On the Existence of Properties of Quark Stars

Title

On the Existence and Properties of Quark Stars

Author

Michael Kent Ph.D. Midshipman USS Copernicus

Abstract

With refined measurements on the compressibility of neutron mater, and growing observational data on black hold masses, a non-negligible gap has opened up between the two regimes previously believed to be joined. A new object must therefore exist to fill this gap and allow for the continuous mass distribution known to exist for stellar masses. These objects must also be supported by a higher order degeneracy neutrons in order to generate the required pressure to support itself against the enormous self-gravitation of these objects. This paper will present the findings for such an object supported by the degenerate free quarks within the stellar structure.  

Introduction

Over the last several years, researchers have worked to refine their calculations for the equation of state and degeneracy pressure of dense neutron matter. These new calculations when applied to the hydrostatic equilibrium equations for neutron stars result in a lowering of the upper bound mass to 2.6 Mʘ (5.2 x 1030 kg). This mass limit is supported by a lack of observation of any such objects above this mass limit, with the heaviest ever neutron star observed at an estimated 2.54 Mʘ.

On the other end of the spectrum, methods for detecting black holes, and measuring their mass have continued to improve. Yet no black hole has yet been detected with a mass less than 3.6 Mʘ. While it is not impossible for black holes to form below this mass, or even at planetary mass scales, the fact that none has yet been found indicates there may be some new phenomenon that is preventing such sized black holes from forming.

Thus a gap between the maximum mass of a neutron star and the minimum mass of a black hole as appeared in the range from 2.6 Mʘ -3.6 Mʘ. Quark stars have long been theorized as a possible outcome of large star collapse; however, there has never been any experimental evidence of such objects, nor any theoretical basis for them, before now. With this gap appearing it seems the conditions may be right to return to an old idea and apply new science to determine if such objects would be stable and what kind of properties they would have.

We will begin the analysis by examining what kinds of properties must be met for such an object to exist stably, such as its composition, density, pressure, and mass. Further properties of these stars will be examined to attempt to find a way to detect them as well as explain why they may not have yet been detected.

Properties for Existence

Composition

A neutron is composed of one up quark (mass 2.15 MeV) two down quarks (mass 4.8 MeV), and have a total mass of 939.6 MeV. Thus it is easily seen that liberated quarks will be relativistic. The speeds of these quarks after liberation will be enormous approaching 0.999976c. Because of the ratios between these high energies and low masses, up-down quark stars are not stable. Such stars will have a degeneracy pressure greater than the gravitational pull of the star and will push itself outward, allowing the quarks to escape, recombine into protons and neutrons and speed away from the star at relativistic speeds until nothing is left.

In order to bring the mass ratio up high enough, a certain percentage of the star must be converted into heaver mass quarks such as strange and charm. The critical mass regime falls between 102 and 305 MeV/c2. The process used to generate these neutrinos is shown in Figure 1. The up-down quarks will scatter generating electron, anti-electron neutrino pairs and positron, electron neutrino pairs. These will later re-scatter generating a strange and charm quark. Thus we know there will be equal parts strange and charm, and since the neutron is composed of two down quarks and one up quarks we know that there will be 2 down quarks for every up quark. Given these ratios we find that the star must become 6.89% to 21.626% heavy quarks. Less than this and the star will evaporate away, more and the star will become too dense and collapse into a black hole.

Calculations using theoretical models for the densities likely to be found within quark stars along with the decay rates of these heavier quarks indicate that quark stars will be able to contain anywhere from 0 to 10% strangeness, indicating that quark star should be able to form; however, even when the masses are right it's not always guaranteed to happen.

Pressure and Density

With the heavy quark ratio solved to give us stable quark star solutions, the objective now is to solve for the density and pressure as a function of radius within the quark star. Finding these functions will allow us to solve the equation for hydrostatic equilibrium and demonstrate a stable solution.

The rough form of the density function is somewhat simple to name. Virtually all non-solid celestial objects behave polytropic equations for the density and pressure, so we start with the simple form of the density equation:

ρ = ρo (1- r/R)n

Where ρo is the central density, R, is the radius of the star, and, n, is the polytropic index of quark-degenerate matter. It remains a problem to find the central density and polytropic index of the star. We know that the index must be less than one-half, the minimum for neutron stars, but will probably be above one-quarter where the densities will start to get too high not to collapse into a black hole.

