Gasdynamical and radiative processes, gaseous halos

Transcription

1 Gasdynamical and radiative processes, gaseous halos Houjun Mo March 19, 2004 Since luminous objects, such as galaxies, are believed to form through the collapse of baryonic gas, it is important to understand the properties of gas in structure formation.

2 General properties In linear evolution, the clustering properties of the are the same as dark matter on scales larger than the Jeans length ( 10 6 M ), but growth of structure is prohibited on smaller scales in the gas component. In non-linear collapse, gas can be heated by shocks, and in the absence of dissipation, gas will establish hydrostatic equilibrium in dark haloes. In the presence of energy dissipation (e.g. radiative cooling), gas can lose pressure support and collapse further in a dark halo potential. The goal: To understand how the gas component behaves in the collapse of cosmic density field

3 Basic principles Fluid equations v t + 1 a P γ 1 ρ t + 3ȧ a ρ + 1 (ρv) = 0, (1) a ( ȧ (v )v + φ + P ), (2) ( t + 1 a v a v = 1 a ) ( ) P ln ρ γ ρ = H C. (3) 2 φ = 4πGa 2 (ρ tot ρ tot ), (4)

4 Ionization Equations The number density for any given species i: n i t + 1 a (n iv) + 3ȧ a n i = j,l k jl n j n l l k il n l n i + Γ γ, j n j Γ γ,i n i, (5) j k jl : rate coefficient for a two-body reaction between species j and l, and Γ γ, j is the photoionization rate for species j. jl is over all j +l i; j over all j +γ i. l is over all two-body reactions involving the annihilation of an i. Γ γ,i n i, the destruction rate of species i.

5 Radiative processes Bound-bound processes. These are the processes by which an electron makes a transition from one bound level to another bound level in an atom (or ion). Such transitions can be made either by the collisions with electrons (collisional exitation and de-exitation) or by the interactions with photons (photon exitation, sponteneous and stimulated decay). Bound-free processes. These processes involve the removal of an electron from a bound orbit. Such removal can happen to an atom (or ion) either when it collides with an electron (collisional ionization) or when it absorbs a photon (photoionization). The reverse process is recombination, by which a free electron recombines with an ion. Free-free processes. These processes involve electrons only in unbound (free) states. When a free electron is accelerated or decelerated, it emits

6 photons through bremsstrahlung. A free electron can also absorb an photon through free-free absorption. Compton scattering. This is the process in which photons are scattered by free electrons (or ions). If the photon energy is high, electrons gain energy (the Compton process). In the opposite case where electron kinetic energy is much higher than the photon energy, energy transfer is from electrons to photons (the inverse Compton process).

7 The radiative cooling processes Compton Cooling: Photons of low energy hν pass through a thermal gas of nonrelativistic electrons with temperature T e (hν kt e m e c 2 ), photons and electrons scatter with the Thomson cross section. Since kt e hν, electrons lose energy to the radiation field. In a thermal background of photons with temperature T γ, the volume cooling rate for the gas is equal to the energy gain in the radiation field: C Comp = du γ dt = 4kT e m e c 2cσ Tn e a r T 4 γ.

8 Free-free emission or bremsstrahlung: At temperatures above 10 6 K primordial gas (consisting of hydrogen and helium) is almost entirely ionized, and above a few 10 7 K enriched gas (which contains also heavier elements) is fully ionized as well. The only significant radiative cooling is then bremsstrahlung due to the acceleration of electrons as they encounter atomic nuclei. For an optically thin gas, bremsstrahlung emissivity the volume cooling rate is obtained from the C ff = ε ff (ν)dν T 1/2 8 ( ne cm 3 ) 2 ergs 1 cm 3, where in the second expression we have assumed Z = 1 and n i n e (valid for a completely ionized hydrogen gas). The total cooling rate is the sum over all species of ions.

9 Other Processes Collisional ionization: Atoms (including partially ionized ones) can be ionized by collision with an electron, removing from the gas an amount of kinetic energy equal to the ionization threshold. Recombination: Electrons can recombine with ions, emitting a photon. Collisional excitation: Atoms can be excited by collision with an electron, thereafter decaying radiatively to the ground state. All these processes depend strongly on T, in the first and second cases because the recombination and (collisional) ionization rates are temperature sensitive, and in the third because the ion abundance depends strongly on temperature.

10 Table 1: H and He cooling rates Process Species Cooling rate/(erg s 1 cm 3 ) COLLISIONAL H e /T (1 + T 1/2 5 ) 1 n e n H 0 EXCITATION He T e /T (1 + T 1/2 5 ) 1 n e n + He COLLISIONAL H T 1/2 e /T (1 + T 1/2 5 ) 1 n e n H 0 IONIZATION He T 1/2 e /T (1 + T 1/2 5 ) 1 n e n He 0 He T 1/2 e /T (1 + T 1/2 5 ) 1 n e n He + RECOMBINATION H T 0.3 ( T 6 ) 1 n e n H + He T n e n He + He T 0.3 ( T 6 ) 1 n e n He ++ DIELECTROIC He T 1.5 e /T ( e 94000/T )n e n He + RECOMBINATION FREE FREE all ions g ff T 1/2 n e (n H + + n He + + 4n He ++)

11 These cooling functions assume ionization equilibrium. For gas not in ionization equilibrium, ion densities, cooling functions should be calculated using non-equilibrium densities. Ionization equilibrium can be achieved if the time scales for radiative processes are much shorter than the dynamical time scale.

