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Mon. Not. R. Astron. Soc. 000, 1–14 (2015) Printed 4 June 2015 (MN LATEX style file v2.2)

On Physical Nature of the Source of Ultraluminous X-ray

Pulsations

G. Ter-Kazarian1⋆

1Ambartsumian Byurakan Astrophysical Observatory, Byurakan, 378433, Aragatsotn District, Armenia

4 June 2015

ABSTRACT

To reconcile the observed unusual high luminosity of periodic source M82X-2 of thefirst NuSTAR ultraluminous X-ray pulsations with the most extreme violation of theEddington limit, and in view that a persistent X-ray radiation fromM82X-2 ultimatelyprecludes the possibility of typical pulsars, we tackle the problem by the implicationsof ”microscopic theory of black hole”, the preceding developments of which are ofvital interest for the physics of ultra-high energy (UHE) cosmic-rays. Replacing acentral singularity by the infrastructures inside event horizon, subject to certain rules,MTBH explains the origin of ZeV-neutrinos which are of vital interest for the source ofUHE- particles. Withal, M82X-2 is assumed to be a spinning intermediate mass blackhole resided in final stage of growth. Then the thermal blackbody X-ray emission,arisen due to the rotational kinetic energy of black hole, escapes from event horizonthrough the vista to outside world that detected as ultraluminous X-ray pulsations.The M82X-2 indeed releases 99.59% of its pulsed radiative energy predominantly inthe X-ray bandpass 0.3−30 keV, while the pulsed radiation over the band 0.3−3 keVwas not detected yet because it is seemingly suppressed by more powerful persistentradiation over this band. We derive a pulse profile and give a quantitative accountof energetics and orbital parameters of the semi-detached X-ray binary containinga primary accretor M82X-2 of inferred mass M ≃ 138.5 − 226M⊙ and secondarymassive,M2 > 48.3−64.9M⊙, O/B-type donor star with radius of R > 22.1−25.7R⊙,respectively.

Key words: black hole physics- accretion: accretion discs - X-rays: binaries- X-rays:individual (M82X-2≡X42.3+5913).

1 INTRODUCTION

The physical nature of the accreting off-nuclear point sourcesin nearby galaxies, so-called the ultraluminous X-ray sources(ULXs), has been an enigma because of their high en-ergy output. Their luminosity ranges from 1039 erg s−1

6

L(0.5− 10keV) 6 1041 erg s−1 (Roberts 2007; Feng & Soria2011), which exceeds the theoretical maximum for spher-ical infall (the Eddington limit) onto stellar-mass blackholes (Roberts 2007; Liu et al. 2013). The ULX sourceshave attracted a great deal of observational and theoreticalattention, in part because their luminosities suggest thatthey may harbor intermediate mass black holes (IMBHs)with an ubiquitous feature of the mass fits of more than102 − 104M⊙ (Feng & Soria 2011; Colbert & Mushotzky1999; Makishima et al. 2000). A ROSAT/HRI catalog ofULXs (Colbert & Mushotzky 1999) provides a clue to fillin the missing population of the most intriguing of X-ray

⋆ E-mail: gago 50@yahoo.com

source classes of IMBHs. The strongest support for ULXscontaining IMBHs comes from X-ray spectroscopy, wherethe most good fits can be obtained to ULX spectra us-ing the same multi-colour thin disk accretion that powersbright Galactic X-ray binaries. The hyperluminous X-raysources with luminosities > 1041 erg s−1 are thought to beamongst the strongest IMBH candidates (Matsumoto et al.2004). Assuming the emission is isotropic, in general,the extreme luminosities of ULXs suggest either thepresence of IMBHs, see e.g. (Komossa & Schulz 1998;Dewangan et al. 2006; Stobbart et al. 2006; Patruno et al.2006; Portegies et al. 2007; Kong et al. 2007; Casella et al.2008; Feng & Kaaret 2010; Liu et al. 2013; Feng & Soria2011; Pasham et al. 2014), or stellar-mass black holes thatare either breaking or circumventing their Eddington limitvia somewhat peculiar geometric beaming of accretionflow (Begelman 2002; King et al. 2001; Okajima et al. 2006;Roberts 2007; Zampieri & Roberts 2009; Stobbart et al.2006; Poutanen et al. 2007). The key physical issue ofwhether ULXs are powered by IMBHs or normal stellar

2 G. Ter-Kazarian

black holes to date is unresolved, primarily because we donot have dynamical mass measurements of the compact ob-jects that power ULXs. Only time will tell whether any ofthese intriguing proposals is correct and which of the hy-pothesized ULX scenario is actually realized in nature.

The most striking is the recent NuSTAR detectionof the first ultraluminous X-ray pulsations with the max-imum luminosity L(3 − 30keV) = 4.9 × 1039 erg s−1, ofaverage period 1.37 s with a 2.5-day sinusoidal modu-lation (Bachetti et al. 2014), coming from an ultralumi-nous source, NuSTAR J095551+6940.8, located nearbystarburst galaxy M82 (NGC 3034). Chandra observa-tions confirm that only the two ULX sources of M82X-1 (CXOU J095550.2+694047) and M82X-2 (also referredto as X42.3+5913) in the Chandra image are sufficientlybright to be the counterpart of NuSTAR J095551+6940.8.The pulsed emission centroid is spatially consistent with thelocation of a variable M82X-2 which further secures the as-sociation of the pulsating source with M82X-2. A currentsituation is quite complicated. On the one hand, it is gen-erally believed that the pulsating X-ray sources are mag-netic neutron stars which are accreting matter from a bi-nary companion (Pringle & Rees 1972; Davidson & Ostriker1973; Lamb et al. 1973; Lamb 1975). Therefore, there isnothing left but M82X-2, which astronomers had thoughtwas a black hole, is the brightest magnetic neutron starsystem ever recorded. Even though, note that the typicalaccreting pulsar model is not without problems. The ac-tual mechanism by which pulsars convert the rotational en-ergy of the neutron star into the observed pulses is poorlyunderstood. Many theoretical models have been proposedthat account for such features, but no single one is com-pelling (Sieber & Wielebinski 1981; Michel 1982). On theother hand, explaining periodic source M82X-2 that obvi-ously has black hole energetics with a ∼ 1.4M⊙ compactobject using typical accreting pulsar models is extremelychallenging. The NuSTAR team discovery is the most ex-treme violation, i.e. the pulsed luminosity of M82X-2 reachesabout ∼ 26.9 times brighter than the theoretical thresholdfor ∼ 1.4M⊙ stellar-mass black holes where the outwardpressure from radiation balances the inward pull of gravityof the pulsar, the so-called Eddington limit. The accretion isinhibited once radiation force is equal or grater than gravityforce. Anyhow, the difficulty is brought into sharper focusby considering the association with M82X-2, which is fea-tured with high luminosity (≃ 1.8 × 1040 erg s−1) of addi-tional persistent continuous broad X-ray radiation observedearlier from its active state (Bachetti et al. 2014). This morecompelling argument in somehow or other implies the lu-minosity ∼ 100 times if compared to the Eddington limit.Therefore, uncomfortably high collimation (∼ 100) wouldbe needed to explain M82X-2 as beamed radiation fromneutron star or stellar black hole binaries. This ultimatelyprecludes the possibility of typical pulsars also because thecentroid of the persistent X-ray emission is between M82X-2 and M82X-1 while the ULX source M82X-1 is the bestknown candidate for an IMBH. Thus, this is completely un-expected and inexplicable in the context of conventional pul-sar model. The fraction of ULXs powered by neutron starsmust now be considered highly uncertain and many detailsof this scenario remain poorly understood. Added to thiswas the fact that the NuSTAR finding of an ultraluminous

