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# EoS in Neutron stars 2- Various models 1

Posted Jul29-08 at 02:38 AM by stevebd1
Updated Aug22-08 at 07:20 PM by stevebd1

(continued from part 1)

Recognised models and alternative states of matter for neutron stars

Part 1 looked at the EoS of a neutron star composed of pure neutron matter, below are models which allow for various other states and phases of matter that might be found in a neutron star with various nucleon-nucleon interactions that would contribute to the overall energy density, (for instance, proton-neutron interaction energy is higher than that of neutron-neutron interaction which would add to the overall energy density, adjusting the EoS, it also makes the star more stable at smaller radii). Key equation of states (pressures versus energy density) are expressed as above.
http://arxiv.org/PS_cache/arxiv/pdf/...705.2708v2.pdf page 3 & 5

It's worth noting that a physically correct EoS can neither be ultrabaric i.e., the pressure cannot be greater than the density (EoS cannot be > 1) or superluminal i.e., the change in pressure cannot be greater than the change in density, otherwise the matter would be providing an increase in pressure before it had the required energy density to provide it. This is described as superluminal communication where information travels faster than light which contradicts relativity. The pure neutron matter model doesn't seem to comply with this after pressure = 450 MeV/fm^3 where the pressure begins to increase more than the energy density. This appears to be one reason why the pure neutron matter model isn't the ideal model for neutron stars.

Neutron, proton star ('traditional' Neutron star)
At approx. 0.28 x 10^15 g/cm^3, nuclei begin to break down into a liquid of nucleons, this could range from just 10% protons, 10% electrons and 80% neutrons up to equal amounts of protons, neutrons and electrons (though the electron only weighs about one 2000th of a proton or neutron and contributes very little mass, it's required to keep the charge of the liquid neutral).

Schroedinger based n, p model
(2.2 sol mass, 10 km radius, 2 - 2.2 billion tonnes/cm^3 core density)

Pe / ρe = Pm / ρm = EoS --> Pf (see key below)

5 / 200 = 0.0089 / 0.356 = 1 / 40 --> 0.080 (about 2 km down)

100 / 600 = 0.178 / 1.068 = 1 / 6 --> 1.602 (about 5 km down)

700* / 1200 = 1.246 / 2.136 = 1 /1.71 --> 11.214 (within the core)
*extrapolated from the graph

Pe / ρe = pressure / density as energy (both in MeV/fm^3), Pm / ρm = pressure / density as mass (both in 10^9 tonnes/cm^3), EoS = Equation of state, Pf = pressure as force (10^34 N/m^2). (100 Mev/fm^3 = 0.178x10^9 tonnes/cm^3. Pressure as kg mass x c^2 = pressure in N)

Neutron, proton, hyperon star (Hyperon star)
At about 0.8x10^15 g/cm^3, understanding of the neutron/proton liquid begins to break down and hyperons can be introduced as an alternative. Hyperons are baryons (protons and neutrons) which contain strange quarks, a strange quark is a second generation down quark with mass of about 80 - 130 MeV compared to the up & down quark which are 1.5 - 4 MeV and 4 - 8 MeV respectively. This can increase the mass of the baryon from approx. 939 MeV to between 1115 - 1321 MeV. There are 2 models-

DD-RBHF n, p, H model
(2 sol mass, 12 km radius, 2 - 2.2 billion tonnes/cm^3 core density)

Pe / ρe = Pm / ρm = EoS --> Pf (see key below)

5 / 200 = 0.0089 / 0.356 = 1 / 40 --> 0.080 (about 2 km down)

110 / 600 = 0.196 / 1.068 = 1 / 5.45 --> 1.764 (about 6 km down)

500* / 1200 = 0.890 / 2.136 = 1 / 2.40 --> 8.010 (within the core)
*extrapolated from the graph

RMF n, p, H model
(2 sol mass, 12 km radius, 2 - 2.2 billion tonnes/cm^3 core density)

Pe / ρe = Pm / ρm = EoS --> Pf (see key below)

10 / 200 = 0.0178 / 0.356 = 1 / 20 --> 0.160 (about 2 km down)

