arXiv:1611.06023v1 [astro-ph.HE] 18 Nov 2016 1998 age l 2013 al. et Wang iharcrec ieo prxmtl – er.The years. 3–4 approximately of time recurrence a with Suzaku like 2009 2015 2001 2012 observed far, so outbursts and eight time shown high has It (see members review). known a 18 all of ( ( 1996 Hz in 401 at spinning is that tadsac f35kc( kpc 3.5 of distance a at A J1808.4 SAX e.g., otnn&Gierli´nski 2003 & Poutanen or ( hours A 90A,Diglo h Netherlands the Dwingeloo, AA, 7900 m,Uiest fLietr ecse E R,UK 7RH, LE1 Leicester Leicester, of University omy, otapo O71J UK 1BJ, SO17 Southampton rebl,M 07,USA 20771, MD Greenbelt, 54C teh,teNetherlands the Utrecht, CA 3584 cec ak94 08X,Asedm h Netherlands The Amsterdam, XH, 1098 904, Park Science edn h Netherlands The Leiden, h crtn ilscn -a usr(AMXP) pulsar X-ray millisecond accreting The D rpittpstuigL using typeset Preprint lessandro atversion raft 7 6 5 4 3 2 1 RXTE n riigis0.05-0.08 its orbiting and ) ,isaeidctmn aiblt ( ( ; properties variability spectral X-ray timing and ) aperiodic , its ), hoeia srpyisGop eateto hsc and Physics of Department Group, Astrophysics Theoretical nvriyo otapo,Sho fPyisadAstronomy and Physics of School Southampton, of University srpyisSineDvso,NS odr pc lgtC Flight Space Goddard NASA Division, Science Astrophysics RNNtoa nttt o pc eerh Sorbonnelaan Research, Space for Institute SRON-National no anke nttt o srnm,Uiest fAms of University Astronomy, for Institute Pannekoek Anton SRN h ehrad nttt o ai srnm,Pos Astronomy, Radio for Institute Netherlands the ASTRON, ednOsraoy ednUiest,NisBhwg2 23 2, Bohrweg Neils University, Leiden Observatory, Leiden ata ta.2008 al. et Hartman AI US ERHADXRYMNTRN FSXJ1808.4 SAX OF MONITORING X-RAY AND SEARCH PULSE RADIO arn ta.2009b al. et Patruno a loe hruhsuyo h ustos(see pulsations the of study thorough a allowed has 2003 hkaat ogn1998 Morgan & Chakrabarty eas n htteobto A 10.-68cniusevol continues J1808.4-3658 SAX of orbit the that find also We erpr h eut fade ai usto erha GHz 2 at search pulsation radio deep a of results the report we ujc headings: Subject ayn asqarpl nue yete tog( G fie kG) (1 strong a either by induced mass-quadrupole varying hvo fohraceigadnnaceigbnre.W d ( We inertia of binaries. moment non-accreting large and a accreting other to observa of limit bhevior Chandra strongest 30-ks the place a and Chandr of pulsations, with radio analysis outburst no timing 2015 detect the X-ray of the monitoring present X-ray an and 2014 eeaigms-oswt ne an with mass-loss generating ikdt irradiation. to linked n’ ade l 1998 al. et Zand ’t in usr icmtneta ih rvd ikbtenAMX quie between in link when a that, provide might suggested that been circumstance also beh a conservativ has pulsar, this by It for predicted evolves. responsible than system mechanism pace underlying faster the much identify a at proceeds , h crtn ilscn -a usr(MP A J1808.4 SAX (AMXP) pulsar X-ray millisecond accreting The XMM-Newton ; P − atruno N 68i naceignurnsa located star neutron accreting an is 3658 / ovember rseta eouino -a telescopes X-ray of resolution spectral or arn ta.2009c al. et Patruno .Ti orewsdsoee by discovered was source This ). A T E 1. 1,2 tl mltajv 5 v. emulateapj style X A , INTRODUCTION 1 2016 21, mruta ; , ). udr ta.2009 al. et Burderi iais eea tr:idvda SXJ1808.4 (SAX individual stars: — general binaries: INTEGRAL ; in—Xry:bnre -as tr tr:pulsar stars: — stars X-rays: — binaries X-rays: — tion akt ta.2009 al. et Cackett n stebs tde AMXP studied best the is and ) alwy&Cmig2006 Cumming & Galloway J aodand arn at 2012 Watts & Patruno M I inns&vndrKlis der van & Wijnands / 2 > / ⊙ ; 2,3 11 ; irink ta.2002 al. Gierli´nski et , ffi 1 opno n2.01 in companion L eoee l 2008 al. et Deloye Chandra ut&vndrKlis der van & Bult . inyo 0 rtednrsa sudron us-ylcva quasi-cyclic undergoing is star donor the or 40% of ciency ucien 7 x 10 ; innse al. et Wijnands 45 K ; W arn tal. et Patruno ait tal. et Papitto uiper cm g , ijnands BeppoSAX rf eso oebr2,2016 21, November version Draft Swift 4 2 P , n saltn h oo b sn t pndw oe)thu power) spin-down its using (by donor the ablating is and ) Astron- terdam, 3CA, 33 ,NL- 2, bs2, tbus EVOLUTION? 3 eter enter ABSTRACT and M , for ) ; ; cilvnder van ichiel , B ult 3,5 -a raito oigfo h crto disk accretion the from coming irradiation X-ray ( ( ako togsi pdrn h ubrt ( outbursts the during up spin strong a of lack enseuae htapla idipnigo h donor the on impinging wind ( pulsar a surface that speculated been optbewt antcdpl nryls sraemag- (surface loss energy field dipole netic magnetic with compatible am-a onepr fSXJ1808.4 SAX of counterpart gamma-ray 2008 J1808.4 SAX for 2009 fe the but behavior, this explain to required is J1808.4 irradiation SAX of counterpart optical the J1808.4 SAX that evidence observational indirect also 2008 2008 and systems estab- pulsar well radio now redback is period and black-widow orbital luminosity the in excess at lished observed modulation the tical for responsible be 2001 oeigo the of modeling ( quiescence during surface mode brown-dwarf non-irradiated ( a to respect far with luminous so no although detected quiescence, been during have pulsar radio pulsations a radio as on turns es ihu ai ustosbigosral;te a b may they observable; th e.g., see being in th material; operating pulsations intra-binary possible mechanism radio also pulsar is without radio it least the on, without turning active pulsar is radio a for evidence ati ofimd bu 0 ftesi-oneeg strans- is energy spin-down the of 30% about confirmed) is part rtne l 2013 al. et Breton 2001 Chakrabarty & Bildsten aeo h usdrdoflxdniyo n AMXP. any of density flux radio pulsed the on date J , 8 h oeettmn fteplain a eeldthe revealed has pulsations the of timing coherent The vo eas tcnhl oudrtn o this how understand to help can it because avior ason sustoseais ihrtenurnsa has star neutron the either scenarios: two iscuss lhuhterqieeto usrwn sotnmentioned often is wind pulsar a of requirement the Although eOn ihlie al. et O˜na Wilhelmi de do ysm ndnie ehns probably mechanism unidentified some by or ld ; has It luminosity. donor the for account cannot ) ; ; , cne A 10.-68trso saradio a as on turns J1808.4-3658 SAX scence, sadbakwdwrdoplas nti work this In pulsars. radio black-widow and Ps iayeouinmdl.I sipratto important is It models. evolution binary e ineeue uigte21 ubrt We outburst. 2015 the during executed tion age l 2013 al. et Wang 2009 ata ta.2009 al. et Hartman − K udr ta.2003 al. et Burderi iga atpc n ecmaei othe to it compare we and pace fast a at ving H ,Sit n NERL npriua,we particular, In INTEGRAL. and Swift, a, sn h re akTlsoei August in Telescope Bank Green the using 68sosapcla ria vlto that evolution orbital peculiar a shows 3658 lis essels udr ta.2003 al. et Burderi B 3 n osatsi-oni uecneta is that quiescence in spin-down constant a and ) − ≈ 68 tr:nurn—sas rota- stars: — neutron stars: — 3658) − 2,3 10 68 HTCUE T ORBITAL ITS CAUSES WHAT 3658: FERMI C , − 8 68todrn uecne( quiescence during too 3658 n oehn iia a enfound been has similar something and ) hristian ,see G, .I eetworks, recent In ). adn ta.2016 al. et Jaodand / A aaipyta i h counter- the (if that imply data LAT ; ata ta.2008 al. et Hartman ; K ( apn ta.2004 al. et Campana 2016 oe ta.2001 al. et Homer nigge arn ta.2012 al. et Patruno ; uigti hs.Asuc of source A phase. this during ) apn ta.2004 al. et Campana aeietfidapossible a identified have ) 6 A itosdet a to due riations ndrew ) − 68adspectral and 3658 .K R. ige al. et Xing − 68i over- is 3658 ; ; ata tal. et Hartman ing iSloe al. et Salvo di tapla wind pulsar a at .Teeis There ). / enee al. et Heinke eoee al. et Deloye ytm(rat (or system e oe tal. et Homer eto star neutron bcrdby obscured e .Indeed, ). 7,1,3 s might ) sindirect as 8 R , Op- . − ( 2015 udy 3658 eble ) l 2 Patruno et al. formed into gamma-rays, providing further evidence in favor tion: does SAX J1808.4−3658 really turn on as a radio pul- of this scenario, although no gamma-ray pulsations have been sar during quiescence? A detection could help to solve two found so far. problems: the first is that it would allow a continuous mon- The orbital evolution of SAX J1808.4−3658 shows an in- itoring of the orbital evolution not only during outbursts but crease of the orbital period on a relatively short timescale also during quiescence. The second is that it can allow the pre- of ∼70 Myr (Hartman et al. 2008; di Salvo et al. 2008) and cise measurement of the spin-down power of the pulsar which an acceleration of the rate of expansion of the orbit up might play a fundamental role in the ablation of the compan- to 2011 (Patruno et al. 2012). The driving mechanism for ion. The X-ray monitoring campaign was made with the Swift the evolution of a binary with a ∼2 hr orbital period like X-Ray Telescope (XRT), the INTEGRAL and Chandra tele- SAX J1808.4−3658 is expected to be angular momentum scopes during the 2015 outburst. We used the Chandra tele- loss due to gravitational wave emission and the expected or- scope to monitor the pulsations of the source and track the bital evolution timescale in this case is ∼4Gyr(Hartman et al. long term orbital evolution of SAX J1808.4−3658. The ques- 2008), which is almost two orders of magnitude longer than tion we seek to answer is whether the determination of a new the observed one. This behavior might require that an addi- orbital solution that includes also the 2015 outburst can pro- tional mechanism has a strong influence on the orbit, beside vide hints on the exact mechanism behind the rapid orbital the gravitational wave emission. evolution of the system. Anomalously fast orbital evolution is not unique to SAX J1808.4−3658. Several other low-mass X-ray bina- 2. X-RAY OBSERVATION AND DATA REDUCTION ries (LMXBs), comprising both neutron star and black To construct the outburst light curve we analyzed all hole accretors, are also observed to show a faster evolu- pointed Swift/XRT observations taken between April 1st, tion than expected (see Section 6 for an in-depth discus- 2015 (MJD 57113) and August 26, 2015 (MJD 57260). Dur- sion). There is also a similar behavior in many (non- ing this period 62 observationswere taken (ProgamIDs33737, accreting) binary radio millisecond pulsars with orbital pa- 33737 and 33801) in either window-timing mode (1.76-ms rameters similar to SAX J1808.4−3658, known as “black- resolution) or photon-counting mode (2.5-s). We extracted X- widows” (BWs), where the rotational power emitted in form ray count rates averaged per spacecraft orbit using the online of wind and radiation by the pulsar is impinging and ablat- Swift/XRT data products generator (Evans et al. 2007). To es- ing the semi-degenerate9 donor companion (Nice et al. 2000; timate the count rate to flux conversion ratio we also used this Doroshenko et al. 2001; Lazaridis et al. 2011). These systems tool to create and fit energy spectra (Evans et al. 2009), using have a companion star with a typical mass of . 