The use of genetic algorithms to find an optimum MHD fluid solution as an approximate description of black holes relativistic jets

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Faculteit der Natuurwetenschappen, Wiskunde en Informatica Graduate School of Informatics, MSc Computational Science

Universiteit van Amsterdam

Department of High Performance Computing In Multidisciplinary Research Faculty of Informational Technologies and Programming

Saint Petersburg National Research University ITMO

the use of genetic algorithms to find an optimum

mhd fluid solution as an approximate description

of black holes relativistic jets

Klim Mikhailov 10668756

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Abstract

This work mainly focuses on near-conical relativistic steady-state jets that originate from accreting black holes: stellar-mass black holes in X-ray binaries (BHBs) and their supermassive counterparts in active galactic nuclei (AGN). A new approach for ap-plying genetic algorithms (GAs) in order to fit observational data with the theoretical predictions provided by the semi-analytical outflow dominated jet model is consid-ered. For data fitting tasks, it is very crucial to find either from physical background or simply from common sense a suitable initial set of given model parameters. The reason is to avoid falling into the local minimum or even into a program error caused by either interpolation or integration subroutines to obtain model spectrum: this prob-lem, for instance, applies to Interactive Spectral Interpretation System (ISIS), which is widely used in the tasks of such kind. We test the supposition that for GAs there is no need for finding model parameters initial approximations which might also increase computational efficiency compared to most local optimisers. An important detail for this specific algorithm construction is that all its genetic operators should incorporate a portion of physics underlying in the task domain in order to conserve one-to-one map-ping carried through by the operators themselves. This incorporation results in dealing with real number genes as well as their tolerance regions during the algorithm perfor-mance. The fitness function was chosen to be reduced linear χ2 measure of difference between model estimations and observational data. The implemented algorithm itself contains all traditional GAs operators (selection, crossover, mutation, and reproduc-tion) with relation to the considered problem. Namely, we use roulette-wheel selection principle together with population sifting; one-point crossover with a certain proba-bility of its application; dynamically adjusted mutation operator that is applied to all individuals relative to their fitness functions; partial reproduction together with elite members maintenance. The algorithm provides reasonably good fits for GX 339-4 and GRS 1915+105 X-ray binaries and can easily be done in parallel.

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Table of constants (in CGS units)

G gravitational constant: G ≈ 6.674 × 10−8cm3· g-1_·s-2

c speed of light: c ≈ 2.997 · 1010_cm·s-1

h Planck’s constant: h ≈ 6.626 · 10−27erg·s kB Boltzman’s constant: kB ≈ 1.38 · 10−16erg·K-1

σSB Stefan–Boltzmann’s constant: σSB ≈ 5.67 · 10−5 erg·s-1·cm-2·K-4

M Solar mass: M≈ 1.989 · 1033g

L Solar luminosity: L ≈ 3.839 · 1033erg·s-1

Jy Jansky (flux density unit): 1 Jy = 10−26W·m-1_·Hz-1

kpc kiloparsec (astronomical distance unit): 1 kpc ≈ 3.086 · 1021cm σT Thompson cross section: σT ≈ 6.652 · 10−25cm2

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Chapter 1 Introduction

Why do the accretion discs surrounding certain astronomical objects, such as the nuclei of active galaxies, emit relativistic jets along their polar axes?

«List of unsolved problems in physics» (Ginzburg, 2001)

When looking closely at the depths of the clear space sky through the powerful as-tronomical photometers or reflectors as located, for example, in Chandra X-ray Obser-vatory (Kaastra et al., 2000), one can find huge bright blobs spreading in both directions from the central object emitting them. Initially described by H. D. Curtis during the observation of Messier 87 (Messier, 1781, hereafter M87) elliptical galaxy as «a curious straight rays....apparently connected to the nucleus» (Curtis, 1918; Sparks et al., 1992), they now serve role as the carriers of matter throughout the ever-growing Universe. From their observational appearance, these lighted knots embody highly collimated streams of plasma matter ejected directly from the astronomical central object – this is exactly what is called jets nowadays. A pronounced diversity in the total spectra obtained from observations (see Fig. 1.1 as a concrete example) gives an indication that jet behaviour is very far from trivial. Gradually improved instruments as well as com-puting equipment in total allow to obtain more qualified data which, in turn, let us construct more sophisticated models to describe jets in more details. Nevertheless, even today, despite the fact that we already have a lot of estimated information relative to jets origin, at the same time, however, there are still a lot of unanswered questions concerning the details of their physics left.

1.1 Why are jets scientifically interesting?

It is natural to ask yourself - why do we care so much about these outlying creations of our Universe? Are they really such an important response to our daily lives? Ob-viously, they cannot guarantee incommensurable profit on the stock market or ensure our longevity, but if thinking larger, they are able to provide some explanation of what

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K. Mikhailov 1.1 Why are jets scientifically interesting?

Figure 1.1: An example of X-ray Binary A 0620-00 spectrum represented as a multiwavelentgh spectral fit with an outflow dominated jet model. Each line corresponds to a particular con-tribution to the total envelope spectrum denoted by dark blue line. Additional highlighted red solid line represents detector space model fit. Figure reproduced from Gallo et al. (2007), further details concerning spectrum components and the model itself will be provided in Chapters 2 and 3. [See the electronic edition of the Thesis for a color version of this figure.]

happened during the early stages of our Universe and is happening with it at the time of writing this thesis.

From a global point of view, relativistic jets serve as carriers providing the transfer of matter. This results in carrying such conservative quantities as plasma energy and (angular) momentum on relatively large distances, depending on central object mass. For instance, supermassive black hole (SMBH) jets are capable of performing this trans-fer on very large distances due to their huge sizes - it is known that these objects may have a length of more than a thousand of light years (Uchiyama et al., 2006). Therefore, the local matter density of the Universe regions located relatively far from jets origin might suddenly be enhanced by energetic particles carried by these jets. And this, in turn, may induce new star formations - theoretically, this might happen when a jet col-lides with the intergalactic cloud, thus providing a subsequent collapse of the latter (van Breugel et al., 2004). Therefore, if we want to understand the reason why our Universe appears to what we can see today, jets are a good thing to deal with. Unfortunately, we are still not able to see much - for example, theoretical deductions are pointing on the presence of gravitational waves (Einstein, 1915; Shapiro & Teukolsky, 1983) or different types of dark matter (Oort, 1932; Zwicky, 1933; Trimble, 1987), but there are

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still no strong observational evidences in their actual existence attitude 1_{. This}

pro-vides a strong reason why people always tend to construct more powerful instruments with more huge measurement ranges as well as higher instruments resolution: such tools as, for instance, Einstein Telescope (ET, Punturo et al. 2010) and Square Kilometre Array (SKA, Jarvis & Rawlings 2004) for binary black holes as well as pulsars detec-tion or Evolved Laser Interferometer Space Antenna (eLISA, Amaro-Seoane et al. 2013) and a third generation of Background Imaging of Cosmic Extragalactic Polarisation (BICEP, Keating et al. 2003) for gravitational waves measurement are already in the pipeline and supposed to greatly enrich our knowledge.

