On the Maximum Accretion Luminosity of Magnetized Neutron Stars

Home , Eddington luminosity

arXiv:1506.03600v3 [astro-ph.HE] 30 Dec 2015 cetd21 etme .Rcie 05Ags 4 norigi in 24; August 2015 Received 6. September 2015 Accepted 2 M MNRAS ybrki&Suye 1988 Syunyaev & Lyubarskii a crto,telmnst srsrce ytewl know well spheri the by of luminosity restricted case is Eddington the luminosity In the factors. accretion, different cal by this limited in reached is be high-energy process can sys- producing that binary of luminosity a way the However, in efficient photons. object extremely compact an a is to tems on matter of Accretion INTRODUCTION 1 n uiPoutanen Juri and ( ⋆ luminosities super-Eddington flux Eddington the gravity exceed of can instead flux significantly. pressure corresponding magnetic the strong and by here anced rcin.O h te ad h uioiyo ihymag- highly exceed of can luminosity NS the hand, netized other the On fraction). edclrt h noigflwo h aeilwihis which material ( the column of accretion flow per- the direction to incoming the funnelled in the escape to can Thom- radiation pendicular the the below and much value be son can cross-section interaction the lxne .Mushtukov, A. Alexander ultraluminous and sources pulsars X-ray X-ray magnetized connecting of stars: luminosity neutron accretion maximum the On c 4 3 2 1 × aa Vlargo)FdrlUiest,Kelvkj str Kremlevskaja University, Federal region) (Volga Kazan ukv bevtr fteRsinAaeyo cecs Sai Sciences, of Academy f Russian Institut the of Observatory Pulkovo uraOsraoy eateto hsc n srnm,U Astronomy, and Physics of Department Observatory, Tuorla -al [email protected] E-mail: 05TeAuthors The 2015 1 = 10 oetasetXryplaside produce indeed pulsars X-ray transient Some 38 . 4M r s erg 000 rAtooi n srpyi,Universit Astrophysik, und Astronomie ur ¨ ⊙ , − (here 1 1 – o yia aso h eto tr(NS) star neutron the of mass typical a for , 11 21)Pern 1Dcme 05Cmie sn NA L MNRAS using Compiled 2015 December 31 Preprint (2015) m H L shdoe asand mass hydrogen is L Edd Edd .Terdainpesr sbal- is pressure radiation The ). 1 4 = usatal o w reasons: two for substantially tr(S antcfil n pnpro.I ssonta h luminosit the that shown is th 10 It of of function period. order a spin the as and of calculated field values is magnetic luminosity (NS) possible model star simplified maximal a using The pulsars X-ray column. luminous of properties study We ABSTRACT Sprmtr hc r bet rvd ihrlmnste ed oth to leads luminosities higher provide to able that are which parameters NS n ogsi eid ( periods spin long and e words: systems. ultra-lumino Key binary of in part fe stars substantial any neutron a show the accreting that not in are conclude does observed can otherwise cut-off we which the luminosities, function with luminosity coincides binaries luminosity ray this Because NS. πGMm ak uye 1976 Sunyaev & Basko uk ta.1976 al. et Lucke L 1 , H ≃ 2 c/ ⋆ 10 [ aeyF Suleimanov, F. Valery σ X T 40 1+ (1 usr:gnrl–satrn tr:nurn–Xry:binarie X-rays: – neutron stars: – scattering – general pulsars: sismass its is tT ¨ at r s erg X bne,Sn ,D706T D-72076 1, Sand ubingen, ¨ − )] 1 ,1,Kzn400,Russia 420008, Kazan 18, ., sago siaefrtelmtn crto uioiyo a of luminosity accretion limiting the for estimate good a is ∼ a om21 ue11 June 2015 form nal n - ; ; P 40 tPtrbr 910 Russia 196140, Petersburg nt iest fTru V Turku, of niversity r s erg ∼ > 1 ieaoeteN ( NS the above rise ae sso stepla uioiyecestecriti- the exceeds luminosity pulsar ( the sur- NS value as the cal at soon spots As hot the face. from originates emission X-ray ( M82 X-ray galaxy ultra-luminous in an X-2 from (ULX) pulsations source the of by discovery observatory revolutionary recent a thermore, 2015 rate emit- accretion of mass configuration the the on that depends known highly ( is region it ting NSs tized for responsible was that energetics. accretion high than rather rotation was 2007 holes al. black et mass Poutanen stellar accreting super-critically the or about of luminosities reach can NS 10 accreting that implies 1996 al. et Kouveliotou lhuhhgl-antzdNshv rvosybeen previously ( have ULXs power to NSs suggested highly-magnetized Although asbakhls10 intermedi holes the black either mass that thought one recently Until limit. ihXrylmnste serve by review (see luminosities X-ray high ak uye 1976 Sunyaev & Basko − . ) h eaienrons fa rao feasible of area an of narrowness relative The s). 5 40 1 rmtesadr hoyo crto nomagne- onto accretion of theory standard the From n rvdsapsiiiyfrtelmnst oexceed to luminosity the for possibility a provides and ) r s erg o h antrlk antcfil ( field magnetic magnetar-like the for − 1 ∼ w reso antd bv h Eddington the above magnitude of orders two , 3 bne,Germany ubingen, ¨ 10 ¨ ais , 4 37 ¨ al egyS Tsygankov S. Sergey ak uye 1976 Sunyaev & Basko ni 0 I250Piikki FI-21500 20, ¨ antie r s erg 2 − r h otpoal ore fsuch of sources probable most the are ) 10 − .Priual,a o uioiythe luminosity low at Particularly, ). 1 ; 5 naceinclm eisto begins column accretion an ) M evdv&Puae 2013 Poutanen & Medvedev sgno ta.2006 al. et Tsygankov ⊙ ( obr uhtk 1999 Mushotzky & Colbert ahtie l 2014 al. et Bachetti eg&Sra2011 Soria & Feng ; sXrysources X-ray us A ,Finland ¨ o, uhuo tal. et Mushtukov T fteaccretion the of trsa lower at atures E ihms X- mass high tl l v3.0 file style X B conclusion e a reach can y ∼ > neutron e 1 s NuSTAR 10 .Fur- ). 14 (e.g. ,it ), ate G) ). ) ) 2 A. A. Mushtukov et al. the Eddington limit. The radiation pressure is balanced by the magnetic pressure in the accretion column, which sup- ports the column. The height of the accretion column increases with the mass accretion rate and can be as high as the NS radius (Lyubarskii & Syunyaev 1988). Even higher accretion columns are probably possible but their luminosity are not expected to grow significantly because of a rapidly decreasing accretion efficiency. In this paper, we develop a simplified model of accretion onto magnetized NSs based on the suggestions by Basko & Sunyaev (1976) and Lyubarskii & Syunyaev (1988). For the first time, we take into account exact Comp- ton scattering cross-sections in strong magnetic field. The structure and the maximal luminosity of the accretion column are calculated. We determine an upper limit on the NS luminosity to be close to 1040 erg s−1, which coincides with the position of a sharp cut-off in the luminosity function of high-mass X-ray binaries (Mineo, Gilfanov, & Sunyaev 2012) indicating that a large fraction of bright X-ray binaries are likely accreting magnetized NSs. A possibility of ˙ 19 −1 extremely high mass accretion rate (M ∼> 10 g s ) onto a Figure 1. The structure of the accretion column. At the NS magnetized NS is discussed. We find that such high M˙ imply surface the cross section of the accretion channel is a thin annular extremely high, magnetar-like magnetic field strength. arc of length l0 and thickness d0. The position of the radiation- dominated shock varies inside the accretion channel, because the radiation energy density drops sharply towards the column edges. As a result the accretion column (i.e. the settling flow) height 2 ACCRETION COLUMN MODEL reaches its maximum inside the channel and decreases towards its borders. 2.1 Analytical estimates Let us consider an X-ray pulsar accreting at a sufficiently high rate. If luminosity exceeds the critical value where M and R are NS mass and radius, h is the height L∗ (Basko & Sunyaev 1976; Mushtukov et al. 2015), the above the stellar surface, ρ is the local mass density in the radiation-dominated shock is formed in the accreting gas. column. This equation gives an upper limit of Prad, since we After the shock, the hot gas settles to the NS surface in the ignore the contribution of the gas pressure and deviations of magnetically confined accretion column, whose height de- the hydrostatic equilibrium due to a subsonic settlement of pends on the accretion rate and the magnetic field strength the gas in the column. This approximation is equivalent to (see Fig. 1). At high accretion rates that are of interest here, a suggestion that the radial radiation flux at every height of the optical depth of the settling flow along and across the the accretion column equals the local Eddington flux magnetic field direction is significantly larger than unity. c GM Our model is based on a description of optically FEdd = 2 , (2) κ|| (R + h) thick accretion column proposed by Lyubarskii & Syunyaev (1988). The cross-section of the accretion column is a thin where κ|| is the opacity along the magnetic field lines. As- annular arc of width d0 which is much less than its length suming constant density and taking a zero boundary con- l0. Then the area of the accretion column base is SD = l0d0. dition (Prad(x = 0,H) = 0) for the hydrostatic equilibrium The gravitational force, which acts on the gas in the equation, we can get a simple analytical expression for the accretion column, will be offset by the radiation pressure radiation pressure distribution along the column height 1 gradient only. The gas pressure becomes important at GM H − h 12 ∼ Prad(x = 0, h) ≈ ρ , (3) virial temperature 10 K at the NS surface, which R + H R + h cannot be reached due to effective electron-positron pair creation and further neutrino cooling at much lower tem- where H is the height of the shock. Assuming further the peratures ∼ 1010 K (Beaudet, Petrosian & Salpeter 1967; optically thick column, we can estimate the flux emergent Basko & Sunyaev 1976). The radiation pressure gradient at from the column side at a given height h using a plane- the centre of the accretion channel can be presented as fol- parallel diffusion approximation for the radiation transfer lows: across the accretion channel:

