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The Phenomenology Of Jets In Astrophysics

The Phenomenology Of Jets In Astrophysics

The phenomenology of jet in astrophysics was studied. Analytical methods were used to obtain an equation for describing jet motion. From the analysis, we understood that βT > 1, where βT is the apparent jet velocity along the observers line of sight. The observed motions of the components show curvatures and changes in velocity. Curved trajectories are due to observed perpendicular acceleration, while variations in velocity are due to changes in apparent parallel acceleration.

Title page i

Certification ii

Dedication iii

Acknowledgements iv

Abstract vi

Table of Contents vii

CHAPTER ONE

INTRODUCTION

1.1 Background of study 1

1.2 Astrophysical Jets 2

1.3 The Family Tree of Astrophysical Jets 4

1.4 Relativistic Jets 6

1.5 Galaxies 7

1.5.1 The Classification of Galaxies 8

1.5.2 Galaxy Formation 12

1.5.2. Evolution of the Galaxy 14

1.6 AGN jets 14

1.7 Active galactic nuclei 18

1.7.1 Classification of AGN 18

1.8 Phenomenology of AGN 20

1.9 Black holes 21

1.10 Accretion Disk 23

1.10.1 Definition and Evidence 23

1.11 Aims and objectives 26

1.11.1 Aim 26

1.11.2 Objectives 26

CHAPTER TWO

LITERATURE REVIEW

2.1 A Brief History of Galaxy Formation 27

2.1.1 Galaxies as Extragalactic Objects 27

2.1.2 Cosmology 29

2.1.3 Structure Formation 33

2.1.4 The Emergence of the Cold Dark Matter Paradigm 38

2.2 Galaxy formation 44

2.2.1 Monolithic Collapse and Merging 44

2.2.2 The Role of Radiative Cooling 47

2.2.3 Galaxy Formation in Dark Matter Halos 48

2.3 The origin of the bright knots 51

2.3.1 YSO jets 51

2.3.2 AGN and microquasar jets 54

2.4 AGN jets composition 56

2.5. Black holes observation 57

2.5.1 Observational Evidence for Black Holes in AGN 57

2.5.1.1 Optical and IR data 58

2.5.1.2 VLBI radio data 58

2.6 X-ray observations 60

2.7 History of Superluminal Motions 63

CHAPTER THREE

ANALYSIS

3.1 superluminal motion 66

3.2 Analysis of the formation of an astrophysical jet 70

3.3 Conclusion 74

CHAPTER FOUR

4.1 Phenomenology 75

4.2 Explanation of the phenomenon 78

4.3 Derivation of apparent velocity 83

4.4 Some contrary evidence 86

4.5 Laser ranging 87

4.6 Special relativity 87

4.6.1 Observational effect 87

CHAPTER FIVE

5.1 Conclusion 91

References93

Astrophysical jets are observed in the Universe in a large variety of environments and under a wide range of sizes and powers. They are generated in active galactic nuclei (AGNs) and YSOs, can travel up to a few thousands of Megaparsecs, and reach the largest powers observed in the Universe (up to ∼1047−48 erg s−1), (Zanni et al., 2003; Godfrey and Shabala, 2013). Astrophysical jets can be found in giant molecular clouds, emanating in the vicinities of young stellar objects (YSOs), and reaching distances of some parsecs (Reipurth and Bally, 2001). They are also located near neutron stars in galactic X-ray binary star systems, such as GRS 1915 + 105 that behave as microquasars generating relativistic jets (Fender, 2004). Astrophysical jets can be found in the asymptotic giant branch (post-AGB) stars as well in pre-planetary and planetary nebulae. Opposite, precessing jets are observed in the SS433 binary source, leading to a peculiar phenomenology (Frank, 2011). A jet-like structure is observed, at X-ray energies, inside the Crab Nebula departing from the embedded pulsar (Hester, 2008). Finally, jets can be at the base of the phenomenology of gamma-ray bursts, observed at the highest radiation energies that are still elusive phenomena because of their extreme distances (Granot, 2007).

