Hot disk of the Swift J0243.6+6124 revealed by Insight-HXMT
V. Doroshenko, S.N. Zhang, A. Santangelo, L. Ji, S. Tsygankov, A. Mushtukov, L.J. Qu, S. Zhang, M.Y. Ge, Y.P. Chen, Q.C. Bu, X.L. Cao, Z. Chang, G. Chen, L.Chen, T.X. Chen, Y.Chen, Y.B. Chen, W. Cui, W.W. Cui, J.K. Deng, Y.W. Dong, Y.Y. Du, M.X. Fu, G.H. Gao, H. Gao, M. Gao, Y.D. Gu, Guan, C.C. Guo, D.W. Han, W. Hu, Y. Huang, J. Huo, S.M. Jia, L.H. Jiang, W.C. Jiang, J. Jin, Y.J. Jin, L.D. Kong, B. Li, C.K.Li, G. Li, M.S. Li, T.P. Li, W. Li, X. Li, X.B. Li, X.F. Li, Y.G. Li, Z.J. Li, Z.W. Li, X.H. Liang, J.Y. Liao, C.Z. Liu, G.Q. Liu, H.W. Liu, S.Z. Liu, X.J. Liu, Y. Liu, Y.N. Liu, B. Lu, F.J. Lu, X.F. Lu, T. Luo, X. Ma, B. Meng, Y. Nang, J.Y. Nie, G. Ou, N. Sai, L.M. Song, X.Y. Song, L. Sun, Y. Tan, L. Tao, Y.L. Tuo, G.F. Wang, J.Wang, W.S. Wang, Y.S. Wang, X.Y. Wen, B.B. Wu, M. Wu, G.C. Xiao, S.L. Xiong, H.Xu, Y.P. Xu, Y.R. Yang, J.W. Yang, S. Yang, Y.J. Yang, A.M. Zhang, C.L. Zhang, C.M. Zhang, F. Zhang, H.M. Zhang, J. Zhang, Q. Zhang, T. Zhang, et al. (16 additional authors not shown)
MMNRAS , 000–000 (0000) Preprint 30 September 2019 Compiled using MNRAS L A TEX style file v3.0
Hot disk of the Swift J0243.6 + V. Doroshenko , , ∗ , S.N. Zhang , , ∗ , A. Santangelo , L. Ji , S. Tsygankov , ,A. Mushtukov , , , L.J. Qu , S. Zhang , M.Y. Ge , Y.P. Chen , Q.C.Bu , X.L. Cao ,Z. Chang , G. Chen , L.Chen , T.X. Chen , Y.Chen , Y.B. Chen , W. Cui , , W.W. Cui ,J.K. Deng , Y.W. Dong , Y.Y. Du , M.X. Fu , G.H. Gao , , H. Gao , , M. Gao ,Y.D. Gu , J.,Guan , C.C. Guo , , D.W. Han , W. Hu , Y. Huang , J. Huo , S.M. Jia ,L.H. Jiang , W.C. Jiang , J. Jin , Y.J. Jin , L.D. Kong , , B. Li , C.K.Li , G. Li , M.S. Li ,T.P. Li , , , W. Li , X. Li , X.B. Li , X.F. Li , Y.G. Li , Z.J. Li , , Z.W. Li , X.H. Liang ,J.Y. Liao , C.Z. Liu , G.Q. Liu , H.W. Liu , S.Z. Liu , X.J. Liu , Y. Liu , Y.N. Liu ,B. Lu , F.J. Lu , X.F. Lu , T. Luo , X. Ma , B. Meng , Y. Nang , , J.Y. Nie , G. Ou ,N. Sai , , L.M. Song , X.Y. Song , L. Sun , Y. Tan , L. Tao , Y.L. Tuo , , G.F. Wang ,J.Wang , W.S. Wang , Y.S. Wang , X.Y. Wen , B.B. Wu , M. Wu , G.C. Xiao , ,S.L. Xiong , H.Xu , Y.P. Xu , , Y.R. Yang , J.W. Yang , S. Yang , Y.J. Yang ,A.M. Zhang , C.L. Zhang , C.M. Zhang , F. Zhang , H.M. Zhang , J. Zhang ,Q. Zhang , T. Zhang , W. Zhang , , W.C. Zhang , W.Z. Zhang , Y. Zhang ,Y. Zhang , , Y.F. Zhang , Y.J. Zhang , Z. Zhang , Z.L. Zhang , H.S. Zhao ,J.L. Zhao , X.F. Zhao , , S.J. Zheng , Y. Zhu , Y.X. Zhu , C.L. Zou Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences,19B Yuquan Road, Beijing 100049, People’s Republic of China Institut für Astronomie und Astrophysik, Sand 1, 72076 Tübingen, Germany University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100049, People’s Republic of China Department of Physics and Astronomy, FI-20014 University of Turku, Turku, Finland Space Research Institute of the Russian Academy of Sciences, Profsoyuznaya Str. 84 /
32, Moscow 117997, Russia Leiden Observatory, Leiden University, NL-2300RA Leiden, The Netherlands Anton Pannekoek Institute, University of Amsterdam, Science Park 904, NL-1098 XH Amsterdam, the Netherlands Department of Astronomy, Beijing Normal University, Beijing 100088, People’s Republic of China Department of Physics, Tsinghua University, Beijing 100084, People’s Republic of China
30 September 2019
ABSTRACT
We report on analysis of observations of the bright transient X-ray pulsar Swift J0243.6 + NuSTAR , and
Swift observatories. We focus on the discovery of a sharp state transition of the timing and spectralproperties of the source at super-Eddington accretion rates, which we associate with thetransition of the accretion disk to a radiation pressure dominated (RPD) state, the firstever directly observed for magnetized neutron star. This transition occurs at slightly higherluminosity compared to already reported transition of the source from sub- to super-criticalaccretion regime associate with onset of an accretion column. We argue that this scenario canonly be realized for comparatively weakly magnetized neutron star, not dissimilar to otherultra-luminous X-ray pulsars (ULPs), which accrete at similar rates. Further evidence for thisconclusion is provided by the non-detection of the transition to the propeller state in quiescencewhich strongly implies compact magnetosphere and thus rules out magnetar-like fields.