dPrad(x = 0, h) GM dPrad(x, h) F⊥(x, h) F⊥,esc(h) 2x = −ρ , (1) = −ρ κ⊥ ≈−ρ κ⊥ , (4) dh (R + h)2 dx c c d0 where x is a distance from the centre of the accretion channel 1 (see Fig.1), κ⊥ is the opacity across the B-field lines, and Photon diffusion time from the centre of the column is tdiff = −5 F⊥,esc(h) = F⊥(d0/2, h) is the escaping flux. The second d0τ/(2c) ≈ 10 s. The matter settling time in the accretion col- −4 equality here was obtained from the condition F⊥(0, h) = 0 umn is tdyn ∼> 7H/vff (H) ∼ 7 × 10 s (for the accretion column of height H = R). As a result, tdiff ≪ tdyn and one can neglect and assuming that the flux increases linearly towards the in the first approximation the effects of plasma movement. edges of the column (where x = d0/2). Then the solution of

MNRAS 000, 1–11 (2015) The maximum luminosity for magnetized neutron stars 3 equation (4) is 2.1.1 Consistency check

τ F (h) 4x2 Let us estimate the values of the basic parameters in or- ≈ 0 ⊥,esc − Prad(x, h) 1 2 , (5) der to justify the proposed model. Let us consider a NS 4c d0 of mass M = 1.4M⊙ and radius R = 10 km accreting at 18 −1 where τ0 = ρ κ⊥d0 is the optical depth across the accretion a rate M˙ ≈ 5 × 10 g s (which gives accretion lumi- channel. This solution allows us to find an emergent flux: nosity L = 1039 erg s−1). We take the column base area 9 2 5 SD = 5×10 cm and its geometrical size are l0 = 5×10 cm 4c Prad(x = 0, h) 4c GM H − h 4 F⊥,esc(h) ≈ ≈ , (6) and d0 = 10 cm, which are expected for a given mass τ0 κ⊥ d0 R + H R + h accretion rate for standard magnetic field of B = 1012 G which is much higher than the Eddington flux for the case (Basko & Sunyaev 1976). We also assume here κ⊥ = κT. of tall (H ≫ d0) accretion columns According to equation (8) the corresponding height of the accretion column is H ≈ 0.75 R. The free-fall velocity at κ|| H − h R + h the height H above the NS surface vff = 2GM/(R + H) ≈ F⊥,esc(h) = 4 FEdd. (7) 10 −1 κ⊥ d0 R + H 1.5 × 10 cm s and the mass densityp for given mass accretion rate ρ ≈ 3.5 × 10−2 g cm−3. Because the mat- The total accretion luminosity of a NS can be estimated by ff ter is decelerated by the radiation-dominated shock at integrating the emergent flux of the surface of two columns: the top of the accretion column (Basko & Sunyaev 1976), H the plasma velocity falls down to v = vff /7 (see, e.g. l0 H c L = 4l0 F⊥,esc(h) dh ≈ 16 f GM (8) Lyubarskii & Syunyaev 1982; Becker 1998), and the mass Z d0 R κ⊥ 0 density goes up to ρ ≈ 0.245 g cm−3. As a result the l0/d0 κT H ≈ 38 f LEdd, Thomson optical thickness across the accretion channel is 50 κ⊥ R τ0 = ρκTd0 ≈ 500, which means that the accretion column is optically thick as it was assumed earlier. where κ ≈ 0.34 cm2 g−1 is the Thomson electron scattering T The radiation pressure and the emergent side flux given opacity of the solar mix plasma and by equations (3) and (6), respectively, near the NS sur- 19 −2 H H H face are Prad ≈ 3.5 × 10 dyncm and F⊥ ≈ 8 × f = log 1+ − . (9) 28 −1 −2 R R R + H 10 erg s cm . Because the column is optically thick we can assume thermodynamical equilibrium deep inside it and The function f(x) grows quadratically f(H/R) ∼ (H/R)2 evaluate the plasma temperature from the radiation field at small column heights H ≪ R and grows logarithmically energy density: at H > R. For the simple analytical estimate (8) we as- 4 εrad aT sumed that the optical thickness across the column is con- Prad ≈ ≈ , (12) 3 3 stant. The dependence of the optical thickness on the height will be taken into account accurately in our numerical model where a is the radiation constant. This relation gives T ≈ 8 (see Section 2.2). The maximum X-ray pulsar luminosity 3.4 × 10 K or kT ≈ 30 keV. Therefore, the gas pressure 15 −2 can be estimated from equation (8) substituting H = R Pg = ρkT/(µsmH) ≈ 5 × 10 dyncm is much smaller (Basko & Sunyaev 1976) than the radiation pressure Prad, as we assumed earlier (here µs = 0.62 is the mean particle weight for fully ionized solar ∗∗ 39 l0/d0 κT M −1 mix plasma). The effective temperature corresponding to the L = L(H = R) ≈ 1.8 × 10 erg s . 4 50 κ⊥ M⊙ escaping flux is determined by the relation F⊥ = σSBT⊥ is 8 (10) T⊥ ≈ 1.9 × 10 K or kT⊥ ≈ 16 keV. This value is close We have suggested earlier that the accretion column to the cut-off energy observed in the spectra of bright X- thickness does not depend on the height above the surface. ray pulsars, especially if the gravitational redshift is taken In fact, the radiation pressure drops down towards the ac- into account (Filippova et al. 2005). We see thus that the cretion channel edges (see equation (5)). As a result, the ra- assumption made constructing the accretion column model diation pressure cannot support the same column height all are reasonable and the model is self-consistent. We consider over the accretion channel. The height has its maximum in this analytical model as a base for further numerical studies. the middle of the channel and decreases towards its edges. In the analytical estimates, we have considered a homo- Therefore, the column thickness has to decrease with the geneous column with a constant cross-section area. For the height as well (see Fig. 1). Taking the radiation pressure numerical model described in the next section, we account Prad(x, h = 0) at the bottom of the accretion column we for the column inhomogeneity and take the exact opacity of can evaluate the column height Hx (i.e. the position of the highly-magnetized plasma. radiative shock) as for a given x using equation (3) where H is replaced with Hx. This height drops from the maximum value H at the centre to the zero value at the column edge: 2.2 Geometry and structure of the accretion column H H 4x2 x − = 1 2 . (11) Using the approach described in Section 2.1, we can con- R + Hx R + H d0 struct a numerical model of the accretion column. Global Therefore, the radial cross section of homogeneous column input model parameters are the NS mass M and radius R, of a small height (H ≪ R) has a roughly parabolic form the mass accretion rate M˙ , and the polar magnetic field (Fig. 1). strength B. The magnetic field is assumed to be dipole with

