Optically thick envelopes around ULXs powered by accreating neutron stars
Alexander A. Mushtukov, Valery F. Suleimanov, Sergey S. Tsygankov, Adam Ingram
aa r X i v : . [ a s t r o - ph . H E ] J a n MNRAS , 1–8 (2016) Preprint 14 October 2018 Compiled using MNRAS L A TEX style file v3.0
Optically thick envelopes around ULXs powered byaccreating neutron stars
Alexander A. Mushtukov, , , ⋆ Valery F. Suleimanov, Sergey S. Tsygankov and Adam Ingram Anton Pannekoek Institute, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands Space Research Institute of the Russian Academy of Sciences, Profsoyuznaya Str. 84/32, Moscow 117997, Russia Pulkovo Observatory, Russian Academy of Sciences, Saint Petersburg 196140, Russia Institut f¨ur Astronomie und Astrophysik, Universit¨at T¨ubingen, Sand 1, D-72076 T¨ubingen, Germany Tuorla observatory, Department of Physics and Astronomy, University of Turku, V¨ais¨al¨antie 20, FI-21500 Piikki¨o, Finland
Accepted ... Received 2016 December; in original form 2016 December .
ABSTRACT
Magnetized neutron stars power at least some ultra-luminous X-ray sources. The ac-cretion flow in these cases is interrupted at the magnetospheric radius and then reachesthe surface of a neutron star following magnetic field lines. Accreting matter movingalong magnetic field lines forms the accretion envelope around the central object. Weshow that, in case of high mass accretion rates & g s − the envelope becomesclosed and optically thick, which influences the dynamics of the accretion flow andthe observational manifestation of the neutron star hidden behind the envelope. Par-ticularly, the optically thick accretion envelope results in a multi-color black-bodyspectrum originating from the magnetospheric surface. The spectrum and photon en-ergy flux vary with the viewing angle, which gives rise to pulsations characterized byhigh pulsed fraction and typically smooth pulse profiles. The reprocessing of radiationdue to interaction with the envelope leads to the disappearance of cyclotron scat-tering features from the spectrum. We speculate that the super-orbital variability ofultra-luminous X-ray sources powered by accreting neutron stars can be attributed toprecession of the neutron star due to interaction of magnetic dipole with the accretiondisc. Key words: pulsars: general – scattering – magnetic fields – radiative transfer –stars: neutron – X-rays: binaries
Accretion onto a highly magnetized (surface magnetic field B & G) neutron star (NS) is governed by the magneticfield of the central object. Namely, the accretion disc is inter-rupted at the magnetosperic radius R m , where the magneticpressure becomes comparable to ram pressure (due to Ke-plerian motion) of accreting material. Then the matter (a)follows magnetic field lines, (b) is accumulated at the mag-netosphere (Siuniaev & Shakura 1977) or (c) is pushed awayfrom the system depending on magnetic field strength, massaccretion rate and NS spin period (Shtykovskiy & Gilfanov2005; Tsygankov et al. 2016a). If the matter continues itsmovement along magnetic field lines, it reaches the NS sur-face at small regions in the vicinity of the magnetic poles.There the matter releases its kinetic energy mostly in X-rays. This gives rise to X-ray pulsar (XRP) phenomenon ⋆ E-mail: [email protected] (AAM) (see Walter et al. 2015 for recent review). The mass accre-tion rate, surface magnetic field structure and strength affectthe geometry of the illuminating region (Basko & Sunyaev1976b; Mushtukov et al. 2015b). Accretion luminosity ofXRPs can be close or even higher than the Eddington lumi-nosity. Recent discoveries of pulsating ultra-luminous X-raysources (ULXs) show that XRP accretion luminosity can beas high as 10 − erg s − , which is a few hundreds timeshigher than the Eddington value for a NS (Bachetti 2014;Israel 2016b,a; F¨urst et al. 2016a).ULXs powered by accreting NSs require special geo-metrical and physical conditions at the emitting region. Itis known that a high mass accretion rate leads to the ap-pearance of an accretion column above the surface of theNS (Basko & Sunyaev 1976b; Mushtukov et al. 2015b). Theaccretion flow is stopped at the top of the column in aradiation-dominated shock and then slowly settles down tothe surface. The column can be as high as the NS radius(Poutanen et al. 2013) and the material is confined there c (cid:13) A. A. Mushtukov et al. by magnetic field. It explains how accretion luminosity canprincipally exceed the Eddington limit (Mushtukov et al.2015a). It is also important that the effective scatteringcross-section and, therefore, radiation pressure are reducedby strong magnetic field (Mushtukov et al. 2016, 2012;Harding & Lai 2006).There are still debates on the magnetic field strength inrecently discovered ULXs powered by magnetized NSs. Themodel of beamed X-ray sources fed at a super-Eddingtonaccretion rate infers relatively