Astrophysical jets are physical conduits along which mass, momentum; energy and magnetic flux are channeled from stellar, galactic and extragalactic objects to the outer medium. Geometrically, these jets are narrow (small opening angle) conical or cylindrical/semi-cylindrical protrusions (Das, 1999). Jets are a ubiquitous phenomenon in the universe. They span a large range of luminosity and degree of collimation, from the most powerful examples observed to emerge from the nuclei of active galaxies (or AGNs) to the jets associated to low-mass young stellar objects (YSOs) within our own Galaxy. In the intermediate scales between these two extremes one finds evidences of outflows associated to neutron stars, massive X-ray binary systems (with SS433 being the best example of this class), symbiotic stars, and galactic stellar mass black holes (or microquasars). Figs. 1 and 2 show some of the finest examples of YSO and AGN jets. Recent reviews of the observational and structural properties of the YSO jets can be found in Reipurth and Bally (2001).

In this work, instead of discussing all the current knowledge about the various classes of astrophysical jets (which would be impossible to cover just in few pages), I focus mainly on the YSO jets, which are the closest and for this reason, excellent laboratories for cosmic jet investigation, and on AGNs jets. Some specific issues about the theory of microquasar jets will be also addressed.

Figure 1.1 Hubble Space telescope images of three YSO (also called Herbig-Haro, or simply HH) jets. (Extracted from the HST website).

Figure 1.2 Images obtained with the Very Large Array (VLA) of jets in a FRI I source (top: 3C31 at the radio frequencies 1.4 GHz and 8.4 GHz), and a FR II source (bottom: 3C175 at 4.9 GHz); extracted from Bridle (1998).

The association of accretion disks and jet production is probably fundamental, the notable exception being the conical jets emitted from rapidly rotating neutron stars. The disks provide both a mechanism to generate the jets and the mass and angular momentum necessary to keep the jet stable. In fact, the example of giant radio galaxies argues that even the jets that appear to be precessing do so in a consistent fashion over time scales on the order of the life time of the jets (i.e., 105 years).

Figure 1.1 shows a possible “family tree” of jet sources. Variants of this figure have been presented in the past (Brinkmann and Siebert, 1999) with regard to AGN, and a number of authors have constructed similar tables. However, Fig. 1.1 includes microquasars; bipolar outflows in the star forming regions of giant molecular clouds; and other jet-like structures. This is done to remark on the association posited above. The fundamental physical principals implicit in the figure are the ubiquity of angular momentum, the enormous mass of compact astrophysical objects, and their energetic and inertial properties.

Figure1.3: The family tree of astrophysical jets (Brinkmann and Siebert, 1999).

Figure 1.4: Accretion disc and jets (Meier, 2003).

Relativistic jets are very powerful jets (Wehrle et al., 2009) of plasma, with speeds close to the speed of light, that are emitted near the central black holes of some active galaxies, notably radio galaxies and quasars, and stellar black holes and neutron stars. Their lengths can reach several thousand (Biretta, 1999) or even hundreds of thousands of light years (Meier, 2003). Because the jet speed is close to the speed of light, the effects of the Special Theory of Relativity are important; in particular, relativistic beaming will change the apparent brightness. The mechanics behind both the creation of the jets (Semenov et al., 2004) and the composition of the jets (Georganopoulos et al., 2005) are still a matter of much debate in the scientific community. Jet composition might vary; some studies favor a model in which the jets are composed of an electrically neutral mixture of nuclei, electrons, and positrons, while others are consistent with a jet primarily of positron-electron plasma (Wardle, 1998).

Massive galactic central black holes have the most powerful jets. Similar jets on a much smaller scale develop from neutron stars and stellar black holes. These systems are often called microquasars. An example is SS433, whose well-observed jet has a velocity of 0.23c, although other microquasars appear to have much higher (but less well measured) jet velocities. Even weaker and less relativistic jets may be associated with many binary systems; the acceleration mechanism for these jets may be similar to the magnetic reconnection processes observed in the Earth’s magnetosphere and the solar wind.

The general hypothesis among astrophysicists is that the formation of relativistic jets is the key to explaining the production of gamma-ray bursts. These jets have Lorentz factors of ~100 or greater (that is, speeds over roughly 0.99995c), making them some of the swiftest celestial objects currently known.

Galaxies are the building blocks of the Universe. They not only are the cradles for the formation of stars and metals, but also serve as beacons that allow us to probe the geometry of space-time. Yet, it is easy to forget that it was not until the 1920’s, with Hubble’s identification of Cepheid variable stars in the Andromeda nebula, that most astronomers became convinced that the many ‘nebulous’ objects cataloged by John Dreyer in his 1888 New General Catalogue of Nebulae and Clusters of Stars and the two supplementary Index Catalogues are indeed galaxies.

