ramses: A numerical N-body and HD code using adaptive mesh refinement (AMR)

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Correlating the AGN power to the Accretion Rate and AGN mode

The radiation power associated with AGN are amongst the greatest in the Universe. The luminosity of typical Seyfert i galaxies is 1044 erg s􀀀1, and for quasars they can even exceed 1046 erg s􀀀1. Figure 2.10 shows the bolometric quasar luminosity function at redshift z 1 to demonstrate the range and probability of different quasar luminosities7. It is assumed that a significant fraction of the accreted rest-mass energy is radiated away close to the event horizon of the BH, leading to an accretion bolometric luminosity given by L = _MBHc2 ; (2.21).
where _MBH is the BH accretion rate, c the speed of light in vacuum, and is the radiative efficiency of the BH. If the accreted matter emits all of its potential energy beyond the innermost stable circular orbit of the BH, then = 0:06 for a non-rotating Schwarzschild BH, and 0:42 for a maximally rotating Kerr BH8. Studies comparing observed AGN luminosities and the inferred BH mass density suggest that, on average, = 0:1 􀀀 0:2 (Soltan, 1982; Fabian & Iwasawa, 1999; Tremaine et al., 2002), a value consistent with moderately spinning black holes.

Radio Galaxies and Jets

Jet activity is accompanied by synchrotron emission from relativistic electrons in the magnetised plasma, that is most prominently observable at radio frequencies and, thus, is the reason why AGN associated with jet activity are usually called radio galaxies or radio-loud quasars. Radio interferometers allowed for the observation and study of these objects in more detail. The subsequent physical model for the double radio sources (Blandford & Rees, 1974; Scheuer, 1974) interpreted the sources as being powered by an AGN via collimated powerful beams (jets).
Morphology Following Fanaroff & Riley (1974) double radio sources are divided into two distinct classed: FR 1 are brightest in the center, i.e. when the separation between the brightest regions on opposite sides of the central galaxy is less than half the total extent of the source. They typically show a bright jet in the center which then decollimates and forms plumes at larger distances. The second class, the FR 2 sources, are brightest at the outer edges, i.e. the separation between the brightest regions is larger than half the total extent. They show dim jets but extended lobes with a bright hotspot at the outer edges.
These morphological properties are correlated with the radio power of the sources, with the FR 1 sources being lower power ( 1025 W Hz􀀀1 at 1.4 GHz; Bridle & Perley, 1984) and the FR 2 showing high radio power ( 1025 W Hz􀀀1). The FR 2 are believed to be able to transfer their power with beams to large distances without dramatic energy losses, while this is not the case for the FR 1 class.
The origin of the dichotomy is still unclear. Possible explanations could depend on the central source, how the jet is formed, or the environment where the jet propagates through. Both jet classes are believed to have relativistic jets in the inner regions. While the FR 2 class are thought to remain also relativistic on large scales, the FR 1 class seems to entrain a significant amount of ambient matter, that slows down the jet on scales larger than a few kpc and cause decollimation. Figure 2.14 shows a schematic representation of an FR 2 source. The bipolar jet is formed in the accretion disc of the active nucleus, that is in radio images visible as a core. While the jet propagates, internal shocks are excited in the beam and the jet plasma passes through regions of rarefaction and compression. Interactions with the ambient gas leads to the formation of a terminal shock and a post-shock hotspot. The shocked plasma leaves the region of very high pressure sideways and forms a backflow that inflates the cocoon. Generated vortices are advected with the flow and inflate the cocoon. Since the ambient gas is much denser than the jet plasma, the jet head propagates much slower than the beam speed which leads to a fast corresponding backflow. The cocoon is over-pressured with respect to the environment and drives a bow shock into the ambient medium, leading to the formation of a thick shell of shocked ambient gas. Because the cocoon pressure is generally higher in the region around the jet head, the axial propagation is faster than the lateral propegation, leading to the bow shock to be elongated. The strong shearing at the contact discontinuity between the jet plasma and the ambient gas causes Kelvin- Helmholz and Rayleigh-Taylor instabilities. These instabilities grow and form fingers that are entrained with the backflow and lead towards the disc and eventually mix with the cocoon gas. Lobes are visible in the radio frequencies in the outer regions of the cocoon, where the synchrotron-emitting electrons are still very energetic.

