The answers to the question asked in my introduction are posed on this page. Contents of this page: Evolution of stars Maximum mass of neutron stars Pressure during the supernova-phase Evolution of binaries Evolution of stars In this part I will discuss how stars with a mass larger than 8 Msun evolve. But why only discuss stars larger than 8 Msun? Because stars with fewer mass will not go supernova, but only drive out a planetary nebula (Audouze et al. 1986). What remains after the seperation of the planetary nebula is a white dwarf. This remnant is the core of the progenitor. The white dwarf will have the most energy at the beginning of its life. Because the gravitational collapse of the core is not enough to turn on the nuclear reactions in the core, the core will loose its energy to its surroundings and after a few million years it will be so faint that it will end up as a sphere of dark matter. And now for stars larger than 8 Msun: This category contains stars where the under-limit is determined by the heaviest stars that can become white dwarfs and the upper-limit by the heaviest stars that still have their hydrogen envelope when the star dies. Usually these are stars that become Type II supernovae. The under-limit (as said above) is about 8 Msun. About the upper-limit people do not agree. E.g. Audouze et al. (1986) said that some people think the upper-limit lies between 20 Msun and 40 Msun, with a favored value around the 40 Msun. The reason why this mass is so uncertain is that it is not sure which stars have their hydrogen envelope at the end of their lives and also because the mass-limit depends on the metallicity of a star. We have to split up this category into 2 theoretical classes: from 8 to 12 Msun, the rest (so > 12 Msun and ~< 40 Msun. These categories are based on the ignition of the burning stages1, 2 (These are the most important ones). The under-limit of the first category are stars that only will ignite carbon. The upper-limit of that category (and so the under limit of the second category) are those stars that will ignite the other burning stages (hydrogen, helium, carbon, neon, oxygen and silicon-burning). For stars with helium cores between ~ 2.2 and ~ 2.5 Msun (is approx. 1/4 of the stars mass) the burning proceeds to nuclear statistical equilibrium while an other proces (rapid electron capture on iron group nuclei) leads to decrement of pressure which leads to a nearly free fall collapse of the core. What remains is not interesting for answering the subquestion. Stars from the mass-region 10 - 12 Msun with helium cores from ~ 2.5 to ~ 3.0 Msun will make an iron core in hydrostatic equilibrium. These stars will give very bright flashes before their core collapses. When the collapse occurs these stars will form neutron stars and eject helium with a trace of heavier elements. Stars heavier than 12 Msun, being a typical red giant, will go supernova and will leave behind a neutron star of approx. 1.2 - 1.4 Msun. With heavier stars you will finally reach a mass of ~ 2.0 Msun for the neutron star. Audouze et al. (1986) said that the maximum mass for neutron stars is about 2.0 Msun (when you have a nuclear equation of state). Therefore stars heavier than 20 Msun will produce a black hole, stars between 10 - 20 Msun will produce a neutron star. NOTICE: For stars heavier than 60 Msun the evolution is difficult, because these stars are almost super-Eddington and thus very unstable. With the table I inserted (Audouze et al. 1986) I want to give a view on the progenitor mass and the result of billions of years of evolution. What I think that has to be added is that under neutron star mass from M = 35 Msun should also stand 'BH' (= black hole). But when you are picturizing the evolution of a star it can be made very simple. Just click here! Take the time to watch it. A bit more detailed movie of the evolution is displayed here (the only thing you need here is a mpeg-player). What you see here is a main-sequence star being born from a dust cloud in the Universe. Then it will be a main-sequence star and evolve into a red giant. The last stage is that the red giant will explode into a supernova and the remnants will be a black hole. Top of this page Maximum mass of neutron stars (Srinivasan 1997) When a star dies it can become a white dwarf. But the maximum mass of a white dwarf is 1.44 Msun due to the Chandrasekar limit. This value is found by taking the relativistic pressure of a gass (Prel = K.pho4/3). Using the equation of hydrostatic equilibrium (dP/dr = -(G.M.pho)/(r2)) we get: M4/3/R5 ~ G.M2/R5 So this relation can only be satisfied for a unique mass, i.e. M = 1.44 Msun, because you can rule out the radius of a star and then the