MNRAS 000,1–16 (2015) Preprint 24 February 2021 Compiled using MNRAS LATEX style file v3.0
Particle acceleration and magnetic field amplification in massive young stellar object jets
Anabella T. Araudo,1,2? Marco Padovani,3 and Alexandre Marcowith4 1ELI Beamlines, Institute of Physics, Czech Academy of Sciences, 25241 Doln´ıBˇreˇzany, Czech Republic 2Astronomical Institute, Czech Academy of Sciences, Boˇcn´ıII 1401, CZ-141 00 Prague, Czech Republic 3INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi, 5 - 50125 Firenze, Italy 4Laboratoire Univers et Particules de Montpellier (LUPM) Universit´eMontpellier, CNRS/IN2P3, CC72, place Eug`ene Bataillon, 34095, Montpellier Cedex 5, France
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT Synchrotron radio emission from non-relativistic jets powered by massive protostars has been reported, indicating the presence of relativistic electrons and magnetic fields of strength ∼0.3−5 mG. We study diffusive shock acceleration and magnetic field am- plification in protostellar jets with speeds between 300 and 1500 km s−1. We show that the magnetic field in the synchrotron emitter can be amplified by the non-resonant hybrid (Bell) instability excited by the cosmic-ray streaming. By combining the syn- chrotron data with basic theory of Bell instability we estimate the magnetic field in the synchrotron emitter and the maximum energy of protons. Protons can achieve maximum energies in the range 0.04 − 0.65 TeV and emit γ rays in their interaction with matter fields. We predict detectable levels of γ rays in IRAS 16547-5247 and IRAS 16848-4603. The γ ray flux can be significantly enhanced by the gas mixing due to Rayleigh-Taylor instability. The detection of this radiation by the Fermi satellite in the GeV domain and the forthcoming Cherenkov Telescope Array at higher energies may open a new window to study the formation of massive stars, as well as diffu- sive acceleration and magnetic field amplification in shocks with velocities of about 1000 km s−1. Key words: acceleration of particles – radiation mechanisms: non-thermal – shock waves – stars: jets – gamma-rays: general
1 INTRODUCTION Carrasco-Gonz´alez et al.(2010) reported on polarized radio emission and confirmed that the radiation is produced by the Stars are formed within dense molecular clouds, accreting synchrotron process. The detection of synchrotron radiation matter onto the central protostar with the formation of a is an evidence that there is a population of mildly-relativistic circumstellar disc and bipolar jets. These ejections are col- or relativistic electrons in the jet. limated flows of disc/stellar matter accelerated by magnetic The cosmic-ray (CR) energy density in molecular clouds field lines (Blandford & Payne 1982; Shu et al. 1994), and −1 is poorly known, but it should be at least of the same or- moving with speeds vj ∼ 300 − 1500 km s into the ambi- arXiv:2102.11583v1 [astro-ph.HE] 23 Feb 2021 der of the CR energy density in the interstellar medium, ent molecular cloud. Molecular matter from the cloud is en- i.e. ∼ 10−12 erg cm−3 (e.g. Ferri`ere 2001)1. Therefore, CRs trained by the jet, forming molecular outflows. Protostellar with energy densities larger than 10−8 erg cm−3 (see Sect.3) jets are thermal radio emitters in most of the cases (Anglada have to be locally accelerated at the jet termination shocks et al. 2018). However, radio emission with negative spectral or/and in the internal jet shocks. This result is in agreement indices has been detected in several protostellar jets suggest- with Padovani et al.(2015, 2016), who have shown that the ing a non-thermal nature of the emission (e.g. Garay et al. large ionization rate estimated from molecular line ratios in 2003; Rodr´ıguez-Kamenetzky et al. 2017; Purser et al. 2016; L1157-B1 (Podio et al. 2014) and OMC2 FIR4 (Ceccarelli Obonyo et al. 2019). In the particular case of the jet as- et al. 2014; Fontani et al. 2017; Favre et al. 2018) can be sociated with the massive protostar IRAS 18162-2048 and its famous Herbig-Haro (HH) objects HH 80 and HH 81,
1 Unless a strong source of CRs contributes to a local enhance- ment, this number is actually an upper limit as CRs are subject ? E-mail: [email protected] to strong ionization losses while propagating in molecular clouds.
© 2015 The Authors 2 A. T. Araudo et al. due to local acceleration of thermal particles to a relativis- Table 1. Non-thermal MYSOs selected from the observational tic regime in protostellar jets. studies carried out by Purser et al.(2016) and Obonyo et al. Diffusive Shock Acceleration (DSA) is the most estab- (2019). From left to right we list the name of the source in the lished mechanism to accelerate particles in sources where Green and IRAS catalogues, the distance d, and the ionized mass shock waves are present, from solar flares (e.g. Chen et al. loss rate M˙ i = XM˙ j, where X is the ionization fraction and M˙ j is 2015) to the outskirts of galaxy clusters (e.g. Petrosian the total jet mass loss rate provided in the literature mentioned 2001). Magnetic turbulence near the shock allows particles before. The values of X and jet velocity fixed by Purser et al. ˙ to diffuse back and forth the shock front gaining energy at (2016) and Obonyo et al.(2019) to compute Mj are X = 0.2 and 0.4, and v = 500 km s−1. The jet kinetic luminosity computed as each crossing cycle (Axford et al. 1977; Krymskii 1977; Bell j L = M˙ v2/2 is also listed. 1978a; Blandford & Ostriker 1978). The maximum energy kin i j that particles can achieve in non-relativistic parallel shocks Source IRAS d M˙ i XLkin −1 −1 is usually determined by radiative losses or advection escape name [kpc] [M yr ] [erg s ] from the acceleration region. Araudo et al.(2007) showed G263.7434 08470-4321 0.7 6.4×10−8 0.2 3.3 × 1033 that completely ionized and fast protostellar jets can accel- G263.7759 08448-4343 0.7 1.3×10−7 0.2 1.1 × 1034 erate particles up to TeV energies if Bohm2 diffusion applies. G310.1420 13484-6100 5.4 1.1×10−6 0.2 9.2 × 1034 However, in a non-completely ionized medium, damping of G313.7654 14212-6131 7.8 4.1×10−6 0.2 3.4 × 1035 −6 35 Alfv´en(or magnetohydrodynamic) waves by ion-neutral col- G339.8838 16484-4603 2.7 5.3×10 0.2 4.4 × 10 G343.1261 16547-4247 2.8 3.5×10−6 0.2 2.9 × 1035 lisions can reduce the particle maximum energy (Drury et al. −7 34 1996; Padovani et al. 2015, 2016). G114.0835 23262+5834 4.2 4.4×10 0.4 3.3 × 10 The detection of molecular emission lines such as Hα, [NII], [SII], and [OIII] in HH objects indicate that pro- tostellar jets are not completely ionized. Emission lines (2019) found that 6 of the 15 massive young stellar ob- are usually associated with internal (low-velocity) shocks jects (MYSOs) observed with the Jansky Very Large Ar- in jets emanating from low-mass protostars. However, in ray (JVLA) show clear evidence of non-thermal emission, sources like HH 80 and HH 81 molecular lines are co-spatial with spectral indices α > 0.42. From this last sample we se- with synchrotron radiation of locally accelerated electrons lect the source IRAS 23262+5834. In Table1 we list the (Rodr´ıguez-Kamenetzky et al. 2019a). Therefore, strong MYSOs selected for our study. They are located at a dis- shocks and scattering waves are present in non-completely tance 0.7 ≤ d/kpc ≤ 7.8 from Earth. The jet ionized mass-loss −8 −1 −5 ionized plasmas. As a result, short wavelength Alfv´en waves rate is 6 × 10 ≤ M˙ i/(M yr ) ≤ 5 × 10 and the kinetic 33 −1 35 are unlikely to scatter particles back and forth the shocks luminosity is 5.3 × 10 ≤ Lkin/(erg s ) ≤ 4.4 × 10 . due to their damping by ion-neutral collisions. Bell(2004) re- The paper is organized as follows: in Sect.2 we describe alized that under certain conditions non-resonant (hereafter the physical properties of protostellar jets and the condition NR) hybrid3 waves can grow faster than Alfv´en(resonant) for them to be non-thermal emitters. In Sect.3 the magnetic waves. The NR (Bell) instability has two main advantages: field and the content of relativistic particles is inferred from (i) the magnetic field is amplified by orders of magnitude and synchrotron emission. In Sect.4 we discuss both the resonant (ii) NR waves are not strongly damped by ion-neutral colli- and non-resonant instabilities in the context of young stellar sions, even if the maximum growth rate is reduced (Reville jets4. In Sect.5 we briefly describe the non-linear saturation et al. 2007). of the amplified magnetic field and determine its strength. In this study we show that the magnetic field in the In Sect.6 we compute the maximum energy of protons and synchrotron emitter of high-mass protostellar jets can be in Sect.7 we estimate the γ-ray flux from our source sample. amplified by the streaming of CRs (Bell 2004), as in super- In Sect.8 we present our conclusions. Gaussian-cgs units are nova remnants (e.g. Vink & Laming 2003) and jets in Active used throughout the paper. Galactic Nuclei (Araudo et al. 2015). By assuming that the number of non-thermal electrons and protons is the same, we find that the energy density in non-thermal protons is large 2 PROTOSTELLAR JET PHYSICAL PROPERTIES enough to drive the Bell instability in the termination region The detection of thermal (free-free) and synchrotron emis- of protostellar jets. In addition, we show that detectable lev- sion from protostellar jets indicates that (i) there are shocks els of gamma rays in the GeV domain are expected from pro- that heat up the plasma and accelerate particles, (ii) the tostellar jets. We consider the sample of non-thermal lobes in regions in the jet where thermal and non-thermal emis- the termination region of high-mass protostellar jets stud- sion is radiated are at least partially ionized, and (iii) mag- ied by Purser et al.(2016) with the Australian Telescope netic fields of about 1-10 mG are needed to explain the de- Compact Array (ATCA) facility. We select the sources with tected synchrotron flux if equipartition arguments apply (see radio spectral index 0.3 < α < 0.8, where the radio flux den- −α Sect. 3.1). In this section we provide some constraints on the sity at frequency ν is Sν ∝ ν . In addition, Obonyo et al. temperature Tu, the ionization fraction Xu, the ion density ni, and the magnetic field Bu upstream of the shock of speed vsh by using the most updated data available in the liter- 2 The Bohm diffusion regime is obtained when the mean-free path ature (see Table2) and conservation equations. We define of the particle imposed by angular scattering is equal to its Lar- Xu = (ne +ni)/nj, where nj = ni +ne +nn, ne and nn are the free mor radius. 3 The term hybrid comes from the nature of the instability which can be described using both magnetohydrodynamic and kinetic 4 We note that this section is plasma-physics oriented. It is writ- theory. ten is a self-contained way. Unfamiliar reader can skip it.