12 Effects of Photoionization on Gas Cooling How photoionization changes cooling: By changing the ionization state: By heating the gas

13 Photoionization Photoionization is the process in which an atom is ionized by the absorption of a photon. H 0 + γ p + e, (6) The photoionization rate: Γ γ,h = ν t cσ pi (ν)n γ (ν)dν. (7) Ionizing flux: N γ (ν) = 4πJ(ν) ch P ν. (8) The photoionization cross sections can be obtained from quantum electrodynamics by calculating the bound-free transition probability of an atom in a radiation field.

14 Photoionization heating When an ionizing photon with energy hν ionizes an electron from an atom with threshold frequency ν i, the surplus energy h(ν ν i ) is transformed into the kinetic energy of the electon, thereby heating the gas. In a static state, photoionization is balanced by recombination. However the loss of energy due to the recombination is smaller than the gain from photoionization, because the rate of recombination is generally higher for lower-energy electrons.

15 The photoionization heating rate is proportional to the densities of ions which can be ionized, and can be written as where H = n i ε i i ε i = ν i 4πJ(ν) hν σ pi,i(ν)(hν hν i ) dν is the mean kinetic energy per photoionization of species i.

16 Radiative cooling by atoms is inefficient at temperature < 10 4 K, where gas is neutral, collisional excitations of atoms by electrons are reduced. Gas at such low temperature can be cooled further if molecules formed in the gas, because of vibrational and rotational quantum states. The energy involved in these transition is much lower than that in an atomic transition, and so molecules can be collisionally excited even in gas with low temperature. Radiative cooling can happen due to energy loss during de-excitations. Molecular Cooling

17 Shocks For a 1-d flow, in the frame moving with the shock front, the Rankine-Hugoniot jump conditions: ρ 2 v 2 = ρ 1 v 1, (9) ρ 2 v P 2 = ρ 1 v P 2, (10) 1 2 v2 2 + P 2 +U 2 = 1 ρ 2 2 v2 1 + P 1 P +U 1, with U = ρ 1 (γ 1)ρ, (11) subscript 1 : upper stream; 2: downstream; v 1 = v sh.

18 In terms of Mach number, M 1 v 1 /c s1 (c 2 s = γp/ρ: ρ 1 = v 2 = γ 1 ρ 2 v 1 γ γ 1 M 2 1, (12) P 2 P 1 = 2γ γ + 1 M2 1 γ 1 γ + 1. (13) If M 1 > 1, gas is compressed: ρ 2 > ρ 1 ; decelerated: v 2 < v 1 ; and heated T 2 > T 1 by the shock. Strong shock: M 1 1, ρ 2 /ρ 1 = (γ + 1)/(γ 1) (= 4 for γ = 5/3), T 2 = [2(γ 1)/(γ + 1) 2 ]µv 2 1 /k (= 3µv2 1 /16k for γ = 5/3),

19 Accretion shocks and the formation of gaseous halos Shock happens at r r vir. Suppose infalling velocity is v in and shocked gas have zero velocity. Then upper stream velocity v 1 = v in + v s ; downstream velocity v 2 = v s, where v s the velocity of the shock front Postshock gas temperature kt 2 = µv2 in 16γ [ ( 2γ 1 + ) ][ ε (γ 1)ε 2ε ( ε) ] + (γ 1) 2, (14) where ( ) 2 4 kt 1 ε γ 1 + γ µv 2. (15) in

20 Postshock density is where g = 1 + γ ( 1 T ) [ (1 γ) T 2 4 ρ 2 = gρ 1, (16) ( ) 2 ( 1 + γ 1 T 1 1 γ T 2 ) 2 + T 1 T 2 ] 1/2. (17) Shock generates entropy: s P ρ T. (18) γ n2/3 Which is conserved if not further shocked and if radiative cooling is not important.

21 The infalling velocity of a baryonic shell: [ 1 2 v2 in = 1 2 v2 ff + W c 2 s1 1 γ 1 ( ρta ρ 1 ) γ 1 ], (19) where c s1 is the sound speed of the preshock gas, ρ ta is the gas density at turnaround, and 1 2 v2 ff GM GM (20) R s R ta is the free-fall energy of the mass shell. Shock and compression heats the gas to roughly the virial temperature of the gas.

22 Hydrostatic equilibrium Pressure gradient balances gravitational force: P(r) = ρ(r) Φ(r), 2 Φ = 4πG(ρ DM + ρ). Under the assumption of spherical symmetry and for an ideal gas, dp dr = d(kt ρ/µ) dr, dφ dr = GM(r) r 2, where M(r) is the total mass within r and µ is the mean molecular weight, [ kt (r)r dlnρ(r) M(r) = + µg dlnr ] dlnt (r) dlnr For a polytropic gas with P = Aρ γ kt (r) = [(1 γ)/γ]µaφ(r), i.e. T exactly follows Φ(r)..