MNRAS 000, 1–11 (2015) 4 A. A. Mushtukov et al. the magnetic ﬁeld strength decreasing with distance as on the local column conditions such as the temperature and the B-ﬁeld strength (see details in Section 2.3). Equation (4) R + h −3 B ≈ B = BS−1, (13) can be transformed to h R h dh/2 where S is the cross-section area of the accretion channel at h 2ρ F⊥,esc(h) 2 F⊥,esc(h) a given height h above the stellar surface. The width and the Prad(x, h) ≈ κ⊥ z dz+ , (17) dh c Z 3 c length of the accretion channel depends on height as follows: x

R + h 3/2 R + h 3/2 where the similar surface boundary condition like in equa- d ≃ d0 , l ≃ l0 . (14) R R tion (16) was used. Because κ⊥ depends on the local temperature which is evaluated from P (x, h) (see eq. (12)), The geometrical parameters of accretion column base l and rad 0 equation (17) could be solved iteratively, if one knows the d can be derived from the global parameters of the model 0 radiation pressure P at the centre of the accretion chan- (see Sect. 2.4). It has to be mentioned that the thickness of rad nel. Then the flux emergent from the wall of the column a sinking region of the accretion column d at some non- h F can be found for the known column thickness d and zero height h is always smaller than its base thickness d ⊥,esc h 0 plasma mass density ρ at every height. The thickness and the because of the parabolic shape of the shock surface. The mass density are found iteratively as well (see the iterative column thus is surrounded by almost free-fall accreting mat- scheme below). ter of geometrical thickness ∼ (d − d )/2, which affects the h The described model could be incorrect for the case of distribution of the emergent radiation over the directions extremely heigh accretion columns (H > R) due to geomet- (Lyubarskii & Syunyaev 1988; Poutanen et al. 2013). ∼ rical reason, namely due to curvature of dipole magnetic field We consider a two-dimensional radial cross section of lines, when the magnetic field and gravity are not aligned the column using two geometrical coordinates h and x and condition given by eq. (1) is not valid. It means that (Fig. 1). The column structure is determined by a set of we cannot find any solution at some high luminosities when equations including local mass conservation law and hydro- H > R. Fortunately, the approximate formulae (8) and (9) static equilibrium. The mass conservation law can be ex- show that the luminosity saturates at H ≈ R and we still pressed as follows: can obtain a sufficiently good estimation of the maximum M˙ X-ray pulsar luminosity for given NS mass and radius and = ρv, (15) 2 SD the magnetic field strength. where ρ and v are local mass density and velocity correspondingly. The exact local values of ρ and v cannot be calculated without solving the full set of radiation- hydrodynamic equations. The velocity profiles in a sinking 2.3 Scattering cross-section region were calculated for particular cases (Wang & Frank In order to compute the accretion column luminosity or the 1981) and can be approximated by power law v ≈ hξ with accretion column height for a given luminosity, we need to the exponent ξ ∼ 1 for relatively low accretion luminos- find the mean opacity across κ and along κ the magnetic ity ∼ 1037 erg s−1 and with higher ξ ∈ [1; 5] depending on ⊥ || field. We assume that the plasma in the column is fully ion- the accretion channel thickness, opacity in a sinking region ized and ignore bremsstrahlung opacity. Thus, the plasma and magnetic field structure. For the case of high accretion opacity is determined by Compton scattering only. Comp- column (H ∼ 3.5R) and σ ≡ σ the velocity profile was cal- T ton scattering cross-section in case of high magnetic field de- culated to be v ≈ h5 (Basko & Sunyaev 1976). Thus, we as- pends strongly on the photon energy, polarization state and sume that the velocity profile in the accretion column below photon momentum direction (Daugherty & Harding 1986; the shock zone is given by a power law and discuss influence Mushtukov, Nagirner & Poutanen 2015). In the case of op- of the parameter ξ on the solutions (see Section 3.2). The tically thick plasma, in diffusion approximation, we only re- velocity at the NS surface is equal to zero, while the plasma quire the Rosseland mean opacities in the two directions: velocity at the top of the column just below the shock is v(H) = vff (H)/7 (see Lyubarskii & Syunyaev 1982; Becker ne σ⊥,|| 1998), where vff (H) = 2GM/(R + H) is a free-fall veloc- κ⊥,|| = σ⊥,|| = , (18) ρ µemH ity at height H above thep NS surface.

According to equation (1), the local radiation pressure where σ⊥,|| are the Rosseland mean Compton scattering Prad can be presented as follows: cross-sections across and along magnetic ﬁeld (see below),

Hx µe = 1.17 is the mean number of nucleons per electron for GM 2 FEdd(Hx) fully ionized solar mix plasma. For the angular-dependent Prad(x, h) ≈ ρ dy + , (16) Z (R + y)2 3 c case, the cross-section along and across the B-ﬁeld are h (Paczynski 1992; van Putten et al. 2013; Mushtukov et al. where we used standard boundary condition for the radia- 2015) tion density εrad = 2F/c and the approximation of thermo- dynamic equilibrium (12). The connection between radiation ∞ 1 dBE(T ) 2 1 pressure Prad at the centre of the accretion channel and the dE dµ 3µ Z dT Z σj (E,µ,T ) emergent radiation ﬂux F⊥,esc is determined approximately 1 0 0 = ∞ , (19) by the radiative transfer equation (4). However, for the accu- σk,j dBE(T ) rate solution one has to account that the opacity κ depends dE ⊥ Z0 dT

