Centrifugal mechanism of acceleration

Acceleration to high energies

It is well known that the magnetospheres of AGN and Pulsars are characterized by strong magnetic fields, which in turn, might force the charged particles to follow the field lines. On the other hand, if the magnetic field lines are rotating (which is the case for the above-mentioned astrophysical objects) the particles will inevitably undergo centrifugal acceleration. In the pioneering work by Machabeli & Rogava was considered a gedanken experiment: a bead moving inside a straight rotating pipe. Dynamics of a particle was analyzed as analytically as numerically and it has been shown that if the rigid rotation is maintained for a sufficiently long time energy of the bead will asymptotically increase. In particular, Rieger & Mannheim, based on the theory developed by Machabeli & Rogava have shown that the Lorentz factor of the bead behaves as

where γ 0 is the initial Lorentz factor, Ω is the angular velocity of rotation, r is the radial coordinate of the particle and c is the speed of light. From this behavior it is evident that radial motion will exhibit a nontrivial character. In due course of motion the particle will reach the light cylinder surface (a hypothetical area where the linear velocity of rotation exactly equals the speed of light), leading to the increase of the poloidal component of velocity. On the other hand, the total velocity cannot exceed the speed of light, therefore, the radial component must decrease. This means that the centrifugal force changes its sign.

As is seen from (1), the Lorentz factor of the particle tends to infinity if the rigid rotation is maintained. This means that in reality the energy has to be limited by certain processes. Generally speaking, there are two major mechanisms: The inverse Compton scattering (ICS) and the so-called breakdown of the bead on the wire (BBW) mechanism. Considering jet-like structures in AGN it has been shown that for a wide range of inclination angles of field lines with respect to the rotation axis, the ICS is the dominant mechanism efficiently limiting the maximum attainable Lorentz factors of electrons γ I C S m a x ∼ 10 8 . On the other hand, it was shown that the BBW becomes dominant for relatively low luminosity AGN L < 8 × 10 40 e r g / s , leading to γ B B W m a x ∼ 10 7 .

The centrifugal effects are more efficient in millisecond Pulsars, since the rotation rate is quite high. Osmanov & Rieger considered the centrifugal acceleration of charged particles in the light cylinder area of the Crab-like Pulsars. It has been shown that electrons might achieve the Lorentz factors γ K N m a x ∼ 10 7 via the inverse Compton Klein-Nishina up-scattering.

Acceleration to very high and ultra high energies

Although the direct centrifugal acceleration has limitations, as analysis shows the effects of rotation still might play an important role in the processes of acceleration of charged particles. Generally speaking, it is believed that the centrifugal relativistic effects may induce plasma waves, which under certain conditions might be unstable efficiently pumping energy from the background flow. On the second stage energy of wave-modes can be transformed into energy of plasma particles, leading to consequent acceleration.

In rotating magnetospheres the centrifugal force acts differently in different locations, leading to generation of Langmuir waves, or Plasma oscillations via the parametric instability. One can show that this mechanism efficiently works in the magnetospheres of AGN and Pulsars.

Considering Crab-like Pulsars it has been shown that by means of the Landau damping the centrifugally induced electrostatic waves efficiently lose energy transferring it to electrons. It is found that energy gain by electrons is given by

where δ r ∼ c / Γ , Γ is the increment of the instability (for details see the cited article), F r e a c ≈ 2 m c Ω ξ ( r ) − 3 , ξ ( r ) = ( 1 − Ω 2 r 2 / c 2 ) 1 / 2 , n p is the plasma number density, m is the electron's mass and n G J is the Goldreich-Julian density. One can show that for typical parameters of the Crab-like Pulsars the particles might gain energies of the order of 100 s of T e V s or even P e V s . In case of millisecond newly born pulsars the electrons might be accelerated to even higher energies 10 18 e V

By examining the magnetospheres of AGN, the acceleration of protons takes place through the Langmuir collapse. As it is shown this mechanism is strong enough to guarantee efficient acceleration of particles to ultra high energies via the Langmuir damping

ϵ p ( e V ) ≈ 6.4 × 10 17 × M 8 − 5 / 2 × L 42 5 / 2 ,

where L 42 ≡ L / 10 42 e r g / s is the normalized luminosity of AGN, M 8 ≡ M / ( 10 8 M ⊙ ) is its normalized mass and M ⊙ is the Solar mass. As it is evident, for a convenient set of parameters one can achieve enormous energies of the order of 10 21 e V , so AGN become cosmic Zevatrons.