weak magnetic field ∼ G(King & Lasota 2016). Assuming torque equilibrium andsolving the torque equation instead leads to B -field strengthof (2 ÷ × G (Ek¸si et al. 2015) or lower ∼ G(Dall’Osso et al. 2015) depending on model assumptions,which are essential near the equilibrium (Parfrey et al.2016). Transitions of ULX M82 X-2 between high andlow states were interpreted as a “propeller”-effect, requir-ing magnetar-like magnetic field ∼ G (Tsygankov et al.2016b). This is in good agreement with predictions of amodel of accretion column (Mushtukov et al. 2015a).The dynamics of the flow in the accretion envelope (cur-tain) were assumed to be unaffected by radiation pressureso far. However, this is not the case for ULXs powered byaccreting NSs. In this case the accretion flow between theinner disc radius and NS surface should be affected by radi-ation pressure. Moreover, simple estimations show that theaccretion curtain at accretion luminosity L & erg s − becomes optically thick and shields the central object com-pletely. It should influence the observational manifestation ofbright XRPs. The possibility of NS shielding of by opticallythick medium was supposed earlier (Ek¸si et al. 2015) in or-der to explain null result of search of pulsations in archivalXMM-Newton observations of several ULXs performed byDoroshenko et al. 2015.In this paper, we discuss properties of accretion cur-tain at the magnetosphere of ultraluminous XRPs. We payattention to its observational manifestation in spectral andtiming properties of ULXs. The accretion disc is interrupted at the magnetospheric ra-dius (Lai 2014), given by R m = 7 × Λ m / R / B / L − / cm , (1)where m = M/M ⊙ is NS mass in units of solar masses, B = B/ G is surface magnetic field strength, L = L/ erg s − is the accretion luminosity, R = R/ cmis the NS radius and Λ is a constant which depends on theaccretion flow geometry with Λ = 0 . r ( λ ) = R m cos λ, (2)where λ is the angular coordinate measured from the accre-tion disc plane and r ( λ ) is a distance from the central objectto a given point at the magnetic dipole line (see Fig. 1).The optical thickness of the flow at the magnetospheric surface is defined by the mass accretion rate ˙ M , local veloc-ity v ( λ ) and mechanism of opacity. For the case of ionizedgas at temperature T > K the opacity is defined byelectron scattering (at lower temperature the Kramer opac-ity dominates with much higher cross-sections and opticalthickness) and local optical thickness is τ e ( λ ) ≈ κ e ˙ Md S D v ( λ ) (cid:18) cos λ cos λ (cid:19) ≈ L / B / β ( λ ) (cid:18) cos λ cos λ (cid:19) , (3)where β ( λ ) = v ( λ ) /c is local dimensionless velocity alongmagnetic field lines, S D ∼ cm is the area of accre-tion channel base at the NS surface, d is the geometricalthickness of the channel at the base, λ ≃ π − ( R/R m ) / isthe coordinate corresponding to the accretion channel baseand κ e = 0 .
34 cm g − is the Thomson electron scatteringopacity for solar abundances.The matter on the B -field lines moves under the influ-ence of gravitational f grav and radiation f rad forces. If thephoton flux is directed from the central object, the directionsof the forces are opposite to each other. If the local magneticfield strength is sufficiently high and is not significantly dis-turbed by radiative force, the dynamics of the material atthe field lines is defined by the longitudinal component of theresulting force. The angle between the radius-vector pointedto a given point at the dipole field line and the tangent tothe field line (see Fig. 1) is χ = arctan(0 . / tan( λ )) . In the case of a rotating NS there is also a centrifugal force f cent in the corotating reference frame, which affects the dy-namics of matter. The centrifugal force is directed perpen-dicular to the spin axis of a NS. Hence, the angle betweenthe tangent to the field line and the centrifugal force is ξ = π + χ − λ. The resulting force gives an acceleration along magneticfield lines a || ( λ ) and the velocity of accretion flow is de-scribed by β d β d λ = R m cos λ (4 − λ ) / c a || ( λ ) , (4)where a || ( λ ) is acceleration along the magnetic field linescaused by the resulting force. This differential equation canbe solved for known acceleration along magnetic field lines,given an initial velocity of material, mass accretion rate,surface magnetic field strength and spin period of the NS. In the case of extremely high mass accretion rate ˙ M & . × Λ / B / m − / , which corresponds to accre-tion luminosity L & . × Λ / B / m / erg s − , (5)the accretion disc becomes radiation dominated and geomet-rically thick (Shakura & Sunyaev 1973; Suleimanov et al.2007). In this case the accretion curtain is closed and ra-diation from the central object cannot leave freely the mag-netosphere. Because the velocity along B -field lines cannot MNRAS , 1–8 (2016) ptically thick accretion envelopes Figure 1.