Key words: accretion,accretion discs–pulsars:general-scattering-stars:magnetic field-stars:neutron-X-rays: binaries c (cid:13) a r X i v : . [ a s t r o - ph . H E ] S e p V. Doroshenko et al
The transient binary X-ray pulsar (XRP) Swift J0243.6 + ∼ × − erg s − cm − .Pulsations with steadily decreasing period of about 9.8 s (Kenneaet al. 2017; Jenke & Wilson-Hodge 2017) implied accretion ontothe neutron star from its Be companion (Kouroubatzakis et al. 2017).Despite the identified counterpart, the distance to the source remainsuncertain. For the rest of the paper we adopt a distance of 6.8 kpc asreported in Bailer-Jones et al. (2018) based on Gaia DR2 parallaxmeasurements of the companion star. We emphasize, however, thateven the lower limit of ∼ . ∼ erg s − (Tsygankov et al. 2018).Another interesting feature of the source was unveiled by radioobservations, which confirmed emission correlated with the X-rayflux close to the peak of the outburst. This was associated by vanden Eijnden et al. (2018b) with an evolving jet, one of the first everobserved from magnetized neutron stars, although radio emissionwas also observed at later stages of the outburst at significantly lowerluminosities (van den Eijnden et al. 2018a,b).Similarly to other ULPs the magnetic field ofSwift J0243.6 + NuSTAR (Jaisawal et al.2018; Tao et al. 2019), nor in the Insight-HXMT observations inthe 2-150 keV energy range (Zhang et al. 2019). Other argumentsthus had to be invoked to estimate the source’s magnetic field.The observed change of the pulse-profile shape in the soft band,and of the pulsed fraction in the hard band prompted Doroshenkoet al. (2018) to conclude that the source switched from the sub-to the super- critical accretion regime (Basko & Sunyaev 1976)at a luminosity of ∼ erg s − , implying that the neutron starhas a magnetic field of ∼ G, i.e., slightly higher than thatusual for accreting pulsars. A similar conclusion was reached byWilson-Hodge et al. (2018) based on the monitoring of the sourcewith NICER and
Fermi / GBM, which revealed peculiar featuresin the dependence of source’s X-ray colours on luminosity bothin the soft and hard band. Both features, despite the significantdi ff erence of the luminosity at which they occured, were associatedwith the transition from the sub- to the super- critical accretionand with onset of an accretion column (Wilson-Hodge et al. 2018).Independently, since the source continued to accrete at a luminosityof ∼ × erg s − without switching to the “propeller” regime(Illarionov & Sunyaev 1975), Tsygankov et al. (2018) estimated anupper limit for the magnetic field of ∼ × G, a value barelyconsistent with other estimates.Here we present the timing analysis of observations ofSwift J0243.6 + NuSTAR mission (Harrison et al. 2013). Based on this analysis, wereport the detection of a striking change of the aperiodic variabilityproperties, pulse profile morphology, and energy spectrum of thesource at a luminosity of L x ∼ . × erg s − , which we associatewith the transition of the inner regions of the accretion disk from the standard gas pressure dominated (GPD) to the radiation pressuredominated (RPD) state. At this point the disc pushes close enoughto the neutron star to make local luminosity of inner disc regionslarge enough to dynamically a ff ect disc structure by radiative pres-sure (Shakura & Sunyaev 1973). This a ff ects the observed X-rayspectrum, aperiodic variability originating within the disc, and cou-pling of the disc with magnetosphere of the neutron star reflectedby change of the observed pulse profiles from the pulsar. In addi-tion, also the already reported transition from sub- to super-criticalaccretion regime is observed.A second important and rather surprising result of our anal-ysis is the discovery of accretion powered X-ray emission fromthe source in deep quiescence, months after the main outburst, at aluminosity as low as ∼ × erg s − . A coherent explanation ofour findings and of the phenomenology already reported in litera-ture unambiguously shows that the accretion disk extends unusuallyclose to the compact object both in quiescence and outburst. Wetherefore conclude that the source’s magnetic field is likely weakerthan previously argued. The rest of the paper is organized as fol-lows: the details of the analysis are presented in section 2 which isfollowed by interpretation of individual observational findings andtheir implications in section 3, and conclusions in section 4. NuSTAR and
Swift / XRT
The source had been observed with
NuSTAR on several occasions(Tao et al. 2019), however, here we focus exclusively on the most re-cent observation after transition to quiescence. The observation wasspecifically aimed to improve the upper limit of the magnetic fieldby Tsygankov et al. (2018), and was conducted following the rapiddecline of the flux revealed by
Swift / BAT monitoring of the sourceon MJD 58557 (80 ks exposure, observation id. 90501310002).The data reduction was carried out using standard proceduresusing the nustardas_06Jul17_v1.8.0 package and most updated cali-bration files. Since the source has been clearly detected in imagesof both
NuSTAR telescopes, to improve counting statistics we com-bined data of both units to perform timing and spectral analysis. Inparticular, we concatenated filtered event lists for timing, while spec-tra from the two units were extracted and modeled independentlyand co-added using the addspec tool for plotting only.To extract source and background spectra and lightcurves, weused circular extraction regions with radii of 80 (cid:48)(cid:48) and 200 (cid:48)(cid:48) centeredon the source and close to the edge of the field of view on thesame detector chip. The source extraction radius was optimizedto achieve the best signal-to noise ratio above 40 keV using theprocedure described in Vybornov et al. (2018). The lightcurves werealso corrected to solar barycenter and for the motion of the binarysystem using the ephemerides provided by Fermi GBM .The main goal of the observation was to determine whetheraccretion continues and the source continues to pulsate. To search forpulsations we used Z statistics (de Jager et al. 1989) on event dataand the procedure described in Doroshenko et al. (2015) to avoidloss of sensitivity due to binning of the lightcurves. A significantpeak with Z ∼ . P ∼ . https://gammaray.nsstc.nasa.gov/gbm/science/pulsars/lightcurves/swiftj0243.html MNRAS , 000–000 (0000) ot disk of the Swift J0243.6 + the corresponding chance detection probability for one harmonicand 10 trial frequencies is ∼ − (de Jager et al. 1989), i.e.,the peak is highly significant. The folded background-subtracted3-80 keV lightcurve reveals a single peaked pulse profile with pulsedfraction of ∼
20% which strongly suggests that the source continuesto accrete (see Fig. 1).The energy spectrum of the source in quiescence remains hardand is well described with a cuto ff power-law ( cutoffpl in Xspec)with no additional components, which also points to continuedaccretion (see Fig. 2). Interstellar absorption was not required bythe fit but was included in the model fixed to interstellar value of9 × atoms cm − for consistency with Swift / XRT . The best-fitstatistics of χ ∼
99 for 104 degrees of freedom indicates the goodquality of the fit. Best fit photon index and cuto ff energy are 1.0(1)and 12.5(2) keV respectively (1 σ confidence level). The bolometricmodel flux turns out to be 5 . × − erg s − cm − , a factor oftwenty smaller compared to previously published limit (Tsygankovet al. 2018).We also monitored the declining phase of the outburst with Swift / XRT, which observed the source at even lower fluxes. The
Swift / XRT lightcurve in 0.5-10 keV band was obtained using theservice provided by the
Swift data center (Evans et al. 2009) byfitting spectra of individual observations (see e.g., Tsygankov et al.2016). In total 44 observations from MJD 58207 to 58748 with totalexposure of ∼
70 ks were considered in this analysis. The absorptioncolumn was not well constrained in all observations, so we fixedit to the interstellar value of 9 × atoms cm − (Willingale et al.2013). This approximation does not significantly a ff ect the fluxestimates since the final bolometric flux estimate was obtained bycomparing the measured source fluxes in the 0.5-10 keV range withthe simultaneous NuSTAR observations. This comparison revealedgood agreement and that the soft band flux accounts for ∼ NuSTAR observation) we assumed the continuum to be the same asrevealed by
NuSTAR and only considered flux as a free parameter.The resulting lightcurve is presented in Fig. 3. As evident from thelightcurve, the source brightness continued to decrease after
NuSTAR observation, reaching as low as ∼ × erg s − , however, thecounting statistics in short XRT pointings (1-2 ks) is not su ffi cient todetect pulsations or even robustly detect potential spectral softening.We can not, therefore, definitively claim that the source continuedto accrete also after NuSTAR observation. On the other hand, theobserved flux variability would be hard to explain otherwise, so itis quite possible that accretion continues even at lower rates thanrevealed by
NuSTAR .Finally, we also obtained the power spectrum of the source tocharacterize variability in quiescence. As shown in Fig.5, the powerspectrum at low luminosities is consistent with a broken powerlawwith a break at 0.19(2) Hz. The observed aperiodic variability thusalso unambiguously shows that the source continues to accrete.Since the observed break frequency is higher than the spin frequencyby a factor of two, we deduce that the timescale of the variabilityrelative to the break is not directly related to the Keplerian timescaleat the inner edge of the disk, expected to be about the spin period atsuch low luminosity. Unfortunately, the available statistics does notallow a more detailed timing analysis, e.g., searching for possibleQPOs or investigating the energy dependence of the pulse profiles.Based on the detection of the pulsations, observed hard energy . . . . . Period, s Z . . . . . Phase . . . . . N o r m . fl ux Figure 1.
