Astrophysical magnetohydrodynamical outflows in the extragalactic binary system LMC X-1
aa r X i v : . [ a s t r o - ph . H E ] S e p Astrophysical magnetohydrodynamical outflows inthe extragalactic binary system LMC X-1
Th V Papavasileiou , ,a , D A Papadopoulos ,b and T S Kosmas ,c Division of Theoretical Physics, University of Ioannina, GR-45110 Ioannina, Greece Department of Informatics, University of Western Macedonia, GR-52100 Kastoria, GreeceE-mail: a [email protected], b [email protected], c [email protected] Keywords : XRBs, relativistic jets, neutrino production, extragalactic,LMC X-1, γ -ray emission Abstract.
In this work, at first we present a model of studying astrophysical flows of binarysystems and microquasars based on the laws of relativistic magnetohydrodynamics. Then,by solving the time independent transfer equation, we estimate the primary and secondaryparticle distributions within the hadronic astrophysical jets as well as the emissivities of highenergy neutrinos and γ -rays. One of our main goals is, by taking into consideration the variousenergy-losses of particles into the hadronic jets, to determine through the transport equationthe respective particle distributions focusing on relativistic hadronic jets of binary systems. Asa concrete example we examine the extragalactic binary system LMC X-1 located in the LargeMagellanic Cloud, a satellite galaxy of our Milky Way Galaxy.
1. Introduction
In recent years, astrophysical magnetohydrodynamical flows in Galactic and extragalactic X-ray binary systems and microquasars have been modelled with the purpose of studying theirmulti messenger emissions (radiative multiwavelength emission and particle, e.g. neutrino,emissions) [1, 2]. For the detection of such emissions, extremely sensitive detector tools arein operation for recording their signals reaching the Earth like KM3NeT, IceCube, ANTARES,etc. Modelling offers good support for future attempts to detect them while in parallel severalnumerical simulations have been performed towards this aim [3, 4, 5].In general, the astrophysical jets, and specifically those coming from microquasars, may wellbe described as fluid flow emanating from the vicinity of the compact object of the binary system.We mention that, microquasars are binary systems consisted of a compact stellar object and adonor (companion) star. Well known microquasar systems include the Galactic X-ray binariesSS433, Cyg X-1, etc., while from the extragalactic ones we mention the LMC X-1, LMC X-3(in the neighbouring galaxy of the Large Magellanic Cloud) [6, 7], and the Messier X-7 (in theMessier 33 galaxy). Their respective relativistic jets emit radiation in various wavelength bandsand high energy neutrinos.Up to now, the SS433 is the only microquasar observed with a definite hadronic contentin its jets, as verified from observations of their spectra (see Ref. [3, 4, 5] and referencestherein). Radiative transfer calculations may be performed at every point in the jet, for a rangef frequencies (energies), at every location [8], providing the relevant emission and absorptioncoefficients. Line-of-sight integration, afterwards, provides synthetic images of γ -ray emission,at the energy-window of interest [8, 9].The relativistic treatment of jets, considers various energy loss mechanisms that occur dueto several hadronic processes, particle decays and particle scattering [1, 2]. In the known fluidapproximation, macroscopically the jet matter behaves as a fluid collimated by the magneticfield. At a smaller scale, consideration of the kinematics of the jet plasma becomes necessaryfor treating shock acceleration effects.In the model employed in this work, the jets are considered to be rather conic along the z-axis(ejection axis) with a radius r ( z ) = z tan ξ , where ξ its half-opening angle. The jet radius atits base, r , is given by r = z tan ξ , where z is the distance of the jet’s base to the centralcompact object. According to the jet-accretion speculation, only 10% of the system’s Eddingtonluminosity ( L k = 1 . × M erg/s , M in solar masses) is transferred to the jet for accelerationand collimation through the magnetic field given by the equipartition of magnetic and kineticenergy density as B = p πρ k ( z ) (see Ref. [1, 2, 8, 9].In the hadronic models assumed in this work, a small portion of the hadrons (mainly protons) q r ≈ . t − acc ≃ ηceB/E p due to the 2nd order Fermi accelerationmechanism to nearly relativistic velocities (with η = 0 . N ′ ( E ′ ) = K E ′− GeV − cm − ,where K is a normalization constant.
