Go to previous page Go up Go to next page

4.3 Deep crustal heating

A neutron star crust that is not in full thermodynamic equilibrium constitutes a reservoir of energy, which can then be released during the star’s evolution. The formation and structure of nonequilibrium neutron star crusts has been considered by many authors [41050362186Jump To The Next Citation Point187Jump To The Next Citation Point188Jump To The Next Citation Point178Jump To The Next Citation Point]. Such a crust can be produced by accretion onto a neutron star in compact LMXB, where the original crust built of a catalyzed matter (see Section 3) is replaced by a crust with a composition strongly deviating from that of nuclear equilibrium. However, building up the accreted crust takes time. The outer crust (Section 3.1), containing ∼ 10− 5M ⊙, is replaced by the accreted crust in (104∕M˙ ) −9 y. To replace the whole crust of mass ∼ 10−2 M ⊙ by accreted matter requires 7 ˙ (10 ∕M −9) y. After that time has passed, the entire “old crust” is pushed down through the crust-core interface, and is molten into the liquid core. The time (107∕M˙− 9) y may seem huge. However, LMXBs can live for ∼ 109 y, so that a fully accreted crust on a neutron star is a realistic possibility.

Heating due to nonequilibrium nuclear processes in the outer and inner crust of an accreting neutron star (deep crustal heating) was calculated, using different scenarios and models [186187188Jump To The Next Citation Point]. The effect of crustal heating on the thermal structure of the interior of an accreting neutron star can be seen in Figures 25View Image and 26View Image. In what follows, we will describe the most recent calculations of crustal heating by Haensel & Zdunik [188Jump To The Next Citation Point]. In spite of the model’s simplicity (one-component plasma, T = 0 approximation), the heating in the accreted outer crust obtained by Haensel & Zdunik [188Jump To The Next Citation Point] agrees nicely with extensive calculations carried out by Gupta et al. [178Jump To The Next Citation Point]. The latter authors considered a multicomponent plasma, a reaction network of many nuclides, and included the contribution from the nuclear excited states. They found that electron captures in the outer crust proceed mostly via the excited states of the daughter nuclei, which then de-excite, the excitation energy heating the matter; this strongly reduces neutrino losses, accompanying nonequilibrium electron captures. The total deep crustal heating obtained by Haensel & Zdunik [188Jump To The Next Citation Point] is equal to Qtot = 1.5 and 1.9 MeV per accreted nucleon for Ai = 106 and Ai = 56, respectively.

View Image

Figure 22: Heat sources accompanying accretion, in the outer (upper panel) and inner (lower panel) accreted crust. Vertical lines, positioned at the bottom of every reaction shell, represent the heat per accreted nucleon. Based on Haensel & Zdunik [188Jump To The Next Citation Point]. Figure made by J.L. Zdunik.

In Figure 22View Image, we show the heat deposited in the matter, per accreted nucleon, in the thin shells where nonequilibrium nuclear processes occur. Actually, reactions proceed at a constant pressure, and there is a density jump within a thin “reaction shell”. The vertical lines, whose height gives the heat deposited in matter, are drawn at the density of the bottom of the reaction shell. The number of heat sources and the heating power of a single source depend on the assumed Ai. In the case of Ai = 56 the number of sources is smaller, and their heat-per-nucleon values Qj are larger, than for Ai = 106.

An important quantity is the integrated heat deposited in the crust in the outer layer with bottom density ρ. It is given by

∑ Q (α)(ρ) = Q (α), (74 ) j j(ρj<ρ)
where (α) labels the crustal heating model (specific Ai, Zi, etc.). The quantity Q (α)(ρ ) for two models of compressional evolution is plotted in Figure 23View Image. The second model illustrates the effect of switching off pycnonuclear reactions. This was done by artificially blocking pycnonuclear fusion until the nuclear charge went down to Zmin = 4, which occurred at 13.25 −3 ρpyc = 10 g cm. And yet, (2) Q for 13 − 3 ρ > 10 g cm is very similar to that obtained in the first scenario, which was the most advantageous, as far as crust heating was concerned. Heating by pycnonuclear fusion at ρ > 1013 g cm −3 is insignificant. Heat from pycnonuclear fusions at ρ ∼ 1012 g cm −3 is to a large extent replaced by an additional heat release associated with electron captures and neutron emissions within the density range 1012 – 1013 g cm–3. The values of Q (1) and Q (2) saturate above 1013.6 g cm–3, where 80% of nucleons are in neutron gas phase. All in all, for two scenarios with Ai = 56, the total deep crustal heat release is 1.8 – 1.9 MeV/nucleon. For Ai = 106, these numbers are lowered by about 0.5 MeV/nucleon.
View Image

