a range of values of the effective temperature. Note that as the effective temperature declines the temperature gradient steepens giving two solutions with the same core radius, eg, the curves for 15,500 and 4000 K. As the effective temperature continues to decline, the envelope becomes fully convective, and the temperature gradient becomes more shallow, providing the third solution. A model with effective temperature between 3110 and 3140 K would also have the same core radius as the 15,500 and 4000 K models.

Many of the models demand that the temperature at some mass cut above the helium core be in excess of the ignition temperature of hydrogen, Th, and hence, the structure is self-inconsistent. For a helium core mass of 4 M , models with total mass in the range 4.05 to 14 M , are excluded. This conclusion does not depend sensitively on the specific value adopted for Th, since the temperature profiles tend to rise so steeply in the excluded range. Similar conclusions follow for models with larger luminosities and... 

Figure 4. The temperature in units of 1000 K is given as a function of Lagrangian mass for models with L = 55,000 L , Teff = 15,500 K, R = 33.4 R and envelope composition X = 0.7, Y - 0.295, Z = 0.005, fora range of total masses. This luminosity corresponds to original main sequence mass of about 15 M and a helium core mass of 4 M . Note that for smaller total mass and hence smaller envelope mass the temperature rises rapidly to exceed the hydrogen ignition temperature well outside the core.

Three different structures are found for stellar models with identical luminosity, total mass, core mass, and core radius as the effective temperature is varied. Two of these solutions can correspond to effective temperatures in the range pertinent to SK -69 202. This implies that models for the progenitor of SN 1987A will prove very sensitive to physical assumptions and numerical treatment in structural and evolutionary calculations. 

An estimate of the iron core mass before collapse is found as follows. We need 1.6 to 1.7 M0 to explain the 12 second signal and a few tenths of a solar mass for the accretion phase luminosity. From a series of stellar evolution calculations of stars producing different iron core masses by Weaver and Woosley (private communication), we find that for models with iron core masses less than 1.5 M0 the density exterior to the core falls so rapidly with radius that appreciable accretion could occur not in a few seconds. For high mass iron cores the density doesn t fall off... 

The core characteristics do not change appreciably in these diverse models with central densities near 1010 g/cm3, a central electron fraction Yf 0.42, and temperature Tc 0.5 MeV. The central entropy per baryon rises and falls with the core mass, with high values inimical to healthy shocks. 

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