Neutron stars, with their strong surface gravity, have interestingly short
timescales for the sedimentation of heavy elements. Recent observations of
unstable thermonuclear burning (observed as X-ray bursts) on the surfaces of
slowly accreting neutron stars ($< 0.01$ of the Eddington rate) motivate us to
examine how sedimentation of CNO isotopes affects the ignition of these bursts.
We further estimate the burst development using a simple one-zone model with a
full reaction network. We report a region of mass accretion rates for weak H
flashes. Such flashes can lead to a large reservoir of He, the unstable burning
of which may explain some observed long bursts (duration $\sim 1000$ s).Comment: 6 pages, 2 figures, submitted to the proceedings of the conference
  "The Multicoloured Landscape of Compact Objects and Their Explosive
  Origins'', 2006 June 11--24, Cefalu, Sicily (Italy), to be published by AI

Brown, Edward F.

Peng, Fang

Truran, James W.

English

arXiv

Neutron stars, with their strong surface gravity, have interestingly short timescales for the sedimentation of heavy elements. Recent observations of unstable thermonuclear burning (observed as X-ray bursts) on the surfaces of slowly accreting neutron stars (< 0.01 of the Eddington rate) motivate us to examine how sedimentation of CNO isotopes affects the ignition of these bursts. We further estimate the burst development using a simple one-zone model with a full reaction network. We report a region of mass accretion rates for weak H flashes. Such flashes can lead to a large reservoir of He, the unstable burning of which may explain some observed long bursts (duration ~ 1000 s)

