REVISTA MEXICANA DE FISICA 49 SUPLEMENTO 4, 139–144 AGOSTO 2003

The fate of matter on accreting neutron stars

M. Bearda,b and M. WiescherbaDepartment of Physics, University of Surrey, Guildford, UK

bDepartment of Physics, University of Notre Dame, Notre Dame, Indiana 46556

Recibido el 12 de marzo de 2003; aceptado el 22 de marzo de 2003

X-ray bursts have been identified as thermonuclear explosions on the surface of accreting neutron stars. During the thermonuclear runawaythe initially hydrogen and helium rich accreted material will be rapidly converted by the rp-process towards heavy elements in the massA=100 range. The evolution and time-scale as well as the final abundance distribution in the ashes of the thermonuclear runaway dependscritically on the nuclear decay and reaction rates along the rp-process path. The paper discusses the subsequent fate of the matter due toelectron capture and pycnonuclear reaction processes in the deeper layers of the neutron star crust. The necessary conditions for the ignitionof pycnonuclear reactions will be presented and nuclear physics input information for a reliable determination of pycnonuclear reaction rateswill be identified.

Keywords:Neutron stars; pycnonuclear reactions.

Las rafagas de rayos X han sido identificadas como explosiones termonucleares en la superficie de estrellas de neutrones en proceso deacrecion. Al desencadenarse el proceso termonuclear el material acumulado, inicialmente rico en hidrogeno y helio, sera rapidamenteconvertido por el proceso rp a elementos pesados en la region de masas alrededor de A=100. La evolucion y la escala temporal, ası como ladistribucion final de abundancias en las cenizas del proceso termonuclear, dependen crıticamente de las razones de reaccion y decaimientonuclear a lo largo de la trayectoria del proceso rp. Este artıculo discute el destino subsecuente de la materia debido a procesos de capturaelectronica y reacciones picnonucleares en las capas mas profundas de la corteza de la estrella de neutrones. Se presentaran las condicionesnecesarias para la ignicion de reacciones picnonucleares y se identificara la informacion relevante de fısica nuclear necesaria para unadeterminacion confiable de las razones de reaccion picnonuclear.

Descriptores:Estrellas neutronicas; reacciones picnonucleares.

PACS: 26.60.+c; 97.10.-q; 26.60.+c; 98.80.Ft

1. Introduction

Nuclear physics plays an important role in stellar evolutionand stellar explosion. The sequential phases in stellar evolu-tion are driven and controlled by nuclear reaction sequencesbetween stable nuclei which determine the life-span of theburning stage and define its final isotopic and elemental abun-dance distributions. Laboratory studies of these processes re-quires the measurement of low energy (≤100keV) nuclear re-actions between stable nuclei at sub-picobarn cross sections.Rapid energy release and temperature increase in stellar envi-ronments is triggered by the evolution of shock fronts throughstellar matter, such as in type II supernovae, or by thermonu-clear runaway processes at highly electron degenerate condi-tions in stellar material. Rapid increase in temperature causesexponential increase of reaction rates which become fasterthan competingβ-decay rates. This drives the reaction pathaway from stability like in the r-process in neutron rich en-vironments [1,2] or the rp-process in hydrogen rich environ-ments [3, 4]. To address these questions in the laboratoryrequires the measurement of nuclear reactions with unstableshort-lived nuclei at radioactive beam facilities. A third veryimportant question in nuclear astrophysics is the fate of mat-ter when exposed to stellar high density conditions such aswhite dwarf or neutron star environments. At these condi-tions nucleosynthesis is dominated by electron capture pro-cesses [5] and at extreme densities even by pycnonuclear re-actions [6]. Accreting neutron stars provide the perfect stel-

lar laboratory environment for probing such processes at theextreme density conditions in the neutron star crust [7]. Inthis paper we will discuss the sequence of nuclear processeswhich determine the fate of the accreted material in the neu-tron star atmosphere and the neutron star crust. We will focuson pycnonuclear reactions which constitute one of the finalstages of nuclear processes in the neutron star crust environ-ment.