23 For an isothermal halo, T is indepndent of r: ( ρ(r) = ρ 0 exp Φ ), c 2 c 2 s = kt s µ, and the Poisson equation gives [ ( 1 d r 2 dr r2dφ dr = 4πG ρ DM (r) + ρ 0 exp Φ )]. c 2 s This can be solved for given ρ DM (r). If ρ DM = 0, the above equation reduces to the Lane-Emden equation. Assuming ρ(r) to be a power law, the solution is Φ(r) = 2kT µ ln r, ρ(r) = 2kT r 0 µ 1 4πGr 2, 2kT M(r) = µ r G.

24 Defining the circular velocity of the halo as V c [GM(r)/r] 1/2, we have Vc 2 = 2kT µ, ρ(r) = V c 2 4πGr, M(r) = V c 2 r 2 G. These are the properties of a singular isothermal sphere. It is clear that if ρ DM has a 1/r 2 profile, the above solution still holds, except that ρ and M are now the total density and total mass, respectively. If we define the limiting radius of a dark halo to be r 200 within which the mean mass density is 200ρ crit, then the total mass of the halo is M = V c 2 r 200 G = V c 3 10GH(z).

25 For the NFW dark halo profile, ρ DM = δ 0 ρ crit (r/r s )(1 + r/r s ) 2, isothermal gas with ρ ρ DM ) has the following profile: ρ(r) = ρ crit δ 0 e b ( 1 + r r s ) brs /r ; b = 4πGµρ 0r 2 s kt.

26 In general, if both gas and dark matter are in static equilibrium within the same potential well Φ(r), then 1 i P = 1 ρ gas ρ DM j j [ρ DM v j v i ]. This relation follows from eliminating the gravitational term Φ in the hydrostatic equation and in the second Jeans equation in static form. If the velocity distribution of dark matter particles is isotropic, i.e. v i v j = σ 2 δ i j, and if the halo is isothermal so that σ 2 and gas temperature T are independent of r, then ρ gas ρ β DM, β µσ2 kt. (which is the ratio of specific kinetic energies in dark matter and gas). This is usually called the β-model

27 More general cases Self-similarity solutions (Bertschinger 1985) Numerical simulations

28 Cooling in Gaseous Haloes Consider a uniform gas cloud with gas density n, radius R, and temperature T. The cooling time is t cool = 3nkT 2n 2 H Λ(T ) T 6 n 3 Λ 23 (T ) yr. This time scale should be compared to the collapse time scale: R 3 t coll = π GM, where M is the total mass of the cloud.

29 For gas in a dark halo at redshift z, where f gas = M gas /M, δ 200. n f gas (1 + δ)(ω 0 h 2 )(1 + z) 3, Virial theorem applied to a uniform cloud gives 3kT µ = 3GM 5R = 3 GM gas 5 f gas R. M gas T 6 f gas (1 + δ) 1/2 (1 + z) 3/2 M. t coll f 1/2 gas n 1/2 3 yr.

30 Effective cooling: t cool < t coll Note that for a given temperature t cool (1 + z) 3 and t coll (1 + z) 3/2 ; in the absence of a strong UV background, protogalaxies with 10 4 K < T < 10 5 K (where Λ 23 10) have t cool much smaller than t coll. So cooling is more effective at higher redshifts and in smaller protogalaxies

31 Overcooling problem In hierarchical models smaller haloes form at higher redshifts. According to the Press-Schechter formaula, the fraction of mass in halos with masses exceeding M is [ δc (t) F(M) = efrc ], 2σ(M) which goes to 1 for M sufficiently small M. Thus, if nothing prevents the cold gas from forming stars, then almost nothing is left to make big galaxies at later time or to make the intergalactic observed today. Possible solutions: A strong UV background suppresses cooling in halos with virial temperature T < 10 5 K, Energy feedback from stars prevents gas from cooling fast.

32 Cooling in an Isothermal Sphere Consider a singular isothermal sphere which initially contains gas in hydrostatic equilibrium at temperature T 0 and with density profile n(r) r 2. At later times a cooling radius, r cool, can be defined by t cool (r cool ) = 3kT 0 2n(r cool )Λ(T 0 ) = t, where t is the age. cooling grows with t. Note that r cool t 1/2 : the region affected significantly by If r cool > r virial, all gas in the halo can cool within a Hubble time; If r cool < r virial, only the region within r cool can cool effectively.

33 The mass accumulation rate at the center of a halo: If r cool < r virial : Ṁ cool = 4πρ gas (r cool )rcool 2 dr cool dt If r cool > r virial : Ṁ infall f gasm t = 3 f gasvc 3 20G An approximation: Ṁ(V c,z) min(ṁ cool,ṁ infall )

34 More detailed models of the formation of gaseous halos Self-similarity models without cooling Spherical models with cooling Cooling flow models (multiphase gas) Numerical Simulations: Results not certain in the presence of cooling, because cooling is a two-body problem and so requires high resolution to get reliable results.