Because of enormously high speeds inherent in jets, their treatment cannot avoid principles of special relativity theory establishing the relations between space and time for fast moving and rest observational frames. Moreover, due to the fact that jets usu-ally originate from an object which possesses huge compactness 2_{, there should be a}

huge depth of the potential well in the vicinity of this massive object. For this reason, we should deal with curved spacetime and thereby with general relativity theory treat-ing gravity as intrinsic geometrical property of four-dimensional space-time, in order to correctly investigate jets mainly near the origin of their formation, at small distances from the central object emitting them. In this regard, these two fundamental postu-lates historically established by Einstein (1905, 1915) can again be tested to prove their viability - jets provide a good chance to perform that.

The variety of different objects emitting jets is very large. It extends from fading white dwarfs, pulsar wind nebulae and young stars to heavy neutron stars or stellar black holes in X-Ray Binaries as well as supermassive black holes hidden in the galactic centres (see, e.g., Körding & Fender, 2006; Laing & Bridle, 2014). A typical depiction of stellar black hole scooping the mass from its companion star, thus resulting in jets launching, is presented in Fig. 1.2.

Therefore, jets appear to be a very common feature, nevertheless also having differ-ences: thus, the most powerful and as a result better observable of them are provided by deep potential well of compact objects. Moreover, a large amount of data achieved either by observations or simulations point on a strong coupling between jets and sev-eral significant but still not fully understood phenomena: highly luminous Gamma Ray Bursts (GRBs, Piran 2004; Gomboc 2012), catastrophic and very powerful Tidal Disrup-tion Events (TDEs, Rees 1988; Gezari 2013), high-energy cosmic rays (CR, Hess 1928) etc.3 _{Therefore, studying jets from various cosmic objects might also let us understand}

their fundamental differences. Roughly speaking, we have a global problem - in order to understand jets, we also need to grasp what entangles all the objects ejecting them. And since the more global the problem, the more efforts one should spend on it, jets need to be placed in rank of treatment task of higher priority.

1_{It is worth noting that in the latter case a good indication might serve gravitational lensing of the}

Bullet Cluster 1E 0657-558 which is a product of two galaxy clusters collision: its lensing profile provides excellent separation of interstellar gas from potential dark matter (Clowe et al., 2006).

2_{Stellar object compactness is treated as a ratio between object mass and its radius, see Chapter 2.} 3_{Details will be provided in Chapter 2 describing main properties of possible jet sources.}

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Figure 1.2: Artist’s impression of black hole NGC300 X-1 in orbit around its Wolf-Rayet star. Figure obtained from Astronomy Central (http://astronomycentral.co.uk/). After grab-bing enough mass by means of companion star, black hole (or probably its accretion disc) starts emitting jets on either side of it. This process will last until the moment when the star gas becomes completely exhausted and disappear in the depths of black hole, thus destroying the present binary system, or when the star itself undergoes supernova explosion and breaks the system apart.

Because the internal composition of jets is still not well understood, it is of huge sig-nificance to find out what exactly feeds these elongated plasma matter streams. Indeed, if we are able to understand the content of jets, it will be possible to estimate the inner structure of their transmitters. This will give us great opportunities to get a feeling of how this jet plasma matter can subsequently influence adjacent Universe regions.

One can also argue that the role jets play in modern science might be seen from a position of multidisciplinary research. Strong reasons are because jets implicitly define multiple astronomical components by incorporating their characteristics inside inner structure as well as giving rise to the countless set of modern technologies and numer-ical approaches. But furthermore, this can be considered from the fact that in order to precisely describe jet behaviour, one needs to apply computational efforts due to its highly strong complexity. This is not so surprising: nowadays it is quite hard to find a discipline in which computers would be of no use. Conversely, without these powerful digital calculators it will be practically impossible to deal with most of the tasks requiring abundant computations. In the context of astrophysics, it will always require computational methods since it contains a mixture of large scale properties such as stars dynamics together with microscopic properties such as radiation theory or gas particles propagation. Thus, to investigate jet properties we need a huge amount of computational help since the equations obtained from the theory are very unlikely to

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K. Mikhailov 1.2 The project

be solved analytically (Meier et al., 2001). Therefore, we need to refer to numerical so-lutions - this is where computers come into game. But even with them and with lots of approximations, finding correct values is not so easy - things get quite time consuming, leading to the growth of computing power. For this reason, scientists decided to build even more powerful calculators, unifying them into groups called computer clusters. Thus, from «tiny» personal computers that are common in everyone mode of life, we now came to large numerical aggregates known as supercomputers that let us gradu-ally reduce the amount of time required for doing computations by performing them in parallel. But not only building powerful machines is needed for performing such wide scale tasks like jets investigation. A good example of unifying computations is creating World Wide Web service by CERN in order to faster share numerical results between different scientific groups (Berners-Lee & Fischetti, 1999). Also a significant role plays an algorithm which we chose for our computations: some of them might spend a lot of time for the search and still find only local solution that we generally do not prefer. In any case, it seems that jets are exactly one of these objects that encourage development of new technologies and means with which it is possible to deal with those hard nuts to crack.

To sum up, jets appear to be a very promising candidate to open simultaneously a number of problematic doors to various cosmological pieces of the Universe highlighted above. Even if this scale task is quite hard to explain, all in all, such tight connections with cutting-edge astronomical puzzles as well as with state-of-the-art computational technologies make jets being one of the modern problems which deserve close attention of many scientific groups.

1.2 The project

The goal of this project is to precisely consider jets from a multidisciplinary point of view: namely, to fit the spectra provided by different XRBs or AGN sources using a given number of parameters from the unified semi-analytical jet model (see, e.g., Fal-cke & Biermann, 1995; FalFal-cke & Markoff, 2000; Markoff et al., 2001, 2005; Gallo et al., 2007; Migliari et al., 2007; Markoff et al., 2007, 2008; Maitra et al., 2009a,b, 2011; Nowak et al., 2011), which is based on the sources fundamental principles described below (see Chapter 2), and find such parameters using genetic algorithms. The precise list of ini-tial parameters that are used in the model as well as their tolerance regions will be considered in Chapter 3. Moreover, there will also be discussed various types of ob-servational regions used for data fitting with the model. The spectral fitting is usually applied using a linear χ2 function as a fitness measure, which we also utilise in this work. The general theory of genetic algorithms (GAs) together with their possible opti-misations as well as parallel implementations will be considered in Chapter 4. Since it is in principle possible to start with random initial parameter values during the search of the fittest parameter set, defining them due to their sensible tolerance will increase the chance of finding the fittest set faster. Moreover, this will also make the search safer

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K. Mikhailov 1.2 The project

since the values beyond parameter tolerance regions might have no physical meaning and as a result might lead to an integration or interpolation error during the model spectrum calculation. In addition, modern studies showed that it is in general of huge importance to consider task specifics and implement GAs according to it (Chen et al., 2011; Tomoiaga et al., 2013). Hence, there is a need to construct such genetic operators that will be able to place newly achieved genes inside the same appropriate tolerance region, thus making the mapping being one-to-one. Such operators will be discussed in Chapter 5. After that in Chapter 6, the algorithm will be compared with traditional X-ray spectral fitting modules such as X-ray spectral-fitting program (XSPEC, Arnaud 1996) and Interactive Spectral Interpretation System (ISIS, Houck & Denicola 2000) to compare their computational strengths and weaknesses. Finally, in Chapter 7 the appro-priate conclusions concerning algorithm performance and efficiency as well as physical significance of the obtained fittest parameters will be made.

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Chapter 2 What are jets?