MNRAS 000, 1–11 (2015) The maximum luminosity for magnetized neutron stars 5 and edge depends on the B-field strength and the luminosity. ∞ π 1 We evaluate Hd using the Shakura & Sunyaev (1973) model dBE(T ) 3 2 1 with a slightly different vertical averaging and with the ac- dE dϕ dµn µn Z dT Z Z π σj (E,µ,T ) curate Kramer opacity (Suleimanov, Lipunova, & Shakura 1 0 0 0 = ∞ , (20) 2007). The disc can be divided into three zones according σ dBE ⊥,j dE to the dominating pressure and opacity sources. The gas Z dT 0 pressure and the Kramer opacity dominate in the outer re- respectively. Here T is the local electron temperature, BE(T ) gions (C-zone), the gas pressure and the electron scattering is the Planck function, E is a photon energy, µ is a cosine dominate in the intermediate zone (B-zone) and the radia- of the angle between photon momentum and the B-field dition pressure may dominate in the inner zone (A-zone). The rection and µn is a cosine between the momentum and per- boundaries between A- and B- and between B- and C- zones − 2 8 16/21 −3/7 16/21 pendicular to the field direction (µ = 1 µn cos ϕ). Index are expected to be at rAB ≈ 2.2×10 L39 m R6 cm j correspond to a given photon polarizationp state (j = 1 for 8 2/3 −1/3 2/3 and rBC ≈ 9 × 10 L39 m R6 cm, respectively. The disc X- and j = 2 for O-mode). In case of mixed polarization the structure at the inner edge is defined then by the magneto- effective cross-section can be estimated as follows spheric radius Rm. A relative disc scale-height at radius r for the C-zone, 1/σ = f/σ1 + (1 − f)/σ2, (21) where accretion disc is expected to be interrupted in case of where f is a fraction of radiation in the X-mode relatively low mass accretion rate, is (Mushtukov et al. 2015). We assume f = 1 for further calculations. It is a rather good approximation because Hd −1/10 3/20 −21/40 3/20 1/8 ≈ 0.056 α L39 m R6 r8 . (23) most of the radiation is expected to be in X-mode due r C to the effective swapping of photons from O- to X-mode Assuming that matter leaves the disc at magnetosphere ra- (Miller 1995; Mushtukov et al. 2012). In case of sufficiently dius and follows the dipole magnetic field lines down to the strong magnetic field the typical photon energy E can be polar caps of the NS we get the ratio l0/d0 and the area SD. much lower than the cyclotron one Ecycl ≃ 11.6 B12 keV. For the case of a truncation in the C-zone: In case of E < Ecycl the scattering cross-sections can ≃ 2 ≃ l0 Rm −1/8 71/140 −1/14 −4/35 be approximated as σ⊥(E) σT(E/Ecycl) and σk(E) ≈ 2π ≃ 95 Λ m B12 L39 , (24) 2 2 2 d0 Hd σT[(1 − µ )+ µ (E/Ecycl) ] for X- and O-mode respectively C (Canuto et al. 1971). As a result the effective cross-section and corresponding area of the accretion channel base for extremely strong B-field tends to be significantly lower (C) ≈ × 9 −7/8 2/5 −1/2 −13/20 19/10 2 that the Thomson value. SD 6.6 10 Λ L39 B12 m R6 cm . (25) The computed values of σ(⊥,||)(1,2) were tabulated for A relative disc scale-height in a relatively narrow B-zone is 40 values of the magnetic field strength distributed logarithmically in the range 1011 − 1015 G. The opacities were Hd −1/10 1/5 −11/20 1/5 1/20 ≈ 0.064 α L39 m R6 r8 , (26) calculated for 300 values of the temperature in the range r B 106 − 109 K for every magnetic field strength B. The vari- and the ratio l0/d0 can be estimated as follows able steps were specially adopted for more detailed description in the temperature region near corresponding electron l0 −1/20 19/35 −1/35 −13/70 ≈ 80 Λ m B12 L39 . (27) cyclotron energy. d0 B The accretion channel base area in this case is

2.4 Accretion column base geometry (B) 9 −19/20 19/70 −41/70 −24/35 129/70 2 SD ≃ 7 × 10 Λ L39 B12 m R6 cm . It is assumed that the accretion flow is interrupted at (28) the magnetospheric radius due to the sufficiently strong In the case of sufficiently high luminosity, L39 ∼> magnetic field pressure. The inner radius of the accretion 21/22 6/11 6/11 0.34 Λ B12 m , the structure of the accretion disc depends on the magnetic field strength and its struc- disc zone A is defined mostly by the radiation pres- ture, mass accretion rate and the feeding mechanism of the sure and the disc scale-height is independent of radius: pulsar. It can be estimated with the following expression (Shakura & Sunyaev 1973; Suleimanov et al. 2007): (Ghosh & Lamb 1978, 1979, 1992) 3 M˙ 7 L39R6 × 7 1/7 10/7 4/7 −2/7 Hd,A ≈ κT ≃ 10 cm (29) Rm = 7 10 Λ m R6 B12 L39 cm, (22) 8π c m where Λ is a constant depending on the accretion flow geom- and therefore etry (Λ = 1 for the case of spherical accretion and Λ < 1 for l0 Rm 8/7 3/7 4/7 −9/7 the case of accretion through the disc, with Λ = 0.5 being ≈ 2π ≃ 44 Λ m R6 B12 L39 . (30) d0 A Hd a commonly used value), m = M/M⊙ and we use notations x in cgs units for other variables Q = Qx10 . The corresponding area of the accretion channel base Accretion channel geometry is defined by the interac- (A) 10 −2 11/7 −8/7 −9/7 8/7 2 S = 1.3 × 10 Λ L B m R cm . (31) tion between the accretion disc and NS magnetosphere. The D 39 12 6 thickness of the accretion channel depends on the penetra- If the magnetic dipole is inclined with respect to the tion depth δ of the accretion disc into the NS magnetosphere. orbital plane, the expression for the hot spot area and the It is expected to be of the order of the disc scale-height at ratio (l0/d0) would be different and the spot would have a ∗ the inner edge Hd. The accretion disc structure at the inner shape of an open ring. We use an additional parameter l0 /l0,