Some galaxies have smooth light profiles with elliptical isophotes, others have spiral arms together with an elliptical-like central bulge, and still others have irregular or peculiar morphologies.

Based on such features, Hubble ordered galaxies in a morphological sequence, which is now referred to as the Hubble sequence. Hubble’s scheme classifies galaxies into four broad classes:

(i) Elliptical galaxies: These have smooth, almost elliptical isotopes and are divided into sub-types E0, E1, ···, E7, where the integer is the one closest to 10(1−b/a), with a and b the lengths of the semi-major and semi-minor axes.

(ii) Spiral galaxies: These have thin disks with spiral arm structures. They are divided into two branches, barred spirals and normal spirals, according to whether or not a recognizable bar-like structure is present in the central part of the galaxy.

On each branch, galaxies are further divided into three classes, a, b and c, according to the following three criteria:

The fraction of the light in the central bulge;

The tightness with which the spiral arms are wound;

The degree to which the spiral arms are resolved into stars, HII regions and ordered dust lanes.

These three criteria are correlated: spirals with a pronounced bulge component usually also have tightly wound spiral arms with relatively faint HII regions, and are classified as Sa’s. On the other hand, spirals with weak or absent bulges usually have open arms and bright HII regions and are classified as Sc’s. When the three criteria give conflicting indications, Hubble put most emphasis on the openness of the spiral arms.

(iii) Lenticular or S0 galaxies: This class is intermediate between ellipticals and spirals. Like ellipticals, lenticular have a smooth light distribution with no spiral arms or HII regions.Like spirals they have a thin disk and a bulge, but the bulge is more dominant than that in a spiral galaxy. They may also have a central bar, in which case they are classified as SB0.

(iv) Irregular galaxies: These objects have neither a dominating bulge nor a rotationally symmetric disk and lack any obvious symmetry. Rather, their appearance is generally patchy, dominated by a few HII regions. Hubble did not include this class in his original sequence because he was uncertain whether it should be considered an extension of any of the other classes. Nowadays irregulars are usually included as an extension to the spiral galaxies (de Vaucouleurs, 1974).

Ellipticals and lenticular together are often referred to as early-type galaxies, while the spirals and irregulars make up the class of late-type galaxies. Indeed, traversing the Hubble sequence from the left to the right the morphologies are said to change from early-to late-type. Although confusing, one often uses the terms ‘early-type spirals’ and ‘late-type spirals’ to refer to galaxies at the left or right of the spiral sequence. We caution, though, that this historical nomenclature has no direct physical basis: the reference to ‘early’ or ‘late’ should not be interpreted as reflecting a property of the galaxy’s evolutionary state. Another largely historical nomenclature, which can be confusing at times, is to refer to faint galaxies with ɱB ≥-18 as ‘dwarf galaxies’. In particular, early-type dwarfs are often split into dwarf ellipticals (dE) and dwarf spheroidal (dSph), although there is no clear distinction between these types-often the term dwarf spheroidal is simply used to refer to early-type galaxies with ɱB ≥−14.

Since Hubble, a variety of other classification schemes have been introduced. A commonly used one is due to(de Vaucouleurs, 1974). He put spirals in the Hubble sequence into a finer gradation by adding new types such as SOa, Sab, Sbc (and the corresponding barred types). After finding that many of Hubble’s irregular galaxies in fact had weak spiral arms, de Vaucouleurs also extended the spiral sequence to irregulars, adding types Scd, Sd, Sdm, Sm, Im and I0, in order of decreasing regularity. (The m stands for ‘Magellanic’ since the Magellanic Clouds are the prototypes of this kind of irregulars). Furthermore, de Vaucouleurs used numbers between −6 and 10 to represent morphological types (the de Vaucouleurs’ T types). Table 1.1 shows the correspondence between de Vaucouleurs’ notations and Hubble’s notations-note that the numerical T-types do not distinguish between barred and unbarred galaxies.

Table 1.1 Galaxy Morphological Types (de Vaucouleurs, 1974).