0.1 M⊙ and the 0.3 − 10 keV energy and the default event grades. One several (but not all) of them have orbital periods of about 1–3 Type I (i.e., thermonuclear) X-ray burst was detected and ex- hours. Most BWs have very short binary evolution timescales cluded from our analysis (see Figure 1). too, orders of magnitude shorter than the expected theoretical We also used data recorded with the INTEGRAL space- values for their secular evolution. These short timescales vari- craft, which carries three high-energy instruments: a high- ations are believed to reflect some short-term effects rather angular resolution imager IBIS, a high-energy resolution spec- than the secular evolution of the binary. trometer SPI and an X-ray monitoring instrument JEM-X. For SAX J1808.4−3658 di Salvo et al. (2008) proposed a These instruments are equippedwith coded aperture masks en- scenario in which the pulsar wind, powered by the rota- abling image reconstruction in the hard X-ray/soft γ-ray band. tional spin-down of the neutron star in quiescence, causes Driven by sensitivity considerations, we used only data from the ejection of the gas flowing through the inner Lagrangian the INTEGRAL Soft Gamma-Ray Imager ISGRI (Lebrun et al. point L1 (radio-ejection scenario; see also Burderi et al. 2001, 2003), the upper detector layer of IBIS (Ubertini et al. 2003), 2009). Hartman et al. (2008, 2009) and Patruno et al. (2012) sensitive to photons with energies in the range ∼20 keV – 1 proposed an alternative mechanism where the binary evolu- MeV (effectively ∼300 keV). Typical integration times are tion is not necessarily driven by the matter expulsion, but in the range 1800–3600 s. We used the imaging software it is rather a quasi-stochastic process due to the develop- tools (Goldwurm et al. 2003) of the Offline Scientific Anal- ment of a significant mass quadrupole in the donor star that ysis (OSA) package version 10.1 distributed by the INTE- results in a coupling between the donor spin and the or- GRAL Science Data Centre (ISDC, see e.g. Courvoisier et al. bital period of the binary. Patruno et al. (2012) in particular 2003). INTEGRAL had the Galactic center/bulge region of- showed that the so-called Applegate mechanism (Applegate ten in its field of view during the April 2015 outburst of 1992; Applegate & Shaham 1994), which assumes that a mass SAX J1808.4−3658 and the initial part of the outburst was quadrupole develops in the donor star due to quasi-periodic properly sampled. The lightcurve for the 20-100 keV range magnetic cycles, seems to be a promising candidate surviving during the early phase of outburst is shown in Figure 2. Each the observational scrutiny. However, there is not yet any con- data point represents the averaged count rate of a combination clusive evidence about the exact operating mechanism behind of typical 3–5 Science Windows. The onset of the outburst, the orbital evolution of SAX J1808.4−3658. somewhat before MJD 57121, is clearly visible in this figure. Motivated by these facts we have conducted a deep radio For the timing analysis we use Chandra data taken with the pulse search of SAX J1808.4−3658 during quiescence and High Resolution Camera (HRC) with the HRC-S detector op- an X-ray monitoring during the last 2015 outburst. The deep erating in timing mode. The observation started on May 24, radio pulse search was done with the Greenbank Radio Tele- 2015 at 22:23:18 UT (MJD 57166.9) and ended on May 25, scope in August 2014 (during quiescence) to answer the ques- 2015 at 07:15:30 UT (MJD 57167.3) for a total exposure time of 29.6-ks. In this configuration the data are collected with 9 In this paper we use the term “semi-degenerate” instead of brown-dwarf a time resolution of 16 µs and very limited energy resolution. because the mass transfer process alters significantly the internal structure The Chandra data were processed with the CIAO software (v of these stars which are quite different from isolated brown dwarfs (Tauris 4.6) and were barycentered with the faxbary tool by using the 2011). most precise optical position available (Hartman et al. 2008) 17 years of evolution SAX J1808.4−3658 3 and the JPL DE405 solar system ephemeris. The pulse pro- were summed in quadrature, providing only total intensity. files are generated by folding the data in stretches of ∼ 2000s We acquired 60/30-min integration on 2014 August 9 and 22, in pulse profiles composed by 32 bins. respectively. The observational setup and offline data analysis The folding procedure uses the ephemeris reported in (see §4) were tested using the millisecond pulsar PSR J1824– Patruno et al. (2012) and extrapolates the solution to the time 2452A (M28A). Radio pulsations from M28A were easily re- of the Chandra observation. Given the low signal-to-noise covered at the known pulsar spin frequency of 327.4 Hz and of the observations we only measured the time of arrivals DM = 119.9pccm−1. of the fundamental pulse frequency (ν) which also prevents that pulse shape variability affects the fiducial point defining 4. RADIO DATA ANALYSIS the pulse time of arrival (ToA; see Hartman et al. 2008 and We began the data analysis by sequentially combining Patruno et al. 2010 for details of the procedure). To follow groups of three observational sub-integration of 322s each the evolution of the orbit and the pulsar spin we fit the ToAs (except last sub-ints in both 9th and 22nd August observations with the software TEMPO2 (Hobbs et al. 2006) and after ob- which lasted for 55 s and 187s, respectively) . This resulted taining a new ephemeris we re-fold the data and repeat the in four independentraw data sets of ∼16min each, i.e. 13% of procedure until convergence of the solution. SAX J1808.4−3658’s binary orbital period of 2.01hr in each We also created power density spectra of the Chandra data. case. Two raw datasets towards the end of 9th and 22nd Au- No background subtraction was applied to the data before gust observations were ∼11 and ∼13 mins long, respectively. calculating the power spectra. The Poissonian noise level The total integration time was sub-divided in this way in order was measured by taking the average power between 3000 to enable linear acceleration searches (see, §4.1), and because and 4000 Hz, a region dominated by counting statistics noise of the potential for eclipsing, which in analogy with the black alone. After obtaining the mean Poissonian value we sub- widow systems could last for at least 10% of the orbit. The ob- tracted it from the power spectra. servation start times, duration, and the corresponding orbital We used 128-s long segments to calculate the power spec- phases of SAX J1808.4−3658 are summarized in Table 1. tra so that our frequency boundaries are 1/128 Hz and 4096 Initial data preparation and periodicity searching was real- Hz. The powers were normalized in the rms normalization ized using PRESTO, a comprehensive pulsar processing soft- (van der Klis 1995) which gives the power density in units of ware developed by Scott Ransom (for details see, Ransom (rms/mean)2 Hz−1. We define the fractional rms amplitude 2001; Ransom et al. 2002, 2003). Radio frequency interfer- between the frequencies ν and ν as: 1 2 ence (RFI) was excised using an RFI mask generated with ν2 1/2 rfifind. Given that the DM towards SAX J1808.4−3658 rms = P(ν)dν (1) is unknown, we used prepsubband to generate RFI-masked, " # Zν1 barycentered, and de-dispersed time series over trial DMs −3 and calculate the errors from the dispersion of the data points ranging from 0–1000pccm (using a DM step size of −3 −3 in the power spectra. 0.1pccm up to a DM of 500pccm and a step size of 0.3 from 500–1000pccm−3, resulting in 6671 time series in to- 3. RADIO OBSERVATIONS tal). For each time series we created a corresponding Fourier power spectrum using realfft. The residual intra-channel SAX J1808.4−3658 was observed on two different occa- DM smearing was 41–81 µs (i.e. 1.6-3.2% of the pulse pe- sions, 2014 August 9 and 22 (MJD 56878 and MJD 56891, riod) for DMs of 100 − 200pccm−3, which corresponds to an respectively), using the 110-m Robert C. Byrd Green Bank approximate distance range of 3–6kpc in the NE2001 model. Telescope (GBT), in West Virginia. During this period, it was As described below, we searched the dedispersed time se- known to be in X-ray quiescence, with the previous outburst ries for pulsations using both a blind Fourier-based periodicity having ended in 2011, and the next outburst starting 2015 search and by directly folding the data using an X-ray-derived April. The data were recorded using the Green Bank Ulti- rotational and orbital ephemeris. mate Pulsar Processing Instrument (GUPPI) backend. This combination of GBT with GUPPI provides a high sensitivity to faint millisecond radio pulsations – arguably the deepest 4.1. Blind Fourier-based periodicity search search that can be done with current radio telescopes, given We first performed a blind periodicity search in the event that the source is well outside the Arecibo-visible declination that the X-ray derived ephemeris was inaccurate and to check range. the possibility for a serendipitous, and unrelated radio pulsar The distance to the source (∼3 kpc) suggests a relatively along the line of sight. high expected dispersion measure (DM& 100pccm−3, based The apparent rotational period of binary pulsars is Doppler on the NE2001 model of Cordes & Lazio 2002, 2003). Fur- shifted by their binary motion. This results in spreading of thermore, there is the potential for radio eclipses from intra- spectral power over multiple Fourier bins as z = aT 2/cP, binary material – in analogy with the rotation-powered black where z is the number of Fourier bins drifted, T is the inte- widow and redback millisecond pulsar systems, where the gration length, c is the speed of light, and P is the spin period. eclipse duration is typically longer at lower radio frequencies, For SAX J1808.4−3658, the maximum orbital acceleration e.g. Archibald et al. (2013). Hence, our observations were is a ≈ 14m/s (companion mass Mc∼0.05 − 0.08 M⊙), cor- conducted at a relatively high central observing frequency of responding to a drift of z = 18 bins in 16-min observations. 2 GHz to mitigate these effects, while still maintaining sen- As demonstrated by Ransom et al. (2001, 2002), such a sig- sitivity to the typically steep spectra of radio pulsars ( f −1.4, nal can be successfully recovered by searching over multiple where f is the frequency of the electromagnetic radiation; see linear frequency derivatives. We employed this technique of Bates et al. (2013)). GUPPI provided 800MHz of bandwidth, Fourier-based acceleration searches, using accelsearch and with 61.44 µs samples and 0.391MHz channels recorded as 8- searched zmax = 100 for all the 6671 Fourier power spectra bit samples in psrfits format. The orthogonal polarizations (§4) in each 16-min sub-integration. 4 Patruno et al.