2. Interaction mechanisms and Energy loss rates in hadronic astrophysical jets
In the recent literature, three are the main interaction mechanisms for relativistic protons.These include interaction with a stellar wind, a radiation field composed of internal and externalemission sources and, finally, the cold hadronic matter of the jet. In this work we focus on thelast interaction because it dominates over the other two.The p-p interactions initialise a reaction chain leading finally to neutrino and γ -rayproduction, that can reach as far as the Earth where they are being detected by underseawater and under-ice detectors such as KM3NeT, ANTARES and IceCube. The aforementionedreaction chain begins with the inelastic p-p collisions of the relativistic protons on the cold onesinside the jet which result to neutral ( π ) and charged ( π ± ) pion production. Neutral pionsdecay into γ -ray photons while the charged ones decay into muons and neutrinos. Subsequently,muons also decay into neutrinos. These are the main reactions feeding the neutrino and gamma-ray production channel in our models. Furthermore, high energy emission spectra can be moreemphatically explained by leptonic models where the energy is transferred by leptons (electrons)instead of hadrons, whereas hadronic models are more suitable for neutrino production processes.All particles that take part in the neutrino and gamma-ray production processes lose energywhile travelling along the acceleration zone which could be due to different mechanisms discussedbelow. At first, the particles can be subjected to adiabatic energy losses due to jet expansionalong the ejection axis with a rate t − ad = 2 υ b z (1)where υ b is the jet’s bulk velocity. Particles can also lose energy because they collide with thejet’s cold matter with rate given by t − ip = 12 n ( z ) cσ inelip ( E ) , n ( z ) = (1 − q r ) L k Γ m p c πr ( z ) υ b . (2)In the latter expression, n ( z ) denotes the cold protons density and Γ is the Lorentz factorcorresponding to the jet’s velocity. The factor 1 / i -index represents the different particles that take part in the collisions with thecold protons. These, due to small muon mass can be mainly relativistic protons and pions.The inelastic cross section for the p-p scattering is given in [10] which equals the cross sectionregarding the π − p scattering with a 2/3 factor [3, 4, 5].In addition, particles accelerated by the magnetic fields emit synchrotron radiation. Thus,gradually they lose part of their energy with a rate t − sync = 43 (cid:16) m e m (cid:17) σ T B πm e c γ (3)where γ = E/mc and σ T = 6 . × − cm , the known Thomson cross section.Finally, protons and pions can lose energy interacting through X-ray, UV and synchrotronradiation due to photo-pion production, while smaller particles such as muons transfer partof their energy to low-energy photons due to inverse Compton scattering. However, suchcontributions can be ignored compared to those mentioned above.
3. Calculation of the particle distributions
In the steady-state model, the jet’s particle distributions obey the transfer equation as [3, 4, 5] ∂N ( E, z ) b ( E, z ) ∂E + t − N ( E, z ) = Q ( E, z ) , (4)where N ( E, z ) is the particle number per unit of energy and volume (
GeV − cm − ) while Q ( E, z ) is the particle source function representing the corresponding production rate (in
GeV − cm − s − ). The energy loss rate, b ( E ) = dE/dt , contains all the cooling mechanismsdiscussed before so that b ( E ) = − Et − loss .Moreover, t − corresponds to the rate at which the number of particles decreases, eitherbecause of escaping from the jet or because of decaying so that t − = t − esc + t − dec . The escaperate is t − esc = c/ ( z max − z ), with z max − z being the length of the acceleration zone.The general solution of the differential equation (4) is N ( E, z ) = 1 | b ( E ) | Z E max E Q ( E ′ , z ) e − τ ( E,E ′ ) dE ′ . (5)It is worth mentioning that, since Eq. (4) holds for protons, pions and muons, a system of threecoupled equations is required to be appropriately solved in order to find the distributions