Cosmic Structure Formation on Supercomputers (and laptops)

Transcription

1 Cosmic Structure Formation on Supercomputers (and laptops) Lecture 4: Smoothed particle hydrodynamics and baryonic sub-grid models Benjamin Moster! Ewald Puchwein 1

2 Outline of the lecture course Lecture 1: Motivation & Initial conditions! Lecture 2: Gravity algorithms & parallelization! Lecture 3: Hydro schemes! Lecture 4: SPH, Radiative cooling & heating, Subresolution physics! Lecture 5: Halo and subhalo finders & Semi-analytic models! Lecture 6: Getting started with Gadget! Lecture 7: Example cosmological box! Lecture 8: Example galaxy collision 2 Benjamin Moster Numerical Structure Formation 4 IoA,

3 Outline of this lecture Smoothed particle hydrodynamics (SPH)! Idea & smoothing kernel! Equations of motion! Artificial viscosity! Advantages and problems (fluid-mixing, SPH blobs)! Modern SPH techniques (Pressure-entropy, artificial diffusion, etc.)! Baryonic sub-grid physics! Radiative Cooling and heating! Star formation! Stellar feedback (Multi-phase model, blastwave feedback)! AGN feedback 3 Benjamin Moster Numerical Structure Formation 4 IoA,

4 Smoothed particle hydrodynamics Idea: treat hydrodynamics in a completely mesh-free fashion Use: set of sampling particles to represent the fluid! Gas cloud of mass M g represented by N particles of mass m = Mg / N! Each particle i interacts with each other particle j through the symmetric SPH Kernel W(rij,h)! Depends on inter-particles distance and on smoothing-length h (defines Kernel width)! Most commonly the cubic spline Kernel is used: 4 W (r, h) = 8 h 3 r = ~x i ~x j 8 >< 1 6(r/h) 2 + 6(r/h) 3 0 apple (r/h) apple 1/2 2(1 r/h) >: 3 1/2 < (r/h) apple < (r/h) Benjamin Moster Numerical Structure Formation 4 IoA,

5 Smoothed particle hydrodynamics The density in SPH is given by smoothing the particle masses: (~x) = X j For any field F, we can define a smoothed interpolated version: F s (~x) = Z And the derivative is given by: ~rf s (~x) = X j mw(~x ~x j,h) d~x F(~x 0 ) W (~x ~x 0,h)= X j Usually smoothing length h is not constant:! Pressure is related to the density via: P i = A i i =( m j j F j ~ rw (~x ~xj,h) 1) i u i m j j F j W (~x ~x j,h) 4 3 h3 = N ngbm i i 5 Benjamin Moster Numerical Structure Formation 4 IoA,

6 Equations of motion Equations of motion derived from Lagrangian:! From we get where ~x i = ~ r j i d~v i m i dt = L = X i X j 1 2 m i~v i 2 m i u i m j P j 2 i and h is constant in the 1st term 6 Benjamin Moster Numerical Structure Formation 4 IoA,

7 Equations of motion Equations of motion derived from Lagrangian:! From we get where ~x i = ~ r j i Differentiation of j h 3 j = const : d~v i m i dt = L = X i X j 1 2 m i~v i 2 m i u i m j P j 2 i and h is constant in the 1st j = 1+ h 1 j ~ri i 3 j 7 Benjamin Moster Numerical Structure Formation 4 IoA,

8 Equations of motion Equations of motion derived from Lagrangian:! From we get where Differentiation of j h 3 j = const :! Using ~x i = ~ r j i ~r i j = m i ~ ri W ij (h j )+ ij X X j d~v we get: i dt = m j with f i = 1+ h i 3 i 1 f i P i 2 i d~v i m i dt = L = X i 1 2 m i~v i 2 m i u i, mass & energy automatically constant 8 Benjamin Moster Numerical Structure Formation 4 IoA, X j m j P j 2 i and h is constant in the 1st j = 1+ h 1 j ~ri i 3 j k m k ~ ri W ki (h i ) ~r i W ij (h i )+f j P j 2 j ~r j W ij (h j )!

