Quantum Magnetic Collapse of a Partially Bosonized Npe-Gas: Implications for Astrophysical Jets

Quantum magnetic collapse of a partially bosonized npe-gas: Implications for astrophysical jets

R. GonzálezFelipe,1, ∗ A. PérezMart´ınez,2, † H. PérezRojas,2, ‡ and G. Quintero Angulo3, § 1ISEL - Instituto Superior de Engenharia de Lisboa, Instituto Politécnico de Lisboa, Rua Conselheiro Em´ıdio Navarro, 1959-007 Lisboa, Portugal CFTP - Centro de F´ısica Teórica de Part´ıculas, Instituto Superior Técnico, Universidade de Lisboa, Avenida Rovisco Pais, 1049-001 Lisboa, Portugal 2Instituto de Cibernética, Matemática y F´ısica (ICIMAF), Calle E esq a 15 Vedado 10400 La Habana Cuba 3Facultad de F´ısica, Universidad de la Habana, San Lázaro y L, Vedado, La Habana 10400, Cuba We study a possible mechanism for astrophysical jet production from a neutron star composed by a partially bosonized npe-gas. We obtain that the expulsion of a stable stream of matter might be triggered by the quantum magnetic collapse of one or various components of the gas, while its collimation is due to the formation of a strong self-generated magnetic field.

