Draftversion February1,2008 PreprinttypesetusingLATEXstyleemulateapjv.04/21/05 INTERMITTENT MILLISECOND X-RAY PULSATIONS FROM THE NEUTRON-STAR X-RAY TRANSIENT SAX J1748.9–2021IN THE GLOBULAR CLUSTER NGC 6440 D. Altamirano1, P. Casella1, A. Patruno1, R. Wijnands1, M. van der Klis1 Draft versionFebruary 1, 2008 ABSTRACT We report on intermittent X-ray pulsations with a frequency of 442.36 Hz from the neutron-star X-ray binary SAX J1748.9–2021in the globular cluster NGC 6440. The pulsations were seen during both 2001 and 2005 outbursts of the source, but only intermittently, appearing and disappearing on 8 timescales of hundreds of seconds. We find a suggestive relation between the occurrence of type-I 0 X-ray bursts and the appearance of the pulsations but the relation is not strict. This behavior is 0 very similar to that of the intermittent accreting millisecond X-ray pulsar HETE J1900.1–2455. The 2 reasonforthe intermittence ofthe pulsationsremainsunclear. Howeveritisnowevidentthatastrict n division between pulsating and non-pulsating does not exist. By studying the Doppler shift of the a pulsationfrequencywedetermineanorbitwithaperiodof8.7hrsandanprojectedsemimajoraxisof J 0.39lightsec. The companionstar mightbe a main–sequenceora slightlyevolvedstarwith a massof 7 ∼1 M⊙. Therefore, SAX J1748.9–2021has a longer period and may have a more massive companion star than all the other accreting millisecond X-ray pulsars except for Aql X-1. ] Subject headings: binaries: general–pulsars: individual (SAX J1748.9–2021)–stars: neutron h p - o 1. INTRODUCTION Theexactformationmechanismsbehindthepulsations r of these three sources remains unknown. The existence Accreting millisecond pulsars (AMPs, Alpar et al. t s 1982; Backer et al. 1982) are transient low mass X-ray ofintermittentpulsationswithasmalldutycycleimplies a thatmanyotherapparentlynon-pulsatingLMXBsmight binaries (LMXBs) that show X-ray pulsations during [ bepulsating,bridgingthegapbetweenthesmallnumber their outbursts. A total of nine AMPs out of 100 of AMPs and the large group of non-pulsating LMXBs. 2 non-pulsating LMXBs have been found to date. The v reason why only this small subgroup of binaries pul- We are performing a detailed analysis of all RXTE 6 sates is still unknown. The first seven AMPs discov- archivaldata of neutron-star LMXBs to search for tran- 1 ered showed persistent X-ray pulsations throughout the sientpulsationsintheirX-rayflux(seealsoCasella et al. 3 2007). Inthis Letter wepresentthe results ofoursearch outbursts. Recently Kaaret et al. (2006) discovered the 1 onthe three X-rayoutbursts observedfromthe globular AMP HETE J1900.1–2455, which has remained active 8. for more than 2 years2 but showed pulsations only in- cluster NGC 6440. 0 termittently during the first ∼ 2 months of activity 2. THENEUTRON-STARTRANSIENTSAXJ1748.9–2021 IN 7 (Galloway et al. 2007). From the transient source Aql NGC6440 0 X-1 pulsations were detected (Casella et al. 2007) only v: for ∼ 150 sec out of the ∼ 1.3 Msec the source has (so NGC 6440 is a GC at 8.5 ± 0.4 kpc (Ortolani et al. 1994). Bright X-ray outbursts from a LMXB were i far) been observed with the Rossi X-ray Time Explorer X reported in 1971, 1998, 2001 and 2005 (Markert et al. (RXTE). 1975; in ’t Zand et al. 1999; Verbunt et al. 2000; r Gavriil et al.(2006,2007)recentlyreportedonthe de- a tectionof∼442.3Hz pulsations inanobservationofthe in’t Zand et al. 2001; Markwardt & Swank 2005). in’t Zand et al. (2001) from X-ray and optical obser- 2005outburstofatransientsourceintheglobularcluster vations concluded that the 1998 and 2001 outbursts (GC) NGC 6440. The pulsations followed a flux decay were from the same object, which they designated observed at the beginning of the observation and were SAX J1748.9–2021. reminiscent of those observed during superbursts; how- ever,asGavriil et al.