UNSTABLE NONRADIAL OSCILLATIONS on HELIUM-BURNING NEUTRON STARS Anthony L

The Astrophysical Journal, 603:252–264, 2004 March 1 # 2004. The American Astronomical Society. All rights reserved. Printed in U.S.A. UNSTABLE NONRADIAL OSCILLATIONS ON HELIUM-BURNING NEUTRON STARS Anthony L. Piro Department of Physics, Broida Hall, University of California, Santa Barbara, CA 93106; [email protected] and Lars Bildsten Kavli Institute for Theoretical Physics and Department of Physics, Kohn Hall, University of California, Santa Barbara, CA 93106; [email protected] Received 2003 September 18; accepted 2003 November 17 ABSTRACT Material accreted onto a neutron star can stably burn in steady state only when the accretion rate is high (typically super-Eddington) or if a large flux from the neutron star crust permeates the outer atmosphere. For such situations we have analyzed the stability of nonradial oscillations, finding one unstable mode for pure helium accretion. This is a shallow surface wave that resides in the helium atmosphere above the heavier ashes of the ocean. It is excited by the increase in the nuclear reaction rate during the oscillations, and it grows on the timescale of a second. For a slowly rotating star, this mode has a frequency !=ð2Þð20 30 HzÞ½lðl þ 1Þ=21=2, and we calculate the full spectrum that a rapidly rotating (330 Hz) neutron star would support. The short-period X-raybinary4U1820À30 is accreting helium-rich material and is the system most likely to show this unstable mode, especially when it is not exhibiting X-ray bursts. Our discovery of an unstable mode in a thermally stable atmosphere shows that nonradial perturbations have a different stability criterion than the spherically symmetric thermal perturbations that generate type I X-ray bursts. Subject headings: accretion, accretion disks — stars: individual (4U 1820À30) — stars: neutron — stars: oscillations — X-rays: stars 1. INTRODUCTION oscillation modes. The criterion for thermal versus mode stability is mathematically different (as we explain in xx 2.2 Type I X-ray bursts are caused by unstable nuclear burning and 3.3), but it is not yet clear how this contrast will manifest on accreting neutron stars (NSs) (see recent reviews by Bildsten itself observationally. Nonradial oscillations in burning NS 1998; Strohmayer & Bildsten 2003) in low-mass X-ray binaries envelopes have been studied before (e.g., McDermott & Taam (LMXBs). They have rise times of seconds with decay times 1987; Strohmayer & Lee 1996), but for atmospheres that are ranging from tens to hundreds of seconds depending on the thermally unstable, leaving our question unanswered. thickness and composition of the accumulated material. These To simplify this initial work, we assume that the NS is ac- bursts repeat every few hours—the timescale to accumulate an creting pure helium and burning this fuel at the same rate it unstable amount of fuel on a NS. Evidence that nuclear accretes. This steady state burning proceeds if the NS envelope burning powers the bursts is the observed ratio of the time- is thermally stable (and no type I X-ray bursts are occurring). It averaged accretion luminosity to the time-averaged burst is stable when either the temperature sensitivity of triple- luminosity (40 for hydrogen-rich accretion and 125 for reactions has weakened because of a high envelope tempera- pure helium accretion). The thermonuclear instability that ture (k5 Â 108 K), or the thermal balance of the burning layer triggers these bursts is usually due to the extreme temperature has been fixed by a high flux from deeper parts of the NS sensitivity of triple- reactions(Hansen&VanHorn1975; (Paczyn´ski 1983a). We then find the eigenfrequencies and test Woosley & Taam 1976; Maraschi & Cavaliere 1977; Joss the stability of g-modes in such an envelope. The one unstable 1977; Lamb & Lamb 1978). mode we find is a shallow surface wave in the helium layer Ever since the original work of Hansen & Van Horn (1975), that is supported by the buoyancy of the helium/carbon inter- the approach to assessing the stability of burning on a NS has face. It has a nonrotating frequency of !