2. Accretion induced nucleosynthesis

There is a wide range of neutron star accretion mechanisms.This includes the fall-back process of supernova shock drivenmaterial onto the surface of the newly born neutron star [8],slow accretion of single neutron stars in interstellar mate-rial [9], and rapid accretion in neutron star binary systemswhich eventually triggers X-ray bursts or X-ray pulsars [10].These scenarios differ mainly in the abundance distribution ofthe accreted material and in the accretion rate, which causessubstantially different subsequent nucleosynthesis scenarios.

In the following we will concentrate on the conditions forrapid accretion onto the surface of a neutron star in close bi-nary systems at rates ofM ≤ 10−2M¯/s. Ignition of theaccreted hydrogen and helium rich matter triggers a ther-monuclear runaway at the electron degenerate conditions atthe neutron star surface. The associated energy release is ab-served as type-I X-ray burst [11–13]. The ignition of the ac-

140 M. BEARD AND M. WIESCHER

creted material is caused by the triple alpha process [14] andthe break-out from the hot CNO cycles [15]. The thermonu-clear runaway proceeds through the rp-process [4] heating theaccreted material within≈1 s to peak temperatures of≈2.5GK. The initial light ion abundance distribution in the ac-creted material is converted to56Ni and subsequently, withinthe few hundred seconds lasting cooling period, into a massdistribution in below the A=100 range [16]. Fig. 1 showsthe mass distribution in the ashes of the thermonuclear run-away. It can be clearly seen that most of the initial mass hasbeen converted to A=64, 68, 72, and 104 which are corre-lated with the N=Z waiting nuclei64Ge, 68Se,72Kr [4] andwith the endpoint nucleus104Sn [16]. Only a small fractionof the initial abundance remains in4He and in12C.

3. Neutronization of matter at increasing den-sity

Because of the high acretion rates the nuclei in the ashes ofthe thermonuclear runaway experience a continuous increasein density, raising the level of electron gas degeneracy in thematerial. The increase of density at a surface layer with ra-dius R depends on the accretion rateM which in turn willdetermine the timescale of all subsequent nuclear processes.The Fermi energy for a completely degenerate electron gasdepends directly on the densityρ

EF =~2

2me(3π2NA)2/3(ρYe)2/3 (1)

Raising the Fermi energy level supplies electrons at high en-ergies causing electron capture reactions that are prohibitedby their negativeQec values at lower density conditions.

At high accretion rates in neutron star binary sys-tems, such electron capture reactions drive the abundancedistribution with continually decreasing Z towards theneutron dripline as illustrated in Fig. 2. For densities

FIGURE 1. Shown is the abundance distribution after the ther-monuclear runaway on the crust of an accreting neutron star. Thebulk of the produced elements are in the mass range between A=64and A=104.

of ρ ≈ 6.6 × 1011 g/cm3, the drip line is reached andsubsequent electron capture processes lead to neutron emis-sion which breaks down the relatively heavy mass distribu-tion along the neutron drip line to light Z and light A iso-topes. Already at densities higher thanρ=109g/cm3 the nu-clei freeze out in a bcc lattice structure. With the furtherincrease of density and the parallel reduction of Z the dis-tance between neighboring isotopes is significantly reduced,the nuclear wave functions begin to overlap, the deflectiveCoulomb forces decrease due to the reduction of Z and alsodue to the screening effects of the degenerate electron gasin the lattice. This can lead to the ignition of pycnonuclearreactions between neighboring nuclei in the lattice.

