An Observational Signature of Sub-Equipartition Magnetic Fields in the Spectra of Black Hole Binaries
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An Observational Signature of Sub-equipartition Magnetic Fields in the Spectra of Black HoleBinaries
John Wallace and Asaf Pe’er Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel
Submitted to ApJABSTRACTA common assumption used in the study of accretion disks is that the magnetic energy density andthe kinetic energy density should be in equipartition. This assumption relies on the faster growth rate ofthe magnetic field strength against the kinetic energy of the particles in the flow, for decreasing radius,combined with a dissipation mechanism that tends towards equipartition. In this paper, we examinethis assumption by modeling the radio, mm and optical spectra of several black hole binaries in theirquiescent state. We use a standard two-component disk model, consisting of an inner geometricallythick and optically thin disk, emitting thermal synchrotron radiation, along with an outer, thin disk,which radiates as a multicolor blackbody. We find that at the low accretion rates typical of thequiescent state, the spectral shape is qualitatively reproduced using magnetic fields that are between0.1% and 1% of the equipartition value, considerably smaller than previously thought. We discuss ourfindings in view of (1) the launching of jets in these objects, which is commonly believed to rely onthe presence of a strong magnetic field in the central region of the disk; and (2) the role of magneticdissipation in the structure of the inflow.
Keywords:
X-ray binary stars; Jets; Stellar mass black holes; Magnetic fields INTRODUCTIONBlack hole binaries (BHBs) exhibit a number of dis-tinct spectral states, which are classically defined bythe hardness of the X-ray emission and the bolomet-ric luminosity of the system in each state. The differ-ent states are in turn associated with separate modes ofaccretion. These are generally divided into four mainstates as follows, from higher accretion rates to lower:The high/soft state (HSS), very high state/intermediatestate (VHS/IS), low/hard state (LHS) and quiescentstate (QS) (Esin et al. 1997; Zdziarski & Gierli´nski 2004;Remillard & McClintock 2006; Narayan & McClintock2008).The HSS, corresponding to accretion rates ˙ M (cid:38) . M Edd , is a high luminosity state, consisting of a cool,geometrically thin disk extending down to the innermoststable circular orbit (ISCO) (Shakura & Sunyaev 1973;Esin et al. 1997; Remillard & McClintock 2006; Narayan
Corresponding author: John [email protected], [email protected] & McClintock 2008). Such a disk is optically thick andradiates as a multicolor blackbody disk (MCD) from itssurface (Frank et al. 2002).At lower accretion rates, the inner regions of the diskdo not radiate efficiently, resulting in a flow which is ad-vection dominated, hot and optically very thin (Narayan& Yi 1994). The inner regions of the disk begin to “in-flate”, resulting in a roughly spherical flow, extendingfrom the ISCO out to some radius r tr , the truncationradius, where the flow can once again radiate efficientlyand return to the thin disk structure (Narayan et al.1996; Esin et al. 1997; Homan & Belloni 2005). For ac-cretion rates ˙ M (cid:46) . M Edd , the system is said to bein the intermediate state (VHS/IS), where the centralhot flow is still small in size and the flow remains radia-tively quite efficient. This state is also associated witha strong jet (Fender et al. 2004; Narayan & McClintock2008; Fender et al. 2009).As the accretion rate decreases, the inner edge of thethin disk recedes further outwards from the black hole(i.e. r tr increases), resulting in a larger inner flow whichis very hot ( T e ∼ − K). For ˙ M ∼ − ˙ M Edd , a r X i v : . [ a s t r o - ph . H E ] J a n Wallace & Pe’er the system is in the LHS, named for its low luminosityand hard X-ray spectrum. The jet persists in this state,albeit at a lower luminosity compared to the VHS/IS.Below ˙ M ∼ − ˙ M Edd , the system enters the quiescentstate — a very low luminosity state with a weak jet,where the system spends the majority of its time (Bel-loni 2010).One common assumption that is used in the study ofaccretion flows is that there should be equipartition be-tween the magnetic and kinetic energy densities. Thisassumption is made on the basis of scaling and energeticsarguments, whereby the magnetic energy density growsfaster than the kinetic energy, reaching equipartitionquickly (see, for example, Shvartsman 1971; Meszaros1975). Maintaining this equipartition then relies onthe existence of dissipation mechanisms in the disk,which subsequently act to keep the magnetic field atits equipartition value.The validity of this assumption has not been testedin-depth, although it is generally relied on in many ar-eas of accretion disk study (see, for example, Narayanet al. 1996; Melia 1992; Narayan et al. 1997; Quataert& Gruzinov 2000; Yuan et al. 2003; Bower et al. 2005;Beskin & Karpov 2005; Marrone et al. 2007; Kuo et al.2014, among others). An initial attempt at abandon-ing equipartition by Scharlemann (1983) found that do-ing so would introduce an instability to