EDP Sciences
Free Access
Issue
A&A
Volume 603, July 2017
Article Number L4
Number of page(s) 4
Section Letters
DOI https://doi.org/10.1051/0004-6361/201730749
Published online 13 July 2017

© ESO, 2017

1. Introduction

Black-hole X-ray binaries (BHXB) always exhibit a compact radio jet when they are in one of three spectral states: quiescent, hard, and hard intermediate (Fender et al. 2004, 2009; Fender & Gallo 2014; Gallo et al. 2014). For a classification of the spectral states of BHXB see Belloni et al. (2005).

For the formation of the jet, two mechanisms have been proposed: plasma gun/magnetic tower (Contopoulos 1995; Lynden-Bell 1996) and centrifugal driving (Blandford & Payne 1982). Both of them require a strong, large-scale, poloidal magnetic field. Such a magnetic field can either originate from a large distance from the black hole and the advecting flow carries it to the inner region and amplifies it (Igoumenshchev 2008; Lovelace et al. 2009; Tchekhovskoy et al. 2011), or it can be produced locally by the Cosmic Battery (Contopoulos & Kazanas 1998; see also Contopoulos et al. 2006, 2009, 2015; Christodoulou et al. 2008). We favor the Cosmic Battery, because jets in BHXB are destroyed and re-created within hours, when the sources cross the so-called jet line (Fender et al. 2004; Miller-Jones et al. 2012). We consider it highly unlikely that the sources anticipate the destruction of their jet and its subsequent re-formation so as to “request” a magnetic field from far away, which should arrive at the inner part of the flow at the time that it is needed. Instead, we think that the observations require the strong poloidal magnetic field to be produced locally, at the right place and the right time. The formation and destruction of jets in the context of the Cosmic Battery, as well as the relevant timescales, have been discussed in Kylafis et al. (2012). An explanation of the rich phenomenology during an outburst of a BHXB has been offered by Kylafis & Belloni (2015a,b).

The spectra of BHXB from the radio to the near infrared are flat to slightly inverted, i.e. flux density Sννα, with 0<~α<~0.5 (Mirabel & Rodriguez 1999; Fender et al. 2000, 2001; Fender 2001; Russell et al. 2006; Corbel et al. 2013; Russell & Shahbaz 2014; Tetarenko et al. 2015). A characteristic frequency in this spectrum is the break frequency νb, where the partially optically thick jet becomes optically thin. This is an important frequency, because Sνb is an indication of the total power emitted by the jet. The frequency νb varies from source to source (Russell et al. 2013a) and also for the same source as a function of time (Russell et al. 2013b, 2014). Above νb, the jet is optically thin and its spectrum falls with an index −1 ≤ α ≤ −0.5 (Fender 2001).

In 1979, synchrotron radio spectra were calculated for both, a Maxwellian distribution of electrons (Jones & Hardee 1979) and for a power-law one Blandford & Königl (1979). In subsequent years, the power-law model became the standard one and the research efforts concentrated on explaining how a power-law distribution of electron energies can be produced.

Two mechanisms can produce power-law distributions of electron energies: shocks (Heavens & Meisenheimer 1987; for a review see Drury 1983) and magnetic reconnection (Spruit et al. 2001; Drenkhahn & Spruit 2002; Sironi & Spitkovsky 2014; Sironi et al. 2015; for a review see Kagan et al. 2015). The question then arises: are shocks and/or magnetic reconnection guarranteed to be present in the entire jet, from its base (where the frequency νb is determined) to the top? One can envision shocks, due to an uneven flow in the jet, and magnetic reconnection, due to partially turbulent magnetic fields, but it is hard to imagine that these mechanisms operate in the entire jet.

In recent years, it has been accepted (Fender 2006; Fender & Gallo 2014; Kylafis et al. 2012; Kylafis & Belloni 2015a,b) that the jets in BHXB originate in the geometrically thick, optically thin, hot, inner flow around black holes, where the temperature of the electrons is large (hundreds of keV; Ichimaru 1977; Narayan & Yi 1994, 1995; Abramowicz et al. 1995; Blandford & Begelman 1999; Narayan et al. 2000; Qataert & Gruzinov 2000; Yuan et al. 2005). Outside the hot flow, the accretion disk is radiatively efficient and geometrically thin, i.e. Shakura-Sunyaev-type. The transition radius between the two types of flow decreases with increasing mass accretion rate and the Shakura-Sunyaev disk extends all the way to the inner stable circular orbit at high accretion rates, when the sources are in the so called soft state and no jet is present (Fender et al. 1999; Russell et al. 2011).

