Abstract: The current understanding of the formation of powerful bi-directional jets in systems such as radio galaxies and quasars is that the process involves a supermassive black hole that is being fed with magnetized gas through an orbiting accretion disc. In this paper we discuss the dynamics of the jet powered by rotating black holes, in the presence of a magnetic field, including the scaling of the jet length and their typical time scales. We consider a unified picture covering all phenomena involving jets and rotating black holes ranging from gamma ray bursts to extragalactic jets and discuss the relevant scaling laws. We have also discussed the acceleration of the particles in jets and consequent synchrotron and inverse Compton radiations. Accelerated protons from jets as possible sources of high energy cosmic rays are also discussed.

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1. Introduction Objects as diverse as X-ray binaries, radio galaxies, quasars, and even the galactic centre, are powered by the gravitational energy released when surrounding gas is accreted by the black hole at their cores. The combination of the strong gravity of the black hole, the rotation in the in falling matter, and the magnetic field are believed to be the key ingredients to jet creation. Apart from copious radiation, one of the manifestations of this accretion energy release is the production of jets, collimated beams of matter that are expelled from the innermost regions of accretion discs. These jets shine particularly brightly at radio frequencies.1, 2

An accretion disc is matter that is drawn to the black hole. In rotating black holes, the matter forms a disc due to the mechanical forces present. In a Schwarzschild black hole, the matter would be drawn in equally from all directions, and thus would form an omni- directional accretion cloud rather than a disc.3

Jets form in Kerr black holes that have an accretion disc. The matter is funnelled into a disc-shaped torus by the black hole’s spin and surrounding magnetic fields, but in the very narrow regions over the black hole’s poles, matter can be energized to extremely high temperatures and speeds, escaping the vicinity of the black hole in the form of high- speed jets.3, 5

2. Dynamics of the Jet For a Kerr black hole, the horizon is given by:6

2 2 a m m r − ±

… (1) Where, m is the geometric mass and a is the geometric angular momentum. From the condition that r should be real, the limiting case is given by, a m = . That is:
2 2 Mc J c GM Max

… (2) From this, the maximum angular momentum is given by,

c G M J Max 2

… (3)

2 From the classical expression for the angular momentum associated with a jet of length l, assuming the particles to be travelling at near speed of light, the expression becomes

mcl J = Considering a conical jet with base radius r and density ρ , the mass of the jet is given by, ρ π l r m 2 3 1

Then the angular momentum becomes:
ρ π c r l J 2 2 3 1

… (4) From the geometry of the jet, we can relate the length of the jet to the radius r as . Here we have assumed the small opening angle of the jet to be 5°. Assuming the number density of the jet to be of the order of , which is consistent with observations, the length of the jet is given by 0 5 tan l r = 3 3 10 − cm

( ) 4 1 2 2 2 5 tan 3 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛

c GM l πρ

… (5) For the billion solar mass black holes, it works out to

kpc m l 2 10 6 19

×

The scaling of the length of the jet with the mass of the central black hole is given by:7

( ) 2 1 4 1 2 2 5 tan 3 M c G l ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛

πρ

… (6)

M l 5.0 ≈

M (in solar mass) l (parsec) 102 2 104 20 106 200 108 2000 Table 1: Scaling of the length of the jet with the mass of the black hole

3 The behaviour of the length of the jet as a function of the mass of the black hole is shown in the following graph based on equation (6).

Fig 1: Length of the jet with BH mass

In the above discussion we have considered the variation of the length of the jet with the mass of the black hole. For a given black hole, the length of the jet depends on the density of the particles emitted out along the jet. From equation (5) we have ( ) 4 1 4 1 2 2 2 1 5 tan 3 ρ π ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛

c GM l

For a 30 solar mass black hole, with number density of , the length of the jet is of the order of,

3 20 10 −

cm n m l 10 10

Number density, n(m-3) Length of the jet, l 103 1kpc 105 300pc 107 100pc 1020 0.01pc Table 2: Variation of jet length with number density (for billion solar mass black holes)

4 3. Jets in the presence of Magnetic Fields The above discussion did not consider the presence of magnetic fields. We could also include the effects of

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