Magnetic flux stabilizing thin accretion disks
MMNRAS , 000–000 (0000) Preprint 5 November 2018 Compiled using MNRAS L A TEX style file v3.0
Magnetic flux stabilizing thin accretion disks
Aleksander S ˛adowski , (cid:63) MIT Kavli Institute for Astrophysics and Space Research, 77 Massachusetts Ave, Cambridge, MA 02139, USA Einstein Fellow
ABSTRACT
We calculate the minimal amount of large-scale poloidal magnetic field that has to thread theinner, radiation-over-gas pressure dominated region of a thin disk for its thermal stability. Sucha net field amplifies the magnetization of the saturated turbulent state and makes it locallystable. For a 10 M (cid:12) black hole the minimal magnetic flux is 10 ( ˙ M / ˙ M Edd ) / G · cm . Thisamount is compared with the amount of uniform magnetic flux that can be provided by thecompanion star – estimated to be in the range 10 − G · cm . If accretion rate is largeenough, the companion is not able to provide the required amount and such a system, if stillsub-Eddington, must be thermally unstable. The peculiar variability of GRS 1915 + Key words: accretion, accretion discs – black hole physics
According to the standard model, radiatively e ffi cient, radiationpressure supported accretion disks are thermally and viscously un-stable (Lightman & Eardley 1974; Shakura & Sunyaev 1976; Piran1978). This prediction is in apparent disagreement with the proper-ties of most black hole (BH) X-ray binaries. Except for two sources(GRS1915 +
105 and IGR J17091-3624, see Belloni et al. (2000)and Altamirano et al. (2011)), all other BH transients stay in theirthermal, high / soft states for days / months without any sign of un-stable behavior. The question arises – what phenomena does thestandard model not account for?One possiblity for explaining the stability of radiation pres-sure dominated thin disks is the presence of a strong magnetic fieldwhich provides additional pressure support and prevents the run-away heating or cooling that would occur without it. This idea hasbeen investigated in recent years by Oda et al. (2009) and Zhenget al. (2011), and very recently verified numerically by Sadowski(2016). Recently, Li & Begelman (2014) have shown that magneticfields may help stabilize the disk also through magnetically drivenoutflows which decrease the disk temperature and thus help the diskbecome more stable at a given accretion rate. The stabilizing e ff ectof strong fields on disk thermal instability was also discussed byBegelman & Pringle (2007).How to make an accretion disk magnetized enough to pre-vent thermal runaway? Magnetorotational instability (MRI, Balbus& Hawley 1991) in an isolated box is known to saturate at a totalto magnetic pressure ratio β = p tot / p mag ≈
10 (e.g., Turner 2004;Hirose et al. 2009). However, if large scale magnetic field threadsthe box, either vertically or radially, the saturated magnetic field is (cid:63)
E-mail: [email protected] (AS) much stronger. In particular, Bai & Stone (2013) have shown thatthe presence of a weak net vertical magnetic field characterized by β = ffi cient accretion flows with radiation pres-sure dominating over thermal pressure. We compare this quantitywith rough estimates of the magnetic flux that can be provided bythe companion stars, and with the amount of flux required for themagnetically arrested (MAD) state.Our work is organized as follows. In Section 2 we calculatethe magnetic flux required for disk stabilization. The flux requiredfor the magnetically arrested state is calculated for comparison inSection 3. In Section 4 we estimate how much uniform magneticfield a companion star can provide. The discussion is given in Sec-tion 5 and our work is summarized in Section 6. In this Section we estimate the