MNRAS 000,1–16 (2015) MFA in jets from MYSOs 3 electrons and neutral densities, respectively, and we assume knots (and HH objects) are thermal emitters (however, see ne = ni hereafter. Rodr´ıguez-Kamenetzky et al. 2017; Osorio et al. 2017, for some exceptions). Non-thermal emission has been detected in the outer 2.1 Ionization fraction and jet temperatures lobes of a handful of protostellar jets, which is more con- sistent with the presence of a strong adiabatic shock at the Bacciotti & Eisloffel(1999) estimated the ionization frac- ¨ location of the non-thermal emitter where synchrotron emit- tion, X, in a sample of low-mass protostellar jets. They found ting electrons are accelerated. In the jet termination region, that X generally decreases almost monotonically along the where the jet impacts against the external medium (see jet, hence the effects of the shocked materials over X seems to Fig.1), the bow shock moves into the molecular cloud at be minor and we may identify X with X . Typical values of X √ u u a speed v ∼ v /(1+1/β χ), where χ ≡ n /n is the jet (n ) between 0.02 and 0.4 have been derived by the authors. Mau- bs j j mc j to molecular cloud (n ) density contrast, and β ≡ R /R , rri et al.(2014) analyzed Hubble Space Telescope arc-second mc h j being R and R the radius of the jet beam and head, respec- images of the DG Tau jet, hence at a much closer location j h tively (e.g. Raga et al. 1998; Hartigan 1989; Chernin et al. to the central source than Bacciotti & Eisloffel(1999). They ¨ 1994). The reverse shock (or Mach disc) in the jet moves at found a low ionization fraction (≈ 0.02) at the base of the vrs = v − 3v /4. In “heavy” jets (χ > 1), the bow shock is jet, then increasing to a plateau value of ≈ 0.7 at a distance j bs faster than the Mach disc5, whereas in “light” jets (χ < 1), of 300 from the central source. The ionization fraction seems the reverse shock is faster than the bow shock. In particular, then to decrease again. By modeling the optical [OII]/[OI] √ vrs ∼ vj and vbs ∼ vj χ when χ 1, whereas vbs ∼ vj when line ratio in shocks at the base of the HN Tau jet, Har- −1 χ 1. As an example, vrs and vbs are ∼750 and 250 km s , tigan et al.(2004) find Xu ≈ 0.7. Te¸sileanu et al.(2012) de- −1 veloped a magnetohydrodynamic (MHD) model of low-mass respectively, when vj = 1000 km s and χ = 0.1. Proper −1 protostellar jets and adapted it to the source RW Aur. They motions of about 300 − 1400 km s have been observed in protostellar jets (e.g. Marti et al. 1993; Masqu´eet al. 2012), found that Xu ≈ 0.1 − 0.2 in the central spine correctly fits the surface brightness fluxes in several forbidden lines. They and therefore we can expect jet velocities comparable to or attributed this ionization fraction to irradiation by X-rays larger than these values. from the protostellar magnetic corona. The detection of GeV photons from classical novae The temperature of the unshocked gas is difficult to (Ackermann et al. 2014) indicates that radiative shocks −1 constrain as line brightness or line ratios are only weakly with velocities ≤ 1000 km s can accelerate particles dependent on it. Garcia et al.(2001) investigated low-mass up to ∼10 GeV (see e.g. Vurm & Metzger 2018). The MHD jets launched from accretion discs including ambipolar present study concentrates, however, on adiabatic (i.e. non- diffusion heating and several sources of ionization (X-rays radiative) shocks. The condition for a shock with velocity and UV radiation as well as collisions). They found a typical vsh to be adiabatic is q ≡ lth/Rj > 1 (Blondin et al. 1990), 4 where we define R as the radius of the section of the jet at jet temperature Tu ≈ 10 K. j the position of the hotspot. The thermal cooling length is In high-mass protostars both Xu and Tu are barely known or even constrained. Following Garcia et al.(2001) we −1 9 lth 16 nu vsh 2 can reasonably expect the jets from high-mass protostars to ' 6.9 × 10 (1) cm 104 cm−3 1000kms−1 have higher mass loss rates and to be denser and cooler than −4 (e.g. Raga et al. 2002), where n is the density upstream of jets from low-mass protostars. Values of Xu as small as 10 u 4 the shock. The condition q > 1 can be rewritten as v > v , are found for jets with Tu ≈ 10 K. However, other sources of sh sh,ad heating like turbulence, MHD wave dissipation, magnetic re- where 2 connection, or weak shock heating have not been considered 2 9 vsh,ad nu 9 Rj in this work. In a very recent study, Fedriani et al.(2019) ' 650 . (2) km s−1 104 cm−3 1016 cm found that the ionization fraction in the jet of the high mass protostar G35.20-0.74N is ∼ 0.1, similar to the values for The reverse- to bow-shock cooling parameter ratio is −3.25 low-mass protostars. Purser et al.(2016) assumed Xu = 0.2 qrs/qbs = lrs/lbs ∼ χ indicating that the termination re- for jets powered by high-mass protostars. For our study we gion of light jets (χ < 1) is composed by an adiabatic Mach 4 select 0.1 ≤ Xu ≤ 1 and Tu = 10 K in MYSO jets. Hotter disc and a radiative bow shock. (colder) jets likely have higher (lower) ionization fraction. The jet ion density can be estimated as
n M˙ v −1 R −2 M˙ ≈ i j j −3 150 −6 −1 −1 16 2.2 Shocks and densities cm 10 M yr 1000 kms 10 cm (3) Internal shocks due to a variable ejection velocity is the most popular model to explain the chain of knots in protostellar (e.g. Rodr´ıguez-Kamenetzky et al. 2017). Owing to the large jets such as HH 111 (Raga et al. 1990). In most of the cases, uncertainties in the different parameters we neglect the con- knots are bright Hα and [SII] emitters in the optical band tribution of helium in Eq. (3). In Fig.2 we plot n ˙ for −1 M indicating shock speeds of few tens of km s , much smaller Rj = R/2 and R/6, where R is the average linear size of the than typical proper motions with speeds of hundreds km s−1. Since velocity perturbations are of the order of 10% of the jet speed, vj (Reipurth & Bally 2001), internal shocks are 5 This situation is very similar to the cloudlet model (Norman & not fast enough to accelerate particles up to very high en- Silk 1979) where a fast cloud moves into a molecular cloud (see ergies, which is consistent with the fact that most of the e.g. Hartigan 1989) or into a slow jet (Yirak et al. 2012).
MNRAS 000,1–16 (2015) 4 A. T. Araudo et al.
-42 51 50 1e7
(1) (2) 14.9 GHz 1e6 (3,4)
(11) 55 ] 1e5 -3
N-1 1e4 (9,10)
Density [cm 1e3 52 00 n (R = R/2) (5,6) jet M j n (R = R/6) (7,8) 100 M j
nff (Rj = R/2)
nff (Rj = R/6) 10 05 1e-7 1e-6 1e-5 -1 Ionized mass loss rate [Msun yr ] Central Source DECLINATION (J2000) Figure 2. Jet ion density, n , for the sources listed in Table2 as 10 M˙ a function of the ionized mass loss rate assuming Rj = R/2 (green jet triangles up) and R/6 (green triangles down). Red squares and circles indicate the value of nff when Rj = R/2 and R/6, respectively.