The structure of accretion flow in ULX powered by accreting NS. The geometrically thick accretion disc is interrupted at themagnetospheric radius and the accretion flow forms optically thick envelope. The inner part of dipolar magnetosphere is filled with hotthermolised photons. The observer detects the photons originated from the outer surface of magnetosphere. exceed the free-fall velocity, the optical thickness of the ac-cretion curtain due to electron scattering τ e ( λ ) > τ e (0) = L / B − / according to (3). Hence the accretion curtain isoptically thick if L & B / . At luminosity L & . × Λ / B / m / R / erg s − (6)accretion disc thickness at the magnetospheric radius R m iscomparable to the size of NS magnetosphere (Lipunov 1982).The photons from the central object are more likelyreprocessed and reflected back into the cavity formed bythe dipole magnetosphere than to immediately penetratethrough the envelope. Each photon undergoes a number ofscattering before it leaves the system. Hence, the radiationfield is in equilibrium inside the magnetosphere and can beroughly described by black-body radiation of certain tem-perature T in , which is determined by the total accretion lu-minosity, the size of magnetosphere and its optical thickness.Let us consider a simplified problem of a point sourceof luminosity L ps surrounded by spherical envelope of ra-dius R sp and optical thickness τ sp ≫
1. Then the outertemperature of the spherical envelope is related to the in-ner temperature T in as T out ≈ T in τ − / . Because the to-tal luminosity is conserved, the outer temperature is re-lated to the luminosity and geometrical size of the envelope: σ SB T = L ps / (4 πR ). Then the internal temperature is T in = [ τ sp L ps / (4 πσ SB R )] / . In the case of a central sourcesurrounded by a dipole surface, the internal temperature de-pends on the distribution of optical thickness over the sur-face.The radiation force due to black-body radiation in themagnetospheric cavity is directed perpendicularly to themagnetic field lines. As a result, the dynamics of the ac-cretion flow along B -field lines is determined by gravity andcentrifugal force only and described by the equation β d β d λ = GM (4 − λ ) / c R m cos λ (7) × " cos χ + (cid:18) ωω ∗ K (cid:19) cos λ cos ξ . Despite high optical thickness of the accretion curtainthe photons penetrate through it after a number of scat-terings. The time of photon diffusion through the accretioncurtain is defined by local optical and geometrical thickness: t diff ≈ τ ( λ ) d ( λ ) c ∼ × − L / B / d ( λ ) β ( λ ) (cid:18) cos λ cos λ (cid:19) sec . (8)If the photon diffusion time is much smaller than the dynam-ical time scale, the local flux at the outer side of a curtain isdefined by the internal temperature and local optical thick-ness (Ivanov 1969) T out ( λ ) ≃ T in τ − / ( λ ) . (9)Because the temperature at the outer side of the curtainvaries depending on the optical thickness, the whole accre-tion curtain radiates multi-color black-body radiation. Thetypical outer temperature can be roughly estimated as T out ∼ (cid:18) Lσ SB πR (cid:19) / ≈ . L / B − / m − / R − / keV . (10)The internal temperature is higher and has to be found nu-merically. We see that the temperature in the envelope is & E is defined by the tem-perature distribution over the visible parts of the envelopeand given by F E = 2 E h c R D Z d ϕ Z d λ cos i ( ϕ, θ ) cos λ (4 − λ ) / exp h EkT out ( λ ) i − , where the integrals are taken over the visible part of mag-netic dipole surface, i denotes the angle between the line MNRAS , 1–8 (2016)