(2007)suggest,theycouldalsobea 3. OBSERVATIONS,DATAANALYSISANDRESULTS detection from a new intermittent accreting millisecond We used data from the RXTE Proportional pulsar. Kaaret et al.(2003)reporta409.7Hzburstoscil- Counter Array (PCA, for instrument information lationinanX-raytransient(SAXJ1748.9–2021)located see Jahoda et al. 2006). Up to July, 2007, there were also in NGC 6440 and this GC harbors at least 24 X- 27 pointed observations of SAX J1748.9–2021, each ray sources (Pooley et al. 2002), so Gavriil et al. (2007) covering 1 to 5 consecutive 90-min satellite orbits. concluded that the burst oscillations and the pulsations Usually, an orbit contains between 1 and 5 ksec of were probably coming from different X-ray transients in useful data separated by 1–4 ksec data gaps due to the same GC. Earth occultations and South Atlantic Anomaly pas- 1Email: [email protected] ; Astronomical Institute, “An- sages. Adopting a source position (α = 17h48m52s.163, ton Pannekoek”, University of Amsterdam, and Center for High δ = −20o21′32′′.40; J2000 Pooley et al. 2002) we con- Energy Astrophysics, Kruislaan 403, 1098 SJ Amsterdam, The verted the 2–60 keV photon arrival times to the Solar Netherlands. 2 Atthetimeofsubmittingthisletter,thesourceisstillactive. Systembarycenterwiththe FTOOLfaxbary,whichuses the JPL DE-200 ephemeris along with the spacecraft 2 0.3 0.3 1998 2001 2005 0.25 X−ray Bursts −−> 0.25 ab) 0.2 0.2 Cr sity ( 0.15 0.15 n e nt 0.1 0.1 I Pulsations −−−−−−−−−−−−−−> 0.05 0.05 0 0 51045 51055 52140 52160 52180 52200 52220 53500 53520 53540 53560 53580 Time (MJD) Time (MJD) Time (MJD) Fig. 1.—Intensity(2.0–16.0keV)normalizedbyCrabvs. timeofthethreeoutbursts. Graysymbolsshowthe16-secaveragedintensity duringthe pointed PCA observations. The continuous lineshows the ASM lightcurve. Black marks at the top markthe times of type-I X-ray bursts. Black marks at the bottom mark the times when we detect significant pulsations. Years of the outburst is shown in each panel. TABLE 1 Timing parametersforNGC6440 Parameter Value Orbitalperiod,Porb(hr) ................................ 8.764(6)hr Projectedsemimajoraxis,axsini(lightsec.)............. 0.39(1) Epochof0o meanlongitude1,T0 (MJD/TDB) .......... 52190.047(4) Eccentricity,e .......................................... <0.001 Spinfrequencyν0 (Hz) .................................. 442.361(1) Pulsarmassfunction,fx (×10−4M⊙).................... ≃4.8 11883400 1MTinhiemmuemancolmonpgaitnuiodnemata0sso,iMnca(cMirc⊙u)la.r..o.r.b.i.t..c.o.r.r.e.s.p.o.n.d..s &0.1 Rate (c/s) 1111788890120000 totheascendingnodeoftheorbit. 1780 1770 1760 ephemeris and fine clock corrections to provide an 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Pulse phase absolute timing accuracy of 5-8 µs (Rots et al. 2004). We performed a Fourier timing analysis using the high-time resolution data collected in the Event (E 125us 64M 0 1s)and the GoodXenon modes. Power Fig. 2.— Top: Dynamicalpowerspectrumofobservation60035- spectra were constructed using data segments of 128, 02-03-00 showing intermittent pulsations (contours). In the light 256 and 512 seconds and with a Nyquist frequency of curve(line)threeX-rayburstsareseen. Thepulsefrequencydrifts 4096 Hz. No background or dead-time corrections were due to orbital Doppler modulation. The lowest contour plotted correspondstoLeahypower13andthehighestto55. Thecontours made prior to the calculation of the power spectra, but were generated from power spectra for non-overlapping 128 sec all reported rms amplitudes are background corrected; intervals of data. Bottom: Leahy normalized (Leahyetal. 