15
S-1 hotspot also listed in Table2. The jet mass loss rate of ion- ized matter M˙ i ∝ vj is assumed to be constant along the jet and then nM˙ turns out to be independent of the jet velocity. −2/9 20 By inserting Eq. (3) in Eq. (2) we find that vsh,ad ∝ Rj . In Table2 we list nM˙ and vsh,ad assuming Rj = R/2. We note that n is a rough estimation of the jet ion density given the 16 58 18.0 17.8 17.6 17.4 17.2 17.0 16.8 16.6 16.4 M˙ RIGHT ASCENSION (J2000) uncertainties in the values of M˙ i and Rj. In particular, light adiabatic jets form a cocoon and therefore the size of the non-thermal lobe at the jet head is expected to be Rh > Rj. 2 Krause(2003) found that (Rj/Rh) ∼ 0.1 − 1 when χ > 0.01; this gives an increment in the jet density by a factor of ∼ 10 reverse jet when Rj = R/6 instead of R/2 (see Fig.2). Nevertheless, the shock derived jet density is likely lower than the typical density Vrs≈Vj values of ∼ 105 cm−3 in the molecular clouds where mas- sive stars form (Hennebelle & Falgarone 2012) giving χ < 1. Therefore qrs > qbs and we should rather expect non-thermal sources in light jets (χ < 1), where the reverse shock is faster than the bow shock. In order to check if synchrotron emission comes from contact the reverse-shock rather than from the bow-shock down- discontinuity stream region, we compare the synchrotron (εsynchr,ν ) and free-free (εff,ν ) emissivities. The temperature of the plasma immediately downstream of the shock with compression ra- tio r = 4 is T ∼ 2.3 × 107 (v /1000kms−1)2 K. The free- Vbs≪Vj d sh free emissivity of the shocked plasma emitted at frequency bow −1 2 shock ν kBTd/h ∼ 480(vsh/1000kms ) GHz is ε , ff ν ≈ 1.4 × 10−33g(ν,T ) Molecular cloud ergcm−3s−1Hz−1 d n 2 v −1 × e sh (4) Figure 1. Top: The radio image of the source IRAS 16547-4247 104 cm−3 1000 kms−1 (G343.1261) at 14.9 GHz (adapted from Rodr´ıguez et al. 2005). (Lang 1974), where k and h are the Boltzmann and Bottom: Sketch of the jet termination region in light protostellar B Planck constants, respectively. The Gaunt factor is g(ν,Td) ∼ jets. Here, Vbs, Vrs, Vj refer to the bow shock, the reverse shock, and the jet velocity, respectively. 0.54logΛ and v 2 ν −1 Λ ≈ 1.1 × 109 sh (5) 1000kms−1 GHz 5 when 2πν ωp and Td > 3.6 × 10 K. Here, ωp ∼ 5.6 ×
MNRAS 000,1–16 (2015) MFA in jets from MYSOs 5
6p 4 −3 −1 10 ne/10 cm rad s is the electron plasma frequency. difficult to infer the magnetic field orientation and strength −1 We note that g(ν,T) ∼ 4.3 when vsh = 1000 km s and in the central jet itself where the origin of the magnetic field ν ∼ 10 GHz. is more likely connected to the ejection process from the On the other hand, the synchrotron emissivity of a accretion disk. Lee et al.(2018) reported on the detection of 2 source located at distance d is εsynchr,ν = 4πd Sν /Ve, where Sν SiO line polarization in the HH 211 protostellar jet, with an 3 estimated magnetic field of about 15 mG at ∼300 AU from is the synchrotron flux measured at frequency ν and Ve ∼ Rj is the volume of the synchrotron emitter (see Eq. A1). By the central protostar. imposing εsynchr,ν > εff,ν , we find that the density of thermal Theoretical studies of magnetically-accelerated jets re- electrons downstream of the shock has to be smaller than quire the action of a poloidal component to accelerate the nff, where plasma and a toroidal component to confine it (Casse & Keppens 2002). The toroidal component evolves with the 1 3 − 1 n d S 2 R 2 v distance z from the star along the jet as Bφ ∝ 1/z. At the ff ≈ 1.4×105 ν j sh 2 cm−3 kpc mJy 1016cm 1000kms−1 jet base, on the edge of the disk, typical values of the magnetic field strength are obtained from the local beta (6) plasma parameter βp = Pg/Pm, where Pg and Pm are the for the non-thermal emission to dominate at frequency ν (see gas and magnetic pressures, respectively. Jet launching re- Henriksen et al. 1991). In Fig.2 we plot nff and in Table2 quires βp ≥ 1 (Casse & Keppens 2002) giving an upper limit −1 we list the values of nff for the case vsh = 1000 km s and for the magnetic field strength at the base of the jet of 10 −3 3 Rj = R/2. We can see that molecular clouds with densities 0.3(ndisc/10 cm )(Tdisc/10 K) G where ndisc and Tdisc are ∼ 105 cm−3 will not (or marginally) satisfy the condition the gas density and temperature in the disc, respectively. nmc < nff for hosting the synchrotron emitter in the shocked Using the model of Combet & Ferreira(2008) we can region downstream of the bow shock. On the other hand, infer typical values of the vertical magnetic field strength nM˙ nff for all the sources in the sample. in jet emitting disks Bz ∼ 0.1–10 G at 1 AU from the star Hereafter, we will consider light jets (χ < 1) for which depending on the central mass of the object and the accre- the synchrotron radiation dominates over the free-free emis- tion rate. We will henceforth consider typical magnetic field sion and identify the reverse shock with jet speed (vj = vrs). strengths at the base of the jet in the range 10 mG–10 G In order to simplify the notation we will name it vsh and use to be conservative. By using a 1/z dilution factor, we ob- −1 6 1000 km s as a characteristic value . The jet ion density tain typical magnetic field strengths of Bj ∼ 10 µG–10 mG √ 3 will be nM˙ ≤ ni ≤ nff with a mean value hnii = nM˙ nff (see at distances of the jet termination shock of ∼ 10 AU or ten 4 Table2). We note that hnii is similar to nM˙ when Rj = R/6. times smaller at a distance of 10 AU. By considering the In Fig.1 we show a sketch of the scenario considered in this flux-freezing condition, Hartigan et al.(2007) showed that study. the magnetic field strength of variable MHD jets can be de- −3 0.85 Setting constraints on ni and vsh is highly important in scribed by the relation (Bj/15 µG) ∼ (ni/100cm ) giving our study given that the jet kinetic energy density values in agreement with the above estimate. However, at distances larger than ∼ 103 AU from the protostar, densi- U 1.7 × 10−4 n v 2 kin i sh ties and magnetic fields are mostly influenced by shocks and −3 ' 4 −3 −1 , (7) ergcm Xu 10 cm 1000kms rarefaction waves. and the jet kinetic luminosity The magnetic field topology at the jet termination shock in high-mass protostars is difficult to assess, but it is likely 37 2 L 2.1 × 10 n v 3 Rj helical due to a combination of a toroidal and a poloidal kin ' i sh , −1 4 −3 −1 16 ergs Xu 10 cm 1000kms 10 cm component (Cerqueira et al. 1997; C´ecere et al. 2016), hence (8) the orientation of the magnetic field with respect to the nor- mal of the termination shock front is likely oblique. On the represent the energy budget to accelerate particles at the other hand, numerical studies of non-relativistic and mag- shock. Another very important parameter in our study is netically driven jets show that kink instabilities destroy the the magnetic field. ordered helical structure of the magnetic field. Moll(2009) showed that at z & 2–15RA, the toroidal field is dissipated and the poloidal component dominates. Here, RA is the Alfv´en 2.3 Magnetic fields radius, where the poloidal jet and the Alfv´enspeeds are Maser emission provides information about the orientation equal. For the objects of interest, this length scale is typi- and strength of magnetic fields in the molecular outflows cally less than a few tens of AU (Pelletier & Pudritz 1992). associated to the central jet. Goddi et al.(2017) derived A dominant poloidal field is also supported by laboratory magnetic field strengths between 100 and 300 mG in the experiments (Albertazzi et al. 2014). protostellar jet W3(H2O), which likely results from strong In this work, acknowledging for the above uncertainties, gas compression behind shocks associated with the outflow we adopt Bj = 10 µG as a characteristic value of the jet expansion in the ambient molecular cloud. However, it is magnetic field in the termination region. The Alfv´enspeed in the jet is given by
6 This value probably overestimates the effective vrs but, on the other hand, vj deduced from proper motions is underestimated 1 given that it corresponds to the velocity projected on the plane v B n − 2 A ' 0.2 j i , (9) of the sky. kms−1 10 µG 104 cm−3
MNRAS 000,1–16 (2015) 6 A. T. Araudo et al.
Table 2. Observed and derived parameters of non-thermal lobes associated with MYSOs. From left to right we list the name of the source and the non-thermal component in the jet, the observed frequency ν, the measured flux density Sν , the radio spectral index α, the major (θmaj) and minor axis (θmin) deconvolved dimensions of the component and its average linear size R, the synchrotron emissivity εsynchr, the lower limit on the shock velocity vsh,ad, the density nM˙ , the upper limit on the lobe density given by Eq. (6), and the average density −6 hnii. We define R = (Rmaj +Rmin)/2 as the characteristic size of the hotspot, where Rmin,maj = 4.8×10 d θmin,maj and d is the source distance given in Table1.