4. Pycnonuclear reaction rates

The pycnonuclear reactions occur between nuclei frozen in adense bcc lattice structure which resembles the lowest ener-getically possible configuration. With increasing density thewave functions of neighboring nuclei overlap and the degen-erate electron gas neutralizes the deflective Coulomb forcescausing an effective enhancement of the tunnel probabilitythrough the Coloumb-barriers despite relatively low temper-ature conditions. The pycnonuclear reaction rates thereforemainly depend on the density and are nearly independentof temperature. The first estimates of pycnonuclear reactionrates were calculated using the WKB approximation for thenuclear wave function at the lattice position. The calculationsadopted a strong screening potential and assumed a perfectlyhomogeneous lattice structure [6]. The rates depend sensi-tively on the potential shape but are rather insensitive to lat-tice imperfections. Temperature dependency of the rates canbe approximated by including interaction between nuclei inthermally excited vibrational states calculated in an harmonicoscillator model. In the following we will refer to these ratesas SVH. An improved treatment of pycnonuclear reaction

FIGURE 2. Shown is the reaction path of the rp-process; thewaiting point isotopes with the highest abundance are marked inblack. Schematically indicated is the subsequent neutronization ofits ashes through electron capture and electron capture induced neu-tron emission.

Rev. Mex. Fıs. 49 S4(2003) 139–144

THE FATE OF MATTER ON ACCRETING NEUTRON STARS 141

rates included polarization effects in the surrounding latticewhich leads to a more effective screening of the Coulomb po-tential while enhancing the effective mass of the interactingnuclei [17]. We will therefore base our calculations on thismodified formalism which we refer to as SK.

The pycnonuclear reaction rate for one single species ofisotopes in a bcc lattice depends critically on the charge num-berZ, the atomic mass numberA, the densityρ (in g/cm3),the astrophysical S-factorS for the nuclear reaction process(in MeV-barn), and the distance parameterλ

r = 1.06× 1045SρA2Z4λ7/4

× exp(−α2 − α1λ−1/2)cm−3s−1 (2)

The distance parameterλ is defined as the ratio between thenuclear Bohr radius

r∗ = 29.03/(AZ2)[fm] (3)

and the lattice spacing [17]

a =3A

4πρNA(4)

and can be expressed in terms of densityρ (in units g/cm3)

λ =

√34

r∗

a= 2.45× 10−4A−4/3Z−2γ−1/3ρ1/3 (5)

with γ=2 as number of ions in a unit cell of a bcc lattice. Theparameterα1 andα2 characterize different approximationsfor calculating the tunnel probability through the screenedCoulomb wall in the WKB approximation. For a static lat-tice these parameters are given asα1=2.639 andα2=-6.305,for a relaxed lattice taking into account polarization and vi-brational degrees of freedom, the parameters areα1=2.517andα2=-6.754 in a bcc lattice. For slightly higher tempera-tures the lattice structure is not homogeneous anymore. It hasbeen shown that at temperatures of≥1000 K fcc lattice con-figurations appear which require a slight modification in thereaction rate formula to accommodate for the different latticestructure.

In most astrophysical cases pycnonuclear processes in-volve different kind of isotope species. This requires a gen-eralization of the above formalism. Pycnonuclear reactionrates for binary isotope mixtures have been developed beforeby modifying the mono isotope formalism of SVH throughcomposition scaling for determining the mean ion separationdistance in the lattice [18]. This affects the distance factorλi,j . The reaction rate itself must be corrected for the factthat interaction between isotopes of different chargeZi, Zj

and mass numbersAi, Aj need to be considered. We adoptedthis approach [18] and modified the SK rates for the calcula-tion of pycnonuclear reaction ratesri,j (in units cm−3s−1)between different isotopes accordingly

ri,j =1.061045

(1 + δi,j)(Xi + Xj)(AiAj)

(Ai + Aj)2(ZiZj)2Si,jρλ

7/4i,j

× exp(−α2 − α1λ−1/2i,j ) (6)

The Kronecker symbolδi,j accounts for cases of pycnonu-clear fusion between identical isotopesi = j. The distanceparameterλ also has to be modified to accommodate for themean distance between various isotope pairs. The mean Bohrradius is given by

r∗i,j =29.03(Ai + Aj)

2AiAjZiZj[fm] (7)

the mean lattice distance isrm,ij = 1.76/2(ai + aj).