=ð2Þð20 30 HzÞ been to apply a spherically symmetric thermal perturbation to ½lðl þ 1Þ=21=2, depending on the accretion rate. a spherical model (Schwarzschild & Ha¨rm 1965). This is done This mode is driven unstable by the -mechanism (i.e., either numerically through a time-dependent code (e.g., pumped by a perturbation in nuclear burning) and grows in Woosley et al. 2003), analytically with a one-zone model (e.g., amplitude on the timescale of a second. This is very different Fujimoto, Hanawa, & Miyaji 1981), or using an eigenvalue from other observed stellar pulsations, such as Cepheid varia- approach (e.g., Narayan & Heyl 2003). These results roughly bles, which are driven by the -mechanism (i.e., pumped by agree and find a thermal runaway once a critical amount of changes in opacity). Unlike the damped, higher radial order fuel has accumulated. These calculations also find the accre- g-modes, the unstable mode suffers little radiative damping in tion rate above which burning is stable. the outer atmosphere. Rapid rotation, like that seen from many We are now taking an alternative approach to this problem. accreting NSs (200–700 Hz), will yield multiple oscillation Using a NS envelope that is stable to spherically symmetric frequencies with a well-defined pattern. If such a frequency thermal perturbations (i.e., not bursting), we ask whether the pattern were seen from a NS it would give strong evidence that envelope is unstable to a nonaxisymmetric perturbation. Since a mode has been excited. the immediate response to such a perturbation is a wave, the In x 2 we describe the physics of a NS star envelope accreting proper way to proceed is through a study of the nonradial and burning helium in steady state. The conditions required for 252 OSCILLATIONS ON He-BURNING NEUTRON STARS 253 dest prod steady state accretion are reviewed. Nonradial adiabatic oscil- where ri and ri are the particle destruction and production lations in a plane-parallel envelope are discussed in x 3, and rate, respectively, Ai is the baryon number of species i,andv ¼ although much of this has been described in previous papers, Àm˙ = is the downward flow velocity (Wallace, Woosley, & we include additional details and strategies that are relevant for Weaver 1982; Bildsten, Salpeter, & Wasserman 1993). We as- NSs. The oscillatory equations are derived in x 3.1, and in x 3.2 sume steady state burning and set @Xi=@t ¼ 0. For the triple- we calculate the radial eigenfunctions and find the g-mode reactions we use the energy generation rate from Fushiki & frequencies for a nonrotating NS. The stability of these modes Lamb (1987a). At larger depths, helium is burned in -captures is investigated in x 3.3, including a look at excitation and to form heavier elements. For the 12Cð; Þ16O reaction we damping times. In x 3.4 we find the frequency of the unstable use the reaction rate from Buchmann (1996, 1997), with the mode on a rapidly rotating NS. The LMXB 4U 1820À30 is fitting formula that gives S ¼ 146 keV barns at 300 keV discussed in x 4 along with other candidates that might show the center-of-mass energy. Rates from Caughlan & Fowler (1988) excited mode we have found. We conclude in x 5bysumma- are used for the reactions 16Oð; Þ20Ne, 20Neð; Þ24Mg, rizing our findings and discussing future directions for this and 24Mgð; Þ28Si. Possible subsequent burning of silicon work. does not occur for the accretion rates we consider. We follow Schmalbrock et al. (1983) and set the uncertainty factor in the 2. STEADY STATE HELIUM ACCRETION 20Neð; Þ24Mg and 24Mgð; Þ28Si reaction rates to 0.1; In solving for a NS envelope, we closely follow the model screening is included, following Salpeter & Van Horn (1969). described by Brown & Bildsten (1998). We assume that pure Since helium-rich accreting material is most likely from a helium is accreting onto a NS and burning into heavier ele- helium white dwarf companion of the NS, a more realistic ments. It initially burns into 12C via the triple- process and model should include a fraction of 14N in the composition. then into heavier elements (16O, 20Ne, 24Mg, and 28Si) via This is the most abundant heavy nucleus in the accreting subsequent -captures. The envelope consists of three layers: material, as this nucleus is the pile-up point during the cata- a layer of pure accreted helium, which is the ‘‘atmosphere’’; a lytic CNO cycle in the progenitor of the helium white dwarf. thin layer where the burning occurs; and a layer of heavy- Its abundance could be as large as 2%, and the only available element ashes, the ‘‘ocean.’’ Below this is the rigid NS crust, fusion reaction in the accreting NS envelope is 14Nð; Þ18F, which provides the bottom boundary for the modes. which releases 4.415 MeV. Using the rates from Caughlan & Fowler (1988) ( see also Couch et al. 1972), we integrated the 2.1. Finding the Envelope Profile continuity equation for 14N, finding that this element is 18 7 À2 We assume that the NS has a mass of M ¼ 1:4 M and a quickly converted to F at a shallow depth of 10 gcm . radius of R ¼ 10 km. Neglecting general relativity, the sur- Unless another capture occurs, the 18F will decay by pos- face gravitational acceleration is then g ¼ GM=R2 1:87 Â itron emission to 18O with a half-life of 109.8 minutes, creating 14 À2 10 cm s . The pressure scale height is h ¼ P=g 200 cm aslightgradientine (the mean molecular weight per elec- at the burning layer.

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