9 Artificial viscosity Up to now: entropy is conserved along the flow (for each particle)! However, in shocks the entropy must increase! Artificial viscosity! Friction force:! d~v i dt vics = X j m j ij ~ ri Wij Since antisymmetric in i and j: (angular) momentum preserved To conserve total E: du i dt vics = 1 2 X m j ij ~v ij ri ~ Wij j 9 Benjamin Moster Numerical Structure Formation 4 IoA,

10 Artificial viscosity Up to now: entropy is conserved along the flow (for each particle)! However, in shocks the entropy must increase! Artificial viscosity! Friction force:! d~v i dt vics = X j m j ij ~ ri Wij Since antisymmetric in i and j: (angular) momentum preserved To conserve total E:! Usual choice: with and To prevent viscosity in shear flows (and conserve angular momentum): Balsara switch: du i dt vics = 1 2 with 10 Benjamin Moster Numerical Structure Formation 4 IoA, X j m j ij ~v ij ~ ri Wij ij =( c ij µ ij + µ 2 ij)/ ij µ ij = h ij~v ij ~x ij x ij 2 + h 2 ij ij =0 for! ~v ij ~x ij < 0 ij! f ij ij f AV i = ~ r ~v i ~ r ~v i + ~ r ~v i

11 Advantages and problems Advantages of SPH:! Numerically very robust! excellent conservation properties! Lagrangian nature trace flow & Galilean-invariant! Inherently adaptive (no ad-hoc refinement needed)! Couples properly to N-body gravity methods!! Problems?! SPH doesn t mix fluids (well)!,kaufmann-blobs form! Impacts accretion rates, SFRs disc sizes, gas fractions, etc Benjamin Moster Numerical Structure Formation 4 IoA,

12 Advantages and problems Advantages of SPH:! Numerically very robust! excellent conservation properties! Lagrangian nature trace flow & Galilean-invariant! Inherently adaptive (no ad-hoc refinement needed)! Couples properly to N-body gravity methods!! Problems?! SPH doesn t mix fluids (well)!,kaufmann-blobs form! Impacts accretion rates, SFRs disc sizes, gas fractions, etc Benjamin Moster Numerical Structure Formation 4 IoA,

13 Advantages and problems Advantages of SPH:! Numerically very robust! excellent conservation properties! Lagrangian nature trace flow & Galilean-invariant! Inherently adaptive (no ad-hoc refinement needed)! Couples properly to N-body gravity methods!! Problems?! SPH doesn t mix fluids (well)!,kaufmann-blobs form! Impacts accretion rates, SFRs disc sizes, gas fractions, etc Benjamin Moster Numerical Structure Formation 4 IoA,

14 Advantages and problems Advantages of SPH:! Numerically very robust! excellent conservation properties! Lagrangian nature trace flow & Galilean-invariant! Inherently adaptive (no ad-hoc refinement needed)! Couples properly to N-body gravity methods!! Problems?! SPH doesn t mix fluids (well)!,kaufmann-blobs form! Impacts accretion rates, SFRs disc sizes, gas fractions, etc Benjamin Moster Numerical Structure Formation 4 IoA,

15 The mixing problem P u ρ! Contact discontinuity in hydrostatic eq.: P = cst.! Left: u1 ρ1, Right: u2 ρ2 u1 = ρ2/ρ1 u2! SPH: smooth density gradient Energy not smoothed! Pressure blib introduced surface tension prevents mixing 12 Benjamin Moster Numerical Structure Formation 4 IoA,

16 Two gas clouds collide Density increases Energy decreases Density is smoothed Energy isn t smoothed Pressure blip Pressure minimum causes nearby particles to collapse blob The blob problem 13 Credit: A. Hobbs & J. Read Benjamin Moster Numerical Structure Formation 4 IoA,

17 Pressure-Entropy SPH Classic SPH: derive EoM from Lagrangian smoothing density:! h i Instead of using density, use pressure: P i = y i = h PN i j=1 m j A 1/ j W ij (h i )! # Automatically guarantees that pressure is single-valued removes pressure blip and surface tension! Equations will be well-behaved so long as pressure is smooth True by definition in contact discontinuities Not true at shocks, but neither is the density constant Do not lose any desirable behaviors of the density formulation 14 Benjamin Moster Numerical Structure Formation 4 IoA,