Caltech Authors - Main

Type I X-ray Bursts at Low Accretion Rates
Fang Peng∗, Edward F. Brown† and James W. Truran∗∗,‡
∗Theoretical Astrophysics, California Institute of Technology, Pasadena, CA 91125
†Department of Physics & Astronomy, Michigan State University, East Lansing, MI 48824
∗∗Department of Astronomy & Astrophysics, University of Chicago, Chicago, IL 60637
‡Physics Division, Argonne National Laboratory, Argonne, IL 60439
Abstract. Neutron stars, with their strong surface gravity, have interestingly short timescales for the
sedimentation of heavy elements. Recent observations of unstable thermonuclear burning (observed
as X-ray bursts) on the surfaces of slowly accreting neutron stars (< 0.01 of the Eddington rate)
motivate us to examine how sedimentation of CNO isotopes affects the ignition of these bursts. We
further estimate the burst development using a simple one-zone model with a full reaction network.
We report a region of mass accretion rates for weak H flashes. Such flashes can lead to a large
reservoir of He, the unstable burning of which may explain some observed long bursts (duration
∼ 1000 s).
Keywords: diffusion — stars: neutron — X-rays: binaries — X-rays: bursts
PACS: 97.60.Jd, 98.70.Qy
INTRODUCTION
An ionized plasma in a gravitational field develops an electric field sufficient to levitate
the ions and ensure overall charge neutrality. When there is more than one species of ion
present, the ions will experience a differential force: lighter ions float upward (defined
by the local gravitational field) and heavier ions sink downward.
Accreting neutron stars, with their strong surface gravity ≈ 2.0×1014 cms−2, are an
ideal place to look for the effects of sedimentation. The sedimentation of heavy elements
and resulting nucleosynthesis in the envelope of isolated neutron stars cooling from
birth was first described by Rosen [1, 2] and has been studied in detail by Chang and
Bildsten [3, 4]. For accreting neutron stars, the rapid stratification removes heavy nuclei
from the photosphere for accretion rates ˙M  10−12Myr−1 [5]. Deeper in the neutron
star envelope, the differentiation of the isotopes can alter the nuclear burning, namely
unstable H/He burning and rapid proton-capture process (rp-process), that powers Type
I X-ray bursts.
Recent long-term monitoring of the galactic center with BeppoSAX led to the discov-
ery of nine “burst-only sources” [see 6, and references therein]. These sources did not
have persistent fluxes detectable with the BeppoSAX/WFC, thus must have extremely
low accretion luminosities (LX < 1036 ergss−1  0.01LEdd). If the accretion is not con-
centrated onto a small surface area, so that the local accretion rate is m˙ ≈ ˙M/(4πR2),
then the sedimentation timescale for CNO nuclei, defined as the time required to move
a scale height relative to the center of mass of a fluid element, is less than the accretion
timescale, defined as the time for a fluid element to reach a given depth. An estimate of
this accretion rate was noted earlier by Wallace et al. [7]. As a result, sedimentation in
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the accreted envelope must be considered in treating the unstable ignition of hydrogen
and helium for these low-m˙ sources. At somewhat higher accretion rates, LX ∼ 0.01LEdd,
several bursts have been observed with longer durations,  1000s, that are intermediate
between mixed H/He bursts and superbursts.
Motivated by these observations, we explore the unstable ignition of hydrogen and
helium at low accretion rates and pay particular attention to the regime where the mass
accretion rate is less than the critical rate needed for stable H burning [8]. We also study
how the sedimentation of CNO nuclei affects the unstable ignition of H and He in an
accreted neutron star envelope. For a full reporting of these results, please see [9].
THE ROLE OF SEDIMENTATION ON IGNITION CONDITIONS
To study the accumulation of H and He, we construct models of a plane-parallel at-
mosphere in hydrostatic equilibrium, and for simplicity we neglect the terms coupling
thermal and particle diffusion [see 10, and references therein]. We follow the treatment
of Burgers [11] and construct for each species an equation of continuity and momen-
tum conservation. Collisions are described by a resistance coefficient Kst = nsnt〈σstvst〉,
with vst the center-of-mass relative velocity between particles of types s and t and σst
the cross-section between these particles.
As an illustration of how sedimentation changes the structure of the envelope, consider
