Modeling Jet Interactions with the Ambient Medium

Recent high-resolution (see, e.g., [13]) observations of astrophysical jets reveal complex structures apparently caused by ejecta from the central engine as the ejecta interact with the surrounding interstellar material. These observations include time-lapsed “movies” of both AGN and microquasars jets which also show that the jet phenomena are highly time-dependent. Such observations can be used to inform models of the jet–ambient-medium interactions. Based on an analysis of these data, we posit that a significant part of the observed phenomena come from the interaction of the ejecta with prior ejecta as well as interstellar material. In this view, astrophysical jets interact with the ambient medium through which they propagate, entraining and accelerating it. We show some elements of the modeling of these jets in this paper, including energy loss and heating via plasma processes, and large scale hydrodynamic and relativistic hydrodynamic simulations.

Synopsis of the work I will present (albeit briefly): 1. Energy deposition of jets into the ambient medium via plasma processes associated with caviton formation. Plasma processes are the dominant energy loss mechanism in most scenarios.
2. Momentum transfer from jet to ambient medium. We are working on article-in-cell modeling of caviton formation and dynamics in order to estimate momentum transfer from jet to ambient medium via plasma processes 3. Hydrodynamic modeling using parallelized version of VH-1 code to estimate ram-pressure momentum transfer. I will show results for a 512^3 run. We have successfully completed a 1024^3 run and are now running a 2048^3 simulation.
4. Initial results from a relativistic MHD run for an astrophysical jet using the PLUTO code (Mignone et al.). We plan to benchmark the PLUTO code by comparing its results to the VH-1 code results for a supersonic, hydrodynamic jet..

Modeling the Interaction of AGN and Microquasar Jets with the Ambient Medium:
We are developing a multi-scale code which uses the energy deposition rate and momentum transfer rate generated by a jet that generates strong plasma turbulence.
We estimate the effect of this turbulence using the LSP PIC code simulations. We then "benchmark" a code that solves a wavepopulation model. Ultimately we will use this wave population code to give inputs (dE/dt, dE/dx, dp/dt and dp/dx) for a highly parallelized hydrodynamic code currently running on the NRL SGI Altix.
n.b.: The wave-population code is benchmarked using a PIC (Particle-In-Cell) code in parameter ranges where a PIC code approach is feasible. The primary energy loss mechanism for the electron-positron jet is via plasma processes: -The two stream instability -The oscillating two stream instability -Ion-acoustic waves These instabilities set up waves in the plasma which produce regions of high electric field strength and relatively low density, called "cavitons" (after "solitons" or solitary waves), which propagate like wave packets in quantum mechanics.
These mix, collapse, and reform, depositing energy into the ambient medium, transferring momentum to it, and entraining (i.e., dragging along and mixing) the ambient medium into the jet.

Jet-Ambient-Medium Interaction (continued)
In order to determine the energy deposition, momentum transfer, and heating, we model the plasma interaction as a system of coupled differential equations.
The solution to these equations gives a normalized wave energy. This wave energy density is then used to determine: -the energy deposition rate of the jet into the ambient medium, -the propagation length, -the heating of the ambient plasma, and -the momentum transfer rate.
For hadronic (p, e+/e-, etc.) the energy extracted by plasma processes slows the e+/e-component first. When the jet e+/ecomponent slows, the hadrons drag the along e+/e-component. The hadrons thus provide the inertia and energy transport necessary to produce the extended jets.
Two-stream instability growth rate for a cold beam: Two-stream instability growth rate for a warm beam: Landau damping for plasma waves in a twotemperature gas where K=    D , T c is the cold temperature and T h are the hot temperatures for gas, f= n h /(n c +n h ); n c and n h are the cold and hot number densities of the gas, and b = 1-f h +T h /T c ), and f c and f h are the cold and hot fractions of the gas. The plasma wave interactions are similar to a predator-prey system

Rabbits Foxes
The environment that the jets propagate through is very complex.
The range of densities involved in the interaction varies over many orders of magnitude, and the jet itself clearly evolves by a number of mechanisms: -Energy loss by various loss mechanisms* -Entrainment of the ambient medium contributes to the changes in the jets' constitution through losses and acceleration of particles.