Source ν Sν α θmaj × θmin R εsynchr vsh,ad nM˙ nff hnii [GHz] [mJy] [arcsec2] [cm] [erg cm−3s−1Hz−1] [km s−1] [cm−3] [cm−3] [cm−3] (1) G263.7434 N 9 0.56 0.45 1.23 × 0.36 4.0 × 1015 4.7 × 10−30 317 1.0 × 103 1.6 × 105 1.2 × 104 (2) G263.7759 NW 17 2.16 0.78 1.64 × 0.34 5.0 × 1015 9.4 × 10−30 353 1.3 × 103 2.2 × 105 1.7 × 104 (3) G310.1420 A4 9 0.93 0.70 1.10 × 0.89 3.3 × 1016 8.4 × 10−31 373 2.5 × 102 6.6 × 104 4.1 × 104 (4) D 9 1.93 0.46 1.38 × 1.05 4.8 × 1016 1.2 × 10−30 377 2.7 × 102 8.0 × 105 4.7 × 104 (5) G313.7654 A2 9 0.14 0.68 0.85 × 0.38 6.6 × 1016 2.3 × 10−31 495 8.6 × 102 3.4 × 104 5.4 × 103 (6) D 5.5 0.15 0.32 0.80 × 0.41 3.4 × 1016 2.6 × 10−31 497 8.9 × 102 3.5 × 104 5.6 × 103 (7) G339.8838 NE 9 2.36 0.39 0.47 × 0.38 8.1 × 1015 1.3 × 10−28 842 7.9 × 104 8.5 × 105 2.6 × 105 (8) SW 9 1.50 0.72 5.33 × 0.85 6.5 × 1016 5.6 × 10−32 463 3.7 × 102 1.7 × 104 2.5 × 103 (9) G343.1261 N4 17 1.80 0.67 0.65 × 0.35 1.0 × 1016 3.2 × 10−29 674 1.6 × 104 4.2 × 105 8.2 × 104 (10) S1 17 4.72 0.45 0.64 × 0.35 1.0 × 1016 3.8 × 10−29 633 9.2 × 103 4.6 × 105 6.5 × 104 (11) G114.0835 B 1.5 0.21 0.42 2.50 × 1.20 1.1 × 1017 2.3 × 10−32 271 0.3 × 102 1.0 × 104 6.1 × 102 Average values 0.55 1.46 × 0.57 3.3 × 1016 1.8 × 10−30 455 1.2 × 103 1.0 × 105 1.6 × 104 Observational data Section 2.2
and the Alfv´en Mach number is where κ ≥ 1 indicates the departure from the Bohm diffu- p 4 sion regime, F = max[Xu/ Tu,4,(1−Xu)], and Tu,4 = Tu/10 K −1 1 vsh vsh Bj ni 2 (Drury et al. 1996). For typical parameters in jets from MA = ' 4800 −1 4 −3 . vA 1000 kms 10 µG 10 cm high mass protostars, and Xu > 0.1 neither ionization nor (10) Coulomb losses quench the acceleration at any energy. We consider that the spectrum of accelerated particles is given To keep the calculations simple, we derive most of estimates by Eq. (B1). in the parallel shock configuration (i.e. with a purely poloidal magnetic field), but we discuss in Sect. 4.1 the impact of the magnetic field obliquity on our results. 3.1 Non-thermal electrons
Synchrotron emission at frequency ν in a magnetic field Bs is mostly produced by electrons with energy 3 RELATIVISTIC PARTICLES AND SYNCHROTRON E ν 1/2 B −1/2 RADIATION e ' 475 s , (12) MeV GHz mG The energy distribution of particles downstream of a shock indicating that synchrotron photons in the range 100 MHz- is a Maxwellian (thermal) distribution with a power-law 10 GHz correspond to relativistic electrons with energies in (non-thermal) tail at a high energies. The transition en- the range 150 MeV-1.5 GeV, when Bs = 1 mG. The energy ergy between both distributions is unknown. By using hy- density in non-thermal electrons following a power-law dis- brid simulations of parallel non-relativistic shocks, Caprioli −s tribution Ne = KeEe is Ue,tot = Ke fe, where fe is defined in & Spitkovsky(2014a) have shown that immediately behind Eq. (B2) and s = 2α + 1. For the present study we select the shock there is a bridge of supra-thermal ions smoothly non-thermal lobes with 0.3 < α < 0.8 and then 1.6 < s < 2.6. connecting the thermal peak with the power-law, while far The normalization factor Ke can be determined from the downstream there is a sharp boundary between thermal and measured synchrotron flux S at a particular frequency , 2 ν ν non-thermal protons at Einj,p ∼ 4Eth, where Eth = mpvsh/2 ∼ as described in AppendixA. By combining Eqs. (A5) and −1 2 5.2(vsh/1000kms ) keV, in agreement with Bell(1978b). (B4) we find that Park et al.(2015) have shown that the typical energy needed s−1 for electron injection into the shock acceleration is compara- Ue,tot −8 fe ν 2 −3 ≈ 4.8 × 10 ξK(s) × (13) ble with Einj,p. Achieving such injection energy Einj is much ergcm 10 GHz more difficult for electrons, as their initial momentum is a − (s+1) εsyn,ν Bs 2 factor mp/me smaller. In dense and non-completely ionized , −30 −1 −3 −1 mG medium, ionization and Coulomb losses can suppress the ac- 10 erg s cm Hz celeration of low-energy particles. Ionization and Coulomb where ξK(s) is given in Eq. (A6) and Bs is the magnetic −4 2 losses are almost constant at energies below E0 = 10 mpc field in the synchrotron emitter, i.e. mostly coming from the −6 4 2 7 and Eth = 4 × 10 (T/10 K)mpc , respectively, and therefore shock downstream medium . The magnitude of Bs is un- they cannot quench the acceleration at the injection energy known. It is commonly assumed that Bs is in equipartition if
n 2.5 × 107 B v 2 1 7 Depending of the properties of the turbulence around the shock, i < j sh , (11) cm−3 κ 10µG 1000kms−1 F the magnetic field strength in the synchrotron emitting region has
MNRAS 000,1–16 (2015) MFA in jets from MYSOs 7
Up,tot/Ukin. These protons can excite various types of insta- 1e-4 ni = nMdot bilities at different scales (see e.g. Marcowith et al. 2016, (10) n =
e,tot 1e-7 Bosch-Ramon et al. 2010; Padovani et al. 2015). However, U (3) (1) we will see that, if the jet (unperturbed) magnetic field (4) B is smaller than a certain value and the driving param- (1) j 1e-8 eter (related to the CR current in the upstream medium) ζ = 2ηpvsh/c is high enough, the dominant instability is non- 1e-9 resonant. The acceleration efficiency of protons with par- 0.01 0.1 1 10 ηp ticular energy E and energy density U in Eq. (B4) is defined Magnetic field in the shock downstream region Bs [mG] p p as ηp = Up/Ukin = ηp,tot/ fp, where fp is defined in Eq. (B2) and plotted in Fig. B1.
Figure 3. Non-thermal electron energy density (Ue,tot) needed to explain the synchrotron flux at the observed frequency ν in the sources listed in Table3. The magnetic field strength in the syn- chrotron emitter is Bs ≤ Beq. Orange and violet circles indicate the 4 COSMIC-RAY STREAMING INSTABILITIES values of Ue,tot and Bs that satisfy the condition Unt = Ukin. The current jp = npevsh of relativistic protons with energy Ep and a number density np = ηpUkin/Ep in Eq. (B3) can drive MHD turbulence due to the force j × B added in the with non-thermal particles, satisfying the minimum energy p momentum equation (Bell 2004, 2005). The parameter requirement to explain the synchrotron emission. By setting ηp represents the fraction of the kinetic energy imparted into U = B2/(8π) we find that the magnetic field in equiparti- e,tot s protons with energy E driving the instabilities. We consider tion with relativistic electrons is p a parallel shock in the z-direction with a small perturbation 2 Beq,e fe s+5 in the plasma. Wave solutions ∝ exp(ikz −ωt) of the first or- ≈ ξ (s) × mG eq 10 der MHD perturbed equations lead to the dispersion relation
2 s−1 s εsyn,ν +5 ν s+5 , (14) 2 10−30 erg s−1 cm−3 Hz−1 GHz 2 2 2 vsh ω = k vA + εkζ (σ(x,s) − 1), (16) rg 2/(s+5) where ξeq(s) ' (1.2ξK(s)) is plotted in Fig. A1. where ε = ±1, −1 rg Ep Bj 3.2 Non-thermal protons ' 3.3 × 1011 (17) cm GeV 10µG The energy density in protons with energy E ≥ m c2 p i is the proton Larmor radius and σ(x,s) = 0 when x ≡ kr >> 1 is U = aU , where a = (m /m )(s−1)/2 f / f (see Ap- g p,tot e,tot p e p e (Bell 2014). The solutions of the dispersion relation in pendixB). The non-thermal energy density is U = U + nt p,tot Eq. (16) are plotted in Fig.4 for s = 2, v = 300 and U = (1 + a)U , and then the magnetic field in equipar- sh e,tot e,tot 1000 km s−1 and n = 103 and 104 cm−3. We plot Im(ω) tition with non-thermal electrons and protons is i and Re(ω) for ε = 1 (Bell or NR branch, bottom panel) 1 and ε = −1 (Alfv´enor resonant branch, top panel). Growing Beq = (1 + a) 2 Beq,e. (15) modes correspond to Im(ω) > 0. Im(ω) and Re(ω) are nor- 2 In Table3 we list the values of s, a, and Beq for all the malized to 1/(3tk), where tk = rgc/3vsh is the time it takes a sources in Table2 and considering Ee,max = 0.1 TeV in fe. parallel shock to travel through the layer within which pro- 2/(s+5) However, we note that Beq ∝ fe and therefore the de- tons of speed ∼ c are confined and the upstream diffusion pendence on Ee,max is negligible. We also point out that Beq coefficient is κBohm = rgc/3 (Bohm diffusion). More generally, is an upper limit for the magnetic field in the synchrotron this layer is controlled by the upstream diffusion coefficient emitter. In Fig.3 we plot Ue,tot for Bs ≤ Beq. Orange and κu which depends on the shock obliquity and we define the 2 violet circles corresponds to the case where the total accel- advection time as tadv = κu/vsh (see Sect. 4.1). eration efficiency in non-thermal electrons and protons is We can see in Fig.4 (top panel) that resonant waves −1 Unt/Ukin = 1 and we fixed vsh = 1000 km s and ni = hnii are unstable when krg ≈ 1 and therefore this instability is resonant. On the other hand, the bottom panel shows that and nM˙ , respectively, to compute Ukin. We define the total proton acceleration efficiency ηp,tot = in the NR regime (krg > 1) Im(ω) Re(ω) and then the NR mode is almost purely growing. The maximum growth rate Γmax ≡ max(Im(ω)) moves to the resonant regime (krg ≈ 1) to account for the residence times of the particles downstream and when vsh decreases. However, even when vsh is as small as upstream. The exact expression is given by Eq. (18) in Parizot 300 km s−1, the NR mode is still dominant due to the large et al.(2006). ion density of protostellar jets. Both modes are present in the
MNRAS 000,1–16 (2015) 8 A. T. Araudo et al.
Table 3. From left to right we list the source names, the relativistic electrons and protons energy power-law slope s = 2α + 1, the total protons-to-electrons energy density ratio (a, for the case Ee,max = Ep,max = 1 TeV), the equipartition magnetic field (Beq), the amplified field in the shock downstream region (Bsat,NR), the total energy density in protons (Up,tot) and their acceleration efficiency (ηp,tot), the protons maximum energy (E ), and the electrons (K ) and protons (K ) normalization constants. We note that η and E are computed p,max √ e p p,tot p,max for ni=nM˙ (Eq.3) and ni=hnii = nM˙ nff (Eqs.3,6).