Figure 23: Integrated heat released in the crust, Q (ρ) (per one accreted nucleon) versus ρ, assuming initial ashes of pure 56Fe. Solid line: HZ* model of Haensel & Zdunik [188Jump To The Next Citation Point], with Ai = 56 . Dash-dotted line: with pycnonuclear fusion blocked until Z = Zmin = 4. Based on Haensel & Zdunik [188Jump To The Next Citation Point]. Figure made by J.L. Zdunik.

The quite remarkable weak dependence of the total heat release in the crust, Qtot, on the nuclear history of an element of matter undergoing compression from ∼ 108 g cm–3 to ∼ 1013.6 g cm–3 deserves an explanation [188Jump To The Next Citation Point]. One has to study the most relevant thermodynamic quantity, the Gibbs free energy per nucleon (baryon chemical potential). Its minimum determines the state of thermodynamic equilibrium. Moreover, its drop at reaction surface P = Pj yields the total energy release Qj per one nucleon [341]. In the T = 0 approximation, we have μb(P ) = [ℰ(P ) + P]∕nb (P ) = enthalpy per nucleon. Minimizing μb(P ), at a fixed P, with respect to the independent thermodynamic variables (A,Z, mean free neutron density ¯nn, mean baryon density nb, size of the Wigner–Seitz cell, etc.), under the constraint of electro-neutrality, ¯n = ¯n p e, we get the ground state of the crust at a given P. This “cold catalyzed matter” (Section 3) corresponds to (0) μ b (P ). All other (α) μb (P ) curves displaying discontinuous drops due to nonequilibrium reactions included in a given evolutionary model (α) lie above the μ (0)(P ) b; see Figure 24View Image.

View Image

Figure 24: Baryon chemical potential at T = 0, μ = (ℰ + P)∕n b b vs. pressure for different models of neutron star crust. Black solid line: (0) μb (P ) for the ground-state crust. Lines with discontinuous drops: μ(bα)(P ) for four evolution models in the accreted crust with Ai = 56. Upper four smooth lines, which nearly coincide: (α) ∑ (α) μ b (P ) + j(P<Pj)Q j ≈ ¯μb(P ). For an explanation see the text. Based on Haensel & Zdunik [188Jump To The Next Citation Point]. Figure made by J.L. Zdunik.

This makes visual the fact that noncatalyzed matter is a reservoir of energy, released in nonequilibrium processes that move the matter closer to the absolute ground state. In spite of dramatic differences between different μ(bα)(P) in the region where the bulk of the heating occurs, P = (1030– 1031.5) erg cm − 3, the functions μ(α)(P) b tend to μ(0)(P ) b for P ≳ 1032 erg cm −3. The general structure of different (α) μ b (P ) is similar. At the same P, their continuous segments have nearly the same slope. What differs between μ(α)(P) bs are discontinuous drops, by Q (αj), at reaction thresholds (α) P j. The functions (α) μ b (P ) can therefore be expressed as (see Haensel & Zdunik [188Jump To The Next Citation Point])

(α) ∑ (α) μb (P ) ≈ ¯μb(P ) − Q j , (75 ) j(P<Pj)
where ¯μb(P ) is a smooth function of P, independent of (α). For 33.5 −3 P > 10 erg cm the values of Qj are negligibly small, and all μ (αb)(P ) come quite close to the ground state line. This implies that the sum ∑ Q(α) j j must be essentially independent of (α).
  Go to previous page Go up Go to next page