The early history of the Vogt-Russell theorem is retraced following its route starting at the realization of a correlation between mass and luminosity of binary and pulsating stars, through the embossing of this observation into a theorem, and finall
y to the emerging first signs of its failure to serve as a theorem in the strict mathematical sense of the word.
Very-low-mass stars can develop secularly unstable hydrogen-burning shells late in their life. Since the thermal pulses that go along are driven at the bottoms of very shallow envelopes, the stars luminosities and effective temperatures react strongl
y during a pulse cycle. Towards the end of the Galaxys stelliferous era, the hydrogen-shell flashing very-low-mass single stars should inflict an intricate light-show performed by the large population of previously inconspicuous dim stars. Unfortunately, this natural spectacle will discharge too late for mankind to indulge in. Not all is hopeless, though: In the case of close binary-star evolution, hydrogen-shell flashes of mass-stripped, very-low mass binary components can develop in a fraction of a Hubble time. Therefore, the Galaxy should be able put forth a few candidates that are going to evolve through a H-shell flash in a humanity-compatible time frame.
In stars with $M_ast lesssim 2 M_odot$, nuclear burning of helium starts under degenerate conditions and, depending on the efficiency of neutrino cooling, more or less off-center. The behavior of the centers of low-mass stars undergoing core helium i
gnition on the $logrho - log T$ plane is not thoroughly explained in the textbooks on stellar evolution and the appropriate discussions remain scattered throughout the primary research literature. Therefore, in the following exposition we collect the available knowledge, we make use of computational data obtained with the open-source star-modeling package MESA, and we compare them with the results in the existing literature. The line of presentation follows essentially that of Thomas (1967) who was the first who outlined correctly the stellar behavior during the off-center helium flashes that lead to central helium burning. The exposition does not contain novel research results; it is intended to be a pedagogically oriented, edifying compilation of pertinent physical aspects which help to emph{understand} the nature of the stars.
Stars in the narrow mass range of about 2.5 and 3.5 solar masses can develop a thermally unstable He-burning shell during its ignition phase. We study, from the point of view secular stability theory, these so called thermal micropulses and we invest
igate their properties; the thermal pulses constitute a convenient conceptual laboratory to look thoroughly into the physical properties of a helium-burning shell during the whole thermally pulsing episode. Linear stability analyses were performed on a large number of 3 solar-mass star models at around the end of their core helium-burning and the beginning of the double-shell burning phase. The stellar models were not assumed to be in thermal equilibrium. The thermal mircopulses, and we conjecture all other thermal pulse episodes encountered by shell-burning stars, can be understood as the nonlinear finite-amplitude realization of an oscillatory secular instability that prevails during the whole thermal pulsing episode. Hence, the cyclic nature of the thermal pulses can be traced back to a linear instability concept.