The Pressure can similarly be developed relatively quickly. A detailed analysis of the Fermi-energy of the free-quarks is all that is required. It is important to note that these quarks will have 24-fold degeneracy with 4 distinguishable quark flavors, 3 quark colors, and 2 quark spins. We must also include additional terms from interactions with virtual W bosons, Z bosons, and gluons that will be exchanged freely and often within the quark star. The results of the calculation below show that the Pressure is proportional to the density to the 10/3.

P(r)= (ℏ c)/(9 m_q ) ∛((π/4)^2 ) ρ^(10⁄3)

Since the pressure, P ~ ρ n+1 / n we can see that n = 3/7. Solving for our polytropic index. This is clearly within the range of expected values, and is only slightly lower than for neutron stars; however the lower mass of the quarks and the exponential nature of the index will provide much higher degeneracy pressure, indicating that this solutions should be stable for a much higher mass range than for neutron degeneracy pressure.

To get the value of the central density, is a simple matter of integrating the density over the volume of the star and setting that equal to the mass. Doing this we find:

ρ_o= (510 M)/(343 π R^3 )

This gives us a provisional solution for ρo in terms of both the mass and radius. We will want to eliminate at least one of these parameters, but for now this should be useful in the next step.

Size and Mass

We can start by applying our density and pressure profiles to the equation for hydrostatic equilibrium. This equation must be satisfied for the star to be stable, so we can use this with the equations we have to solve for the radius as a function of the mass.

The results of the calculation are in line with what we would expect. The radius will decrease as mass increases, and the range of radii are smaller than those of neutron stars.

R(M)= (4 (ℏ c)^4)/(3(G m_q )^3 M^0.25 )

Plugging this back into our expression for ρo we find:

ρ_o=(2295(G m_q )^9 M^(7/4))/(10976 π (ℏc)^12 )

If we plot the radius of our Quark stars against the Schwartzchild Radius of equal massed objects we find that there is a max mass for Quark stars that then flow directly into Black Hole masses. As shown in Figure 3 this mass transition point is at 3.459 Mʘ.

At this point we have accomplished the primary goal. We have found the stable mass range of Quark Stars: 2.6 Mʘ - 3.459Mʘ. We have the radius of such stars as a function of the mass with decreases smoothly until it reaches the schwartzchild radius, giving us a grounded theoretical lower limit for the mass of black holes. We have the density and pressure profile outlined in Figure 2. At this point we now look at some of the other features of the Quark Star.

Quark Star Detection

Luminosity

Like Neutron stars, Quark stars are incredibly cool. When Quark stars form, the temperature of the star drops as thermal energy is used to break the strong nuclear bonds between the quarks. This drops the temperature of Quark stars down around 4000K.

Quark Star spectrum peaks in the NIR at a wavelength of 1.2 μm. This is below the visible threshold, but with the tail of the emission profile in the low visible. Quark Stars would have a soft red color.

The diminished size and low temperature of the Quark Stars reduce their luminosity however to around 7 x 10-11Lʘ. Quark Stars are so faint that they fall below the threshold for vision only 33.31AU.

There is one way to detect Quark stars from the background. The up and down quark scattering outlined in Figure 1 generate electron-positron pairs. In the outer layers of the star, the neutrinos escape, and the electrons and positrons interact with the neutrinos generated in the core giving rise to the generation of strange and charm quarks. In the core of the star however, the electron-positron pairs are left alone long enough to meet and self-annihilate. This generates a strong 0.511 MeV signal from these stars. Detection of this signal would be difficult as it is even fainter than the star's normal luminosity; however the high energy of the photons allows it to travel without extinction over large distances, and a positive signal would indicate a high probability of a Quark Star.