Hubble E E-SO SO SO-Sa Sa Sa-b Sb Sb-c Sc Sc-Irr Irr

deV E SO-SO0 SO+ Sa Sab Sb Sbc Scd Sdm Im

T-5-3-2 0 1 2 3 4 6 8 10

The Hubble classification and its revisions encompass the morphologies of the majority of the observed galaxies in the local Universe. However, there are also galaxies with strange appearances which defy Hubble’s classification. From their morphologies, these “peculiar” galaxies all appear to have been strongly perturbed in the recent past and to be far from dynamical equilibrium, indicating that they are undergoing a transformation. A good example is the Antennae (Fig. 1.5:) where the tails are produced by the interaction of the two spiral galaxies, NGC 4038 and NGC 4039, in the process of merging.

Figure 1.5: The peculiar galaxy known as the Antennae, a system exhibiting prominent tidal tails (the left inlet), a signature of a recent merger of two spiral galaxies. The close-up of the center reveals the presence of large amounts of dust and many clusters of newly formed stars. [Courtesy of B. Whitmore, NASA, and Space Telescope Science Institute].

The study of galactic formation and evolution attempts to answer questions regarding how galaxies are formed and their evolutionary path over the history of the universe. Some of the theories in this field have now become accepted, but it is still an active area in astrophysics (Fabian, 1999).

Current cosmological models of the early universe are based on the Big Bang Theory. About 300,000 years after this event, atoms of hydrogen and helium began to form an event called recombination. Nearly all the hydrogen was neutral (non-ionized) and readily absorbed light, and no stars had yet formed. As a result, this period has been called the ‘Dark Ages’. It was from density fluctuation (or anisotropic irregularities) in this primordial matter that large structures begin to appear. As a result masses started to condense within cold dark mater halos. These primordial structures would eventually become the galaxy we see today.

The detailed process by which early galaxy formation occurred is a major open question in astronomy. Theories could be divided into two categories: top-down and bottom up.

Top-down theories (such as the Eggen-Lynden-Bell-Sandage [ELS] Model), protogalaxies formed in a large scale simultaneous collapse lasting about one hundred million years.

In Bottom-up theories (such as the Searle-Zinn [SZ] Model), small structures such as globular clusters form first and then a number of such bodies accrete to form a large galaxy. Modern theories most be modified to account for the probable presence of large dark halos.

Ones protogalaxies begin to form and contract, the first halo stars (called population 111 stars) appeared within them. These were composed almost entirely of hydrogen and helium, and may have been massive. If so, these huge stars would have quickly consume their supply of fuel and become supernovae, releasing heavy element into the interstellar medium. This first penetration of stars re-ionized the surrounding neural hydrogen, creating expanding bubbles of space through which light could readily travel(Fabian, 1999).

The evolution of galaxies could be significantly affected by interactions and collisions. Mergers of galaxies were common during the early epoch, and the majority of galaxies were peculiar in morphology. Given the distance between the stars, the great majority of stellar systems in colliding galaxies will be affected. However, gravitational stripping of the interstellar gas and dust that make up the spiral arms produces a long stain of stars known as tidal trails. Example of these formation can be seen in NGC4676 or the Antannae Galaxies (Massaglia, 2003).

These are observed to emerge from the nuclei of active galaxies, like Seyfert galaxies, distant quasars, and radio galaxies, and may extend for distances up to few mega-parsecs into the intergalactic medium. Some of them are considered to be the largest single coherent structures found in the Universe.

None of the basic parameters, like the jet velocity, the Mach number, or the jet to the ambient density ratio can be directly constrained by observations in the case of AGN jets. Observers must therefore interpret their data relying upon statistical analyzes and at an assumed basic model (Fabian, 1999).

In contrast to YSO jets, the main difficulty one faces when trying to comprehend the nature of jets from Active Galactic Nuclei (and also the jets from galactic black holes) is the absence of lines in the radiation spectrum of these objects. Their emission is typically continuous and non-thermal in a wide frequency range that goes from the radio to the X-ray bands (and is due to synchrotron and inverse Compton emission). This is basically the reason why after almost five decades since the discovery of the radio emission from the jet in the galaxy Cygnus A (Jennison and Das Gupta, 1953) observers and theorists are still debating about very basic questions such as the extragalactic jets composition, i.e., whether they are made of ordinary matter (electron-proton) or of electron-positron pairs (Massaglia, 2003).