the flow, whichcould lead to an increased dissipation rate of the mag-netic field. The study was carried out for purely spher-ical accretion flows and relied on a number of simplify-ing assumptions and the theory was not developed formore general applications. Kowalenko & Melia (1999)discuss the effect of relaxing the equipartition require-ment, with the aim of incorporating turbulence in theflow into models of the dissipation of magnetic field in aplasma. The authors find that at large radii, the mag-netic field may be suppressed to below its equipartitionvalue, while at small radii, the field may exceed equipar-tition, outstripping the dissipation mechanism. Despitethe finding that removing an equipartition prescriptionmay have important implications for the magnetic prop-erties of the flow and for the heating of particles in thedisk, this avenue of investigation has not been pursuedfurther, to our knowledge.Numerical simulations by Igumenshchev & Narayan(2002); Igumenshchev et al. (2003) find that the mag-netic field may, in fact, be maintained at slightly sub-equipartition and that it is dependent on the nature ofthe field supplied by the inflowing matter. They alsofind that magnetic dissipation has a significant effect onthe structure of the flow, given that strong magneticdissipation would contribute to electron heating, as well as causing turbulence in the flow. It is clear then thatthe assumption of equipartition may have significant ef-fects on how the flow evolves, in terms of heating andmagnetic flux build-up.Of the four main observational states in BHBs, threeof the states (VHS/IS, LHS and QS) are associated withthe formation or presence of jets (Narayan & McClintock2008). In the intermediate and low/hard states, thesejets are powerful and dominate the emission in the ra-dio/optical bands. In the quiescent state, the jet poweris substantially weaker and may no longer outshine thedisk emission. A central part of the study of relativisticjets is to understand the role played by the magneticfield in launching the jets seen in these states (Bland-ford & Znajek 1977; Blandford & K¨onigl 1979; Casella& Pe’er 2009; Tchekhovskoy et al. 2011). It is thereforevital to have a clear understanding of how the magneticfield behaves in the inner regions of an accretion disk.In order to study the magnetic fields within the flow, weturn to the quiescent state, where any contribution of ajet to the spectrum is most suppressed and thus allowsfor closer examination of the properties of the flow itself.Previous studies have considered mainly the higher en-ergy portions of the spectrum, at and above 10 Hz inparticular. This was generally due to a lack of observa-tional data at lower frequencies (around 10 − Hz),where we expect to find synchrotron emission from thedisk. Early attempts at modelling BHB sources usinga two-component accretion disk suggest that the X-rayspectrum is formed by Comptonized disk photons and isfound to be insensitive to the exact magnetic propertiesof the disk (e.g. Narayan et al. 1996, 1997; Esin et al.1997; McClintock et al. 2003). As a result, it was notpossible to constrain the magnetic field of the disk usingthese observations.More recent observations from ALMA, VLA andSpitzer have given more complete coverage of the spec-trum of some BHB sources, in particular at frequenciesbelow the peak of the thin disk emission, which occurs ataround 10 Hz (Narayan & McClintock 2008). Sourcessuch as V404 Cyg, XTE J1118+480 and A0620-00 havebeen observed extensively at these frequencies (Galloet al. 2007; Froning et al. 2011; Bernardini et al. 2016;Din¸cer et al. 2018; Gallo et al. 2019), which makes themideal candidates to test for disk emission.The exact source of the emission across the broadbandspectrum is still the subject of some debate. Most cur-rent models favour a jet-dominated or disk-jet emissionscenario to explain the broadband spectrum of the bi-nary system, since this can explain an observed correla-tion between radio and X-Ray luminosities (Gallo et al.2003; Yuan & Cui 2005), while also providing an expla- quipartition in Black Hole Binaries r tr in order to probe the magnetic properties of the accre-tion disk. We want to find an observational signature ofthe magnetic field within the flow and to this end, weaim to measure the contribution of thermal synchrotronemission from the inner regions of the quiescent diskto the overall spectrum of the source. We apply thismodel to three sources, A0620-00, V404 Cyg and XTEJ1118+480 in their quiescent states, in order to examinethe effect of changing the fraction of equipartition, ε B ,which represents the ratio of the magnetic energy den-sity to the kinetic inflow energy density. We study thedisk component of the emission in the microwave, in-frared and optical portions of the spectrum, which showevidence of some excess emission (Gallo et al. 2007),commonly attributed to a jet or outflow. In particu-lar, we show for the first time that the magnetic fieldpresent in the accretion flow is significantly weaker thanpreviously assumed, which conflicts with current theo-ries of jet-launching and suggests very strong magneticdissipation in the flow. MODEL r tr Figure 1.