I. INTRODUCTION it is believed that all jets are produced and maintained by similar mechanisms. Although the physics behind jet production and maintenance is still under debate, the Astrophysics is living a golden period after the appear- general consensus is that magnetic fields play an impor- ance of new tools to explore the Universe and, in particu- tant role [5–7]. lar, among its more interesting components, the neutron stars (NS). The direct detection of gravitational waves in Theoretical studies on jet formation seek for space-time the past four years [1–3] has opened the door to listening metrics that allow to construct gravitomagnetic jets [8– to our Universe. At the same time, the progressive im- 12], as well as for mechanisms to explain their origins [13]. provement of the observational techniques as well as the Laboratory experiments are also being conducted with launch of new spatial observatories like the Neutron Star the aim of reproducing jets [14, 15]. Our current un- Interior Composition Explorer (NICER) Mission1 com- derstanding of relativistic jet formation is mainly based plement this new scenario and will allow to obtain more on 3D simulations performed in the context of general accurate constraints on NS observables. In the years to relativistic magneto-hydrodynamics [16, 17]. Some at- come, the results of these observations and their interpre- tempts to simulate numerically astrophysical jets, for in- tations will contribute to the challenge of understanding stance, the M87 from giant elliptical galaxy (a.k.a Virgo the physics of the matter in the interior of neutron stars, A, NGV4486, and 3C274), have also been made in the the densest objects in the Universe, as well as some others framework of specific gravity theories [18]. exotic phenomena like astrophysical jets. In this paper, we propose a possible mechanism for as- Astrophysical jets are streams of collimated matter trophysical jet formation and maintenance based on two that might be ejected by several objects (stars, proto- properties of magnetized quantum gases, namely, quan- stars, protoplanetary nebulae, compact objects, galaxies, tum magnetic collapse and self-magnetization [19–24]. etc.) [4, 5]. Depending on the source, jets may have dif- In most dense and cold quantum gases, the presence of ferent size, scale and velocities. Nevertheless, they all a magnetic field produces a decrease in the pressure in bear two outstanding features that make them different the direction perpendicular to the magnetic axis. If the from any other object in the Universe: their elongated magnetic field strength and densities are high enough, form and the fact that the matter composing them moves the perpendicular pressure component may become neg- away from the source without dispersing. The elongated ative, pushing the particles toward the magnetic axis and arXiv:1911.09147v1 [astro-ph.HE] 20 Nov 2019 form of jets is particularly intriguing, if one recalls that producing an instability that results in a cigar shaped most astronomical objects are spherical or oblate shaped, structure. Such an instability, known as quantum mag- due to the combination of the angular momentum con- netic collapse [19], might perfectly be the ultimate cause servation and the central symmetry of the gravitational that originates the jet. On the other hand, the huge force. That is the reason why, regardless of their origins, magnetic fields required for the occurrence of the magnetic collapse might also result in the phenomenon of self- magnetization, produced by the alignment of the magnetic moment of the particles that compose the jet. ∗ Electronic address: [email protected] To explore the possibility that both effects, quantum †Electronic address: [email protected] ‡Electronic address: [email protected] magnetic collapse and self-magnetization, can actually §Electronic address: gquintero@fisica.uh.cu be behind jet formation, we shall assume that the jet is 1 https://heasarc.gsfc.nasa.gov/docs/nicer/ originated inside a neutron star. In addition, we consider 2 a simple scenario of a NS composed by an ideal gas of the expressions neutrons, protons and electrons in which a fraction of the  q nucleons is paired forming composite bosons [25, 26]. We p2 + 2qBl + m2, charged fermion,  k refer to such a mixture of gases as a partially bosonized εf = q npe-gas.  p 2 2  pk + p⊥ + m − κB, neutral fermion, The paper is organized as follows. In Sec. II, we study (2) the magnetic collapse of a partially bosonized npe-gas and and a heuristic argument in favor of this physical phe-  q nomenon is given. Section III is devoted to the genera- p2 − qBl + m2, charged scalar boson,  k tion of the magnetic field inside the NS, and the study εb = q of the equations of state (EoS) of the collimated matter.  2 p 2 2  p + p⊥ + m − κB, vector neutral boson, Finally, our concluding remarks are given in Sec. IV. A k (3) brief summary of the EoS of each npe-gas component is presented in Appendix A. where m denotes the mass of the particles, q is the elec- tric charge, and κ the magnetic moment. In the above equations, the fermion spectrum is obtained by solving the Dirac equation for charged and neutral particles [22, 23], while the boson spectrum is derived from II. PARTIALLY BOSONIZED npe-GAS UNDER the Klein-Gordon equation [27] in the case of scalar par- THE ACTION OF A MAGNETIC FIELD ticles, and from the Proca equation in the case of the vectorial ones [24]. We consider a neutron star composed by a mixture of Once Ωi is determined, the EoS of each gas is obtained free neutrons n, protons p, and electrons e - gas npe -, through the usual expressions for magnetized gases [20], and paired (bosonized) neutrons nn and protons pp. We namely, assume that there are no interactions between these particles, i.e. all species are considered as ideal relativistic Pi k = −Ωi,Pi ⊥ = −Ωi − MiB, gases. We also assume that particles are under the action E = −Ω + µ N + TS , (4) of a locally uniform and constant magnetic field, directed i i i i i along the z axis, i.e., B = (0, 0,B). ∂Ωi ∂Ωi ∂Ωi Ni = − , Mi = − ,Si = − , The thermodynamic description of each gas in the mix- ∂µi ∂B ∂T ture is given in Appendix A, where the EoS are obtained starting from the general expression of the thermody- where Pi k and Pi ⊥ are the pressures of the gas along namical potential and perpendicular to the magnetic field direction, respectively; Mi is the magnetization, Ei is the internal energy, Ni is the particle number density, and Si is the entropy. T X Ω (B, T, µ) = dp d2p Note that the above EoS contains the anisotropy in the i 4π k ⊥ s pressures (Pi k 6= Pi ⊥) induced in the system by the pres- h i ence of the magnetic field. −(εi−µi)/T −(εi+µi)/T × ln 1 + e 1 + e , (1) The total thermodynamic quantities of the gas mix- T T ture, i.e., the pressures Pk and P⊥ , the magnetization where T is the temperature, s are the spin projections, MT and the energy ET , are then calculated as and pk and p⊥ are the particle momentum components T X T X along and perpendicular to the magnetic field direction, Pk = Pi k,P⊥ = Pi ⊥, respectively; εi is the energy spectrum, µi is the chemi- i i cal potential, and the index i denotes the particle species, T X M = Mi, (5) i.e. i = e, p, n, nn, pp. In what follows, we will work in i the T = 0 limit for fermions, while keeping tempera- T X ture effects for bosons. The difference in the treatment is E = Ei. justified by the fact that, although in stellar conditions i both kind of gases can be considered strongly degener- We will also consider that the partially bosonized npe- ated, bosons remains very sensitive to temperature ef- gas is in stellar equilibrium, so that the following condi- fects due to the drastic changes on the gas entropy and tions are satisfied: pressure caused by Bose-Einstein condensation. T T The effects of the magnetic field are included in the Nn + Np = NB, baryon number conservation, thermodynamical potential through the spectrum of the T N = Ne, charge neutrality, (6) particles. The spectrum of the unpaired and paired nu- p cleons in the presence of a magnetic field B are given by µp + µn = µe, β-equilibrium, 3