18 Physical Diffusion should be included in simulations (shouldn t include metal diffusion without energy diffusion! du/dt = P/ρ v! Diffusion removes the energy minimum pressure smooth! Surface tension vanishes Fluids can mix / no blobs form! However: Unclear how much diffusion to be added Artificial Diffusion 15 Benjamin Moster Numerical Structure Formation 4 IoA,

19 Kelvin Helmholtz Instability Clouds Saturn 16 Benjamin Moster Numerical Structure Formation 4 IoA,

20 Blob test Classic SPH Modern SPH 17 Benjamin Moster Numerical Structure Formation 4 IoA,

21 Isolated discs in hot halo Classic SPH Modern SPH 18 Benjamin Moster Numerical Structure Formation 4 IoA,

22 Outline of this lecture Smoothed particle hydrodynamics (SPH)! Idea & smoothing kernel! Equations of motion! Artificial viscosity! Advantages and problems (fluid-mixing, SPH blobs)! Modern SPH techniques (Pressure-entropy, artificial diffusion, etc.)! Baryonic sub-grid physics! Radiative Cooling and heating! Star formation! Stellar feedback (Multi-phase model, blastwave feedback)! AGN feedback 19 Benjamin Moster Numerical Structure Formation 4 IoA,

23 How to form a galaxy Up to now we have included Gravity (N-body), and Hydrodynamics! All other processed happen below the scale we can resolve! Include effective models to describe these physics:! Radiative cooling and heating! Star formation! Feedback from supernovae! Feedback from AGN! Feedback from other sources?! All processes are in general rather poorly understood 20 Benjamin Moster Numerical Structure Formation 4 IoA,

24 Radiative Cooling & Heating Stars can only form when gas is dense enough (i.e. cold enough)! Gas can cool via a number of mechanisms:! Inverse Compton scattering of CMB photons by electrons in the hot gas. Long cooling time (only important in the early Universe).! Excitation of rot. & vib. levels in molecular hydrogen through collisions, and subsequent decay (important in low mass haloes).! Photon emission after collisions between atoms and electrons, excited levels decay radiatively (important in int. mass haloes).! Bremsstrahlung: electrons are accelerated in an ionized plasma (important in massive clusters)! Gas is heated by a background of high energy UV photons from quasars and massive stars 21 Benjamin Moster Numerical Structure Formation 4 IoA,

25 Radiative Cooling & Heating Each particle (or cell) carries information on density and temperature! Need to compute abundances of the different ionic species of the gas Assume: gas optically thin, & Ionisation equilibrium! Compute cooling rate:!! = X i i (n i,t) Heating rate from photo- ionisation: H = X n i i i Katz Benjamin Moster Numerical Structure Formation 4 IoA,

26 Change in thermal energy With the cooling rate Λ and the heating rate H, we can compute the change in thermal energy:! du i dt = H i Up to now, only primordial cooling! In practice, cooling is computed on an element-by-element basis: where Λi is tabulated for nh, T, and z! Need to follow the creation of elements in SNe (usually 11 elements are tracked) i i = H,He + X i>he i 23 Benjamin Moster Numerical Structure Formation 4 IoA,

27 Star formation Stars should form in a gas cloud if: - the gas density is high enough (threshold density nmin) - the region is Jeans unstable! d dt = g t sf where ε is an efficiency and tsf is a characteristic timescale! The star formation rate is given by Often the dynamical timescale is used for t sf! Implemented in the code in a stochastic way: p = m g 1 e t/t sf m Draw random number r: if r < p create new star particle! No feedback gas can form stars on dynamical time overcooling 24 Benjamin Moster Numerical Structure Formation 4 IoA,

28 Stellar feedback: Multiphase model Springel & Hernquist (2003) developed a multi-phase model! Each gas particle is treated as a two-phase medium: cold and hot! Gas density is ρ = ρc + ρh with specific energy ε = εcρc + εhρh! d dt =(1 \epsilon_h where β is the fraction of short-lived stars and t* is a timescale! Star formation: cold clouds are converted to stars: ) c t SNe release energy and move gas from cold to hot phase. Cloud mass evaporated proportional to SN mass:! Cold clouds grow via radiative cooling d c dt RC = d h dt d c dt EV RC = A c t = net( h, h ) h c 25 Benjamin Moster Numerical Structure Formation 4 IoA,