minimal amount of poloidal ver-tical flux required for thermal stability of an accretion disk. Weassume that the disk is locally stable if at least 50% of the pres-sure is provided by the magnetic field (Sadowski 2016). Such ahighly magnetized state is obtained when disk is threaded by netvertical magnetic field satisfying β (cid:46) c (cid:13) a r X i v : . [ a s t r o - ph . H E ] J un A. S ˛adowski (Bai & Stone 2013; Salvesen et al. 2016). The minimal amount ofmagnetic flux required for stability of the entire disk is obtained byintegrating the net vertical flux over the whole otherwise unstableregion, i.e., where radiation pressure exceeds gas pressure. Doingso we also assume that no magnetic flux has accumulated at theBH. In the standard α -disk model (Shakura & Sunyaev 1973) thevertically integrated total pressure P at radius R is determinedsolely by the angular momentum conservation,2 π R α P tot = ˙ M (cid:16) (cid:112) GM BH R − (cid:112) GM BH R in (cid:17) , (1)where ˙ M is the accretion rate, M BH is the mass of the BH, G isthe gravitational constant, α is the disk viscosity parameter, and R in = R G = GM / c is the location of the inner edge of the disk .In cgs units it equals, P tot = × (cid:32) ˙ M ˙ M Edd (cid:33) (cid:32) RR G (cid:33) − / (cid:18) α . (cid:19) − J erg / cm , (2)where ˙ M Edd = . × M BH / M (cid:12) g / s is the Eddington accretionrate (which, according to this definition, corresponds to a thin diskemitting the Eddington luminosity), and J = − √ R in / R .The strength of the net vertical field that is required to providehighly magnetized saturated turbulent state is given with respectto the equatorial plane pressure, not the vertically integrated one.This may be estimated knowing the disk half-thickness H whichfor the radiation pressure dominated regime of the standard thindisk solution equals (Shapiro & Teukolsky 1983), H = × (cid:32) M BH M (cid:12) (cid:33) (cid:32) ˙ M ˙ M Edd (cid:33) J cm . (3)The equatorial plane total pressure p tot now equals p tot = P tot / H .According to our assumptions, half of the total pressure comesfrom the magnetic field. The remaining amount is the sum of theradiation and thermal pressure. The strength of the net vertial field, B z , that was needed to enhance the magnetization of the saturatedstate, is, B z = (cid:114) π p tot β = (4) = × (cid:32) M BH M (cid:12) (cid:33) − / (cid:32) RR G (cid:33) − / (cid:18) β (cid:19) − / (cid:18) α . (cid:19) − / G . The total required flux is obtained by integrating B z over oth-erwise unstable region. The radiation pressure dominates over gaspressure in the inner region up to a critical radius R max (Shapiro &Teukolsky 1983), R max / R G = × (cid:32) M BH M (cid:12) (cid:33) / (cid:32) ˙ M ˙ M Edd (cid:33) / (cid:18) α . (cid:19) / . (5)Performing the integral one obtains Φ = × (cid:32) M BH M (cid:12) (cid:33) / (cid:32) ˙ M ˙ M Edd (cid:33) / (cid:18) β (cid:19) − / (cid:18) α . (cid:19) − / G · cm . (6)When calculating the minimal amount of large scale magneticflux required for stabilization of the disk we have implicitly as-sumed that the accretion flow, which at the accretion rates of in-terest is geometrically thin, was able to advect this net magneticfield inward into the inner region. Whether or not the large-scale In this work we ignore, for simplicity, the BH rotation. · cm = field can be advected depends on the balance between the advec-tion and di ff usion of the field. Standard geometrically thin disksdrag vertical magnetic field ine ffi ciently (Lubow et al. 1994; Ghosh& Abramowicz 1997), and therefore are unlikely to drag significantamount of magnetic field on to the BH. If magnetic field accumu-lates there it exerts significant outward pressure on the accretionflow (see the discussion of the magnetically arrested state in Sec-tion 3). The net magnetic field that we require for stabilization doesnot need to accumulate on the BH – it is enough if it threads thedisk itself and its pressure is a factor β lower that the total pres-sure. The radial gradient of such net-field pressure is negligible inthin accretion disks when compared with gravitational and centrifu-gal forces. In other words, it is only the magnetic tension that hasto be overcome, not the radial gradient of pressure (as in the MADstate). Whether or not sub-Eddington disk are able to drag evensuch a small amount of field inward, is still debated (see e.g., Guilet& Ogilvie 2012, 2013; Avara et al. 2015). Our work bases on theassumption that it is possible. Equation 6 gives the minimal amount of magnetic flux of uni-form polarity that has to be provided to thermally stabilize radia-tivelly e ffi cient, radiation-over-gas pressure dominated thin disks.This amount depends on the BH mass ( M BH ) and the accretion rate( ˙ M ), as well as two other parameters ( β and α ) that result fromnon-linear evolution of MRI and are likely to have rather weak orno dependence at all on the former two. Therefore, the amount ofrequired magnetic flux scales mostly with the BH mass and the ac-cretion rate.There are ∼
20 low-mass X-ray binaries with existing dynam-ical estimates of the compact object mass indicating towards BHs(e.g., Özel et al. 2010). The masses of the transient objects rangefrom ∼ ∼ M (cid:12) . Most such systems undergo transitions fromthe quiescent states to outbursts when they reach significant frac-tions of the Eddington luminosity (Dunn et al. 2010) and enterthe radiation-over-gas pressure dominated, presumably unstable,regime.To estimate how much magnetic flux of uniform polarity is re-quired for stabilization of each source one needs some measure ofthe accretion rate. For this purpose we take the luminosity at out-burst maximum as given in Table 2 of Steiner et al. (2013) . In Ta-ble 1 we show the masses, accretion rate estimates and the requiredfluxes (Eq. 6, obtained assuming the fiducial values of β and α ) forthe five BH X-ray binaries with well established BH masses and ex-isting estimates for the peak luminosities (compiled from Steiner etal. (2013) and Fragos & McClintock (2015)). The amount of mag-netic flux required for their stabilization ranges from 6 . × to5 . × G · cm . The source that requires by far most of uniformlarge scale magnetic flux (almost three times more than the sec-ond one) is GRS 1915 + +
105 is at the same time the only source with well es-tablished parameters that shows long duration, very rapid and shorttimescale variability near the Eddington luminosity(e.g., Belloni et Because the thin disk geometry may not apply to magnetic pressure domi-nated disks, we adopt their L Peak that was obtained assuming isotropic emis-sion. MNRAS , 000–000 (0000) agnetic flux stabilizing thin accretion disks Table 1.
Magnetic fluxes required to stabilize particular BH X-ray binariesName M BH / M (cid:12) L Peak / L Edd Φ [G · cm ]GRS 1915 +
105 12 . . . × XTE J1550-564 9 . .
53 1 . × GRS 1124-683 7 . .
61 1 . × A0620-00 6 . .
47 1 . × GRO J1655-40 6 . .
34 6 . × Masses and luminosities compiled from Steiner et al. (2013),Reid et al. (2014) and Fragos & McClintock (2015). al. 2000) that results, presumably, from thermal instability of its ac-cretion disk. One should note, however, that the exceptionally highmass and luminosity of GRS 1915 +
105 are not the only propertiesmaking it stand out. At the same time it has the longest orbital pe-riod, the largest accretion disk, a giant companion star (Greiner etal. 2001) and presumably also the largest BH spin (McClintock etal. 2006).