pulsations has implications for understanding that they mayindeed not be rare in the ULX population. If so, it will sig-nificantly enhance our understanding of the ULX Universewhich currently we do not have. In the future astronomersalso will look at more ULXs, and it is possible they couldprove an expected ubiquitous feature of even more energeticULX pulsations as being a common phenomena in the lo-cal Universe. This will support the idea that more ULXsbeat with the pulse of black holes rather than magnetizedneutron star systems. With this perspective in sight, it isbecoming increasingly important that in the case at handthe conventional pulsar model be tested critically. Puttingapart the discussion of inherent problems of the mass scal-ing of the black holes in ULXs, which is beyond the scope ofthis report, we have focused on black hole rather than typicalaccreting pulsar models. To reconcile the observed unusualhigh pulsed luminosity with the above mentioned violationof the Eddington limit, we examine the physics which is atwork in ultraluminous pulsations by assuming that M82X-2is being a spinning intermediate-mass black hole (SIMBH).A black hole of intermediate mass is certainly an attractivescenario. Fore example, Kong et al. (2007) suggest that theM82X-2 is a binary system with a black hole accretor. As-suming the persistent emission is isotropic, the X-ray lumi-nosity ≃ 1040 erg s−1 implies that the compact object is a> 100M⊙ IMBH in the low/hard state. The standard phe-nomenological black hole model (PBHM), of course, is ruledout to achieve our goal because in this framework the blackholes do not have a way to create the pulses due to the twoprinciple features as follows: 1) black holes do not radiate;2) the spinning black holes are axisymmetric and have no in-ternal structure on which to attach a periodic emitter. Anymechanism depending on accretion would not be periodicto such fantastic precision. Orbital motion, whether modu-lating some emission mechanism directly or exciting short-period pulsations, would decay very quickly due to gravita-tional radiation. Inspired by the discovery of the rare andmighty pulsations from the ULX in M82 with NuSTAR, werevisited the MTBH (see Sect.2), which completes PBHMby exploring the most important processes of spontaneousbreaking of gravitation gauge symmetry and rearrangementof vacuum state at huge energies. One of the purposes of thisreport is to motivate and justify the further implications ofMTBH framework to circumvent the alluded obstacles. Wewill proceed according to the following structure. To startwith, the physical nature of the problem is further discussedin Subsection 1.1. Section 2 deals with a brief review of keyobjectives of the MTBH framework of relevance to IMBHseeds as a guiding principle to make the rest of paper un-derstandable. In Section 3 we set out to examine the ultralu-minous pulsations powered by M82X-2. We derive a generalprofile of pulsed luminosity of M82X-2, give a quantitativeaccount of a potential dynamical mass scaling of M82X-2and other energetics, estimate the mass of companion andthe orbit parameters of the mass-exchange binary, discussthe measured spin-up rate, and calculate the torque addedto M82X-2 per unit mass of accreted matter. Concludingremarks are given in Section 4.

On Nature of Ultraluminous X-ray Pulsations 3

1.1 Further motivation and more details

The most bright in persistently X-rays source in regions ofnuclear and galaxy’s disk of the galaxy M82 is M82X-1 whichcan reach L(0.3 − 10keV) ∼ 1041 erg s−1. This source is thebest known candidate for an IMBH, for which most evidencestacks up through a combination of its extreme luminosity,the 62.0±2.5 day periodicity in the X-ray source, co-locationwithin the young, dense stellar cluster MCG-11 with an ageof 7-12 Myr (McCrady et al. 2003). The second brightest isbeing a transient, M82X-2, X-ray luminosity of which variesfrom below the detection level 2.5×1038 erg s−1 to its activestate L(0.3−10keV) ≃ 1.8×1040 erg s−1 and it was turned offtwice in 1999 and 2000 indicating a factor of > 50 variabil-ity (Kong et al. 2007). These two sources are separated by5”, and so can only be clearly resolved by the Chandra X-raytelescope. The X-ray spectra of some Chandra observationsof M82X-2 are best fitted with an absorbed powerlaw modelwith photon index ranging from 1.3 to 1.7. These spectra aresimilar to those of Galactic black hole binary candidates seenin the low/hard state except that a very hard spectrum wasseen in one of the observations. Detection of coherent pulsa-tions, a binary orbit, and spin-up behaviour indicative of anaccretion torque unambiguously, allow to feature the M82X-2 as mass-exchange binary that contain an accreting IMBHprimary and a nondegenerate secondary donor star. Theirsecondaries are expected to be massive O/B-type stars. Aflux F (0.5− 10keV) = 4.07× 10−12 erg cm−2, s−1 is derivedfor M82X-2 during the Chandra observation. Eventually, inaddition to the orbital modulation, about an evident lin-ear spin-up of the pulsar was reported by (Bachetti et al.2014), with p ≃ −2× 10−10s/s over the interval from modi-fied Julian days 56696 to 56701 when the pulse detection ismost significant. Phase connecting the observations enablesdetection of a changing secular spin-up rate over a longertimespan as well as erratic variations. All accreting pulsarsshow stochastic variations in their spin frequencies and lumi-nosities, including those displaying secular spin-up or spin-down on long time scales, blurring the conventional distinc-tion between disk-fed and wind-fed binaries (Bildsten et al.1997). Pulsed flux and accretion torque are strongly corre-lated in outbursts of transient accreting pulsars, but uncor-related, or even anticorrelated, in persistent sources. Theobserved secular spin-up rate can be accounted for quanti-tatively if one assumes the reduction of the torque on therapidly spinning neutron star. But the NUSTAR measuredrate of secular spin-up is unexpected and difficult to ex-plain. The NuSTAR team notably points out that if thespin-up results from the torque applied by accreting mate-rial threading onto the magnetic field, is likely in spin equi-librium given the short spin-up timescale, P/P ∼ 300yr.The spin-up supports an accretion rate that is only a fewtimes higher than Eddington, independent of any assump-tion about the pulsar magnetic field. For an equilibrium spinperiod of 1.37s and M ∼ MEdd, the implied magnetic fieldis B ∼ 1012 Gauss, and too low to have an appreciable effecton LEdd. As also noticed by these authors, a fan beam ge-ometry (Gnedin & Sunyaev 1973) viewed at a favorable an-gle may be controversial possibility to produce the observedpulse profile and provide the requisite moderate collimation.Transient behavior of M82X-2 with a massive companiondonor star likely requires an IMBH, even though the physi-

cal interpretation of the latter is still controversial. It is notclear to date how such black holes would form. On the onehand, to produce an massive IMBH of 102−104M⊙, the corecollapse of an isolated star in the current epoch is not a vi-able process, which is how the stellar black holes are thoughtto form, because of the lack of very massive stars. On theother hand, their environments lack the extreme conditions,i.e. high density and velocities observed at the centers ofgalaxies, which seemingly lead to the formation of supper-massive black holes. There are two conventional scenariosfor the formation of IMBHs. The first is the merging of stel-lar mass black holes and other compact objects by meansof gravitational radiation. The second one is the runawaycollision of massive stars in dense stellar clusters and thecollapse of the collision product into an IMBH. But, in fact,most ULX host galaxies do not even have stellar clusterssufficiently massive and compact to satisfy the requirementsfor runaway core collapse. These objects could be formedinside clusters that have since dispersed. However, the evap-oration timescale of such clusters would be too long to ex-plain the observed association of many ULXs with young(6 20Myr) stellar populations. A metal-free population IIIstars formed in the very early Universe could reach massesof a few hundred M⊙, above the pair-instability limit, andthus may have collapsed into IMBHs (Madau & Rees 2001).It was then expected that in young and dense star clus-ters, dynamical friction could lead to massive stars sinkingtowards the center and undergoing runaway collisions andmergers on timescales 106 yr. Dynamical evidence for IMBHswith masses ∼ 104M⊙ has been proposed for a few globularclusters, e.g., G1 in M31 (Gebhardt et al. 2005), althoughthe issue is still controversial (Anderson, & van der Marel2010). There are also several other exotic scenarios on forma-tion of black hole seeds: In massive and/or compact globularclusters, a central seed black hole may grow by up to a fac-tor of 100 via accretion of gas lost by the first generation ofcluster stars in their red-giant phase (Vesperini et al. 2010);or, IMBHs may wander in the halo of major galaxies, af-ter tidal stripping of merging satellite dwarfs that containednuclear BHs (King & Dehnen 2005; Bellovary et al. 2010).Although this seems the most plausible explanations for themost extreme ULX found to date, HLX-1 in ESO243-49,in general, currently we do not have a complete theoreti-cal interpretation of physical nature of growth of black holeseeds. MTBH may become of eminent physical significancefor tackling this problem. For the benefit of the reader, there-fore, a brief outline of the key ideas behind the MTBH as aguiding principle will be given in the next section to makethe rest of paper understandable. In particular, we shall seehow a IMBH seed is thought to form. There are several im-portant topics, not touched upon here, which will eventuallybenefit from a proposed gravitation theory.