The comparison between the results of the such modi-fied SK formalism and modified versions of the SHV formal-ism are shown in Fig. 3 for the case of12C+16O [18]. Goodagreement is obtained with the predictions for the generalizedSVH relaxed lattice rates. Comaprison with other predictions(IOV) in the literature show less agreement, in particular to-wards lower densities [18]ρ ≤109g/cm3. The deviations aremainly due to the methodological discrepancies in treatingthe interaction potential and the screening potential withinthe lattice. At higher densities however, where pycnonuclearreactions are expected to play an important role, all rate pre-dictions show agreement within one order of magnitude.

The density dependence of the distance parameterλi,j

causes a steep increase in reaction rate for pycnonuclear pro-cesses, however, the absolute strength is given by the astro-physical S-factorSi,j which basically describes the fusionprobability in units MeV-barn. The S-factor is a critical pa-rameter which needs to be determined for reliably describingthe pycnonuclear reaction rates. While for fusion reactionsbetween stable nuclei the S-factor can be directly taken fromlow energy fusion cross section measurements, for neutronrich radioactive nuclei no experimental fusion data are avail-able. The determination of the S-factor therefore depends onthe development of reliable theoretical methods for its calcu-lation.

5. The astrophysical S-factor

The astrophysical S-factor is a well-known quantity in thedetermination of thermonuclear reaction rates. It has beenintroduced to correct for the penetrability of s-wave chargedparticles in the reaction cross sectionσi,j(Ecm) as a functionof the center of mass energy Ecm [19]

Si,j(Ecm) = σi,j(Ecm)Ecme(2πη−gEcm) (8)

Rev. Mex. Fıs. 49 S4(2003) 139–144

142 M. BEARD AND M. WIESCHER

FIGURE 3. Shown is the comparison between different models for calculating the pycnonuclear reaction rate for12C+16O. The upper partof the figure shows the reaction rates as a function of density, the lower part shows the ratios of reaction rates calculated with the formalismoutlined in reference 20 and the modified rate given by equation (6).

with the Gamow termη = ZiZjµe2/(~√

2Ecm) and thesize factorg = 0.122(µR3/2ZiZj)1/2 [20] with µ as re-duced mass and R the interaction radius. For reactions be-tween stable nuclei like12C+12C or 12C+16O the cross sec-tion σi,j(Ecm) can be determined experimentally but for re-actions between short-lived radioactive nuclei the cross sec-tion needs to be derived by theoretical means.

We have applied the Hauser-Feshbach model for calcu-lating the cross section, using the code SMOKER [4] to de-termine the various fusion evaporation channels at very lowenergies.

σi,j(Ecm) =πλ2

a

(2Ii + 1)(2Ij + 1)

∑

J

(2J + 1)TaTb∑

i Ti(9)

Ij and Ij represent the spins of the interacting particles,λa isthe Compton wavelength for the entrance channel, and Ta, Tb

characterize the specific transmission coefficients for the en-trance and exit channels. The individual particle transmissioncoefficients Ti, are calculated solving the Schrodinger equa-tion for a parametrized optical model potential. A typicalpotential depth of 460 MeV was used for most calculationssince it showed best agreement with the available experimen-tal data [21] (see also Fig. 4). The sum over all reaction chan-nels gives the total fusion cross section. In comparison withthese results we used a simple barrier penetration model us-ing the code CCFUS to estimate the total fusion cross sectiondirectly [22]. The latter approach shows less agreement withthe experimental data towards lower energies. Figure 4 showsthe S-factor calculated for the reactions16O+16O, 16O+16Oand16O+18O in comparison with experimental data [20,23].It can clearly be seen that the Hauser-Feshbach based S-factorpredictions agree fairly well (within a factor of 3) with the ex-perimental data.

For calculating the pycnonuclear reaction rates for lightnuclei near the neutron drip-line we therefore have used theHauser-Feshbach model for determining the S-factor. The S-factors for fusion processes for neutron rich oxygen nucleiare shown in Fig. 5. The figure clearly demonstrates a signif-icant increase of the S-factor with the Z of the fusioning

FIGURE 4. Theoretical predictions for the S-factors of13C+13C,16O+16O and16O+18O using the Hauser-Feshbach model formal-ism. The results are shown in comparison with experimental data.

isotopes. This is clearly a simplistic approach since it is basedon an optical model approximation which does not includeany potential distorting effects due to neutron skin or neutronhalo configurations. Detailed neutron and proton density cal-culations are presently being performed to include possiblehalo effects in the optical potential predictions [24].