Top:
A periodogram for event arrvial times for
NuSTAR ob-servation 90501310002 (80 (cid:48)(cid:48) extraction radius, 3-80 keV band).
Bottom:
Background-subtracted lightcurve for
NuSTAR observation 90501310002 in3-80 keV band folded with best fit period. The pulse profile is plotted twotimes for clarity. spectrum, and aperiodic variability properties, we conclude thus thatthe source continues to accrete at very low luminosity.
The Hard X-ray Modulation Telescope (HXMT) named “Insight”after launch on June 15 2017 from Jiuquan launch centre, is China’sfirst X-ray astronomy satellite (Li 2007b; Zhang et al. 2014). Thethree main instruments on-board are the high energy X-ray telescope(HE) boasting e ff ective area of ∼ between 20-250 keV,the medium energy X-ray telescope (ME) operating in 5-30 keV(e ff ective area 952 cm ), and the low energy X-ray telescope (LE)operating in 1-15 keV with e ff ective area of 384 cm (Zhang et al.2014).Insight-HXMT provides an unprecedented view of the phe-nomenology of the source at high luminosity. With a total expo-sure of ∼
835 ks accumulated in 98 pointings over the period fromMJD 58033 to MJD 58112, observations of Swift J0243.6 + MNRAS000
835 ks accumulated in 98 pointings over the period fromMJD 58033 to MJD 58112, observations of Swift J0243.6 + MNRAS000 , 000–000 (0000)
V. Doroshenko et al Energy, keV − − k e V ( P ho t on s c m − s − k e V − ) Figure 2.
Broadband energy spectrum of the source as observed by
NuSTAR in quiescence unfolded with best-fit model and multiplied by energy squared.
Time, MJD L X , e r g s − Figure 3.
Long-term bolometric lightcurve of Swift J0243.6 + Swift / BAT (blue line),
Swift / XRT(blue points), and
NuSTAR (red crosses). The dimmest
NuSTAR observationmarks a hard upper limit on luminosity of “propeller” state, however, accre-tion likely continues even at lower rates as follows from the observed fluxvariability revealed by
Swift / XRT at later stages. of the HE instrument. With a total e ff ective area of 5100 cm be-tween 20 and 250 keV, it is the main instrument of Insight-HXMTo ff ering high time resolution (0.012 ms) and low dead-time evenfor very bright sources (Zhang et al. 2014). This makes the HE anideal tool for timing studies, so we used it for timing analysis. Wehave verified, however, that similar results can be obtained with MEdetector, although quality of resulting power spectra is lower due tothe lower e ff ective area and shorted good time intervals associatedwith higher in-orbit background. For more details on HXMT andperformance of individual instruments please refer to (Zhang et al.2014). The data analysis was performed with hxmtdas v2.01 follow-ing the recommended procedures in the user’s guide . More detail on data analysis for Swift J0243.6 + Swift / BAT and
NuSTAR is presented in Fig. 3. The fluxes forInsight-HXMT are estimated based on the broadband spectral analy-sis 2-150 keV energy range of individual observations by Zhang et al.(2019) and adopted here from that publication. The values appear tobe in line with the
Swift / BAT count-rate, which can be robustly con-verted to luminosity using the multiplicative factor of ∼ . × (for a distance of 6.8 kpc). Note that we omit errorbars for BATpoints in Fig. 3 for clarity, but those are rather large, i.e. the light-curve is in excellent agreement (within uncertainties) with otherinstruments throughout the outburst. The NuSTAR and
Swift / XRTfluxes deduced from the spectral analysis in 3-80 keV and 0.5-10 keVenergy range are also consistent with those of Insight-HXMT oncethe bolometric correction estimated based on the best-fit model istaken into the account. For instance, for the nearly simultaneousobservation close to the peak of the outburst (i.e. on MJD 58067)the flux measured by
NuSTAR and HXMT agree to within ∼ ff ects of the orbital motion ofthe satellite and for the binary motion assuming the ephemeridesobtained by the Fermi GBM pulsar team . For each observation, wesearched for pulsations, and determined the period value using thephase-connection technique (Deeter et al. 1981). The spin evolutionobserved by Insight-HXMT was found to be consistent with thatrevealed by the Fermi GBM (Zhang et al. 2019).To investigate the evolution of the pulse profile shape withluminosity, the obtained profiles were arranged as a function ofthe flux (Zhang et al. 2019) and aligned with each other using theFFTFIT routine (Taylor 1992). Note that the significant evolution ofthe pulse profiles with flux implied that we had to use an iterativeprocedure, going from low to high fluxes and vice versa, until thealignment presented in Fig. 4 was obtained. We emphasize thetwo major changes of the observed pulse profile shape observed atluminosities of ∼ . − . × erg s − , in line with findings byWilson-Hodge et al. (2018).High counting statistics also allowed to investigate aperiodicvariability in the source. Here we followed the same approach asRevnivtsev et al. (2009) to suppress the pulsations and enable anal-ysis of the aperiodic noise. In particular, taking into considerationthe average length of good time intervals and the source spin period,we split lightcurves from each observation in segments of ∼
300 scorresponding to 30 spin cycles. Each segment was then folded withthe spin-period determined for a given observation to obtain an aver-age pulsed lightcurve, which was then subtracted from the observedlightcurve. The power spectra of the resulting lightcurves was thenobtained using the powspec program by averaging the power spectraof individual segments. Examples of representative power spectraat di ff erent luminosities are shown in Fig 5. The luminosity depen- https://gammaray.nsstc.nasa.gov/gbm/science/pulsars/lightcurves/swiftj0243.html MNRAS , 000–000 (0000) ot disk of the Swift J0243.6 + Time, MJD L X , e r g s super-critical, GPD disksub-critical, GPD disksuper-critical, RPD disk . . . . . . Phase . Frequency, Hz HighfrequencybreakLowfrequencybreakQPO
Figure 4.