2 THE MTBH, REVISITED:THE

IMPLICATIONS FOR IMBHS

With typical bolometric luminosities∼ 1045−48 erg s−1, theactive galactic nuclei (AGNs) are amongst the most lu-minous emitters in the Universe, particularly at high en-ergies (gamma-rays) and radio wavelengths. From its his-torical development, up to current interests, the efforts in

4 G. Ter-Kazarian

the AGN physics have evoked the study of a major un-solved problem of how efficiently such huge energies ob-served can be generated. This energy scale severely chal-lenges conventional source models. The huge energy releasefrom compact regions of AGNs requires extremely high ef-ficiency (typically > 10 per cent) of conversion of rest massto other forms of energy. This serves as the main argumentin favour of supermassive black holes (SMBHs), with massesof 106 − 1010M⊙, as central engines of massive AGNs. Theextreme conditions, i.e. high density and velocities observedat the centers of AGNs, seemingly lead to the formationof SMBHs. In high luminosity AGNs the large-scale inter-nal gravitational instabilities drive gas towards the nucleusthat trigger big starbursts, and the coeval compact clus-ter just formed. It seemed they have some connection tothe nuclear fueling through mass loss of young stars as welltheir tidal disruption, and supernovae. Therefore, a complexstudy of AGN evolution requires a comprehensive under-standing of the important phenomenon of black hole growth.Most SMBH growth happens in the AGN phase. A signifi-cant fraction of the total black hole growth, 60%, happensin the most luminous AGN, quasars (Treister et al. 2010).In an AGN phase, which lasts ∼ 108 years, the central su-permassive black hole can gain up to ∼ 107−8M⊙, so eventhe most massive galaxies will have only a few of theseevents over their lifetime. These ideas gather support es-pecially from a breakthrough made in recent observationaland computational efforts on understanding of coevolutionof black holes and their host galaxies, particularly throughself-regulated growth and feedback from accretion-poweredoutflows, see e.g. (Natarajan & Treister 2009; Vestergaard2004; Volonteri et al. 2008; Volonteri & Natarajan 2009;Volonteri 2010; Shankar et al. 2009; Kelly et al. 2010;Natarajan 2011; Treister & Megan 2012; Willott et al. 2010;Devecchi & Volonteri 2009). Most aspects of the modelsthat describe the growth and accretion history of supermas-sive black holes, and evolution and assembly history of thisscenario have been explored in detail by (Volonteri et al.2008; Volonteri & Natarajan 2009). In these models, at earlytimes the properties of the assembling SMBH seeds are moretightly coupled to properties of the dark matter halo as theirgrowth is driven by the merger history of halos. A conven-tional theoretical interpretation of aforementioned physicalscenarios is based on a general belief, reinforced by state-ments in textbooks, that a longstanding standard PBHMcan describe the growth of accreting black hole seeds. Inthe framework of genereal relativity (GR), PBHM impliesthe most general Kerr-Newman black hole model, with pa-rameters of mass (M), angular momentum (J) and charge(Q), still has to put in by hand. But such beliefs are suspectand should be critically re-examined. PBHM cannot be cur-rently accepted as convincing model for addressing the blackhole growth, because in this framework the very source ofgravitational field of the black hole is a kind of meaninglesscurvature singularity at the center of the stationary blackhole, which is hidden behind the event horizon. The the-ory breaks down inside the event horizon which is causallydisconnected from the exterior world. Either the Kruskalcontinuation of the Schwarzschild (J = 0, Q = 0) metric,or the Kerr (Q = 0) metric, or the Reissner-Nordstrom(J = 0) metric, show that the static observers fail to ex-ist inside the horizon. Any object that collapses to form

a black hole will go on to collapse to a singularity insidethe black hole. Any timelike worldline must strike the cen-tral singularity which wholly absorbs the infalling matter.Therefore, the ultimate fate of collapsing matter once it hascrossed the black hole surface is unknown. This, in turn, dis-ables any accumulation of matter in the central part and,thus, neither the growth of black holes nor the increase oftheir mass-energy density could occur at accretion of out-side matter, or by means of merger processes. But how canone be sure that some hitherto unknown source of pres-sure does not become important at huge energies and haltthe infinite collapse. To fill the void which the standardPBHM presents, one plausible idea to innovate the solu-tion to alluded key problems would appear to be the frame-work of MTBH. It was originally proposed by (Ter-Kazarian2001) and references therein, and thoroughly discussed in(Ter-Kazarian 2014, 2015a). Here we briefly recount someof the highlights of the MTBH. The MTBH was extensionof PBHM and rather completes it by exploring the mostimportant processes of spontaneous breaking of gravitationgauge symmetry and rearrangement of vacuum state at hugeenergies (Ter-Kazarian 1997, 2001, 2010). Proposed gravita-tion theory is constructed in the framework of spacetimedeformation/distortion theory (Ter-Kazarian 2015a) (alsosee Ter-Kazarian (2011, 2015b)). We develop a distortiongauge induced fiber-bundle formulation of gravitation. Toinvolve a drastic revision of the role of gauge fields in thephysical concept of the spacetime deformation/distortion,we generalize the standard gauge scheme by exploring adistortion gauge field which acts on the external spacetimegroups. The fundamental field is distortion gauge field and,thus, all the fundamental gravitational structures in fact -the metric as much as the coframes and connections - ac-quire a distortion-gauge induced theoretical interpretation.The distortion fields are treated in the Maurer-Cartan non-linear structure equations as the Goldstone fields. In thisreport we will not be concerned with the actual details ofthis comprehensive framework, but only use it as a back-drop to validate the theory with more observational tests.For details, the interested reader is invited to consult theoriginal papers. Discussed gravitation theory is consistentwith GR up to the limit of neutron stars. But MTBH man-ifests its virtues applied to the physics at huge energies.Whereas a significant change of properties of spacetime con-tinuum, so-called ”inner distortion” (ID), arises simultane-ously with the strong gravity. We are interested in simplecase of the spherical-symmetric gravitational field, in pres-ence of spherical-symmetric space-like ID-field. A distortionof the basis vectors in the ID regime yields the transforma-tions of Poincare generators of translations which, in turn,allow to derive the laws of phase transition for individualparticle found in the ID-region. Consequently the wholematter undergoes phase transition of second kind which sup-ply a powerful pathway to form a stable super-dense proto-matter core (SPC) inside the event horizon. Consisting of theproto-matter core and the outer layers of ordinary matter,the SPC-configuration is the spherical-symmetric distribu-tion of matter in many-phase stratified states. A layering ofSPC-configurations is a consequence of the onset of differ-ent regimes in equation of state. The simulations confirm inbrief the following scenario. The energy density and internalpressure have sharply increased in proto-matter core, with

On Nature of Ultraluminous X-ray Pulsations 5

Figure 1. Left panel: Phenomenological model of non-spinningblack hole. The meaningless singularity occurs at the centerinside the black hole. Right panel: Microscopic model of non-spinning black hole, with the central stable SPC. An infallingmatter with the time forms PD around the SPC. In final stageof growth, a PD has reached out the edge of the event horizon.Whereas a metric singularity inevitably disappears and UHE neu-trinos may escape from event horizon to outside world throughvista - a slim belt area S = 2πRgd - with opening angle θν .Accepted notations: EH=Event Horizon, AD=Accretion Disk,SPC=Superdense Proto-matter Core, PD=Proto-matter Disk.

respect to corresponding central values of neutron star, pro-portional to gravitational forces of compression. This coun-teracts the collapse and equilibrium holds even for up tothe masses ∼ 1010M⊙. The most important question of sta-bility of SPC was a central issue of this research. Minimiz-ing the total energy gives the equilibrium configurations.The second derivative of total energy gives stability infor-mation (Ter-Kazarian 2014). Although a relativity tends todestabilize configurations, however, a numerical integrationsof the stability equations of SPC clearly proves the stabilityof resulting cores. Due to it, the stable equilibrium holdsin outward layers too and, thus, an accumulation of matteris allowed now around the SPC. The phenomenological andmicroscopic black hole models have been schematically pre-sented in Fig. 1, to guide the eye. One of the most remarkablefeatures of MTBH is that a central singularity cannot occur,which is replaced now by finite though unbelievably extremeconditions held in the SPC where, nevertheless, static ob-servers are existed. The SPC surrounded by the accretiondisk presents the microscopic model of AGN. The SPC ac-commodates the highest energy scale up to hundreds ZeV incentral proto-matter core of suppermasive black holes, whichaccounts for the spectral distribution of the resulting radia-tion of galactic nuclei. An external physics of accretion ontothe black hole in the earlier half of its lifetime is identical tothe processes in Schwarzschild’s model. However, a crucialdifference in the model context between the phenomenolog-ical and microscopic black holes is arisen in the second halfof their lifetime. The black hole seeds might grow driven bythe accretion of outside matter when they were getting mostof their masses. An infalling matter with time forms a slimproto-matter disk around the proto-matter core tapering offfaster at reaching out the event horizon. The thickness ofproto-matter disk at the edge of event horizon is of linear sized. Whereas a metric singularity inevitably disappears andthe ZeV-neutrinos produced via simple or modified URCAprocesses in deep layers of SPC and proto-matter disk mayescape from event horizon to outside world through a slimbelt area S = 2πRgd, even after the strong neutrino trap-ping. The neutrinos are collimated in very small opening

angle θν ≃ εd = d2Rg

≪ 1. The ”trapping” is due to the fact

that as the neutrinos are formed in proto-matter mediumat super-high densities they experience greater difficulty es-caping from it before being dragged along with the matter,i.e. the neutrinos are ”trapped” comove with matter. In thisframework we have introduced a notion of the pre-radiationtime (PRT) of black hole which is referred to as a lapse oftime TBH from the birth of black hole till neutrino radiation- the earlier half of the lifetime:

TBH =Md

M. (1)

Here Md is the mass of proto-matter disk, M is an appro-priately averaged mass accretion rate. For example, the re-lation between typical PRT versus bolometric luminosity ofsuppermassive black holes becomes

TBH ≃ 0.32Rd

rOV

(MBH

M⊙

)21039W

Lbolyr, (2)

where Rd is the radius of the proto-matter core, rOV =13.68 km. At times > TBH , the black hole no longer holds asa region of spacetime that cannot communicate with the ex-ternal Universe. In this framework, we computed the fluxesof ZeV-neutrinos from plausible accreting supermassiveblack holes closely linking with the 377 AGNs (Ter-Kazarian2014, 2015a). In accord, the AGNs are favored as promisingpure neutrino sources because the computed neutrino fluxesare highly beamed along the plane of accretion disk, andpeaked at high energies and collimated in very small openingangle θν ∼ εd ≪ 1. While hard to detect, the extragalacticZeV-neutrinos may reveal clues on the puzzle of the origin ofUHECR, as they have the advantage of representing uniquefingerprints of hadron interactions and, therefore, can ini-tiate the cascades of UHECR particles with huge energiesexceeding 1.0×1020 eV (comprehensive reviews can be foundin Castellina & Donato (2012); Letessier-Selvon & Stanev(2011); Sigl (2011); Kotera & Olinto (2010); Semikoz(2010)). Consequently, we have studied a growth of proto-matter disk and derived the mass

MSeedBH

M⊙≃ MBH

M⊙

(1− 2.305

Rd

rOV

MBH

M⊙

), (3)

and initial redshift

zSeed ≃ z +H0 TBH , (4)

of black hole seed (Ter-Kazarian 2015a), where H0 is theHubble’s constant. Whereas interpreting the redshift as acosmological Doppler effect, and that the Hubble law couldmost easily be understood in terms of expansion of the uni-verse, we are interested in the purely academic question ofprinciple, to ask what could be the initial redshift, zSeed, ofseed black hole if the mass, the luminosity and the redshift,z, of black hole at present time are known. Equation (3)

holds at Rd

rOV6 0.023

Rsg

rOV, where Rs

g is the gravitationalradius of black hole seed. We have undertaken a large seriesof numerical simulations with the goal to trace an evolutionof the mass assembly history of 377 accreting supermassiveblack hole seeds in AGNs to the present time and examinethe observable signatures today. Given the redshifts, massesand luminosities of these black holes at present time col-lected from the literature, we compute the initial redshifts

6 G. Ter-Kazarian

and masses of the corresponding black hole seeds. Thus, hav-ing gained some insight into the supermassive black holephysics, let us now comment briefly on the implications forIMBHs. Equation (3) shows that the seed of IMBH can beformed in the stellar mass black hole region if

Rd

rOV6 0.434

M⊙

MBH

(1− M⊙

MBH

). (5)

A creation of IMBHs may be possible in the young stellarpopulations that we generally find ULXs co-habiting with.To render our discussion here a bit more transparent, wenote that IMBH with mass, say MBH ≃ 1.0 × 103M⊙,may originate from the seed with stellar mass of ∼ 1M⊙

at Rd/rOV ≃ 4.3×10−4, and would reach to the finale stateafter a lapse of growing time of order TBH ≃MBH/M .

3 THE MASS-EXCHANGE BINARY

CONTAINING M82X-2

As in the case of neutron stars, we expect that accretingblack holes are fast spinning objects. For the self-containedarguments, we need to extend the preceding algorithmof non-spinning MBHM to its spinning counterpart. Thestandard calculations of rapidly rotating relativistic bod-ies in astrophysics can be found in, e.g. (Carter 1970;Bardeen 1970; Komatsu et al. 1989; Cook et al. 1992, 1994;Bonazzola et al. 1993; Bonazzola 1994; Stergioulas 2003))and references therein. The non-spinning SPC is static andspherically symmetric. So, we need to consider a more gen-eral geometry which can describe rotating SPCs which areaxisymmetric. The principle foundation of the spinning con-figurations first comprises the following additional distinc-tive features with respect to non-spinning ones: 1) Rapidrotation causes the shape of the SPC to be flattened by cen-trifugal forces - flattened at poles and buldged at equator(oblate spheroid, which is second order effect in the rota-tion rate). 2) A rotating massive SPC drags space and timearound with it. The local inertial frames are dragged by therotation of the gravitational field, i.e. a gyroscope orbitingnear the SPC will be dragged along with the rapidly rotatingSPC (also see Ter-Kazarian (2012)). Beside the geodetic pro-cession, a spin of the body produces in addition the Lense-Thirring procession. To look into the future, measurementof the gyrogravitational ratio of particle would be a fur-ther step, see e.g. (Ni 2010) and references therein, towardsprobing the microscopic origin of gravity. Let the world co-ordinate t(= x0) be the time (in units of c), and φ(= x1) bethe azimuthal angle about the axis of symmetry. Moreover,a metric of a two-space , (x2, x3), can always be diagonal-ized. Since the source of gravitational field has motions thatare pure rotational about the axis of symmetry, then theenergy-momentum tensor as the source of the metric willhave the same symmetry. Namely, the space M4 would beinvariant against simultaneous inversion of time t and az-imuthal angle φ. The 3+1 formalism is the most commonlyused approach in which, as usual, spacetime is decomposedinto the one parameter family of space-like slices - the hy-persurfaces Σt. The study of a dragging effect is assisted byincorporating with the soldering tools in order to relate localLorentz symmetry to curved spacetime. These are the lin-ear frames and forms in tangent fiber-bundles to the external

general smooth differential manifold, whose components areso-called tetrad (vierbein) fields. Whereas, the M4 has ateach point a tangent space, TxM4, spanned by the anholo-nomic orthonormal frame field, e, as a shorthand for thecollection of the 4-tuplet (e0 = exp(−ν) (∂t + ω∂φ), e1 =exp(−ψ) ∂φ, e2 = exp(−µ2) ∂2, e3 = exp(−µ3) ∂3), whereea = e µ

a ∂µ. This is called a Bardeen observer, locally non-rotating observer, or the local Zero Angular-Momentum Ob-servers (ZAMO), i.e. observers whose worldlines are nor-mal to the hypersurfaces defined by constant coordinatetime, t = const, also called Eulerian observers. Here weuse Greek alphabet (µ, ν, ρ, ... = 0, 1, 2, 3) to denote theholonomic world indices related to M4, and the first halfof Latin alphabet (a, b, c, ... = 0, 1, 2, 3) to denote the an-holonomic indices related to the tangent space. The framefield, e, then defines a dual vector, ϑ, of differential forms,

ϑ =

ϑ0 = exp ν dtϑ1 = expψ (dψ − ωdt)

ϑ2 = expµ2 dx2

ϑ3 = expµ3 dx3

, as a shorthand for the

collection of the ϑb = ebµ dxµ, whose values at every point

form the dual basis, such that ea ⌋ϑb = δba, where ⌋ denot-ing the interior product, namely, this is a C∞-bilinear map⌋ : Ω1 → Ω0 with Ωp denotes the C∞-modulo of differen-tial p-forms on M4. In components e µ

a ebµ = δba. The normds of infinitesimal displacement dxµ on M4, describing thestationary and axisymmetric spacetimes, reads

ds : = e ϑ = eµ ⊗ ϑµ ∈ M4. (6)

Therefore, the holonomic metric on the space M4 can berecast in the form

g = gµν ϑµ ⊗ ϑν = g(eµ, eν)ϑ

µ ⊗ ϑν , (7)

with the components gµν = g(eµ, eν) in dual holonomic basisϑµ ≡ dxµ. Hence

ds2 = − exp(2ν)dt2 + exp(2ψ)(dφ− ωdt)2+exp(2µ2)(dx

2)2 + exp(2µ3)(dx3)2,

(8)

where the five quantities ν, ψ, ω, µ2 and µ3 are only func-tions of the coordinate x2 and x3. In the case at hand, themetric function ω is the angular velocity of the local ZAMOwith respect to an observer at rest at infinity. Thereby theredshift factor α ≡ exp ν is the time dilation factor betweenthe proper time of the local ZAMO and coordinate time talong a radial coordinate line, i.e. the redshift factor for thetime-slicing of a spacetime. In accord, all the geometrical ob-jects are split into corresponding components with respectto this time-slice of spacetime. In particular, the splittingof manifold M4 into a foliation of three-surfaces will inducea corresponding splitting of the affine connection, curvatureand, thus, of the energy-momentum tensor which can as wellbe written in terms of the energy density E = T (n,n) =Tµνn

µnν measured by an adapted Eulerian observer of four-velocity nµ, the momentum flow Jα = −γµ