The intrinsic physics of jets origin and their successive propagation still contains a considerable amount of gaps in its total description. The most unknown details are lo-cated at the exact beginning of their formation, in the vicinity of central massive object that is supposed to play one of the key roles in giving jets rise. It is still not known precisely what the main mechanism of jet launch is - there exist a few standard theo-ries concerning this phenomenon, though none of them are able to explain it in every detail. The same situation takes place with a couple of distinctive jets characteristics -acceleration and collimation. Besides, since a large diversity of heterogeneous objects provide plasma outflows, one should expect similar physics unifying all of them. This chapter provides a literature review of what is already known about these objects char-acteristics as well as jets intrinsic properties themselves. There are plenty of good books about physics related to jets and massive objects launching them, the main five used as inspiration for this chapter are Rybicki & Lightman (1979, hereafter RL79), Frank et al. (2002, hereafter FKR02), Guthmann (2002, hereafter G02), Rosswog & Brüggen (2007, hereafter RB07), and Meier (2012, hereafter M12).

2.1 Related physics

As was previously stated in Chapter 1, highly collimated streams of plasma matter or jets, ejected directly from the astronomical central object, are a quite widespread phenomenon. In the mean time, it is almost incorrect to study jets based only on classical (Newtonian) assumptions. To give a specific example, the propagation of jets happens with very high speeds close to the speed of light c1_{. Thus, the characteristic}

speed measure, bulk Lorentz factor γ2_{, might increase to a value of ∼ 30 or even more,}

which corresponds to a highly luminal motion of about 99.94 % of c (RB07), so that jets should be treated based on special relativity principle. In addition, it is also set from observations that the initial velocity of jets propagating from the central object

1_{This and below constant values are given in the appropriate table of constants, see p. iv.} 2_{Lorentz factor is defined as γ = 1/p1 − v}2_/c2_.

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K. Mikhailov 2.1 Related physics

might be described in terms of escape velocity treated as a ratio between object’s mass and the distance between it and escaping body: vesc = p2GM/r, thereby signifying

its dependence on central object mass. This, for instance, results in a fact that we also need to take general relativity into account starting from 10 rShnear the SMBH (Gezari,

2013) if we consider AGN jets, where rSh =

2GM

c2 (2.1)

is a Schwarzschild radius rSh, beyond which all the infalling to the non-spinning black

hole matter ceases to be visible and as a result open for investigation (Schwarzschild, 1916). Luckily, the latter is of huge need only at the vicinity of massive bodies, namely within their sphere of influence which is defined as a region inside which a considered massive object has more dominant gravitational potential than the other surrounding objects (RB07). Nevertheless, the presence of such a large amount of mass in the centre of considered non-spherically symmetric system might also allow us to treat it as a potential source of gravitational waves (de Freitas Pacheco, 2010).

Since jets include highly relativistic plasma inside, the main types of elementary particles proceeded to outflow are protons, electrons, and positrons. Despite the fact that early hydrodynamical models (see, e.g., Falcke & Biermann, 1995) propose only leptonic content of relativistic ejecta, this appeared to be inconsistent with the total jet power distribution for energetic reasons, since protons due to their much bigger mass compared to leptons should carry jet intrinsic kinetic energy, while relatively lightweight electrons or their opposite sign twins positrons might be responsible for the emanating jet radiation (Markoff et al., 2005). Furthermore, recently it became known about baryons as well as ordinary atoms included in jets relativistic matter (Díaz Trigo et al., 2013). Knowing jets structure will provide new edges of understanding what exactly served as a reason of stars and other different astronomical objects formation. However, since leptons sustain fast cooling, they appear to be the main light generators compared to baryons that are often not directly visible. One way or another, it is not so essential for the purpose of the current project whether the content of jets is mainly baryonic or leptonic.

Throughout the history of jets observational investigation, it was outlined that there are mainly four components playing essential role in jets formation. Without going too much into details concerning each component, let us briefly introduce and discuss them in order to have a clear image of the total system containing the outflows we are mainly interested in. First of all, there should be an accretion process of gathering interstellar matter by either compact or non-compact central object like white dwarf, neutron star or black hole. This process eventually forms accretion disc in the central plane of mas-sive object because of the mixture of angular momenta from several inclined matter or-bits (RB07). Initial theory of a steady-state geometrically thin, optically thick stationary accretion disc rotating around black hole by means of Keplerian orbits as well as provid-ing multi-temperature black body radiation with temperature profiles T ∝ R−3/4 and

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T ∝ M1/4 _{was developed in Shakura & Sunyaev (1973); Novikov & Thorne (1973)}3_{. In}

a nutshell, the matter either from interstellar medium (AGN) or from companion star (XRB) serves by means of its angular momentum as a raw material to heat disc surface due to viscous processes and provide subsequent thermal radiation. Accordingly, mat-ter coming from the disc reaches high enough temperature to turn into plasma state which suddenly moves with the flow - the link between jet and accretion disc power provisions is established through the «jet-disc symbiosis» (Falcke & Biermann, 1995; Fender, 2010). Based on its provided relation between temperature and accretion disc radius, this simple «Shakura-Sunyaev» disc model is able to explain different types of radiation provided, for instance, in case of X-Ray Binaries and galactic centres: since XRB discs are usually thousands of kilometres in radius compared to AGN discs of bil-lion km length, their radiation should have higher temperature and, as a result, higher energy. Therefore, we often have X-ray emission in XRB and only ultraviolet/visible radiation in AGN, resulting in much cooler discs in the latter case (Netzer, 2006). How-ever, the model lacks strong explanation of viscosity profile which should play crucial role in driving and heating accreted matter, as well as changing accretion behaviour for different X-ray states of a particular source. More complicated models like, for in-stance, Advection Dominated Acretion Flow (ADAF, Narayan & Yi 1994, 1995a,b) treat disc faintness as a consequence of its low radiative efficiency due to weak Coulomb coupling between electrons which is responsible for electromagnetic radiation together with gravitational energy carried by protons due to their bigger masses. The advection takes place because of protons being attracted by central object, thus carrying off gravi-tational energy and leading to disc radiative inefficiency and, as a result, faint emission often pointed by observations. The model more clearly explains jets behaviour with relation to different (more precisely, low luminosity) accretion states, pointing on disc thermal emission and outflows power law radiation: when accretion rate is low, the density of advection flow gets smaller, thus rejecting jets. Inversely, with accretion pro-cess getting higher, ADAF obtains higher density and bigger probability for jets to be launched (Esin et al., 1997). There exist various modifications to ADAF, in total known as Radiatively Inefficient Accretion Flows (RIAFs, FKR02, Trump et al. 2011) where: Advection Dominated Inflow-Outflow Solutions (ADIOS, Blandford & Begelman 1999; Beskin & Division 2003, M12) supposes that the radiative inefficient flow produces near-zero net energy flux, resulting in positive accretion energy which due to its small frac-tion gathered by the hole itself should be driven away by means of some external pro-cesses like winds or jets; Convection-Dominated Accretion Flow (CDAF, Igumenshchev & Abramowicz 1999; Narayan et al. 2002) treats the stability of inefficient flow due to the rate of viscous processes happening in accretion disc; Magnetically-Dominated Ac-cretion Flow (MDAF, Meier 2004, 2005) takes magnetic field into account and considers its influence as most dominant during the formation of inefficient accretion flow and the following jet-disc coupling.