In this picture, it is natural to expect that the electrons in the jet, at least at its base, should be thermal or close to thermal. Therefore, even out of curiosity, we examine what type of radio spectrum is produced by a thermal distribution of electrons in the jet.

As mentioned above, most of the theoretical work on the radio emission from jets has assumed a power-law distribution of electron energies (see however Falcke & Markoff 2000, who considered also a thermal jet model for Sgr A*). An extensive study of jet radio spectra using a power-law distribution was done by Kaiser (2006). An also extensive study, using both power-law and thermal distributions, was done by Pe’er & Casella (2009). In order to obtain analytic results, both of these studies calculated spectra in the direction perpendicular to the jet (θ = π/ 2). However, as we show below, the spectral index α depends strongly on the observation angle θ.

In this Letter we consider a simple jet model, compute the radio spectrum as a function of θ, and demonstrate that two completely different electron energy distributions (thermal and power-law) result in similar spectra. In subsequent work, we will explore different models to see if it is possible to infer the electron energy distribution from observations.

In Sect. 2 we describe our model, in Sect. 3 we compute the radio emission from the jet, in Sect. 4 we remark on some aspects of our calculations, and in Sect. 5 we present our conclusions.

2. The model

2.1. Characteristics of the jet

As a demonstration, we assume a parabolic jet, which has been used extensively before (see Sect. 4). In other words, we assume that the radius of the jet as a function of distance from the center of the compact object is (1)where R0 is the radius at the base of the jet and z0 is the height of the base of the jet.

For simplicity, we assume that the jet is accelerated close to its launching region and that it has constant velocity v = 0.8c. From the continuity equation we infer that the number density of the electrons in the jet as a function of distance is (2)where n0 is the number density of the electrons at the base of the jet.

For the magnetic field in the jet, we also make the simple assumption that it is nearly parallel to the z-axis and that its strength is determined by flux conservation along the jet: B(z)πR2(z) = const. This implies that (3)where B0 is the strength of the magnetic field at the base of the jet.

2.2. Electron energy distribution

For the energy distribution of the electrons in the rest frame of the jet, or equivalently for the distribution of the Lorentz factor γ since Ee = γmec2, we assume either a relativistic Maxwellian, i.e. a Maxwell-Jüttner distribution, (4a)where , Θ = kTe/mec2, and K2 is the modified Bessel function of the second kind, or a power-law distribution (4b)in the range γminγγmax.

An analytic treatment of cooling in a jet was described in Kaiser (2006). For the values of the parameters that we use, the synchrotron timescale at, say z = 10z0, is about 1 s, while the flow timescale there is 0.003 s. Thus, synchrotron cooling can be neglected. For simplicity, we also neglect diabatic expansion cooling, because γ(z) ∝ (z0/z)1/3 (Pe’er & Casella 2009).

In Fig. 1 we plot the Maxwell-Jüttner distribution for Θ = 0.4 as a line with stars and a power law distribution with p = 4 from γmin = 1 to γmax = 10 as a line with diamonds. Both distributions are normalized to unity. Despite the fact that the distributions ot tesimptqmonds">(4rs an">(4ns ar-eql citavetely differens, in the range Θ = kTe/mec2p = 4γmin = 1, γmax = 10 rtr>Opeion witDEXTER"> >

Thl distribution of electronm as a function of zγ is tion where ne(z> in ginmptqmods"2)n, and (γ> in ginmptqmonds">(4r fo">(4et.