αTµνnν and the

corresponding stress tensor Sαβ = γµαγ

νβTµν (S = Sα

α), that

is Tαβ = Enαnβ + nαJβ + Jαnβ + Sαβ. Here nα is theunit orthogonal vector to the hypersurface Σt, whereas thespacetime metric g induces a first fundamental form withthe spatial metric γαβ on each Σt as γαβ = gαβ + nαnβ .The form (8) includes one gauge freedom for the coordinatechoice. For the spherical type coordinates x2 = r and x3 = θ,for example, so-called quasi-isotropic gauge corresponds to

On Nature of Ultraluminous X-ray Pulsations 7

γrθ = 0 and γθθ = r2γrr. Then, one may define the secondfundamental form which associates with each vector tan-gent to Σt, and the extrinsic curvature of the hypersurfaceΣt as minus the second fundamental form. Aftermath, onecan define the usual Lorentz factor W = −nµu

ν = αut fora perfect fluid with conventional stress-energy tensor T µν =(ρ+P )uµuν +Pgµν , where ρ is the energy density and P isthe pressure. Hence E =W 2(ρ+P )−P and J i = (E+P )vi,where the fluid three-velocity vi(i = 1, 2, 3) implies ui =W (vi − βi/α). Thereby the resulting stress tensor can bewritten Sij = (E + P )vivj + Pγij . The four-velocity for ro-tating fluid reads u = ui(∂/∂t) + Ω∂/∂φ, where Ω = uφ/ut

is the fluid angular velocity as seen by an inertial observerat rest at infinity. It is convenient to give the equations inthe isotropic gauge, exp(2µ2) = exp(2µ3) = exp(2µ), in thecylindrical coordinates d = dx2, dz = dx3, or in pseu-dospherical coordinates dr = dx2 and r dθ = dx3, wherethe cylindrical radius is written exp(ψ) = r sin θ B exp(−ν),with B denoting a function of r and θ only. Therefore, themetric (8) becomes

ds2 = − exp(2ν)dt2 +B2r2 sin2 θ exp(−2ν)(dφ−ωdt)2 + exp(2µ)(dr2 + r2dθ2).

(9)

Consequently, the components of the energy - momentumtensor of matter with total density ρ and pressure P aregiven in the nonrotating anholonomic orthonormal frame asT ab = eaµe

bνT

µν , T 00 = W 2(ρ + PV 2), T 11 = W 2(ρ +PV 2), T 01 = W 2(ρ + P )V and T 22 = T 33 = P , with itstrace T = −ρ + 3P , where V is the velocity (in units of

c) with respect to the Bardeen observer V = B(Ω−ω)

α2 , soW = 1√

1−V 2. However, at this point we cut short and, in

what follows, we will refrain from providing further lengthydetails of the mathematical apparatus of proposed gravita-tion theory at huge energies and rigorous solution of theextended equations describing the spinning MBHM, whichis beyond the scope of this report. The latter include thegravitational and ID field equations of the time-slices withabove result of stress-energy tensor, the angular momentumequation with the momentum flow determining the frame-dragging potential, the hydrostatic equilibrium equation andthe state equation specified for each domain of many lay-ered spinning SPC- configurations. Theoretical evolutionarypaths of this type which are suitable for rigorous compari-son with the behavior of observed sources will be separatetopic of comprehensive investigation elsewhere. But someevidence for a simplified physical picture, without loss ofgenerality, is provided in the rest of this section where weextend preceding developments of MTBH in concise form,without going into the subtleties, as applied to the initiallyrigid-body spinning IMBH configuration of angular velocityΩ. If this were the case, eventually the spinning proto-mattercore and a slim co-spinning proto-matter disk driven by ac-cretion would be formed. The evolution and structure of aproto-matter disk is largely determined by internal friction.Before tempting to build a physical model in quest, the otherfeatures of SIMBH configuration also need to be accounted.The fact that the rotational energy has a steeper dependenceon the radius of the compact object than the internal energyin the relativistic limit is quite significant. Equilibrium canalways be achieved for massive configurations with nonzeroangular momentum by decreasing its radius. Also, there are

two characteristic features that distinguish a spinning rel-ativistic SPC-configuration from its non-spinning counter-part: 3) The geodetic effect, as in case of a gyroscope, leadsan accretion stream to a tilting of its spin axis in the plain ofthe orbit. Hence a proto-matter disk will be tilted from theplane of accretion on a definite angle δ towards the equator.4) Besides the UHE neutrinos, produced in the deep interiorlayers of superdense proto-matter medium as in case of non-spinning model, the additional thermal defuse blackbody ra-diation is revealed from the outer surface layers of ordinarymatter of spinning SPC and co-spinning proto-matter slimdisk. All of the rotational kinetic energy is dissipated asthermal blackbody radiation. This is due to the physicalcondition that these layers optically thick and, eventually,in earlier half of the lifetime of spinning black hole, at times< TBH , the strict thermodynamic equilibrium prevails forthis radiation because there would be no net flux to out-side of event horizon in any direction. That is, the emissionfrom the isothermal, optically thick outer layers at surfaceis blackbody, which is the most efficient radiation mecha-nism. This radiation is free of trapping because it is emittedfrom the thin outer layers. Such a picture will guide us inrelatively simple way toward first look at some of the as-sociated physics and can be quick to estimate the physicalcharacteristics of mass-exchange X-ray binaries. Examiningthe pulsations revealed from M82X-2, as a working modelwe assume the source of the flashes to be a self-gravitatingSIMBH resided in the final stage of growth. In this stage, aslim co-spinning proto-matter disk has reached out the edgeof the event horizon, where a metric singularity inevitablydisappears and the energy is carried away to outside worldthrough a slim belt area S by the thermal defuse blackbodyX-ray radiation. As M82X-2 spins, we see pulses because ofthe axial tilt or obliquity. Hence the X-ray beams interceptEarth-like lighthouse beacons. The orbital motion causes amodulation in the observed pulse frequency. The SIMBHmodel of M82X-2 in binary system is schematically plottedin Fig. 2. No eclipse condition holds. The parameters of abinary system is viewed in the orbital plane.

3.1 Basic geometry: Implications on the pulse

profile, mass scaling and energetics

A knowledge of the dynamical mass measurements of thecompact objects that power ULXs is a primarily necessaryprerequisite to the derivation of a complete picture aboutthe physical nature of ULXs. Keeping in mind aforesaid, weare now in a position to derive a general pulse profile de-pendent upon the position angles, and give a quantitativeaccount of a potential dynamical mass scaling of M82X-2and other energetics. The most reliable method is to mea-sure the mass function through the secondary mass and or-bital parameters, which can be measured only if the sec-ondary donor star is optically identified. In the absence ofdirect mass-function measurements from phase-resolved op-tical spectroscopy, we still have to rely on X-ray spectraland timing modeling and other indirect clues. In case of thefirst ultraluminous pulsar, only the X-ray mass function ismeasured (Bachetti et al. 2014), when the optical secondaryis unknown and most of the orbital parameters are yet tobe measured. However, exploring the key physical charac-teristics of a SIMBH model, let us consider the space-fixed

8 G. Ter-Kazarian

Figure 2. A schematic SIMBH model of M82X-2 constituting mass-exchange binary with the O/B-type donor star. The angle i is thebinary inclination with respect to the plan of the sky. No eclipse condition holds. In final stage of growth, PD has reached out the edgeof the EH. The thermal defuse blackbody X-rays beams may escape from SIMBH through a slim belt area S = 2πRgd to outside worldthat sweep past Earth like lighthouse beacons. Parameters of a binary system is viewed in the orbital plane. The picture is not to scale.Accepted notations: EH=Event Horizon, SPC=Superdense Proto-matter Core, PD=Proto-matter Disk.

Cartesian coordinate system labeled (z,x,y), with zx as aplane-of sight, and the axis s of the M82X-2-fixed frameas the rotation axis. A schematic plot is given in Fig. 3.Here and throughout we now use following notational con-ventions. The angles θ and φ are spherical polar coordinates.The observed pulses are produced because of periodic varia-tions with time of the projection dzx(t) of vector ~d(t) aligned

with the ~n(t) (~d(t) = d ~n(t)|n(t)|

) on the plane-of sight, where

~n(t) is the normal to the plane of the proto-matter disk atthe moment t. The ~n(0) lies in the plane of zs. triangle”.That is, given a unit sphere, a ”spherical triangle” on thesurface of the sphere is defined by the great circles connect-ing three points u, v, and w on the sphere (shown at top).

The lengths of these three sides are α = (s, n) (from u to

v), the axial tilt θ = (z, s) (from u to w), and β = (z, n)(from v to w). The angles of the corners u and e opposite βequal u = e = Ωt. The proto-matter disk was shifted fromthe orbital direction on angle δ = θ−α towards the equator.The projection dzx(t) is written

dzx(d, θ, φ, α, t) =√d2z(d, θ, α, t) + d2x(d, θ, α, φ, t), (10)

where

dz(d, θ, α, t) ≡ ~d(..., t) · ~ez = d cosβ(θ, α, t), (11)

and

dx(d, θ, α, φ, t) ≡ ~d(..., t) · ~ex =d sin β(θ, α, t) cos φn(θ, α, φ, t).