Non-thermal Electron Tails in the Maxwell-Boltzmann Distribution: Evolution Examined Theoretically
J =2q ∫ v f e (v)dv thus has the form of an integral ∫ x e -x^2 dx = 1/2e, from 1 to infinity, where we have ignored the effect of the potential of the caviton on the distribution function.
The volume of the caviton = ( D ) 3 , and the caviton density is equal to the normalized wave energy density. Therefore, de/dt = 2.7x10 -7  3 T 4 W 1 /n e 3/2 ergs/cm 3 s This energy deposition rate can power the BLR in a typical AGN from the two-stream instability processes.

Shock Heating
If we consider the effect on heating from strong shocks, the classical result is: where the polytropic index,  = 5/3, for a monatomic, ideal gas.
(see e.g., Manami Sasaki's talk on Wednesday) Since the temperature for an ideal gas follows the density (assuming constant pressure), the temperature differential due to shock heating is T 2 /T 1 ~ 4/1.
Note that the estimates of heating of the gas due to plasma processes is roughly T 2 /T 1 = 10 5 K/10 4 K. i.e., an order of magnitude greater than shock heating..
We have confirmed the Newman et al. (Newman:1990) 2D PIC collapse simulations.
• 2D PIC simulations were compared directly with 2D Zakharov equation model. • 2D PIC simulations used an ion density "depression" to drive Langmuir wave packets.
-Despite relatively modest spatial and particle resolutions, good agreement between the two models was obtained.
• Below, we describe LSP simulations that mimic the basic simulation model presented in [Newman:1990].

Slide
Case with a 5.5% density depression: • The initial fields closely resemble the example given in Fig 1(b)  3. Simulation of AGN jet using highly parallelized version of VH-1 hydrodynamic code n.b.: Code yields estimates of temperature, density, velocity (v x ,v y and v z ), and cooling rate for each volume in the simulation. The highly parallelized VH-1 code is running on the NRL SGI Altix supercomputer. Kinwah Wu and Curtis Saxton (MSSL, UK) are collaborators on this part of the project.

Concluding Remarks:
Particle-in-cell simulations of electron-proton jet interacting with an ambient medium yield: -energy deposition rate -momentum transfer rate consistent with wave-population code model.
Plasma processes are the dominant energy loss mechanism for most astrophysical scenarios.
We need to determine whether or not plasma processes are the dominant momentum transfer mechanism. This seems likely, and we are proceeding on this project.
We have underway a multi-scale code effort to incorporate plasma processes as energy and momentum drivers for the evolution of astrophysical jets modeled by a parallelized VH-1 hydrodynamics code and the PLUTO relativistic magneto-hydrodynamics code.
We are exploring caviton formation and dynamics with fully kinetic particle-in-cell (PIC) simulations: • The calculations presented here model relativistic electron-positron beam propagation through an electron-proton background plasmas • The PIC simulations follow the growth and saturation of an initially cold electron-positron beam undergoing unstable two-stream oscillations. • Coupling to the background plasma electrons and ions through three-wave processes leads to the formation of ion waves and cavitons, beginning at the time of the initial saturation of the twostream mode, and continuing thereafter. • We are analyzing these simulations to study several issues related caviton dynamics, including momentum transfer from the cavitons to the background thermal plasma and energetic electron and proton acceleration.
PIC simulation model of jet showing caviton formation and momentum transfer: • 1D, length L = 6x10 6 cm • Periodic boundaries.
Beam-plasma interaction converts a cold beam into a thermal e + -estream interpenetrating the background plasma Field energy vs. t: Electron and positron total kinetic energy: 2. Particle-in-cell simulation of electron-proton jet interacting with a fixed magnetic field Jet-plasmon-solenoid movie goes here!