Whenever a relativistic charged particle encounters a magnetic field line it spirals around it and emit polarized synchrotron radiation at meter and centimeter (radio) wavelengths. Since the electrons are much lighter and therefore, much faster moving than, e.g., the protons, they are responsible for most of the observed synchrotron radiation from the AGN jets. Inverse Compton is the radiation that is produced (generally at X-rays wavelengths) by photons from the environment which are scattered by relativistic electrons to higher frequencies. These mechanisms are both non-thermal in nature because there is no link between the emission they produce and the temperature of the radiating object. Their radiation spectrum is therefore, not described by a black-body curve. Radiation is an evidence of the presence of both charged particles (possibly mostly electrons moving away from the source at relativistic velocities) and magnetic fields (Massaglia, 2003).

At parsec scales, close to the source, several AGN jets exhibit a series of bright components (or knots) which travel from the core with apparent superluminal motion (i.e., a velocity apparently greater than the light speed).

This apparent effect is interpreted as a consequence of relativistic motions in a jet propagating at a small angle to the line of sight with flow velocity as large as ~ 99% of the speed of light (and in some cases, even beyond). The same phenomenon is detected in galactic microquasars. These inferred jet velocities close to the speed of light suggest that jets are formed within a few gravitational radii of the event horizon of the black hole (Jennison and Das Gupta, 1953).

Based on their morphology at kilo-parsec scales, the extragalactic jets observed in radio galaxies have been historically classified into two categories (Fanaroff and Riley 1974): a first class of objects, preferentially found in rich clusters of galaxies and hosted by weak-lined galaxies, shows jet-dominated emission and two-sided jets and was named FR I; a second one, found isolated or in poor groups and hosted by strong emission line galaxies, presents lobe-dominated emission and one-sided jets and was called FR II (or “classical doubles”).

Apparently, the morphology and dynamics of the jets at kilo-parsec scales are dominated by the interaction of the jet with the surrounding extragalactic medium that tends to decelerate the flows to non-relativistic velocities, although some jets (FRII) seem to remain relativistic even on these large scales (Jennison and Das Gupta, 1953).

A clear indication of supersonic speeds in these jets at kilo-parsec scales is the presence of shocks that are resolved in the transverse direction, and the signature of shocks is given by the behavior of the polarization vector of the synchrotron radiation. Well-resolved jets are often highly linearly polarized and their magnetic fields are thus partially ordered. Detailed polarization maps obtained in the optical and radio bands of the jets of M 87 (the closest AGN) (Perlman et al., 1999), for example, show that the magnetic field is mainly parallel to the jet axis but becomes predominantly perpendicular in the regions of emission knots, indicating a shock compression of the field lines along the shock front. Polarization can be also detected in the prominent radio lobes and hot spots that form in some jets at the region where they impact with the intergalactic medium. Again, the field direction is oriented along the lobe’s border, i.e., perpendicular to the jet axis indicating compression by shock (Fanaroff and Riley 1974).

Another parameter to constrain is the jet-to-ambient density ratio. For this parameter one has to rely upon analytical and numerical modeling and compare qualitatively the spatial density distribution, resulting from the calculations, with the observed brightness distributions. Simulations of supersonic, under dense jets are able to reproduce the overall picture of the observed FRII jets (Massaglia, Bodo, and Ferrari 1996). They are, therefore, under dense with respect to the ambient medium, with h ~ 10-5 in the case of pair plasma (proton + electron) jets (Massaglia, 2003).

We notice that the basic reason why shocks lead to non-thermal radiation in AGN jets and thermal emission in YSO jets is the 6 to 8 orders of magnitude ratio in the ambient density. YSO jets propagate in a high density ambient medium of molecular clouds and their shocks can, therefore, heat the high density matter that will in turn suffer efficient radiative losses (which are α ρ2, where ρ is the gas density) through line emission. On the other hand, extragalactic jets propagate in a tenuous intergalactic or intra-cluster medium and the shocks can be considered with good approximation adiabatic. They will have the main effect of accelerating the particles to relativistic velocities via a first-order-Fermi process that, in turn, will yield synchrotron radiation in the shocked ambient magnetic field and the only possible signature that we observe of these shocks is the reorientation of the polarization vector of the radiation (Perlman et al., 1999).

They are particularly luminous compact regions at the centers of some galaxies. Their activity is inferred to be due to the accretion of material onto the supermassive black hole at the galaxy’s center. The jets that are emitted from the poles of the accretion disk around the black hole can extend over distances of millions of light years (Perlman et al., 1999).