A. A. Mushtukov et al. of sight and local normal to the dipole surface and D isthe distance from the NS to observer. If the typical time ofphoton diffusion through the accretion curtain (8) is compa-rable to the dynamical time-scale (which can be the case athigh luminosity & few × erg s − ), the final spectrum isdisturbed.The total luminosity of the accretion curtain can beobtained by integration of the flux over the dipole surface: L = 4 πh c R ∞ Z d EE λ Z λ d λ cos λ (4 − λ ) / exp h EkT out ( λ ) i − ≃ . × R , × ∞ Z d E keV E λ Z λ d λ cos λ (4 − λ ) / exp h EkT out ( λ ) i − R m , = R m / cm and E keV = E/ R m (1) canbe roughly estimated as T disc ≃ . L / B − / m − / R − / keV , (11)which is close to the temperature expected from the ac-cretion envelope (10). The energy spectrum of the ac-cretion disc is likely given by milti-color black-body(Shakura & Sunyaev 1973), but can be disturbed due toadvection. As a result, one can expect an additional non-pulsating component component in spectrum of ULXs pow-ered by accreting NSs.The magnetic field strength at the magnetospheric sur-face should be high enough to confine the accretion flowaffected by internal radiation pressure P rad ≈ aT /
3. Thenthe magnetic pressure P mag ( λ ) = B ( λ )8 π should exceed P rad : P rad /P mag .
1. The magnetic field pres-sure can be estimated from below by P mag ( λ = 0) ≃ . × Λ − B − / L / m − / R − / . Using estimation of the outer temperature (10) we get P rad P mag . × − τ L − / B / m / R − / , (12)which turns to P rad /P mag . . L m / R − / in the caseof accretion flow of free-fall velocity. Therefore, the accre-tion envelope can be unstable at accretion luminosity above10 erg s − , but this is beyond the scope of the presentwork. XRPs usually show hard energy spectra well fitted bya broken power law and complicated pulse profiles (seeWalter et al. 2015 for recent review). Accreting NSs in nor-mal XRPs are surrounded by an accretion curtain of inter-mediate optical thickness and relatively low temperature.The principal mechanisms of interaction between photonsand accretion curtain are photoionization of heavy elementsand Compton scattering with the energy recoil effect whenthe photon energy E ≫ kT e (Syunyaev 1976). The photonsoriginating from the emitting region (hot spots or accre-tion column) are reprocessed partially. The photons expe-rience ∼ τ scatterings on the average, where τ e is the op-tical thickness (Ivanov 1969; Illarionov & Syunyaev 1972).The initial energy of photons will be reduced due to the re-coil effect from E ≫ kT e to ∼ max(3 k B T e , m e c /τ ), whichleads to effective energy absorption by the accretion curtain(Syunyaev 1976). The absorbed energy is re-emitted in thesoft X-ray energy band. Because the plasma layer occupiesonly some part of the magnetospheric surface, it affects X-ray pulsations and can cause the complicated structure ofpulse profiles (Basko & Sunyaev 1976a). A detailed discus-sion of effects arising from accretion curtain in normal XRPsis beyond the scope of this paper. XRPs commonly show fast aperiodic variability of X-rayflux over a broad frequency range (similar variability isshown by accreting black holes, see Ingram 2016). The vari-ability is explained by the propagating fluctuations model(Lyubarskii 1997). The initial mass accretion rate variabil-ity is produced all over accretion disc due to MHD processes(Balbus & Hawley 1991) and results in mass accretion ratevariability at the magnetospheric radius (Lyubarskii 1997).In the case of normal XRPs the observer detects photonsdirectly from the emitting regions near the NS surface andvariability of the mass accretion rate is directly imprinted invariability of observed X-ray flux (Revnivtsev et al. 2009),which provides the possibility of diagnostics of accretiondisc. In case of extremely high mass accretion rate typical forULXs the optically thick envelope screens the direct X-rayflux originating from the vicinity of the NS and reprocessesit. Observed variability is defined by variability of the massaccretion rate at the NS surface and variability of the localmass accretion rate and optical thickness at the magneto-spheric envelope. As a result, aperiodic variability of ULXsis expected to be suppressed on time scales shorter than thetimescale of matter travel from the disc to the NS surface.