Magnetic flux that is advected across the inner edge of the disk ac-cumulates on the BH. If advection is e ffi cient the accumulated mag-netic field exerts radial pressure large enough to dynamically a ff ectthe infalling gas. When this outward magnetic pressure roughly bal-ances tha radial gravitational force, the disk enters the magneticallyarrested state (MAD), where accretion is possible only because theinterchange instability allows the gas to penetrate the accumulatedfield by breaking into clumps or filaments (Narayan et al. 2003;Tchekhovskoy et al. 2011).The amount of magnetic flux accumulated at the BH which re-sults in the MAD state extending up to a given radius was estimatedby Narayan et al. (2003) (their Eq. 2). Taking the horizon radius asthe limit of the extent of the MAD regime, one obtains the minimalamount of flux required to provide the saturated magnetic field atthe BH (see also Yuan & Narayan 2014), Φ MAD = × (cid:32) M BH M (cid:12) (cid:33) / (cid:32) ˙ M ˙ M Edd (cid:33) / (cid:18) (cid:15) − (cid:19) − / G · cm , (7)where (cid:15) is the ratio of the gas radial velocity to the free-fall velocity.If only such an amount is advected on the BH, the innermost part ofthe flow will be in the MAD state, and the BH itself, if rotating, wille ffi ciently generate relativistic magnetic jets (Blandford & Znajek1977). The companion stars in most BH X-ray binaries overflow theirRoche lobes and transfer gas to the compact object. The expelledmatter forms an accretion disk and gradually approaches the blackhole, sometimes in a rather violent way due to the ionization in-stability modulating the flow (Lasota 2001). The gas drawn fromthe stellar surface brings magnetic field with it which triggers tur-bulence in the accretion disk. One may expect, that such magneticfield will have some coherence which will determine the amount ofpoloidal magnetic flux of uniform polarity available in the accretiondisk. Below we very roughly estimate that amount.It is reasonable to expect that the companion stars in low mass
Figure 1.
Schematic picture of the advection of magnetic tubes from thesurface layers of the companion star towards the inner region of the accre-tion disk in an X-ray binary.
X-ray binaries are tidally locked to the rotation of the binary sys-tem. This fact has significant consequences. Firstly, the gas thatoverflows the Roche lobe comes from exactly the same substellarspot on the companion star surface. Therefore, the magnetic fieldadvected with the gas towards the compact object will simply re-flect the magnetic field in the surface layers of the star, and willnot be a ff ected by sweeping through the stellar surface due to ro-tation. Secondly, tidal locking modifies the rotational period of thecompanion star, which is one of the major factors determining thee ffi ciency of magnetic field generation in the stellar interior.For obvious reasons the magnetic field of the Sun is knownbest (see e.g., Schrijver & Zwaan 2000). It exhibits 11 year longactivity cycles over which the polarity of the dipolar componentflips. Over that period Solar activity changes as well. During the ac-tivity periods multiple flares, coronal mass ejections, and sunspotsoccur on the surface. These phenomena are related to the emer-gence of magnetic field, either in the form of large closed loops oropen field lines, which is generated within the convective envelopeof the Sun through the dynamo process (Parker 1955). Emergingmagnetic field probes the magnetic field properties below the stellarsurface. In particular, one may expect that the magnetic field belowthe surface forms magnetic field loops containing similar magneticflux as the loops penetrating the stellar surface. The magnetic tubesin the surface layers will be advected with the gas and may domi-nate the large-scale properties of the magnetic field in the accretiondisk (Fig. 1).The magnetic field near the active regions in the Sun canreach and exceed ∼ .