">3. Raitavele traftter

is the Readsk is>Refrized tReigan et al. 2013Glliniulos et al. 2004Glliniulo. 2005; Kylafis et al. 20082015, 2611)f the equatioy for the trafttpe of radiphottrons in the jeinhe directioe n alon, whicltrength imeasufered bd > in ginmptqn where (&57;,s7;>(&57;,s7;>

Fomildtela relativistie magneti: plas>)f thgyrosyynchrotron emissihe forlisrum is thgporoprediatw one. For simplicity>whert we ute thextremela relativistie forlisrns, i, whice the emissiocoy efficieum ie (semods"6.36)pe oRybickoni &Lheigelman 1979n where is t A chgure of the electrn>, &6647; is t esparch angle of the electrn>, where is the modified Bessel function of the second kib).

Thdbsorumptiocoy efficieum ie (semods"6.50)pe oRybickoni &Lheigelman 1979n ; for f distthe obserr,ce is where bb νb2009; fon otheg possiblA characteristic frequeies)et, which is determined bm the cdimption than the jet ha, opticn> dwitg-eqlel td one at its bais 0(1)where R0 is the radiue of the jet at its baie (semods"1))st. Thimeansct that the entire jet is optically th. D due te coexamanation of the radio spectrn>, νb < < E emissiocoy efficieue (&57;,i>z>zν = 1)12. Tht electron energy distributionSha:nd therma(h sta),he power-lan with ˩ ≤= 10 rtr>Opeion witDEXTER"> >

3. Radio emission from the jet

<. For thpurropoiue os a demonstraveur calculatin>, we aaliis>Refcienhe valuey to the volervee paramete s. Thus, we make R0 = 04; Rb, where Rb =0.5, z0 54; Rbe n0 = 1)1e B0 2i> 215; = 1)1, Θ = 0.4e p = 4, γmin = 1, and γmax ; "b>). For thesparch angl, we makr the val>, &6647; 3 10

For ths>Refcienhe values of the parameter, wf kiy, usinmods"10,b) tha, νb 2.7i> 215; = 1)νb = 1) to γmax = 10νb =.6i> 215; = 1)).

In Fig. 1 we shoe the emissiocoy efficieue (&57;,i>z>z; fok frequency ν = 1)12. Thsym="bonSha:nh stars (therm)n, anpl Rusaiep power-la. If 4 we distctor the power-law distributio> to γmax = 101γmin = 1γmax = 10 We havsolervenumeetricallexe prissio(9)n, a,le for ths>Refcienhe values of the parameteus, we have found ave lloiewine valueo for the spectral indet: α =1)o f>kkθ =1)). Fo (θ = π/ 2α = 0i>α = 0i>)f thlleasd ong ine gratmenn witGlliniulor (2005 to α = 0i>2. The val>n of αθ =1) to θ = π/ 22. The rediction of αθ&1647>α 20 ± 0.05 ≤.05Also,le there inois neal to examinr direction, &6960;/ b>&l">θπ

2709; for a revin>).ereus, we havd demonstradas thag t thermal jei>As an celbulend “corona”n for thups Battolins os sofphottro4). Ie fa,a; Couttizdictins in the jee can explais noonteln the observee power-lat hi-n energe spectral index Eb, bud also)th thhepedcienht of the timlagrk oFouriefok frequenc(Reigan et al. 2013),)th thnarroiewint of thautre rrelquatioy functios with increasinphottron energ(Glliniulos et al. 2004)c)th thhependcienht ox 2008, and)or ths>lquatios betweef Eb2015As an examp4). Ie fa,ar > it wan one of the Rstietn fochoossting afis examp4).

Conclusioon

Contrary to common belict that ths flat to slightly inverted radio specaon from jet simyke a power-law energy distribution of the electrons if the jet, we havd deonstrada where thag t thermaw distribution of electron energien produce Besnartiallidesepticnr radio spectra.

Wf kiyitrd intquesting ajeo the index α2014
). It ito tonearle to infer that the energy distribution of the electronalintmanesos between thermal and power lds, but it iicentgusties.

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Refeble fon ua/fto cotmeoal ansuggquestis>). Wd alse thinD sitcuritGlliniuloe fon ua/ftd discuctioal ano cotmeo.p? Onn ou i(NDK)se this TomRstp; Bello, Rob>; Fend, Serap; Markon>, anD hav; Russele fon ua/ftd discuctioes.

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