A. Quantum magnetic collapse 10-1 The magnetic collapse of quantum gases have been previously discussed as a mechanism that provokes one of the pressure components to become negative [19, 24]. If -2 10 the perpendicular pressure becomes less than zero, this collapse would be transverse to the magnetic axis and the particles of the gas would be pushed toward it. This gives rise to the formation of an elongated and axisym- 10-3 metric structure with a cigar shape, supporting the idea that the ejection of mass out of the star could be due to the transverse magnetic collapse of one or more quantum gas species. 10-4 In order to verify that the transverse magnetic collapse 1010 1011 1012 1013 1014 1015 1016 indeed occurs inside the NS, the following condition for the perpendicular pressure of at least one gas component must be satisfied: FIG. 1: Proton fraction as a function of the neutron mass 17 18 density for B = 0, B = 5 × 10 G, and B = 5 × 10 G. The P (B,N ) = −Ω (B,N ) − M (B,N )B = 0, (8) curves shift to left as the magnetic field intensity increases. i ⊥ i i i i i together with the stellar equilibrium conditions given in Eqs. (6). The results are illustrated in Fig. 2. The solid T T lines correspond to Pi ⊥ = 0, which can be well approxi- where Np = Np + 2Npp,Nn = Nn + 2Nnn are the to- 13 0 3/2 mated by the analytical fits Np ' 4 × 10 N (B/Bc) tal number of protons and neutrons, respectively; NB p T and N ' 7N 0(B/B )3/2 for protons and neutrons, re- is the baryonic number density; xn = 2Nnn/Nn and n n c T spectively, with N 0 ' N 0 ' 2.7 × 1039 cm−3 and the xp = 2Npp/Np correspond to the fractions of total p n 20 bosonized neutrons and protons, respectively. critical magnetic fields given by Bc ' 1.49 × 10 G and 20 Bc ' 1.56 × 10 G for protons and neutrons, respec- The fractions of paired nucleons, xn and xp, depend in general on the density and the temperature. How- tively. The colored area corresponds to the density and ever, such a dependence is presently unknown, since we magnetic field values for which Pi⊥ < 0 and the gas is are unable to reproduce the astrophysical conditions in unstable. Above the solid lines, Pi⊥ > 0 and the gas terrestrial laboratories. Therefore, hereafter we will take is stable. The horizontal dashed lines delimit the ap- them as free parameters that are set externally to the proximate range for the typical densities of each particle system. For simplicity, in our numerical calculations, we species inside the NS. shall assume x = x = 0.5. The possibility of the magnetic collapse of the gas n p of electrons, protons and neutrons becomes evident in Under the stellar conditions given in Eqs. (6) and ne- Fig. 2. As can be seen in these plots, given a value of the glecting the effect of bosonization in the β-equilibrium magnetic field, there is a critical density for electrons, condition, the fraction of protons (and electrons) for a protons and neutrons below which the gas is unstable. given neutron density reads [28, 29] This is due to the fact that the parallel pressure (−Ω) of a Fermi system increases as the density increases, which helps to balance the “magnetic pressure term” (−MB) T 2 2 2 2 4 3/2 Np 1 1 + 4Q/(mnyn) + 4(Q − me)/(mnyn) and stabilizes the gas. The critical densities of electrons T = 2 , Nn 8 1 + 1/yn and protons are of the order of those expected in NS and, (7) in particular, for values of the magnetic field higher than 10 2 2 1/2 10 G are always larger. In the case of neutrons, the where Q = mn −mp, yn = pF /mn and pF = (µ −εn) critical densities are lower than those estimated for the is the Fermi momentum of neutrons. crust and core of neutron stars, except when the mag- 16 In the presence of the magnetic field, the proton and netic field is very high (B & 3 × 10 G). electron fractions depend on its intensity B through pF . Unlike what happens for fermions, the critical density This dependence can be seen in Fig. 1, where the pro- of paired neutrons is quite sensitive to the temperature, ton fraction given in Eq. (7) is plotted as a function of as can be observed in Fig. 3, where the instability region 8 9 10 the neutron mass density ρn = mnNn for different values is plotted for T = 10 K, T = 10 K, and T = 10 K of B. Increasing the magnetic field shifts the minimum (from lighter to darker gray regions). Notice that the of the proton fraction to the left and, consequently, to phase diagram is reversed in comparison to the case of the region of lower densities. Also, an increase of the fermions (cf. Fig. 2), i.e., the gas becomes stable be- magnetic field increases the fraction of protons (and elec- low the critical particle density. This occurs because the trons) inside the star in the high density region. magnetization and magnetic pressure −MiB increase as 4