AGN are the most powerful steady sources of luminosity in the universe. Unlike normal galaxies, whose spectra are dominated by starlight in the optical or dust in the IR, AGN have extremely broad spectra with large amounts of energy emitted from the radio or infrared all the way through X-rays or even gamma-rays (Fabian, 1999). AGN are often characterized by variability on timescales as short as a few days, which implies that emission must come a small region of order light days in size. AGN can be classified into different categories based on phenomenological features:

Seyfert galaxies have relatively slow, weak, poorly collimated flows (Marscher, 2010). These galaxies have modest luminosities and are radio-quiet (i.e., radio emission is weak), yet they are well-studied because they generally lie near to us (Fabian, 1999). Seyfert spectra show prominent emission lines of highly ionized atoms that cannot be produced by stars (Rosswog and Brüggen, 2007).

Radio galaxies have spectra similar to normal elliptical galaxies, except that they exhibit an excess amount of energy in radio wavelengths. The category of radio galaxies includes Fanaroff-Riley type I galaxies, which have strong jets with relativistic speeds, and Fanaroff-Riley type II galaxies, which have some of the most luminous, highly focused and relativistic beams (Marscher, 2010).

Quasars are extremely luminous galactic nuclei that outshine their host galaxies. Their spectra are nearly featureless, and about 10% of quasars are radio-loud, a feature that is generally associated with having a collimated relativistic outflow (Fabian, 1999).

Blazars are AGN where the jet happens to be aligned (or nearly so) with our line of sight. Despite their cosmological distances, blazars are easily detected because their radiation has been boosted by special relativistic effects. Blazars emit polarized light with a featureless, non-thermal spectrum typical of synchrotron radiation (radiation emitted when charged particles are accelerated radially); this synchrotron emission generally swamps any absorption or emission lines in the spectrum. Blazars are extremely luminous and highly variable, and are characterized by large X-ray and gamma-ray luminosities (Rosswog and Brüggen, 2007). The brightest blazars are further sub-classified as flat-spectrum radio quasars (FSRQs) and BL Lacertae (BL Lac) objects, which both radiate most of their energy in MeV and GeV gamma-rays (Fossati et al., 1998).

AGN are powered by the energy released as matter falls down the potential well of the central black hole. Because this matter carries angular momentum, it settles into a disk on scales much smaller than a pc around the black hole, and viscous stresses then transport angular momentum outwards, releasing energy as the material is moved inwards. On larger scales, where angular momentum doesn’t dominate the dynamics and temperatures are lower, molecular gas and dust is thought to form into a toroidal structure generally referred to as the dusty torus.

There exist two additional regions of material:

The Broad Line Region (BLR), an area consisting of fast-moving clouds orbiting fairly close in to the central black hole, extending approximately up to 0:1 pc for bright AGN (Kaspi et al., 1996).

The Narrow Line Region (NLR), an area consisting of slower-moving clouds orbiting further out from the central black hole, extending approximately 10 ~1000 pc out (Alexander and Hickox, 2012). A jet extending from the poles of the accretion disk in radio-loud galaxies such as blazars can interact with the clouds in both of these regions.

Almost everyone has heard of black holes. They occupy a special place in the public imagination and rightfully so, for they are among the most exotic and interesting objects in nature. However, what exactly are black holes, and how do we know they exist?

Perhaps the simplest and most intuitive definition of a black hole is an object whose gravity is so strong that nothing can escape, even at the speed of light. To understand this better, we can consider the idea of escape velocity. Imagine we are standing on the Earth and throwing a ball up in the air. The faster the initial speed of the ball, the higher the ball will go before it comes back down. Based on our understanding of gravity, as two objects move apart the forces between them decreases proportional to the square of the distance — this is given by the famous equation which many may remember from their physics education: F = GMm/R2 (where F is the force, G is the gravitational constant, M and m are masses of the objects, and R is the distance between them). So as we throw the ball higher and higher, the pull of gravity on the ball becomes weaker and weaker, and we can imagine throwing the ball with such high speed that it leaves the gravitational field of the Earth completely and flies far into space. The speed at which this happens is called the escape velocity; for the Earth it is approximately 11 kilometres per second, which is why we need such powerful rockets to launch vehicles into deep space (Fossati et al., 1998).