mass only depends on a few constants. But what if the mass is higher? A neutron star will be formed. Chandrasekhar (1939) said about neutron stars: "If the degenerate cores attain high densities (as is possible in these stars) the protons and electrons will combine to form neutrons. This would cause a sudden diminution of pressure resulting in the collapse of the star to a neutron core giving rise to an enormous liberation of gravitational energy. This may be the origin of the supernova phenomenon." This prediction implicates that the masses of neutron stars should be close to 1.44 Msun. What is then the maximum of neutron stars on the analogy of white dwarfs? Very important for this question is the hydrostatic equilibrium. That equilibrium depends very much on the equation of state of neutron star matter an thus on the pressure on density (P = P(rho)). To treat this subject we have to look to the equation of state of neutron star matter in two regimes: densities below and above the neutron drip (pho = 4.3 1011 gr cm-3). Firstly the equation of state below neutron drip. In the plot you see at the right you see the equations of state of a few theories. Only the important lines will be discussed. The line labeled with Ch (of Chandrasekhar) used the equation of state of the ideal degenerate electron gas model. HW stads for the first equation of state ever that was based on the 'liquis drop' model of the nucleus. The BPS line is almost baste on the same equation of state as HW. They only differ that the BPS used lattice energy (the net Coulomb interaction energy of all the nuclei and the electrons). Also the equation of state of an ideal elctron gas with a nuclear composition of pure Fe56 is shown, as well as that of the ideal n, p, e- gass (n,p,e-). Secondly the equation of state above the neutron drip. This category will also be divided into two regions: between neutron drip and nuclear density (rho0 = 2.8 1014 gr cm-3; at this point nuclei begin to dissolve into a fluid core) and above. The methode to create the BBP line (in the left figure) involves not only the semi-empirical mass formula, but also results from manybody calculations. The basic point is as following: when the density is larger than the neutron drip, there are neutrons outside the nuclei. This has the following effects that neutrons exert pressure on the nuclei and when the free neutron density becomes comparable to the density in the nuclei, the matter in- and outside the nuclei are very similar and this reduces greatly the nuclear surface energy. The HW line is the line which is based on the 'liquid drop' model and is only given for comparison. What you need for densities higher than the nuclear density to ~ 1015 gr cm-3 is the nuclear potential for the nucleon interaction and techniques for solving the manybody problem. The effect of the solving of the manybody problem is that the equation of state will be more 'stiffened' (see figure on the right). These theories are needed to give a clear view of how a neutron star is built up. Because we need a stable neutron star we use the equation of state for hydrostatic equilibrium, but this time using general relativity. To get the mass and radius of the neutron star we have to integrate te equation of hydrostatic equilibrium using the assumed equation of state. To calculate a model of a neutron star you have to take three tings into account: the central density (rhoc) given by the equation of state, when you go outwards in the neutron star from the core, the equation of state will give you the new value of the pressure when another density is reached and when you are on the crust of the neutron star the pressure has to be 0 (and the mass of the neutron star is contained within that crust). Chandrasekhar calculated with his equation of state that the maximum mass of a neutron star is ~ 5.37 Msun. More recent models (as you can see left) show that the maximum mass of neutron stars is ~ 2.0 Msun. Because of uncertainties in the equation of state above the neutron drip density this value is not very sure, but as you can see in the graph: most modern models will converge around ~ 2.0 Msun. Until so far the theoretical side of the maximum mass of neutron stars. Now let us turn our view to nature. Neutron stars 'in the wild' have a maximum mass of approx. 