Schwarzschild criterion

At sufficiently high densities (e.g. cores of upper main-sequence stars), the > sign virtually becomes an equality (adiabatic stratification), but at lower densities (e.g. envelopes of the Sun and cooler stars) an exact calculation is very difficult and in most models a crude approximation based on mixing-length theory is used. In a situation where the chemical composition changes with depth, Eq. (5.24) (known as the Schwarzschild criterion) needs to be replaced by more complicated considerations. 

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 convective core size in very massive H-burning stars may well be approximated by the Schwarzschild criterion. 

Saio, Kato, and Nomoto (1988) recently examined under what conditions a massive star undergoes a blue-red-blue evolution. The evolution of a star of initial mass 20 M0 star in the HR diagram is shown in Figure 1 from the zero-age main-sequence through carbon ignition at the center. The metallicity in the envelope was assumed to be Z = 0.005 and the Schwarzschild criterion was adopted. The star shows the three types of evolutionary path (A, B, C) depending on the mass loss, metallicity, and the change in the helium abundance Y in the envelope. 

Table 4. Nucleosynthesis results for 15 massive star models computed up to silicon ignition (Langer Henkel 1995). The symbols have the following meanings Mi is the initial stellar mass, and Z the metallicity. asc is the semiconvective mixing parameter, with asc = 0 corresponding to the Ledoux criterion, asc = oo to the Schwarzschild criterion for convection. Mf is the final stellar mass, Mco the final CO-core mass and Mrem is the assumed remnant mass. Me and Mo are the total mass of carbon and oxygen ejected by stellar wind mass loss and by the supernova explosion (initially present amounts are not subtracted). The values /13. .. /is designate production factors for 13C, 14N, 170, and lsO, and AY/ AZ is the ratio of the net yields of helium to metals.

Figure 4. Comparison of the oxygen yield Mo as a function of the initial stellar mass Mzams for models with and without stellar wind mass loss, and assuming a remnant mass of the order of 2Mq. Upper panel Models with slow semiconvective mixing. Weaver Woosley s (1993 WW93) models are computed without mass loss, Langer Henkel (1995 LH95) included mass loss. Lower panel Models with fast semiconvective mixing resp. Schwarzschild criterion for convection. Woosley Weaver (1995 WW95) neglected mass loss, Maeder (1992 M92) explored models with standard and excessive mass loss.

The Schwarzschild criterion for convection Vrad > Vad can thus be expressed as... 

However, even for convection zones close to the stellar surface, where convection becomes strongly non-adiabatic and Lconv —> 0, T (r) = T(r) > 1 is not a criterion for instability. This can be seen when the Schwarzschild criterion is written in entropy formulation as < 0. Since oc yv iy )/y (with a being the adiabatic sound speed), we have for adiabatic convection (4s- =0) As in a hydrostatic situation... 

For the same reason, no convective core overshooting was invoked. This has been applied in many massive star calculations in the recent years in order to obtain a wider main sequence band (cf., e.g., Schaller et al. 1992). However, as rotationally induced mixing has a very similar effect (e.g., Langer 1992, Fliegner et al. 1996), the interpretation that the main sequence widening is due to rotation and thus that the convective cores of non-rotating stars are not extended over their sizes predicted by the Schwarzschild criterion was adopted. 

We may annotate here that the adiabatic expansion in the gravity field plays a certain role for the stability of stars. The Schwarzschild criterion for stability is based on the adiabatic expansion of a gas in the interior of a star under the influence of gravity. 

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