29 Stellar feedback: Multiphase model The rates at which the hot and the cold phases evolve are:! d c dt =! d h! dt c t The star formation timescale t* is chosen to reproduce the observed Schmidt-Kennicutt relation: t ( ) =t 0 Free parameters are chosen such that - KS-relation reproduced A c + net( h, h ) t - threshold density ρth corresponds to observed one h c = c + A c net ( h, h ) t t th h 1/2 c 26 Benjamin Moster Numerical Structure Formation 4 IoA,

30 Galactic winds Feedback is still very inefficient (overcooling)! Introduce mechanical feedback given by galactic winds! The mass-loss rate is proportional to the SFR: η is called mass loading factor, observations suggest η = 2! Wind carries fixed fraction of SN energy: Set equal to the kinetic wind energy:! Particle added to the wind if random number is below p w =1 exp (1 ) c t t SN Ṁ Ṁ w = Ṁ v w = and! ~v 0 = ~v + v w ~n Numerical trick: decouple winds from hydro for short amount of time 27 Benjamin Moster Numerical Structure Formation 4 IoA, s 2 SN (1 )

31 Blastwave feedback Thermal feedback is usually quickly radiated away (in dense gas)! Stinson et al. (2006) implemented feedback based on a SN blast wave! In each SN explosion a fraction of the SN energy is added to the ISM! Cooling is temporarily switched of in neighboring particles within a radius of: for a time: R E = E 0.32 SN n 0.16 P 0.2 pc t E = E 0.31 SN n 0.27 P 0.64 yr based on work by Chevalier (1974) and McKee & Ostriker (1977)! Neighboring particles will become hot, reducing the SFR! High temperature leads to high pressure and drives a galactic wind 28 Benjamin Moster Numerical Structure Formation 4 IoA,

32 AGN feedback In massive haloes, SN feedback cannot reduce the SF efficiency Potential well too deep to drive massive winds! Energy released from accretion of gas onto BHs can be considerable! Relevant scales for the BH cannot be resolved, hence effective model:! Accretion rate given by Bondi-Hoyle rate: limited by Eddington rate.! Ṁ B = 4 G2 M 2 BH (c 2 s + v 2 ) 3/2 Some fraction εf of the radiated luminosity can couple thermally to surrounding gas: Ė FB = f L r = f r Ṁ BH c 2 (εf: M-E conversion)! Usually εf = % of the accreted rest mass energy is available 29 Benjamin Moster Numerical Structure Formation 4 IoA,

33 AGN feedback Implementation: BH particles are colissionless sink particles though compute gas density & temperature for each BH! For each gas particle around BH compute: p = wṁbh with w = kernel weight. If random number < p absorb gas particle in BH (including momentum)! t Add energy kernel-weighted to the thermal energy of the surrounding gas particles.! Merge BHs if they are within one smoothing length of each other and if the relative velocity is smaller than the sound speed 30 Benjamin Moster Numerical Structure Formation 4 IoA,

34 Results of feedback In low mass haloes SN feedback is very efficient! In massive haloes AGN feedback can reduce the conversion efficiency! To simulate realistic massive galaxies, a combination of both is needed 31 Benjamin Moster Numerical Structure Formation 4 IoA,

35 Text Books:! Final notes Cosmology: Galaxy Formation and Evolution (Mo, vdbosch, White)! Papers:! Hopkins (2013), MNRAS, 428, 2840! Hu et al. (2014), MNRAS, 443, 1173! Springel & Hernquist (2003), MNRAS, 339, 289! Stinson et al. (2006), MNRAS, 373, 1074! Springel et al. (2005), MNRAS, 62, 79! Gadget and N-GenIC website: 32 Benjamin Moster Numerical Structure Formation 4 IoA,

36 Up next Lecture 1: Motivation & Initial conditions! Lecture 2: Gravity algorithms & parallelization! Lecture 3: Hydro schemes! Lecture 4: Radiative cooling, photo heating & Subresolution physics! Lecture 5: Halo and subhalo finders & Semi-analytic models! Lecture 6: Getting started with Gadget! Lecture 7: Example cosmological box! Lecture 8: Example galaxy collision 33 Benjamin Moster Numerical Structure Formation 4 IoA,

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38 Extra Slides