1983) deadtime corrections are negligible. powerspectrumof512secofdatacentered∼7ksecafterthestart of this observation. Maximum Leahy power is 102, corresponding 3.1. Colors, light curves and states toasingle-trialprobabilityof∼7·10−23givenPoissonianstatistics in the photon arrival times (vanderKlis 1995). Inset: The 2–60 FromtheStandard2data(Jahoda et al.2006),wecal- keVlightcurvefoldedatthe2.26-msperiod. Twocyclesareplotted forclarity. Thepulseprofileissinusoidal,witha95% upper limit culatedcolorsandintensitieswithatimeresolutionof16 of0.4%(rms)ontheamplitudeofthesecondharmonic. seconds and normalized by Crab (e.g Altamirano et al. 2007). ThePCAobservationssamplethreedifferentout- the significance for their detection to be .2.5σ. bursts(seeFig.1). Thecolor-colordiagramsshowapat- 3.2. Pulsations tern (not plotted) typical for atoll sources. The power spectralfitsconfirmtheidentificationofthesestates(see We inspected each power spectrum for significant fea- also Kaaret et al. 2003). We looked for kHz QPOs, but tures. We found several, at frequencies ∼ 442.3 Hz found none. in 7 observations: 60035-02-02-04/05/06, 60035-02-03- No thermonuclear bursts were detected in the first 00/02/03duringthesecondoutburstand91050-03-07-00 outburst, sixteen during the second (Kaaret et al. 2003; duringthethirdoutburst(seealsoGavriil et al.2007,for Galloway et al. 2006) and four during the third one. We a detailed analysis of this observation). in’t Zand et al. searchedforburstoscillationsduringallburstsinthe15– (2001)concludedthat the 1998and2001outburstsfrom 4000 Hz frequency range but found none. Kaaret et al. theLMXBinNGC6440werefromthesamesource(Sec- (2003)reporteda∼4.4σ burstoscillationat∼409.7Hz. tion 2). Since pulsations are detected in both the 2001 Wefindtheseauthorsunderestimatedthe numberoftri- and2005outbursts,wecannow concludethat these two als by a factor of at least 180, as their estimate did not outbursts are also from the same source. Hence, all out- take into account the number of X-ray bursts analyzed burstsobservedfromNGC 6440overthe lastdecade are and the fact that a sliding window was used to find the from SAX J1748.9–2021. maximum power. Moreover, we also found that the dis- The pulsations are detected intermittently, appearing tribution of powers is not exponential as these authors and disappearing on time scales of hundreds of seconds. assumed. Taking into account these effects we estimate The appearance of pulsations seems to be related to the 3 442.4 0.05 2 0.04 442.39 0.03 442.38 )⊙ 1.5 0.02 R Frequency (Hz) 444444222...333567 - 000..0011 Residuals (Hz) mpanion radius R (c 1 32° 16.8° 11.9° 9.5° 8.0° 7.0° 6.3° -0.02 Co 0.5 442.34 SAX J1748.9-2021 -0.03 ZAMS 8 Gyrs 442.33 -0.04 12 Gyrs 0.01 Gyrs 0 442.32 -0.05 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 Companion mass Mc (M⊙) Phase Fig. 3.— Pulse frequency as a function of orbital phase. The Fig. 4.— Mass–radius relationship for a Roche lobe-filling plot has been obtained by folding all data between the first and companion (continuous line), isochrones of 0.01, 8 and 12 Gyrs lastpulsedetection in2001. Pulsations weredetected during6of with solar metallicity (triangles, crosses and squares, respectively the 18 orbital cycles covered. Drawn curve is the best-fit orbital ; Girardietal. 2000) and theoretical Zero-age main sequence model,measuredfrequenciesandpost-fitresidualsareshown. The (ZAMS,dashed line;Toutetal.1996). Black circlesmarkthe in- residuals’r.m.s. is1.2×10−3 Hz. clination of the system as estimated by the mass function of this system. occurrenceoftype-IX-raybursts,butthe relationisnot strict. The first two bursts were observedin an observa- 4. DISCUSSION tiononOctober8th 2001;thefirstpulsationsadaylater. We have discovered intermittent pulsations from the During the third outburst we detect four bursts; pulsa- neutron-star LMXB SAX