(25)

make Equation (24) dimensionless and show that a scattering length at least as large as the neutron reduced Compton length produces a last term at least times larger than the first term and the left side of the equation. Neglecting these gives

(26)

Evaluating Equation (19) and applying the semiclassical approximation, we obtain

(27)

Substituting Equation (22) into (27) gives

(28)

(29)

Evaluating Equation (17) and taking the mean gives

(30)

(31)

(32)

(33)

Dots indicate derivatives with respect to time. We will now show that this energy-momentum density behaves like a perfect fluid. The density of a perfect fluid isand its pressure is. Ruffini and Bonazzola showed that, as long as kg/m^{3}, the terms with behave like a perfect fluid and reduce to the form [16],

(34)

(35)

where kg/m^{3}. The maximum density of a stable OV star easily meets Ruffini and Bonazzola’s conditions (it is kg/m^{3}), so these reductions are valid.

We consider, the exact form of which we obtain shortly, to be the effective field that describes a large collection of neutrons. Substituting Equations (23), (34) and (35) into (30) through (33) gives

(36)

(37)

(38)

Substituting Equations (25) gives dimensionless forms showing that the term with is negligible because it is times the other terms. Neglecting this term and substituting Equation (26) gives

(39)

(40)

(41)

The only two diagonal energy-momentum tensor components we need are and since the star's pressure is now isotropic.

Substituting Equations (39) and (41) into (28) and (29) gives

(42)

(43)

3. Boundary Conditions and Condition of Chemical Equilibrium

Beyond the radial distance where, the condensate density becomes negligible. We will show that this is also the star’s radius,.

At the star’s radius, the interior solutions for and must match the vacuum Schwarzschild solution. In particular, in the limit as approaches from the left,

(44)

Oppenheimer and Volkoff showed that the enclosed mass is related to the metric by [1]

(45)

Substituting and using l’Hôpital’s rule gives

(46)

The condition for chemical equilibrium mathematically defines the effective neutron field; Equation (2) must hold for every point within the star.

The chemical potential of a degenerate ideal gas in flat spacetime is equal to its Fermi energy [17] and the presence of a gravitational field does not change this relationship [18]. The Fermi momentum and are related by

(47)

Combining this with the relativistic relationship, , gives

(48)

The chemical potential of the condensate is

(49)

Substituting Equations (20) with the index, (23), (26), (48), and (49) into (2) gives

(50)

The inverse hyperbolic cosine is not defined when its parameter is less than one. As increases, so will until at the star’s radius. This would give

(51)

Since, there must be a radial distance, , such that. As a result, type II fermicon stars share a feature with type I fermicon stars: each has a Bose-Einstein condensate halo. The difference is that type I fermicon stars have a purely neutron core while type II fermicon stars have a core that is a mixture of neutrons and a Bose-Einstein condensate. Since our model requires a mixture of the neutrons and the condensate, this is an indication of its limits. Nevertheless, we expect results which have negligible halos or halos that dominate the star to be valid.

4. Macroscopic Quantities

Solving Equation (45) for the enclosed mass gives

(52)

The density and pressure are

(53)

(54)

5. Solutions

We found numerical solutions for the differential equations of the metric components, Equations (42) and (43), by providing a value for at the center of the star and searching for a boson ground-state energy, , that satisfied the limiting condition, Equation (44), for at the outer border. We abandoned solutions with masses less than 0.1 M_{⊙} since neutrons decay into protons, electrons, and antineutrinos in these cases. Stellar radii are measured using the Schwarzschild radial coordinate for comparison with the event horizon the star would have had if it collapsed.

In both cases, there is a maximum mass and a range of masses where two or more radii exist. Oppenheimer and Volkoff showed that the equilibrium solution with a lower density is the stable one and the one with the higher density is unstable to perturbations [1].

For example, Figure 2 compares solutions for OV stars and for type II fermicon stars whose bosons have a repulsive scattering length of one femtometer and that are formed from “strongly-bound” () pairs () of neutrons. The critical masses are 0.710 M_{⊙} for OV and 0.428 M_{⊙} for fermicon stars. OV and fermicon stars with radii less than 9.11 km and 4.10 km, respectively, are unstable. Note the loop at the far left of the fermicon star plot, which indicates that three equilibrium states exist between 0.241 M_{⊙} and 0.283 M_{⊙}. For each mass in this range, only the state with the largest radius is stable.

The speed of sound in a medium is

Figure 2. Q = 0.9, s = 2, a = 1 fm solutions compared with OV solutions.

(55)

We can find this speed, since Equations (53) and (54) give expressions for the density and pressure. The maximum speed of sound for a type II fermicon star with a scattering length of a femtometer is. The solutions obey causality.

Figure 3 shows another way to compare the two sets of solutions in Figure 2 by plotting mass-vs-core density. OV and fermicon stars with densities greater than 4.01 kg/μm^{3} and 18.3 kg/μm^{3}, respectively, are unstable. To see why the fermicon stars have higher densities than OV stars, consider the asymptotic behavior of fermicon density and pressure as the scattering length gets very small or very large:

(56)

(57)

(58)

(59)

We found that the boson ground-state energy stays close to the rest energy of a boson and that does not vary much as initial parameters are changed. All else being equal, the density and pressure of fermicon stars with small scattering lengths get larger as the scattering length gets smaller. Recall that the motivation for exploring fermicon stars was that we sought equations of

Figure 3. Q = 0.9, s = 2, a = 1 fm mass-vs.-core density.

state with lower pressure to reduce the gravitational field and increase stability. We therefore would expect lowscattering length fermicon stars to have lower critical masses than OV stars.