Positive or Negative Feedback

As discussed above, AGN feedback is advocated to suppress star formation in massive galaxies in order to reproduce the observed high-end tail of the galaxy mass function. And, indeed, recent large-scale cosmological simulations (e.g., Dubois et al., 2014; Vogelsberger et al., 2014; Khandai et al., 2015a; Schaye et al., 2015), employing strong AGN and stellar feedback, produced good agreement with the observations. However, despite negative feedback from AGN being an important ingredient in the formation and evolution of massive galaxies, the details remain vague, primarily because the coupling mechanism between the AGN and the ISM of the host galaxy is still unknown.
From an observational point of view the impact of AGN feedback on star formation is still not fully understood, mostly because of the difficult nature of their observations, particularly at higher redshift. While there are several detailed observations available showing the interaction of the AGN with interstellar gas, both for jets and winds, it is still  unclear what their actual importance on the evolution of the host galaxy is and how they affect the star formation within the galaxy.
Both recent observations (Zinn et al., 2013; Cresci et al., 2015b,a; Salomé et al., 2015) and simulations (Gaibler et al., 2012; Wagner et al., 2012) suggest that local interactions between jets and cold gas may not only result in suppressed star-formation but may also trigger star formation (positive feedback). Both modes can coexist in a single galaxy during different or overlapping phases. This paints a more complicated picture that will be discussed in this Section along with observational evidence and theoretical models for both positive and negative feedback.


Different Approaches to solve the Fluid Equations

There is a huge range of different numerical methods for HD simulations used within astrophysics. They can mostly be divided into three fundamental methods:
Grid-based (or Eulerian) Method: The Euler equations (Eq. 3.20) are numerically evolved on a discrete spacial mesh, where the differential operator is calculated using a second-order finite difference approximation. The fluid quantities are then updated at each point of the grid. To capture the large density ranges often found in galaxy formation simulations or around shocks, an adaptive method is used which recursively refines high density cells (See Section 3.3 for more information). The main (cosmological) AMR codes today are Art (Kravtsov et al., 1997), Orion (Klein, 1999), Flash (Fryxell et al., 2000), ramses (Teyssier, 2002), and Enzo (O’Shea et al., 2004).
Smoothed Particle Hydrodynamics (SPH) (or Lagrangian) Method: Traces the fluid elements, such as density and pressure, with particles. It is a Lagrangian scheme as the geometry of the flow is closely followed by the particles. An average of a particular quantity such as mass, energy, pressure, or density is calculated considering the local neighbourhood within a smoothing length of each particle. Mathematically this is done by a kernel function (see Dehnen (2001) for more information). The most widely used examples of SPH codes are Gasoline (Wadsley et al., 2004) and Gadget (Springel, 2005).
Moving-Mesh Method: Replaces the smoothing kernel used in SPH codes with a grid computed via Voronoi tesselation. The method tries to retain the high accuracy of mesh-based HD methods for shocks while at the same time using the advantages of Galilean-invariance and geometric flexibility of the SPH codes. Hence the principal idea for achieving such a convergence is by allowing the mesh itself to move and deform (see (Springel, 2010) for further detail). This has been implemented into Arepo (Springel, 2011).
In theory all methods will give the same answer, provided they both solve the same underlying equations. In practice, since most treatments of numerical fluid dynamics are approximations that have different numerical error, each method has its own advantages and disadvantages.

Discretising the Fluid to solve it on a Grid

In order to be able to solve the fluid equation (Eq. 3.20) numerically we discretise the equations and with this the underlying fluid into individual chunks (grid cells) and compute the continuity equations at each grid point by computing the partial differential equations only with respect to the neighbouring grid cells. With U in equation (3.20) consisting of conserved quantities such a problem is also called the Riemann Problem and naturally arises due to the discreteness of the grid where for each grid cell an individual Riemann Problem has to be solved. More precisely, the Riemann problem for equation (3.20) is an initial value problem with left and right initial data at the discontinuity. The solution to this Riemann problem is made of three waves: rarefaction waves propagating towards the denser medium, a rapid shock wave propagating towards the less dense medium, and a slow contact wave to the least dense medium. One can rewrite equation (3.20) into quasi-linear form as @U @t + Jr U = 0.

