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4.4 Thermal structure of accreted crusts and X-ray bursts

A thermonuclear flash is triggered by an instability in the thermonuclear burning. The relevant quantities are the local heating rate due to thermonuclear fusion, ˙ɛnuc, and cooling rate, ˙ɛcool, resulting from heat diffusion, volume expansion, and neutrino emission. A steady state of an accreting neutron star corresponds to ɛ˙ = ˙ɛ nuc cool. It depends on the accretion rate, composition of accreted plasma, structure and physical properties (thermal conductivity, neutrino emissivity, etc. of the stellar interior. Examples of the steady thermal structure of accreting neutron stars are shown in Figures 25View Image and 26View Image.
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Figure 25: Temperature (local, in the reference frame of the star) vs. density within the crust of an accreting neutron star (soft EoS of the core, M = 1.24M ⊙) in a steady thermal state, with standard cooling of the core (no fast cooling of the direct Urca type). Upper solid curve – ˙ −9.96 −1 M = 10 M ⊙ y. Lower solid curve – ˙ − 11 −1 M = 10 M ⊙ y. H and He burning shells are indicated by asterisks. Deep crustal heating is included. Dashed line – temperature profile without deep crustal heating. Based on Figure 3b of Miralda-Escudé el al. [292Jump To The Next Citation Point].
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Figure 26: Same as in Figure 25View Image, but for fast neutrino cooling due to pion condensation in the inner core. Based on Figure 3c of Miralda-Escudé et al. [292].

A stable steady state of an accreting neutron star satisfies

˙ɛ = ˙ɛ , ∂ɛ˙nuc < ∂-˙ɛcool. (76 ) nuc cool ∂T ∂T
The inequality guarantees that any thermal perturbation will be damped by a self-regulated cooling. Under the increasing weight of accreted matter, an element of the crust in the He burning shell moves in the ρ − T plane, in the direction of increasing ρ. At some moment, after crossing the “ignition line” characterized by
∂-˙ɛnuc ∂ ˙ɛcool ∂T = ∂T =⇒ ρign(T ), (77 )
burning becomes unstable, and a self-accelerating thermonuclear flash is ignited because
∂ ˙ɛnuc ∂ɛ˙cool ∂T > ∂T . (78 )
In the standard picture, it is the instability in the helium burning via 12 3α −→ C, which triggers an X-ray burst (see, e.g., [152]). The total energy release in a burst can then be easily estimated (neglecting the general relativistic correction) via
Eburst ≈ Qnuc Mburn ∕mu , (79 )
where Qnuc is the mean energy per nucleon released in the thermonuclear flash (2 – 8 MeV, depending on composition of burnt material), Mburn = Menv(ρign) is the mass of the envelope burnt in the flash (determined by the ignition density ρign), and mu is the atomic mass unit. The value of Menv(ρign) can be read from Figure 39View Image. A good model of X-ray bursts should yield 40 Eburst ∼ 10 erg with a recurrence time of hours. This can be satisfied with ignition of helium flashes at ∼ 107 g cm–3.

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