MNRAS 000, 1–11 (2015) 6 A. A. Mushtukov et al. which shows what part of the full ring length l0 is filled with the function Hx we finally get the dependence of a sinking ∗ accreting gas. In numerical calculations we take l0 /l0 ≥ 0.5. region width dh on the height h. ∗ It is also natural to assume that the ratio l0 /l0 is higher for (v) As soon as the central radiation pressure Prad(0, h) higher mass accretion rates. and the thickness of the sinking region dh are known at any height, we use equation (17) and get the emergent flux F⊥,esc(h) as a function of h. 2.5 Computation scheme (vi) We get the resulting X-ray pulsar luminosity for a given accretion column height H and the temperature struc- A general computation scheme consists of a few steps. The ture main task is to find the central accretion column height H, H which corresponds to a given X-ray pulsar luminosity L (i.e. 3/2 ′ R + h ˙ L ≃ 4 l0 F⊥,esc(h)dh. (32) to a given mass accretion rate M). The parameters of the Z R model are NS mass M and radius R, surface magnetic field 0 strength B, parameter ξ, which determines the velocity pro- (vii) Using computed Prad(x, h) we get new temper- file in the sinking region, the polarization composition of ature distribution in the accretion column T (x, h) = radiation given by parameter f (see Section 2.3) and the 1/4 ∗ (3 Prad(x, h)/a) . Then we return to step (ii), recalculate ratio l0 /l0, which gives what part of the full ring is filled the luminosity L′ for a given column height H and new tem- with the accreting matter (see Section 2.4). The accretion perature T (x, h) and continue this procedure until we get the column base geometry (l0, d0) is defined by L and B (see accretion luminosity L′(H) with the relative accuracy of 1 Section 2.4). The accretion column height and the structure per cent. are calculated by double iteration scheme. The external it- (viii) If the luminosity L′(H) does not coincide with the eration finds the accretion column height for a given set given accretion luminosity L we get the next guess of the of parameters by the dichotomy method, while the internal accretion column height and return to step (i). We continue iteration calculates the luminosity for current accretion col- the iteration procedure until the relative accuracy of 1 per umn height and given base geometry. We make a first guess cent is reached. of the accretion column height using approximation (8) and assuming κ⊥ = κT. The steps of the iterative scheme are: Finally, we get the height and the structure of the accretion column for a given mass accretion rate. (i) The temperature inside the sinking region T (x, h) is calculated analytically using equations (3), (5), (6) and (12) as a first guess and recalculated iteratively further (see below). 3 NUMERICAL RESULTS (ii) For the current temperature distribution we compute 3.1 Accretion column properties the opacity κk,⊥(x, h) (see Section 2.3). (iii) For the current accretion column height H we com- There are four parameters (in addition to the NS mass M pute the central radiation pressure (x = 0) using formula and radius R) defining the accretion column height and (16) as a function of height h inside the sinking region, i.e. structure. These are the surface polar NS magnetic field ˙ we get Prad(0, h) for h ∈ [0; H]. Formally, formulae (15)– strength B, the mass accretion rate M or the accretion lu- (16) lead to the infinite values of the mass density and the minosity L = GMM/R˙ , and the geometrical sizes of the radiation pressure at h → 0. It means that formula (16) is accretion column base, d0 and l0. Here we investigate how inaccurate at the very bottom of the accretion column. The these parameters affect the properties of the accretion col- total pressure becomes higher than the magnetic pressure umn. In order to separate the effects caused by different 2 (Prad + Pg) > B /4π and plasma spreads over the NS sur- reasons, we will assume a fixed accretion column base ge- 5 face. The height h0, where it happens, is taken as the base ometry with the following parameters l0 = 7.7 × 10 cm 2 4 of the accretion column. d0 = 1.3 × 10 cm in this section only. The self-consistent (iv) Then we find a dependence of the thickness of a results, where the dependence of accretion channel geome- sinking region dh on height h. For that, we first compute try on the mass accretion rate is taken into account, will be F⊥,esc(h = h0) using formula (17) from the already known discussed in Section 3.2. Prad(0, h0), current opacity κ⊥(x, h) and known thickness at Two main parameters that determine the global proper- the base dh0 = d0. Then using the same formula (17) and ties of the accretion column are the local mass accretion rate ˙ already known F⊥,esc(h = h0) we compute the radiation M/2SD and the effective scattering cross-section across the pressure at the base h = h0 from the middle of the column accretion channel σ⊥. The vertical temperature structure in to the edge, i.e. we get Prad(x, h = h0) for x ∈ [0; d0/2]. The the centre of sinking region of the accretion column for a evaluated radiation pressure at the accretion column base given column height depends on the local mass accretion Prad(x, h = h0) is used to get the column height Hx at dis- rate only. The temperature structure across the accretion tance x from the center of the channel (see equation (16)). channel and, therefore, the emergent flux at given height is Then we get Prad(x, h) using again equation (16). Reversing determined by the local effective scattering cross-sections d0 σ⊥(1,2) (see Section 2.3) and the column density 0 ρdz across the channel. The stronger the field, the smallerR the 2 scattering cross-section and the smaller the optical thick- The height h0 depends on the mass accretion rate, B-field strength and velocity profile in the sinking region and takes on ness across the column. As a result, the higher the B-field, a value ∼ (0.1 − 100) cm, which is ≪ d0 and does not affect the the higher the emergent flux F⊥ for the fixed temperature in luminosity calculated for given accretion column height. the column centre and the column density. Global properties

MNRAS 000, 1–11 (2015) The maximum luminosity for magnetized neutron stars 7

102 luminosity as higher column at relatively low field strength. On the other hand, the increased leaking of thermal energy a makes the central column temperature lower in case of high B-field (Fig. 2 a) and supporting of the accretion column is less effective. Then the accretion column is also smaller in case of higher field strength for fixed mass accretion rate. 1 10 We note, that the significant dependence of accretion col- [keV] T B=3*1013 G umn properties on the magnetic field strength takes place for sufficiently high magnetic field strengths (B > 1012 G) only, B=1013 G when most energy is radiated below the electron cyclotron B 12 =3*10 G energy Ecyc = 11.6 B12 keV and the effective electron scat- 0 10 tering cross-section drops significantly below σT (Fig. 2 c). 0.01 0.1 1 As B < 1012 G, the properties of sufficiently bright accre- h/R ∼ tion columns become almost independent on the magnetic 102 field strength. The effective temperature of the accretion column is not b a monotonic function of height and reaches its maximum value above the NS surface (Fig. 2b). On the contrary, the internal temperature decreases monotonically with height B=3*1013 G (Fig. 2a), according to the basic principles described above. 101 [keV] B 13 The value of internal temperature on the top of the column eff =10 G

T is defined by the boundary condition in equation (16). The B 12 =3*10 G internal central temperature is higher than the effective temperature at a given height and this difference is defined by the optical thickness across the accretion column. The op- 100 tical depth across the column decreases with the height be- 0.01 0.1 1 h/R cause of decreasing of geometrical thickness (see Section 2.1) and overcompensates the drop of the internal temperature. This causes the increase of the effective temperature with height. 0 B 12 10 =3*10 G We note that the difference between the effective and the internal temperatures is larger for a relatively low B- B=1013 G field strength at a fixed luminosity, because of higher elec- T 10-1 σ tron scattering opacity. These temperatures become close to / 13 σ B=3*10 G each other as soon as the optical thickness across the column becomes close to unity. It happens when the accretion lumi- -2 10 nosity is close to the critical luminosity value L∗ for a given c magnetic field strength (Mushtukov et al. 2015), which defines the minimum luminosity when the accretion column 10-3 0.01 0.1 1 exist. h/R It was shown above that the column reaches its maximum luminosity when the column height H ≈ R. Therefore, Figure 2. Theoretical dependences of (a) the internal tempera- a stronger magnetic field allows X-ray pulsars to reach higher ture, (b) the effective surface temperature, and (c) the effective luminosities, i.e. the accretion column of a fixed height will 39 −1 scattering cross-section on the h/R ratio for L = 10 ergs be more luminous when the magnetic field is stronger. We 5 and fixed accretion column base geometry: l0 = 7.7 × 10 cm 4 illustrate this statement by a few accretion columns with d0 = 1.3 × 10 cm. Red solid, green dotted, and blue dashed lines 12 13 13 the same accretion column base geometry and different lu- correspond to B = 3 × 10 , 10 and 3 × 10 G, respectively. 13 6 minosities, for two fixed magnetic field strengths: 10 and Parameters: M = 1.4M⊙, R = 10 cm, ξ = 1, f = 1. 1014 G (see Fig. 3). It is clear that the accretion columns of similar heights but different field strength produce luminosities that may differ by an order of magnitude. of a self-consistent accretion column model, which have to The dependences of column properties on the geomet- be constructed in an iterative way, are determined by these rical parameters are much simpler and we do not illustrate facts. it in details. Decrease of the column base thickness d0 and Let us focus on the dependence of accretion column the increase of column base length l0 lead to lower accre- properties on the magnetic field strength and consider action column heights at a given mass accretion rate and the cretion column models corresponding to three different sur- B-field strength. In fact, the geometrical parameters are not face B-field strength and fixed accretion luminosity L = independent, they depend on the magnetospheric radius Rm 1039 erg s−1 (see Fig. 2). The higher field strength leads to and the accretion disc scale-height at its inner edge. Both of lower effective scattering cross-section (Fig. 2 c). Therefore, them depend on the mass accretion rate and the magnetic 4 the emergent flux at a given height (F⊥ = σSBTeff ) is higher field strength on the NS surface. We consider this problem (Fig. 2 b). Smaller column at high B-field provides the same in detail in the next section.