FIGURE 5. Theoretical predictions for the S-factor of22O+24O,24O+24O, 24O+34Ne, and24O+42Mg using the Hauser-Feshbachformalism.

Rev. Mex. Fıs. 49 S4(2003) 139–144

THE FATE OF MATTER ON ACCRETING NEUTRON STARS 143

Figure 6 shows the reaction rates for a series of pycnonu-clear fusion processes between neutron-rich oxygen, neon,and magnesium isotopes. The reaction rates for neutronrich oxygen isotopes22O+24O, 24O+24O, 24O+34Ne, and24O+42Mg are shown in the upper part of the figure while thelower part displays the reaction rates for neutron rich neoninduced pycnonuclear fusion processes such as24O+34Ne,34Ne+36Ne, 36Ne+36Ne, as well as34Ne+42Mg. The reac-tion rates show a steep increase with density from≤10−70

cm−3s−1 at ρ ≤109 g/cm3 to ≥1070 cm−3s−1 at ρ ≥1013

g/cm3. Higher densities are not considered since that wouldrequire an improved treatment for shielding effects betweenextended charge distributions [17]. The reaction rates be-tween isotopes of identical Z show only minor differenceswhich are entirely due to the difference in S-factor. For thesecases a more accurate calculation including halo effects forisotopes near the drip-line is clearly desirable since thosecan cause considerable enhancement of the fusion probabil-ity [25]. At density conditionsρ ≤1012 g/cm3 the reac-tion rates for fusion processes between nuclei with higherZ-values rapidly decrease due to the strong Z-dependence.This is only insufficiently compensated by large increase of

FIGURE 6. Pycnonuclear reaction rate predictions for22O+24O,24O+24O, 24O+34Ne, 24O+42Mg (upper part) and24O+34Ne,34Ne+36Ne,36Ne+36Ne,34Ne+42Mg (lower part).

the S-factor value with Z. Again, for very neutron rich iso-topes the effects of extended halos need to be taken into ac-count for a more reliable calculation of the S-factor.

6. Conclusion

Pycnonuclear reactions can have a strong impact on the fateof matter at density conditionsρ ≥1011 g/cm3. At these con-ditions the rates become comparable with competing electroncapture processes. This causes a drastic change in nucleosyn-thesis pattern. Electron capture reactions and electron captureinduced neutron emission systematically reduce the chargedistribution towards lower Z, causing a gradual neutroniza-tion of matter. At high density conditions pycnonuclear reac-tions can compete with the electron capture and can rapidlyshift the average charge distribution towards higher (≈ dou-ble) Z value. This can have drastic effects on the thermal andelectrical conductivity of the neutron crust matter [7,26]. Py-cnonuclear reactions themselves represent a significant heatsource within the crust of a neutron star. The amount of re-leased heat depends directly on the reaction rates

ε =∑

i,j

ri,jQi,j (10)

with Q-values ofQi,j ≈25-35 MeV for near drip-line fu-sion processes [27]. Crucial for the cooling process is thedepth, where the heat release takes place. There exists a kindof “watershed” given byd ln κ/d ln P = −1, (P is the lo-cal pressure andκ the opacity, taking into account both thephoton diffusion and the electron conduction). Is the heatreleased at the low density side of that watershed it will beirradiated away towards the surface of the neutron star. Thetime scale for heat diffusion towards the surface is set by thethermal conductivity for a given density profile and is there-fore directly correlated with the Z distribution in the crustmaterial. Pycnonuclear heat released in deeper layers will betransported inwards and will be stored for a certain period inthe core of the neutron star. The subsequent cooling dependson the state of matter in the core.

Acknowledgments

Work was supported by the National Science Foundation andby funding through the University of Notre Dame.

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