1040 1040

1038 1038

1036 1036

1034 1034

1032 1032

1030 1030 108 1010 1012 1014 1016 1018 108 1010 1012 1014 1016 1018

40 10 FIG. 3: Phase diagram for the transverse magnetic collapse of paired neutrons inside a neutron star. In the shaded gray

38 regions, P⊥(B,N) < 0 and the gas is unstable. The horizontal 10 lines delimit typical ranges of neutron densities inside the star.

1036 −MB is positive and it increases the perpendicular pressure, thus preventing the transverse magnetic collapse of 34 10 this gas. Let us summarize the results of this section. Under our 1032 assumptions, we have shown that a transverse magnetic collapse is possible for the gases of electrons, protons, neutrons and paired neutrons in a NS configuration. At 30 10 this point, it is worth remarking that the conditions for 1015 1016 1017 1018 1019 1020 such a collapse are assumed within the interior of the star. The NS is considered stable as a whole, although FIG. 2: Phase diagrams for the transverse magnetic collapse somewhere in its core some of the gases that compose it of electrons and protons (upper plot), and neutrons (lower could become unstable. Therefore, in what follows, we plot) inside a neutron star. In the colored region, P⊥(B,N) < assume that the densities and magnetic field values are 0 and the gas is unstable. The horizontal lines delimit the such that the neutron gas component never collapses. We typical ranges for the density of each particle inside the star. shall also consider that the conditions for the collapse of the remaining gases are local inside the star, or in other words, that the interior of the star is not homogeneous the density of a vector boson gas increases, enhancing the neither in density nor in the magnetic field. appearance of the Bose-Einstein condensate that dimin- ishes the parallel pressure −Ωi. The sum of both effects is a decrease in the parallel pressure that for high enough B. Ejection of matter from the neutron star densities makes the gas unstable. However, for magnetic 17 fields B ≥ 10 G, a minimum appears and the critical In the previous section we have found that electrons, curve starts increasing with B. At such strong magnetic protons, and bosonized neutrons in the interior of a NS fields the vacuum pressure term −Ωvac becomes signifi- can produce an instability in the transverse pressure. cant and helps to stabilize the gas. The critical densities However, the fact that these gases can collapse does not of the paired neutrons are also in the range of those ex- automatically imply that matter is ejected from the NS. 12 pected within a NS, in particular for B ≥ 10 G, con- For the ejection to take place, the collapsed gases must firming also the possibility of the magnetic collapse of the have a parallel pressure greater than the gravitational paired neutron gas. pressure of the star. This condition would in principle We have not shown the phase diagram for the paired guarantee a way in which matter could escape from the protons. Unlike the other gases, the paired proton gas star. has a diamagnetic√ behavior, since the magnetization For the study of the matter ejection mechanism de- M = −eN/(2 m2 + eB) < 0 (see Eq. (A6d) of Ap- scribed above, we shall compare the parallel pressures pendix A). Consequently, the magnetic pressure term of the collapsed gases with some “heuristic” value of the 5 gravitational pressure. A rough estimate can be obtained 4 considering the average gravitational pressure obtained 10 for a compact object with a typical NS mass M in the limit of compactness, 1.4M . M . 2.1M , where M 102 is the mass of the Sun. The compactness solution corresponds to the analyti- 0 cal solution of the Tolman-Oppenheimer-Volkoff (TOV) 10 equations, assuming that the inner density of the compact object is constant [28]. The radius and the pressure 10-2 inside the star are given as functions of the total mass M, the radius R of the star and its constant energy density E [28], 10-4