Source s a Beq Bsat,NR Up,tot ηp,tot Ep,max Ke Kp [mG] [mG] [erg cm−3] [TeV] [ergs−1 cm−3] [ergs−1 cm−3] n =n n =hn i n =n n =hn i i M˙ i i i M˙i i i (1) G263.74 N 1.90 18.81 4.79 0.55 3.6 × 10−6 0.43 0.07 0.13 0.04 2.5 × 10−8 7.5 × 10−7 (2) G263.7759 NW 2.56 5.11 7.70 0.92 1.0 × 10−5 0.93 0.08 0.13 0.06 4.3 × 10−10 1.5 × 10−7 (3) G310.1420 A4 2.40 8.90 3.27 0.40 1.8 × 10−6 0.86 0.06 0.38 0.14 3.1 × 10−10 5.9 × 10−8 (4) D 1.92 19.24 3.65 0.42 2.1 × 10−6 0.91 0.10 0.88 0.21 1.3 × 10−8 4.1 × 10−7 (5) G313.7654 A2 2.36 10.14 2.25 0.28 8.7 × 10−7 0.12 0.02 0.16 0.08 2.0 × 10−10 3.3 × 10−8 (6) D 1.64 10.14 2.01 0.22 5.7 × 10−7 0.08 0.06 0.65 0.26 2.0 × 10−8 2.3 × 10−7 (7) G339.8838 NE 1.78 15.06 14.6 1.63 3.2 × 10−5 0.05 0.04 0.28 0.15 4.9 × 10−7 9.2 × 10−6 (8) SW 2.43 7.80 1.60 0.19 4.4 × 10−7 0.14 0.02 0.17 0.09 5.2 × 10−11 1.2 × 10−8 (9) G343.1261 N4 2.34 10.80 10.2 1.22 1.7 × 10−5 0.13 0.03 0.17 0.09 4.8 × 10−9 7.4 × 10−7 (10) S1 1.89 63.07 9.93 1.13 1.5 × 10−5 0.20 0.06 0.43 0.16 1.1 × 10−7 3.2 × 10−6 (11) G114.0853 B 1.84 17.13 0.82 0.09 1.0 × 10−7 0.35 0.05 0.43 0.10 1.1 × 10−9 2.5 × 10−8 Average values 2.13 5.52 0.19 7.7 × 10−6 0.38 0.05 0.55 0.20 3.8 × 10−9 2.4 × 10−7 Section3 Section5 Section6 Section7
2 plasma, although the dominance of one over the other de- In the limit ζMA < 1, the Alfv´eninstability dominates 2 2 2 pends on the competition between the Alfv´en (that contains and the dispersion relation in Eq. (16) reduces to ω = k vA. vA) and the driving (that contains ζ) terms in Eq. (16). We The maximum growth rate of resonant modes occurs for√ par- note that the NR mode exists when 1/rg ≤ k ≤ kmax, where allel propagating modes (Achterberg 1981) at kmax,R ≈ 3/rg kmax = Γmax/vA. Therefore, the condition for the growth of and therefore kmax,Rrg ≈ 1, as shown in Fig.4 (top panel). 2 the NR mode is kmaxrg > 1, which can be written as ζMA > 1, The growth rate reaches its maximum value at Γmax,R ∼ where Γmax,NR (e.g. Pelletier et al. 2006; Amato & Blasi 2009). Alfven instabilities dominates over Bell instabilities when −2 ηp n Bj v 3 ζM2 ' 3072 i sh . A . 4 −3 −1 0 02 10 cm 10µG 1000kms − 1 − 1 vsh ηp 2 Bj ni 2 (18) < 128 . (22) kms−1 0.02 10µG 104cm−3 It is clear from Eq. (18) that the NR mode is more easily This regime was studied by Padovani et al.(2015, 2016) in 2 triggered by the fastest shocks. The condition ζMA > 1 can the context of low mass protostars, where the jet typical ve- 2 be written also as Up(vsh/c) > Bj /(8π), or equivalently locities are ∼ 100 km s−1. The important aspect of destabi- lizing perturbations in resonance with CRs, i.e. when kr ≈ 1, U B −2 v −1 g p −9 j sh is that the scattering process is more efficient. However, this −3 > 1.2 × 10 −1 . (19) ergcm 10µG 1000kms does not guarantee to have a strong amplification. In the 2 present paper we mostly focus on the NR instability. In the limit ζMA > 1, the NR instability dominates. In the regime kmax,NR ≥ k ≥ 1/rg, σ ≈ 0 and the dispersion rela- tion in Eq. (16) reduces to 4.1 Magnetic field obliquity 2 2 2 2 vsh In the perpendicular shock case the wave speed in Eq. (20) ω = k vA − kζ . (20) rg now includes a contribution of the compression modes char- The maximum growth rate of the NR instability excited by acterized by the local sound speed cs (Bell 2005; Marcowith et al. 2018). If cs < vA the non-resonant growth rate is protons with energy Ep and carrying a current of strength unmodified (Bell 2005; Matthews et al. 2017). If c > v j is Γ = 0.5( j /c)pπ/n m , giving s A p max p i i the growth rate drops because of thermal effects (see Sec- Γ ηp v 3 tion 4.2.1). max,NR ' 2.6 × 10−5 sh s−1 0.02 1000kms−1 The necessary condition for the streaming instabilities 1 −1 to grow is also modified. In perpendicular shocks, the precur- ni 2 Ep × 4 −3 . (21) sor size is shorten because the particle transport is controlled 10 cm GeV 2 by the perpendicular diffusion κu = κ⊥ = κBohmκ/(1 + κ ) We note that Γmax,NR is independent of Bj, however the (Forman & Gleeson 1975) where κ = κk/κBohm. If angu- 2 wavenumber kmax,NR = Γmax,NR/vA = ζMA/rg of the fastest lar diffusion proceeds at a smaller rate than Bohm then growing modes does depend on Bj (see Eq. 18). We also κ 1 and κ⊥ ' κBohm/κ. We derive the general expres- b note that as Γmax,NR and kmax,NR decrease with vsh. Reso- sion Γmax,NRtadv ∼ ηp(vsh/vA)κ /24, where b = 1 and −1 (with nant modes should take over as it is the case in low mass κ > 1) for parallel and perpendicular shocks, respectively. In YSOs (Padovani et al. 2016). perpendicular shocks, Γmax,NRtadv > 1 gives ηp > κ/MA. High
MNRAS 000,1–16 (2015) MFA in jets from MYSOs 9
-1 3 -3 v =1000 km s , n =10 cm (Zweibel & Everett 2010). This leads to the condition 100 sh i -1 4 -3 v =1000 km s , n =10 cm −1 sh i Tu ηp Ep -1 4 4 v =300 km s , n =10 > 1.1 × 10 sh i K 0.02 GeV
c] 3 3 g 10 − − v 3 Bj ni 2 /r ω) sh 2 × (23) sh Re( 1000kms−1 10 µG 104cm−3 ) [v
ω 2 2 1 in which case an extra term ∝ k Tu rgω needs to be added in the dispersion relation in Eq. (16). The maximum growth ), Re( ω rate of the thermally modified Bell instability is
Im( Im(ω 0.1 ) 2 2 np 3 v 3 Γ ' ω sh , (24) max,th ci n v ε = -1 i i
0.01 where ωci = eBj/(mpc) is the ion cyclotron frequency (Reville 0.1 1 10 100 1000 et al. 2007). For typical values in protostellar jets, k [1/rg] 1 1 − 1 -1 3 -3 − 3 3 v =1000 km s , n =10 cm Γmax,th ηp 3 Ep Tu 100 sh i ' 1.9 4 -1 4 -3 Γ , 0.02 GeV 10 K vsh=1000 km s , ni=10 cm max NR -1 4 v =300 km s , n =10 −1 − 1 sh i vsh Bj ni 2 × −1 4 −3 . (25)
c] 1000kms 10 µG 10 cm
g 10
/r ω) 2 Im( sh We stress that Γmax,th < Γmax,NR when the temperature is
) [v within the range where the condition in Eq. (23) is satis- ω 1 fied. The thermally modified Bell instability is damped when
), Re( Γmax,thtadv < 1, i.e. when
ω Re( ω )
Im( 2 Tu 7 ηp Ep 0.1 > 4.5 × 10 , (26) K 0.02 GeV ε = 1 where we have assumed tadv = tk. When thermal effects are 0.01 taken into account, they can reduce the growth rate of the 0.1 1 10 100 1000 NR instability but we do not expect strong damping by k [1/rg] thermal effects over the development of the NR streaming modes, unless the proton acceleration efficiency ηp is unrea- Figure 4. Dispersion relation ω(k) for the resonant (upper and sonably small in which case the NR modes are not destabi- lower panels for krgm ≤ 1) and non-resonant (lower panel only for lized. krgm ≥ 1) modes destabilized by CRs following a power-law energy distribution with s = 2. Re(ω) – dashed lines – and Im(ω) – solid −1 3 4.2.2 Partially ionized medium lines – are plotted for vsh = 300 and 1000 km s and ni = 10 and 4 −3 10 cm . We fix Bj = 10 µG and ηp = 0.01. In a partially ionized protostellar jet (i.e. Xu < 1) the friction arising between charged and neutral particles can quench the growth of CR driven instabilities at shock precursors, κ values probably quench the development of fast streaming and therefore DSA is less efficient (e.g. Drury et al. 1996; instabilities in the configuration of a weakly CR modified Reville et al. 2007). When ion-neutral collisions are taken into account in the MHD equations, the dispersion relation shock. In the mean time, if κ > 100 then ηp > 0.1 and the destabilization of streaming instabilities occurs in the regime of the CR-driven instability becomes of strongly CR modified shocks, a case beyond the scope of 2 ! 3 2 1 vsh 2 2 the present linear analysis. ω + iω νin + ω εζ k(1 − σ) − k vA + 1 − Xu rgm 2 ! Xu vsh 2 2 iνin εζ k(1 − σ) − k vA = 0, (27) 1 − Xu rgm 4.2 Reduction of the NR growth rate due to environmental conditions where the ion-neutral collision frequency is given by Different effects may reduce the growth rate of the NR in- ν T 0.5 n in ' 8.9 × 10−4 u n , (28) stability. In this paper we discuss thermal effects and ion- s−1 104 K 105cm−3 neutral collisions in non-completely ionized jets. in the shock upstream region with Tu > 140 K and neutral density nn = (1/Xu − 1)ni (Jean et al. 2009). We point out that in a completely ionized plasma Xu = 1, nn = 0 and then = 0. In such a case the dispersion relation in Eq. (27) is 4.2.1 Thermal effects νin identical to Eq. (16). We solve Eq. (27) for ε = 1 and dif- 3 Thermal effects are important when (vA/vi) < ferent values of Xu. As pointed out by Reville et al.(2007), 2 (np/ni)(vsh/vi), where the ion speed is vi = kBTu/mp ion-neutral collisions are unable to stabilize the Bell modes
MNRAS 000,1–16 (2015) 10 A. T. Araudo et al.