J1748.9–2021. Pulsations ap- tions wereonlydetected afterthe third one. We alsode- pearanddisappearontimescalesofhundredsofseconds. tectedpulsationswithnoprecedingburst. Thestructure Although we find a suggestive relation between the ap- of our data does not allow us to tell if pulsations and/or pearance of the pulsations and the occurrence of type-I other bursts occurred during data gaps. Figure 2 (top) X-ray bursts (the pulsations appearing after a burst), illustrates the relation between pulsations and bursts. the relation is not strict. We find bursts with no subse- The amplitude of the pulsations varies strongly between quent pulsations and pulsations with no preceding burst ∼2%and(often)undetectable (0.3%rmsamplitudeup- (although a burst could have occurred in the preceding per limit at the 95% confidence level). Pulsations are datagaps). FromtheDopplershiftsonthepulsationswe seen right after the occurrence of the first and the third determinethatthesystemisinanear-circularorbitwith burst,butinthemiddleofthedatapulsationsarepresent periodof 8.7hours andprojectedradius of0.39lightsec. without the detection of a preceding burst (although a Thestability ofthe pulsations(after correctingfor the burst could have happened just before the start of this binary orbit) strongly suggests that the pulsation fre- data segment). In Figure 2 (bottom) we show a power quency reflects the neutronstar spin frequency and that spectrum and corresponding 2–60 keV pulse profile (in- SAXJ1748.9–2021isanaccretingmillisecondX-raypul- set). In these data the pulsation is relatively hard; the sar. The characteristics of the pulsations are reminis- rmsamplitudeincreaseswithenergyfrom∼1%at3keV cent of the those found in HETE J1900.1–2455: in both to ∼3% at 13 keV. sources the pulsations were only intermittently detected The 2–10 keV luminosity during the observations in and a possible relation between burst occurrence and which we detected pulsations was between 3 and 4 × pulse amplitude exists. However, there are differences: 1037 ergs s−1 (assuming a fixed NH = 8.2×1021 cm−2; in HETE J1900.1–2455 the pulsations were only seen in’t Zand et al.2001). Otherobservationsatsimilarflux during the first two months of the outburst and their and those at higher ( up to ≃ 5×1037 ergs s−1 in ob- amplitude decreased steadily on timescales of days after servation 91050-03-06-00) and lower fluxes do not show the bursts which might have caused them to reappear pulsations(seeFig.1). From16,32,64and128secaver- (Galloway et al.2007). InSAXJ1748.9–2021wefindthe age colors we found no significant changes in the energy pulsations in the middle of the 2001 and 2005 outbursts spectra correlated with the pulse-amplitude variations. and not in the beginning. Furthermore, the amplitude Westudiedthepulsefrequencydriftsusingpowerspec- of the pulsations behaves erratically, switching between tra of 128, 256 and 512 sec data and find a clear 8.7 detection and non-detection on time scales of hundreds hours sinusoidal modulation which we interpret as due ofseconds. Despite thesedifferences,the behaviorofthe to Doppler shifts by binary orbital motion with that pe- pulsationsinbothsourcesissosimilarthatweconsiderit riod. Inordertoobtainanorbitalsolution,weperformed likelythatthesamemechanismcausestheintermittency aχ2scanontheorbitalparametersusingthemethodde- of the pulsations in both. scribedbyKirsch et al.(2004)andPapitto et al.