3M 1/3 R = , (9) 10-6 4πE 1011 1012 1013 1014 1015

p1 − 2GM/R − p1 − 2GMr2/R3 P (r) = E , (10) 104 p1 − 2GMr2/R3 − 3p1 − 2GM/R where G is the gravitational constant, r is the internal 102 radius of the star, and E is determined by the EoS of the partially bosonized npe-gas given in Eqs. (5). The average pressure inside the star is deﬁned as 100

1 Z R PGAV = P (r)dr . (11) 10-2 R 0

In Fig. 4, we show the comparison of PGAV with the 10-4 parallel pressures of the electron, proton and bosonized neutron gases. For the magnetic fields and densities considered, the three gases have already collapsed. The neu- 10-6 11 12 13 14 15 tron gas component is not included in this analysis be- 10 10 10 10 10 cause we assume that the stability of the NS is sustained by the non-negative neutron pressure. For values of the FIG. 4: Parallel pressure of electrons, protons and bosonized magnetic field around 1014 G, the average gravitational neutrons for typical neutron densities at the inner of a neutron pressure can be overcome by the parallel pressures of the star, for B = 1014 G and B = 1016 G. The shaded gray region collapsed protons and electrons in the nucleus of the star delimits density values between the crust (ρ ' 4×1011 g/cm3) 14 3 (ρ ∼ 2.7 × 10 g/cm [30]). As the magnetic field in- and the core (ρ ' 2.7 × 1014 g/cm3) of the star [30]. creases, the parallel pressures of the Fermi gases decrease. Instead, for the bosonized neutron gas, the parallel pressure remains almost constant over the range of magnetic III. SELF-GENERATED MAGNETIC FIELD field intensities, depending essentially on the tempera- AND COLLIMATED MATTER ture T , as can be seen in the figure. Nevertheless, for typical NS temperature and density ranges, Pnnk never exceeds PGAV . Once matter leaves the star and the jet is formed, a Therefore, depending on the values of the magnetic natural question arises: how does it remain collimated? field and the particle density, the collapsed gases could The elongated shape of the jet could be easily explained overcome gravity and trigger the ejection of matter. In if one or more gas components inside the star remain in this sense, the origin of the jet will be different depending a collapsed state, decreasing the perpendicular pressure on the density and the magnetic field, that is, depend- and letting gravity to compress matter toward the jet ing on the region of the NS from which it originates. axis. This is only possible if the magnetic field in the jet Since the NS as a whole is considered a stable object, continues to be as intense as the one produced by the NS. jet production will be set by local inhomogeneities of the From the observational point of view, it has been magnetic field and density inside the star. shown that the magnetic field of the jet is of the or- In the following, we will assume that the jet (or ex- der of the magnetic field of its source [6, 31]. Therefore, pelled matter from the star) has the same composition the question to be answered is whether matter can re- as the neutron star, that is, it will be also formed by a main magnetized once it leaves the star. To address this partially bosonized npe-gas, but in which there is at least question, we compute the magnetization of the partially one collapsing gas component. bosonized npe-gas as a function of the magnetic field, and 6

1020 1018

1018 1017

1016 1016

1014

1015 1012

1014 1010

108 1013 1014 1015 1016 1017 1018 1011 1012 1013 1014 1015

24 10 FIG. 6: The self-generated magnetic ﬁeld as a function of the neutron mass density. The shaded gray region corresponds to 1022 the typical density values between the crust and the core of the NS.