Schematic of the black hole and its surroundingaccretion disk, showing the structure of the model used inthis paper. The spherical gray area is the hot, inner flow,while the long, black regions on the outside represent thethin disk. The truncation radius, r tr is the point at whichthe flow switches from thick to thin. Typical values of r tr are 10 − r g . The overall system size is ∼ r g . Our model consists of two components, namely an in-ner, hot flow and and an outer, thin disk (a schematic of this model is shown in figure 1). The inner flow radiatesvia thermal synchrotron and bremsstrahlung emission,while the thin disk is modelled as an MCD. This modelis supported by observational evidence, which suggeststhat the thin disk does indeed retreat from the inner re-gions in the lower luminosity states, particularly in thequiescent state, where there is no evidence for the softblackbody emission at X-Ray wavelengths that would beexpected from the inner regions of a thin disk (Zdziarski& Gierli´nski 2004; Done et al. 2007).2.1.
Thin Disk
The spectrum of the outer disk is constrained by themass of the central black hole in the system and theaccretion rate. For a given frequency ν , the specificemitted flux, F ν , is (Frank et al. 2002): F ν = 4 πh cos( i ) ν c d (cid:90) r out r in r d r exp( hν/k B T ( r )) − , (1)where: T ( r ) = (cid:40) GM ˙ M πr σ (cid:20) − (cid:18) rr s (cid:19)(cid:21)(cid:41) / . (2)Here, i is the inclination of the thin disk to the lineof sight, M is the mass of the central object, ˙ M is theaccretion rate, r s is the Schwarzschild radius and d isthe distance from the source to the observer. G is thegravitational constant, h is Planck’s constant, σ is theStefan-Boltzmann constant and c is the speed of light.The value of M can in general be determined inde-pendently of the spectrum (e.g. by dynamical measure-ment), so for sources where this is the case, it is a fixedparameter of this model.For sources with independent mass measurements, theaccretion rate, ˙ M , is generally constrained by the con-tribution of the thin disk to the spectrum — since thetemperature of the disk is dependent only on mass andaccretion rate. Matching the observed flux can thereforeset the constraint on the mass inflow rate, which is alsoimportant for the emission from the thick disk, since thethin disk feeds directly into the inner flow. The otherparameters that affect the spectrum of the thin disk arethe values of r in , r out and the inclination of the thin diskto the line of sight, i . r in in this case corresponds to the truncation radius, r tr . Typical values of r tr are expected to be a few 10 r g to ∼ r g (Narayan & McClintock 2008). Values of r out for these systems generally lie at around 10 r g , dueto the fact that for an accreting binary system, the in-falling material must come from the companion star andso the size of its accretion disk cannot be larger than theorbital size of the system. Wallace & Pe’er
The value of the inclination is in many cases less cer-tain, although there are some observational constraintsfor a limited number of sources.2.2.