(2005). A related system might be Aql X-1 in which a short- Our best estimates are listed in Table 1. The combi- lived(∼150s)andveryrare(dutycycleof0.03%)episode nation of data gaps and intermittency of the pulsations of strong pulsations at the neutron-star spin frequency yielded aliases, which are taken into account by the re- has been detected (Casella et al. 2007). In this source, portederrors. InFigure3weplotthe pulsefrequencyas no X-rays bursts were seen in the ∼ 1400 s before the a function of orbital phase. pulsations, making it unlikely that they were triggered 4 by a burst. It is unclear if the pulsations in Aql X-1 Hydrodynamic flows in the surface layer of the neu- wereaccretion–drivenor due to unusualnuclear burning tron star may screen the magnetic field (see review by episodes; the same applies to SAX J1748.9–2021. The Bhattacharya 2002, and references within); perhaps vi- extreme rarity of the pulsations in Aql X-1 could indi- olent processes like bursts temporarily affect such flows, cate that the mechanism behind them is different from diminishing screening and enhancing the channeling. that responsible for the pulsations in HETE J1900.1– Alternatively, variations in a scattering or screen- 2455 and SAX J1748.9–2021. Nevertheless, irrespective ing medium may cause the pulse amplitude modu- of the mechanisms behind the pulsations in these three lation (see e.g. discussions in Psaltis & Chakrabarty sources,itisclearthatastrictdivisionbetweenpulsating (1999), Titarchuk et al. (2002), G¨og˘u¨¸s et al. 2007, and non-pulsating sources cannot be made anymore. It Titarchuk et al. 2007, Casella et al. 2007 and references is possible that all sources pulsate occasionally although within). Forourresults,thepropertiesofsuchamedium the recurrence times could be very long. shouldchangeontimescalesofhundredsofseconds;note Assuming a constant dipolar magnetic field, following that we did not detect spectral changes associated with Psaltis & Chakrabarty(1999)(i.e.,assumingageometri- pulse strength modulation. cally thin disk and neglecting inner disk wind mass loss, With anorbitalperiodof ∼8.7 hours,this binary sys- radiationdragandGReffects)weestimatethemagnetic tem is clearly not an ultra-compact binary as usually field to be B & 1.3×108 Gauss. This assumes a 10 km found in globular clusters and in fact, SAX J1748.9– radius 1.4M⊙ neutron star and M˙max, the highest ac- 2021 is the AMSP with the longest orbital period cretion rate at which pulses are detected, of 0.28 of the after Aql X-1, which has an orbital period of ∼ Eddington critical value as derived from the luminosity 19 hrs (Chevalier & Ilovaisky 1991; Welsh et al. 2000). observedatthetimeusingabolometricfluxcorrectionof The mass-radius relation for a low-mass Roche lobe- 1.4 (Migliari & Fender 2006). In the standard magnetic filling companion in a binary (Eggleton 1983) is Rc = channelingscenario,thequestionremainsofwhatcauses 0.24M1/3 q2/3 (1+q)1/3 P2/3/(0.6q2/3+ℓog(1+q1/3)), NS hr the large variations in pulse amplitude. with P the orbital period in hours, M the mass hr NS Comparisons between HETE J1900.1–2455, of the neutron star, R radius of the companion and c SAX J1748.9–2021 and the other 7 AMSPs can q = M /M , the mass ratio. Given the mass function c NS provide clues to understand the pulse-strength vari- and the orbital period and assuming a 1.4M⊙ neutron ations. In SAX J1748.9–2021 and Aql X-1 the time star,we plotinFigure4 the mass-radiusrelationshipfor scales on which the pulse amplitude can fluctuate are the companion star. Given that the age of the Globular as short as ∼ 102 s, too short for the properties of the ClusterNGC 6440is 10±2Gyrs(Santos & Piatti2004) neutron star core to change (Galloway et al. 2007). So, anditsmetallicityisapproximatelysolar(Ortolani et al. these changes must originate in the disk or the outer 1994), in Figure 4 we also plot the isochrones for stars layers of the neutron star envelope. Galloway et al. with ages of 8 and 12 Gyrs and solar metallicity. Stars (2007) suggested (for HETE J1900.1–2455) that the with a Mc < 0.85M⊙ cannot fill the Roche lobe while accumulation