MNRAS 000, 1–11 (2015) 8 A. A. Mushtukov et al.

2 14 13 10 B=10 G B=10 G 100 2.0 T

-1 1.3 σ /

10 0.8 1 σ [keV] 10 13.1 T B 13 16.5 0.8 17.2

=10 G -2 17.2 10 1.3 14 16.5 2.0 B

=10 G 13.1

100 10-3 0.01 0.1 1 0.01 0.1 1 h/R h/R

Figure 3. The internal temperature and the effective scattering cross-section as a function of height. Different curves correspond to various accretion column height: 0.25R, 0.5R, R and different luminosities, which depend on the field strength. The accretion column 5 4 base geometry is fixed: l0 = 7.7 × 10 cm, d0 = 1.3 × 10 cm. Red solid lines correspond to the case of surface magnetic field strength B = 1013 G, while blue dashed lines correspond to B = 1014 G. The corresponding luminosities (in units of 1039 ergs−1) are marked next to each curve. The higher the accretion column, the higher the internal temperature, the higher effective scattering cross-section 6 for a given height above the NS surface. Parameters: M = 1.4M⊙, R = 10 cm, ξ = 1, f = 1.

1041 The velocity profile inside the sinking region, which is 1015 G defined by the parameter ξ, does not affect much the luminosity dependence on the accretion column height: its vari- 40 10 ations within the interval 1 ≤ ξ ≤ 5 lead to variations of accretion luminosity by a factor ∼< 2 for a given column ] 39 height. The higher the ξ, the lower the calculated lumi- -1 10 nosity value. If one takes scaled velocity structure obtained 1011 G 38 by Basko & Sunyaev (1976) for the accretion column with [erg s 10 L constant scattering cross-section σ = σT, the luminosity is smaller by a factor < 2 than that given by ξ = 1. Thus, we 1037 take ξ = 1 in our calculations. Theoretical dependences of X-ray pulsar luminosities on 1036 the accretion column height for various B-field strength are 0.01 0.1 1 10 shown in Fig. 4. All dependences start at relatively small H/R accretion column heights corresponding to the critical luminosity L∗, which depends on the B-field strength as well (Mushtukov et al. 2015). The lowest height is expected to Figure 4. The accretion column luminosity as a function of its be comparable with the thickness d0 of the accretion chan- height. Red solid lines correspond to various surface magnetic field nel near the NS surface. Below the critical luminosity, the strength: B = 1011, 3×1011, 1012, 3×1012, 1013, 3×1013, 1014, 3× energy is emitted by a hotspots at the NS surface. The lu- 1014 and 1015 G. The red circles at the right end of the curves indicate the maximum accretion luminosity values and the corre- minosity dependences on the height of the accretion column sponding accretion column height. The red squares indicate the are close to the approximate analytical dependence L(H/R) minimum critical luminosity L∗ when the accretion column ex- given by equation (9) for the case of relatively low magnetic ists. The saturation of the luminosity is caused by two different field strength (B < 3 × 1012 G). The deviations of the com- factors: a switch of the accretion disc behaviour on its inner ra- puted dependences from the analytical one can be explained 13 dius for B ∼< 10 G and by increasing of the effective scattering by changes of accretion channel geometry, which is given 13 cross-section for B ∼> 10 G, which makes the column growth by the ratio l0/d0 assumed to be constant in the approxi- ineffective at some point. The black dotted line shows the approx- mate relation (9). The dependence of l0/d0 on the accretion imate dependence (9) of the luminosity on the accretion column rate varies according to the region where the accretion disc height for κ⊥ = κT. The parameters used in the calculations are is interrupted by the NS magnetic field (see Section 2.4). M = 1.4M , R = 106 cm, Λ = 0.5, l∗/l = 1, ξ = 1, f = 1. ⊙ 0 0 If the accretion rate is relatively low and the disc is interrupted in the C-zone, the accretion column luminosity L ∝ f 35/39(H/R). However, if M˙ becomes sufficiently high 3.2 Maximum luminosity and the disc is disrupted in the A-zone, the luminosity grows 7/16 Using the accretion column model described above we inves- with the column height much slower as L ∝ f (H/R). tigate the maximum accretion luminosity of magnetized NS This transition from the C- and B-zones to the A-zone is as a function of the surface magnetic field strength. Higher clearly seen in Fig. 4. The transition luminosity depends on tr ≈ 6/11 accretion luminosity corresponds to a taller accretion col- the magnetic field strength and is given by L39 0.2 B12 . umn. The luminosity dependences on the accretion column

MNRAS 000, 1–11 (2015) The maximum luminosity for magnetized neutron stars 9

13 height L(H) for B > 10 G are shifted to higher values 1041 because of a strong reduction of the effective electron scattering cross-section. The averaged scattering cross-section 40 falls quickly at high B and, therefore, the factor κT/κ⊥ 10 grows substantially. We see, however, that the luminosity does not grow as much with the column height as it hap- ] 39 -1 10 pens at lower B. This is related to the fact that with in- ˙ creasing M the optical thickness across the column grows 38 [erg s 10 leading to a higher inner temperature and correspondingly L higher scattering cross-section (because the photon peak becomes closer to the resonance) which leads to a reduction of 37 10 10 s the escaping flux. Thus at some point the further growth of 1.37 s the column is not effective any more and luminosities higher 1036 than some maximum luminosity (for given B-field strength) 1012 1013 1014 1015 cannot be supported by accretion column of any height. B [G] The dependence of the maximum luminosity on the surface magnetic field is shown in Fig. 5 by a thick solid line. We limited the maximal accretion luminosity for low B-field Figure 5. The maximum accretion luminosity L∗∗ as a func- strength by hand at H/R = 1, because of a much slower lu- tion of the magnetic field strength is given by thick solid line. 13 15 minosity grow at H/R > 1 and a questionable validity of our In the interval 10 2) there is a lower limit on the l0 = 4 × 10 cm and d0 = 1.7 × 10 cm. Parameters: M = 1.4M⊙, ∼ 6 ∗ 4/3 R = 10 cm, Λ = 0.5, l0/l0 = 1, ξ = 1, f = 1. magnetic field: B12 ∼> 4 L39 .

