File Name: black holes white dwarfs and neutron stars shapiro .zip
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We implement the quantum hydrodynamic equations and carry out numerical simulations. As a model of a white dwarf star we consider a zero temperature droplet of attractively interacting degenerate atomic bosons and spin-polarized atomic fermions. Such mixtures are investigated experimentally nowadays. We find that the white dwarf star is stripped off its mass while passing the periastron.
Due to nonlinear effects, the accretion disk originated from the white dwarf becomes fragmented and the onset of a quantum turbulence with giant quantized vortices present in the bosonic component of the accretion disk is observed.
We find a charged mass, falling onto a black hole, could be responsible for recently discovered ultraluminous X-ray bursts. The simulations show that final passage of a white dwarf near a black hole can cause a gamma-ray burst. White dwarf WD stars are ubiquitous in the Universe. They can be found as companions in various binary systems, including those with ordinary stars, giant stars, or compact objects as another white dwarfs, neutron stars NSs , or black holes BHs e.
The dynamics of a white dwarf in the field of a black hole, in particular its tidal disruption TD , has been modeled for years. Typically a smoothed particle hydrodynamics SPH simulations are performed to monitor the behavior of a white dwarf.
In this simulations a white dwarf is represented by a collection of SPH particles according to the Helmholtz equation of state. The self-gravity of the white dwarf is included but with adaptive gravitational softening 6 , 7.
The other group of numerical approaches to binary systems is based on general relativistic hydrodynamic simulations. The hydrodynamic equations, enriched by the polytropic equation of state of stellar matter, has been already used to study the TD of the main-sequence star by the BH 8 or the merger of the WD and NS 9.
Here, we are studying the dynamics of a model WD in the field of a BH incorporating quantum hydrodynamics. As a model of cold WD star we propose to consider the Bose—Fermi droplet consisting of ultracold bosonic and fermionic atoms. Such systems have been recently predicted theoretically 10 , An atomic Bose—Fermi droplet can exists because of subtle interplay of two effects.
The first one is related to the attraction between bosons and fermions. If it is strong enough then bosons start to effectively attract each other 12 and the droplet becomes unstable against collapse. Then all particles in the droplet attract each other just like particles in the WD star attract themselves gravitationally.
This collapse can be stopped by the fermionic component of the system. Its quantum pressure, like the pressure of degenerate electrons in the WD, is able to counteract the collapse and, to some extent, stabilize the system.
It results in the very high Fermi temperature, a few orders of magnitude larger than the WD temperatures. Indeed, it is save to treat electrons as a gas at zero temperature.
Hence, we propose here to consider bosonic component of a WD as a gas at zero temperature as well, although the description of bosons including thermal fraction is already well known Let us mention that the possibility of formation of the Bose—Einstein condensation in helium white dwarf stars was already discussed in 15 , 16 , 17 , To describe Bose—Fermi mixtures we use the formalism of quantum hydrodynamics One of the first attempts to discuss fermionic gases within this framework was already done many years ago, see Ref.
Here, we follow this reasoning and apply quantum hydrodynamic equations both for fermionic and bosonic clouds in a droplet. Now we place the Bose—Fermi droplet in the field of an artificial black hole. We assume a non-rotating black hole described by the Schwarzschild space-time metric. A motion of a test particle in the Schwarzschild metric conserves both the energy and the orbital angular momentum.
The energy of a test particle can be, as in the Newtonian case, divided into kinetic and potential energies. This term also depends on the angular momentum of a test particle. There exists, however, a surprisingly well working approximation to the radial potential, proposed by Paczynsky and Wiita The pseudo-Newtonian potential gives efficiency factors in a good agreement with the true solution.
Since it does not depend on the angular momentum, now the motion of a test particle can be considered as a motion in a three-dimensional space with the Newtonian potential replaced by the pseudo-Newtonian one. Then, the equations of motion for the Bose—Fermi droplet moving in the field of a fixed black hole can be put in the form which generalizes Eq. We solve numerically Eq. Such mixtures are studied intensively experimentally 12 , 23 , To find the densities of the Bose—Fermi droplet far away from the black hole we solve Eq.
Then the droplet is located at some distance far away from the horizon from the artificial black hole and pushed perpendicularly to the radial direction with some initial velocity. As shown in Ref. The stability condition just given is correct only for the Bose—Fermi droplet being in a free space, i. It slightly changes when the droplet is put in any external potential, in particular, the one originating from the artificial black hole.
We consider three cases related to open and closed orbits of the WD. We are interested in the final stages of the evolution of black hole-white dwarf binary, when tidal forces become damaging.
It happens when the white dwarf star itself gets larger than its Roche lobe. Only then the white dwarf starts to loose its mass through the inner Lagrangian point L1. The simulations show that the white dwarf circulates the black hole only a few times. It is significantly stripped off the mass after each periastron passage, many orders of magnitude stronger than the estimation given in The mass loss at the third passage is extremely large Fig.