of theparticles involved in the reaction chain (protons, pions, muons). Afterwards, the calculations ofneutrino and γ -ray emissivities can be found through the source functions Q ( E, z ). As discussed previously, a realistic source function Q ( E, z ) for the relativistic protons is a power-law distribution. In the jet’s rest frame, due to the Fermi mechanism combined with the time-independent continuity equation this is written as Q ( E ′ , z ) = Q (cid:16) z z (cid:17) E ′− , Q = 8 q r L k z r ln ( E maxp /E minp ) , (6)where Q is related to K (see the Introduction above), and E minp = 1 . GeV is the minimumproton energy. The maximum energy is assumed to be equal to E maxp ≃ GeV . The aboveinjection function transforms to the observer’s reference frame as described in Ref. [4, 5] .1.1. Pion distribution
The pion source function is obtained by the product of the abovedistribution and the total number of p − p collisions as Q π ( E, z ) = cn ( z ) Z EEmax N p (cid:18) Ex , z (cid:19) F π (cid:18) x, Ex (cid:19) σ inelpp (cid:18) Ex (cid:19) dxx (7)where x = E/E p and F π ( x, E/x ) denotes the pion mean number produced per p − p collision[10]. As can be implied from Eq. (7), the proton distribution is entering the integrand of ther.h.s. in order to provide the pion source function. In a similar manner, the mean right handed and left handed muonnumber per pion decay is integrated in the total injection function considering the CP invarianceand also provided that N π ( E π , z ) = N π + ( E π , z ) + N π − ( E π , z ) as [1, 2] Q µ ± R ,µ ∓ L ( E µ , z ) = Z E max E µ dE π t − π,dec ( E π ) N π ( E π , z ) N ± µ Θ( x − r π ) (8)where N ± µ represent the positive (negative) right (left) handed muon spectra, respectively.Furthermore, in Eq. (8) x = E µ /E π , r π = ( m µ /m π ) and Θ( y ) the Heaviside function. Wemention that, the pion decay rate is t − π,dec = (2 . × − γ π ) − s − which implies that, the piondistribution is important for calculating the muon distribution. From the above discussion we see that, neutrinos are produced directly from pion decay as wellas from their decay-products, i.e. charged muons ( µ ± ). Thus, the total emissivity considersboth contributions as Q ν ( E, z ) = Q π → ν ( E, z ) + Q µ → ν ( E, z ) (9)The first term gives the neutrino injection originating from pion decay as Q π → ν ( E, z ) = Z E max E t − π,dec ( E π ) N π ( E π , z ) Θ(1 − r π − x ) E π (1 − r π ) dE π (10)( x = E/E π ) while the second term gives Q µ → ν ( E, z ) = X i =1 Z E max E t − µ,dec ( E µ ) N µ i ( E µ , z ) (cid:20) − x + 43 x + (3 x − − x ) h i (cid:21) dE µ E µ (11)( x = E/E µ ). In the latter equation, t − µ,dec = (2 . × − γ µ ) − s − , and h = h = − h = − h =1. From the four different integrals of the latter summation, the first and second represent theleft handed muons of positive and negative charge, respectively, while the third and fourth standfor the corresponding right handed ones [6]. Finally, one may evaluate the neutrino intensity byintegrating the emissivity over the acceleration zone [2, 1] I ν ( E ) = Z V Q ν ( E, z ) d r = π ( tanξ ) Z z max z Q ν ( E, z ) z dz (12)Such calculations will be presented elsewhere [6]. able 1. Model parameters describing geometric characteristics of the extragalactic LMC X-1,in the Large Magellanic Cloud, and the Galactic Cygnus X-1 binary systems.Description Parameter LMC X-1 Cygnus X-1Jet’s base z × [ cm ] 1 × [ cm ]End of acceleration zone z max × [ cm ] 5 × [ cm ]Mass of compact object M BH M ⊙ [11] 14.8 M ⊙ [12]Angle to the line-of-sight θ ◦ [11] 27.1 ◦ [12]Jet’s half-opening angle ξ ◦ ◦ Jet’s bulk velocity υ b
4. Results and discussion