1.5 Msun (Brown et al. 1996), as you can see in the figure at the right. In this figure (Brown et al. 1996) the masses of low mass X-ray binaries (LMXB's) are plotted. The graph displays the measured mass of 18 compact objects. X-ray binaries at the top, radio pulsars and their companions at the bottom. The vertical dashed line indicates a Mcritic of 1.5 Msun. The reason why binary systems are used is that the mass of the separate stars in a binary system can be very easily and accurately deduced. Using also the conclusion that follows from Chandrasekhar's statement (see above) that the mass of neutron stars lies around 1.44 Msun, the theories above and the evidence that nature gives us you can say that the maximum mass of a neutron star lies between 1.44 and 2.0 Msun. Most values of the maximum mass of neutron stars you see in literature are within this range. E.g. Audouze et al. (1986) (as I told earlier) chose 2.0 Msun as Mcritical, because of a 'nuclear equation of state', Zampieri et al. accept 1.5 Msun as Mcritic. Why discuss the maximum mass of neutron stars? Beecause there are already detected neutron stars and we also now that neutron stars evolved from main-sequence stars of approx. 10 - 20 Msun. But we also now that there are main-sequence stars with a larger mass. What will be the remnants of those stars? To answer that question you need the maximum mass of a neutron star. Because when you have stars with a larger mass than 20 Msun you expect another end-product at the end of the evolution of such a star. One thing you can think about is a black hole. (This whole topic is very important in the case of SN 1987A, because people have predicted that the mass would be in the region of the Mcritic for neutron stars.) Top of this page Pressure during the supernova-phase At the end of their lives heavy stars can do 3 things (Fryer 1999): go supernova and become a neutron star, or go supernova and become a neutron star for a while and after accreting mass from the surroundings (e.g. its own supernova remnants) collapse under its own gravity and become a black hole, or collapse directly under its gravity to become a black hole. What my plan was was to calculate the pressure during the supernova-phase using the pressure of the supernova and the force that gravity exerts onto the matter that will be ejected when the star goes supernova. When you do this I think you can calculate the mass that will become a black hole when the star directly collapses. What I was thinking about was to look at the moment the star could go supernova. Then there you have radiation-pressure, convective-pressure, shockwave-pressure (this will begin to build up) and gravity as main forces. I wanted to simplify this by only using shockwave-pressure and gravity. This is not possible with my current knowledge of Astronomy and physics. What I can give to you is the expected value: about 1.5 Msun or higher. Top of this page Evolution of binaries As I told in my introduction, according to Brown & Bethe (1994) there would be ~ 109 low-mass black holes in our Galaxy. They have based this on their own scenario for evolution of stars and on the fact that Van den Heuvel (1992) said that there have to be at least 109 neutron stars in our Galaxy. Van den Heuvel estimated that the ratio of neutron stars to black holes (= f) to be 10 < f < 45. Brown & Bethe said that f ~ 1 and that therefore the number of black holes should be ~ 109. It seems that f ~ 1, 109 neutron stars and 109 black holes are very rough estimates. When you have 1011 stars in our Galaxy, 1% should be a neutron star and 1% should be a black hole, of which we haven't noticed one (except for two candidates SN 1987A and 4U 1700-37). So there must be something wrong in the evolution-theory of stars. Recently Brown et al. (1996) proposed that the evolution of binaries should be different from the evolution of normal main-sequence stars. They propose to change it this way: First of all you have to assume that binary systems are created according to the standard model. At the end of its main-sequence life the primary star will begin to expand and transfer its outer layer, the hydrogen envelope, to its companion. What remains of the primary star is a naked helium core. This helium star will not evolve like the helium core of a normal main-sequence star with a hydrogen envelope. The secondary star, that got the hydrogen envelope of the primary star, will evolve just like a normal main-sequence star, except for the rejuvenation that this star got from its companion. Therefore the star will live longer before it comes to the end of its life. So the final mass of the primary star when it has gone supernova will be lower and thus the chance of becoming a black hole is far smaller then when the star would have evolved as a normal single main-sequence star. Therefore the number of black holes in our Galaxy should be much lower than 109.

You can also see with my presentation of 28-9-1999 (DUTCH, text only). To go there click here.

Conclusions
References
Acknowledgments
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