(12)

Here the ~ez and ~ex denote unit vectors along the axes z andx, respectively, φn = φ+A is the azimuthal angle of vector~n(t) (see Fig. 3). The vertex angle opposite the side α is A.To the extent that all of the rotational energy of M82X-2is dissipated as thermal defuse X-ray blackbody radiation,this may escape from the event horizon to outside world onlythrough a slim belt area S. The radiation arisen from perarea of surface is σT 4

s , where Ts is the surface temperature,σ is the Stefan-Boltzmann constant. Therefore, the pulsed

Figure 3. A schematic plot explaining the spherical trianglesolved by the law of cosines. The space-fixed Cartesian coordi-nate system is labeled (z,x,y), with zx as a plane-of sight. Axis s

of the M82X-2-fixed frame is rotation axis. The angles θ and ϕ arespherical polar coordinates. The line of nodes N is defined as theintersection of the equatorial and proto-matter disk planes. It isperpendicular to both the z axis and vector ~n(t), where ~n(t) is thenormal to the plane of proto-matter disk at moment t. The ~n(0)lies in the plane of zs. The lengths of three sides of a ”spherical

triangle” (shown at top) are θ = (z, s), α = (s, n) and β = (z, n).The vertex angle of opposite β is Ωt.

luminosity L will be observed if and only if the projectionof the belt area Szx = 2πRgdzx(d, θ, φ, α, t) on the plane-ofsight zx is not zero. So, pulsed luminosity reads

On Nature of Ultraluminous X-ray Pulsations 9

L(Rg, d, Ts, θ, φ, α, t) = Szx σT4s =

2πRgdzx(d, θ, φ, α, t) σT4s ≡

L0(M,d, Ts) Φ(θ, φ, α, t),

(13)

where its amplitude and phase, respectively, are

L0(M,d, Ts) ≃ 1.05× 104 (erg s−1)M

M⊙

d

m

T 4s

K4,

Φ(θ, φ, α, t) ≡√

1− sin2 β sin2(φ+A).

(14)

The ”spherical triangle” is solved by the law of cosines

cos β(θ, α, t) = cos θ cosα+ sin θ sinα cos Ωt,

cosA(θ, α, t) =cosα− cos θ cos β

sin θ sin β.

(15)

Consequently, the pulsed flux can be written in the form

F (Rg, d, θ, φ, α, t) = F0(M,d) Φ(θ, φ, α, t). (16)

where, given the distance D ≃ 3.6Mpc to the galaxyM82 (Bachetti et al. 2014), the flux amplitude is

F0(M,d) = cos iL0(M,d)

4πD2≃

6.8 × 10−48 (erg s−1cm−2) cos iM

M⊙

d

m

T 4s

K4,

(17)

where i is the inclination angle. Thus, the theoreticalmodel of periodic source M82X-2 left six free parameters:(M,d, Ts, θ, φ, α). The figures Fig. 4 reveals the diversityof the behavior of characteristic phase Φ(θ, φ, α, x ≡ Ωt)profiles versus the time, viewed at given position angles(θ, φ, α). At hard look, the position angles can be ad-justed from rigorous comparison with the behavior of ob-served pulsed light curve of M82X-2 by solving inverseproblem. This will be discussed separately elsewhere. Nowaccording to (13), for maximum value of pulsed luminos-ity either at β = πs or φ + A = πs (s=0,1,2,...), we

have L0(M,d, Ts) = L(3 − 30 keV) = 4.9 × 1039 erg s−1

or F0(M,d) = F (3 − 30 keV) ≃ 1.58 × 10−12 erg s−1cm−2

(cos i > 1/2 (Bachetti et al. 2014)). Hence the surface tem-perature scales ∝ T−4

s with the black hole mass:

M

M⊙=

4.66 × 1035K4

T 4s

m

d. (18)

That is, a cooler radiation surface implies a bigger blackhole. Next, assuming the persistent emission -L(0.3 −10 keV) = 1.8 × 1040 erg s−1 - from M82X-2 is isotropic, wemay impose a strict Eddington limit on the mass transferrate that can be accepted by the black hole, L < LEdd ≃1.3× 1038 M

M⊙erg s−1. This imposes stringent constraint on

the lower limit of black hole mass

M

M⊙> 138.5, Rg > 408.6 km, (19)

because for an accreting of ∼ 10 per cent of the Eddingtonlimit - a fairly typical accretion rate for a high-state blackhole - this points towards a rough estimate of upper limitof mass M < 1385M⊙. For the knowledge of more accurateupper mass limit, a good progress can still be made byestablishing a direct physical connection between massesof M82X-2 and M82X-1, and then rely on the availablemass estimates of the latter, see e.g. (Okajima et al.2006; Mucciarelli et al. 2006; Dewangan et al. 2006;

Stobbart et al. 2006; Patruno et al. 2006; Portegies et al.2007; Roberts 2007; Casella et al. 2008; Feng & Kaaret2010; Feng & Soria 2011; Pasham et al. 2014). The con-troversy, however, is with their mass range. As one mayenvisage, a different mass estimates of M82X-1 may yielddifferent mass values of M82X-2. So, we should be careful inchoosing the most accurate black hole mass measurementto-date. As the centroid of the persistent emission isbetween M82X-2 and M82X-1 which indicates that M82X-2harbors an IMBH, as M82X-1 does, we suppose that accre-tion onto a black hole is well approximated by the relationLacc = ηc2M = GM

RM . This gives M1/M = L1/L ≃ 5.556.

According to MTBH, we have M ∝ M2 for both thecollisionless and the hydrodynamic spherical accretionsonto black hole (Ter-Kazarian 2014, 2015a). Making use ofthese relations, we then obtain

M1 ≃ 2.357M. (20)

The M82X-1 is a good candidate for hard state ULXswhich may be one of the very few ULXs that changetheir spectral state during outbursts, switching from ahard to a thermal state (Feng & Soria 2011). The type-C low frequency quasi-periodic oscillations (QPOs) andbroadband timing noise, detected in the two XMM-Newtonobservations in 2001 and 2004, in the central regionof M82 (Strohmayer & Mushotzky 2003; Dewangan et al.2006; Mucciarelli et al. 2006) and later confirmed to orig-inate from the M82X-1 (Feng & Kaaret 2007), suggest thatthe ULX harbors a massive black hole. The mass estimateby (Dewangan et al. 2006) is based on the assumption thatM82X-1 also follows well established relation of the pho-ton spectral index versus QPO frequency, Γ− νQPO, foundfor the Galactic X-ray binaries in their high or intermedi-ate states. The resulting IMBH mass for M82X-1 is in therange of 25−520M⊙. However, there may be systematic er-rors in the photon indices measured with XMM-Newton andRossi X-ray Timing Explorer (RXTE) due to contaminationfrom nearby sources, as indicated by large apparent changesin the effective absorption column. Another mass estimateby (Casella et al. 2008) is inferred from the correlationswith the X-ray luminosity and type C QPO frequency. Thismethod is based on the correlation between characteristicfrequencies, on the fundamental plane and on the variabil-ity plane of accreting black holes. Exploring this method, theblack hole masses inferred from the characteristic frequen-cies are all about 103−104M⊙ indicating that ULXs containIMBHs (Casella et al. 2008; Feng & Soria 2011). But, theseresults were not without problems, notably pointed out bythe same authors. Such mass estimates are based on scalingrelations which use low-frequency characteristic timescaleswhich have large intrinsic uncertainties. In particular, it wasunclear whether these mHz oscillations are indeed the Type-C analogs of stellar mass black, and both the Type-C and themHz oscillations are variable, resulting in a large dispersionin the measured mass of 25 − 1300M⊙. Positive identifica-tion of the emission states requires both timing and spectralinformation. Consequently, with simultaneous observationsexploiting the high angular resolution of Chandra to isolatethe ULX spectrum from diffuse emission and nearby sourcesand the large collecting area of XMM-Newton observationsof M82 to obtain timing information, Feng & Kaaret (2010)found surprisingly that the previously known QPOs in the

10 G. Ter-Kazarian

Figure 4. Characteristic phase profiles vs. the time (x ≡ Ωt), viewed at the following position angles (θ, α, φ): (1) (450, 300, 00);(2) (450, 300, 900); (3) (450, 300, 1350); (4) (450, 300, 600); (5) (450, 900, 900); (6) (450, 180, 900); (7) (450, 600, 900); (8) (450, 78.30, 900);(9) (900, 900, 1800); (10) (900, 0.60, 720); (11) (900, 300, 600); (12) (720, 300, 600); (13) (600, 900, 600); (14) (600, 450, 600);(15) (600, 1530, 600); (16) (1.20, 900, 600).