Thick Disk
The contribution of the inner disk is mainly in theform of synchrotron emission, where a thermal popula-tion of electrons is assumed. We tie the magnetic fieldin the flow to the kinetic energy of the infalling materialby assuming that the magnetic energy density is somefraction of equipartition, ε B with the kinetic infall en-ergy (following Marrone et al. 2006; Shvartsman 1971;Melia 1992, for example).The resulting model of the inner accretion flow is asfollows: The radial density profile is assumed to followa power law: n ( r ) = n (cid:18) rr s (cid:19) − β , (3)where here n is the number density at the Schwarzschildradius. The value of β is determined by the structureof the accretion flow. For free-falling gas (Blandford& Begelman 1999), ˙ M ( r ) ∝ r p , and then β = − p .For spherical accretion (Bondi 1952) or an advectiondominated accretion flow (ADAF) (Narayan & Yi 1994), β = , while for a convection-dominated accretion flow(CDAF), β = . In this work, the value of β is taken tobe 1.5, corresponding to an ADAF/Spherical flow.Some fraction of equipartition is assumed betweenmagnetic, kinetic and gravitational energy (e.g. Shvarts-man 1971; Melia 1992). This allows the number densityof particles n e to be tied to the magnetic field strength B . Then making use of the equation for the density asa power law and assuming pure hydrogen in the plasmagives: ρ = m H n (cid:16) rr s (cid:17) − β . Equating the energy densi-ties results in: B π = ε B ρu , where u is the infall speed,with u ∼ c √ rrs (e.g. Narayan & McClintock 2008). Thusthe magnetic field strength is: B = (cid:112) ε B πc m H n (cid:18) rr s (cid:19) − ( β +1)2 . (4)The temperature at the inner edge of the accretionflow is a free parameter of the model, with typicalelectron temperatures of ∼ − K for a two-temperature flow (Narayan & McClintock 2008).We use the angle-averaged synchrotron emissivityfrom Wardzi´nski & Zdziarski (2000): j ν = 2 / π / e n e ν / cK (1 / Θ) v / exp (cid:34) − (cid:18) v (cid:19) / (cid:35) , (5) with Θ = kT /mc and v = ν/ν c Θ , ν c is the cyclotronfrequency. Here, K is the modified Bessel function ofthe second kind (e.g. Abramowitz & Stegun 1970).For thermal electrons, the absorption coefficient isgiven by Kirchhoff’s Law: α ν = j ν /B ν (Rybicki &Lightman 1979). To find the specific intensity, we solvenumerically the radiative transfer equation along theline of sight: d I ν ( s )d s = − α ν ( s ) I ν ( s ) + j ν ( s ) . (6)To calculate the specific flux from a spherical source,(see Rybicki & Lightman 1979, figure 1.6), we get: F ν = π (cid:90) ss I ν ( s (cid:48) ) sin θ c ( s (cid:48) ) d s (cid:48) , (7)where: θ c ( s ) = sin − ( s/d ) (for an observer at distance d ). Thus the final expression for the specific flux is: F ν = πd (cid:90) ss I ν ( s (cid:48) ) s (cid:48) d s (cid:48) . (8) MODELLED SOURCES3.1.
A0620-00
A0620-00 is a soft X-Ray transient source, consistingof a 6 . M (cid:12) compact object and a 0 . M (cid:12) K-type com-panion, with an orbital period of 7.75 hr (McClintock& Remillard 1986; Cantrell et al. 2010). It is one of themost extensively observed BHB systems to date. The in-clination of the system is estimated at: i = (51 ± . ◦ ,while the distance from Earth is d = (1 . ± .
12) kpc(Tetarenko et al. 2016).In constructing the broadband SED of A0620-00, weuse the above parameters for the central black hole,along with an accretion rate of 2 . × − M (cid:12) peryear. This rate is initially based on previous estimates,e.g. Narayan et al. (1996), and chosen to match the thindisk contribution to the spectrum.Furthermore, we adopt an outer disk radius, r out ,of 10 r g , on the basis that the orbital period impliesa system size of around 10 cm, which correspondsto ∼ r g . The value of r tr is varied along a few10 − r g , which is in line with standard choices byother modellers (e.g. ). The value of r tr is less well con-strained and is generally treated as a free parameterof most models. Narayan et al. (1996) offer a possiblemethod for choosing a reasonable value, based on com-parisons with the H α line, although the authors find thatan accurate determination using this method is trickyand maintain r tr as a free parameter.We compare with observations as reported in Galloet al. (2019, and references therein), which give broad- quipartition in Black Hole Binaries ε B in theflow. 3.2. XTE J1118+480
XTE J1118+480 is another soft X-Ray transientsource, consisting of a 7 . M (cid:12) compact object and a0 . M (cid:12) companion, with an orbital period of 4.1 hr. Theinclination of the system is in the range: 68 ◦ < i < ◦ ,while the distance from Earth is d = (1 . ± .