Table 1 Green Bank Telescope Summary of Observations Integration Orbital Phase Obs No. Sub-int No. Obs. Start Obs. Start Date MJD Time (Mins.) Coverage S-band observations 1 1 2014-08-09 56878.149456 16.1 0.97-0.10 1 2 2014-08-09 56878.160641 16.1 0.10-0.24 1 3 2014-08-09 56878.171826 16.1 0.24-0.37 1 4 2014-08-09 56878.183011 11.6 0.37-0.46 2 5 2014-08-22 56891.022211 16.1 0.39-0.53 2 6 2014-08-22 56891.033396 13.8 0.53-0.65

We then identified the best candidates from the above accel- Each of the 6671 dedispersed time series for every 16-min eration searches using the ACCEL sift subroutine of PRESTO, sub-integration were then folded using prepfold and these which groups candidates found at different trial DMs. AC- 2 × 600 ephemerides. Moreover, the folding operation was CEL sift did not identify any candidates with a rotational pe- conducted in two additional ways: by allowing prepfold to riod close to that of SAX J1808.4−3658. Nonetheless, in case optimize the S/N in a narrow range of spin period and spin there was a serendipitous pulsar along the same line of sight, period derivative around the nominal ephemeris prediction for each of the ACCEL sift candidates, we folded the corre- and only allowing an optimization in spin period derivative. sponding de-dispersed time series and selected those folds Hence, at the end of these ephemeris-based searches we ob- showing a reduced-χ2 > 2 (this is used as a proxy for signal- tained 2 × 2 × 6671 × 600 = 16, 010, 400 folded profiles. We to-noise) to also fold the raw data. We then inspected the filtered the profiles by creating histograms of the S/N of the candidates by eye and used parameters such as signal to noise folds in each 16-min sub-integration and choosing only candi- ratio, measured DM, pulse profile and, converged period and dates above a certain threshold to inspect by eye. We found period derivative solution from prepfold output plot to make no candidate profiles with sufficient S/N that clearly peaked an informed selection. This inspection did not reveal any con- in both trial DM and ∆Tasc. vincing pulsar candidate from the blind search. 5. RESULTS 4.2. Direct Folding Search with X-ray-derived Ephemeris 5.1. X-Ray Lightcurve With a priori knowledge of the spin and orbital parameters, it is possible to perform a deeper search for radio pulsations SAX J1808.4−3658 was detected in outburst with compared to the blind search discussed above. Previous coher- Swift/BAT on April 9th, 2015 (MJD 57121, Sanna et al. 2015). ent timing analysis of SAX J1808.4−3658, enabled by its X- During the closest previous Swift/XRT observation, which ray pulsations during outbursts, provides such an ephemeris. occurred on April 3th (MJD 57195), SAX J1808.4−3658 However, the short-orbital-period black widow and red- was still in quiescence (Campana et al. 2015). The 0.3–10 back millisecond pulsar binaries are known to show non- keV X-ray lightcurve of SAX J1808.4−3658 (see Figure 1) deterministic orbital variations (see, Patruno et al. 2012; shows the very typical evolution that was also observed in the Breton et al. 2013; Archibald et al. 2013) and such variations other outbursts. The outburst has started after approximately should also be expected in the case of SAX J1808.4−3658, 3.5 years since the previous one, in line with the typical meaning that any previously derived ephemeris may not ex- recurrence time of 3–4 years. trapolate well to future observations. X-ray pulsation searches The Swift/XRT started monitoring SAX J1808.4−3658 af- in the redback transitional millisecond pulsar PSR J1023 + ter a Type I X-ray burst on April 11th (MJD 57123). The 0038 (e.g., Archibald et al. 2015; Jaodand et al. 2016) have source showed the same evolution seen in previous out- bursts, with an observed 0.3–10 keV peak flux of ≈3 × established that one can successfully account for such non- −9 −1 −2 deterministic orbital variations by searching over a small de- 10 erg s cm that (assuming a distance of 3.5 kpc; viation in the time of ascending node (T ). Therefore, when Galloway & Cumming 2006) corresponds to a luminosity of asc − folding the GBT radio data with X-ray derived ephemerides, 4 × 1036 erg s 1. The outburst showed an initial near exponen- we searched both over DM and a ∆Tasc value compared to the tial decay (slow decay) lasting about 15 days. It then tran- fiducial ephemeris value. sitioned into a faster linear decay for about 5 days when the Given the integration times of 1 and 0.5hr respectively dur- source reached a luminosity of ≈1035 erg s−1, before entering ing the first and the second observation epochs, we could a prolonged outburst reflaring tail that lasted another ∼ 100 ensure significant orbital coverage of SAX J1808.4−3658’s days. ∼2hr orbit (see Table 1). We used two known orbital During the outburst reflaring tail, typical of all previ- ephemerides: the one obtained from coherent timing analy- ous outbursts (van der Klis 2000; Wijnands et al. 2001, 2003; sis up to 2011 (Patruno et al. 2012) and the one obtained by Campana et al. 2008; Patruno et al. 2009c, 2016), the lumi- also including the 2015 outburst (§4.2). In addition, we varied nosity of the source oscillates between very faint states close 32 −1 36 −1 Tasc over a range of ±30s in stepsof0.1 s, resulting in 2 × 600 to 2×10 erg s and relatively brighter ones of ∼10 erg s . trial ephemerides per DM trial. Two bright reflares were seen in 2015 on May 13th (MJD 17 years of evolution SAX J1808.4−3658 5

57155) and May 18th (MJD 57164) after which several pro- Table 2 gressively weaker reflares followed on a cadence of five to ten SAX J1808.4−3658 Timing Solution for the 2015 Outburst days. The Chandra observation we report in this work took place during the second bright reflare. Parameter Value Stat. Error Syst. Error The power spectra of the Chandra data show no relevant ν [Hz] 400.9752067 1.1 × 10−6 ∼10−6 −6 feature at any frequency. We exclude the presence of a Tasc [MJD] 57167.025002 7 × 10 1 Hz modulation (similar to that observed in several previ- e < 0.003 (95% c.l.) ∗ −3 a1 sin i (lt-ms) 62.812 2 × 10 ous outbursts) with rms amplitude larger than 10% at the 95% ∗ −6 P b (s) 7249.156980 4 × 10 confidence level. This upper limit is derived by looking at ∗ −4 the power in the 0.05–10 Hz range as done for example in Eccentricity e (95% c.l.) < 1.2 × 10 Patruno et al. (2009c). Epoch (MJD) 52499.9602472 ∗ these values are taken from Hartman et al. (2009) and are kept fixed during the fit. 5.2. X-Ray Pulsations The X-ray pulsations are very clearly detected in each data 2008) and fit the time of passage through the ascending node segment at the 4-8σ level, where we define the significance as (which is equivalent to orbital phase zero) together with the the ratio between the pulse amplitude and its statistical error. measurements of the previous outbursts. The reference point The sinusoidal fractional amplitude of the pulsations is, on Tasc,ref is taken from Table 1 of Hartman et al. (2009), and average, around 2% (sinusoidal amplitude or semi-amplitude) we use the quantity ∆ Tasc = Tasc,i − Tasc,ref + NPb , where and does not show any significant variation during the dura- Tasc,i refers to the i–th outburst and N is the closest integer
to tion of the observations. The pulse time of arrivals are fitted Tasc,i − Tasc,ref /Pb. The reference orbital period Pb can also with a constant pulse frequency plus a Keplerian circular orbit be found in Table
1 of Hartman et al. (2009). and the statistical errors on the fitted parameters are obtained Up to the 2008 outburst, the ∆ Tasc evolution showed a trend with standard χ2 minimization techniques. that was compatible with a quadratic polynomial representing Since we do not see any significant timing noise in the data an orbital expansion at a constant rate (Hartman et al. 2008; (at the timescales of the observations) and the variance of the di Salvo et al. 2008). Indeed, the time of passage through the pulse ToAs is compatible with that expected from measure- ascending node can be expressed as a polynomial expansion: ment errors alone, we take our statistical errors as a good 1 2 representation of the true statistical ones. In previous work Tasc(N) = Tasc,ref + PbN + PbP˙ bN + ... (2) (e.g., Hartman et al. 2008; Patruno et al. 2012) it was shown 2 that when observing the pulsations of SAX J1808.4−3658 a By adding the 2011 outburst data points, Patruno et al. 2012 strong timing noise is always observed on timescales of the showed that a quadratic polynomial was insufficient to de- order of hours to days. Part of this noise is correlated to scribe the observed behavior of the Tasc variations, which X-ray flux variations and introduce systematic errors on the were instead successfully described by a cubic polynomial. determination of the spin frequency of the order of 10−8– The physical interpretation given was that, on the observed 10−7 Hz. These systematic errors are particularly pronounced baseline of 13 years, the orbit was expanding at an acceler- during the reflares when strong pulse shape variability is ated rate. observed (Hartman et al. 2008, 2009). The magnitude of We now add the 2015 outburst data (see Figure 3) and we such systematic errors can be estimated by looking at long first try to fit the Tasc data points with a quadratic polynomial, data stretches that are longer than the typical timing noise which corresponds to the solution found in Hartman et al. timescales. However, since the pulsations available in our (2008, 2009); di Salvo et al. (2008). The fit is statistically analysis refer only to a short data span (≈ 30-ks) we can- poor with a χ2 of 492 for 4 degrees of freedom (dof). A not determine the size of the systematic errors in our analysis. cubic polynomial is also a poor description of the data with Indeed, Hartman et al. (2008, 2009) and Patruno et al. (2012) χ2/ dof = 314/3. To obtain a p-value above the canonical 5% estimated the average systematic error on the pulse frequency threshold we need to fit the data with a fifth-order polynomial over the entire baseline of the observations, which lasted for (χ2/ dof = 3.4/1, p-value 6%), which suggests that either the weeks/months. Here, instead, the much shorter data span im- observed variability is governed by a stochastic process or, if plies that the systematic effect of timing noise can be substan- a periodicity is present, it must be significantly longer than the tially larger than average. For example, by looking at Fig- observational baseline10. We stress that the concavity of the ure 1 in Hartman et al. (2009) we see that on timescales of 5th order polynomial curve changes sign around 2011, which few hours the timing noise can induce pulse phase shifts of implies that the orbit has started to shrink after that time. the order of 0.1–0.3 cycles. For this work such a phase shift Next, we tried to fit the data with a sinusoid that could rep- would translate in systematic errors on the determination of resent the effect of a slightly eccentric orbit with periastron the pulse frequency of up to a few 10−6Hz. advance. From our previous work (Patruno et al. 2012) we Even if our statistical errors on the pulse frequencyare large already know that a sinusoid is a statistically poor fit to the (∼ 10−6 Hz) we cannot neglect the effect of systematic errors, data. Indeed we find a formally bad fit with a χ2 = 83.9 for although we can only use a rough estimate of its magnitude 2 dof. Furthermore the fit requires an eccentricity of about by looking at the behavior of the pulsations recorded during 0.004 which is much larger than the best upper limits avail- − previous outbursts. The orbital and pulse frequency solution able on SAX J1808.4−3658 (e < 1.2 × 10 4, Hartman et al. is reported in Table 2. 2009). Finally we attempted to fit the data with a Keplerian

10 5.3. Orbital Solution As a cautionary test we also try to remove the 2011 point (assuming it is an outlier, even if there is no evidence or reason to believe that this is the To determine the orbital evolution we follow the procedure case) and fit the data again with a quadratic polynomial. The data give also a already used in Patruno et al. (2012); Hartman et al. (2009, poor fit with χ2 = 11.1 for 3 degrees of freedom (and p-value < 1%) 6 Patruno et al.


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Figure 1. X-ray lightcurve (0.3–10 keV) of SAX J1808.4−3658 obtained with the Swift/XRT telescope. The cross marks the occurrence of a Type I X-ray burst, whereas open circles are non-detection. The arrow identifies the time of the Chandra observation.

−1 20 vorb ≈ 50ms with variations along the orbit due to the large eccentricity. To get a first order orbital acceleration we use −7 −2 e = 0 and we get aorb ≈ 6 × 10 m s . This would imply a 15 pulse frequency derivative of (Joshi & Rasio 1997): a · n ν˙ = ν orb (3) p s c 10 where n is a unit vector along the line of sight and νs = 401 Hz is the spin frequency of SAX J1808.4−3658. This gives ν = × −13 θ θ Count rate (ct/s) 5 ˙ p 8 10 cos ,where is the angle between the accel- eration and line of sight vectors. Since we know from previ- ous observations that SAX J1808.4−3658 is spinning down −15 −1 0 at a relatively constant rate ofν ˙s ≈ 10 Hz s we can confi- dently exclude this scenario.