Lightcurve of the source as observed by Insight-HXMT (black points) with color-shading showing the di ff erent states of the source. The relevanttransition luminosities marked with horizontal lines in all panels. Transition from the sub- to super-critical luminosity is marked by a dramatic change ofthe pulse profile shape (2nd panel, slices at fixed luminosity are pulse profiles scaled to the same amplitude to emphasize shape evolution). Transition of theaccretion disk from GDP to RDP state besides a change in pulse profile shape is also accompanied by a change in power spectrum where amplitude of theaperiodic variability increases in 1-10 Hz frequency range (3rd panel, slices at given luminosity are power spectra multiplied by frequency and scaled as squareroot to emphasize changes in shape). dence of the power spectrum and pulse profiles are better illustratedin Fig. 4 where two distinct regions can be identified.At low luminosities the power spectrum is well described by abroken power law as typical for magnetic accretors (Revnivtsev et al.2009; Doroshenko et al. 2014). Besides that, some observationsreveal low-frequency quasi-periodic oscillations with frequency of0.1-0.2 Hz, as already reported by Wilson-Hodge et al. (2018). Thedependence of the QPO frequency on the flux has not previouslyreported, likely due to the shorter duration of NICER observationsand more complex pulse profile shape which complicate subtrac-tion of the pulsations and detection of QPOs. On the contrary, thepower spectra obtained with Insight-HXMT reveal weak QPOs withluminosity-dependent frequency from ∼
50 mHz to ∼
200 mHz. Wenotice, however, that in several of the observations it’s di ffi cult todistinguish the feature from the remaining variability associatedwith the imperfect subtraction of pulsations which makes assess-ment of the QPO significance rather complicated. On the other hand,without subtraction of the pulsations QPOs are not detected at all asthe power spectrum is completely dominated by pulsed flux. We cannot exclude, therefore, that apparent presence of QPOs is associatedwith imperfect subtraction of the pulsations. The dynamical powerspectrum (see Fig. 4) reveals, however, that the QPO frequencychanges with luminosity, which points to the physical nature ofthe feature. Extrapolation of QPO frequency observed by HXMTto lower luminosities L X ∼ erg s − where similar feature at50-70 mHz was reported by Wilson-Hodge et al. (2018) also givesconsistent values as can be seen in Fig. 6 although no significantQPOs could be detected at this flux level with HXMT. We note thatsimilar features have been reported for other X-ray pulsars (Fingeret al. 1996).At higher luminosities, i.e., at high accretion rates, the observedpower spectra are di ff erent from those at lower luminosities. Inparticular, a double, rather than single, broken power law shape isnecessary to model the power spectra spectra as shown in Fig. 4,5. We emphasize that this striking change occurs at a luminosity coinciding with the transition of the observed pulse profile shape,i.e., at ∼ × erg s − .To quantify the evolution of the observed power spectrum,and particularly of the break frequencies with luminosity, all powerspectra were converted to a format readable by xspec (see Ingram &Done (2012) for details). The spectra were then approximated usingthe double broken powerlaw model implemented as the bkn2 modelin xspec . Subtraction of the pulsations altered the expected whitenoise level, which was thus determined by including an additionalpowerlaw component with flat power index in the model. For eachobservation, we started the modeling by fixing the first break of the bkn2 model to the spin frequency, and the second break to 10 kHz(i.e., far above the Nyquist frequency) to mimic the single brokenpowerlaw typically observed from other pulsars. We also includeda lorentzian line with width fixed to 0.1 of its central frequencyand arbitrary normalization to account for possible QPOs in allobservations. The relative width of the feature was fixed to a valuechosen based on the analysis of several observations where the widthof the QPO feature could be well constrained. The central frequencyof the QPO was then searched between 50 and 200 mHz, and thefeature was finally included in the fit if the fit statistics improvedsignificantly (i.e. by more than 2 σ as calculated using the f-test)for any of the trial frequencies. The same procedure was repeatedfor the high-frequency break, which was searched in range between0.1 Hz and 1 kHz. The results are presented in Fig 6.We note that the QPO frequency is below the lowest breakfrequency, by a factor of ∼ − f QPO ∝ L / X is also similar, and in fact, consistentwith what predicted from theory if one assumes that the QPO isassociated with the Keplerian timescale, or with the beat frequencybetween the Keplerian frequency at the inner disk edge and theneutron star’s spin frequency (Finger et al. 1996).We note that in some of the observations it is di ffi cult to con-strain the frequency of the low frequency break due to the presence MNRAS000
200 mHz. Wenotice, however, that in several of the observations it’s di ffi cult todistinguish the feature from the remaining variability associatedwith the imperfect subtraction of pulsations which makes assess-ment of the QPO significance rather complicated. On the other hand,without subtraction of the pulsations QPOs are not detected at all asthe power spectrum is completely dominated by pulsed flux. We cannot exclude, therefore, that apparent presence of QPOs is associatedwith imperfect subtraction of the pulsations. The dynamical powerspectrum (see Fig. 4) reveals, however, that the QPO frequencychanges with luminosity, which points to the physical nature ofthe feature. Extrapolation of QPO frequency observed by HXMTto lower luminosities L X ∼ erg s − where similar feature at50-70 mHz was reported by Wilson-Hodge et al. (2018) also givesconsistent values as can be seen in Fig. 6 although no significantQPOs could be detected at this flux level with HXMT. We note thatsimilar features have been reported for other X-ray pulsars (Fingeret al. 1996).At higher luminosities, i.e., at high accretion rates, the observedpower spectra are di ff erent from those at lower luminosities. Inparticular, a double, rather than single, broken power law shape isnecessary to model the power spectra spectra as shown in Fig. 4,5. We emphasize that this striking change occurs at a luminosity coinciding with the transition of the observed pulse profile shape,i.e., at ∼ × erg s − .To quantify the evolution of the observed power spectrum,and particularly of the break frequencies with luminosity, all powerspectra were converted to a format readable by xspec (see Ingram &Done (2012) for details). The spectra were then approximated usingthe double broken powerlaw model implemented as the bkn2 modelin xspec . Subtraction of the pulsations altered the expected whitenoise level, which was thus determined by including an additionalpowerlaw component with flat power index in the model. For eachobservation, we started the modeling by fixing the first break of the bkn2 model to the spin frequency, and the second break to 10 kHz(i.e., far above the Nyquist frequency) to mimic the single brokenpowerlaw typically observed from other pulsars. We also includeda lorentzian line with width fixed to 0.1 of its central frequencyand arbitrary normalization to account for possible QPOs in allobservations. The relative width of the feature was fixed to a valuechosen based on the analysis of several observations where the widthof the QPO feature could be well constrained. The central frequencyof the QPO was then searched between 50 and 200 mHz, and thefeature was finally included in the fit if the fit statistics improvedsignificantly (i.e. by more than 2 σ as calculated using the f-test)for any of the trial frequencies. The same procedure was repeatedfor the high-frequency break, which was searched in range between0.1 Hz and 1 kHz. The results are presented in Fig 6.We note that the QPO frequency is below the lowest breakfrequency, by a factor of ∼ − f QPO ∝ L / X is also similar, and in fact, consistentwith what predicted from theory if one assumes that the QPO isassociated with the Keplerian timescale, or with the beat frequencybetween the Keplerian frequency at the inner disk edge and theneutron star’s spin frequency (Finger et al. 1996).We note that in some of the observations it is di ffi cult to con-strain the frequency of the low frequency break due to the presence MNRAS000 , 000–000 (0000)