001 of the solar hemisphere area (Harvey& Zwaan 1993). Multiplying these two numbers (magnetic fieldstrength by the estimate of its coherence area) one gets the estimateof the maximal amount of magnetic flux in an active region (reflect-ing roughly the amount of magnetic flux in coherent regions belowthe surface) – Φ (cid:12) (cid:46) G · cm . Linsky & Schöller (2015) give therange of magnetic fluxes in individual active regions on the Sun as10 to 10 . G · cm , roughly consistent with the previous estimate.Direct measurements of magnetic flux contained in singlemagnetic tubes in distant stars is in most cases impossible. Mag-netic fields on stars are implied from the spatially-unresolved stel-lar light of nearby stars which provides information only about theintegrated (a ff ected by cancelations of magnetic field of opposingpolarity) magnetic field. Even Zeeman Doppler imaging (e.g., Do-nati & Landstreet 2009) can measure only the net (again a ff ectedby cancellations) magnetic field in active regions.Companion stars in X-ray binaries di ff er from the Sun in manyaspects (for rather comprehensive list BH X-ray binaries properties MNRAS , 000–000 (0000)
A. S ˛adowski see Fragos & McClintock 2015). They show di ff erent masses andspectral types. Most are K or M type dwarfs with masses between0 . . M (cid:12) . Some (e.g., the companion of GRS 1915 + . ∼
24 d period). Only one BH X-ray binary rotates with a longer pe-riod than the Sun (GRS 1915 + ∼
34 d), but this rate of rotationis still exceptional for a giant star. It is unclear to what level and forhow long the tidally synchronized companions retain di ff erentialrotation of their interiors.Like in the Sun, the stellar magnetic fields are assumed to re-sult from the dynamo activity in their di ff erentially rotating con-vective zones. This assumption is supported by observations show-ing that the activity indeed scales with rotation, in agreement withthe dynamo theory (Reiners 2012). This relation can be character-ized by the Rossby number, R – the ratio of the rotational periodof the star and its convective turnover time (e.g., Stepien 1994).Reiners et al. (2009) have shown that the magnitude of the meansurface magnetic field of K and M dwarfs saturates around 3000 Gfor R < − . , and scales like 1 / R for larger values. The starswith saturated magnetic fields strength are the fastest rotators (or-bital periods (cid:46) days) – quite similar to most of the BH X-ray binarycompanions. The Rossby number of the Sun is of the order of unity(Reiners 2012). Therefore, if the Sun was rapidly rotating and Sun-like stars followed the same dependence on the Rossby number,its mean magnetic field would be 10 to 100 times stronger thanit is presently. If the size of the active regions stayed the same (aconservative assumption), the magnetic flux contained in a singlemagnetic tube would increase by a similar factor.We conclude that the amount of the magnetic flux of uniformpolarity that the companion star provides depends strongly on thestar properties, most importantly its rotation, but also mass and evo-lutionary history. A rapidly rotating Sun would likely provide atmost 10 − G · cm . It is not possible to give a comparableestimate for any of the companion stars in X-ray binaries due toour lack of understanding of magnetic properties of distant stars,especially ones tidally locked and significantly a ff ected by evolu-tion. We therefore take the range specific for the fast rotating Sunonly as the ballpark for what the companion stars can provide. In the previous sections we estimated magnetic fluxes required forthe thermal stability of a thin accretion disk, for the MAD state, andprovided by the companion stars. All three of them turn out to havethe same order of magnitude for near-Eddington accretion rates andstellar mass BHs. It is somewhat surprising, especially when com-paring the first two with the magnetic flux provided by the stellarcompanion which is determined by the e ffi ciency of stellar dynamowhich knows nothing about the properties of the inner regions ofaccretion disks.Figure 2 shows how the fluxes required for stabilization andfor the magnetically arrested state change with accretion rate forthe fiducial parameters ( M BH = M (cid:12) , β = α = .
1, and (cid:15) = . Φ , blue line) grows faster. At roughly 0 .
01 ˙ M Edd they have the same value, ∼ G · cm .The shaded region in the background reflects the rough esti- Φ Φ MAD M / M Edd Φ [ G c m ] Figure 2.
Minimal magnetic flux required for stabilization of a thin disk( Φ , blue line), as a function of normalized accretion rate. Red line de-notes the minimal amount of magnetic flux required for the magneticallyarrested (MAD) state ( Φ MAD ). These fluxes were calculated assuming M BH = M (cid:12) , β = α = .
1, and (cid:15) = .