For simplicity and in order to get qualitative results we con-sider the accretion disc to be aligned with the equatorialplane of magnetic dipole. Inclining the magnetic dipole fieldchanges the equations, but the main physical ideas remainthe same.We consider a Keplerian accretion disc, where the an-
MNRAS , 1–8 (2016) ptically thick accretion envelopes gular velocity is given byΩ K = (cid:18) GMR (cid:19) / = 11 . (cid:18) mR (cid:19) / rad s − . (13)The angular velocity of the magnetosphere is Ω = 2 π/P . Ifthe mass accretion rate and accretion luminosity are higherthan the limiting values corresponding to “propeller”-effect(see, e.g., Tsygankov et al. 2016a): L lim ≃ × Λ / B P − / m − / R erg s − , (14)then the magnetospheric radius is smaller than the corota-tional radius and Keplerian angular velocity is higher thanthe angular velocity of NS magnetosphere. The accretingmaterial looses its kinetic energy due to interaction with themagnetosphere. The lost energy is going into heat. The re-leased energy is defined by the difference between Keplerianand magnetospheric angular velocities: W th ∝ (Ω K − Ω) .The temperature caused by dissipation of kinetic energy is T = 12 ( γ − T vir (cid:20) − ΩΩ K (cid:21) , (15)where γ is the adiabatic index and the virial temperature isdefined as T vir = GMµR R , where R is the gas constant and µ the mean atomic weightper particle (Spruit 2010). Hence, the temperature can beestimated as min( T, T disc ) , where T ≈ γ − − X Λ − m / L / B − / R − / (cid:20) − ΩΩ K (cid:21) keV , where X is the hydrogen mass fraction. The typical dimen-sionless thermal velocity of protons β th = v th /c at the innerdisc radii is given by β th = 1 c (cid:18) k B Tm p (cid:19) / ≈ . (cid:18) γ −
11 + X (cid:19) / Λ − / m / × L / B − / R − / (cid:12)(cid:12)(cid:12)(cid:12) − ΩΩ K (cid:12)(cid:12)(cid:12)(cid:12) , (16)We take this velocity as the initial velocity of the accretionflow along the magnetic field lines.The accretion disc has a certain geometrical thickness,which is defined by hydrostatic balance in the vertical direc-tion (Shakura & Sunyaev 1973) and affected by the originof opacity (Suleimanov et al. 2007). The disc can be dividedinto three zones according to the dominating pressure andopacity sources. Gas pressure and Kramer opacity dominatein the outer regions of the accretion disc (C-zone). The rel-evant disc scaleheight is given by (cid:18) Hr (cid:19) C = 0 . α − / L / m − / R / r / , (17)where r = r/ cm is the dimensionless radial coordinate.Gas pressure and electron scattering dominate in the inter-mediate zone (B-zone), where (cid:18) Hr (cid:19) B = 0064 α − / L / m − / R / r / , (18) and radiation pressure may dominate in the inner zone (A-zone) where (cid:18) Hr (cid:19) A = 0 . L R m r − . (19)The boundary between B- and C-zones is at r BC ≈ × L / m − / R / cm, while the boundary between A- andB-zones is at r AB ≈ . × L / m − / R / cm.At mass accretion rates above few × g s − (seeequation (5)) the disc is interrupted at the radiation domi-nated A-zone. According to (19) geometrical thickness canbe comparable to the magnetospheric radius at L & × Λ / B / m / R / (Mushtukov et al. 2015a; Lipunov1982), but increase of accretion disc thickness can be stoppedif accretion disc becomes advection-dominated (Lasota et al.2016). Hence, the geometrical thickness of accretion discgiven by (19) can be overestimated.The geometrical thickness gives the initial coordinate λ from which the accretion flow starts its motion along mag-netic field lines. We assume that the accretion flow startswith typical thermal velocity defined by interaction betweenthe accretion disc and the magnetoshpere at R m (16). Itmight be important for the case of extremely high mass ac-cretion rates, when the disc is interrupted at its radiationdominated zone (A-zone) and geometrically thick there. We solve equation (7) numerically for a given NS mass, spinperiod and surface magnetic field strength, taking the initialvelocity to be equal to the thermal velocity of protons at theinner radius of accretion disc (16). The initial coordinate isdefined by the geomerical thickness of accretion disc λ ini =atan(0 . H/R ) at the magnetospheric radius.The typical velocity profiles and corresponding opticalthickness distributions over the magnetospheric surface aregiven by Fig. 2a,b. The mass accretion rate and optical thick-ness define the surface temperature T out (see Fig. 2c). Therotation of NS affects the surface temperature only slightly.According to numerical solutions of equation (7) the typi-cal time scale of motion of material between the magneto-spheric radius and NS surface is ∼ (0 . ÷
1) sec. The accre-tion flow receives the major part of final kinetic energy in thevery vicinity of a NS, where the influence of radiation andcentrifugal forces is negligible. As a result, the final veloc-ity of matter near NS surface is affected by mass accretionrate only slightly and earlier theoretical constructions onthe structure of X-ray flux forming region (Basko & Sunyaev1976b; Mushtukov et al. 2015b,a,c) are not influenced by dy-namics in the accretion curtain.The internal temperature T in depends on the mass ac-cretion luminosity and magnetic field strength (see Fig. 3).The temperature is affected by the spin period of the NSbecause rotation influences the velocity (and therefore thelocal optical thickness) of accretion flow due to the centrifu-gal force.Because of the geometry of the accretion flow, the opti-cal thickness can be high enough to cause advection of pho-tons. The advection process becomes important when thetime of photon diffusion (8) becomes comparable to the timeover which the material passes a distance equal to the local MNRAS , 1–8 (2016)