V. Doroshenko et al . Frequency, Hz P o w e r × F r e qu e n c y Figure 5.
Representative power spectra above (red) and below (black) thetransition from GPD to RPD state as observed by Insight-HXMT (arbitraryscaled and multiplied by the frequency to highlight the changes in shape).The low luminosity power spectrum is obtained by combining observationsin L X = − × erg s − to improve statistics. The second power spectrumis from a single observation slightly above the transition. Estimated whitenoise and pulsed flux had been subtracted in both cases as described in thetext. Note the QPOs feature around 0.1-0.2 Hz. Finally, the power spectrumof the source in quiescence as observed by NuSTAR is also shown (bluepoints). In the latter case pulsations are not subtracted but do not contributesignificantly to the power spectrum. of the QPOs, and QPO harmonics at similar frequencies. This isparticularly true for brighter observations where QPOs are moreprominent. The scatter of the lower frequency break fit values athigh luminosities shown in Fig 6 might thus be simply related tothe stability of the fit rather than to some physical variability. Thefrequency of the break is also poorly constrained by Insight-HXMTat low fluxes since the relatively low statistics makes it hard to distin-guish the break from the variable QPO reported by (Wilson-Hodgeet al. 2018) at 50 −
70 mHz. Finally, it is clear that in the transitionregion we misidentify the two breaks for some observations.Given these limitations, it is hard to draw any robust conclu-sions regarding the luminosity dependence of the low frequencybreak, whether it stays the same throughout the outburst, or dropsto lower frequencies above the transition. What is clear, however, isthat a sharp change in the observed aperiodic variability properties,coincident with the change of the observed pulse profile shape, oc-curs at L X ∼ − × erg s − (see Fig. 4), so we have to concludethat the two transitions are physically related. In this section we summarize and interpret the observational resultspresented above and in the literature. In particular, we argue thatnon-detection of the propeller transition in quiescence implies arelatively low magnetic dipole field for the pulsar. In this case theaccretion disk, at higher accretion rates, can extend deep into themagnetosphere and has a small inner radius. The energy releasewithin the disk must in this case be substantial, and in fact, su ffi cientfor the transition of the inner disk regions to the RPD state. Thetransition takes place around MJD 58045 and 58098 in rising anddeclining parts of the outburst respectively, and thus is likely respon-sible for the observed power and energy spectra, and the pulse profile L X , erg s − f b / Q P O , H z Figure 6.
Frequency of the break in the power spectra of the aperiodicvariability as function of flux, based on the 20-40 keV Insight-HXMT HElightcurves. The black points indicate either the single break (at lower lumi-nosities), or the lower frequency break when the double broken powerlawmodel is required to fit the data. The red points indicate the location ofthe high frequency break where present. The blue points indicate the QPOfrequency if detected. To guide the eye we included the red and blue lines toindicate powerlaws with index 6 / / changes. On the other hand, changes in source hardness and pulseprofile shape reported by Tsygankov et al. (2018) and Wilson-Hodgeet al. (2018) at slightly lower luminosity (i.e. around MJD 58035and 58139 respectively) can in this case be readily associated withthe onset of accretion column. We show that both transitions, thelimit on “propeller” transitional luminosity, and observed spin-uprate can only be reconciled if the magnetic field of the neutron staris comparatively weak. Below we discuss our interpretation in moredetails. As demonstrated above, the source does not enter the “propeller”regime even in quiescence and continues to accrete at fluxes downto at least 5 . × − erg s − cm − . For the accretion to continue,the source luminosity must be larger than the propeller luminosity(Tsygankov et al. 2018): L prop ≤ × k / B erg s − (1)for standard neutron star parameters. The coupling constant k hereaccounts for the e ff ective magnetosphere size compared to theAlfvèn radius. For the assumed distance of 6.8 kpc the accretion fluxobserved by NuStar implies L prop ≤ × erg s − , a factor 20 lowerthan reported by (Tsygankov et al. 2018). The lowest luminosity ob-served by is another factor of five lower with L prop ≤ × erg s − .Under the same assumptions (i.e. k = . B ∼ . − .
4, which is rather low comparedto other pulsars. Of course, as already mentioned above, the XRTdata do not allow detection of the pulsations, so that conclusionscomes with a caveat. Moreover, as discussed by Tsygankov et al.(2018), this estimate depends strongly on the rather uncertain valueof the coupling constant, which has thus to be considered as a pa-rameter. In the context of current work it is, however, more relevant
MNRAS , 000–000 (0000) ot disk of the Swift J0243.6 + to determine the condition for the onset of the propeller stage fromthe comparison of the co-rotation radius R c = ( GM /ω ) / with thee ff ective magnetospheric radius R m = kR A ∝ ˙ M / . This implies R m ≤ R c (cid:39) . × cm in quiescence, so the magnetosphere mustbe smaller than R m ≤ . × ( L x / L prop ) − / ∼ − × cm atluminosities 1.5-4.5 × erg s − where transitions in pulse profileshape and power spectrum take place, and ∼ − × cm close tothe peak of the outburst. This conclusion has important implication since local temperatureand energy release rate within the accretion disk increase closerto the compact object. For highly magnetized neutron stars themagnetosphere normally truncates the disk far away from the com-pact object, so energy release within the disk can be ignored, thegas pressure dominates and the disk remains thin. The situationin Swift J0243.6 + ff erent due to the ex-tremely high accretion rate and small magnetosphere. The boundarybetween gas pressure and radiation dominated zones can be esti-mated from the balance between gas and radiation pressures in thedisk (Mönkkönen et al. 2019): R AB = m / ˙ M / α / ∼ . × cm , (2)close to the peak of the outburst. Here α (cid:46) m is the mass of the neutronstar in units of solar mass, and ˙ M is the accretion rate in units of10 g s − . Given the estimate of the magnetosphere size obtainedabove, it is thus clear that substantial part of the disk is likely to bein the RPD regime at least close to the peak of the outburst.Indeed, the accretion rate corresponding to RPD transition canbe estimated by equating the transition radius to the magnetospheresize estimate obtained above, which yields ˙ M ∼
14, i.e. signifi-cantly below the peak accretion rate. We can also explicitly compareit with the standard magnetosphere radius R m (Lamb et al. 1973;Frank et al. 2002; Andersson et al. 2005; Mönkkönen et al. 2019): R m = . × k m / R / ∗ , B / L − / cm , (3)which results in (Mönkkönen et al. 2019): L AB = × k / α − / m / R / B / erg s − . (4)Dependence of the transitional luminosity on k , B and α is com-paratively weak, so for any meaningful values the transitional lumi-nosity to an order of magnitude is L AB ∼ × erg s − , i.e. belowthe outbursts peak luminosity.Moreover, given that the propeller transition was actually notdetected, the estimate of the magnetospheric radius presented aboveis only an upper limit, i.e. the accretion disk can, in fact, extendeven closer to the neutron star when radiative pressure becomes dy-namically important. The corresponding characteristic spherizationradius is given by (Shakura & Sunyaev 1973): R sp = ˙ M σ T π m p c ∼ ˙ M cm (5)or ∼ . × cm close to the peak of the outburst. While some-what smaller than the aforementioned lower limit on the expectedmagnetosphere size at outbursts peak, it is clear that as the innerdisk radius approaches the spherization radius, its thickness can notbe neglected anymore. Furthermore, the estimate above neglectsadditional energy input associated with irradiation of the disk bycompact object and interaction of the accretion flow with the mag-netosphere, so changes in disk structure can be anticipated for larger . . . . BAT (15-50 keV), counts s − M AX I( - e V ) , c oun t ss − Figure 7.