1) kpc(Tetarenko et al. 2016).We compare with observations as reported in Galloet al. (2007, and references therein). Again focusingon explaining the excess of emission observed at OIRfrequencies, which is exhibited by XTE J1118+480 asin A0620-00.Here, the orbital period again implies a system size ofaround ∼ cm, which corresponds to ∼ r g . Thevalue of r tr is chosen to be 5 × r g , along with anaccretion rate of 3 × − M (cid:12) yr − .3.3. V404 Cygni
V404 Cyg consists of a 7 . M (cid:12) compact object anda 0 . M (cid:12) K-type companion, with an orbital period of155 hr (Bernardini et al. 2016). The inclination of thesystem is estimated at: i = (81 ± . ◦ , while the dis-tance from Earth is: d = (1 . ± .
12) kpc (Tetarenkoet al. 2016). The accretion rate in V404 Cyg is less cer-tain, however it can be estimated using the shape of theblackbody spectrum (as discussed in section 2.1). Usingthis procedure to set the accretion rate leads to a valueof 10 − M (cid:12) per year.We adopt an outer disk radius, r out , of 10 r g , asthe orbital period again implies a system size of around10 cm, which corresponds to ∼ r g for a 7 M (cid:12) blackhole. The value of r tr is varied as before, along a few10 − r g .V404 Cyg also displays the excess of emission at OIRfrequencies below the thin disk peak, which we arguecan again be explained as thermal synchrotron emissionfrom the inner flow. The lack of spectral coverage atwavelengths between the radio and the OIR means thatit is not possible to confidently constrain the magneticfield in V404 Cyg’s inner accretion flow, although models with sub-equipartition fields can produce the necessaryexcess emission and so further observations (e.g. withALMA) could place stronger constraints on the field. RESULTSIt is important to note that in each of these cases,we are not attempting a statistical best fit, but ratherto generate a representative spectrum which visually re-produces the main features of each spectrum and illus-trates the magnitudes of the quantities involved. Theseresults should therefore not be taken as an attempt todetermine the exact parameters of the sources in ques-tion, but rather to demonstrate the properties of themagnetic field in ourmodel in a theoretical sense, mak-ing use of values already available in the literature, suchas accretion rate and inclination. -1 ν F ν ν [Hz]SED of A0620-00 in Quiescent State Thin diskSynchrotronTotal Figure 2.
SED of A0620-00, along with observational datafrom Gallo et al. (2019, and references therein). The ex-cess of emission near 10 Hz is well described by thermalsynchrotron from a weakly-magnetized, hot flow. The ra-dio data are not explained by the synchrotron portion ofthe spectrum, suggesting that some contribution from a ra-dio emitting jet or outflow would still be necessary to fullyexplain the observed spectrum of A0620-00.
In the case of A0620-00, we find good qualitativeagreement between our model and the data for a veryweak magnetic field (see figure 2). For the parametersgiven in section 3.1, we find that an accretion disk with atemperature of 6 × K at the inner edge of the accre-tion disk and a truncation radius of 4 × r g can wellexplain the observed excess radiation with an equiparti-tion ratio ε B = 10 − .Figure 3 clearly illustrates that the excess emissionat OIR frequencies is naturally explained by a sub-equipartition model. We disfavour models with valuesof ε B close to equipartition, since these models overpro-duce emission at all frequencies between 10 Hz and10 Hz.
Wallace & Pe’er ν F ν ν [Hz]Variation of ε B in A0620-0010.50.10.010.0010.0001 Figure 3.
The effect of changing the value of ε B , showingthat an equipartition magnetic field overproduces emissionand dominates the contribution from the thin disk. Movingto lower values of ε B results in the characteristic “bump” inthe spectrum at around 10 Hz. Clearly, the value of themagnetic field must be significantly below equipartition inorder to explain the observed excess emission.