1020

1018

1016 shown in Fig. 6, in which the self-generated magnetic

14 field BSG is plotted as a function of the neutron mass 10 density. We notice that, depending on the density, the values of the self-generated magnetic field can be high 12 10 enough (B ≥ 1013 G) to keep the gases that form the 1014 1015 1016 1017 1018 SG jet in the collapsed regime. In the region of the typical density values between the crust and the core of the NS, 13 FIG. 5: Upper panel: the magnetization of the different the self-generated magnetic field is 8 × 10 G . BSG . 16 components of the bosonized npe-gas as a function of B for 4 × 10 G. 14 3 a neutron mass density ρn = 10 g/cm . Lower panel: the total magnetization of the bosonized npe-gas as a function of B for several values of the neutron mass density ρn. We have established that the transverse magnetic collapse is a viable mechanism for the ejection of matter from a NS, and that the expelled matter can produce a magnetic field high enough to keep it collimated. Let us look for a self-generation of the magnetic field, i.e., for now proceed to characterize the EoS of the jet through the existence of a solution of the equation MT = B/4π. the pressure dependence on the internal energy density In the upper panel of Fig. 5, the magnetization of the and the magnetic field intensity. This is shown in Fig. 7, different components of the bosonized npe-gas are plot- where the total parallel and perpendicular pressures of ted as functions of the magnetic field B for a neutron the jet are presented as functions of the internal energy 14 3 mass density ρn = 10 g/cm . As can be seen from this for B = 1014 G and B = 1016 G. As before, we assume an figure, for lower values of B the magnetization is dom- equal fraction of bosonized baryons, i.e., xn = xp = 0.5. inated by the electron and proton contributions, while We note that the difference between the parallel and per- for larger values of B the magnetization of paired neu- pendicular pressures of the jet can reach up to three or- nn trons M turns out to be the most significant. For ders of magnitude in the region of high energy densi- the range of magnetic field values considered, the lat- ties, being this difference a function of B (and xn). For ter quantity remains almost constant and is given by lower densities, both pressures are approximately equal. nn 14 M ' κnnρn/(4mn) ' 2.9 × 10 G. From the gravitational equilibrium viewpoint, the differ- In the lower panel of Fig. 5, we present the total ence in the pressures is related to a difference in the jet magnetization of the npe-gas for several values of the dimensions [32] that could account for its elongated form. neutron mass density. The dashed line corresponds to Therefore, this EoS could describe an axially symmetric B/4π and it intersects the curves indicating that the structure of the jet as far as its interior densities are in self-magnetization condition might be fulfilled. This is the region where the pressures are anisotropic. 7

The mechanism advocated here for astrophysical jet production and maintenance is based on the properties of 104 strongly magnetized quantum gases. The EoS was studied assuming that the source is a neutron star composed

2 of a partially bosonized npe-gas, in which at least one of 10 the gas species in the mixture collapses. Nevertheless, the proposed mechanism could also be applied to other 100 compact objects as long as they contain gases susceptible to suﬀer a transversal magnetic collapse. Therefore,

-2 it could be useful for explaining the physical phenomenon 10 of matter ejection in other astrophysical environments.

Finally, an important issue to be addressed is the grav- 10-4 itational stability of the jet. The search for stable solutions of Einstein’s equations with an anisotropic EoS 10-1 100 101 102 103 104 of matter requires numerical methods to integrate Ein- stein’s equations in an axisymmetric space-time. This is a nontrivial problem that is still unsolved. Some attempts

104 at solving this problem have been carried out in the context of other physical situations. Yet, those results can- not be applied to the description of such elongated struc- 102 tures as astrophysical jets. For instance, in Ref. [33], a search for inner solutions of Einstein’s equations assuming an almost flat-symmetric metric was performed, since 100 this type of space-time adapts to the symmetry that the magnetic field imposes on the energy-momentum tensor. 10-2 However, the use of this type of space-time imposes conditions to the pressures and energy that are only fulfilled

-4 in very restrictive situations (e.g., when the pressures and 10 energy of the vacuum dominate over those of matter).