57120 57123 57126 57129 57132 57135 57138 5.4. Radio Pulse Search Exhaustive searches using both a blind Fourier-based Figure 2. INTEGRAL/IBIS lightcurve of the 2015 outburst in the 20–100 periodicity search, and folding with a range of per- keV energy band. The onset of the outburst is detected around MJD 57121. turbed ephemerides, failed to find radio pulsations from The data points have typical integration times of 1800 – 3600 s. The horizon- tal error bars (larger than the symbols only for a few points) define the time SAX J1808.4−3658 for any trial DM or ∆Tasc §4. interval over which the data has been integrated. In the absence of detectable radio pulsations, we can place a stringent upper limit on pulsed radio emission from SAX J1808.4−3658, with the notable caveat that an active ra- orbital delay curve that could represent the effect of orbital dio pulsar could in principle be enshrouded by intra-binary motion caused by a third body in an eccentric orbit. We find material for a large fraction of the time (e.g. Archibald et al. this can fit the data (with χ2 = 1.1 for 1 dof) if the third body 2015). In analogy with the black widow systems, however, hasa mass of about8 Jupiter masses and is in a relativelywide it is reasonable to assume that SAX J1808.4−3658 would (≈ 5 AU) orbit with an eccentricity of about 0.7 and an orbital only be eclipsed for ∼10% of its orbit at 2GHz observing fre- period of about 17.4 years. We can test this scenario by look- quency. ing at the pulse frequency derivative of SAX J1808.4−3658 To set an upper limit on the flux density, we use since the pulsar would be accelerated along the orbit. Given the modified radiometer equation (see Dewey et al. 1985; the fitted orbital parameters, the orbital velocity would be Bhattacharya 1998; Lorimer & Kramer 2012): 17 years of evolution SAX J1808.4−3658 7

1998-2011 Orbital Solution 60 1998-2008 Orbital Solution 5th-order 50

40

Tasc [s] 30 ∆ 20

10

0 0.6 0.3 0 -0.3 Resid. [s] -0.6 1998.3 2000.1 2002.8 2005.4 2008.7 2011.8 2015.3 Outburst Year

Figure 3. Orbital evolution of SAX J1808.4−3658 over 17 years. The ∆ Tasc cannot be fitted with a cubic (solid black line) or a quadratic (dotted orange line) polynomial. A 5-th order polynomial (solid blue line) is necessary to obtain a statistically acceptable fit, which suggests a complex behavior of the orbit. The panel in the bottom shows the residuals with respect to the 5-th order polynomial fit.

6. DISCUSSION S N β Tsys W We have conducted the deepest radio pulse search for S min = (4) SAX J1808.4−3658 during its quiescent phase in August   r P − W G np tobs △ f 2014. No radio pulsations have been detected, setting the p strongest possible upper limit (30 µJ at 2 GHz) on the pres- / We used the S-band receiver frontend (Rcvr2 3) at GBT. ence of radio pulsations that exist (to date) for any AMXP. For this receiver the system noise temperature Tsys is 22K −1 The presence of a radio pulsar turning on during quiescence and the gain of the telescope G is 1.9KJy . Here, △ f is the cannot be excluded with the present upper limits, but if a ra- 800MHz bandwidth, the correction factor β is assumed ideal dio pulsar signal is present it has to be quite weak at high and close to 1, the number of polarisations np is two and fi- radio frequencies (2 GHz), substantially scattered by the in- 11 nally the integration time tobs corresponds to 16min . With tervening interstellar medium, or perpetually eclipsed to be the assumption that the pulse duty cycle is ∼ 10% and signal still compatible with the current constraints. The beam width to noise ratio S for candidate identification by eye is 8, of millisecond radio pulsars is very large (typical values of N min ◦ we obtain a maximum  flux density of 30 µJy at 2GHz (equiv- ∼100 , e.g. Lorimer 2008) so that missing the pulsar because alently ∼ 50 µJy at 1.4GHz, for an assumed spectral index of of beaming, although possible, is unlikely. The strongest ev- α = −1.4). idence for a large beaming angle comes from X-ray obser- This limit can be used as an important input for future ra- vations of globular clusters, where very few unidentified X- dio searches, and a point of comparison in the event that ray sources have spectral properties compatible with unknown SAX J1808.4−3658 becomes a detectable radio pulsar in millisecond pulsars whose radio beam is not pointing towards the future. We note that of the 106 Galactic field pulsars Earth (Heinke et al. 2005). Even if SAX J1808.4−3658 is an (outside of globular clusters) in the ATNF catalog (acces- active radio pulsar in quiescence, there is still a good chance sible at http://www.atnf.csiro.au/research/pulsar/psrcat/; See that eclipses might appear for 10–50% of the orbit due to free- also Manchester et al. 2005) with quoted flux densities at free absorption by intra-binary material, a common occur- 1.4GHz, and with spin period < 10ms, less than 10% have rence in black widow and redback pulsars (Nice et al. 2000; comparably low flux density to the upper limit we set on Roberts 2013). To avoid this problem we have observed at SAX J1808.4−3658. These low-flux-density millisecond pul- a high-enough radio frequency that a long eclipse duration is sars are predominantly recent discoveries from the PALFA unlikely. We have also observed at a wide range of orbital pulsar survey with Arecibo (Scholz et al. 2015; Lazarus et al. phases, when the neutron star is not behind its companion. 2015). After 17 years of X-ray monitoring, the orbital period evolu- tion of SAX J1808.4−3658 shows a non-predictablebehavior. The statistical fit to the data show that neither a parabolic nora 11 While in principle we could quote a 2x deeper limit by coherently fold- ing the full 1-hr 9 August 2014 data set, we choose not to do so because some cubic polynomialcan describe the data correctly. We found an fraction of this integration is during orbital phases in which any radio pulsar ambiguity in the interpretation of the long term trend of Tasc, is likely to be eclipsed. We therefore prefer to set a more conservative flux since the observations can be explained in two ways. Either density limit using the 16-min sub-integration, which together span a wide the orbit is expanding throughout the 17-years long observa- range of orbital phases. 8 Patruno et al. tional window, with some fluctuations around the mean P˙ b, its efficiency depends on a number of poorly understood (and or the orbit has expanded until ∼2011 followed by a shrink- difficult to measure) stellar parameters (see e.g., Knigge et al. age (i.e., the fifth-order polynomial curve changes concavity). 2011 for a discussion). It also remains rather speculative This is not a surprising behavior since many binary systems whether the single-star braking laws can be extended, unal- have shown a similar orbital evolution. However, identifying tered, in binary systems. Given these uncertainties the mag- the precise short-term mechanism responsible for such orbital netic braking timescale can vary by up to an order of magni- evolution is a relatively difficult task. tude and indeed different recipes have been given in the liter- In the following we will proceed by first discussing some ature (Skumanich 1972; Rappaport et al. 1983; Stepien 1995; fundamental properties of binary evolution, then we will com- see also Tauris & van den Heuvel 2006 and Appendix A in pare SAX J1808.4−3658 to other known binaries that show Knigge et al. 2011 for a review of several magnetic braking anomalous orbital evolution and finally we will review possi- models). Nonetheless, moderately wide binaries where mag- ble mechanisms to explain such an anomaly. netic braking is dominant, are thought to lose angular momen- tum on a shorter timescale than those compact binaries where 6.1. Binary Evolution Timescales angular momentum loss is dominated by gravitational wave Looking at the binary evolution, it is useful to define a emission. The orbital parameters of SAX J1808.4−3658 im- Pb ply that it should be a typical gravitational waves driven bi- timescale τev = that can be compared to the expected evo- P˙b nary, since magnetic braking is believed to turn off (or be- lutionary timescales from theoretical models. Differentiating come less efficient) once the donor becomes fully convective the third Kepler law and assuming that all mass lost by the and/or semi-degenerate (Spruit & Ritter 1983; see however, companion is accreted by the primary, one obtains the well Wright & Drake 2016 for recent results that suggest that a dy- known equation (see e.g., Frank et al. 2002): namo process might still occur in fully convective stars). It is interesting at this point to compare what is observed a˙ 2J˙ −2M˙ = + c (1 − q) (5) in compact radio pulsar binaries (black widows and redbacks, a J Mc see e.g. Roberts 2013) as well as other LMXBs and accret- ing white dwarfs which have similar orbital parameters as where q = Mc/MNS is the mass ratio between the compan- SAX J1808.4−3658 and have a measured P˙b. The reason ion (Mc) and neutron star mass (MNS), a is the orbital sepa- ration, J is the angular momentum and the dot refers to the why these systems might be relevant in this context is two- first time derivative. In general the angular momentum loss fold: other LMXBs might be behaving in a similar way as of the binary (J˙) can be decomposed in four terms (see e.g., SAX J1808.4−3658 since the same mechanisms might be at Tauris & van den Heuvel 2006): play, whereas black widows and redbacks are non-accreting systems and therefore Eq. (5) simplifies. In the last few years J˙ J˙gw J˙ J˙ J˙ three redback pulsars have transitioned to an accreting LMXB = + mb + ml + soc (6) J J J J J state (Archibald et al. 2009; Papitto et al. 2013; Bassa et al. 2014; Roy et al. 2015). Therefore it is still possible that more where the subscripts gw, mb, ml and soc refer to gravita- of these systems (if not all) could display the same behavior tional wave emission, magnetic braking, mass loss and spin- and therefore the assumption that redbacks are always non- orbit coupling, respectively. When the binary is relatively accreting might be invalid. As yet, no black widow system compact, (orbital period of less than ∼ 1 day), the evolu- has been observed to transition from a rotation-powered to an tion of the system is believed to be driven by angular mo- accretion-powered state. mentum loss (encoded in the J˙ term in the expression above; see e.g. Frank et al. 2002) rather than the nuclear evolution 6.2. Comparison with Other Interacting Binaries of the donor star. In ultra-compact (Pb < 80 min) and com- pact binaries (80 min< P . 3.5 hr) the angular momentum A large number of interacting binaries have a measured or- b bital period evolution which is too fast to be explained with loss is believed to be mainly due to emission of gravitational ˙ ˙ waves (J˙ ) which becomes very efficient at short orbital sep- simple binary evolution models involving only Jgw and Jmb gw and requires a number of additional effects. These systems in- arations (generally when Pb . 3hr, van der Sluys 2011). If there is no mass loss from the system, then the loss of angular clude binary pulsars, cataclysmic variables and LMXBs with momentum via gravitational waves drives the mass transfer neutron star and black hole accretors. Among the binary pul- and the orbital period changes according to the following ex- sars, we discuss below only the cases of the black widows pression (Rappaport et al. 1987; Verbunt 1993; di Salvo et al. and redbacks because all binaries with a white-dwarf/neutron 2008): star companion are following the predictions of general rela- tivity with exquisite precision (e.g., Taylor & Weisberg 1989; −14 −1/3 −5/3 P˙ b = −1.4 × 10 MNS Mc M P × Weisberg & Taylor 2002). We also exclude from the sample b,hr those radio pulsars with a B-type star companion (not to be × (ζ − 1/3)/(ζ + 5/3 − 2q) (7) confused with the Be X-ray binaries, which are accreting sys- tems and a subset of high mass X-ray binaries) since very where all masses are expressed in solar units, Pb,hr is the orbital period in hours and ζ is the effective mass-radius different mechanisms involving the short nuclear evolution ζ timescale of the massive companion need to be considered. index of the donor star (R ∝ M ; see for example c c This is also the reason why we do not include high mass X- van Teeseling & King 1998). ray binaries in our sample. When the orbital period of the binary is wider, in the range of 3.5hr . P . 0.5–1 day, the dominant mechanism driv- b 6.2.1. Black Widows and RedBacks ing the binary evolution is thought to be angular momentum loss via magnetic braking (J˙mb). There is currently consider- In black widows and redbacks the companion star is be- able uncertainty about the details of magnetic braking because ing ablated by the pulsar wind and high energy radiation, 17 years of evolution SAX J1808.4−3658 9