Comparison of count-rates in the soft (2-20 keV) and hard (15-50 keV) energy bands based on daily lightcurves by the
Swift / BAT and MAXImissions. At lower fluxes the two bands are well correlated (blue dotted line),however, this is not the case at higher luminosities, where the brightness inthe soft band substantially increases. The vertical lines correspond to thetransitional luminosities as observed by Insight-HXMT. radii (Chashkina et al. 2017). We conclude, therefore, that non-detection of the propeller transition in quiescence directly impliesthat the accretion disk transitions to RPD state during the outburst,which must have some implications on structure of the accretiondisk and associated observables.
From an observational point of view, the transition to RPD state andthickening (or eventual spherization) of the accretion disk can beexpected to a ff ect the velocity of matter within the disk and its innerradius (Chashkina et al. 2017). These factors can be expected toa ff ect the geometry of the accretion flow, and, as a consequence, theobserved X-ray spectrum, pulse profiles, and aperiodic variabilityproperties of the pulsar. Moreover, at this stage the X-ray emissionfrom the disk itself may become observable. Below we illustrate thatthis indeed appears to be the case, and thus argue that observationsconfirm presence of a thick RPD disk in Swift J0243.6 + Spectral transitions, thermal emission from the disk, and on-set of the accretion column
Spectral transitions during the out-burst have been discussed by Wilson-Hodge et al. (2018) who usedNICER and GBM hardness ratios to argue for spectral changes. Inparticular, the transition to the super-critical accretion regime hasbeen suggested to occur at L X ∼ erg s − based on the observedturn-over in the hardness-intensity diagram and pulse profiles inthe soft band (Wilson-Hodge et al. 2018, see, i.e. discussion andFig. A1). Same authors also noted slightly higher transitional lumi-nosity when Fermi / GBM colors are considered, which was attributedthis to complex dependence of the source spectrum and consideredboth events as single transition from sub- to super-critical accretionassociated with onset of an accretion column.We note, however, that both transitions turn out to be coincidentin luminosity with the pulse profile changes detected by Insight-HXMT in the hard band at at luminosities of ∼ − ∼ − × erg s − . The Insight-HXMT observes both transitions in a MNRAS000
Spectral transitions during the out-burst have been discussed by Wilson-Hodge et al. (2018) who usedNICER and GBM hardness ratios to argue for spectral changes. Inparticular, the transition to the super-critical accretion regime hasbeen suggested to occur at L X ∼ erg s − based on the observedturn-over in the hardness-intensity diagram and pulse profiles inthe soft band (Wilson-Hodge et al. 2018, see, i.e. discussion andFig. A1). Same authors also noted slightly higher transitional lumi-nosity when Fermi / GBM colors are considered, which was attributedthis to complex dependence of the source spectrum and consideredboth events as single transition from sub- to super-critical accretionassociated with onset of an accretion column.We note, however, that both transitions turn out to be coincidentin luminosity with the pulse profile changes detected by Insight-HXMT in the hard band at at luminosities of ∼ − ∼ − × erg s − . The Insight-HXMT observes both transitions in a MNRAS000 , 000–000 (0000)
V. Doroshenko et al single energy band, which rules out suggestion by Wilson-Hodgeet al. (2018) and implies that both events actually took place. Thisconclusion is confirmed by the observed spectral evolution of thesource strongly and di ff erence in the observed variability propertiesdescribed below.Let us first discuss the spectral evolution. A prominent softcomponent was reported by Tao et al. (2019) based on the analy-sis of NuSTAR spectra close to the peak of the outburst. Tao et al.(2019) attributed this component to the emission from the accretioncolumn and an outflow, presumably powered by accretion fromsuper-Eddington accretion disk. Tao et al. (2019) did not estimatethe luminosity corresponding to the appearance of this component,however, it can be estimated by direct comparison of the soft andhard light-curves as observed by
Swift / BAT and MAXI monitors in15-50 keV and 2-20 keV energy bands. As illustrated in Fig. 7, thecount-rate in the soft band substantially increases above certain lumi-nosity. Given the evolution of the spectrum with luminosity reportedby Tao et al. (2019), observed spectral softening is clearly relatedto the enhancement of the soft component identified in broadbandspectral analysis.We note that the observed temperature and luminosity (i.e.,emission region size) of the soft component reported by Tao et al.(2019) are consistent with blackbody-like emission of a thick super-Eddington disk truncated at ∼ cm from the neutron star, inagreement with the estimates of the inner disk radius discussedabove. The appearance of the soft component coincides with thehigher-luminosity transition in pulse profile shape and aperiodicvariability properties revealed by Insight-HXMT, and therefore hasto be related to changes of the accretion disk structure. Given thatthe only change expected at this luminosity is the transition of thedisk to the RPD state, we conclude that the observed evolutionof spectral and timing properties ∼ − × erg s − is indeedassociated with such transition.The lower luminosity transition can then be readily associatedwith onset of an accretion column as suggested by Wilson-Hodgeet al. (2018); Tsygankov et al. (2018), and can be used to estimatemagnetic field of the neutron star. We note that no changes of thepower spectrum associated with this transition are observed, i.e. itis likely related to the emission region itself rather than the disk.The corresponding transition luminosity can be estimated ∼ . × erg s − based on evolution of the pulse profiles of the source inhard band revealed by Insight-HXMT. That is slightly higher, butconsistent with 0 . − . × erg s − reported by based on thehardness evolution in the soft band. Features and origin of the observed power spectrum
The ob-served flux variability is induced by local accretion rate fluctuationsoccurring throughout the accretion disk at timescales related to localKeplerian timescale (Lyubarskii 1997) and results in the observedpowerlaw type spectra for flux variability when integrated over thedisk. The disruption of the disk by rotating magnetosphere imposes abreak with frequency correlated with accretion rate and proportionalto the Keplerian frequency at the magnetosphere, which can be usedto estimate the inner disk radius (Revnivtsev et al. 2009). At lowerluminosities the power spectrum of Swift J0243.6 + ∼ . − + L X ∼ − × erg s − , the power spectrum changes qualitatively, and a second break athigher frequencies appears.As already mentioned, the transition is also accompanied by adramatic change of the observed pulse profile shape around thesame luminosity. Given that the aperiodic variability originatesin the accretion disk, it is clear that simultaneous change of thepulse profile shape and of the power spectrum must be triggeredby a major change in the accretion disk structure rather than thatof emission region, i.e., by the disk transition to the RPD state.Detailed modeling of the observed evolution of the power spectrumis beyond the scope of the present work. Below we only discussit qualitatively in context of the aperiodic