The equations of gravity tell us that the escape velocity for a spherical object (such as a star or planet) is given by;

vesc = √GM/R.

Thus, if we make an object more massive (larger M) or compress it (smaller R), then we increase the escape velocity. Taking this to its extreme, we can imagine taking an object as massive as the Sun and compressing it down to a radius of only 3 km, which implies an enormous density (a teaspoonful of this material on Earth would weigh many billions of tones). This object would then be so massive and yet so small that the escape velocity at its surface would be equal to the speed of light. Since nothing in the Universe can travel faster than light, we infer that nothing can escape this object’s strong gravitational field. Such a remarkable entity is what we call a black hole (Fossati et al., 1998).

The concept of escape velocity gives us a clear and intuitive way of thinking about black holes, but unfortunately does not provide a full description of the physics behind these objects. For this we require a more complete description of gravity, which was provided by Einstein in his theory of general relativity in the early 20th century. Einstein’s remarkable insight was that gravity is not simply a force between two objects that acts at a distance, but instead represents a fundamental curvature in the fabric of space and time in the Universe. Massive objects curve the space time around them, and the motion of objects follows straight lines in this curved space; thus the Moon orbits the Earth because it is following the curvature of space induced by the Earth’s gravity. This is an extraordinary idea, but it makes some robust predictions, chief among which is that light rays (which have no mass, and so traditionally were not thought to experience gravity) are deflected as they pass close to the Sun. The observation of this effect provided among the first experimental proof of Einstein’s theory.

In light of general relativity, to fully understand black holes we must think of them as objects for which the strong gravity bends space so much that even light cannot escape. In this sense, black holes truly are black — if we could see a black hole directly, we would see a black sphere, surrounded by images of background objects that had been deflected or lensed by the strong gravity of the hole (Gillessen et al., 2009).

When the molecular cloud contracts to form a star because of the presence of angular momentum, the contraction will not be entirely spherical. While the central protostar grows roughly spherically by accreting matter from the envelope, the outer part of the cloud tends to be mainly distributed into a disk due to the action of the gravity and centrifugal forces. In presence of rotation the orbit of least energy is circular.The matter located in the disk is rotating with a Keplerian motion around the central star. The Keplerian angular velocity of a particle orbiting in the disk is given by;

ΩK= (GM*/r3)1/2

Where G is the universal gravitational constant, M*is the mass of the star, and r is the distance between the centre of the star and the particle.

As a result, the angular velocity decreases outwards, while the specific angular momentum that scales as h = r2Ω increases outwards (the specific angular momentum is the angular momentum per unit of mass). If the disk is viscous, the inner parts that are rotating faster than the outer parts shear past the outer parts. A transport of angular momentum is established from the inner part to the outer part. According to Lynden-Bell and Pringle (Lynden-Bell et al., 1974),a simple energy balance shows that a viscous disk can lower its energy by transporting mass towards lower radius and angular momentum towards larger radius. The Keplerian disks surrounding young stars are therefore very likely accreting mass from the outer parts to the inner parts of the disk, while angular momentum is evacuated towards the outer part of the disk (Lynden-Bell et al., 1974).

Figure 1.6CO(3-2) emission from the disk of the T Tauri star TW Hya (a) and the Herbig Ae star HD 163296 (b). The colors represent the velocity field of the gas with respect to us and show clearly matter coming towards us (in blue) and moving away from us (red) consistent with a Keplerian rotation of material in a disk that would be inclined with respect to the line of sight (Williams and Cieza, 2011).

Evidence of material in Keplerian rotation within elongated structure has been obtained by many authors by measuring the radio emission from the molecular gas in the environments of young stars (Fig. 7.5). Evidence of hot disks has also been obtained using various observing techniques such as chronography (Augereau et al., 2001), adaptive optics (Pantin et al., 2000), interferometry (Eisner et al., 2005), or IR imagery (Okamoto et al., 2009). From these observations it seems now evident that the material situated from small AU to large AU is distributed into a disk.

This work is aimed at using analytical method to derive a mathematical relation that may be able to describe the phenomenology of jet in astrophysics.

The following are some important steps to achieve the aim.

Some equations in electrodynamics and fluid mechanics will be collected.

These equations will be used to derive a final equation for the phenomenology of jet.

The final equation will be compared with those in the literature for validity.


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