Table 3 676, MXB 1658–298, SAX J1748.9–2021, AX J1745.6–2901 Black Widows and Redbacks with measured P˙ b and SAX J1808.4−3658) and persistent neutron star LMXBs (Her X–1, 2A 1822–37, XB 1916–053). Furthermore, some ˙ Name (PSR) Type Pb(hr) Pb Error of these LMXBs are accreting pulsars and have orbital periods J1023+0038 RB 4.8 −7.3 × 10−11 0.06 × 10−11 determined via timing of their pulsations (e.g., Her X–1, 2A J1227-4853 RB 6.9 −8.7 × 10−10 0.1 × 10−10 1822–37, SAX J1808.4−3658 SAX J1748.9–2021) whereas −9 −9 J1723-2837 RB 14.8 −3.5 × 10 0.12 × 10 others are eclipsing systems and their period is determined J1731-1847 BW 7.5 −1.08 × 1010 0.07 × 10−10 −11 −11 via X-ray and/or optical photometry. No other orbital period J2051-0827 BW 2.4 −1.6 × 10 0.08 × 10 derivative has been measured so far for any other LMXB. In + × −11 × −11 J1959 2048 BW 9.2 1.47 10 0.08 10 Table 4 we list the LMXBs sample with their orbital period derivatives and the main proposed explanation given in the thus producing potential mass loss. Observational evidence of this phenomenon comes from the fact that radio pulsa- literature. The binary EXO 0748–676 is an eclipsing binary with a tions are very often eclipsed by intra-binary material that in- 0.4 M⊙ donor that shows sudden variations in P˙ b, which were duces free-free absorption of the pulsed signal. The orbital ff parameters of SAX J1808.4−3658 are compatible with those proposed to be due to spin-orbit coupling (Wol et al. 2002, 2009). Wolff et al. (2002) analyzed two segments of X-ray of a BW, and its 0.05-0.08 M companion is also a semi- ⊙ data (1985–1990 and 1996–2000) and showed that the period degenerate star (Bildsten & Chakrabarty 2001; Deloye et al. 2008; Wang et al. 2013). The only difference between BWs had increases by about 8 ms. However, the period increase shows jitters and cannot be fit with a constant P˙ . Further and SAX J1808.4−3658 is that in the latter system the com- b work by Wolff et al. (2009) extended the analysis until 2008 panion is in Roche lobe overflow whereas BWs are thought, at least in some cases, to be detached systems Breton et al. and observed a similar behavior. The eclipsing binary 2A 1822–37 is an accreting pulsar (2013). with a ∼0.3 − 0.4 M companion and it has a relatively In black widows, as well as in redback pulsars, the short- ⊙ term effects on the orbital evolution do occur on timescales steady increase of the orbital period measured over a base- line of 30 years (see e.g., Burderi et al. 2010; Iaria et al. 2011; which are generally orders of magnitude shorter than the pre- Chou et al. 2016). This system is an accretion-disk corona dicted (secular) ones from angular momentum loss due to gravitational waves and/or magnetic braking. The six BWs system showing extended partial eclipses of the central X-ray source. Burderi et al. (2010) and Iaria et al. (2011) have sug- and RBs which have a measured (and publicly available12) gested that the binary contains an Eddington limited accreting P˙ b show orbital evolution timescales from 100 Myr down to less than 1 Myr (see Table 3). Five of them have nega- neutron star whose irradiation of the donor is inducing severe mass loss that can explain the large orbital period derivative tive orbital period derivative (the orbit is shrinking), whereas observed. The positive sign of P˙b is ascribed to the response only one (PSR B1957+20) has a positive value (the orbit is ζ expanding). Similarly, in the case of SAX J1808.4−3658, of the radius of the donor to mass-loss, with Rc ∝ Mc and Pb n < 1/3. Burderi et al. (2010) suggests that the donor in 2A τev = ˙ ≈ 70 Myr, whereas from Eq. (7) one would have Pb 1822–37 has a deep convective envelope with ζ = −1/3 (see expected a timescale of a few Gyr (varying slightly with the e.g., Rappaport et al. 1982) thus justifying the positive P˙ b. exact neutron star and companion mass chosen) thus meaning The source AX J1745.6–2901 is an eclipsing binary with ˙ that the Jgw term is not the dominant one. It is important to an accreting neutron star and a negative P˙ (Ponti et al. 2015). stress that these apparent orbital period derivatives are epoch b The donor mass is constrained to be Mc . 0.8 M⊙. A strong dependent, and change on year-long timescales. mass loss is also suggested in this case, but since the system is shrinking the mass-radius index needs to be n > 1/3. The 6.2.2. Other Low Mass X-Ray Binaries data on the Tasc collected over a baseline of about 30 years In the few compact LMXBs where an orbital period deriva- show significant scatter of up to several tens of seconds. tive can be measured, we see a roughly equally distributed XB 1916–053 is an ultracompact (Pb ≈ 50min) persistent sign (6 positive, 4 negative signs and three upper-limits ob- dipping source monitored for over 37 years by X-ray satel- served so far). The magnitude of the orbital period deriva- lites. Hu et al. (2008) studied the first 24 years of data and tive is always much larger than expected from conservative found that a quadratic function (i.e., a constant P˙b) was able binary evolution (J˙ = 0) and/or from angular momentum to describe the data correctly. However, Iaria et al. (2015) loss via gravitational waves and/or magnetic braking. In the found that, when considering the entire 37 years of observa- literature a number of different mechanisms have been sug- tions, a quadratic function was unable to fit the orbital evolu- gested to explain the large P˙ b. These include mass loss from tion and a model with a sinusoidal variation in addition to the the companion (Burderi et al. 2010; Ponti et al. 2015), en- quadratic component was required. A third body with mass hanced magnetic braking (Gonz´alez Hern´andez et al. 2014), of ∼ 0.06 M⊙ was invoked to explain the observations with an the presence of a third body (Iaria et al. 2015), spin-orbit orbital period of ≈ 26 years. It is instructiveto notice that a de- coupling (Wolff etal. 2009; Patruno et al. 2012) and in viation from a quadratic function was not apparent in the first some cases even modified theories of gravity (Yagi 2012). 24 years of data, which suggests that a very long timescale The group of LMXBs appears to be the most heteroge- periodicity (or quasi-periodicity) might still be present even neous among the different binaries that we are consider- in binaries where a constant P˙b is observed over baselines of ing here. Indeed this group comprises transient LMXBs a few decades. with black hole accretors (XTE J1118+480 and A0620–00), The globular cluster source SAX J1748.9–2021 is instead transient LMXBs with neutron star accretors (EXO 0748– an intermittent AMXP (Altamirano et al. 2008; Gavriil et al. 2007; Patruno et al. 2009a; Sanna et al. 2016) and it has been 12 For a complete list of binary pulsars we refer to the ATNF pulsar catalog observed in outburst five times. Its companion star is likely http://www.atnf.csiro.au/people/pulsar/psrcat/ 10 Patruno et al. to be a ∼0.8 M⊙ star close to the turnoff mass of the globu- systems, where the P˙ b is detected by looking at the eclipse lar cluster NGC 6440, although much smaller masses down times in optical data, some selection effects might be present. to 0.1 M⊙ cannot be excluded (Altamirano et al. 2008). The This means that those systems with a P˙b in line with the theo- orbital evolution has been studied by looking at the orbital retical predictions might be more difficult to measure/detect ephemeris calculated with coherent timing in a way similar to and therefore the reported values are invariably skewed to- what has been donein this work. Sanna et al. (2016) describes wards large/anomalous P˙b values. Something similar applies the orbital evolution with a quadratic function although the fit also to most LMXBs, with the exception of those systems shows large deviations of the order of 100 seconds from the where the P˙ b is measured via pulsar timing, in which case best fit function (which translated into a poor χ2 of 78.4 for 1 the sensitivity of the timing technique potentially allows the dof). These authors interpret the large orbital expansion with detection of values orders of magnitude smaller than in the a highly non-conservativemass loss scenario where the binary CVs and Algol binaries. For the CVs there is ample litera- is losing more than 97% of the mass flowing through the inner ture on the topic and several different systems with a large Lagrangian point L1. P˙ b are reported. In this paper we include only the T Pyxidis Her X–1 shows instead a steady decrease of the orbit and IM Normae systems which are the two best studied cases whose value is however compatible with both a conservative and have the highest orbital period variation (Patterson et al. and a non-conservative mass transfer scenario (Staubert et al. 2015, 2016). We also include NN-Serpentis (Brinkworth et al. 2009). 2006) which is an eclipsing post common-envelope binary Finally, the orbital evolution of the two transient where no mass transfer is currently ongoing. We summarize black hole LMXBs XTE J1118+480 and A0620– the information on the orbital period evolution of all the bi- 00 (Gonz´alez Hern´andez et al. 2012, 2014) was measured naries discussed in this work in Figure 4. From the figure, it with radial velocity curves determined via optical spec- is clear that all sources with short orbital periods that should troscopy for a period of time of ∼ 10 and 20 years respectively. be losing angular momentum via gravitational wave emission, These observations were all carried out during the extended are evolving on timescales which are at least an order of mag- periodsof quiescenceof the binaries. The orbit shows a steady nitude shorter than expected. The binaries with wider orbits, shrinkage interpreted as due to enhanced magnetic braking. where magnetic braking should dominate, show also shorter The two binaries have a companion mass of 0.2 M⊙ (XTE evolutionary timescales than predicted, although a larger scat- J1118+480; Gonz´alez Hern´andez et al. 2014) and 0.4 M⊙ tering is observed and some sources are close to the theoreti- (A0620–00; Cantrell et al. 2010; Gonz´alez Hern´andez et al. cal predictions. 2014), respectively. 6.3. Models 6.2.3. Cataclysmic Variables Since a large number of scenarios are invoked in the liter- It is well known that some cataclysmic variables show an ature to explain the orbital evolution of different interacting anomalous orbital period derivative as well, with some of binaries, it appears legitimate to ask whether it is still possi- them proposed to be transferring mass at a higher rate than ble to find a common (and/or perhaps still unknown) mech- expected due to irradiation of the companion (Patterson et al. anism behind the observed behavior. In the following dis- 2016, 2015; Knigge et al. 2000). Even some Algol type cussion we proceed by considering all models proposed in binaries (i.e., a semi-detached system composed by a de- the literature and try to apply each of them to the case of tached early type main sequence star and a less massive sub- SAX J1808.4−3658. with the exception of the third body giant/giant star in Roche lobe overflow) have been reported to model, which has be already excluded (see Section 5.3. evolve on a very short timescale (Erdem & Ozt¨urk¨ 2014). In this case non-conservative mass transfer scenario is expected 6.3.1. Mass Loss Model to take place since the red giant will emit a significant wind. If the companion star experiences severe mass loss, for ex- However, for a few of these systems (i.e., all the converging ample because of ablation due to the irradiation from a pulsar ones) the required mass loss is larger than the highest theoreti- wind or from the X-rays originating close to the compact ob- cal value for wind mass loss in giant stars. These observations ject, then the orbital period of the binary changes dramatically might suggest that short term effects have some influence on (see Section 6.1). the orbital evolution of accreting and non accreting neutron In this case the orbital period derivative will depend on the stars, persistent and transient systems and white dwarf/black amount of wind lost from the companion: hole/main sequence stellar accretors. It is worth noticing that no neutron star + white dwarf binary (both accreting and non- P˙b M˙ c accreting) has been observed (as yet) to evolve on anomalous = −2 (8) P M timescales. The only exception is the ultra-compact LMXB b c 4U 1822–30, which, however, is located in a globular cluster where Mc is the companion mass (see e.g., Frank et al. and therefore its large P˙ b might simply be due to the contam- 2002; Postnov & Yungelson 2014). Applying this model ination induced by the gravitational potential well of the clus- to SAX J1808.4−3658 it is possible to explain the ob- ter (Jain et al. 2010). This suggests that, if there is a common served P˙ b if the donor is losing mass at a rate of about −9 −1 reason behind this behavior for all type of binaries (which is 10 M⊙ yr (di Salvo et al. 2008). of course not necessarily true), it must be related to the type of companion (main sequence or semi-degenerate star) rather For the donor to lose a substantial amount of mass, there than the type of accretor. must be a way to efficiently inject energy into the donor star. Whatever mechanism is chosen, the amount of energy 6.2.4. Caveats necessary to create such a strong mass loss must be consis- A few cautionary words are necessary at this point on the tent with the total energy budget available to the binary. In CV, Algol type binaries and some LMXBs. For these type of SAX J1808.4−3658 it has been proposed that the mass loss 17 years of evolution SAX J1808.4−3658 11

