6.4.1.1 Idealized Models of White Dwarfs and Neutron Stars

Fig. 6.7 Sirius is a binary star system composed by a normal main sequence star, Sirius A, twenty-five times more luminous than the Sun, with about two solar masses and a white dwarf companion, Sirius B, of about 0.6 solar masses that have got compressed into a volume similar to the size of the Planet Earth. Sirius B extinguished its nuclear fuel, went through the stage of Red Giant and collapsed into its actual state of White Dwarf about 120 millions of years ago. Sirius B will steadily cool, as the remaining heat is radiated into space over a period of more than two billion years

White Dwarfs As their name indicates, white dwarfs are stars of very small magnitude but very high superficial temperature, so that their total luminosity is quite faint although the light they emit is extremely white, corresponding to electronic transitions between very high energy levels. For instance, at about 8.6 light years from the Solar System, in the Canis Maior Constellation, there is the most luminous of night stars, well known to Mankind from remotest antiquity, namely Sirius. This latter, named Sirius A, is actually member of a binary system whose other member, Sirius B, is a white dwarf (see Fig. 6.7). With a mass equal to 0.6 solar masses, Sirius B has a radius of about some thousands kilometers, namely it is as big as the Earth. However the superficial temperature of Sirius B is much higher than that of the Sun, namely it is 25,200 K, which makes the faint light it emits so much white. What is the simplest theoretical description of such stars as Sirius B? Imagine that the progenitor which later collapsed into a white dwarf was just a cloud of hydrogen. Gravitation compressed such a cloud until its core reached a sufficiently high temperature to initiate the hydrogen fusion cycle. Protons joined pair by pair into deuterium, then collisions of deuterium nuclei generated tritium and eventually all tritium nuclei fused, pair by pair, into helium nuclei, each time liberating two protons that completed the cycle. For billions of years the fusion cycle went on liberating the energy that made the star luminous and provided the pressure necessary to maintain it in equilibrium against gravitational attraction. Although slow the process is not eternal and eventually it comes to an end when all the hydrogen is fused into helium. At this point of its evolution a main sequence star can

Fig. 6.8 The fusion cycle of hydrogen is the main engine powering middle sized normal stars. When all hydrogen is fused into helium the star starts cooling down and gravitational collapse begins. Considering the minuteness of the electron mass in comparison with that of the baryons, the total weight of a white dwarf is approximately estimated from the number of its electrons. To each of them we are supposed to add two baryons: one proton and one neutron

be idealized as a cloud of helium that starts compressing into smaller and smaller volumes under the effect of gravitational attraction. The contraction continues until all electrons are stripped away from their nuclei and condense into a Fermi gas of extremely high density. Let us evaluate the electron density of such a star. Taking into account the very small weight of electrons compared with that of the proton and of the nuclei we get that the total mass of the star is approximated by the following formula:

where mp is the mass of a nucleon (at this level we can consider the mass of a proton and that of a neutron equal) me is the mass of the electron and N is the total number of electrons. Indeed in a helium atom there are two nucleons for each electron (see Fig. 6.8). The radius R of the star can be estimated from its volume via the formula:

Hence in terms of the total mass and of the radius of the star, the density of the fermionic gas, which in this case is the electronic density, is given by:

Correspondingly the Fermi momentum is given by:

we can say that Mwd is the mass of the white dwarf star measured in units of baryon masses mp . Similarly Rwd is the radius of the same star measured in units of the Compton wave length:

of the particles which compose the high density degenerate Fermi gas, in this case the electrons.

Inserting such information into previous formulae for the high density Fermi gas pressure we obtain the following result:

It is very interesting to see that a completely analogous formula holds true also for neutron stars.

Neutron Stars The neutron particle was experimentally revealed in 1932 by Sir James Chadwick, the British physicist who obtained the 1935 Nobel Premium for this discovery. One year only after this detection, Wilhelm Baade and Fritz Zwicky5 proposed the existence of neutron stars. The two California-based physicists were seeking a theoretical explanation for the enormous energy released in supernova explosions. They argued that what powers supernovae is just the release of the gravitational binding energy of a collapsing normal star. When all thermonuclear fuel is exhausted, a sufficiently large star collapses under gravitational self attraction to such a compressed state where all protons, neutrons and electrons are squeezed into such a small volume and are so much close to each other that inverse β-decay takes place systematically. By capturing an electron and releasing a neutrino, all

5Wilhelm Heinrich Walter Baade (1893–1960) was a German astronomer who emigrated to the USA in 1931 and mostly worked at Mount Wilson Observatory in California. Born in Bulgaria from Swiss parents, Fritz Zwicky (1898–1974), worked most of the time in the USA where he obtained his Ph.D. from the California Institute of Technology. Later he was to be appointed Professor by the same Institute giving important contributions to various areas of Astronomy. He worked in association with the Mount Wilson and Palomar Observatories. Zwicky was also a brilliant engineer and he is considered the father of modern jet propulsion engines. Through his first wife Zwicky become a relative of the US President Frank Delan Roosevelt.

Fig. 6.9 When all protons neutrons and electrons are squeezed into a sufficiently small volume all protons are converted into neutrons by capturing an electron and releasing a neutrino

protons are converted into neutrons (see Fig. 6.9). In this way we can conceive the existence of very compact stars, purely made of neutrons, where, once again, gravitational attraction is compensated by the pressure of a degenerate Fermi gas of very high density. Such an equilibrium state, denominated Neutron Star was predicted by Baade and Fricky as the result of supernovae explosions in a paper of 1934 [7].

Approximately thirty years later, in 1965, Antony Hewish and Samuel Okoye were able to detect strange radio signals coming from the center of the Crab Nebula where in the year 1054 had taken place the most luminous supernova explosion ever recorded in history.6 Hewish (see Fig. 6.10) won the 1974 Nobel Prize in Physics for his role in interpreting the pulsating radio-wave signals coming from the Crab-Nebula as the emissions of a pulsar, namely a rotating neutron star in which the magnetic moment is not perfectly aligned with the rotation axis. Under these conditions, which are the generic ones for a rapidly rotating neutron star, the latter becomes a radio-antenna and its regular pulse signal allows for its own detection. Several neutron stars, both galactic and extra-galactic, were discovered in the following years. The neutron star closest to Earth, named Calvera, which is at a distance of about 250 light years from us, was discovered in 2007.

Let us calculate the Fermi pressure of a neutron star regarding it as a free Fermi gas of neutrons.

6The 1054 supernova was observed by Chinese and Arabic astronomers and its sudden appearance was also recorded in the Chronicles of the St. Gallen Monastery in Switzerland. According to these witnesses SN 1054 was so bright as to be seen in daylight for 23 days and was visible in the night sky for 653 days.