variability properties ofRPD disks already discussed in the literature.The main open problem to interpret aperiodic variability lies inthe not yet understood origin of the timescales on which accretionrate fluctuations occur within the disk. Revnivtsev et al. (2009) sug-gested these occur at local Keplerian timescale. On the other hand,Mushtukov et al. (2019a) suggested that the dynamo timescale, ex-pected to be proportional to the local Keplerian timescale, appearsto be a better justified assumption from a physics point of view. Ineither case, however, the break frequency in the power spectrumoriginating in a truncated disk is expected to correlate with theKeplerian frequency at the magnetosphere. The exact relation be-tween the two frequencies is still unclear and subject of theoreticalinvestigations, which hampers quantitative predictions.Qualitatively, however, it is reasonable to assume that devi-ations from the Keplerian motion within the disk are expected toa ff ect the timescale of fluctuations, and thus the observed powerspectra. Interaction of an RPD disk with magnetosphere has beenconsidered by (Chashkina et al. 2017), who found that indeed therotation law in RPD zone di ff ers from the Keplerian by several per-cent. More importantly, however, transition to RPD zone alters alsothe e ff ective magnetosphere radius, and overall disk structure. Thatis particularly relevant if we consider that fluctuations are likelyto occur on dynamo timescale, which also depends on other diskproperties such as viscosity and vertical scale (Mushtukov et al.2019a). Both are expected to change upon the RPD transition, andso it can be expected to alter the emerging power spectrum.Mönkkönen et al. (2019) used this argument to interpret the ob-served peculiar power spectrum of “bursting pulsar” GRO J1744 − + −
28 has an order ofmagnitude lower field). However, a detailed modeling of the powerspectrum that takes into account di ff erences in magnetic field andspin frequency is required to assess whether this scenario is viable.Alternatively, the appearance of the second break could be re-lated to the propagation of the emission from the pulsar through theoptically thick envelope expected to enclose a large part of the mag-netosphere at high luminosities (Mushtukov et al. 2019b). Finallyinteraction of the disk with the quadrupole field component enabledby change of the disk structure could also explain the emergence ofthe second break (Mönkkönen et al. 2019). The multipole nature ofthe magnetic field in the vicinity of the neutron star’s surface couldalso explain the non-detection of the cyclotron line, which in princi-ple shall fall within the energy range of NuSTAR and
Insight-HXMT .Indeed, higher field at the surface might imply a higher line energy,outside the energy range observed by the HE, or simply could smear
MNRAS , 000–000 (0000) ot disk of the Swift J0243.6 + Figure 8.
The suggested geometry of the accretion disk and the emissionregion for respective states is also sketched for illustration. a lower energy feature due significant gradients of the fields acrossthe emission region expected for quadrupole field configuration. Asimilar scenario has been suggested for some ULPs (Israel et al.2017; Tsygankov et al. 2017; Middleton et al. 2019).
We outline, therefore, the following scenario for the evolution of theaccretion geometry of the source with luminosity. As illustrated inFig. 8, the source first makes a transition (from III to II) from thesub- to the super-critical accretion, an accretion column is formedand the geometry of the emission changes. This change is reflectedin the observed pulse profile shape and luminosity dependenceof soft X-ray colours reported by Wilson-Hodge et al. (2018). Inthe second, higher luminosity transition the disk moves from theGPD to the RPD state (from II to I). The disk thickness and, as aconsequence, the geometry of the accretion flow and the emissionregion geometry change. This transition is reflected in the changeof the power spectrum, appearance of the strong soft excess in theX-ray spectrum associated with inner disk regions (Tao et al. 2019),and in the detection of radio emission correlated with X-ray fluxattributed by van den Eijnden et al. (2018b) to jet formation. Thetransition might also be accompanied by a change of the dominantmode of interaction of the accretion disk with the magnetosphere,i.e., from dipole to quadrupole.The main observational arguments in favor of this scenarioare the non-detection of the propeller transition by
NuSTAR (whichimplies that the magnetosphere must be compact both in quies- cence and outburst), and the detection of the transitions in pulseprofile and power spectra shape by HXMT-
Insight (which can beexplained with RPD transition). So far we tried to keep the discus-sion independent of the assumed magnetic field of the source, andhave not cross-checked the self consistency of the model beyondorder of magnitude comparisons. With the improved upper limiton the propeller luminosity it becomes, however, possible to putstronger constrains on the magnetosphere size both in quiescenceand outburst, and thus the magnetic field of the source.Indeed, as discussed by Doroshenko et al. (2018), the observedhigh spin-up rate at outbursts peak imposes a lower limit on thee ff ective magnetosphere size. On the other hand, the lack of tran-sition to the propeller state in the quiescence, imposes an upperlimit, and so combining the two turns out quite a powerful tool toactually measure the size of the magnetosphere. Indeed, assumingthat magnetosphere size in quiescence R m is close to the corotationradius R c and standard scaling for the magnetospheric radius, wecan limit the size of the magnetosphere at any point as kR m ≤ R c (cid:32) ˙ M ˙ M prop (cid:33) − / ∼ . × (cid:32) ˙ M ˙ M prop (cid:33) − / cm , (6)where ˙ M prop denotes the accretion rate corresponding to the tran-sition to propeller. At the same time, as discussed by Doroshenkoet al. (2018), the magnetosphere must be su ffi ciently large to explainthe observed spin-up rate: kR m ≥ (cid:18) I ˙ ω ˙ M (cid:19) GM , (7)where ˙ ω is the observed spin-up rate at given accretion rate ˙ M . Herewe ignore completely any braking torques, so this is an absolutelower limit on the magnetosphere size. Now we can combine the twoequations inserting the appropriate numerical values. Consideringthe uncertainty in the magnetospheric radius dependence on theaccretion in the RPD state (Chashkina et al. 2017), it makes sense tolimit our comparison to the GPD regime where the highest observedspin-up rate is ˙ ω ∼ . × − rad s − at ˙ M ∼ . × g s − (Doroshenko et al. 2018). We thus obtain for the same accretion rate2 . × cm ≤ kR m ≤ . × cm , (8)and2 . × cm ≤ kR m ≤ . × cm , (9)assuming the lowest NuSTAR and
Swift fluxes as limits for the pro-peller luminosity. In both cases the limits are consistent with eachother, and the e ff ective magnetosphere size is at least factor of fivelower than the RPD transitional radius as per Eq.2. In other words,the observed spin-up rate is consistent with the suggestion that thedisk undergoes the RPD transition. We note also that the accretionluminosity can not be much lower than that observed by XRT, as inthis case transition to propeller is inevitable. It can not be excluded,therefore, that variability observed by XRT actually is associatedwith unstable accretion around the propeller luminosity. Neverthe-less, assuming kR m = × , standard definition of magnetosphericradius, and, as usual, k = .