Table 4 Binaries with Anomalous Orbital Period Derivatives

Name Pb[hr] P˙ b Transient Companion Type Sign Proposed Model References Neutron Star LMXBs EXO 0748–676a 3.8 1.9 × 10−11 Yes MS + SOC Wolff et al. (2009, 2002) 2A 1822–37 5.5 1.51(8) × 10−10 No MS + Mass Loss Burderi et al. (2010); Iaria et al. (2011) SAX J1808.4–3658 2.0 3.5(2) × 10−12 Yes SD + SOC/Mass Loss Patruno et al. (2012); di Salvo et al. (2008) MXB 1658–298 7.1 8.4(9) × 10−12 Yes MS + Unknown Paul & Jain (2010) XB 1916–053 0.8 1.5(3) × 10−11 No SD + Third Body Iaria et al. (2015) SAX J1748.9–2021 8.8 1.1(3) × 10−10 Yes MS/Sub-G + Mass-Loss Sanna et al. (2016) AX J1745.6–2901 8.4 -4.03(32) × 10−11 Yes MS/Sub-G – Mass Loss Ponti et al. (2015) Hercules X–1 40.8 -4.85(13) × 10−11 No MS – Several Staubert et al. (2009) Black Hole LMXBs XTE J1118+480 4.0 -6(1.8) × 10−11 Yes MS – Enhanced MB Gonz´alez Hern´andez et al. (2012, 2014) A0620–00 7.8 -1.9(3) × 10−11 Yes MS – Enhanced MB Gonz´alez Hern´andez et al. (2014) MS = main sequence; sub-G = sub-giant; SD = semi degenerate; MB= magnetic braking; SOC = Spin-Orbit Coupling; a This source shows segments of data where a constant Pb is required. The error on P˙ b is not given and confidence intervals are determined via Maximum Likelihood Method (Wolff et al. 2009). is driven by a pulsar wind and high energy radiation imping- mostly because of the unknown neutron star mass. If we as- ing onto the donor surface (di Salvo et al. 2008; Burderi et al. sumea rangeoftotalbinarymassfrom1.4upto 3M⊙, thenthe 2009). For a circular binary orbit, the total angular momen- variation in a is of the orderof 20%(6−8×1010 cm). Here we 10 tum is (Frank et al. 2002): will assume a = 6.4×10 cm(MNS = 1.4 M⊙, Mc = 0.08 M⊙). The orbital energy is therefore: −1/3 1/3 J ∝ MNS Mc M P (9) b 47 Eorb ∼ 3 × 10 erg (14) where Pb is the orbital period and M = MNS + Mc the total binary mass, and Mc < MNS. The orbital energy Eorb is The semi-major axis of the binary changes according to the 3rd Kepler law: − Eorb ∝ J/Pb (10) 2a a˙ = P˙ (15) 3P b From Eq. 9, an increase in Pb requires Mc to decrease, since b for an isolated system J and M cannot increase. And from The value P˙ is measured from observations and its value is − b Eq. 10 we see that Eorb must decrease, making the orbit ∼3.5 × 10−12. Thereforea ˙ = 2 × 10−5 cm s−1. The orbital less tightly bound. In other words, an orbital period increase energy variation is (we assume M˙ terms are negligible): requires energy injection from somewhere. Marsh & Pringle (1990) show that energy injection by the secondary star is too GM M a˙ E˙ = NS c ∼ 9 × 1031 erg s−1 (16) slow for observed Pb changes as this is governed by the star’s orb 2 a2 thermal timescale13. The only energy source left is the spin energy of the NS. The total spin down power is: This is: 2 2 2 2 E˙sd = Iωω˙ (17) Espin ∼ k MNS vK ∼ k GMNS/RNS (11) where ω = 2π ν,ω ˙ = 2πν˙ and ν andν ˙ are the spin frequency where k is the radius of gyration and R its physical radius, NS × −15 −1 and v = (GM /R )1/2 is the breakup spin velocity. This is (401Hz)and the spin down(1.65(20) 10 Hz s ) observed K NS NS in SAX J1808.4−3658 (Patruno et al. 2012). Numerically, a huge reservoir, since 34 −1 E˙sd ≈ 2.6 × 10 erg s . 2 MNS a We need an efficiency η of at least the ratio between the two Espin/(−E) ∼ k ∼ 400 − 1000 (12) powers: Mc R E˙ 2 orb with k ∼ 0.4. But of course not all of Espin can be used to η = ∼ 0.003 (18) E˙sd drive mass loss from the system and so increase Pb. We can estimate the required minimum efficiency η for spin energy For the parameters assumed above, the Roche lobe radius − conversion into orbit energy for SAX J1808.4 3658. is RL = 0.16 R⊙ (see e.g., Eggleton 1983) and the fraction of 2 The total orbital binding energy of the binary is: intercepted power is f = (RL/2a) which is approximately 0.8%. This value suggests that the donor star must be ex- GMNS Mc Eorb = − (13) tremely efficient in converting the incident power into mass 2a loss since ǫ ∼ η/0.8 ≈ 40%. Campana et al. (2004) estimated If we use the third Kepler law, then we have that the orbital that the irradiating power14 required to explain the bright op- period is very well determined (Pb = 2.01 hr, from the X-ray tical counterpart of SAX J1808.4−3658 (observed with VLT +3 33 −1 pulse timing). The total mass of the binary is ill constrained, data in 2002) amounted to Lirr = 8−1 × 10 erg s (see also

13 This is of course what happens when a very low-mass star expands 14 The authors used a distance of 2.5 kpc, whereas here we rescale the lumi- on mass transfer (Mc > MNS) with Pb ∝ 1/Mc. The stars thermal energy nosity for a distance of 3.5±0.1 kpc as determined by Galloway & Cumming expands it adiabatically. (2006) 12 Patruno et al.

-8

-9

-10 τ=1 Myr

|) -11 τ=10 Myr b .

Log(|P -12 τ=100 Myr

-13 τ=1 Gyr PSR - PSR + LMXBs - -14 τ=10 Gyr LMXBs + CV- CV+ SAX J1808 -15 0.5 1 2 3 4 5 6 7 8 910 12 24 48

Log(Pb) [hours]

Figure 4. |P˙ b| vs. Pb diagram for SAX J1808.4−3658 (blue pentagon), LMXBs (red squares), binary pulsars (BW and RBs, blue circles) and some CVs (purple crosses). The open symbols identify converging systems (negative P˙ b) whereas filled symbols are diverging ones. Only T PyX and IM Nor are plotted for CVs (plus NN-Ser which is drawn with the same symbol used for CVs). The oblique black dotted lines identify evolutionary timescales τ = Pb/|P˙b|. The dashed vertical lines roughly separates the binaries where the dominant angular momentum loss should be gravitational radiation (left) from those where magnetic braking is expected to dominate (right) assuming no mass loss or spin-orbit coupling is present in the binary. The cyan oblique lines are theoretical values of |P˙ b| expected if gravitational wave emission is the main driver of orbital evolution. From top to bottom these lines are valid for a donor mass-radius relation Rc ∝ Mc −1/3 and Rc ∝ Mc respectively, assuming a donor mass of 0.08 M⊙ and a neutron star mass of 1.4 M⊙. A value of 0.08 M⊙ has been chosen as it represents the most likely donor mass in SAX J1808.4−3658 . The orange oblique lines is instead the orbital period derivative as a function of orbital period when the maximum possible magnetic braking effect is considered (see main text for a discussion). A mass-radius index ζ = 1 has been assumed for the magnetic braking case.

Homer et al. 2001 for a similar estimate made with observa- and the spin-down power is linear (see e.g. a discussion in tions taken in 1999). Therefore the optical data require a con- Hartman et al. 2009): version of a fraction ξ = Lirr/E˙sd ≈ 0.2–0.6 of incident power into thermal radiation by the donor star (the range provided 1 2 M˙ c = E˙sd G Mc Rc (19) takes into account the 1 σ error bars on the irradiation lumi- 2a! nosity and the possibility that the distance is 2.5 kpc rather than 3.5 kpc). Since from the observationalconstraint we have Since we have seen that the donor needs to convert the inci- that ξ + ǫ = 0.5 − 1.0 and ξ + ǫ is boundto be equalto 1 by the dent spin-down power into mass-loss with extraordinary effi- conservation of energy, this scenario is energetically plausible ciency, close to 40% (ǫ ∼ 0.4), then the spin-down power is for a range of parameters compatible with the observations. larger than initially estimated either because of a larger mo- If we assume that the mass loss from the donor of ment of inertia of the neutron star or because the spin-down SAX J1808.4−3658 is constant, then one still needs to ex- ω˙ is slightly larger than observed. In the first case one would plain the Tasc of the 2011 outburst. This data point deviated by need I & 1.7 × 1045 gcm2 and this can be used in principle approximately 7 seconds from the predicted value that can be to constrain the equation of state of ultra-dense matter (un- −12 −1 obtained in Eq. (2) by using a constant P˙b = 3.5 × 10 s s . der the assumption that mass-loss is the main mechanism re- The 7-s deviation can be explained if the mass loss rate has in- sponsible for the binary evolution). In Figure 5 we plot an creased by about ∼ 70% during the 2008-2011 period, which illustrative example of the type of constraints that can be ob- would require a proportionally larger spin-down power than tained when using this method for a selection of equations of assumed above. Indeed the relation between the mass loss state (Fortin et al. 2016). 17 years of evolution SAX J1808.4−3658 13

