Eddington limit

Show that the Eddington limiting luminosity, at which the gradient of radiation pressure balances gravity near the surface, is given by... 

However, there are two sets of observations which indicate that AGB stars do indeed reach the AGB limit Mbol = -7.1. Firstly, the results of WBF and Reid, Glass and Catchpole (1987) for the LPVs show that such stars do reach the AGB limit. Secondly, a luminosity function for red stars in the bar of the LMC obtained by Hughes and Wood (1987) shows that AGB stars extend to Mbol = -7.1, although there is a steep fall-off with luminosity above Mboi = -6. One explanation for the rapid fall-off in the number of AGB stars above Mboi = -6 is provided by Wood and Faulkner (1986) who show that envelope ejection will occur at this point in all but the most massive AGB stars due to the luminosity of the star exceeding the Eddington limit at the base of the hydrogen-rich envelope during the surface luminosity peak of a helium shell flash. 

Figure 3 Comparison between observations and evolutionary tracks applicable for extremely helium-rich stars. The helium main sequence (HeMS) is labelled with stellar masses, HL is the Hayashi limit. The hatched line indicates the Eddington limit for pure helium composition. Evolutionary tracks are from Paczynski (1971 dashed lines labelled with M/MQ), Schonberner (1977, M = 0.7 MQ, full drawn line) and Law (1982 M = 1 Mq, dotted line). Stellar symbols are the same as in Fig. 1. 

It is remarkable that He-rich stars appear to have a higher rate of mass loss than stars with solar-type atmospheres with the same Teff and L-values The Wolf-Rayet stars have, on the average, Al-values that are 140 times larger than the values for corresponding 0 and B type stars (De Jager et al., 1987). This may be due to the fact that WR stars, with their large Helium abundance, are relatively closer to their Eddington limit than the most luminous 0-type stars. 

The second observational correlation results from the detailed photo-spheric quantitative spectroscopy by Mdndez et al. (1987), which allows to determine the position of CSPN in the log g, log Te -plane with high precision. By transformation of evolutionary tracks into this plane stellar masses M/MQ and radii R/R0 can be read off directly. The log g, log Te diagram of CSPN can also be used to investigate how the strength of stellar winds depends on the stellar parameters. This is done in Fig. 14, which reveals clearly that the more massive objects closer to the Eddington limit have observable wind features. 

The Eddington limit is not only important in determining the upper mass limit of stars, but it is likely to induce strong mass loss once a massive star comes close to it. I.e., the Eddington limit may shape the initial-final mass relation for the most massive star, with severe consequences for nucleosynthesis of may species. 

Does the Eddington limit apply in the stellar interior ... 

The Eddington limit, in brief, is a limit to hydrostatic stability defined by the condition that the outwards directed force induced by the radiation momentum balances the inwards directed gravity. We want to emphasise that, as a stability limit, the Eddington limit applies only to hydrostatic situations. This does not mean that radiation forces are unimportant in hydrodynamic situations (e.g stellar pulsations) however, a stability limit makes of course no sense in an unstable situation. 

Since the convective flux does not contribute to the radiative acceleration one might think of a modified Eddington limit in the stellar interior according to... 

In summary, the Eddington limit, in whichever form, can not be applied in the stellar interior to predict instability. It only applies at the stellar surface, where density inversions can not occur and the convective flux is negligible. In the next section, however, we show that the Eddington limit by itself should also not applied at the stellar surface. 

It is crucial to realize that for T —> 1 Eq. (5.31) gives vcrit —> 0. Therefore, if we assume a star to evolve towards the Eddington limit (T — 1), no matter what its rotation rate may be, it will arrive at critical rotation well before T = 1 is actually reached. Therefore, one may rather speak of the D-limit instead of the Eddington limit. Note that in this simplified approach gravity darkening is neglected however, Maeder (1999) found this to not change our conclusions for the most massive stars qualitatively. 

However, supermassive stars encounter more stability problems than just the Eddington limit. The almost completely adiabatic structure of radiation dominated stars implies... 

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