A. A. Mushtukov et al. a β λ , deg b τ λ , deg c T ou t , k e V λ , deg Figure 2.
The dimensionless velocity (a), optical thickness (b)and effective outer temperature (c) at the dipole surface of theaccreting NS as a function of coordinate λ . Red solid and dashedblack lines are given for NS spinning with period P = 1 sec andnon-rotating NS respectively. The accretion flow velocity is notaffected significantly by the mass accretion rate and the curves at(a) correspond to accretion luminosity 5 × erg s − . Two setsof curves on (b) and (c) plots are given for accretion luminosity of5 × erg s − (lower) and 10 erg s − (upper) . Magnetic fieldstrength is B = 10 G. The parameters used in the calculations: M = 1 . M ⊙ , R = 10 cm, Λ = 0 . geometrical thickness of the envelope: t ∗ ( λ ) = d ( λ ) / ( cβ ( λ )).It happens at cos λ . λ L / B / . Therefore, the ad-vective zone is located very close to the NS surface even ataccretion luminosity L ∼ ÷ erg s − . We do not takeadvection in the accretion envelope into account.The matter velocity of the accretion flow is significantlysmaller than free-fall velocity all over the envelope except the B = G B = . G P = 1 sec T i n [ k e V ] L [erg s -1 ] Figure 3.
The internal temperature T in as a function of accretionluminosity is shown by red solid lines for NS spinning with period P = 1 sec and by black dashed lines for non-rotating NS. Differentcurves correspond to different magnetic field strength: 10 , × , × , × , . × , . × G (from left toright). The upper ends of red curves are defined by conditionof super-critical disc at the magnetospheric radius (defined bythe disc scale height becoming comparable to the magnetosphericradius). The parameters used in the calculations: M = 1 . M ⊙ , R = 10 cm, Λ = 0 . p r op e ll e r , s ec s u p e r - c r i t i c a l d i s c a c c r e t i o n τ e,min =10 τ e,min =3 τ e,min =1 L [ e r g s - ] B [G] Figure 4.