1994). The model has been then further extended to Roche 3 lobe filling systems like cataclysmic variables (Richman et al. 1994). The model can be briefly summarized as follows. If 2.5 the donor star has internal deformations, then the gravita- tional potential outside of the active star is (terms higher than quadrupolar are ignored here): 2 ) GM 3 xi xk Sun φ(x) = − − GQik (20) 1.5 r 2 r5

where xi and xk are Cartesian coordinates measured from Mass (M 1 the center of mass of the star and Qik is the quadrupole ten- sor (related to inertia tensor). For simplicity one can as- 0.5 sume a circular orbit (as is legitimate to do in a binary like SAX J1808.4−3658), an alignment between the donor spin axis and orbital angular momentum, and that the stellar spin 0 7 8 9 10 11 12 13 14 15 16 and orbit are synchronized. If the Cartesian system is chosen Radius (km) so that the z-axis is the angular momentum axis and the x-axis points from the center of mass to the companion star, then Qik reduces to Qxx = Q. In a circular orbit, the relative velocity Figure 5. Constraints on the mass-radius relation of the neutron star in can be written as: SAX J1808.4−3658 under the assumption that the binary is driven by mass dφ loss. The different curves correspond to different equation of states of ultra- v2 = r (21) dense matter and are taken from Fortin et al. (2016). The thick part of the dt curves marks the segments for which the moment of inertia of the neutron star is > 1.7 × 1045 g cm2 (see main text for a discussion). Therefore from Eq. (20) one can see that the relative veloc- ity v is related to the time varying quadrupole Q. Since v is In conclusion, this mechanism is energetically feasible if: also related to the orbital period of the binary, it is clear that a time varying mass quadrupole term induces a variation of • SAX J1808.4−3658 has a neutron star with a large mo- the orbital period. Applegate (1992) suggests that the cause ment of inertia of the time varying quadrupole Q might be related to mag- netic cycles of period P (in a way analogous to the famil- • the incident pulsar power can be converted into mass mod iar 11 years-long magnetic solar cycles). During these cycles, loss with a ≈ 40% efficiency. the magnetic field induces a redistribution of the angular mo- When looking at the whole sample of pulsar binaries, the mentum in different layers of the star and allows a transition behavior of other black-widow pulsars cannot be explained between different equilibrium configurations. The strength of by this model since at least two of them are shrinking their the surface magnetic field B required to explain a variation ∆P orbit. Therefore if the mass-loss model is correct we need of the orbital period can be written as: ff two di erent mechanisms to explain the population of binary GM2 A 2 ∆P pulsars. Furthermore, in a recent work (Patruno 2016) we 2∼ B 10 4 (22) have studied the orbital evolution of another AMXP, namely R R  Pmod IGR J0029+5934. This system can be considered a “twin” The transport of angular momentum inside the star needs of system of SAX J1808.4−3658 since its orbital and physical course some energy which Applegate (1992) suggests might parameters are extremely similar (see e.g., Patruno & Watts come either from the donor internal nuclear burning reservoir 2012). However, in that case the orbit of the binary is vary- or from tidal heating (Applegate & Shaham 1994). This is ing at a very slow pace (τ > 0.5 Gyr), compatible with a in essence the Applegate model which has been propose as a conservative scenario where the binary evolution is driven by viable way to explain the behavior of many binary systems. gravitational wave emission. In that case we constrained the It seems therefore natural to extend it to the case of LMXBs. efficiency of the pulsar spin-down to mass-loss conversion to For the case of SAX J1808.4−3658 one needs to assume . + be 5%. It is not clear therefore why IGR J0029 5934 is a value for Pmod since the Tasc variations observed so far unable to convert spin-down power into mass-loss whereas do not show a complete cycle. By assuming Pmod = 50 yr, 10 SAX J1808.4−3658 is so efficient. We stress that the obser- R = 0.1 R⊙, M = 0.07 M⊙, A = 7 × 10 cm one finds vations of both systems suggest a donor irradiated by a pulsar B ∼ 1kG. wind/high energy radiation and the donor mass is almost iden- tical. There are, however, two main problems with the Apple- gate model applied to SAX J1808.4−3658 and to many other 6.3.2. Spin-Orbit Coupling compact binaries considered here. The first is that the re- An exchangeof angular momentumbetween the stellar spin quired B field is of the order of 1kG which is much larger and the orbit can generate variationsof the orbitalperiod. This than typical values thought to be present in fully convective variations of the orbital angular momentum are encoded in the stars. However, some isolated low mass stars and brown term J˙soc in Eq.(6). dwarfs have been observed with relatively strong surface B The so-called Applegate model was developed fields (Morin et al. 2010) and in some cases with fields larger by Applegate (1992) and Applegate & Shaham (1994) than 1 kG (Reiners 2012). Furthermore, recent studies have to explain the orbital variability observed in a sample of provided observational evidence for the presence of magnetic eclipsing variables and then later applied to the case of the activity in at least four fully convective stars, suggesting that black-widow pulsar PSR B1957+20 (Applegate & Shaham the dynamo mechanism that produces stellar magnetic fields 14 Patruno et al. operates also through convection despite the absence of the 2015 outburst. We have not detected radio pulsations and tachocline, which is the boundary layer between radiative we place strong constraints on the flux density of the puta- and convective envelope where the magnetic fields are gen- tive radio pulsar which, if really active, needs to be either erated (Wright & Drake 2016). among the 10% dimmest pulsars known or fully obscured by The second problem, which seems more difficult to circum- radio absorbing material (which would also be atypical at the vent, was discussed in a critical review by Brinkworth et al. relatively high radio observing frequencies we searched at). (2006), who found that for very low mass stars like NN- The study of the orbital evolution of the system has been ex- Serpentis (which is a non-accreting post-common envelope tended to include the 2015 outburst, and we find two possi- binary with a 0.15 M⊙ companion and an orbital period of 3 ble interpretations of the data: either the orbit is expanding hours), the internal energy budget of the donor star might be with stochastic fluctuations around the mean or the system is insufficient to generate the required donor distortion. Even shrinking with a change of sign around 2011. if one invokes the tidal heating mechanism proposed in In the first case the pulsar spin-down power is ablating the Applegate & Shaham (1994) there would still be insufficient companion with an efficiency for the conversion of impinging energy available to generate the required stellar distortions power to mass-loss of the order of 40%. (see e.g. Burderi 2015). As was the case for the mass-loss Alternatively the Applegate model can explain the behavior model, the only source of energy left is the spin-down energy of SAX J1808.4−3658 if a strong surface magnetic field of of thepulsar. In this case thereneedsto bea viablemechanism the order of a kG is present. The source of energy that powers to transport energy deeper in the donor star which is able to this field needs to be the spin-down power of the pulsar but generate a varying mass quadrupole. As noted by Applegate there is no evidence that such large fields exist in the donor (1992), if the donor star becomes more oblate then the mass star of SAX J1808.4−3658 or that they can be generated by quadrupole ∆ Q > 0 and the orbital period decreases. The op- the pulsar wind/high energy irradiation. This requires further posite happens if ∆ Q < 0: the orbital period increases. The theoretical investigation. observed behavior of most binaries considered in this work might then be explained if, for some reason, some of them (like SAX J1808.4−3658 and other diverging binaries) have We would like to thank M. Fortin for providing the data for ∆ Q < 0 whereas all other converging binaries have ∆ Q > 0. the mass-radius relations of several neutron star models. We This idea remains highly speculative at the moment since the would like to thank E.P.J. van den Heuvel for interesting dis- problem of what happens in the deep layers of irradiated stars cussions and suggestions. A.P. acknowledges support from has not been investigated yet. an NWO Vidi fellowship. R.W. was supported by an NWO 6.3.3. Enhanced Magnetic Braking Top Grant, Module 1. J.W.T.H. is an NWO Vidi Fellow. A.J. and J.W.T.H. acknowledge funding from the European Re- ˙ This model could explain the sign and strength of Pb in search Council under the European Union’s Seventh Frame- SAX J1808.4−3658 only if the donor magnetic field is suffi- work Programme (FP7/2007-2013) / ERC grant agreement nr. ciently strong. Indeed, the mass lost by the donor cannot be 337062(DRAGNET). Some of the results presented in this pa- larger than the one estimated in Section 6.3.1 since the energy per were based on observations obtained with GBT+GUPPI. budget does not allow it. As an example, we follow the recipe We would therefore like to express our gratitude towards the provided by Justham et al. (2006), in which case the angular National Radio Astronomy Observatory (NRAO) - a facility momentum lost by the binary via magnetic braking is: of the United States National Science Foundation (NSF) - re- sponsible for operating the Green Bank Telescope. Computa- J˙ = −Ω B R13/4 M˙ 1/2(G M )−1/4 (23) mb d s c w c tional support for radio data analysis was provided by super- where Ωd is the angular rotational frequency of the donor computer Cartesius - a service offered by the Dutch SURF- star, Bs is its dipolar magnetic field at the surface and sara. Part of the scientific results reported in this article are M˙ w is the amount of wind loss rate. If we assume that based the observations made by the Chandra X-ray Observa- SAX J1808.4−3658 is changing orbital parameters mainly tory. This research has made use of software provided by because of angular momentum loss, then by rearranging the Chandra X-ray Center (CXC) in the application packages Eq.(5) we obtain: CIAO. 1/2 a˙ G 35 2 −2 REFERENCES J˙ = MNS Mc ≈ 2 × 10 gcm s (24) a1/2  M  Altamirano, D., Casella, P., Patruno, A., Wijnands, R., & van der Klis, M. Assuming that the donor is tidally locked, then Eq.(23) gives 2008, ApJ, 674, L45 the strength of the minimum Bs field required which is of Applegate, J. H. 1992, ApJ, 385, 621 the order of 103–104 G for a maximum wind loss rate of Applegate, J. H., & Shaham, J. 1994, ApJ, 436, 312 −9 −1 Archibald, A. M., Kaspi, V. M., Hessels, J. W. T., et al. 2013, ArXiv e-prints, 10 M⊙ yr , even stronger than the value calculated for the arXiv:1311.5161 Applegate model. This shows that the magnetic braking Archibald, A. M., Stairs, I. H., Ransom, S. M., et al. 2009, Science, 324, 1411 model is unlikely to be the correct one. Furthermore, such ex- Archibald, A. M., Bogdanov, S., Patruno, A., et al. 2015, ApJ, 807, 62 planation cannot work in several other binaries since at least Bailes, M., Bates, S. D., Bhalerao, V., et al. 2011, Science, 333, 1717 Bassa, C. G., Patruno, A., Hessels, J. W. T., et al. 2014, MNRAS, 441, 1825 in some neutron star LMXBs the sign of the observed orbital Bates, S. D., Lorimer, D. R., & Verbiest, J. P. W. 2013, MNRAS, 431, 1352 period derivative is opposite to that expected when magnetic Bhattacharya, D. 1998, in NATO Advanced Science Institutes (ASI) Series braking is the main driver of binary evolution. C, Vol. 515, NATO Advanced Science Institutes (ASI) Series C, ed. R. Buccheri, J. van Paradijs, & A. Alpar, 103 Bildsten, L., & Chakrabarty, D. 2001, ApJ, 557, 292 7. CONCLUSIONS Breton, R. P., van Kerkwijk, M. H., Roberts, M. S. E., et al. 2013, ApJ, 769, 108 We have studied the AMXP SAX J1808.4−3658 at radio- Brinkworth, C. S., Marsh, T. R., Dhillon, V. S., & Knigge, C. 2006, wavelengths during quiescence in 2014 and during its last MNRAS, 365, 287 17 years of evolution SAX J1808.4−3658 15