-1 0 1 2 3 4 In another attempt, structure equations for spheroidal 10 10 10 10 10 10 objects were obtained [34]. The latter equations are re- stricted to the case of small deformations with respect to FIG. 7: The total parallel and perpendicular pressures of the the spherical shape and are not appropriate to describe jet as a function of its internal energy for B = 1014 G and a jet. The gravitational stability of jets is certainly an B = 1016 G. interesting problem that deserves further study.

IV. CONCLUDING REMARKS

In this work, we have studied the transversal magnetic collapse of matter inside a neutron star, considering it V. ACKNOWLEDGMENTS as a partially bosonized npe-gas. We have found that, in typical conditions for the NS interior, the electron, proton, neutron, and paired neutron gases are susceptible The authors thank Gabriel Gil Pérez for useful to suffer a transverse magnetic collapse, while the gas of comments. G.A.Q and A.P.M have been supported paired protons is always stable. by the grant No. 500.03401 of the PNCB-MES, Cuba, Under certain conditions of the magnetic field strength and the grant of the Office of External Activities of and particle densities, the collapsed gases could over- the Abdus Salam International Centre for Theoretical come gravity and trigger the expulsion of matter out of Physics (ICTP) through NT-09. A.P.M. expresses the star. Moreover, the collapsed matter can be self- her gratitude to Fermilab and ICTP for their hospi- magnetized, producing a magnetic field strong enough tality. The work of R.G.F. was partially supported to keep the gases in a collapse configuration once they by Funda¸cão para a Ciência e a Tecnologia (FCT, leave the star. Since the difference between the paral- Portugal) through the projects CFTP-FCT Unit 777 lel and perpendicular pressures of the collapsed matter (UID/FIS/00777/2019), CERN/FIS-PAR/0004/2017 can reach up to three orders of magnitude, the EoS ob- and PTDC/FIS-PAR/29436/2017, which are partly tained in this framework is expected to describe a highly funded through POCTI (FEDER), COMPETE, QREN elongated structure such as a jet. and EU. 8

Appendix A: Equations of state of magnetized 2. Neutral fermions quantum gases

For a gas of neutral fermions (neutrons) the EoS read In this appendix, we present the equations of state of as [23] charged and neutral magnetized gases of fermions and bosons used in our calculations. These are obtained con- E = −P + µN, (A2a) sidering a uniform and constant magnetic ﬁeld in the z k direction, B = (0, 0,B). m4 X µf 3 (1 + ηb)(5ηb − 3)µf P = + k 2π2 12m 24m Hereafter, m is the particle mass, q its charge, κ the η=1,−1 magnetic moment of neutral particles, Bc is the critical (1 + ηb)3L ηbµ3s + − , (A2b) magnetic ﬁeld at which the magnetic energy of the parti- 24 6m3 cle becomes comparable to its rest mass, and b ≡ B/Bc. P⊥ = Pk − MB, (A2c)

m3κ X (1 − 2ηb) M = η f 2π2 6 η=1,−1 (1 + ηb)2(1 − ηb/2) µ3 + L − s , (A2d) 3 6m2 1. Charged fermions m3 X f 3 ηb(1 + ηb) ηbµ2 N = + f − s , (A2e) 2π2 3 2 2m2 η=1,−1 The EoS for a gas of charged fermions (electrons and protons) at low temperatures are [22]

where η denotes the spin states, Bc = m/κ and the functions f, L and s are deﬁned as

2 lmax m b X 2 µ + pF E = 2 gl µ pF + El ln , (A1a) p 2 2 4π El µ − ε(η) l=0 f = , (A3) m lmax p ! m2b µ + p 1 µ + µ2 − ε(η)2 X 2 F L = ln , (A4) Pk = 2 gl µ pF − El ln , (A1b) 4π El 1 + ηb m l=0 π m 4 2 lmax s = − arcsin (1 + ηb) , (A5) m b X µ + pF 2 µ P⊥ = 2 gl l ln , (A1c) 2π El l=0