In order to establish such accretion process, there should exist a massive central

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object, e.g. stellar black hole in XRB case with typical masses of 5 − 10 M or its

supermassive counterpart in AGN case with typical masses of 106 − 1010 _M

. Since

in principle non-compact objects are also able to produce accretion discs, one needs to distinguish between compact and non-compact objects in order to better characterise the physics in each cases. A good measure of this object property is a compactness parameter ζ, which is defined as

ζ = GM

r , (2.2)

where G is the gravitational constant, M is a mass of the considered object, and r is its radius. Furthermore, based primarily on temperature versus sources masses de-pendence given before, it is of course also possible to outline the difference between corresponding accretion discs radiation.

One needs also to take into account a strong magnetic field whose lines establish the direction of jet matter propagation. Nowadays there is still no detailed explanation about where exactly this magnetic field originates. Despite the fact that the same can be said concerning jet origin, nonetheless there have been formed two main principles describing the formation of relativistic outflows with the presence of curling magnetic field lines. One of them states that jets launch by means of energy gained from black holes rotation (Blandford & Znajek mechanism). The idea is that the rotational energy can be extracted from BH spinning around its rotational axis (Kerr black hole, Misner et al. 1973) by means of the drain caused by forming torsional waves. The basic ingredi-ents for this mechanism to gain strength are the angular momentum, magnetic field as well as black hole itself. Strong enough values of angular momentum as well as mag-netic field strength cause multitudinous production of electron-positron pairs from elec-trostatically accelerated charged particles non-thermal radiation which in turn leads to the vacuum instability. This mechanism is applicable not only for supermassive black holes in galactic nuclei, but also for their lighter stellar duplicates.

An alternative approach proposes that matter acceleration process appears strictly along the magnetic field lines that originate from accretion disc (Blandford & Payne mechanism). According to this mechanism angular momentum as well as rotational energy are transferred outwards due to disc viscous and magnetic torques, which usu-ally appear from gravitational interactions with stellar companion in XRB case or in-terstellar medium in AGN case. The plasma matter subsequently moves along the lines of magnetic field distribution, simultaneously experiencing rotation perpendicular az-imuthal axis. The approach is also suitable for black hole as well as neutron star bina-ries. It is also important to note that both these mechanisms may co-exist simultane-ously (McKinney et al., 2012).

One should also not forget to consider gravitational effects, especially in the near origin of jet launch since it is affected by central object gravitational field. Nowadays there are already steps been made to correct the dynamics of outflow ejection taking this latter point into account (Polko et al., 2013, 2014).

To give a full picture of properties, one needs also to mention jets acceleration as 11

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K. Mikhailov 2.2 Typical spectrum contributions

a sure sign of outflows presence. Such acceleration is capable of providing velocities very close to the speed of light c. Main reasons for this process are of either thermal or magnetic origin (G02). Most known of them are: «uncoiling spring» model (Uchida & Shibata, 1985) which explains acceleration by means of the net force from toroidal magnetic field which pushes plasma upwards and thereby increases its velocity; ther-mal driving model (Livio, 1999) which takes disc corona as well as magnetosphere into account and treats acceleration from particles dissipation point of view. Jet acceleration can also be described by the denoted above Blandford & Znajek and Blandford & Payne mechanisms: the first treats acceleration based on rotational energy of the central black hole extracted by intrinsic magnetic field, while the second provides possible explana-tion due to plasma co-rotaexplana-tion along the lines of poloidal magnetic field. Nevertheless, the true nature of accelerative movement is still under question.

Initially being accelerated in the nozzle region and afterwards gaining velocity through adiabatic expansion due to pressure gradients (Falcke & Biermann, 1995), jets also attain another inherent characteristic called collimation, which differentiates them from other types of inner system emission (e.g. uncollimated solar wind) as well as makes them relatively easy to observe. One has a reason to believe that this phe-nomenon is caused due to the pinch effect according to which magnetic field gener-ated by jet current produces magnetic pressure gradient which itself collimates the outflow (see Begelman et al., 1984, for a review). Nevertheless, there are some other possible reasons for outflows collimation (G02). One of them, thermal pressure confine-ment, is based on pressure difference between the outflow and its surrounding interstel-lar environment which then coerces jets to collimate. Initially established within the nozzle-like «twin exhaust model» by Blandford & Rees (1974), which was later called in question due to inconsistency with observations, it was then modified by Fabian & Rees (1995) by taking external, surrounding jet gas pressure into account. Another pos-sible theory by Kim & Ostriker (2000) claims that jets are able to collimate according to magneto-rotational instabilities (MRI, Balbus & Hawley 1991) which create magne-tohydrodynamic turbulence of accretion disc inherent to the system of consideration.

2.2 Typical spectrum contributions

The total spectrum obtained from the «disc-jet» system represents a mixture of contributions at various wavelengths. Schematic view over the whole multiwavelength range for the X-Ray Binary star system XTE J1118+480 consisting of stellar black hole and its companion star is shown in Fig. 2.1.4_.

The radio emission is caused by the flat to inverted power law of the form Fν = να (Blandford & Königl, 1979), where the spectral index α obtains values from

0.0 to 0.3 (Markoff, 2010). The contribution responsible for emission in a radio region

4_{We consider flux density I}

ν in Jansky (Jy) versus frequency ν in Hz. The product of flux to the

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is mostly linearly polarised synchrotron radiation, which is a result of the linearly po-larised emission by relativistic electrons moving helically along magnetic field lines of strength B with the characteristic relativistic gyrofrequency (RL79)

ω = qeB γmec

, (2.3)

where qeis electron charge and meis electron mass.

Characteristic features of such emission along the whole jet distance serve an opti-cally thick self-absorbed regime ν5/2at lower frequencies due to neighbouring electrons radiation and optically thin regime at higher frequencies. The latter regime depends on the power index of electrons energy distribution. Namely, if a distribution of electrons energies can be expressed as a power law of the form

dn(E) = CE−pdE, (2.4)

where p is a distribution power index, optically thin contributions will also have a power law behaviour with index (p − 1)/2. Since jet magnetic field strength and mat-ter density decrease along its propagation as functions of its distance: r−1 and r−2, respectively (Hjellming & Johnston, 1988; Falcke & Biermann, 1995), each synchrotron spectrum turns out to be shifted, thus resulting in a near flat total radiation of this type. This results in a much wavelength spectrum which is a convolution of several synchrotron contributions. Such phenomenon is also known as a core shift (see, e.g., Markoff, 2010). The difference between two optical regimes denoted above is achiev-able in a certain point named as thick-to-thin break usually happening in near infrared (NIR) region (Markoff et al., 2005). This is a theoretical point, where an abrupt transition from supersonic to subsonic movement, or shock, happens, thereby exceeding entropy as well as cardinally changing jet synchrotron power law inclination. The synchrotron emission covers broad range of frequencies starting from low radio and ending with hard X-ray or even gamma range5_.

Thermal radiation _{created by multicoloured accretion disc itself obeys normal} multi-temperature black body law ranging from infrared (IR) to ultraviolet (UV) rel-ative to disc characteristic parameters: inner and outer radii of the disc as well as its temperature. Thermal seed photons provided by the disc might then undergo In-verse Compton _{(IC, see below) scattering and be dissipated by synchrotron electrons,} therefore producing high energy power law tails which exceed to ultraviolet or big-ger frequencies. Besides, additional black body radiation might be also obtained from companion star in XRB case. For AGN, there is also an IR component from dusty torus surrounding its central part (Pier & Krolik, 1992, 1993). Moreover, it is possible to reveal the presence of iron emission lines in jets spectrum (Migliari et al., 2002).