a trace species, labeled 2, in a background of a species labeled 1, i.e., n1	 n2 and species
1 has velocity w1 = 0. The electric force is eE = mugA1/(Z1 + 1) where the electrons
are non-degenerate and eE = mugA1/Z1 where the electrons are degenerate. In these
expressions, mu is the atomic mass unit. Substituting E into the equation of motion
for species 2 then determines the sedimentation velocity, wsed = w2 = n2(A2mug−
Z2eE)/K12. We chose the sign of wsed to be positive if species 2 moves downward.
Using the Stokes-Einstein relation to determine K12 from the drag coefficient for
a liquid sphere ([12]), and a nonrelativistic electron equation of state, we find the
sedimentation velocity of a trace nucleus [see 13]
wsed = 2×10−3g14
T 0.37
ρ0.65
(A2Z1−Z2A1)A0.11
Z2.31 Z
0.3
2
cms−1, (1)
where we use the common shorthand g14 = g/(1014 cms−2), T7 = T/(107 K), and
ρ5 = ρ/(105 gcm−3). In a pure H plasma, the sedimentation velocity1 is greater than the
mean velocity u= m˙/ρ for m˙< 400g cm−2 s−1T 0.37 ρ0.45 (A2−Z2)/Z0.32 ; under conditions
at which H ignites (T7 ∼ 5;ρ5 ∼ 4) this corresponds to m˙ < 0.02m˙Edd and m˙ < 0.05m˙Edd
for a trace 4He and 12C nucleus, respectively. Here m˙Edd = 2mpc/[(1+ XH)σTHR] =
8.8×104 g cm−2 s−1 is the local Eddington mass accretion rate for a solar composition,
σTH is the Thomson scattering cross section, and XH is the hydrogen mass fraction.
We numerically solve for the evolution of the accreting envelope prior to unstable
ignition by coupling the diffusion equations to the equations for temperature T and heat
1 Throughout this paper we use a Newtonian metric and assume a neutron star mass 1.4 M and radius
10 km, so that g14 = 1.9.
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flux F ,
∂T
∂y =
F
ρK ,
∂F
∂y =CP
(∂T
∂ t + m˙
∂T
∂y
)
−CPT m˙
y
∇ad− εnucl , (2)
where dy = −ρdz is the column density, CP is the specific heat at constant pressure,
εnucl is the nuclear heating rate per unit mass, and ∇ad ≡ (∂ lnT/∂ lnP)S. For detailed
description of the numerical method in solving these equations, please refer to [9].
We take the composition of matter accreted from the companion star to be roughly
solar composition in 1H and 4He, and distribute the remaining mass evenly between 12C
and 16O: X(1H) = 0.71, X(4He) = 0.27, X(12C) = 0.01, and X(16O) = 0.01. Below the
fuel layer are ashes from previous X-ray bursts; we set the ash composition to 64Zn,
consistent with the findings of recent one-dimensional calculations of repeated X-ray
bursts [14, 15]. We use a radiative-zero outer boundary, and for the inner boundary, at
the bottom of the ash layer, we set the flux to a constant value, Fc = 0.1 MeV m˙/mu [16].
This flux is set by reactions in the deep crust—electron captures, neutron emissions, and
pycnonuclear reactions.
We explore the evolution of the accreting neutron star envelope for accretion rates
10−4–0.6 m˙Edd (10–5×104 g cm−2 s−1 for accretion of a solar composition mixture). For
each accretion rate, we ran calculations both with and without isotopic sedimentation,
and we follow the thermal and chemical evolution of the mixture until the envelope
becomes thermally unstable. We determine this ignition point using a simple one-zone
criterion, (∂εnucl/∂T )P > (∂εcool/∂T )P, where εcool = ρKT/y2 is an approximation of
the local cooling rate. The mass fraction of H and He at the column where either H or
He unstably ignites is plotted in fig. 1, as a function of mass accretion rates. When
sedimentation is included, the abundance of H at the base of the accreted envelope
is depressed. It is tantalizing that the accretion rate at which mixed H/He ignition
occurs is increased by a factor of 2 when sedimentation is taken into account, and we
speculate that this might alleviate the discrepancy between the predicted transition in
burst duration [8] and recent observations [see, for example 17]
BURSTS AT LOW ACCRETION RATES
When the temperature in the neutron star envelope is sufficiently low, the CNO cycle be-
comes temperature dependent; as a result, the ignition of H becomes thermally unstable
at low accretion rates [8]. As a first investigation of the unstable ignition of H when the
atmosphere is stratified, we perform several calculations of the burst nucleosynthesis.
We approximate the cooling by a one-zone finite differencing over the envelope,
CP
dT
dt = εnucl− εcool, (3)
where εcool = ρKTy−2 and we evaluate εnucl from a reaction network. Included in this
network are 686 isotopes covering proton-rich nuclei up to Xe [see 18]. The reaction