Bult, P., & van der Klis, M. 2015, ApJ, 806, 90 Nice, D. J., Arzoumanian, Z., & Thorsett, S. E. 2000, in Astronomical Burderi, L. 2015, ’Is Radio-Ejection ubiquitous among Accreting Society of the Pacific Conference Series, Vol. 202, IAU Colloq. 177: Millisecond Pulsar?’, EWASS Symposium 11 Tenerife, Spain, June 25-26, Pulsar Astronomy - 2000 and Beyond, ed. M. Kramer, N. Wex, & 2015. R. Wielebinski, 67 Burderi, L., Di Salvo, T., D’Antona, F., Robba, N. R., & Testa, V. 2003, Papitto, A., di Salvo, T., D’Aı, A., et al. 2009, A&A, 493, L39 A&A, 404, L43 Papitto, A., Ferrigno, C., Bozzo, E., et al. 2013, Nature, 501, 517 Burderi, L., Di Salvo, T., Riggio, A., et al. 2010, A&A, 515, A44 Patruno, A., Altamirano, D., Hessels, J. W. T., et al. 2009a, ApJ, 690, 1856 Burderi, L., Riggio, A., di Salvo, T., et al. 2009, A&A, 496, L17 Patruno, A., Bult, P., Gopakumar, A., et al. 2012, ApJ, 746, L27 Burderi, L., Possenti, A., D’Antona, F., et al. 2001, ApJ, 560, L71 Patruno, A., Hartman, J. M., Wijnands, R., Chakrabarty, D., & van der Klis, Cackett, E. M., Altamirano, D., Patruno, A., et al. 2009, ApJ, 694, L21 M. 2010, ApJ, 717, 1253 Campana, S., D’Avanzo, P., Baglio, C., Zelati, F. C., & Kennea, J. 2015, The Patruno, A., Maitra, D., Curran, P. A., et al. 2016, ApJ, 817, 100 Astronomer’s Telegram, 7379 Patruno, A., Rea, N., Altamirano, D., et al. 2009b, MNRAS, 396, L51 Campana, S., Stella, L., & Kennea, J. A. 2008, ApJ, 684, L99 Patruno, A., Watts, A., Klein Wolt, M., Wijnands, R., & van der Klis, M. Campana, S., D’Avanzo, P., Casares, J., et al. 2004, ApJ, 614, L49 2009c, ApJ, 707, 1296 Cantrell, A. G., Bailyn, C. D., Orosz, J. A., et al. 2010, ApJ, 710, 1127 Patruno, A., & Watts, A. L. 2012, ArXiv e-prints, arXiv:1206.2727 Chakrabarty, D., & Morgan, E. H. 1998, Nature, 394, 346 Patterson, J., Monard, B., Warhurst, P., & Myers, G. 2015, Journal of the Chou, Y., Hsieh, H.-E., Hu, C.-P., Yang, T.-C., & Su, Y.-H. 2016, ArXiv American Association of Variable Star Observers (JAAVSO), 43, 259 e-prints, arXiv:1608.04190 Patterson, J., Oksanen, A., Kemp, J., et al. 2016, ArXiv e-prints, Cordes, J. M., & Lazio, T. J. W. 2002, ArXiv Astrophysics e-prints, arXiv:1603.00291 astro-ph/0207156 Paul, B., & Jain, C. 2010, in The First Year of MAXI: Monitoring Variable —. 2003, ArXiv Astrophysics e-prints, astro-ph/0301598 X-ray Sources, 26 Courvoisier, T. J.-L., Walter, R., Beckmann, V., et al. 2003, A&A, 411, L53 Ponti, G., De, K., Munoz-Darias, T., Stella, L., & Nandra, K. 2015, ArXiv de O˜na Wilhelmi, E., Papitto, A., Li, J., et al. 2016, MNRAS, 456, 2647 e-prints, arXiv:1511.02855 Deloye, C. J., Heinke, C. O., Taam, R. E., & Jonker, P. G. 2008, MNRAS, Postnov, K. A., & Yungelson, L. R. 2014, Living Reviews in Relativity, 17, 391, 1619 arXiv:1403.4754 Dewey, R. J., Taylor, J. H., Weisberg, J. M., & Stokes, G. H. 1985, ApJ, 294, Poutanen, J., & Gierli´nski, M. 2003, MNRAS, 343, 1301 L25 Ransom, S. M. 2001, PhD thesis, Harvard University di Salvo, T., Burderi, L., Riggio, A., Papitto, A., & Menna, M. T. 2008, Ransom, S. M., Cordes, J. M., & Eikenberry, S. S. 2003, ApJ, 589, 911 MNRAS, 389, 1851 Ransom, S. M., Eikenberry, S. S., & Middleditch, J. 2002, AJ, 124, 1788 Doroshenko, O., L¨ohmer, O., Kramer, M., et al. 2001, A&A, 379, 579 Ransom, S. M., Greenhill, L. J., Herrnstein, J. R., et al. 2001, ApJ, 546, L25 Eggleton, P. P. 1983, ApJ, 268, 368 Rappaport, S., Joss, P. C., & Webbink, R. F. 1982, ApJ, 254, 616 Erdem, A., & Ozt¨urk,¨ O. 2014, MNRAS, 441, 1166 Rappaport, S., Ma, C. P., Joss, P. C., & Nelson, L. A. 1987, ApJ, 322, 842 Evans, P. A., Beardmore, A. P., Page, K. L., et al. 2007, A&A, 469, 379 Rappaport, S., Verbunt, F., & Joss, P. C. 1983, ApJ, 275, 713 —. 2009, MNRAS, 397, 1177 Reiners, A. 2012, Living Reviews in Solar Physics, 9, arXiv:1203.0241 Fortin, M., Providˆencia, C., Raduta, A. R., et al. 2016, Phys. Rev. C, 94, Richman, H. R., Applegate, J. H., & Patterson, J. 1994, PASP, 106, 1075 035804 Roberts, M. S. E. 2013, in IAU Symposium, Vol. 291, IAU Symposium, ed. Frank, J., King, A., & Raine, D. J. 2002, Accretion Power in Astrophysics: J. van Leeuwen, 127–132 Third Edition (Accretion Power in Astrophysics, by Juhan Frank and Roy, J., Ray, P. S., Bhattacharyya, B., et al. 2015, ApJ, 800, L12 Andrew King and Derek Raine, pp. 398. ISBN 0521620538. Cambridge, Sanna, A., Riggio, A., Pintore, F., et al. 2015, The Astronomer’s Telegram, UK: Cambridge University Press, February 2002.) 7364 Galloway, D. K., & Cumming, A. 2006, ApJ, 652, 559 Sanna, A., Burderi, L., Riggio, A., et al. 2016, MNRAS, 459, 1340 Gavriil, F. P., Strohmayer, T. E., Swank, J. H., & Markwardt, C. B. 2007, Scholz, P., Kaspi, V. M., Lyne, A. G., et al. 2015, ApJ, 800, 123 ApJ, 669, L29 Skumanich, A. 1972, ApJ, 171, 565 Gierli´nski, M., Done, C., & Barret, D. 2002, MNRAS, 331, 141 Spruit, H. C., & Ritter, H. 1983, A&A, 124, 267 Goldwurm, A., David, P., Foschini, L., et al. 2003, A&A, 411, L223 Staubert, R., Klochkov, D., & Wilms, J. 2009, A&A, 500, 883 Gonz´alez Hern´andez, J. I., Rebolo, R., & Casares, J. 2012, ApJ, 744, L25 Stepien, K. 1995, MNRAS, 274, 1019 —. 2014, MNRAS, 438, L21 Tauris, T. M. 2011, in Astronomical Society of the Pacific Conference Hartman, J. M., Patruno, A., Chakrabarty, D., et al. 2009, ApJ, 702, 1673 Series, Vol. 447, Evolution of Compact Binaries, ed. L. Schmidtobreick, —. 2008, ApJ, 675, 1468 M. R. Schreiber, & C. Tappert, 285 Heinke, C. O., Grindlay, J. E., Edmonds, P. D., et al. 2005, ApJ, 625, 796 Tauris, T. M., & van den Heuvel, E. P. J. 2006, Formation and evolution of Heinke, C. O., Jonker, P. G., Wijnands, R., Deloye, C. J., & Taam, R. E. compact stellar X-ray sources (Compact stellar X-ray sources), 623–665 2009, ApJ, 691, 1035 Taylor, J. H., & Weisberg, J. M. 1989, ApJ, 345, 434 Hobbs, G. B., Edwards, R. T., & Manchester, R. N. 2006, MNRAS, 369, 655 Thorsett, S. E., Arzoumanian, Z., Camilo, F., & Lyne, A. G. 1999, ApJ, 523, Homer, L., Charles, P. A., Chakrabarty, D., & van Zyl, L. 2001, MNRAS, 763 325, 1471 Ubertini, P., Lebrun, F., Di Cocco, G., et al. 2003, A&A, 411, L131 Hu, C.-P., Chou, Y., & Chung, Y.-Y. 2008, ApJ, 680, 1405 van der Klis, M. 1995, in The Lives of the Neutron Stars, ed. M. A. Alpar, Iaria, R., di Salvo, T., Burderi, L., et al. 2011, A&A, 534, A85 U. Kiziloglu, & J. van Paradijs, 301–+ Iaria, R., Di Salvo, T., Gambino, A. F., et al. 2015, A&A, 582, A32 van der Klis, M. 2000, ARA&A, 38, 717 in ’t Zand, J. J. M., Heise, J., Muller, J. M., et al. 1998, A&A, 331, L25 van der Sluys, M. 2011, in Astronomical Society of the Pacific Conference Jain, C., Paul, B., & Dutta, A. 2010, MNRAS, 409, 755 Series, Vol. 447, Evolution of Compact Binaries, ed. L. Schmidtobreick, Jaodand, A., Archibald, A. M., Hessels, J. W. T., et al. 2016, ApJ, 830, 122 M. R. Schreiber, & C. Tappert, 317 Joshi, K. J., & Rasio, F. A. 1997, ApJ, 479, 948 van Teeseling, A., & King, A. R. 1998, A&A, 338, 957 Justham, S., Rappaport, S., & Podsiadlowski, P. 2006, MNRAS, 366, 1415 Verbunt, F. 1993, ARA&A, 31, 93 Knigge, C., Baraffe, I., & Patterson, J. 2011, ApJS, 194, 28 Wang, Z., Breton, R. P., Heinke, C. O., Deloye, C. J., & Zhong, J. 2013, ApJ, Knigge, C., King, A. R., & Patterson, J. 2000, A&A, 364, L75 765, 151 Lazaridis, K., Verbiest, J. P. W., Tauris, T. M., et al. 2011, MNRAS, 414, Weisberg, J. M., & Taylor, J. H. 2002, ApJ, 576, 942 3134 Wijnands, R., M´endez, M., Markwardt, C., et al. 2001, ApJ, 560, 892 Lazarus, P., Brazier, A., Hessels, J. W. T., et al. 2015, ApJ, 812, 81 Wijnands, R., & van der Klis, M. 1998, Nature, 394, 344 Lebrun, F., Leray, J. P., Lavocat, P., et al. 2003, A&A, 411, L141 Wijnands, R., van der Klis, M., Homan, J., et al. 2003, Nature, 424, 44 Lorimer, D. R. 2008, Living Reviews in Relativity, 11, arXiv:0811.0762 Wolff, M. T., Hertz, P., Wood, K. S., Ray, P. S., & Bandyopadhyay, R. M. Lorimer, D. R., & Kramer, M. 2012, Handbook of Pulsar Astronomy 2002, ApJ, 575, 384 Manchester, R. N., Hobbs, G. B., Teoh, A., & Hobbs, M. 2005, AJ, 129, Wolff, M. T., Ray, P. S., Wood, K. S., & Hertz, P. L. 2009, ApJS, 183, 156 1993 Wolszczan, A. 1994, Science, 264, 538 Marsh, T. R., & Pringle, J. E. 1990, ApJ, 365, 677 Wolszczan, A., & Frail, D. A. 1992, Nature, 355, 145 Morin, J., Donati, J.-F., Petit, P., et al. 2010, MNRAS, 407, 2269 Wright, N. J., & Drake, J. J. 2016, Nature, 535, 526 Xing, Y., Wang, Z., & Jithesh, V. 2015, ArXiv e-prints, arXiv:1502.00733 Yagi, K. 2012, Phys. Rev. D, 86, 081504