5_{The synchrotron time-signal obtained experimentally subject to narrow emission area, that is caused}

by relativistic beaming, represents delta function which provides a constant broad frequency range after applying Fourier transform (see e.g. RB07).

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Figure 2.1: An example of a total multiwavelength spectrum obtained from the Galactic source XTE J1118+480 (Markoff et al., 2001). The contributions are marked with corresponding line types and notes: post-shock optically thick region (solid black line at relatively smaller frequen-cies on the left), post-shock optically thin region (solid black line at relatively higher frequenfrequen-cies on the right), pre-shock jet region (dark blue dash line), accretion disc thermal emission (purple dotted line) and inverse Compton visible/ UV contribution (blue dotted line). Also given sev-eral sets of measurements for certain frequency regions which are matched by proper acronyms corresponding to different telescopes and observatories. Top right: sketch of the system model used to obtain this total spectrum. The common structure of the multipart jet - disc assembly consists of the following elements: rotating accretion disc appeared at the origin of the system around massive central object; initial nozzle region, accelerating shock region and finally long collimated outflow aspirant up to the interstellar medium (ISM) or even further to the inter-galactic medium (IGM) in the form of lobes and hotspots (Falcke & Biermann, 1995). More detailed image of jet-disc geometry is shown on Fig. 3.2 in Chapter 3. [See the electronic edition of the Thesis for a color version of this figure.]

Another distinctive feature of the jet, which is usually expressed as a bump in its total spectrum (see again Fig. 2.1), is a thermal Comptonisation emission. This contribu-tion is determined by Inverse Compton effect which is expressed by up-scattering seed (usually soft/thermal) photons emitted either from the disc or from interstellar medium (companion star, host Galaxy) by means of encountering with accelerated and thereby highly energetic jet electrons and gaining energy from them. The threshold defining whether the energy should be transferred from electrons to photons or vice versa is

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K. Mikhailov 2.3 Possible jets progenitors

defined by an average photon energy (RB07) h4Eγ

Eγ

i ∼ 4kBTe− Eγ

mc2 , (2.5)

which is positive for energy transfer to photons and negative in the opposite case. Another characteristic feature of this effect are the series of multiple energy distribu-tions among highly energetic particles during synchrotron process. This phenomenon is known as synchrotron self-comptonisation (SSC) process (Arbeiter et al., 2005).

All in all, the frequency range highlighted by the latter two contributions spans from visible and UV to X-ray. The electrons instead are losing energy, so that their energy distribution becomes minor compared to higher energy photons that now begin dominating in the spectrum (RL79).

2.3 Possible jets progenitors

The origin of jets launch may represent practically every compact astronomical object, starting from neutron stars or black holes in X-Ray Binaries (XRBs, Mirabel & Rodríguez 1994) and ending with a large classification of AGN types. But not only compact objects play crucial role in jets formation – such astronomical elements as white dwarfs and young stellar objects are also responsible for producing relativistic outflows 6. In order to understand their principal differences, let us briefly describe main of them.

XRBs usually represent coupled systems with either a neutron star (NS) or stellar black hole (BH) as a first object (Oppenheimer & Volkoff, 1939) and a companion star as a second object (Lewin & van der Klis, 2006). The size of companion star defines whether the XRB is a Low-mass, an Intermediate-mass, or a High-Mass binary. The most dominant radiation emitted by XRBs is X-ray radiation that is provided by gravi-tational potential energy of stellar matter infalling to the central object which, in turn, accumulates it by means of the accretion process. Alternatively, X-rays might be cre-ated in an optically thick region loccre-ated close to the jet launch zone, where photons either from the disc or from jet itself are able to increase their energy by means of collisions with highly relativistic electrons, or even directly contributed by the jet it-self. This region is known as disc corona but still lacks much of understanding (see Poutanen, 1998, for a review). In addition to the total spectrum of XRBs, there are also radio contribution as well as thermal (mainly, optical) emission, the latter is provided by companion star. The geometry of the coupled BS/BH-companion star binary system can be well described by Roche potential that sets the gravitational balance between two bodies in this system (FKR02, RB07): when a coupled object exceeds the boundary line of this potential, mass transfer begins. Some XRBs are also known as microquasars

6_{However, due to large radii their accretion luminosities become very small so that only the objects}

close to our planet could be observed (RB07).

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according to their possible radio-jet emission identification (M12). Nowadays, dynami-cal studies of the orbits are able to provide a lower limit on the mass of the central object and thereby distinguish whether XRB contains neutron star (NS) or stellar black hole (BH) in its centre. In addition, there exist some differences concerning XRB gravita-tionally predominant object such as pulsations, cyclotron lines or thermonuclear bursts in favour of neutron stars and their lack in black holes case. Nevertheless, however, it is in general hard to distinguish whether we have NS or BH in the centre of considered binary system, especially due to their strong gravitational fields and, as a result, high capability of catching light. Therefore, we will treat both NS and BH XRBs together. But since this work is based mainly on black holes, we will refer to black hole X-ray Binaries (BHBs) only. The first discovered BHB was Cygnus X-1 in our Galaxy which represents binary system with a stellar BH of about 15 solar masses together with a blue supergiant companion star feeding it by means of its stellar wind (Bolton, 1972). Nowa-days, there are more than 300 binaries known that contain massive central objects in our Galaxy (Chaty, 2011).

Another characteristic property of BHBs are their various X-ray states (Tananbaum et al., 1972) usually lasting from around 1 year to approximately 50 years. There are two most distinguishable states: thermal dominated spectrally steep soft state and non-thermal dominated spectrally flat hard state, happening at exactly the same accretion rate, together with the appropriate intermediate states for both cases, respectively. Rel-ative to jets, this distinction manifests itself in the presence of the Hardness Inten-sity Diagram (HID, Fender et al. 2004) that represents a plot of X-ray intenInten-sity versus double-energy «hardness ratio» meaning the fraction of X-ray radiation compared to other different types of it. Typical HID for BHBs is shown on Fig. 2.2: its right region characterises hard X-ray state with radio jets availability, whereas area on the left de-scribes soft X-ray state with the absence of jets. Both these states are separated by the characteristic «jet line», showing from where it is possible for a jet to occur. An impor-tance of such diagram is provided by the fact that it clearly shows disc-jet coupling.

The older brothers of microquasars, quasars, are included into a group of highly powerful representatives of active galactic centres, with a supermassive black holes originated in them. This «AGN Zoo» (Falcke 1996a, 1998, G02, M12) contains besides them Seyfert galaxies, radio galaxies, blazars etc. The difference between all these AGN is mainly due to their emission lines type as well as their orientation according to our observation capabilities: type 1 AGN possess strong optical-UV type of broad emission lines consisting mostly of fast-moving gas, whereas type 2 AGN have narrow emission lines with slow gas and are more dominant in X-ray emission (Netzer, 2006). Nevertheless, it is also possible to characterise them due to their radio compared to op-tical emission: radio-loud AGN responsible for jet launch, while radio-quiet AGN lack this ability (Urry & Padovani, 1995). Starting with discoveries of M87 Active Galaxy, astronomers subsequently found that even our own Galaxy has an enormous super-massive gravitational attractor Sagittarius A* (Sgr A*, Balick & Brown 1974; Melia & Falcke 2001, M12), in its center located more than 8 kpc far from us. There is also a