rates are taken from the compilation REACLIB [see 19, 20, and references therein], and
consist of experimental rates and Hauser-Feshbach calculations with the code NON-
SMOKER [20]. The initial temperature, column density, and composition are taken from
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10-4 10-3 10-2 10-1 100
m /mEdd
0.0
0.2
0.4
0.6
0.8
1.0
X
(1
H
)
H ignition
He ignition
• •
FIGURE 1. Mass fraction of hydrogen at the column where either H (open symbols, solid lines) or He
(filled symbols, dotted lines) unstably ignites, as a function of mass accretion rates. We show results for
which sedimentation is ignored (circles), and for which it is included (squares). From [9] and reproduced
by permission of the AAS.
the values of the quasi-static calculation at the ignition point. For the unstable ignition of
H, the temperature in the accreted envelope is set by the flux emergent from the deeper
ocean and crust. As a result, the accretion rate at which the burst behavior changes will
depend on assumptions about the heating in the crust.
We shall first describe the outcome of these one-zone calculations (eq. [3]). The left
panel of Figure 2 shows the post-ignition evolution for accretion rates 9.1×10−4 m˙Edd
(80 g cm−2 s−1; solid lines), 1.1× 10−3 m˙Edd (100 g cm−2 s−1; dotted lines), and
2.3×10−3 m˙Edd (200 g cm−2 s−1; dashed lines). We plot three different quantities: the
evolution of temperature (top panel), the heat flux, Fcool = yεcool, normalized to the
accretion flux (middle panel), and the mass fractions of hydrogen and helium (bottom
panel). We then repeat this calculation at higher mass accretion rates (Fig. 2, right panel),
5.7×10−3 m˙Edd (500 g cm−2 s−1; solid lines) and 1.1×10−2 m˙Edd (103 g cm−2 s−1; dot-
ted lines).
It is immediately clear that there are two very different outcomes for the H ignition.
At lower accretion rates, the rise in temperature following H ignition is sufficient to
trigger a vigorous He flash with a decay timescale ∼ 10 s (Fig. 2, middle-left panel). At
higher accretion rates, however, the flash is too weak to ignite He. As an interpretation
of these results, recall that the ignition curve for the 3α reaction has a turning point
at yturn ≈ 5×107 gcm−2, which terminates the unstable branch [21, 22]. At the higher
accretion rates (Fig. 2, right panel) the ignition of H occurs at y < yturn, and the local
rise in temperature does not trigger unstable He ignition. In fact, the temperature rise is
not even sufficient to initiate convection, as the radiative temperature gradient needed to
carry Fcool is d lnT/d lnP≈ 0.25 < (∂ lnT/∂ lnP)s. In contrast, at lower accretion rates
(Fig. 2, left panel), the ignition of H occurs at y > yturn, and the rise in temperature will
ignite the triple-α reaction.
If subsequent H flashes do not ignite the underlying He, then a large layer of nearly
516
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0 20 40 60 80 100
Time [s]
0.01
0.10
1.00
X(
1 H
), X
(4 H
e)
1H
4He
1
10
100
1000
F c
o
o
l /
F a
cc
108
109
T 
[K
]
0 200 400 600 800 1000
Time [s]
0.01
0.10
1.00
X(
1 H
), X
(4 H
e)
1H
4He
0.01
0.10
1.00
10.00
100.00
F c
o
o
l /
F a
cc
108
109
T 
[K
]
FIGURE 2. One-zone burst calculation following unstable H ignition. Left: for three mass accretion
rates, m˙/m˙Edd = 9.1×10−4 (solid lines), 1.1×10−3 (dotted lines) and 2.3×10−3 (dashed lines), respec-
tively. Right: for two mass accretion rates, m˙/m˙Edd = 5.7× 10−3 (solid lines) and 0.011 (dotted lines),
respectively. Top panel: temperature evolution; Middle panel: the ratio of one-zone cooling flux to the
accretion flux; Bottom panel: mass fraction of hydrogen (thin lines) and 4He (thick lines), respectively.
Diffusion and sedimentation is included. From [9], and reproduced by permission of the AAS.
pure He will accumulate. Because our system is not burning H steadily, the temperature
in the He layer is colder than if the burning were in steady-state. As a result, a large
He layer should accumulate. To get the thermal structure of an accumulated pure He
layer, we integrated equations (2), with ∂/∂ t → 0, from the column where H ignition
occurred, and set the temperature there to that found in the accumulating model. For
m˙/m˙Edd = 5.7× 10−3 and 0.011, the He ignition column density varies from y ≈
2× 109–3× 1011 gcm−2 (recurrence time ≈ 0.06− 19 yr) for Fcmu/m˙ varies from 1.0
to 0.1. Our one-zone approximation gives a cooling time τ ≈ 400 s[y/(1010 gcm−2)],
which roughly agrees with the long burst duration.
CONCLUSION