Table of contents :

Contents List of Figures
List of Tables
1 Introduction 
2 The Formation of Galaxies 
2.1 Dark Matter Statistics
2.1.1 Spherical Top-Hat Collapse Model
2.1.2 The Halo Mass Function
2.2 N-body Simulations
2.2.1 Comparing N-body Simulations with Observations
2.3 Including the Baryons into the Picture
2.3.1 Physics of the Interstellar Medium Radiative Cooling and Heating Star Formation
2.3.2 Feedback Processes
2.3.3 Comparing Hydrodynamical Simulations with Observations
2.4 Active Galactic Nuclei
2.4.1 Classification of AGN
2.4.2 Correlating the AGN power to the Accretion Rate and AGN mode .
2.4.3 Radio Galaxies and Jets
2.4.4 Quasars
2.4.5 Positive or Negative Feedback Observations Theoretical Work
3 Numerical Modeling of Galaxies 
3.1 Collisionless N-body systems
3.1.1 Particle-Mesh Method
3.2 Collisional Systems
3.2.1 Deriving the Fluid Equations
3.2.2 Different Approaches to solve the Fluid Equations
3.2.3 Discretising the Fluid to solve it on a Grid
3.3 ramses: A numerical N-body and HD code using adaptive mesh refinement (AMR)
3.3.1 Adaptive Mesh Refinement structure
3.3.2 Time-stepping Scheme
3.4 Sub-grid physics to study galaxy formation and evolution
3.4.1 Radiative Cooling and Heating
3.4.2 Polytropic Equation of State
3.4.3 Star Formation
3.4.4 Supernova Feedback
3.4.5 Black Hole Feedback
3.5 Radiation-Hydrodynamics (RHD)
3.5.1 The Radiation-Hydrodynamics equations
3.5.2 Moments of the RT equation
3.5.3 The RHD equations
3.5.4 Closing the Moment Equations
3.6 Ramses-RT: An RHD extension to ramses to model propagation of photons
4 External pressure-triggering of star formation 
4.1 Jet Propagation
4.2 Playing with Positive Feedback: External Pressure-triggering of a Starforming Disk Galaxy
4.2.1 Introduction
4.2.2 Simulation Set-up
4.2.3 Results Disc fragmentation and star formation history Mass Flow Rate
4.2.4 Conclusions
4.3 External pressure-triggering of star formation in a disc galaxy: a template for positive feedback
4.3.1 Introduction
4.3.2 Simulation Set-up Basic simulation scheme Application of external pressure
4.3.3 Results Qualitative differences Disc fragmentation Star formation history Clump properties The galaxy’s mass budget The star formation rate The Kennicutt-Schmidt relation
4.3.4 Conclusions
4.3.5 Appendix Bipolar pressure increase Effects of supernova feedback Convergence Studies
5 Feedback from Radiatively-driven AGN Winds 
5.1 Setting up the initial two-phase density distribution
5.2 Outflows Driven by Quasars in High-Redshift Galaxies with Radiation Hydrodynamics
5.2.1 Introduction
5.2.2 Methods Initial Gas Density Distribution
5.2.3 Radiation Hydrodynamics Modeling the Quasar
5.2.4 Results Effects of Different Cloud Sizes Qualitative Effects of Cloud Sizes Effects of Different Photon Groups on the Cloud Evolution Efficiency of the Photon-Gas Coupling
5.2.5 Evolution of the Optical Depth Effects of the Quasar Position Comparison between Different Luminosities
5.2.6 Discussion
5.2.7 Conclusions
5.2.8 Appendix
6 Conclusions and Perspectives 
6.1 Conclusions
6.2 Future Prospects


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