This 0.511 MeV signal will be gravitationally redshifted by the high gravitation of the Quark Star. The expressions for the magnitude of the shift can get very complex given the rate of rotation, the charge pockets that develop and the extreme densities with the surface of the object existing very near to the Schwarzschild radius. In the equatorial plane (θ = π/2) for a mid-range mass, the redshift comes out to around z = 6.135, corresponding to a detection wavelength of around 72 keV or a wavelength of around 20 pm in the hard X-rays. This is still a high enough energy to punch through interstellar gas and dust with little problem; however the signal would be very weak and difficult to detect.

Spin and Jets

Like Neutron Stars, Quark stars have the potential to rotate extremely rapidly. While some fraction are able to pass their angular momentum during the supernova to the exploding gas, most retain the majority of their angular momentum, resulting in extremely high rotational velocities. Similarly, many of these quark stars will retain the magnetic fields from the original stars. The combination of this high magnetic field and fast rate of rotation is the origin of pulsars.

The problem becomes the lack of observation of any such object. The answer to this is the fact that quarks carry net electric charge, while neutrons do not. The rapid rotation of the quarks magnetic field generates an electric field. For pulsars this electric field accelerates protons and electrons from the surface. In quarks in the quark stars however, are capable of moving with the electric field, opposing it, and weakening it. The strong forces binding the quarks together also work to prevent the remaining electric field from ripping quarks away from the surface of the star and into the beam.

Pulsars are the easiest way to detect neutron stars, thus with Quark stars unable to form the same radiation jets, it begins to explain why Quark Stars have never been directly observed.

Exotic Matter Production

Penta-quark Formation

The densities inside the core of a degenerate quark star are tremendous, nearly two orders of magnitude greater than the densities of an atomic nucleus. With these densities, the ultra-relativistic speeds involved, and the pressures at the core, it is possible for the generation of penta-quarks in the cores of these objects. With up, down, strange, and charm quarks as well as their anti-particle counterparts in trace amounts there are 140 different combinations of penta-quark particle that could be created in the cores.

Penta-quarks are highly unstable particles under normal conditions, and generally have tremendous binding energies. Penta-quark decays can be extremely energetic, containing a higher energy density than even antimatter. If one can stabilize the penta-quarks long enough to control the decay rate as well as harness the energy from the resulting reactants, Penta-quark energy could provide a new generation in energy supply rendering scarcity an obsolete idea anywhere the technology could be constructed.

Neutronium

In addition to the penta-quarks in the core, the surface layers of a Quark Star contain a high density of neutronium. As shown in figure three there is a sharp rise in the density of the stellar material at the surface. This sharp rise layers from free neutron crust to a high-density neutronium sub crust before giving away to the free quarks that form the bulk of the star.

It is probable that all species that have harnessed the use of neutronium have used quark stars as their source. While the cores of these compact objects are difficult to reach due to the high densities, and interference patterns from the matter above it. The neutronium sub-crust on quark stars is just a few dozen meters under the surface, and is much more accessible.

Conclusion

The gap between the upper limit on neutron star masses and the minimum observed black hole has been filled by the existence of quark degenerate matter. This form of degenerate matter has been shown to be stable in the required mass range. The inherent properties of these objects explain why they haven't yet been discovered. These include the rare circumstances when the quark-degenerate matter will compact enough to remain stable. Approximately 60% of the time, these objects fail to attain a high enough strangeness causing rapid evaporation of the object. On the occasions that these objects do form they are incredibly dim, and peak in wavelengths that are easily extinguished by the interstellar gas and dust.

Quark stars might be identifiable by their relatively strong 0.511 MeV signal from electron-positron annihilation. This signal is extremely high energy, and can penetrate gas and cloud layers with little affect to the photons. Far from the object this will appear as a faint 20 pm signal in the X-ray band due to the extreme gravitational redshift around these objects.

Detection of these objects would have tremendous scientific benefit for the purposes of studying large quantum systems, high density physics, quark-gluon plasmas, and the effects of extreme warping of space-time similar to that found around black holes. Additionally there is the possibility of the next generation of power source.

Theoretically there is still a lot of work to be done in determining how these objects form from the beginnings of the core-collapse supernova; and how their stability requirements come into play. The structure of these stars and electrodynamics of the dense quantum state are also areas of potential future research.