-1 We note that Bsat,NR does not depends on Bj. vsh = 1000 km s 4 -3 Once the magnetic field strength reaches the saturation n = 10 cm i 2 4 value Bsat,NR, the Alfv´envelocity becomes larger and ηpM T = 10 K A u decreases. At the same time the Larmor radius of particles
c] s = 2 g decreases down to values r ∼ 1/k and then the NR Bell /r ω) g 2 Im( sh instability becomes subdominant. Caprioli & Spitkovsky
) [v (2014b) proposed a refined model for the growth of the ω 10 non-resonant modes including a non-linear stage (when en-
), Re( ergy densities in the turbulent and background magnetic ω x = 1 u fields become similar). They find a maximum magnetic field Im( √ xu = 0.8 strength Bmax/Bj ∼ M/ 2, where M = vsh/cs is the sonic xu = 0.6
xu = 0.4 Mach number and cs is the sound speed. In our case we x = 0.2 u have moderate M with a typical range 5-50, which makes the result obtained in Eq. (30) likely optimistic. It should 100 1000 be also noted that Caprioli & Spitkovsky(2014b) con- k [1/rg] duct their study in the regime of beta plasma parameter 2 βp = 2(cs/vA) ∼ 1 whereas MYSO jets have rather βp > 1 if Figure 5. Growth rate for different values of the ionization fraction not 1 because they are relatively hot and dense with a upstream of the shock. weak magnetic field. although the maximum growth rate decreases with Xu, as 5.1 The magnetic field in the shock downstream region we illustrate in Fig.5. This behaviour for Γmax,inc (the max- imum growth rate in the incompletely ionized jet) indi- The perturbations produced by the NR instability can lead cates that longer times are required to satisfy the condition to various effects in the shock downstream region (Giacalone Γmax,inctadv > 1 to excite the NR modes. & Jokipii 2007). As NR modes are not normal modes of the plasma, they are expected to be damped rapidly once the source of excitation disappears. Weibel instabilities would also contribute to produce magnetic field fluctuations up- 5 MAGNETIC FIELD AMPLIFICATION stream of the shocks. However, Weibel modes are expected Magnetic fields in the synchrotron emitter in protostellar to decay over a few plasma skin depth downstream, so over jets have strengths Bs ≈ 0.1–10 mG (see Sect.3). These val- much smaller scales than Rj. Conversely, dynamo processes ues are larger than the expected magnetic field in the jet can further amplify the magnetic field produced at the shock termination region, Bj ≈ 1–100 µG, requiring amplification front (Bell 2004; Pelletier et al. 2006; Inoue et al. 2009; Mar- of the magnetic field, given that compression by the shock cowith & Casse 2010; van Marle et al. 2018; Tzeferacos et al. does not produce strong enough fields. In this study we con- 2018), even if the upstream region is partially ionized (Xu & sider the magnetic field amplification by the Bell instability Lazarian 2017). This aspect involves a model of non-linear discussed in the previous section, but in the non-linear case evolution of the plasma and it is beyond the scope of this (Bell 2004, 2005). paper. For simplicity, we consider that the amplified mag- In the linear regime analyzed in Sect.4, the magnetic netic field in the shock upstream region is compressed by field increases exponentially with time until it reaches a the shock and maintained on a spatial scale ∼ R (the size of value B ∼ 2Bj, after that the amplification enters in a non- the synchrotron emitter). linear regime and the magnetic field growth becomes linear In the shock downstream region, the (turbulent) with time. At a particle energy Ep, the amplified magnetic isotropic upstream random magnetic field is compressed by p 2 field saturates at the shock by a factor rB = (2r + 1)/3 ≈ 3.3 when the 2 3 shock compression ratio is r = 4 (e.g. Parizot et al. 2006; Bp nimiv v 8 ∼ η sh = U sh (29) Zirakashvili & Ptuskin 2008) . Therefore, the amplified field p c p c 8π downstream of the shock is Bsat,d = rBBsat,NR. On the other (Bell 2004; Pelletier et al. 2006; Zirakashvili & Ptuskin hand, the magnetic field in the synchrotron emitter Bs, i.e. 2008). The above expression refers to magnetic field am- mostly coming from the shock downstream region, is a func- plification by protons with energy Ep. However, the distri- tion of the energy density in non-thermal electrons. In Fig.6 bution of relativistic protons driving the current spans from we plot Up,tot = aUe,tot (green-solid lines), where Ue,tot is given 2 ≈ 1 GeV to Ep,max ≈ 1 TeV (see Sect.6) and therefore the (to- by Eq. (13), and Up,tot = (c/vsh)Bsat,NR/(8π) by Eq. (30) (red- tal) saturated magnetic field Bsat,NR immediately upstream dashed line). By assuming that Bsat,d = Bs we find that of the shock is better estimated by considering the total pro- 2 ton population with energy density U and acceleration s+5 2 p,tot Bsat,d 2 a vsh vsh s+5 efficiency η , which gives = rB ∼ 0.38ξsat(s) −1 p,tot Beq 1 + a c 1000kms 1 1 (31) B U 2 v sat,NR ≈ . p,tot sh 2 0 3 −6 −3 −1 mG 10 ergcm 1000kms (30) 1 1 3 ηp,tot 2 ni 2 vsh 2 ≈ 1.2 . 8 0.1 104 cm−3 1000kms−1 Hereafter we use r = 4 and rB = 3.3.
MNRAS 000,1–16 (2015) MFA in jets from MYSOs 11
6 MAXIMUM ENERGIES 1e-3 (10) (9) -1 6.1 Protons
1e-4 If the non-resonant streaming instability is important and (6) = 1000 km s (5) v sh ] the magnetic field is amplified, the maximum energy of pro- -3 tons is determined by the amount of protons that escape 1e-5 (7) from the shock upstream region (Zirakashvili & Ptuskin [erg cm (2) 2008; Bell et al. 2013). Given that only the most energetic
p,tot 1e-6 (1) protons can penetrate far upstream from the shock and am- U (4) (3) plify the magnetic field in the shock precursor, the avail- able time to accelerate these particles is ∼ 5/Γ (E ), 1e-7 max,NR p,max (8) where Γ (E ) is the maximum growth rate of NR ni = nMdot max,NR p,max n =
MNRAS 000,1–16 (2015) 12 A. T. Araudo et al. energy Ee,max exceeds Ep,max, then the turbulence expe- Fermi (10 yr) rienced by electrons will be in the small-scale turbu- 1e-14 nlobe = 4nM (7) dot lence regime and the diffusion coefficient would be κu ' (10) κD (E ,B )(E /E )2. This rapid increase with the Bohm p,max s e p,max (9) ] 1e-15 (8) energy would limit Ee,max to Ep,max unless radiative losses -1
s (2)
dominate and limit it to smaller values. Then, one should -2 expect to have Ee,max ≤ Ep,max. (3) (4) By inserting κu into the acceleration timescale (tacc ∝ 1e-16
2 2 [erg cm (6) κu/v ) and the diffusive loss timescale (R /6κu) we find (1) sh j ,pp
γ (5) a maximum electron energy Ee,max,diff to be within a fac- F 1e-17 tor of a few of the maximum electron energy fixed by syn- (11) chrotron losses Ee,max,rad for the different sources in our sam- ple. Then, one can argue that the statement Ee,max ≤ Ep,max 1e-18 is reasonably verified accounting for the uncertainties on 0.0001 0.001 0.01 0.1 1 the parameters controlled by the microphysics of turbu- Eγ [erg] lence generation at fast shocks, namely `c, κ, Bs, and on the macroscopic jet parameters vsh and Rj. Hereafter we as- sume that Ee,max = Ep,max.(Ee,max,rad < Ep,max never happens Figure 7. π0-decay for all the sources in our sample for the case of in our source sample with the derived values of the magnetic ni = nM˙ . Black dashed lines indicates the Fermi sensitivity for 10 yr field strength Bs.) We left to a future work more precise cal- of observation. Numbers indicate the source name (see Table3). culation of Ee,max.