It may be smaller by a factor of a few if one accounts for the 4 DISCUSSION energy advection (Poutanen et al. 2007). Using the condi- 4.1 An admissible accretion luminosity range tion (33) we get an approximate upper limit for the accretion luminosity: We have shown in the previous section that a tall accretion 40 −1 max 7/9 4/9 8/9 1/3 column can radiate in excess of 10 erg s for the magnetar L39 ≃ 5.7 Λ B12 m R6 (35) like magnetic fields B > 1014 G. However, we have not ad- ∼ shown by a dotted line in Fig. 5. We see that the theoret- dressed the question if a necessary high accretion rate is con- ical maximum column luminosity is always below the limit sistent with the presence of the column. The problem is that coming from this spherization constraint and therefore our at high accretion rate the accretion disc may become locally model is self-consistent. super-Eddington outside of the magnetosphere producing In addition to the upper limit on the luminosity, there strong outflows (Shakura & Sunyaev 1973; Poutanen et al. should exist a lower limit too. At low accretion rates, the 2007) and reducing the NS luminosity. Additional problem spin frequency of the NS may exceed the Keplerian fre- arises because in this case the disc becomes very thick at R m quency at the magnetospheric radius. Then the accretion leading to a nearly spherical accretion, which is limited by of matter may be inhibited by the centrifugal force exerted the Eddington luminosity. The correction of the opacity at by the rotating magnetosphere and the matter then may high B will not play any role here just because the magnetic be expelled from the system due to the “propeller effect” field drops rapidly with the distance and at some radius the (Illarionov & Sunyaev 1975). This dramatically reduces the cyclotron resonance will appear in the X-ray range where accretion luminosity. The propeller starts to operate when most of the photons are escaping. the magnetospheric radius Rm becomes larger than the coro- The condition for the absence of the outflows outside tation radius the magnetosphere can be written as 2 1/3 GMP 8 1/3 2/3 Rsp < Rm, (33) RC = ≃ 1.5 × 10 m P cm, (36) 4π2 where Rsp is the spherization radius where the radiation force is still balanced by gravitation. Its simplest estimate where P is a NS spin period in seconds. This happens at is based on equating the scale-height of the radiation- luminosity dominated disc to the radial distance (Shakura & Sunyaev min −2 7/2 −2/3 5 2 −7/3 L39 ≃ 7 × 10 Λ m R6B12P , (37) 1973): 3 κ L R which defines the likely lower limit on X-ray luminosity of a R = T M˙ = 7.5 × 106 39 6 cm. (34) sp 8π c m binary system.

MNRAS 000, 1–11 (2015) 10 A. A. Mushtukov et al.

During the accretion process a NS in a binary sys- It was shown above that the accreting NS may have tem is spun up to the period determined by the condi- accretion columns radiating in excess of 1040 erg s−1 for 14 tion Rm ≃ RC, when the accretion luminosity is given by magnetar-like magnetic fields B0 ∼> 10 G (see Fig. 5). equation (37). For the Eddington accretion rate relaxation In other words, some ULXs can actually be extreme X-ray 1/3 −4/3 happens very rapidly, within τrel ≈ 300 I45m µ30 yr, pulsars as indeed was recently observed for X-2 source in 45 2 where I45 = I/10 g cm is the NS moment of inertia and galaxy M82 (Bachetti et al. 2014). Its high pulsating lumi- 30 3 39 −1 µ30 = µ/10 G cm is the NS dipole magnetic moment nosity ∼ 4.9 × 10 erg s implies according to our model 14 (Lipunov 1982). Of course, the luminosity may exceed the the magnetic field B ∼> 10 G. The observed pulsation pe- minimum value (37) over short time intervals when the ac- riod of 1.37 s makes the magnetic field of ∼ 1014 G the only cretion rate increases. possible option for this particular ULX, because the higher We conclude that the X-ray pulsar luminosity is likely field strength leads to the propeller effect (see Section 4.1). to be bound from below by Lmin and from above by L∗∗ ≈ Because the ULX is very likely to be close to the equilibrium 3/4 0.35B12 . It is clear from Fig. 5 that the stronger the mag- state (Bachetti et al. 2014; Lipunov 1982), an estimation of netic field, the narrower the possible luminosity range for a its magnetic field based on the analysis of the period P and given NS spin period. A likely maximum accretion luminos- the period derivative P˙ is not valid since one need to know ity given by the interception of the two constraints is then exact physics of disc and magnetosphere interaction and 2/5 5/4 −3 −21/10 the estimations of magnetic field strength become strongly L ≈ 1.6 m P R Λ , (38) 39 6 model dependent near the equilibrium (Parfrey et al. 2015; corresponding to the magnetic field Dall’Osso et al. 2015; Ek¸si et al. 2015). Transition between 8/15 28/15 −4 −14/5 accretion and propeller regime has been detected recently B12 ≈ 7.5 m P R Λ . (39) 6 in M82 X-2 (Tsygankov et al. 2015) confirming our estima- Thus the luminosities of X-ray pulsars in excess of tion of the magnetic field strength of ∼ 1014 G. A high ac- 1040 erg s−1 would definitely require an extremely strong B- cretion rate needed to power high ULX luminosities (L ∼ field (B > 1014 G) and a large spin period (P > 1 s). 1039 − 1041 erg s−1) from pulsars imply a large thickness of the accretion disc on its inner edge. This results in repro- cessing of a significant part of pulsar radiation in the disc, 4.2 X-ray pulsars and ULXs which might lead to spectral deviations comparing to the All known X-ray pulsars can roughly be divided into two big intrinsic spectrum of the accretion column. groups: persistent and transient sources. Among the brightest persistent sources we mention here Cen X-3, SMC X-1 and LMC X-4. The only source in this group with the known 5 SUMMARY magnetic field is Cen X-3, which shows the cyclotron line around 28 keV (Nagase et al. 1992) that corresponds to the We have constructed a simplified model of the accre- corrected for the redshift magnetic field B ∼ 3.2 × 1012 G. tion column on a magnetized NS. We based the model The brightest X-ray pulsars in the Magellanic Clouds do not on previous suggestions by Basko & Sunyaev (1976) and show any cyclotron features in their spectra and hence the Lyubarskii & Syunyaev (1988) that the column is vertically magnetic fields there are unknown. The maximal observed supported by the Eddington flux. We use diffusion approxi- luminosities in this group range from ∼ 5 × 1037 erg s−1 for mation to evaluate the radiation flux through the sides of the Cen X-3 to ∼ 1039 erg s−1 for SMC X-1 and LMC X-4 during column. For the first time, we use the exact Compton scat- outbursts. tering cross-sections in strong magnetic field and find the Another group of transient sources mainly consists of column temperature structure and the cross-sections self- NS in binary systems with the Be companions. They are consistently via an iterative procedure. As a result we have known to exhibit rare non-periodic type II outbursts with obtained the accretion column luminosity as a function of the peak X-ray luminosity reaching the Eddington value (for the magnetic field strength and the column height. We also a review see Reig 2011). The brightest sources in this group evaluate for a given surface magnetic field the maximum pos- are 4U 0115+63 and V 0332+53 with the peak luminosi- sible column luminosity that is reached when the column ties as high as a few ×1038 erg s−1 and a clear evidence of height becomes comparable to the NS radius. We showed 40 −1 the high accretion column (see e.g., Tsygankov et al. 2006, that the column may radiative in excess of 10 erg s in 14 2007; Tsygankov, Lutovinov & Serber 2010; Poutanen et al. the case of the magnetar-line magnetic field ∼> 10 G. 2013; Postnov et al. 2015). The magnetic fields are known We also evaluate the possible range of accretion lumi- from the cyclotron lines: ∼ 1.3 × 1012 G for 4U 0115+63 nosity in X-ray pulsars. It is limited from above by the con- (White, Swank & Holt 1983) and ∼ 3.2 × 1012 G for V dition that the accretion disc does not spherize outside the 0332+53 (Makishima et al. 1990). magnetosphere. This condition, however, gives the limiting Apart of these two groups of X-ray pulsars we can men- luminosity above our upper limit thus confirming the con- tion a unique transient source GRO J1744–28 showing both sistency of our calculations. The pulsar luminosity may also X-ray pulsations and unusual short bursts. The broadband be limited from below by the propeller effect. observations of this source revealed a cyclotron absorption The range of possible pulsar luminosities becomes nar- line in the energy spectrum at E ≈ 4.5 keV (D’A`ıet al. rower for stronger magnetic fields. The surface magnetic field 2015; Doroshenko et al. 2015), which corresponds to B ≃ strength can be limited from below if one knows the lumi- 11 4/3 5 × 10 G. The maximum luminosity during the outbursts nosity of the object: B12 ∼> 4 L39 . Thus the X-ray pulsar lu- was ∼ (3 ÷ 4) × 1038 erg s−1. The observed luminosities of minosity exceeding a few 1039 erg s−1 requires magnetic field 13 40 −1 all these pulsars are in good agreement with our limits. ∼> 10 G, while the ULX luminosities of 10 erg s would