source disappeared. The light curve was no longer highlyvariable and the power spectrum was consistent with thatof white noise. The energy spectrum also changed dramati-cally from a straight power-law to a disk-like spectrum. Thedisappearance of QPOs and the low noise level suggest thatthe source was not in the hard state. All results are well con-sistent with that expected for the thermal state. The moni-toring data from RXTE indicate that these Chandra andXMM-Newton observations were made during the sourceoutbursts, suggesting that M82X-1 usually stays in the hardstate and could transition to the thermal state during out-bursts. The spectral fitting suggests that the ULX contains aclose to Eddington (Ldisk/LEdd ∼ 2) rapidly spinning IMBHof 200 − 800M⊙ masses. The thermal dominant states areall found during outbursts. Nonetheless, modeling of X-rayenergy spectra during the thermal-dominant state using afully relativistic multi-colored disk model has large uncer-tainties owing to both systematic and measurement errors.In addition to the large mass uncertainty associated withthe modeling, the same authors also found that the energyspectra can be equally well-fit with a stellar-mass black holeaccreting at a rate of roughly 160 higher than the Edding-ton limit. Also, the X-rays from this source are known tomodulate with an orbital periodicity of 62 days, which indi-cates to an intermediate-mass black hole with mass in the

range of 200−5000M⊙ (Patruno et al. 2006; Portegies et al.2007). But, a recent study finds that this periodicity mayinstead be due to a precessing accretion disk in which casea stellar-mass black hole will suffice to explain the appar-ent long periodicity (Pasham, D. R., & Strohmayer 2013).Thus, the mass measurements above have large uncer-tainties. This makes black hole masses obtained by thismethod at the very least questionable. In what follows,therefore, we adopt an alternative and less ambiguous massdetermination for intermediate-mass black holes suggestedby (Pasham et al. 2014) which seems to be a more reli-able determinant of the mass of M82X-1. These authorssearched RXTE’s proportional counter array archival datato look for 3 : 2 oscillation pairs in the frequency range of1 − 16 Hz which corresponds to a black hole mass rangeof 50 − 2000M⊙. In stellar-mass black holes it is knownthat the high frequency quasi-periodic oscillations that oc-cur in a 3 : 2 ratio (100 − 450 Hz) are stable and scale in-versely with black hole mass with a reasonably small disper-sion (McClintock & Remillard 2006; Remillard et al. 1999,2002a,b; Strohmayer 2001a,b). Pasham et al. (2014) reportstable, twin-peak (3:2 frequency ratio) X-ray quasi-periodicoscillations from M82X-1 at the frequencies of 3.32 ± 0.06Hz (coherence, Q = centroid frequency (ν)/width(ν) > 27)and 5.07 ± 0.06 Hz (Q > 40). The discovery of a stable

On Nature of Ultraluminous X-ray Pulsations 11

3 : 2 high-frequency periodicity simultaneously with thelow-frequency mHz oscillations allows for the first time toset the overall frequency scale of the X-ray power spec-trum. This result not only asserts that the mHz quasi-periodic oscillations of M82X-1 are the Type-C analogs ofstellar-mass black holes but also provides an independentand the most accurate black hole mass measurement to-date. Assuming that one can scale the stellar-mass relation-ship, they estimate the black hole mass of M82X-1 to be428 ± 105M⊙. They also estimate the mass using the rel-ativistic precession model, which yields a value of 415 ±63M⊙. Combining the average 2-10 keV X-ray luminos-ity Pasham, D. R., & Strohmayer (2013); Kaaret & Feng(2007) of the source of 5 × 1040 ergs s−1 with the measuredmass suggests that the source is accreting close to the Ed-dington limit with an accretion efficiency of 0.8±0.2. Makinguse of the mass values 428± 105M⊙ with (20), we provide,therefore, the mass estimate for M82X-2:

M ≃ 138.5 − 226M⊙, Rg ≃ 408.6 − 666.7 km, (21)

Rotation speed at surface of M82X-2 with upper limit mass226M⊙ as rigidly spinning IMBH configuration of angularvelocity Ω equals v = RgΩ ≃ 3.06 × 108 cm s−1. We mayalso slightly improve the lower mass limit 323M⊙ of M82X-1 given by (Pasham et al. 2014), to now be 326.5M⊙. Com-bining (21) and (18), we obtain then

2.06 × 1033 <T 4s

K4

d

m< 3.34 × 1033. (22)

In our setting, we adopt the rather concrete proposal of non-spinning black holes (Ter-Kazarian 2014, 2015b), i.e. theneutrino flux from spinning M82X-2 might as well be highlybeamed along the plane of proto-matter disk and collimatedin very small opening angle. For the values (21), this yields

7.5 × 10−7 d

m< θν ∼ εd < 1.2× 10−6 d

m. (23)

The εd is expected to be very small for M82X-2, about anorder of magnitude ∼ 10−5. In accord,

d ≃ 61− 100m, εd ≃ (4.6− 7.5) × 10−5. (24)

This, together with (22), give Ts ≃ 7.6 × 107 K. Thus,M82X-2 indeed releases 99.59% of its radiative energy pre-dominantly in the X-ray bandpass of 0.3 − 30 keV. How-ever, its studies in other wavelengths well give us use-ful information on its physical nature and environment.From Wien’s displacement law we obtain the wavelengthλmax ≃ 0.381nm at which the radiation curve peaks, whichcorresponds to energy hνmax ≃ 3.2 keV. It seemed thatthe pulsed radiation over the band 0.3 − 3 keV was notdetected yet because it is suppressed by more powerfulpersistent radiation over this band. Assuming the emis-sion arisen from accretion is isotropic we are able to in-fer the most important ratios of the pulsed and persistentluminosities to the isotropic Eddington limit for M82X-2: L/LEdd ≃ 0.17 − 0.28, L/LEdd ≃ 0.63 − 1.03, respec-tively, where LEdd ≃ (1.75−2.85)×1040 erg s−1. This showsthat the accretion would be in the usual sub-Eddingtonregime. Given a dynamical mass (21) and angular velocityΩ = 2π

P(P = 1.37 s), we may calculate the rotational kinetic

energy Erot = 12IΩ2 of M82X-2, where I = 2

5MR2

g is themoment of inertia if M82X-2 is regarded as the rigidly spin-

ning ”canonical” configuration of mass M and radius Rg.Hence

Erot ≃ (3.72− 16.17) × 1051 erg. (25)

When all energy thermalized, radiation emerges as a black-body. A significant fraction of the accretor M82X-2 surfaceradiates the accretion luminosity at temperature

Tb =

(Lacc

4πR2g

)1/4

≃(LEdd

4πR2g

)1/4

, (26)

such that Tb ≃ (3.88 − 5.37) × 106K. The gravitational en-ergy of each accreted electron-proton pair turned directlyinto heat at (shock) temperature Tsh: 3kTsh =

GMmp

Rg,

so Tsh ≃ 1.8 × 1012K. Hence typical photon energies ofpersistent radiation lies between kTb ≃ 0.34 keV 6 hν 6

kTsh ≃ 180MeV. So, M82X-2 is persistent X-ray and pos-sibly gamma-ray emitter. Also, given the mass of the mostbrightest source M82X-1 of persistent X-ray radiation, typ-ical photon energies of persistent radiation lie in rangekTb(X1) ≃ 0.3 keV 6 hν 6 kTsh ≃ 180MeV.

3.2 The mass of companion star and orbit

parameters

Once the mass scaling of M82X-2 is accomplished, this canpotentially be used further to quantify the association be-tween the M82X-2 and the optical secondary donor star inX-ray binary. The orbital period Por is a key parameter fordynamical mass measurement. From Fig. 2, the separationof the two masses is a, and their distances from the centerof mass are a1 and a2. The highly circular orbit, combinedwith the mass function f(M, M2, i) = 2.1M⊙, the lack ofeclipses and assumption of a Roche-lobe- filling companionconstrain the inclination angle to be i < 60o (Bachetti et al.2014). These alow to determine the mass M2 of donor star:

M2

M⊙>

48.3, for M = 138.5M⊙,64.9, for M = 226M⊙.

(27)

Thus, optical companion is a typical O/B supergiant, whichevolves away from the main sequence in just a few millionyears. The binary separation can be computed with Kepler’sLaw a3 = G(M+M2)

4π2 P 2or. The Doppler curve of the spectrum

of NuSTAR J095551+6940.8 shows a Por = 2.5-day sinu-soidal modulation arisen from binary orbit (Bachetti et al.2014). All these give the projection of the orbital velocity ofM82X-2 along the line of sight v1 = 2π

Pora1 sin i ≃ 200.9 km

s,

and hence a1 sin i ≃ 9.9R⊙. The absence of eclipses impliescos i > R

a, where R is the radius of companion donor star.