1e-13 Fermi (10 yrs) 7 GAMMA-RAY EMISSION Synchrotron TeV electrons and protons can emit gamma-rays by their 1e-14
interaction with ambient cold protons through relativis- ]
-1 proton-proton
tic Bremsstrahlung and proton-proton (pp) collisions. In- s verse Compton scattering is a mechanism contributing to -2 gamma-ray emission as well. In the jet termination region, 1e-15 4 Rel. Bremsstrahlung at ∼ 10 AU from the central protostar, the stellar pho- [erg cm γ ton field is very diluted and therefore the Inverse Compton F scattering is not expected to be very relevant. However, we 1e-16 IRAS 16547 N4 (9) stress that thermal photons from the bow-shock downstream IRAS 16547 S1 (10) region can be boosted to the γ-ray domain. The γ-ray flux IRAS 16484 NE (7) 1e-17 by up-scattering of photons from the bow shock will be com- 1e-16 1e-14 1e-12 1e-10 1e-08 1e-06 0.0001 0.01 1 puted in a following paper, together with the diffuse emission Eγ [erg] in the molecular cloud. Proton-proton and relativistic Bremsstrahlung cooling timescales are comparable. However, the number of rela- Figure 8. Spectral energy distribution from radio to gamma-rays tivistic protons per unit energy is larger than that of elec- for the three hot spots: IRAS 16547 N4, IRAS 16547 S1, and IRAS trons, i.e. Np > Ne, given that Kp > Ke (see AppendixB and 16848 NE. Black squares indicate synchrotron data (Purser et al. Fig. A1) and we assume that the slope is the same in both 2016) and the black dashed line represent the Fermi sensitivity electrons and protons energy distributions. In particular, for 10 years of observation. the distribution of relativistic protons in the jet hotspot is −s (s−1)/2 Np = KpEp , where Kp = Ke(mp/me) and Ke is computed from Eq. (A5) by fixing Bs = 3.3Bsat,NR. In Table3 we list can see that the predicted emission in G339-8838 NE (7), the values of Ke and Kp. As a consequence, the emission in G34301261 N4 (9), and S1 (10) is detectable by Fermi- the γ-ray domain will be dominated by hadrons. LAT after 10 years of observation (with 5σ-confidence) when nlobe = 4nM˙ . For these three sources we plot in Fig.8 the spectral energy distribution (SED) including also syn- 7.1 Proton-proton collisions chrotron emission and relativistic Bremsstrahlung and using In the one-zone model approximation, the specific luminos- the formulation in Blumenthal & Gould(1970). We assume ity of gamma rays due to π0-decay can be written as Ee,max = Ep,max as it is calculated in Eq. (33). We note that Fγ,pp ∝ nlobe and therefore the interaction 2 Eγ Lγ,pp = Eγ qγ (Eγ )nlobeVe, (35) of relativistic protons with clumps denser than the jet will increase the γ-ray flux. where qγ is the emissivity (see Eqs. (16) and (17) in Araudo et al.(2007)). We assume that the volume of the γ-ray emit- ter is the same as the synchrotron emission volume (V ) and e 7.2 Density enhancement in the jet termination region that the thermal ion density in the lobe is nlobe = 4nM˙ . In 2 Fig.7 we plot the flux Fγ,pp = Eγ Lγ,pp/(4πd ) for all the The termination region of non-relativistic light jets (nj < sources from our sample and the Fermi-LAT sensitivity. We nmc) is expected to be a combination of an adiabatic reverse
MNRAS 000,1–16 (2015) MFA in jets from MYSOs 13 shock and a radiative bow shock, leading to a large density Table 4. Average values of the most relevant parameters in our ratio at the contact discontinuity between both shocks (see study (see Tables2 and3). e.g. Rodr´ıguez-Kamenetzky et al. 2019b). The density of the plasma downstream of the radiative bow shock is From observations Radio spectral index α = 0.55 0 −1 n v 2 T −30 −3 −1 −1 mc = bs (36) Synchrotron emissivity εsyn,ν = 2 × 10 erg cm s Hz 500 −1 4 16 nmc 100kms 10 K Size of the non-thermal lobe R = 2.9 × 10 cm (e.g. Blondin et al. 1990) when the plasma is cooled down Assumed values −1 to a temperature T, making the density contrast at the con- Shock speed vsh = 1000 km s 0 0 Jet magnetic field Bj = 10 µG tact discontinuity χ = 4nj/nmc χ. As a consequence, the 4 Jet temperature Tu = 10 K contact discontinuity is unstable to dynamical and thermal Computed values 0 instabilities. A dense layer of density nmc located at distance 3 −3 ni = nM˙ = 1.2 × 10 cm l downstream of the bow shock (see Eq.1) fragments into Jet ion density 4 −3 th ni = hnii = 1.6 × 10 cm −1 several clumps (e.g. Calder´on et al. 2020). In this case, the Minimum shock speed vsh,ad = 455 km s effective density downstream the reverse shock will increase Equipartition magnetic field Beq = 7.25 mG 0 up to a value ∼ fv,clumpnmc, where fv,clump ≤ 1 is the volume Downstream magnetic field Bs = 0.84 mG −5 −3 3 Protons total energy density Up,tot = 1.5 × 10 erg cm filling factor of clumps in the lobe with volume ∼ Rj . How- ever, we draw attention to the fact that the component of Protons total acceleration efficiency ηp,tot = 0.05 Protons maximum energy Ep,max = 0.24 TeV the magnetic field in the ambient molecular cloud parallel −9 s−1 −3 Electrons normalization constant Ke = 3.8 × 10 erg cm to the bow shock front (Bmc,⊥) limits the compression factor −7 s−1 −3 Protons normalization constant Kp = 2.4 × 10 erg cm to a maximum value (Blondin et al. 1990) n0 n 1 v B −1 max mc 2 bs mc,⊥ situ accelerated. We consider a sample of 11 non-thermal ∼ 100 4 −3 −1 . (37) nmc 10 cm 100kms 0.1mG lobes in MYSOs as referenced by Purser et al.(2016) and The magnetic field in molecular clouds is Bmc ∼ mG Obonyo et al.(2019). From the observational data in the (Crutcher 2012) and we have assumed that Bmc,⊥ ∼ 0.1Bmc. above mentioned papers, the lobes selected for our study √ p In the case of light jets, vbs ∼ vj χ = vj nj/nmc (see Sect. 2.2) have an average radio spectral index α = 0.55, emissivity −30 −3 −1 −1 and therefore Eq. (37) can be written as εsynchr = 2.1 × 10 erg cm s Hz , and linear size R = 16 0 1 −1 2.9 × 10 cm (see Table4). Magnetic fields Bs ∼ 0.3 − 5 mG nmax nj 2 vj Bmc,⊥ ∼ 1000 4 −3 −1 . (38) are needed to explain the synchrotron radio flux. These large nmc 10 cm 1000kms 0.1mG values of Bs are difficult to explain by a simple compression This indicates that significant enhancement in the plasma in an adiabatic strong shock at the jet termination region, density downstream of the reverse shock is feasible if insta- at ∼ 0.1 pc from the central protostar. We note that strong bilities grow fast enough to fragment the dense shell and compression of the magnetic field in the molecular cloud by form the clumps. However, we note that even when clumps the radiative bow shock could explain values of Bs in the are not formed, protons accelerated at the jet reverse shock range 0.3 − 5 mG , and this scenario will be analysed in a can diffuse down to the dense shell and radiate there. future study. In the present paper we focus on magnetic field amplification by streaming instabilities. 7.2.1 Kelvin-Helmholtz and Rayleigh-Taylor instabilities CR streaming instabilities can generate magnetic field perturbations necessary for the particles to diffuse back and Kelvin-Helmoltz (KH) and Rayleigh-Taylor (RT) instabili- forth the shock. The CR streaming can amplify resonant ties can grow in the contact discontinuity due to the velocity Alfv´enwaves with wavelength of the order of the CR Lar- shear and the force exerted by the downstream material of mor radius rg but also excite the NR (Bell) instability which the reverse shock on that of the bow shock. If the bow shock produces perturbations at scales much smaller than rg (Bell is radiative, the formation of a shell much denser than the 2004). Bell’s instability can amplify small perturbations in jet (nj) and the molecular cloud (nmc) at the contact discon- the magnetic field up to values much larger than the unper- tinuity makes the working surface unstable even in the case turbed jet magnetic field Bj if ζMA > 1 where of light jets. Following the analysis in Blondin et al.(1990), the acceleration of the dense shell with a width w can be −2 shell 2 Up Bj vsh 2 0 ζM = 7000 written as a ∼ njvrs/nmcwshell. For a characteristic dynamical A 10−7ergcm−3 10µG 1000kms−1 timescale t = R /v we find dyn j j (40) − 1 tRT T 2 vj is an equivalent expression for Eq. (18). We remark that even ∼ 0.05 4 −1 , (39) tdyn 10 K 1000kms in the case where ζ is small due to the slow speed of the √ shock (compared e.g. with supernova remnants), the large where tRT ∼ 1/ ak is the growing time of RT instabilities and we have assumed that k ∼ 1/2w and w ∼ R /3. values of jet densities makes MA significantly large to satisfy shell shell j 2 the condition ζMA 1 over a large range of parameters. However, large ion densities in a non-completely ionized jet would increase the ion-neutral collisions. Alfv´en waves can 8 SUMMARY AND CONCLUSIONS be damped by ion-neutral collisions (e.g. Drury et al. 1996). The detection of synchrotron emission from jets powered Conversely, Bell modes are not as heavily damped, although by high-mass protostars indicates that electrons are in- the maximum growth rate decreases (Reville et al. 2007).
MNRAS 000,1–16 (2015) 14 A. T. Araudo et al.