01. The shaded regionreflects the rough estimate of the maximal magnetic flux provided by thecompanion star (10 − G · cm , Section 4). mate of the amount of the uniform magnetic flux that can be pro-vided by the companion star, 10 − G · cm (Section 4). It isclear that for a fixed value of that quantity, i.e., for a given com-panion star, there is a critical accretion rate above which the fluxprovided by the companion is not enough to stabilize the unstableinner region of the accretion flow. Therefore, each X-ray binarysystem becomes unstable if this critical accretion rate (specific foreach system) is exceeded. One may expect that this critical value isof the order of the Eddington accretion rate.If the companion star is capable of providing plenty of mag-netic field of uniform polarity, then such a critical accretion ratewould exceed the Eddington one and a given system will never be-come unstable since super-Eddington accretion flows are stabilizedby advection of heat (see Abramowicz et al. 1988).Qualitatively similar conclusion applies to the amount of fluxrequired for the magnetically arrested disk – even if the advec-tion of magnetic field allows for the field accumulation on the BH,there exists a critical, near-Eddington accretion rate above whichthe MAD state cannot be sustained. In other words, highly super-Eddington accretion flows cannot be magnetically arrested and e ffi -ciently produce magnetically-driven, relativistic jets (although theyare likely to generate radiative jets, see Sikora & Wilson 1981;Narayan et al. 1983; Sa¸dowski & Narayan 2015).If only the companion star provides the large scale poloidalmagnetic flux and this critical accretion rate for stabilization is in-deed significant, then one would expect that X-ray binaries withmost massive BHs and accreting at largest, but sub-Eddington, ac-cretion rates will be most di ffi cult to stabilize. Out of the bestknown BH X-ray binaries, GRS 1915 +
105 is such an example, withmost massive BH and near-Eddington outburst luminosity, and it isindeed the only one unstable. It has to be mentioned, however, thatits companion star is very evolved and presumably significantly af-fected by binary evolution (Fragos & McClintock 2015), and there-fore peculiar within the set of other companion stars. Thus, the es-timate of the magnetic flux available from the companion that wederived basing on Solar and dwarf star magnetic properties may notbe accurate (Stepien 1994).Thermal instability is not specific to BH accretion flows. Simi-lar phenomenon is expected to take place in radiation pressure dom-inated, radiatively e ffi cient disks around neutron stars (NSs). Simi- MNRAS000
105 is such an example, withmost massive BH and near-Eddington outburst luminosity, and it isindeed the only one unstable. It has to be mentioned, however, thatits companion star is very evolved and presumably significantly af-fected by binary evolution (Fragos & McClintock 2015), and there-fore peculiar within the set of other companion stars. Thus, the es-timate of the magnetic flux available from the companion that wederived basing on Solar and dwarf star magnetic properties may notbe accurate (Stepien 1994).Thermal instability is not specific to BH accretion flows. Simi-lar phenomenon is expected to take place in radiation pressure dom-inated, radiatively e ffi cient disks around neutron stars (NSs). Simi- MNRAS000 , 000–000 (0000) agnetic flux stabilizing thin accretion disks larly to the BH case, large scale magnetic field may play importantrole in stabilizing them. In the case of NS systems, however, themagnetic field of the NS itself may provide extra stabilizing e ff ect.In addition, low mass of NSs would suggest small amount of mag-netic flux required for stabilization, relatively easier to provide bythe companion star. Out of all NS systems known, only the RapidBurster showed (twice in 16 years) light curves that resemble thoseof GRS 1915 +
105 (Bagnoli & in’t Zand 2015) what seems to beconsistent with the picture presented in this Letter.Several questions, however, arise. We assumed that the mag-netic flux is provided by the companion star, and the magnetic tubesreaching its surface layers near the substellar point are e ffi cientlydragged into the inner region of the accretion disk. To provide theobserved stability of outbursts of most X-ray binaries, the durationof which is determined by the propagation of viscous instabilitythrough the whole accretion disk and is often of the order of months(Lasota 2001), one would have to make sure that magnetic flux ac-cumulated in the inner region is not canceled out by a flux of op-posite polarity during this period. That would require either that asingle magnetic