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Figure 2.2: Typical representation of Hardness Intensity Diagram for black hole binaries. The upper plot illustrates X-ray intensity growing upwards versus spectral hardness growing to the right dependence. The evolution of jet-disc coupling states (from I to IV ) is moving counter-clockwise during BHB lifecycle based on X-ray intensity - hardness plane or from right to left according to hardness scale only. The conventional names for X-ray states are: LS = low/hard state, VH/IS = very high/intermediate states, HS = high/soft state. Main components describing all these states are: accretion disc (red), corona (yellow), jet (blue) and its internal shock region (dark blue). The lower plot shows changes in accretion disc inner radius (red line) as well as jet bulk Lorentz factor Γ (blue line) with respect to spectral hardness. We see that at some point characterised by «jet line» Γ rises steeply, thereby disabling further jet formation. Diagram image taken from Fender et al. 2004 (see also Remillard & McClintock, 2006, for details).

perception that each galactic centre contains its own SMBH (Zheng, 2012). Since active galactic centres lack any of permanent companion stars, typical processes occurring near SMBH usually include different kinds of ionisation as well as radiative recombina-tion in nuclei ionised gas accreted from interstellar medium by means of surrounding stars winds, thereby giving rise to collisionally excited plasma which is then transferred upwards as a jet. Moreover, X-ray emission might as well be explained by still insuf-ficiently explored corona layer (Narayan & Yi, 1995b). It is also important to note that since general relativity predicts similarity in BH behaviour with respect to their masses scale, it is theoretically supposed that the same type of HID plot (see Fig. 2.2) is also applicable to AGN with the only difference that it should take much longer time pe-riod for the system with SMBH inside to produce the whole states cycle (Merloni et al., 2003). Typical components embodying AGN structure apart from central BH with its corresponding accretion disc and jets flowing outwards from the central vicinity

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pict characteristic feature of galactic centres. The Broad Line Region (BLR, see, e.g., Peterson, 2006) represents relatively hot ionised supersonic flow placed closer to the central source medium and providing broad optical emission lines. Usual BLR sizes are several tenths of parsecs. The Narrow Line Region (NLR, see, e.g., Heckman et al., 1981), which is identified mostly in Seyfert galaxies as an axisymmetric dusty as well as gaseous region, transfers contained in it particles radially regarding galactic centre. It also provides much narrower emission lines compared to BLR, though it contains bigger sizes: from 50 pc to 200 kpc. The Dusty Torus (DT, Jaffe et al. 2004) usually extends around the central object for 10 − 100 pc and contains optically thick cold gas, thereby enabling only X-ray emission from central object, but also provides infrared thermal re-emission in terms of blackbody radiation. Such complexity of AGN systems also stresses heterogeneity in the behaviour of emitted afterwards relativistic outflows. Nevertheless, not only interstellar matter from the surrounding galactic nuclei envi-ronment is capable of feeding supermassive central objects lurking within and thereby forming enormous accretion discs. It was pointed by Hills (1975) that mass accretion can also occur in quiescent galactic centres due to the tidal disruption events (TDEs, see also Levan 2012; Gezari 2013). These events are mainly related to the strong grav-itational influence exerted to a star passing too closely to the galactic central black hole. In TDE case the star suddenly got destroyed with its remnants afterwards being accreted, thereby rapidly increasing total mass accretion rate. All this results in ap-pearance of highly luminous tidal disruption flares (TDFs) which due to their similarity with Gamma Ray Bust (GRB, see below) afterglows are able to mislead observations as it was with Sw 1644+57 stellar object which was subsequently supposed to be a blazar (Bloom et al., 2011). The unique way of matter accumulation from the unlucky star passing quite close to the inactive galactic centre also provides specific properties mainly established in the UV-optical as well as soft X-ray emission from the galaxy nuclear centre flares. This emission is supposed to be scalable compared to the pro-vided accretion flow which gives an indication of the tidal disruption dynamics as well as mass properties of SMBH and the disrupted star (Gezari et al., 2012). Since jets are also present in these powerful events, there should be strong evidences to assume both of them are narrowly connected. Moreover, it is also possible to probe black hole spin as well as gravitational waves bursts due to these events (Kobayashi et al., 2004). All in all, tidal events seem to be very complex occasion containing very diversified physics, so that they are believed to serve a good role of observational laboratories not only for relativistic ejecta investigation, but also for probing black holes spin, mass and population density. Planned large-scale visible as well as ultraviolet and X-ray obser-vations (Gezari, 2013) will shed more light to the mysterious behaviour of enormous events happening in the middle of various galaxies.

Large amount of facts indicate that jets are tightly connected with GRBs (see, e.g., Vedrenne & Atteia, 2009) which are frequent 7 _{occasions of extremely powerful}

rel-ativistic flow producing large amount of highly luminous gamma ray emission. This

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results in a non-thermal spectrum, though these bursts are also responsible for the emis-sion of non electromagnetic character such as cosmic rays, neutrinos or gravitational waves (Gomboc, 2012). Being unidentified for a long period of time due to sophisticated relativistic effects, such bursts were first discovered by the USA in the late 1960’s due to their performed nuclear tests (Zhang & Mészáros, 2004). Nevertheless, it was in the short run found that these bursts are quite widespread phenomena and are distributed isotropically throughout the whole Universe. Due to their quick happening as well as large amount of energy carried out, GRBs are thought to usually happen after a huge catastrophic event. For long GRBs, which typically last about several tens of seconds, it is usually supernova explosion occurred after the death of a massive star which then turns into a rapidly rotating black hole or a strongly magnetised (B ≈ 1014− 1015 _G)

rapidly rotating neutron star (magnetar). In this case, jets are usually formed with the participation of compact objects formed before supernova explosion, but afterwards also with the help of expelled stellar matter collapsed into objects newly established accretion discs by means of gravitational forces, strong magnetic field as well as rem-nant stellar mantle pressure collimating it - there are plenty of possible scenarios with no single point of view to date (see, e.g., Piran, 2004). For short GRBs taking about 2 seconds or less to happen (Kouveliotou et al., 1993), a number of various numerical sim-ulations (see, e.g., Rezzolla et al., 2011) report gravitational merges of magnetised binary compact neutron stars, eventually resulting in speedily spinning black hole, as a pos-sible scenario for short bursts central engine. Nevertheless, the true nature of mergers is not yet well established, so other binary candidates such as black hole together with neutron star or even binary system with two black holes are also possible (Buonanno et al., 2007; Foucart et al., 2013). It is believed that jet formation mechanism in case of short GRBs should be similar to their long counterparts, with the more dominant magnetic field influence (Gomboc, 2012). Finally, there are a few bursts establishing very long times of about several tens of minutes and named ultra-long GRBs, yet their origin is not unified and might have traces of supernova occasion as in long bursts case but also TDEs with supermassive black holes and its sacrificial star (Levan et al., 2014). Another GRBs distinctive features are incredibly huge Lorentz factors of γ ∼ 100 or even more that reflect bursts huge energy reservations. Moreover, GRBs are sup-posed to possess a very high luminosity of 1051− 1052 _{ergs/s, thus being very bright}

objects. Signs for that are observations at very high redshifts with the largest ones being z ∼ 8−9 (Gehrels et al., 2004), which means that most of bursts link to the stars born at the early ages of the Universe (Wijers et al., 1998). Two most promising models exist for the explanation of such highly luminous blasts: fireball model (Piran, 1999) diminishes magnetic field energy compared to the energy carried by particles and operates with the re-thermalisation of baryonic kinetic energy mainly caused by shocks, whereas electro-magnetic model (Usov, 1992) does put the emphasis on electro-magnetic field and treat it in a way that it considers optically thick electron–positron plasma producing ultrarelativis-tic highly-magnetised jets. The observational evidence for jets in bursts are supposed to be expressed in a lower energy emission known as (jetted) afterglow, produced after

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K. Mikhailov 2.4 Observational obstacles

initial energy dissipation at larger distances (Piran & Granot, 2001). Such type of emis-sion is thought to be produced by external shock happening due to colliemis-sions between propagating jet and surrounding environmental medium of host galaxies: the afterglow emission is a decreasing function of time and frequency with accordance to power law dependence. It is also known from computer magnetohydrodynamic (MHD) simula-tions that GRBs jets are enclosed by peculiar cocoon-like structure until they reach the interstellar structure out of their restrained objects (Bromberg et al., 2014).