Using a simplified numerical model of an accreting neutron star envelope that allows for
differential isotopic velocities, we have undertaken a first study of the effects of sedimen-
tation and diffusion on the unstable ignition of hydrogen and helium on the surfaces of
accreting neutron stars, and have investigated the outcome of unstable hydrogen ignition
using simple one-zone models. Our principal conclusions are:
1. The effect of sedimentation changes the conditions at H/He ignition even for
m˙ 0.1m˙Edd.
2. There is a range of accretion rates for which unstable H ignition does not trigger
517
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a He flash. This range depends on the flux from the deep crustal heating and the
degree of settling of He and CNO nuclei. We speculate that successive weak H
flashes can lead to the accumulation of a large reservoir of He; this may explain the
long bursts observed from some sources.
ACKNOWLEDGMENTS
This work is supported in part by the Joint Institute for Nuclear Astrophysics under
NSF-PFC grant PHY 02-16783, and by the Department of Energy under grant B523820
to the Center for Astrophysical Thermonuclear Flashes at the University of Chicago.
EFB is supported by the NSF under grant AST-0507456; JWT acknowledges support
from the U.S. DOE, under contract No. W-31-109-ENG-38.
REFERENCES
1. L. C. Rosen, Ap&SS 5, 150 (1969).
2. L. C. Rosen, Ap&SS 5, 92 (1969).
3. P. Chang, and L. Bildsten, ApJ 585, 464–474 (2003).
4. P. Chang, and L. Bildsten, ApJ 605, 830–839 (2004).
5. L. Bildsten, E. E. Salpeter, and I. Wasserman, ApJ 384, 143–176 (1992).
6. R. Cornelisse, J. J. M. in’t Zand, E. Kuulkers, J. Heise, F. Verbunt, M. Cocchi, A. Bazzano, L. Na-
talucci, and P. Ubertini, Nuclear Physics B Proceedings Supplements 132, 518–523 (2004).
7. R. K. Wallace, S. E. Woosley, and T. A. Weaver, ApJ 258, 696 (1982).
8. M. Y. Fujimoto, T. Hanawa, and S. Miyaji, ApJ 247, 267–278 (1981).
9. F. Peng, E. F. Brown, and J. W. Truran, ApJ in press (2006), astro-ph/0609583.
10. C. Paquette, C. Pelletier, G. Fontaine, and G. Michaud, ApJS 61, 177–195 (1986).
11. J. M. Burgers, Flow Equations for Composite Gases, Academic Press, Inc., New York, 1969.
12. L. Bildsten, and D. M. Hall, ApJ 549, 219 (2001).
13. E. F. Brown, L. Bildsten, and P. Chang, ApJ 574, 920–929 (2002).
14. S. E. Woosley, A. Heger, A. Cumming, R. D. Hoffman, J. Pruet, T. Rauscher, J. L. Fisker, H. Schatz,
B. A. Brown, and M. Wiescher, ApJS 151, 75 (2004).
15. J. L. Fisker, E. Brown, M. Liebendoerfer, H. Schatz, and F. K. Thielemann, Nucl. Phys. A 758, 447
(2005).
16. E. F. Brown, ApJ 531, 988 (2000), astro-ph/9910215.
17. P. R. den Hartog, J. J. M. in’t Zand, E. Kuulkers, R. Cornelisse, J. Heise, A. Bazzano, M. Cocchi,
L. Natalucci, and P. Ubertini, A&A 400, 633–641 (2003).
18. H. Schatz, A. Aprahamian, V. Barnard, L. Bildsten, A. Cumming, M. Ouellette, T. Rauscher, F.-K.
Thielemann, and M. Wiescher, Phys. Rev. Lett. 86, 3471 (2001), astro-ph/0102418.
19. F. Thielemann, M. Arnould, and J. Truran, “Thermonuclear Reaction Rates from Statistical Model
Calculations,” in Advances in Nuclear Astrophysics, edited by E. Vangioni-Flam, J. Audouze,
M. Casse, J.-P. Chieze, and J. Tran Thanh Van, Editions Frontières, Gif-sur-Yvette, 1986, p. 525,
URL http://ie.lbl.gov/astro/friedel.html.
20. T. Rauscher, and F. Thielemann, At. Data Nucl. Data Tables 75, 1 (2000), astro-ph/0004059.
21. L. Bildsten, “Thermonuclear Burning on Rapidly Accreting Neutron Stars,” in The Many Faces of
Neutron Stars, edited by A. Alpar, R. Buccheri, and J. van Paradijs, NATO ASI ser. C, Kluwer,
Dordrecht, 1998, vol. 515, p. 419.
22. A. Cumming, and L. Bildsten, ApJ 544, 453–474 (2000).
518
Downloaded 18 Aug 2007 to 131.215.225.9. Redistribution subject to AIP license or copyright, see http://proceedings.aip.org/proceedings/cpcr.jsp


Type I X-ray Bursts at Low Accretion Rates

Fang Peng

Edward F. Brown

James W. Truran

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Type I X‐ray Bursts at Low Accretion Rates

Flow Equations for Composite Gases,

Thermonuclear Burning on Rapidly Accreting Neutron Stars,” in The Many Faces of Neutron Stars, edited by

Thermonuclear Reaction Rates from Statistical Model Calculations,” in Advances in Nuclear Astrophysics,

to 131.215.225.9. Redistribution subject to AIP license or copyright, see http://proceedings.aip.org/proceedings/cpcr.jsp

http://authors.library.caltech.edu/8663/1/PENaipcp07.pdf

Type I X-ray Bursts at Low Accretion Rates

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