Effects of finite temperature on the magnetized equation of state in neutron stars composed of a Bose-Einstein condensate

We study the role of temperature and magnetic field on the equation of state and macroscopic properties of Bose-Einstein condensate stars. These compact objects are composed of a condensed gas of interacting neutral vector bosons coupled to a uniform and constant magnetic field. We found that the main consequence of a finite temperature in the magnetized equations of state is to increase the inner pressure of the star. As a consequence, magnetized hot Bose-Einstein condensate stars are larger and heavier than their zero-temperature counterparts. However, the maximum masses obtained by the model remain almost unchanged, and the magnetic deformation of the star increases with the temperature. Besides, increasing the temperature reduces the number of stable stars, an effect that the magnetic field enhances. The implications of our results for the star's evolution, compactness, redshift, and mass quadrupolar moment are also analyzed.

Figure 1

The phase diagram of Bose-Einstein condensation in the density-temperature plane for $B=0$ and $B={10}^{17}$ G. The BEC exists in the shadowed regions.

Figure 2

Fraction of particles, antiparticles, and particles in the ground state as functions of $\rho$ for several values of the temperature and $B=0$. The vertical lines correspond to the density at which the condensate appears.

Figure 3

The EoS of hot BECS with $B=0$ (a) and $B={10}^{17}$ G (b). In (b), the dashed and solid lines correspond to the parallel and perpendicular pressures respectively, and the horizontal line signals the value of the Maxwell pressure $B^2/8\pi$ for $B={10}^{17}$ G.

Figure 4

The EoS of hot BECS with $B={10}^{17}$ G and no Maxwell contribution $\pm B^2/8\pi$ in the energy and pressures. The dashed and solid lines correspond to the parallel and perpendicular pressures respectively.

Figure 5

The mass vs the central mass density of nonmagnetized BECS with different inner temperatures and $B=0$.

Figure 6

Mass-radius diagram (a) and compactness (b) of BECS at different temperatures and $B=0$; in this case, since $B=0$ the star is spherical and the polar and equatorial radii coincide ($Z=R$). The black dots mark the position on each of the curves of a star with baryon mass of $1.7M_{\odot }$ (composed of around ${10}^{57}$ bosons); they have been joined with a dotted line for a better visualization of the star evolution as the temperature decreases (see the text).

Figure 7

Gravitational redshift (a) and moment of inertia as functions of the mass of the star (b) for several values of the temperature and $B=0$. The black dots mark the position on each curve of the star with baryon mass of $1.7M_{\odot }$; they have been joined with a dotted line for a better visualization of the star evolution as the temperature decreases.

Figure 8

Mass–central-density (a) and mass-mean radius $R_m$ (b) curves for BECS at several temperatures and magnetic field $B={10}^{17}$ G (dashed curves) and $B=0$ (solid curves). The black dots in (b) mark the position on each curve of the star with baryon mass of $1.7M_{\odot }$; they have been joined with a dotted line for a better visualization of the star evolution as the temperature decreases.

Figure 9

The transversal section of the stars with the maximum mass in the polar-equatorial radii ($Z\text{−}R$) plane for several values of $t$ and $B$. Due to their high inner densities, for these stars the magnetic deformation is negligible. The external circle in the plots is not a star but the angular scale.

Figure 10

Compactness (a) and gravitational redshift (b) for BECS at several temperatures and magnetic field $B={10}^{17}$ G (dashed curves) and $B=0$ (solid curves). The black and gray dots mark the position on each curve of the star with baryon mass of $1.7M_{\odot }$; they have been joined with dotted lines for a better visualization of the star evolution as the temperature decreases.

Figure 11

Equatorial (solid) and polar (dashed) radii as a function of the central density for $B={10}^{17}$ G and several values of temperature (a). The mass quadrupolar moment as a function of the star mass (b). The parameter $\gamma$ as a function of the star mass (c). In (a) the $B=0$, $T=0$ is included for reference. The black dots in (b) mark the position on each curve of the star with baryon mass of $1.7M_{\odot }$; they have been joined with a dotted line for a better visualization of the star evolution as the temperature decreases.

Figure 12

BECS with $t=0.001$ and $B=0$, $B={10}^{17}$ G and inner magnetic field intensity dependent on the star particle density $B\left(\rho \right)$: EoS, the solid curves, correspond to the parallel pressure while the dashed ones correspond to the perpendicular pressure (a); mass-radius curves (b); mass–central-density curves (c). Inner magnetic field of the star versus its internal equatorial radius $r$ (in units of the equatorial radius of the star $R$) for stars with $B\left(\rho \right)$, central density $\rho ={10}^{15}\mathrm{g}/{\mathrm{cm}}^3$, and several temperatures (d).