Hence

R >

22.1R⊙, for M = 138.5M⊙,25.7R⊙, for M = 226M⊙.

(28)

As well as a1 > 11.4R⊙, a2 > 32.9R⊙ for M/M⊙ = 138.5;and a1 is the same, a2 > 39.9R⊙ for M/M⊙ = 226. TheRoche lobe radius RL for donor star is

RL >

15R⊙, for M = 138.5M⊙,14.3R⊙, for M = 226M⊙.

(29)

Thus, the M82X-2 and donor star constitute the semi-detached binary, accreting through Roche-lobe overflow.Donor star exceeds its Roche lobe (R > RL), therefore its

12 G. Ter-Kazarian

shape is distorted because of mass transfer from donor starthrough the inner Lagrange point L1 to the M82X-2. Theaccretion stream is expected to be rather narrow as it flowsthrough the L1 point and into the Roche lobe of the primary.

3.3 Spin-up rate and the torque added to M82X-2

Continuing on our quest, we next determine the conditionsunder which ULX pulsed source spins up (Bachetti et al.2014) and hance gains rotational energy as matter is ac-creted, i.e. we discuss the relationship between the proper-ties of the exterior flow and the measured rate of change ofangular velocity dΩ/dt. Although measurements of spin-upor spin-down appears to be the most promising method fordetermining the angular momentum transport by the inflow-ing matter, which in turn, may provide information aboutpattern of material flow outside the event horizon of SIMBH,the extraction of this information from such measurementsclearly requires some care. We explore the relationship be-tween the torque (≡ l) flux through the event horizon, spin-up rate of SIMBH and the rate of change of its rotationalenergy. The rates of change of the SIMBH angular velocityand of the rotational energy can be related to the flux oftorque across the event horizon boundary as follows. Therate of change of the torque is given by (see Ghosh et al.(1977))

d

dt(IΩ) = Ml, (30)

where I is the moment of inertia, and l is the torque addedto the SIMBH per unit mass of accreted matter. Equation(30) gives for the rate of change of angular velocity

dΩ

dt=

M

Lbh

[lΩ− Ω2R2

gir

(M

I

dI

dM

)], (31)

where Lbh ≡ IΩ, and Rgir is the radius of gyration ofSIMBH. The rate of change of the rotational energy is

dErot

dt=

d

dt

(1

2IΩ2

). (32)

To make further progress we recast (31) and (32) into theform

dErot

dt= M

[lΩ− 1

2ΩR2

gir

(M

I

dI

dM

)]. (33)

When MI

dIdM

> 0, which is generally the case, the SIMBH’sbehavior can be conveniently characterized by the dimen-sionless parameter

ζ ≡ l

ΩR2gir

(M

I

dI

dM

)−1

. (34)

Thus the black hole loses rotational energy and spins downfor ζ < 1/2, whereas it gains rotational energy and spinsup for ζ > 1; for 1/2 < ζ < 1 the black hole spins downeven though it is gaining rotational energy. The logarith-mic derivative M

IdIdM

for ”canonical” configuration, i.e. spin-ning uniform-density sphere with mass M and radius Rg , isd ln Id lnM

= dd lnM

ln(25MR2

g

)= 3, so

ζ =l

3ΩR2gir

, (35)

where R2gir = I/M = 2

5R2

g . For the spin up regime of M82X-2 when ζ ≃ 0.073 l

ΩR2

gir

> 1, we obtain l > 2.192R2g s

−1, so

l >

3.7× 105 km2s−1, for M = 138.5M⊙

9.7× 105 km2s−1, for M = 226M⊙.(36)

The time derivative of the angular velocity (31) gives

l =4πR2

g

5P

(3− P

P

M

M

). (37)

Combing M ≃ 6.35 × 10−7M⊙ yr−1, and a linear spin-up

p ≃ −2 × 10−10s/s of the NuSTAR J095551+6940.8 pul-sar (Bachetti et al. 2014), from (37) we obtain the torquesadded to M82X-2 per unit mass of accreted matter, whichsatisfy the spin-up condition (36) of ζ > 1:

l ≃

1.1× 1011 km2s−1, for M = 138.5M⊙,9.8× 1011 km2s−1, for M = 226M⊙.

(38)

4 CONCLUDING REMARKS

The observed behavior of the NuSTAR first ultraluminouspulsations with the energy of about 10 million suns thatobviously hints to black hole energetics is completely unex-pected and inexplicable in the context of the typical pulsarmodels. Identification of this periodic source with the tran-sient M82X-2 as the source of the flashes imposes a stringentconstrains, i.e. a highly luminosity of the additional persis-tent X-ray radiation observed earlier from M82X-2 to itsactive state strongly hints to typical black hole energeticswhich ultimately precludes the possibility of typical pulsars.Whereas, the centroid of the persistent emission is betweenM82X-2 and M82X-1 while M82X-1 is the best known can-didate for an IMBH. So, the M82X-2 seemed harbors anIMBH, as M82X-1 does. The finding of an ultraluminouspulsations has implications for understanding that they mayindeed not be rare in the ULX population. Our model sup-ports the idea that ultraluminous X-ray pulsations will bea common phenomena in the local Universe. In the futureastronomers also will look at more ULXs, and it is possiblethey could validate this idea. Putting apart the discussionof inherent problems of the mass scaling of the black holesin ULXs, we have focused on black hole rather than typicalaccreting pulsar models. It is, of course, impossible to ex-plain the pulsed high luminosity in the framework of widelyaccepted conventional phenomenological black hole modelbecause black holes do not radiate, also the spinning blackholes are axisymmetric and have no internal structure onwhich to attach a periodic emitter. The new conceptual el-ement of the implications of the framework of MTBH intackling this problem is noteworthy. The MTBH exploresthe most important processes of spontaneous breaking ofgravitation gauge symmetry and rearrangement of vacuumstate at huge energies. In the framework of MTBH, we as-sume the source of the flashes M82X-2 to be SIMBH, residedin the final stage of growth. If this were the case, even-tually, a slim co-spinning proto-matter disk driven by ac-cretion would be formed around the spinning proto-mattercore, tilted from the plane of accretion on a definite an-gle δ towards the equator. In this stage, it has reached outthe edge of the event horizon where a metric singularity in-evitably disappears. The energy is carried away then fromevent horizon through a slim belt area to outside world byboth the ultra-high energy neutrinos produced in the su-perdense proto-matter medium - the flux of which is highly

On Nature of Ultraluminous X-ray Pulsations 13

beamed along the plane of proto-matter disk as in the caseof non-spinning model - and the thermal defuse blackbodyradiation revealed from the outer surface layers of ordinarymatter of spinning SPC and co-spinning proto-matter disk.All of the rotational energy of SIMBH is dissipated as ther-mal defuse blackbody X-ray radiation to outside world. AsM82X-2 spins, we see pulses because of the axial tilt orobliquity. That is to say, the M82X-2 is the emitter of bothpersistent continuous broad X-ray radiation and pulsatingX-rays. We derive the general profiles of pulsed luminosityand X-ray flux of M82X-2. Thus, M82X-2 indeed releases99.59% of its radiative energy predominantly in the X-raybandpass of 0.3−30 keV, while the pulsed radiation over theband 0.3 − 3 keV was not detected yet because it is seem-ingly suppressed by more powerful persistent radiation overthis band. Since there is not enough information to test thetheory, i.e. in the absence of direct mass-function measure-ments from phase-resolved optical spectroscopy, we still haveto rely on X-ray spectral and timing modeling and other in-direct clues and, thus, the resulting theoretical model nec-essarily includes a number of poorly known parameters. Wederive a general pulse profile dependent upon position an-gles, and give a quantitative account of all the energeticsand a potential dynamical mass scaling and orbital param-eters of the semi-detached X-ray binary containing primaryM82X-2 and the secondary massive O/B-type donor star,accreting through Roche-lobe overflow. These results have aheuristic value but are far from rigorous. At hard look, theposition angles can be adjusted from rigorous comparisonwith the behavior of observed pulsed light curve of M82X-2by solving inverse problem. This will be discussed in subse-quent paper. Finally it should be emphasized that the keyto our construction procedure of both MTBH in general,and suggested model of ultraluminous X-ray pulsations inparticular, is widely based on the premises of our experi-ence of accretion physics. So, this inherits all the problemsof the scenario of runaway core collapse which has alwaysbeen a matter of uncertainties. However, we note that ourconstructions will be valid as well in the case if some hith-erto unknown yet mechanism in Nature will in somehow orother way produce the superdense proto-matter, away fromthe accretion physics.

ACKNOWLEDGMENTS

I would like to thank H.Pikichyan and A.Beglaryan for usefuldiscussions.

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