The two-component model cannot explain the emis-sion at radio frequencies for any sensible choice of pa-rameters, with a sharp cut-off in the synchrotron emis-sion component occurring at around 10 Hz, meaningthat some contribution from a jet or outflow would haveto be present to account for the lower energy radiation. ν F ν ν [Hz]SED of XTE J1118+480 in Quiescent StateThin diskSynchrotronTotal Figure 4.
SED of XTE J1118+480, along with observa-tional data from Gallo et al. (2007, and references therein).The excess of emission near 10 Hz is once again well de-scribed by thermal synchrotron from a weakly-magnetized,hot flow.
The excess OIR emission in XTE J1118+480 can sim-ilarly be modelled as thermal synchrotron emission com-ing from a weakly magnetized inner flow (see figure 4).Using the values in section 3.2), combined with an innertemperature of 5 × K and a truncation radius of5 × r g , there is good agreement with the observeddata points if we again choose ε B = 8 × − — a veryweak field. ν F ν ν [Hz]SED of V404 Cygni in Quiescent State Thin diskSynchrotronTotal Figure 5.
SED of V404 Cyg, along with observational datafrom Gallo et al. (2007, and references therein). The excessof emission near 10 Hz is well described by thermal syn-chrotron from a weakly-magnetized, hot flow. The parame-ters used in this case are: T = 3 × K, r tr = 3 × r g .The value of ε B is found to be 10 − , which is extremely small,even when compared with that of the previous two sources. In order not to overproduce synchrotron emission inV404 Cyg, we require a value of ε B = 10 − , correspond-ing to a magnetic field strength of around 38 G at 10 r g ,which is far below equipartition and far below the valueof ε B found in the previous two sources (see 5). Indeed,in the case of A0620-00, our model suggests a magneticfield strength of around 2000 G at a distance of 10 r g from the central black hole. For XTE J1118+480, wecalculate a value of 1750 G. We therefore find that themagnetic field in V404 Cyg must be significantly smallerthan that in the other two sources, according to thisanalysis.One possible explanation for this difference betweensources is that the accretion rate in V404 Cyg is muchhigher than in the other two sources ( ˙ M ∼ − vs.˙ M ∼ − ). This higher accretion rate should stillcorrespond to a system in the quiescent state, given thatit represents an accretion rate of around 10 − ˙ M Edd − − ˙ M Edd (see the analysis by Narayan & McClintock2008, for example). It should be noted however, thatthis value does lie on the upper limit of the range ofaccretion rates expected in the quiescent state and sothe case of V404 Cyg may warrant further investigationin order to determine exactly which accretion state it isexhibiting. DISCUSSIONThe observational data for all these sources is com-bined from several different observing campaigns, some-times spanning several years. While all of the datapoints we use correspond to observations of the sourcesin their quiescent states, the nature of BHBs is that theyare variable in a number of parameters, most notably quipartition in Black Hole Binaries M (where the massof the central object is known). This combines with theunderstanding that any variation in the accretion ratewill result in a change in the truncation radius. Themagnitude of this change is not clear, however it is im-portant to bear in mind when assessing these results.Observations of A0620-00 show that the accretion rateappears to remain somewhat stable over time, whichlends weight to the assumption that the overall shape ofthe spectrum will remain largely unchanged over time(as long as it is observed in the same spectral state).Sub-equipartition models provide a good explanationfor the emission from the all three sources, althoughonly in the case of A0620-00 do we have enough datato rule out some of the higher temperature cases. Withthis constraint, the value of the magnetic field falls to aslow as 0.1% of the equipartition value. In all cases, forreasonable choices of parameters, an equipartition mag-netic field would overproduce emission at sub-millimeterwavelengths. As mentioned previously, the lack of si-multaneous data in all of the regions of interest maypresent an obstacle to making firm conclusions, howeversince the accretion rate can be constrained by the thindisk component and since all of the sources show an ex-cess in the OIR region, it is reasonable to assume that itshould exist as a general feature of the spectrum acrosslong time periods.The value of the magnetic field in V404 Cyg was foundto be extremely small, even relative to the small fieldsfound in A0620-00 and XTE J1118+480. Some partof this discrepancy may be explained by the higher ac-cretion rate in V404 Cyg, which lies towards the upperlimit of accretion rates believed to be found in the quies-cent state, at around 0.1%–1% of the Eddington value.This situation may not be well described by the simpletwo-component model used in this work. In reality, thedynamics