2 lmax with ε(η) = m + η κ B. m X 2 µ + pF M = 2 gl µ pF − El ln , (A1d) 4π Bc El l=0

l m2b Xmax N = 2 gl pF , (A1e) 4π2 3. Charged scalar bosons l=0

In this section, we present the EoS for a gas of charged scalar bosons (paired protons),2 following the same pro- where E denotes the internal energy, Pk and P⊥ are the pressure components along and perpendicular to the cedure developed in Refs. [35] and [24]. We have checked magnetic field direction, respectively, M is the magneti- that our results are consistent with those obtained in the zation, and N is the particle number density. very low temperature limit [27]. For a charged scalar gas of bosons, the EoS read in the In Eqs. (A1), the integer numbers l are the Landau 2 2 levels, gl = 2 − δ0,l, lmax = I[(µ − m )/(2qB)] and I[z] denotes the integer part of z. The Fermi momentum is 2 2 1/2 pF = (µ − El ) , with the rest energy given as El = 2 To our knowledge, these equations have not been previously pre- 2 1/2 2 (2qBl + m ) and the quantity Bc = m /q. sented in the literature. 9 weak-field (WF) and strong-field (SF) regimes as 4. Neutral vector bosons

( εN − 3Ω/2, WF, E = (A6a) For a gas of neutral vector bosons (paired neutrons) in εN − Ω/2, SF, the limit of low temperatures the EoS read [24]

0 Pk = −Ω, (A6b) 3 ∂µ E = εN + Ω − Ω − N, (A10a) vac 2 st ∂β P⊥ = Pk − MB, (A6c) e Pk = −Ωst − Ωvac, (A10b) M = − N, (A6d) 2ε P⊥ = −Ωst − Ωvac − MB, (A10c)  3/2 βµ0  3ε Li3/2(e ) κ  , WF, M = √ N, (A10d)  (2πβ)3/2 1 − b N = (A6e) 2 1/2 βµ0 0  3m b ε Li1/2(e ) ε3/2Li (eµ β)  , SF. 3/2  3/2 1/2 N = Nc + √ , (A10e) (2π) β 2π πβ3/2(2 − b) √ 0 2 In Eqs. (A6), µ = µ − ε, ε = 2mp 1 + b, Bc = 2m /e 0 p with β =√ 1/T the inverse temperature, µ = µ − ε, and Lik(x) is the polylogarithmic function of order k. ε = mn 1 − b the rest energy, Bc = mn/(2κ), and The weak (T > mpb) and strong (T < mpb) ﬁeld regimes 3/2 Nc = N 1 − (T/Tc) the density of condensed par- are separated by the condition T = mpb. Furthermore, ticles. Moreover,

 3/2 β(µ−ε) 3ε Li (e ) 0  − 5/2 , WF, ε3/2Li (eβµ )  3/2 5/2 5/2  (2π) β Ωst = −√ (A11) Ω = (A7) 2π πβ5/2(2 − b) 2 1/2 β(µ−ε)  3m bε Li3/2(e )  − , SF,  (2π)3/2β3/2 is the statistical contribution to the thermodynamical potential, where 4  m 2 2 2 " 3/2# Ωvac = − b (66 − 5b ) − 3(6 − 2b − b )  ζ(3/2)T Tc 288π  ε − 1 − Θ(T − Tc), WF,  4π T µ = × (1 − b)2 log(1 − b) − 3(6 + 2b − b2) 4 2 2  m b T  ε 1 − , SF,  8π2N 2 × (1 + b)2 log(1 + b) (A12) (A8) and corresponds to the vacuum term, and 2π N 2/3 Tc = (A9) ε 3 ζ(3/2) " 3/2# 0 ζ(3/2)T Tc µ = − 1 − Θ(T − Tc),(A13) is the condensation temperature and ζ(x) is the Riemann 4π T zeta function. "√ #2/3 We conclude from Eq. (A6d) that the magnetization 1 2π π(2 − b)N T = . (A14) M is always negative for a gas of charged scalar bosons c ε ζ(3/2) and, therefore, a magnetized charged boson gas never collapses magnetically.

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