### Some of the topics of my recent research

- Electron-positron cascades in
pulsars
I am studying electron-positron plasma generation in magnetospheres of pulsars starting from first principles using a hybrid Particle-In-Cell/Monte-Carlo numerical code.

- Force-free pulsar
magnetosphere
I developed what is up to now the most advanced model of the force-free magnetosphere of an aligned pulsar using a high-resolution multigrid numerical code specially developed for this task.

- Magnetar magnetosphere
I and my collaborators developed an analytical model for the structure of the magnetosphere of an oscillating neutron star.

### Electromagnetically driven electron-positron cascades

Electron-positron cascades play a central role in theory of radio pulsars — it is believed that radio pulsar "dies" when its rotation became so slow that cascades in the polar caps stop producing electron-positron plasma in the magnetosphere. Non-stationary regime of electromagnetically driven cascades has been poorly studied and all previous attempts to model time-dependent cascades relied on some simplifying assumptions. I decided to conduct a detailed study of dynamics of electron-positrons cascades starting from first principles and developed a numerical algorithm for self-consistent time-dependent kinetic simulation of electromagnetically driven electron-positron cascades — wherein particle acceleration, pair creation and screening of the electric field are calculated simultaneously — and wrote a hybrid Particle-In-Cell / Monte-Carlo code based on it.

I have already performed exhaustive self-consistent simulations of polar cap (PC) pair cascades in 1D. These simulations studied in detail both classes of pulsar PC cascade models: (i) those that do not allow charges to be extracted from the NS surface (Timokhin 2009, 2010), the model proposed by Ruderman & Sutherland (1975) and (ii) the more popular ones in which particles can freely leave the surface (Timokhin & Arons 2013), the Space Charge Limited Flow (SCLF) regime, first proposed by Arons & Scharlemann (1979).

##### Plasma flow with no particles extraction from the star

For the model when no particles can be extracted from the neutron star the cascade zone easily adjusts to the current density required by the magnetosphere and produces dense electron-positron plasma in accordance with qualitative expectations of the original model. But surprisingly, the pair formation turned out to be very regular, showing a limit cycle behavior with the electron-positron plasma having a thermalized low-energy component. Pair formation happens in bursts and each discharge also excites a coherent superluminal electrostatic plasma wave that might be important for the generation of pulsar radio emission. Formation of such wave is shown on the following video (full resolution video MP4, [6.7 MB]):

##### Plasma flow with abundant particles supply from the star

In the case when particles can be freely extracted from
the neutron star the cascade behavior turned out to
be *qualitatively* different than expected in the
being for a long time "standard" cascade models. The
flow character depends on the ratio of the current
density to the current density
$J_{\rm{}GJ}\equiv\rho_{\rm{}GJ}c$, where
$\rho_{\rm{}GJ}$ is the so-called Goldreich-Julian
charge density - the characteristic charge density in
the magnetosphere. For the field lines where the
current density is smaller that the Goldreich-Julian
current density $0< J/J_{\rm{}GJ}< 1$ no pair plasma is
produced because the accelerating zone is very small due
to an instability of the plasma flow. Plasma flow along
these field lines can be described as a beam of mildly
relativistic particles propagating through a cloud of
trapped particles with near-thermal distribution, the
plasma density is low and equal to the GJ charge
density, $n=\rho_{\rm{}GJ}/e$. Formation of such low
energetic flow is shown on the video below (full
resolution video
MP4, [8.2 MB]):

Pair formation is possible only along field lines where the current density is either larger than the Goldreich-Julian current density (super-GJ flow, $J/J_{\rm{}GJ}> 1$) of has the opposite sign to it (anti-GJ flow, $J/J_{\rm{}GJ}< 0$). Pair creation is highly non-stationary, similar to discharges in the Ruderman-Sutherland model. The video below shows plasma flow for super-GJ current density (full resolution video MP4, [6.1 MB]):

###### Overview of plasma flow regimes for SCLF

##### References

- Timokhin A. N. and Arons J., 2013, MNRAS, 429, 20 [ADS]
- Timokhin A. N., 2010, MNRAS, 408, 2092 [ADS]
- Timokhin A. N., 2009 Fermi Symposium eConf Proceedings C091122 (arXiv:0912.5475)

### Force-free pulsar magnetosphere

Radio pulsars are highly magnetized rapidly rotating neutron stars. Most of pulsar emission is of non-thermal origin and is produced in the magnetosphere. Figuratively speaking, pulsars are shining electrical generators. Pulsar emission is beamed, and because of neutron star rotation the beam(s) crosses the line of sight at regular time intervals. The received radiation is in form of extremely regularly repeating pulses. The pulse peaks are narrow, what points to smallness of emitting regions, and, hence, to smallness of regions where particles are accelerated. The rest of the magnetosphere should be, therefore, "quiet", what together with other physical arguments makes a force-free magnetosphere model, introduced by Goldreich & Julian (1969), a natural "standard model" for pulsar magnetosphere.

I studied in detail properties of the force-free pulsar magnetosphere. I developed a high-resolution multigrid code for solution of the relativistic Grad-Shafranov equation describing force-free cylindrically-symmetric magnetic field of a rapidly rotating highly magnetized star (the so-called "pulsar equation"). I studied properties of differentially rotating pulsar magnetosphere using analytical models too. I found a set of solutions with different sizes of the closed magnetic field line zone. I have explicitly shown that the current density distribution in the force-free magnetosphere of an aligned pulsar in incompatible with the current theories of pair creation in the pulsar polar cap. I also found that the current sheet separating closed and open magnetic field lines has non-zero surface charge density.

On of the important results of these studies is that stationary unidirectional particle flow in the polar cap cascade zone cannot coexist with the the force-free pulsar magnetosphere, and, hence, models of electron-positron plasma generation in pulsar magnetospheres must be revised.

##### Figures below illustrate some of the obtained results

##### References

- Timokhin A. N., 2010, MNRAS, 408, L41 [ADS]
- Timokhin A. N., 2007, MNRAS, 379, 605 [ADS]
- Timokhin A. N., 2007, Ap&SS, 308, 575 [ADS]
- Timokhin A. N., 2006, MNRAS, 368, 1055 [ADS]

### Magnetar magnetosphere

In the power spectrum of Soft Gamma Repeater emission during the tail phase of giant flares one can sometimes see several distinct features which are interpreted as signatures of neutron star oscillations. If these features are indeed due to seismic vibrations of neutron star, this offers an unique opportunity for testing models of neutron star internal structure. To be sure of the driving process it is important to know how oscillations of the neutron star surface modulate the magnetospheric emission.

I and my collaborators suggested a model for the quasiperiodic component of magnetar emission during the tail phase of giant flares. The model invokes modulation of the particle number density in the magnetosphere of magnetar due to changing magnetospheric currents; the magnetospheric currents are modulated by torsional motion of the neutron star surface. I developed an axisymmetric analytical model for the structure of magnetosphere of an oscillating neutron star and calculated the angular distribution of the optical depth to the resonant Compton scattering. The anisotropy of the optical depth may be why QPO are observed only at particular rotational phases.

##### Figures below illustrate some of the obtained results

##### References

- Timokhin A. N., Eichler D. and Lyubarsky Yu., 2008, ApJ, 680, 1398 [ADS]