MNRAS 000, 1–11 (2015) The maximum luminosity for magnetized neutron stars 11

14 be consistent only with the magnetar-like fields ∼> 10 G. Makishima K. et al., 1990, ApJ, 365, L59 On the other hand, the magnetic field cannot be too large, Medvedev A. S., Poutanen J., 2013, MNRAS, 431, 2690 because of the propeller effect. We showed that the estimated Miller M. C., 1995, ApJ, 448, L29 X-ray luminosity of ULX X-2 in galaxy M82 (Bachetti et al. Mineo S., Gilfanov M., Sunyaev R., 2012, MNRAS, 419, 2095 2014) of ≈ 1040 erg s−1 can be consistent with our model Mushtukov A. A., Nagirner D. I., Poutanen J., 2012, Phys. Rev. D, 85, 103002 only for B ≈ 1014 G. Mushtukov A. A., Nagirner D. I., Poutanen J., 2015, Because of the set of constraints it is likely that the 40 −1 arXiv:1512.06681 luminosity L ≃ (1 ÷ 3) × 10 erg s is a good guess Mushtukov A. A., Suleimanov V. F., Tsygankov S. S., Poutanen for the NS maximum possible accretion luminosity. Inter- J., 2015, MNRAS, 447, 1847 estingly, this value coincides with the cut-off observed in Nagase F., Corbet R. H. D., Day C. S. R., Inoue H., Takeshima the X-ray luminosity function of high-mass X-ray binaries, T., Yoshida K., Mihara T., 1992, ApJ, 396, 147 which does not show any features at lower luminosities Paczynski B., 1992, Acta Astronomica, 42, 145 (Mineo, Gilfanov & Sunyaev 2012). Therefore, it is natural Parfrey K., Spitkovsky A., Beloborodov A. M., 2015, to suggest that accreting NSs are not rare among ULXs and arXiv:1507.08627 their fraction is probably close to their fraction among the Postnov K. A., Gornostaev M. I., Klochkov D., Laplace E., Lukin V. V., Shakura N. I., 2015, MNRAS, 452, 1601 high-mass X-ray binaries. Poutanen J., Lipunova G., Fabrika S., Butkevich A. G., Abol- masov P., 2007, MNRAS, 377, 1187 Poutanen J., Mushtukov A. A., Suleimanov V. F., Tsygankov ACKNOWLEDGEMENTS S. S., Nagirner D. I., Doroshenko V., Lutovinov A. A., 2013, ApJ, 777, 115 This research was supported by the Magnus Ehrnrooth Reig P., 2011, Ap&SS, 332, 1 Foundation (VFS), the Jenny and Antti Wihuri Founda- Shakura N. I., Sunyaev R. A., 1973, A&A, 24, 337 tion (VFS), the Russian Science Foundation grant 14-12- Suleimanov V. F., Lipunova G. V., Shakura N. I., 2007, Astron- 01287 (AAM, SST), the Academy of Finland grant 268740 omy Reports, 51, 549 Tsygankov S. S., Lutovinov A. A., Churazov E. M., Sunyaev (JP), and the German research Foundation (DFG) grant R. A., 2006, MNRAS, 371, 19 WE 1312/48-1 (VFS). Partial support comes from the EU Tsygankov S. S., Lutovinov A. A., Churazov E. M., Sunyaev COST Action MP1304 “NewCompStar”. R. A., 2007, Astronomy Letters, 33, 368 Tsygankov S. S., Lutovinov A. A., Serber A. V., 2010, MNRAS, 401, 1628 Tsygankov S. S., Mushtukov A. A., Suleimanov V. F., Poutanen REFERENCES J., 2015, arXiv:1507.08288 Bachetti M. et al., 2014, Nature, 514, 202 van Putten T., Watts A. L., D’Angelo C. R., Baring M. G., Kou- Basko M. M., Sunyaev R. A., 1976, MNRAS, 175, 395 veliotou C., 2013, MNRAS, 434, 1398 Beaudet G., Petrosian V., Salpeter E. E., 1967, ApJ, 150, 979 Wang Y.-M., Frank J., 1981, A&A, 93, 255 Becker P. A., 1998, ApJ, 498, 790 White N. E., Swank J. H., Holt S. S., 1983, ApJ, 270, 711 Canuto V., Lodenquai J., Ruderman M., 1971, Phys. Rev. D, 3, 2303 Colbert E. J. M., Mushotzky R. F., 1999, ApJ, 519, 89 D’A`ıA. et al., 2015, MNRAS, 449, 4288 Dall’Osso S., Perna R., Stella L., 2015, MNRAS, 449, 2144 Daugherty J. K., Harding A. K., 1986, ApJ, 309, 362 Doroshenko R., Santangelo A., Doroshenko V., Suleimanov V., Piraino S., 2015, arXiv:1503.09020 Ek¸si K. Y., Anda¸c I.˙ C., ¸Cıkınto˘glu S., Gen¸cali A. A., Gung¨ ör C., Oztekin¨ F., 2015, MNRAS, 448, L40 Feng H., Soria R., 2011, New Astron. Rev., 55, 166 Filippova E. V., Tsygankov S. S., Lutovinov A. A., Sunyaev R. A., 2005, Astronomy Letters, 31, 729 Ghosh P., Lamb F. K., 1978, ApJ, 223, L83 Ghosh P., Lamb F. K., 1979, ApJ, 232, 259 Ghosh P., Lamb F. K., 1992, in X-Ray Binaries and the Forma- tion of Binary and Millisecond Radio Pulsars, van den Heuvel E. P. J., Rappaport S. A., eds., Kluwer Publ., Dordrecht, pp. 487–510 Illarionov A. F., Sunyaev R. A., 1975, A&A, 39, 185 Kouveliotou C., van Paradijs J., Fishman G. J., Briggs M. S., Kommers J., Harmon B. A., Meegan C. A., Lewin W. H. G., 1996, Nature, 379, 799 Lipunov V. M., 1982, Soviet Ast., 26, 54 Lucke R., Yentis D., Friedman H., Fritz G., Shulman S., 1976, ApJ, 206, L25 Lyubarskii Y. E., Syunyaev R. A., 1982, Soviet Astronomy Let- ters, 8, 330 Lyubarskii Y. E., Syunyaev R. A., 1988, Soviet Astronomy Let- ters, 14, 390

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