All in all, it should be clear that all these jet-containing objects possess similar prop-erties as well as reflect resembling physical principles. For the latter, GRB, case, despite the fact that they are more influenced by interaction with the surrounding environment and initially cover descriptive physics far beyond the scope of jet propagation, it is of huge importance to connect these objects with more cohesive and more tied to black hole vicinity XRBs and AGN systems in order to treat their characteristic similarities or, conversely, distinctions. Therefore, considering such a huge amount of possible trans-fer candidates makes jets being a topic of huge discussions and debates among multiple scientific groups.

2.4 Observational obstacles

Along with a plurality of discoveries and breakthroughs in understanding jets launch mechanism there is still plenty of obstacles in its description. As we have already noticed, jet-disc total spectrum represents highly diverse object with several contributions connected to each other in a varying degree. This fact shows that this complexly organised spectrum should be treated in different frequency regions, more precisely from radio to X-ray or even gamma range. This, however, might be a diffi-culty since to observe the whole spectrum simultaneously we need to launch several telescopes with different frequency capture settings, which is not so easy to perform for organisational reasons. In addition, the huge distances that the light should over-come in order to be captured by astronomical instruments might also affect detection. There are many different reflecting obstacles that might be on its way causing light dispersion: interstellar medium, other gravitational attractors etc. Finally, because of special relativity presence in jets treatment, one needs also to take certain relativistic effects into account in order to obtain the spectrum correctly. Most known effects are: Doppler effect (see Toman, 2013, for a review), causing the shift in frequency between the rest (observer) and moving frames; relativistic beaming (Sparks et al., 1992), placing photons with small angles into a cone characterised by their gamma-factor; achromatic break (Vedrenne & Atteia, 2009), producing abrupt change in light curve behaviour. As a good example of difficulties caused by the listed above effects: because jets speed is highly relativistic, nowadays we still do not know how to directly obtain experimental data from the jet launching backwards. We can only state that they should be present because of the disc plane structure that represents axial symmetry guaranteeing jets in both directions.

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K. Mikhailov 2.5 Outflow dominated model

2.5 Outflow dominated model

The absence of knowledge about jets true mechanism provides a great variety of semi-analytical jet models for various accreting objects. This work is based on outflow dominated jet model that focuses mostly on jet behaviour, making only general assump-tions concerning other parts of the related system (accretion disc, possible companion star etc.). Being semi-analytical, this kind of model tries to predict jet dynamics based on fundamental principles (e.g. laws of energy or angular momentum conservation) as well as fitting with observational data. Productive examples of such model for XRBs can be found in Markoff et al. (2001, 2005), for AGN – in Falcke et al. (1993); Falcke & Markoff (2000); Yuan et al. (2002). Initially based only on hydrodynamic (HD) fluid principles, without mainly taking jets acceleration and collimation into account, mod-ern model principle of jets investigation treats them in terms of MHD flow of general form (M12): ∂ρ ∂t = − ∇(ρv), ∂P ∂t = −γP ∇ v, ∂v ∂t = −(v ∇)v − 1 ρ∇ P + 1 ρc[j × B], ∂B ∂t = ∇ × [v × B] , j = c 4π [∇ ×B] , (2.6)

where ρ is the mass density, v is the bulk plasma velocity, P the plasma pressure, γ is the adiabatic index,j is the current density, and B is the magnetic field. Therefore, plasma particles behave as ionised, neutral, homogeneous fluid, whereas mass flux is closely connected to magnetic flux so that matter moves along magnetic field lines. This allows us to better describe magnetic field as well as reduce the number of free parameters fed into the model. Moreover, MHD solutions appear to be more suitable to different classes of jets containing objects due to their relativistic saturation. In fact, we can also deal with steady states within MHD concept: that is, plasma flow and magnetic field are set as functions not dependent on time. This latter assumption greatly reduces computational problem, leaving us to deal only with one ordinary differential equation. In general case, there is a need to consider nine partial differential equations setting the full dynamics picture of jets propagation. This problem can be solved only with a help of supercomputers or large computer clusters (Meier et al., 2001).

However, even with all these strong suppositions there are still many unanswered questions. Huge amount of experimental data as well as simulations indicate that the weighted portion of radiation that is mostly related to uncollimated disc wind (Fendt, 2009) might be obtained from the region near accretion disc known as corona (Meyer & Meyer-Hofmeister, 1994; Poutanen, 1998). This area might play a significant role in transferring highly heated gas from accretion to the forming outflow. Furthermore, it is still not clear how exactly jets propagate throughout their surrounding medium.

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K. Mikhailov 2.5 Outflow dominated model

The outflow dominated point of view concerning jets dynamics is as follows (Fal-cke & Biermann 1995, see again the sketch on Fig. 2.1): first jets accelerate with high speeds from the nozzle region that appears to be much close to the disc. After that comes adiabatic expansion under pressure with further weak acceleration. Then an intermediate phase with shocks happens, thus gradually changing outflow behaviour: jets become overcollimated, with power-law particles distribution tails spreading for high-energy regions. Finally, in the end, jets reduce their speed from supersonic to subsonic and suddenly cease to exist either in the interstellar medium (ISM) or even in the intergalactic medium (IGM) with lobes (hotspots) fabrication. The trajectory of jets movement contains four characteristic points on its way (Polko et al., 2010): ini-tial (nozzle) sonic point with proper sound speed, modified magnetosonic slow point (MSP) setting boundary conditions for jet further spread, Alfven point with character-istic Alfven speed that for low field strengths B gives υA = B/

√

4πρ, and modified magnetosonic fast point (MFP) representing jet acceleration caused by shock. The lat-ter point is crucial because beyond this point, there is no causal connection with the previous behaviour since the speed gained by jet exceeds the speed of information prop-agation. To describe what is exactly happening both at these points and at the vicinity around them, one needs to solve a specific set of wind type equations (modified Parker solar wind equations, Parker 1958). These equations represent relativistic Bernoulli (en-ergy) equation (Merches & Teodorescu, 1968) providing total energy distribution sup-plied by accretion process, and transfield equation (Beskin & Division, 2003), expressing the force balance throughout longitudinal and perpendicular magnetic field lines.

Due to complexity of all these equations incorporated together in one single model, one needs to turn to numerical schemes in order to solve them. One simplification often used is that the behaviour of magnetic field lines is alike and is possible to describe with only knowledge of one concrete line (self-similarity principle, Li et al. 1992). Nowadays modern papers (see, e.g., Polko et al., 2014) have already started to deviate from this principle, but the clear thing is that with tending to describe the phenomena more precisely, we need more complex equations which are really hard to solve, even with the computational help. That is why we want to use a distributed algorithm capable to use this semi-analytical model, which operates with many free parameters, in order to perform a search in its parameter space and find the most suitable solution for the fit with observational data. All model parameters such as inner disc radius Rdor the mass

of central object M•should be fixed either from observations or from simulations itself.