The stripped mass forms an accretion disk around the black hole. Frames show densities at various times during the first revolution, when mainly bosonic matter contributes to an accretion disk. Three periastron passages marked by 1, 2, and 3 , equally separated in time, can be clearly identified.
Each successive passage is accompanied by increased amount of stripped mass. At the third passage the binary ends its life, see Fig. Figure 1 illustrates the way the white dwarf is stripped of its mass. The pipe connecting the black hole and the white dwarf is open extremely fast when the star is passing through the periastron for the first time, frame 1 b. The charged mass falling onto the black hole becomes a source of powerful radiation.
The time corresponding to the opening of a pipe is simply interpreted as a rise time of a signal, an X-ray burst as argued in 25 , detected by the observer. The mass is stripped off a white dwarf during almost a quarter of the circulation period, see frame 1 c,d, and the accretion disk is created.
Hence, the signal decays and the falling time is estimated to be about 50 times longer than the rise time, the ratio similar to that reported in 26 for NGC and NGC Finally, the pipe is broken frame 1 d and the radiation is seized at the background level until the white dwarf, while circulating the black hole frame 1 e,f , enters periastron region again Fig.
The stripped mass both of bosonic and fermionic type forms a massive accretion disk. Frames show densities during the third revolution. The binary system ends its life and the white dwarf goes away of the black hole. Figure 1 d—f show the accretion disk appearing around the black hole after the white dwarf is stripped off its mass for the first periastron passage.
The size of the accretion disk, i. Indeed, the existence of flat parts in the curve in Fig. Eventually, after a huge loss of mass during the third periastron passage, the white dwarf is expelled out of the neighbourhood of the black hole, see Fig. The binary system ends its life. The matter remained in the accretion disk becomes fragmented due to modulational instability 27 —a nonlinear effect, both classical and quantum, closely connected to the existence of solitary waves, already observed for Bose—Einstein condensates 28 , Although only bosonic component densities are shown in Figs.
Only during the first revolution the accretion disk is mainly formed from bosons. This is because the external field of a black hole changes the stability condition for the white dwarf and relative number of bosons and fermions in the white dwarf must change. In our case, an excess bosonic matter falls on a black hole during the first revolution. However, later on, when the action of a black hole on a white dwarf gets stronger, the white dwarf is stripped equally of bosonic and fermionic matter Fig.
An estimation of the mass of the black hole can be done based on the assumptions that at the periastron the white dwarf overfills its Roche lobe. Atomic white dwarf in the field of the black hole, located at the center of each box. The white dwarf is continuously stripped off its mass and an accretion disk is formed. At the final stage, the fragmentation of the orbiting material is clearly visible.
Also, giant quantized vortices are formed in the bosonic accretion disk see Fig. The white dwarf is continuously stripped off its matter and an accretion disk appears around the black hole, Fig. The orbiting material becomes fragmented due to nonlinear effects and giant vortices are formed in the bosonic component see Fig. Eventually, the white dwarf goes away from the black hole ending the life of the binary system.
Density with streamlines left frames and the phase right frames of the bosonic accretion disk at different times, increasing from top to bottom. The black hole is in the 0,0 position. The quantized vortices are visible as plus and minus signs present within the matter spiraling around the black hole, at the inner and outer edges of the matter spout, as well as in the region of low density.
Figure 5 clearly shows that quantized vortices of a single charge with both signs of vortivity are nucleated in a large number in the accretion disk in bosonic component while the matter is falling onto the black hole.
They are formed at the inner and outer edges of the matter spout and some of them move into the region of high density. Most of the vortices are created in the region of low density, in the neighbourhood of the black hole.
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Я только что выяснил, что ТРАНСТЕКСТ устарел. Все дело в алгоритме, сочинить который оказалось не под силу нашим лучшим криптографам! - Стратмор стукнул кулаком по столу. Сьюзан окаменела. Она не произнесла ни слова. За десять лет их знакомства Стратмор выходил из себя всего несколько раз, и этого ни разу не произошло в разговоре с .
Но, сэр, мутация… - Немедленно! - крикнул Стратмор. Чатрукьян некоторое время смотрел на него, лишившись дара речи, а потом бегом направился прочь из шифровалки. Стратмор повернулся и с удивлением увидел Хейла. Сьюзан поняла, в чем дело: все это время Хейл вел себя тихо, подозрительно тихо, поскольку отлично знал, что нет такой диагностики, в которой использовалась бы цепная мутация, тем более такая, которая занимала ТРАНСТЕКСТ уже восемнадцать часов. Хейл не проронил ни слова. Казалось, вспыхнувшая на его глазах перепалка абсолютно его не касается.
- Это должно быть что-то фундаментальное.
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