of matter on the surface burying the stars with Mc > 0.95M⊙ would have a radius exceed- magnetic field (Cumming et al. 2001) plays a role. Our ing the Roche lobe. This wouldimply a donor star mass results show that this mechanism probably cannot of 0.90±0.05M⊙. However, for masses of 0.95–1.1M⊙, work for SAX J1748.9–2021, as the pulsations are not stars have evolved off the main sequence so binary mass seen in the beginning of the outbursts, but instead transfer can have affected the radius of the donor star, ∼ 3 weeks and ∼ 5 weeks after the start of the 2001 which means we cannot firmly exclude masses of 0.95– and 2005 outbursts, respectively, so after a considerable 1.1M⊙. Therefore a more conservative mass range for amount of matter has already accreted. Interestingly, the donor star is 0.85-1.1M⊙. Intriguingly, this requires we observe pulsations only around a mass accretion theinclinationtobeabout9o,whichhasa.1%apriori rate of ≃ 2 × 10−16 M⊙/sec as inferred from the probabilityforanisotropicsampleofbinaryinclinations. X-ray luminosity, not above or below, indicating that Of course, this estimate is assuming that SAX J1748.9– instantaneous mass accretion rate rather than total 2021 is in a primordial binary. If a different evolution- accreted mass is the important quantity. ary path took place (e.g. dynamical interactions), the In both HETE J1900.1–2455 and SAX J1748.9–2021 mass of the companion might be much smaller (see e.g. the pulsations seem to appear together with bursts al- van Zyl et al. 2004). though the exact connection is complex. This suggests that surface processes may affect the magnetic field. REFERENCES AlparM.A.,ChengA.F.,etal.,1982, Nature,300,728 Gavriil F.P., Strohmayer T.E., et al., Sep. 2006, In: Bulletin of AltamiranoD.,vanderKlisM.,etal.,2007, SubmittedtoApJ the American Astronomical Society, vol. 38 of Bulletin of the BackerD.C.,KulkarniS.R.,etal.,1982,Nature,300,615 AmericanAstronomicalSociety, 336–+ Bhattacharya D., Jun. 2002, Journal of Astrophysics and GavriilF.P.,StrohmayerT.E.,etal.,Nov.2007,ApJ,669,L29 Astronomy,23,67 GirardiL.,BressanA.,etal.,Feb.2000,A&AS,141,371 CasellaP.,AltamiranoD.,etal.,2007,ApJLaccepted Go¨g˘u¨¸s E.,AlparM.A.,GilfanovM.,2007,ApJ,659,580 ChevalierC.,IlovaiskyS.A.,Nov.1991,A&A,251,L11 in’tZandJ.J.M.,VerbuntF.,etal.,1999,A&A,345,100 CummingA.,ZweibelE.,BildstenL.,2001,ApJ,557,958 in’tZandJ.J.M.,vanKerkwijkM.H.,etal.,2001, ApJ,563,L41 EggletonP.P.,1983, ApJ,268,368 JahodaK.,MarkwardtC.B.,etal.,2006,ApJS,163,401 GallowayD.K.,MunoM.P.,etal.,2006,ArXive-prints KaaretP.,ZandJ.J.M.i.,etal.,2003,ApJ,598,481 GallowayD.K.,MorganE.H.,etal.,2007,ApJ,654,L73 KaaretP.,MorganE.H.,etal.,2006, ApJ,638,963 KirschM.G.F.,MukerjeeK.,etal.,Aug.2004,A&A,423,L9 LeahyD.A.,DarbroW.,etal.,1983,ApJ,266,160 5 MarkertT.H.,BackmanD.E.,etal.,1975, Nature,257,32 ToutC.A.,PolsO.R.,etal.,1996,MNRAS,281,257 Markwardt C.B., Swank J.H., 2005, The Astronomer’s Telegram, vanderKlisM.,1995,ProceedingsoftheNATOAdvancedStudy 495,1 Institute on the Lives of the Neutron Stars, held in Kemer, MigliariS.,Fender R.P.,2006,MNRAS,366,79 Turkey, August19-September 12,1993. Editor(s), M.A.Alpar, OrtolaniS.,BarbuyB.,BicaE.,1994,A&AS,108,653 U. Kiziloglu, J. van Paradijs; Publisher, Kluwer Academic, PapittoA.,MennaM.T.,etal.,Mar.2005,ApJ,621,L113 Dordrecht,TheNetherlands,Boston,Massachusetts, 301 PooleyD.,LewinW.H.G.,etal.,2002,ApJ,573,184 vanZylL.,CharlesP.A.,etal.,May2004, MNRAS,350,649 PsaltisD.,ChakrabartyD.,1999,ApJ,521,332 VerbuntF.,vanKerkwijkM.H.,etal.,2000,A&A,359,960 RotsA.H.,JahodaK.,LyneA.G.,2004, ApJ,605,L129 WelshW.F.,RobinsonE.L.,YoungP.,Aug.2000,AJ,120,943 Santos J.F.C.Jr.,PiattiA.E.,2004,A&A,428,79 TitarchukL.,CuiW.,WoodK.,2002,ApJ,576,L49 TitarchukL.,KuznetsovS.,ShaposhnikovN.,2007,ArXive-prints, 706