In the present work, one of our goals is to calculate the cooling rates and energy distributionsof all particles participating in the chain reactions of p − p mechanism that takes place in thehadronic astrophysical jets of binary stars and microquasars. Then, the energy spectra of theproduced high-energy neutrinos and gamma-rays are simulated numerically through the solutionof the corresponding transfer equations.By employing a C-code developed by our group here (it uses the Gauss-Legendre numericalintegration of the GSL library), we concentrated on performing extensive calculations for theGalactic Cygnus X-1 and the extragalactic LMC X-1 binary systems. The parameter values usedfor LMC X-1 (and Cygnus X-1) are listed in Table 1. By using the values of the parameterslisted in Table 1, mostly describing geometric characteristics of these systems, in Fig. 1 wedisplay the proton, pion and muon cooling rates calculated for the aforementioned systems. -4 -2 t - [ s - ] E [GeV]
Proton cooling rates
Synchrotron emissionParticle collisionAdiabatic cooling -4 -2 E [GeV]
Pion cooling rates
Synchrotron emissionParticle collisionAdiabatic coolingDecay -2 E [GeV]
Muon cooling rates
Synchrotron emissionAdiabatic coolingDecay -4 -2 t - [ s - ] E [GeV]
Proton cooling rates
Synchrotron emissionParticle collisionAdiabatic cooling -2 E [GeV]
Pion cooling rates
Synchrotron emissionParticle collisionAdiabatic coolingDecay -2 E [GeV]
Muon cooling rates
Synchrotron emissionAdiabatic coolingDecay
Figure 1.
Cooling rates for the relativistic protons and the secondary particles ( π ± , µ ± )produced after the p-p mechanism takes place in LMC X-1 (1st row) and Cyg X-1 (2nd row).s can be seen, the particle synchrotron losses become dominant for large energies with theparticle mass setting the separation point. In the case of protons, due to their large mass,synchrotron losses are not dominant up to very high energies. This leads to a distribution withstable inclination matching the hot proton power-law exponent. On the other hand, for pionsand muons, we see that the decay dominates the lower energy band and that is why the twocurves in the respective graphs (corresponding to distributions considering energy losses alongwith the decay process only) progress similarly, especially for energies up to the decay ratestabilization. The synchrotron losses take off causing the smoother transition in the solid line’scase (with energy losses) compared to the dashed one (only decay).After calculating all the necessary distributions, the neutrino and γ -ray emissivities as wellas the corresponding intensities may be provided. For the LMC X-1 system, our results haveshown that, the increase of the half-opening angle ξ leads to a decrease in the γ -ray production,which is an expected result since the p-p collision rate drops with the jet’s expansion [6].
5. Summary and Conclusions
Black Hole X-ray binary systems (BHXRBs), consisting of a high mass compact object (blackhole) and absorbing mass out of a companion star that results in an accretion disc formation, havebeen identified through their relativistic magnetohydrodynamical astrophysical flow ejectionperpendicular to the aforementioned disc. This flow is mostly accelerated and collimated by thepresence of a rather strong magnetic field which is initially attached to the rotational disc. Aportion of the hadronic jet’s particles (mostlye protons) are accelerated to relativistic velocitiesthrough shock waves travelling across the jet. Then, a reaction chain takes place stemming fromthe inelastic p − p interactions which leads to production of neutrinos and high energy γ -rays,both detectable at the terrestrial extremely sensitive detectors.In this work, we focus mainly on the mechanisms and phenomena that affect highly theinelastic p − p scattering and the generated secondary particles which participate afterwards inreaction chains leading to the emission of neutrinos and γ -rays. For two concrete examples, theGalactic Cygnus X-1 and the extragalactic LMC X-1 binary systems, we studied in more detailtheir cooling rates and energy distributions. Numerical simulations of neutrino emissivities andneutrino intensities will be publised elsewhere.
6. Acknowledgments
TSK acknowledges that this research is co-financed by Greece and the European Union(European Social Fund-ESF) through the Operational Programme ”Human ResourcesDevelopment, Education and Lifelong Learning 2014- 2020” in the context of the project(MIS5047635).
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