Additional model details connected together with parameter characteristic values and tolerance regions will be considered in the following chapter.

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Chapter 3 Model parameters

As was explained in Chapter 2, semi-analytical jet models contain a large number of parameters due to the complexity of phenomena they are trying to describe. By pa-rameters we mean the values that are served to the program embodying jet model as an input data. For example, the agnjet model program, which is used in the project and which was historically modified to describe the multiwavelength jet behaviour for both XRB and AGN systems, now contains 23 initial parameters. The result of this semi-analytical model is a total spectrum that includes a mixture of contributions: pre-shock and post-pre-shock synchrotron emission, an inverse Compton process radiation, and an optional black body component. Meanwhile, not all of this model parameters are needed to be varied: one part comes from astronomical observations, another part rep-resents either flags or switches to the corresponding regimes. For the genetic algorithm construction only the part of parameters that is free to vary needs to be taken into ac-count. Namely, the parameter tolerance regions from appropriate minimal to maximal values are worth taking into account for the GA initialisation in order not to spend computational time on reaching these regions first or, even worse, not to obtain non-physical properties at the end of the algorithm work. Furthermore, parameter regions will also become useful during the construction of GAs operators in order to make their genes mapping being one-to-one, see Chapters 4 and 5. In this chapter we explain physical meaning as well as tolerance boundaries for all the parameters required for the agnjet _{program. We also discuss which of them should vary more and which should} in principle change a little. Good papers explaining most of the parameters are Falcke & Biermann (1995, hereafter FB95), Markoff et al. (2005, hereafter MN05), Maitra et al. (2009a, hereafter M09). The model itself uses the work of Blandford & Königl (1979) as a background for optically thick plasma outflows description. Its main advantage is that it takes both jet high collimation and acceleration into account.

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K. Mikhailov 3.1 Fixed parameters

3.1 Fixed parameters

It is always possible to obtain some information about the system of interest by looking at its behaviour using different observational tools. Many essential system op-tions were obtained directly from the experiment throughout the history: Avogadro constant by J. Perren with gamboge particles centrifugation, charge of electron by R. Milliken based on the movement of oil droplets in a uniform electric field between two metal plates, Higgs boson mass using CERN Large Hadron Collider (LHC, ATLAS Collaboration 2013) etc. Despite the fact that usually the values of physical quantities are obtained with certain statistical errors, they still remain trustful within the con-fines of its permissible range. Therefore, we can trust the values achieved by means of measurements and treat them as fixed.

The agnjet model operates with steady-state near-conical 1 jets originated either from XRBs black hole or from its AGN supermassive partner. In this case we can take its axial symmetry into consideration and describe parameters only with relation to z - axes perpendicular to the accretion disc plane xy, for instance. As for observational domain, the model provides some initial parameters that have empirical origin. The first parameter of such type is a mass of either XRB or AGN black hole M•. There are

multiple ways to determine the mass of the gravitational singularity lurking in XRB or AGN, but most of them are based on the measurement of the characteristics of orbits inherent to objects rotating around them (e.g. companion stars, radio sources or gas discs). In the simplest and quite frequent case rotation occurs in Keplerian orbits, as evidenced by the proportionality of the rotation speed of the satellite to the square root of the orbits semi-major axis:

V = r

GM

r , (3.1)

where G is the gravitational constant and r is a distance to the gravitational centre. Inverting (3.1), we obtain the desired mass of the central object.

For BHBs, it is possible to evaluate minimal possible mass of a compact object by means of its companion star, namely its radial velocity curve. The value obtained from this curve measurement is the mass function f (M ), which depends mainly on binary compact and companion objects masses M• and M∗, respectively, as well as orbital

inclination angle θ provided by BH accretion disc (Remillard & McClintock, 2006): f (M ) = M•sin

3_θ

(1 + q)2, (3.2)

where q = M∗/M•.

In some AGN cases, when the objects represent satellites in a continuum (gas disc or a dense star cluster) so that its gravity affects the characteristics of the orbit, the

1_{The actual shape of model plasma outflow is determined from the interrelation between its}

longitu-dinal and sideways expansions

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K. Mikhailov 3.1 Fixed parameters

radial distribution of mass in the galaxy core is obtained by solving the collisionless Bernoulli equation (Rehbinder, 1985). Another possibility is to use the reverberation mapping (Peterson & Horne, 2004) by measuring the size of the broad emission-line region (BLR), leading to the following mass estimate:

M = fBR

r4V2

G , (3.3)

which is different from (3.1) by the presence of unit-order scaling factor fBRthat

rep-resents the shape and kinematics of BLR. One needs also to take Doppler effect of the emission lines with radius r into account when obtaining root mean square (RMS) of satellite velocity 4υ. This method is complicated by the lack of knowledge about true geometry and kinematics of BLR as well as technical obstacles in r measurements.

The typical sizes for black holes masses are different for a concrete class of central objects. For XRBs black hole masses typically fall in the range of a few to ≈ 10 M,

for AGN the range of masses is typically 106 − 1010 _M

, where M is a solar mass

serving as a characteristic mass measure in all astronomical observations. For example, the mass for the GX 339-4 XRB source (Bradt & McClintock, 1983) is estimated to be around 7 M (Chen, 2011), the mass for the Sgr A* (Balick & Brown, 1974) is assessed

to be around 4 × 106_M

(Gillessen et al., 2009). The biggest AGN BH mass nowadays

amounts to more than 1010M(van den Bosch et al., 2012), and this is used as an upper

limit for the tolerance region. Lower limit, in turn, might vary a lot, but in any case should be bigger that at least one solar mass.

Next parameter is a distance from the observer to the gravitational central object D. This quantity is often measured by resolving the structure of light velocity along the line of sight using special techniques, e.g. optical and near-infrared (OIR) photometry for XTE J1118+480 (M09). A good measure for describing velocity profile changes is represented by dimensionless measure known as redshift:

z = λobs− λem λem

, (3.4)

where λobs and λem are the observed and the emitted wavelengths, respectively. For

z > 0 wavelengths are shifted toward the red (longer wavelength) region of the spec-trum, whereas for z < 0 wavelengths are shifted toward the blue (shorter wavelength) region of the spectrum. Reasons for this might be caused due to Doppler shift, energy losses in association with gravitational field overcoming or even because of the expan-sion of the Universe (RB07) 2_{. In general, since the object emitting outflows can be}

arbitrarily either far or close dependent on our measurement capabilities, reasonable

2_{The astronomical distance is traditionally set in special units called kilo parsecs (kpc) which are}

treated as a distance from the Earth to the star that has a parallax (angle between two chosen distant stars) of one arc second. To get a feeling of characteristic quantities, some examples of distances to jet-containing central objects: ≈ 7 kpc to GX 339-4 (Maccarone, 2003) and ≈ 8 kpc to Sgr A* (Eisenhauer et al., 2003).

The use of genetic algorithms to find an optimum MHD fluid solution as an approximate description of black holes relativistic jets

the use of genetic algorithms to find an optimum

mhd fluid solution as an approximate description

of black holes relativistic jets

Abstract

Contents

Table of constants (in CGS units)

Chapter 1

Introduction

1.1 Why are jets scientifically interesting?

1.2 The project

Chapter 2

What are jets?