Thermal effects only weakly affect the NR instability growth Rodr´ıguez-Kamenetzky et al. 2017; Rodr´ıguez-Kamenetzky rate (see Fig.5). High magnetic field obliquity can also lead et al. 2019b). to a decrease of the streaming instability growth rates or High sensitivity radio observations in the GHz do- even a complete quenching if the precursor length becomes main are very important to model the synchrotron emission too short (see Sect. 4.1). and constrain the energy density in non-thermal particles By assuming that the large magnetic field in the syn- (Sect.3). To this purpose, the detection of polarized emis- sion will be crucial to disentangle thermal contamination in chrotron emitter (Bs) is due to amplification through the the GHz domain, as well as to investigate the morphology Bell’s instability and fixing v = 1000 km s−1 we estimate sh of the magnetic field near the shock. The Next Generation Bs ∼ 0.4Beq and the energy density in non-thermal electrons Very Large Array (ngVLA) will play a fundamental role on Ue,tot. Then, the energy density in non-thermal protons is that (Galv´an-Madrid et al. 2018; Hull et al. 2018). Moving Up,tot = aUe,tot. Under the assumption that the jet ionized to lower frequencies, the low-energy cutoff of the electrons density is ni = hnii, we estimate the proton acceleration ef- distribution is an important parameter related with the effi- ficiency ηp,tot to be ∼ 0.05. We stress that this method is different with respect to that used in supernova remnants, ciency of shocks to inject electrons from the thermal pool to where the magnetic field is usually estimated by comparing the high energy tail. In this sense, the forthcoming Square the width of X-ray filament profiles with the synchrotron Kilometre Array (SKA) will be extremely important to ob- cooling length. serve the low-energy cutoff (see e.g. Feeney-Johansson et al. 2019). By knowing ηp,tot and Bs we estimate the maximum en- Finally we note that protostellar jets have been well ergy of protons accelerated in the jet reverse shock and the studied through several laboratory experiments (e.g. Nico- γ-ray emission that they produce. By considering the ampli- la¨ı et al. 2008; Liang et al. 2018; Suzuki-Vidal et al. 2012). fication timescale of the magnetic field in the shock upstream In particular, Suzuki-Vidal et al.(2015) have shown that region we find Ep,max ∼ 0.1 TeV. These protons can emit γ laboratory bow shocks formed by the collision of two coun- rays through their collisions with thermal ions. We compute terstreaming and supersonic plasma jets is fragmented due the γ-ray flux Fγ,pp for all the sources in our sample. We note to the rapid growth of thermal instabilities. The formation that Fγ,pp ∝ nlobe and the jet ionized density at the termina- of collisionless shocks (Li et al. 2019), the development of tion region is nM˙ ≤ ni ≤ nff, where nff/nM˙ ∼ 100 (see Fig.2). plasma instabilities, and the acceleration of particles in laser We find that having a density nlobe = 4nM˙ in the non-thermal plasma (Reville et al. 2013) is going to open a fascinating emitter is high enough to reach detectable levels of γ rays era of laboratory astrophysics in synergy with high energy- with Fermi in IRAS 16547 N4, IRAS 16547 S1, and IRAS astrophysics. 16848 NE, as we show in Fig.7. Although there is no claim of detection of these sources by Fermi, we expect that our result will motivate a future study on the Fermi data at the ACKNOWLEDGEMENTS location of these sources. In addition, these sources will be perfect targets for a point-source mode. We also note that The authors thank the anonymous referee for his/her com- mixing due to dynamical instabilities can significantly en- ments that helped us to improve the quality of our pa- hance the density of targets in the lobes. The very dense per. The authors thank C. Carrasco-Gonz´alez and S. Cabrit shell formed downstream of the radiative bow shock is un- for insightful discussions on protostellar jets. A.T.A. thanks stable and fragmented in clumps with density ∼ 100 − 1000 Tony Bell and K. Blundell for encouragement and motivat- times the density of the ambient medium (i.e. the molecu- ing discussions on jets, particle acceleration, and streaming −13 −2 −1 lar cloud). By achieving Fγ,pp ∼ 5 × 10 erg cm s at instabilities. A.T.A. thanks M.V. del Valle and R. Santos- 0.1 TeV, CTA will be of great importance to measure the Lima for their help with numerical calculations. A.T.A. cut-off of the spectrum. Also, the spatial resolution of CTA thanks the Czech Science Foundation under the grant GACRˇ is expected to be better than Fermi. The detection of γ rays 20-19854S titled “Particle Acceleration Studies in Astro- from protostellar jets will open a new window to study stellar physical Jets”. M.P. acknowledges funding from the INAF formation, as well as the efficiency of DSA in the high density PRIN-SKA 2017 program 1.05.01.88.04, by the Italian Min- 3 4 −3 −1 (ni ∼ 10 − 10 cm ) and low velocity (vsh ∼ 1000 km s ) istero dell’Istruzione, Universit`ae Ricerca through the grant regime. In particular, the detection of diffuse gamma-ray Progetti Premiali 2012–iALMA (CUP C52I13000140001). emission in molecular clouds where MYSOs are embedded and by the project PRIN-INAF-MAIN-STREAM 2017“Pro- will be a piece of evidence of proton acceleration in proto- toplanetary disks seen through the eyes of new-generation stellar jets. instruments”. This work has been carried out thanks to the In a following paper we will perform more detailed support of the OCEVU Labex (ANR-11-LABX-0060) and calculations of the multi-wavelength lepto-hadronic spec- the A*MIDEX project (ANR-11- IDEX-0001-02) funded by the ”Investissements d’Avenir” French government program tral energy distribution, from radio to γ rays. The diffuse managed by the ANR. γ-ray emission of particles accelerated in the jet termi- nation shocks and interacting with ions in the molecular cloud where the protostar is embeded will be also mod- eled, as well as the emission of secondary particles. 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Shu F., Najita J., Ostriker E., Wilkin F., Ruden S., Lizano S., e /K ξ 3 1994, ApJ, 429, 781 200 p Suzuki-Vidal F., Bocchi M., Lebedev e. a., 2012, Physics of Plas- K mas, 19, 022708 2
Suzuki-Vidal F., Lebedev S. V., Ciardi A., Pickworth L. A., Ro- 100 driguez R., Gil J. M., Espinosa G., Hartigan P., 2015, ApJ, 1 815, 96 Te¸sileanu O., Mignone A., Massaglia S., Bacciotti F., 2012, ApJ, 0 0 746, 96 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 Tzeferacos P., Rigby A., Bott A. F. A., Bell A. R., Bingham R., Non-thermal particles spectral index s Casner A., Cattaneo F., et al 2018, Nature Communications, 9, 591 Vink J., Laming J. M., 2003, ApJ, 584, 758 Figure A1. Functions ξK (Eq. A6), ξ , and ξ (left y-axis) and Vurm I., Metzger B. D., 2018, ApJ, 852, 62 eq sat Kp/Ke (right y-axis) as a function of the spectral index of non- Xu S., Lazarian A., 2017, ApJ, 850, 126 thermal particles. Yirak K., Schroeder E., Frank A., Cunningham A. J., 2012, ApJ, 746, 133 Zirakashvili V. N., Ptuskin V. S., 2008, ApJ, 678, 939 where Zweibel E. G., Everett J. E., 2010, ApJ, 709, 1412 van Marle A. J., Casse F., Marcowith A., 2018, MNRAS, 473, s+5 3s−1 3s+19 Γ 4 Γ 12 Γ 12 3394 (s) ' −3.55(s−2) . (A6) ξK 10 s+7 Γ 4 (s + 1)
The function ξK(s) is plotted in Figure A1. APPENDIX A: SYNCHROTRON EMISSION The synchrotron emissivity per unit frequency ν emitted by a source with volume Ve located at distance d from Earth is APPENDIX B: PARTICLE ENERGY DISTRIBUTION IN 2 THE SHOCK DOWNSTREAM REGION εsyn,ν = 4πd Sν /Ve, giving Non-thermal particles accelerated in a non-relativistic shock ε 1.2 × 10−30 d 2 S R −3 syn,ν = ν , follow a power-law energy distribution −1 −1 −3 16 ergHz s cm fVe kpc mJy 10 cm ( (1−s) s+1 (A1) 2 2 − 2 2 N = Ke,pme,pc Ee,p Einj < Ee,p < me,pc , e,p −s 2 where Sν is the flux at frequency ν. We have defined Ve = Ke,pEe,p me,pc < Ee,p < Ee,p,max, 3 fVeRj , where Ve is the volume filled in with non-thermal elec- (B1) trons and fVe is the volume filling factor. We consider that where e and p stands for electrons and protons, respectively, ε , = 4πε , where the emission coefficient for synchrotron syn ν ν 2 radiation is and Einj ∼ 2mpvsh. We note that the condition Ne = Np at 2 (s−1)/2 1−s E < mec gives Kp/Ke = (mp/me) (see Fig. A1). The s+ s+1 1 ν 2 2 spectrum is flatter in the non-relativistic regime, and there- εν = c5KehsinΘ 2 iBs , (A2) 2c1 fore non-relativistic particles do not contribute significantly 18 to the total energy density U and hence most of the Θ is the electron pitch angle, c1 = 6.264 × 10 Hz and e,p non-thermal number and energy density is due to particles −23 2 1.866 × 10 3s − 1 3s + 19 with Ee,p me,pc . In the case of negligible energy losses, c = Γ Γ ergG−1 sterad−1 & 5 (s + 1) 12 12 the total energy density stored in non-thermal particles is (A3) Ue,p,tot = Ke,p fe,p(s), where 1 2−s (Arbutina et al. 2012; Beck & Krause 2005). We assume an Ee,p,max if s < 2 2−s isotropic distribution for the orientation of Θ and therefore Ee,p,max fe,p ∼ log m c2 if s = 2 (B2) e,p s+5 1 2 2−s √ Γ s−2 (me,pc ) if s > 2. s+1 π 4 hsinΘ 2 i = . (A4) 2 s+7 We note that the ratio a = Up,tot/Uetot can be written as a = Γ 4 (s−1)/2 (mp/me) fp/ fe, and for s > 2 it can be approximated as By combining Eqs. (A1)-(A4) we find that (3−s)/2 a ∼ (mp/me) . In Figure B1 we plot fp, fe, and a for 2 −3 K ξ (s) d S R 1.6 ≤ s ≤ 2.6 and Ee, = Ep, = 0.1, 1, and 10 TeV. e ≈ . × −9 K ν max max s−1 −3 1 6 10 16 erg cm fVe kpc mJy 10 cm The energy and number density of particles with a cer- 1−s 2−s s+1 tain energy E are n = K E and U = K E , re- s−1 − e,p e,p e,p e,p e,p e,p e,p ν 2 B 2 × s , (A5) spectively. We are particularly interested in protons, given GHz mG that they are responsible for the CR streaming instability.
MNRAS 000,1–16 (2015) MFA in jets from MYSOs 17
10000 25 Emax = 1 TeV 2 ) c Emax = 0.1 TeV e = m Emax = 10 TeV e,min 20 1000
Electrons (E 15 a e,p 100 f
10
10 2 c ) 5 = m p p,min Protons (E 1 0 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 Non-thermal particles spectral index s
Figure B1. Functions fe and fp (left axis) and a (right axis) as a function of the spectral index of non-thermal particles for different values of Ee,max and Ep,max.
The mumber and energy density of protons with energy Ep are 1−s Up,tot (2 − s) Ep if s < 2 Ep,max Ep,max −1 Up,tot 1−s Up,tot Ep np = E ∼ Ep,max 2 if s = 2 p m c2 log mpc fp p mc2 1−s Up,tot Ep 2 (s − 2) 2 if s > 2 mpc mpc (B3) and
2−s Ep Up,tot (2 − s) if s < 2 Ep,max U Up,tot p,tot 2−s if s = 2 Up = Ep ∼ log Ep,max fp mpc2 2−s Ep Up,tot (s − 2) 2 if s > 2, mpc (B4) respectively.
This paper has been typeset from a TEX/LATEX file prepared by the author.
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