tube is accreted for a longer time than the outburstduration, or that the net magnetic field of the tubes hover for sucha time at fixed location in the disk having established the advec-tion / di ff usion equilibrium (analytical solutions of the steady-stateradial distribution of poloidal fields were obtained and studied byOkuzumi et al. 2014). The latter may be pre ff ered if thin accretiondisks are indeed ine ffi cient accretors of the magnetic field.One other question is whether it is indeed the magnetic fieldfrom the active regions of the companion star that dominates thelarge scale magnetic structures in the disk. In principle, plasma-related e ff ects may be operating as well. An example is the Con-topoulos battery which can generate poloidal magnetic flux of uni-form polarity as a result of the Poyinting-Robertson radiative drag(Contopoulos et al. 2015). Although field generated in such a wayis instantenously insignificant, given enough time, it could aggre-gate to provide the amount of magnetic flux relavant in the contextdiscussed in this work. We have calculated the minimal amount of large-scale poloidalmagnetic flux that has to thread the inner part of a thin, radiation-over-gas pressure dominated accretion disk to stabilize it againstthermal instability. We have compared that amount with the mag-netic flux that has to accumulate on the BH to magnetically arrestthe disk, and with the maximal magnetic flux of uniform polaritythat can be advected with the gas from the companion star in X-raybinaries. We summarize our findings below, all of which depend onthe assumptions that magnetic field can be advected inward and thatthe magnetic field coming from the companion star dominates thelarge-scale magnetic properties of the inner accretion disk region.(i)
Magnetic flux required for stabilization: – To stabilize theinner region of a thin accretion disk, where radiation pressure dom-inates over thermal pressure, one has to provide net poloidal flux ofthe order of 10 G · cm for ˙ M = . M Edd and 10 M (cid:12) BH (Eq. 6).Such a magnetic field, although weak when compared with the lo-cal gas and radiation pressures, will enhance the magnetization ofthe saturated state of MRI and lead to a magnetic pressure sup-ported, and therefore stable, state. This critical amount of the large-scale magnetic flux grows with BH mass and the normalized ac-cretion rate. This net magnetic field does not have to accumulate on the BH and therefore can be relatively easily advected into theinner region.(ii)
GRS 1915 + – Out of the BH X-ray binaries with wellestablished BH and binary parameters, GRS 1915 +
105 requiresmost magnetic flux, ∼ × G · cm to be stabilized due to itslarge BH mass and luminosity.(iii) Magnetic flux provided by the companion star – In a tidallylocked X-ray binary the gas accreting towards the compact ob-ject can drag magnetic field from the surface layers of the com-panion star. We estimated the amount of magnetic flux containedin magnetic tubes of rapidly rotating stars to be of the order of10 − G · cm .(iv) Critical accretion rate for stabilization – For a given sys-tem, the amount of magnetic flux required for stabilization abovesome critical, near-Eddington, accretion rate is larger than can beprovided by the companion star. Such systems are expected to bethermally unstable (and GRS 1915 +
105 may be an example), un-less their transfer rate exceeds the Eddington rate, in which casethey are stabilized by the advection, and magnetic contribution isno longer required.(v)
Magnetically Arrested Disk – To saturate the magnetic fluxaccumulated at the BH, and to enter the MAD state resulting ine ffi cient jet production, one has to provide a comparable amount ofmagnetic flux ( ∼ G · cm for 1 ˙ M Edd , Eq. 7). This state, however,requires significant accumulation of the magnetic field at the BHthat exerts outward pressure and therefore requires very e ffi cientadvection of the magnetic field, which may not be the case for thindisks. Even if the advection is e ff ective, when the accretion rateexceeds significantly the Eddington rate, the companion star cannotprovide enough uniform magnetic flux to maintain the magneticallyarrested state. Therefore, one should not expect e ffi cient generationof relativistic jets in super-Eddington accretion flows. The author thanks Jack Steiner, Jean-Pierre Lasota, RameshNarayan, and John Raymond for helpful comments. The author ac-knowledges support for this work by NASA through Einstein Post-doctotral Fellowship number PF4-150126 awarded by the ChandraX-ray Center, which is operated by the Smithsonian AstrophysicalObservatory for NASA under contract NAS8-03060.
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