of the disk are likely to be significantly morecomplex and it stands to reason that the largest devi-ations from this simplified behaviour would occur forextreme values of the quiescent accretion rate.If the disk geometry is significantly altered by a recenttransition from a higher luminosity state (e.g. due to thepresence of a strong jet or outflow, which may disrupt orotherwise alter the inner regions of the accretion flow),numerical simulations of the system may be requiredto constrain the accretion mode and to assess whetheror not the system has fully settled into the quiescentstate. There are some hints that V404 Cyg’s accretion mode and geometry may not be identical to that of theother two sources, given that the accretion rate of V404Cyg is higher, however we require a significantly largertruncation radius in order to fit the blackbody portionof the spectrum. This runs counter to what is usuallyexpected, whereby a higher accretion rate correspondsto a smaller transition radius. This is possibly due tosome effects of state transitions in the system, whichshould coincide with the shutting off of a powerful jet asthe system moves into the quiescent state. Over time, itis also expected that the outer disk will begin to moveinwards, eventually leading to the reappearance of thejet.The large value of the truncation radius used in thiscase will result in the temperature of the inner disk drop-ping quite low. This will have some implications on thestructure of the disk, however the ADAF solution willstill be valid in this region and so should not pose amajor problem for this analysis. The drop in temper-ature does present a larger problem however, which isdiscussed in detail in Wardzi´nski & Zdziarski (2000),namely that as the dimensionless temperature Θ fallsbelow a value of around 0 .
9, the emissivity (equation 5)will tend to overestimate the rate of production of syn-chrotron emission. This issue can be partially alleviatedby the physical argument that the disk emission shouldbe largely dominated by emission from the innermost re-gions, where the temperature, density and magnetic fieldare highest. As such, ignoring the effects of equation 5beyond around 10 r g should not be a drastic depar-ture from the true value. Direct numerical solutions ofthe synchrotron emission, based on numerical disk sim-ulations, will provide a more comprehensive analysis ofextreme cases such as this, without relying on analyticalapproximations of the emissivity to the same degree.The low value of ε B can, however, be verified by com-paring the total synchrotron power of V404 Cyg withone of the other sources. Given that: P tot ∼ nB T (9)(e.g. Wardzi´nski & Zdziarski 2000), comparing the rele-vant quantities in V404 Cyg and A0620-00, we see thata value of ε B = 10 − in V404 Cyg produces: P V P A = n V n A B B T t ∼ . . (10)Plotting the observed fluxes together (see figure 6) in-dicates that the expected power in our calculations iscorrect as expected by standard synchrotron theory.Indeed, we find agreement with the calculations ofDallilar et al. (2017), who find a magnetic field of 33 Gfrom synchrotron fitting, which matches almost exactly Wallace & Pe’er ν F ν ν [Hz]Observations of A0620-00 and V404 CygniV404 CygniA0620-00 Figure 6.
Comparison of the observed fluxes in V404 Cyg(triangles) and A0620-00 (circles), which shows that the dif-ference in synchrotron flux is a factor of a few. The mostrelevant area of comparison based on our calculation is thepoint around 10 Hz‘’, which comes from the synchrotronemission in the inner flow. The higher energy emission comesfrom the outer thin disk in our model, while the lower energyemission is likely to come from a jet or outflow. with the 38 G we find. The authors also agree that thesource is likely to be in a non-equipartition configura-tion. Similarly, Jana et al. (2020) find a magnetic fieldin V404 Cyg of 90–900 G, although their analysis dealswith the outburst phase of the source and uses a slightlydifferent model, in which the thin disk extends all theway to the ISCO and is surrounded by the opticallythin region, rather than replaced by it. In any case, theimportant finding is that the magnetic field strength islikely to be low in this source.A sub-equipartition magnetic field suggests that anydissipation mechanism of the magnetic field is likely tobe quite strong, providing significant heating to the ac-cretion flow. Given that the magnetic field grows fasterthan the kinetic energy, such a dissipation mechanismmust exist and furthermore, must operate efficiently.There are numerous potential instabilities that can existwithin an accretion disk, many of which will work to dis-sipate the field. Previously, the assumption of equipar-tition relied on these mechanisms favouring a magneticfield at its equipartition value and working constantly tomaintain such a scenario. If it is the case that magneticfields do not tend towards equipartition, then there aresome consequences that should