5, this translates to B ∼ .
16, that is anextremely weak field for an X-ray pulsar. This is highly unlikely, andthus we have to conclude that the accretion disk in fact penetratesmuch deeper in the magnetosphere than commonly assumed.This conclusion can be verified, to some extent, by comparingthe observed luminosity of the second transition, i.e. from sub- tosuper- critical accretion regime with theoretical predictions. Asalready mentioned, the observed transitional luminosity is L crit ∼ MNRAS000
16, that is anextremely weak field for an X-ray pulsar. This is highly unlikely, andthus we have to conclude that the accretion disk in fact penetratesmuch deeper in the magnetosphere than commonly assumed.This conclusion can be verified, to some extent, by comparingthe observed luminosity of the second transition, i.e. from sub- tosuper- critical accretion regime with theoretical predictions. Asalready mentioned, the observed transitional luminosity is L crit ∼ MNRAS000 , 000–000 (0000) V. Doroshenko et al . × erg s − , which can be compared with the same relationby Becker et al. (2012) as Wilson-Hodge et al. (2018). Assuming kR m ∼ × cm we arrive thereby to the same value of B ∼ . × − L − / ∼ . k ∼ . ff ective magnetospheresize. On the other hand, theoretical predictions of critical luminosityare rather uncertain on their own, since the critical luminosity isa ff ected by assumed emission region geometry, particularly area ofthe hotspots on the surface of neutron star. This area is defined bythe geometry of the accretion flow, which is believed to be definedby the e ff ective magnetosphere size (Mushtukov et al. 2019a). Thisdependency is ignored by Becker et al. (2012), and here estimate byMushtukov et al. (2015a) is more appropriate. Numerical evaluationusing this model gives slightly lower field of B ∼ − k ∼ . − . k are clearly preferred.The RPD transitional luminosity predicted for the estimatedvalues of k and B appears by factor of two lower than observed,and in fact, more consistent with the observed luminosity of firsttransition L AB ∼ × erg s − , i.e. the transition to RDP regimeoccurs simultaneously with onset of an accretion column, which isa surprising coincidence. In principle, the e ff ects of RPD transitioncould be expected to become apparent at higher luminosities whensubstantial part of the disk transitions to RPD state. Furthermore,we only considered a fixed assumed distance, which is also ratheruncertain and decreasing the distance to lower limit of ∼ The surprise outburst of Swift J0243.6 + L x ∼ . − × erg s − ,transition from the sub- to the super- critical regime and onset of theaccretion column at L x ∼ − × erg s − , transition to the RPDstate at the highest luminosity L x ∼ . × erg s − accompaniedby the appearance of the strong soft excess in X-ray spectrum, andthe observed spin-up rate of the neutron star throughout the outburst.Considering all observables, we conclude that the dipole componentof the magnetic field in Swift J0243.6 + B ∼ − × G, and likely at the lower limit of this range, i.e., inrange typical for accreting pulsars. Even so, the source has been ableto reach ULP luminosity levels while still pulsating. This conclusionis in line with recent field estimates (Tong 2015; Chen 2017; Xu & Li 2017; ? ) for extra-galactic ULPs, where magnetar-like like fieldswere initially suggested (Mushtukov et al. 2015b).We note that our conclusions are confirmed by findings ofJaisawal et al. (2019) who investigated shape of the iron line andthermal emission from the source based on NICER observations.The iron line was found to broaden with luminosity suggesting highvelocities in inner disc regions and inner disc radius as small as ∼ × cm, i.e. consistent with our findings. Thermal emissionfrom the disc similar to that reported by Tao et al. (2019) was alsodetected, although interpretation of the broadband continuum, asdiscussed by Jaisawal et al. (2019), is slightly di ff erent in the twocases.It can be possible due to strong multiple components of themagnetic field (see e.g. Israel et al. 2017). Another possibility isrelated to the geometrical thickness of accretion column: geometri-cally thin accretion columns can support larger accretion luminosity(see approximate equation 10 in Mushtukov et al. 2015b). It is as-sumed that the geometrical thickness of a column is determinedby the penetration depth of accretion disk into the magnetosphere.In the paper by Mushtukov et al. 2015b the penetration depth wastaken to be about a geometrical thickness of a disk at the magne-tospheric radius. Because the accretion disk tends to be radiationpressure dominated and geometrically thick at large mass accretionrates, the geometrical thickness of accretion column was taken tobe large. However, if the penetration depth of the accretion diskinto the magnetosphere is significantly smaller than its geometricalthickness (it might be due to a strong radiation force at the inner diskedge), the geometrical thickness of accretion columns in the paperby Mushtukov et al. 2015b was overestimated, and, therefore, themaximal luminosity of the columns was underestimated. In this casethe enormous accretion luminosity of ULX pulsars can be explainedwithout the hypothesis of magnetar-like magnetic fields.We note also that high magnetic fields inferred for ULPs, andin particular M82 X-2 (Bachetti et al. 2014; Tsygankov et al. 2016)largely stem from the assumption that the disk is truncated at ap-proximately half of the Alfvénic radius (Tsygankov et al. 2016) andmay be overestimated if it is not the case. As discussed above, thee ff ective magnetosphere size in Swift J0243.6 + k ∼ . − . −
3, hasbeen also suggested as a possibility to resolve discrepancy betweenvarious estimates of the magnetic field in ULPs (Tsygankov et al.2017). Observations of Swift J0243.6 + + + NuSTAR , NICER , and VLA observations of this source will likely be key fortackling the problem of ULPs.
MNRAS , 000–000 (0000) ot disk of the Swift J0243.6 + ACKNOWLEDGEMENTS
This work made use of the data from the Insight-HXMT mis-sion, a project funded by China National Space Administration(CNSA) and the Chinese Academy of Sciences (CAS). The Insight-HXMT team gratefully acknowledges the support from the NationalProgram on Key Research and Development Project (Grant No.2016YFA0400800) from the Minister of Science and Technologyof China (MOST) and the Strategic Priority Research Program ofthe Chinese Academy of Sciences (Grant No. XDB23040400). Theauthors thank supports from the National Natural Science Founda-tion of China under Grants No. 11503027, 11673023, 11733009,U1838201 and U1838202. This work was supported by the Rus-sian Science Foundation grant 19-12-00423 (VD, ST, AM). Thisresearch has made use of MAXI data provided by RIKEN, JAXAand the MAXI team. We acknowledge the use of public data andproducts from the
Swift data archive. We also thank the
NuSTAR team for approving the DDT observation of Swift J0243.6 + REFERENCES
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