The accretion luminosities where the minimal opticalthickness of the envelope due to electron scattering is equal 1, 3and 10, are given by red solid lines (spin period of NS is takento be P = 1 sec) and red dashed lines (non-rotating NS). Thegrey dashed-dotted line shows the limiting luminosity due to thepropeller effect for spin period P = 1 sec. The centrifugal forcebecomes essential when the accreting NS is close to the limitingpropeller luminosity: the force slows down the accretion flow andmakes it optically thicker. The grey solid line limits the region ofsuper-critical disc accretion given by (6). The parameters used inthe calculations: M = 1 . M ⊙ , R = 10 cm, Λ = 0 . , 1–8 (2016) ptically thick accretion envelopes regions close to NS surface. As a result, the estimation ofoptical thickness from the assumption of free-fall velocity isunderestimated and the accretion curtain becomes opticallythick already at L & erg s − (see Fig. 4). Therefore,ULXs powered by accretion onto magnetized NS have to besurrounded by the optically thick envelope. The minimal op-tical thickness on the magnetosphere is affected significantlyby centrifugal force if the accretion luminosity becomes com-parable to the limiting propeller luminosity: the centrifugalforce reduces the velocity along magnetic field lines, whichleads to increasing of optical thickness. The accretion onto magnetized NS goes through the accre-tion curtain at the magnetosphere. In the case of extremelyhigh mass accretion rates typical for ULXs, the accretioncurtain becomes closed and optically thick (see Fig. 4). Thisaffects the dynamics of matter in the curtain and the basicobservational manifestation of the accreting NS at luminos-ity above few × erg s − .The closed accretion curtain intercepts and reprocessesthe initially hard radiation from the central object intoblack-body like radiation of temperature which depends onthe local optical thickness and total accretion luminosity(see Fig. 2c). Hence the accretion curtain manifests itselfby multi-color black-body spectrum, which varies with theviewing angle and can be dominated by the curtain temper-ature. In this case the total spectrum can be fitted by dou-ble black-body, where the harder component correspondsto accretion envelope and low energy componenet corre-sponds to an advection dominated accretion disc. Varia-tions of viewing angle due to NS spin lead to variations ofthe observed spectrum and cause observed pulsations. Be-cause the accretion disc provides a non-pulsating low en-ergy component in the spectrum, the pulsed fraction is ex-pected to be higher in the higher energy band. These state-ments are in agreement with observational data: the spec-trum of pulsating ULX-1 in NGC 5907 is well-described bymulti-color black-body (F¨urst et al. 2016b), while the pulsedfraction is, indeed, higher in the higher energy band. It isinteresting, that the spectra of several ULXs are well fit-ted with a double black-body model (Stobbart et al. 2006;Kajava & Rico-Villas 2016).At larger timescales the viewing angle can also vary dueto precession of the magnetic dipole (Lipunov & Shakura1980) giving rise to super-orbital variability. The precessionperiod of the NS can be roughly estimated: t pr ≈ . × µ − I R , P − cos ψ (3 cos ζ −
1) years , (20)where ψ is the angle between the normal to the accretiondisc plane and NS spin axis, ζ is the angle between the spinaxis and magnetic field axis, µ = B R is NS magneticmoment and I is NS moment of inertia. If NS has surfacedipole magnetic field > G the precession period can beabout a year. It is interesting that a few ULXs includingbright pulsating ULX-1 in galaxy NGC 5907 (Israel 2016b)show modulation of a photon flux with typical period of afew months: pulsating M82 X-2 - 55 days (Kong et al. 2016),M82 X-1 - 62 days (Kaaret et al. 2006), pulsating ULX NGC 7793 P13 - 65 days and 3000 days (Hu et al. 2017), pulsatingULX-1 NGC 5907 - 78 days (Walton et al. 2016), NGC 5408X-1 - 115 days and 243 days (Strohmayer 2009; Han et al.2012), HLX ESO 243-39 - 375 days (Servillat et al. 2011).Finally we conclude that(i) Optically thick accretion envelope affects the ob-served spectra of ULXs because of reprocessing of X-rayradiation from the central object. In the case of a closed ac-cretion curtain, the spectrum is described by a multi-colorblack-body, which can vary with viewing angle. It is likelythat the multi-color black-body spectrum of the accretionenvelope is dominated by the curtain temperature.(ii) Optically thick envelope makes cyclotron lines orig-inating from the central object undetectable.(iii) The curtain affects the observed pulse profile. Theobserver see photons, that have been reprocessed by the en-velope and re-emitted from a much more extended photo-sphere of a geometrical size comparable to the geometricalsize of NS magnitosphere. This explains a difference betweentypical pulse profiles of XRPs and pulsating ULXs: in caseof normal XRPs, observer detects photons from the centralobject and the pulse profile is disturbed by the accretioncurtain, while in the case of ULXs the pulse profile is totallydefined by emission from the envelope.(iv) In the case of normal XRPs velocity of the matter inthe accretion curtain is almost unaffected by radiation pres-sure. In the case of ULXs powered by accretion onto a NS,the velocity of material on the magnetosphere is different,but the velocity near the NS surface is still close to free-fallvelocity. As a result, the developed theoretical models of thegeometry of illuminating regions are valid (Basko & Sunyaev1976b; Mushtukov et al. 2015a).
ACKNOWLEDGMENTS
This research was supported by the Russian Science Foun-dation grant 14-12-01287 (AAM) and the German researchFoundation (DFG) grant WE 1312/48-1 (VFS). Partial sup-port comes from the EU COST Action MP1304 “NewComp-Star”.
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