be considered.Firstly, if the value of the ε B is not observed to beconstant over a large number of sources, it is possiblethat the dissipation of magnetic field follows a randomprocess. This is certainly possible, given the turbulentnature of accretion flows, although it is not clear that itwould be energetically favourable.If, on the other hand, ε B is observed to follow a clearrelation across different accreting sources, then the fun- damental idea of an equipartition assumption need notchange. That is, if the magnetic field generally tendsto a particular value of ε B , then the notion of a dissi-pation mechanism within the flow acting to maintain aparticular magnetic configuration would still hold true,albeit for a significantly smaller magnetic field strengththan previously assumed. The energetics of such a sit-uation may suggest the presence of another balancingquantity, but such considerations are beyond the scopeof this paper.If the accretion disk cannot, or rather does not sustaina large magnetic field, this could pose a problem for thelaunching of jets from the inner accretion flow, which isgenerally understood to require the presence of strongmagnetic fields. Given that the two-component modelin this paper cannot adequately explain the observedflat spectrum at low energies, which is characteristic ofa self-absorbed jet or outflow, it seems likely that someform of jet is present to account for this emission. Thisjet would have to be quite weak, with a break coming atrelatively low frequencies, to avoid contaminating thedisk contribution. Avoiding sub-equipartition fields inthe accretion flow possibly requires substantial mass lossbetween the outer edge of the disk and the innermostregions, resulting in a significantly lower accretion rateat the horizon of the black hole, compared to that at r out .It is unclear whether this configuration is unique tothe quiescent state, since it is not possible to directlyprobe disk emission in the LHS and there is no thick diskcomponent in the HSS. Discontinuities in the magneticfield structure across states do not seem likely, giventhe more-or-less continuous behaviour of other systemparameters between states, however the influence of ajet on the overall magnetic structure of the system (andvice-versa) is still not a settled question and may providesome clues as to what happens between states. CONCLUSIONSIn this paper, we have modelled the SEDs of 3 BHBsources in order to examine the common assumption ofequipartition between the magnetic and kinetic energiesin the accretion flow. We use a two-component model,consisting of a hot, optically thin, geometrically thickinner flow, surrounded by a cold, optically thick, geo-metrically thin outer flow to reproduce the main spec-tral features of the sources (we do not perform statis-tical fits for this purpose). We find that an excess ofemission observed at OIR frequencies can be explainedas thermal synchrotron emission coming from the hotinner region of the accretion flow. We argue for a mag-netic field that is sub-equipartition, with values of ε B quipartition in Black Hole Binaries − in the caseof V404 Cyg. This is much lower than often assumedand has implications for the mechanisms of dissipationof magnetic fields within accretion flows, as well as thelaunching of jets that are often associated with BHBsources. In particular, many current jet models rely ona large magnetic field to power the jet. The absence ofsuch a large magnetic field could prove problematic toa number of current theories surrounding jet-formation,which warrants further investigation to reconcile thesepoints.This result underscores the importance of OIR ob-servations of BHBs, which can provide important con-straints on the strength of the magnetic field strengthin the inner accretion flow. Application of this model tofurther sources in the quiescent state may reveal whetherthe low magnetic field holds across a variety of sources,or if there is significant variation in magnetic regimesbetween sources. In either scenario, there are impor-tant implications for our understanding of the role ofthe magnetic field in launching jets.Numerical studies of weakly-magnetized disks shouldprovide insight into whether such configurations are con-sistent with jet-launching. In addition, numerical stud-ies may provide a differentiator between the LHS andQS, which are sometimes understood to be close rela-tives, due to their structure (large, thick, inner disk,surrounded by outer thin disk). If the magnetic con- figuration in the QS proves to be different from that inthe LHS, it may account for the difference in jet powerobserved in each state. If the LHS is well explained witha low magnetic field configuration as in the QS, it maybe that the role of the magnetic field in powering the jetis less important than previously believed.ACKNOWLEDGMENTSThis paper makes use of the following ALMA data:ADS/JAO.ALMA Facilities:
ALMA, CTIO:1.3m (ANDICAM), CXO,Spitzer, VLAREFERENCES