110 / 600 = 0.196 / 1.068 = 1 / 5.45 --> 1.746 (about 6 km down)

370 / 1200 = 0.659 / 2.136 = 1 / 3.25 --> 5.927 (within the core)

Pe / ρe = pressure / density as energy (both in MeV/fm^3), Pm / ρm = pressure / density as mass (both in 10^9 tonnes/cm^3), EoS = Equation of state, Pf = pressure as force (10^34 N/m^2). (100 Mev/fm^3 = 0.178x10^9 tonnes/cm^3. Pressure as kg mass x c^2 = pressure in N)

Hyperons can make up approx. 20% of the neutron stars mass and would appear between 0.8x10^9 tonnes/cm^3 and 2x10^9 tonnes/cm^3 in the following form-
http://arxiv.org/PS_cache/arxiv/pdf/...705.2708v2.pdf page 7

hyperon type - quark combination (u- up, d- down, s- strange) - charge - mass in MeV --> approx. density level (10^9 tonnes/cm^3) (approx. depth in km based on a 12 km star)
lambda_zero (Λ0) - uds 0e - 1115.7 --> 0.8 (approx. 4 km down)
sigma_minus (Σ-) - dds -1e - 1197.4 --> 0.9 (5)
sigma_plus (Σ+) - uus +1e - 1189.4 --> 1.1 (6)
sigma_zero (Σ0) - uds 0e - 1192.5 --> 1.4 (8)
xi_zero (Ξ0) - uss 0e - 1315 --> 1.6 (9)
xi_minus (Ξ-) - dss -1e - 1321 --> 1.8 (10)

As hadron regions prefer to maintain an overall effect of neutrality, delta_plus-plus baryons might also be formed (amongst other delta baryons such as delta_plus (uud) and delta_zero (udd) which are protons and neutrons with the added mass of a pion**). The delta_plus-plus is a baryon with 3 up quarks (uuu) providing a charge of +2e (up quark = charge of +2/3e, therefore 3 x +2/3e = +2e).

delta type - quark combination (u- up, d- down) - charge - combination (or decay to)
All delta baryons have the same mass (1232 MeV)
delta_plus-plus (Δ++) - uuu +2e - proton + pion_plus
delta_plus (Δ+) - uud +1e - neutron + pion_plus or proton + pion_zero
delta_zero (Δ0) - udd 0e - neutron + pion_zero or proton + pion_minus
delta_minus (Δ-) - udd - 1e - neutron + pion_minus

**The pion (or pi-meson) mediates the strong force between protons and neutrons changing a proton into a neutron and a neutron into a proton due to the quark, antiquark pair in the meson. In the ground state of atomic nuclei, the proton and neutron constantly change state due to the transfer of pions but the overall number of protons/neutrons remains the same within the nuclei. As the density in neutron stars increases, the pion as a mediator of the strong force becomes redundant (as the nucleons are held in place by gravity) and may take on a different form such as binding to become kaons or a pi-meson condensate (see K-meson condensate model below).

pion type - charge - mass (MeV) - quark combination
pion_plus ($\pi^+$) +1e - 139.57 - 1 up, 1 antidown quark
pion_zero ($\pi^0$) 0e - 134.97 - 1 down, 1 antidown quark or 1 up, 1 antiup quark
pion_minus ($\pi^-$) -1e - 139.57 - 1 down, 1 antiup quark (antiparticle of the pion_plus)

Another strange hyperon is the omega_minus which is made up of 3 strange quarks, weighing in at 1672 MeV. There appears to be little evidence that this hyperon would be formed in neutron stars. It is also unlikely that 'charmed' hyperons would form in a neutron star due to the fact that the second generation up quark, the 'charmed' quark, weighs in at a hefty 1150 -1350 MeV which is larger than the proton & neutron plus some of the hyperons and requires a density of approx. 10^11 tonnes/cm^3 to be formed.

List of baryons- http://en.wikipedia.org/wiki/List_of_baryons

Hyperon Star Cross-section

(continue to part 3)
Posted in Neutron stars