In contrast, fermicon stars with large scattering lengths have expressions for density and pressure identical to those of OV stars. However, , which represents the presence of neutrons, is smaller in a fermicon star than in an OV star because some of the neutrons have condensed into bosons. We therefore would expect critical mass to increase as scattering length increases and sufficientlylarge scattering lengths to produce more massive stars than OV stars.

These expectations are confirmed in Figures 4 and 5, which show mass-vs-radius graphs for fermicon stars with, , and scattering lengths that vary from a femtometer to 20 picometers. Figure 4 includes the plot for OV stars. Figure 5 includes the solutions for type I fermicon stars with, , and

Figure 4. Type II fermicon stars with femptometer-scale scattering. All of the fermicon solutions have Q = 0.9 and s = 2.

Figure 5. Type II fermicon stars with picometer-scale scattering. All solutions have Q = 0.9 and s = 2.

pm. It only includes the stable solutions for the pm and pm stars because the time to find the unstable solutions was prohibitive.

Although the pm type II fermicon stars have masses higher than OV stars, they are less massive than type I stars. Type I fermicon stars have higher critical masses because they have constant density. The critical mass for a star with a constant density of is [19]

(60)

For kg/μm^{3}, this is a maximum mass of 1.8 M_{⊙}, which is 2.6 times larger than an OV star’s critical mass of 0.71 M_{⊙} at the same core density.

A type II fermicon star with, , and pm has a critical mass of 13.6 M_{⊙}. Adding neutron self-interaction and rotation to the OV star model quadruples its critical mass. If the same increase applies to type II fermicon stars, then pairing of neutrons into a Bose-Einstein condensate within neutron stars could raise their critical masses to about 50 M_{⊙}, preventing collapse to stellar black holes. The maximum speed of sound for the stable solutions is; as the solutions become less dense, the speed of sound decreases.

Returning to type II fermicon stars with one-femtometer scattering lengths, Figure 6 shows the relationship between the star’s radius and its core density. Type II fermicon stars have cores consisting of a mixture of neutrons and condensate surrounded by a halo of condensate. The thickness of this halo diminishes as the core density rises. The fermicon star with a critical mass has a 4.10 km radius, 334 meters of which is the condensate halo. The Schwarzschild radius of a black hole with this star’s mass is 1.26 km.

Figure 7 shows the relationship between the core density and radius for type II fermicon stars with a scattering length of a picometer. These stars have larger radii consistent with their lower densities. Their radii are about 200 times larger than their Schwarzschild radii.

One feature that is difficult to see in Figure 7 is that the low-density stars are purely condensate and that the

Figure 6. Q = 0.9, s = 2, a = 1 fm radius-vs.-core density.

stable stars that have neutrons are still dominated in volume by the condensate. Figure 8 shows radii for densities up to 0.00018 kg/μm^{3}. Below 0.000572 kg/μm^{3}, essentially all of the neutrons group into the condensate (more accurately, from the precision of our solutions, these stars have a core with a radius less than a millimeter). A critical-mass picometer-scaled fermicon star has a core with a radius of nine kilometers and a halo 194 kilometers thick.

Type II fermicon stars with scattering lengths comparable to their reduced Compton lengths do not have the opportunity to condense into esentially-pure condensate stars since such stars have much lower masses (the neutrons in a star below 0.1 M_{⊙} will decay).

If the neutrons pair into bosons with a stronger binding (e.g.,), the boson rest energy will be smaller. As Figures 9 and 10 show, these stars are slightly more massive at lower densities than stars with weaker binding, condense completely into condensate stars at low densities, and are slightly larger than stars with weaker binding.

Neutrons may also find it energetically favorable to associate in groups of four (or higher even numbers) into bosons. Figures 11 through 12 show features of type II fermicon stars formed when four neutrons strongly group into a single boson (and) and these bosons have a repulsive interaction with a scattering length of one femtometer. These stars have lower critical masses at higher densities (0.153 M_{⊙} at 151 kg/μm^{3}) than their

Figure 7. Q = 0.9, s = 2, a = 1 pm radius-vs.-core density.

Figure 8. Q = 0.9, s = 2, a = 1 pm radii at low densities.

Figure 9. Q = 0.8, s = 2, a = 1 fm mass-vs.-core density.

Figure 10. Q = 0.8, s = 2, a = 1 fm radius-vs.-core density.

Figure 11. Q = 0.9, s = 4, a = 1 fm mass-vs.-core density.

Figure 12. Q = 0.9, s = 4, a = 1 fm radius-vs.-core density.

counterparts (0.428 M_{⊙} at 18.3 kg/μm^{3}), and are correspondingly smaller with comparatively thinner condensate halos.

6. Conclusion

If strong pairwise grouping of neutrons into a BoseEinstein condensate of spin-0 bosons with scattering lengths of at least twenty picometers takes place in nonrotating massive neutron stars, these stars will become predominately condensate by volume in chemical equilibrium with their remaining neutrons. They will have a minimum radius roughly half that of the moon and have masses greater than 13 M_{⊙}. If neutron interaction and rotation have the same effect on these stars that they have on the stars modeled by Oppenheimer and Volkoff, then their critical masses would be greater than 50 M_{⊙}, avoiding collapse to black holes.

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