ASTR 610 Theory of Galaxy Formation Lecture 15: Heating & Cooling Frank van den Bosch Yale University, Fall 2020 Heating & Cooling In this lecture we address heating and cooling of gas inside dark matter haloes. After discussing shock heating & hydrostatic equilibrium, we introduce the concept of `virial temperature’, discuss radiative cooling processes and introduce the cooling function. We discuss the link between cooling and galaxy formation, and end with a discussion of photo-ionization heating. Topics that will be covered include: shock heating hydrostatic equilibrium virial temperature radiative cooling cooling function & cooling time ionization equilibrium photo-ionization heating ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University Shock Heating Consider a gas cloud of mass M gas falling into a halo of mass M h with velocity vin At some point the gas is shocked; either close to center, where flow lines converge, or at the accretion shock, which is typically located close to the virial radius. If we assume that the shock thermalizes all the kinetic energy of the gas cloud, so 2 kBTin that v gas 0 after it is shocked (a reasonable assumption), and that v in µmp (so that internal energy of infalling gas can be ignored) then the internal energy of the shocked gas is equal to the kinetic energy of the gas at infall: 3 1 2 Eint,sh = 2 NkB Tsh = 2 Mgasvin where N = M gas / ( µm p ) is the number of gas particles, and we have assumed a mono-atomic gas, for which γ =5/3 µmp 2 Tsh = vin 3kB If the gas falls in from large distance (where Φ ( r ) 0 ), and has negligible, initial velocity, then v v (r )= 2 Φ(r ) in esc sh | sh | ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University Shock Heating 2 GMvir 2 If we assume that r sh = r vir (a common assumption), then vin = ζ = ζVvir rvir Here ζ = (1) is a parameter that depends on the detailed density profile of the halo. O The temperature of the shocked gas in a halo with virial velocity V vir is ζ µmp 2 Tsh Vvir 3 kB The build-up of a virial shock (discontinuity in velocity) at around the virial radius in a collapsing structure. Based on 1D calculations in an expanding Universe... Source: Birnboim & Dekel, 2003, MNRAS, 345, 349 345, MNRAS, 2003, Dekel, & Birnboim Source: ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University Hydrostatic Equilibrium If we assume that gas is non-radiative (cannot cool except via adiabatic expansion, and cannot be heated by radiation), then the shocked gas will settle in Hydrostatic Equilibrium P (r)= ρ Φ(r) ∇ − gas∇ Spherical Symmetry Ideal Gas dΦ GM(r) dP kB d Φ = = P = = (ρT) r dr r2 ∇ dr µmp dr k T (r) r d ln ρ d ln T M(r)=M (r)+M (r)= B gas + gas DM − µm G d ln r d ln r p If one knows T ( r ) and ρ gas ( r ) one can infer the total mass profile Mtot(r) ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University Hydrostatic Equilibrium NOTE: we have made the assumption that P = P thermal . In general, one can also have non-thermal pressure support from magnetic fields, cosmic rays and/or turbulence. When these are present we have that k T (r) r dlnρ dlnT P dlnP M (r)= B gas + + nt nt tot − µm G dlnr dlnr P dlnr p th contribution due to non-thermal pressure Unfortunately, accurate measurements of P nt ( r ) are extremely difficult to obtain.... NOTE: simply stating that the gas is in HE is not sufficient to determine its density or temperature profiles. To make progress one often makes simplifying assumptions. Examples of such assumptions are isothermal gas T (r)=T Pnt =0 polytropic gas P (r) ρΓ ∝ gas ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University Hydrostatic Equilibrium of Polytropic Gas A polytropic gas has an equation of state: P ρΓ Γ = polytropic index ∝ 1 Γ In this case, hydrostatic equilibrium implies kBT (r)= − µmp Φ(r) Γ (temperature profile reflects gravitational potential) Using the ideal gas law, according to which P ρ T , we also have that ∝ 1 Γ 1 Γ ρ(r) [T (r)] − P (r) [ρ(r)] ∝ ∝ An important example of a polytropic gas is an isentropic gas for which Γ=γ For a mono-atomic, isentropic gas, we have that γ =5 / 3 , and thus ρ(r) T (r)3/2 ∝ It can be shown that a polytropic gas cloud in HE is stable if Γ > 4 / 3 . If this criterion is not met, then a small compression of the gas cloud will not result in a sufficient increase in pressure gradient to overcome the increase in the gravitational force; the gas cloud is unable to re-establish HE. It will collapse and fragment... ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University The Virial Temperature In the absence of a full solution for T(r), one can get a rough estimate of the temperature of the gas using the virial theorem: I = moment of inertia 1 d2I K = kinetic energy =2K + W +Σ 2 dt2 W = potential energy Σ = surface pressure The system is said to be in virial equilibrium if 2 K + W + Σ =0 . If the system is not in virial equilibium, it either expands ( d 2 I/ d t 2 > 0 ) or contracts ( d 2 I/ d t 2 < 0 ) The gas in a halo of mass M vir and radius r vir is in virial equilibrium if Mgas GMgasMvir 3 3 kBTvir ζ 4πrclPext =0 µmp − rvir − where we have assumed that the gas is ideal and mono-atomic, and the halo is spherical. If we ignore the external pressure ( P ext =0 ), this defines the virial temperature: ζ µmp 2 virial temperature Tvir = Vvir 3 kB which is exactly the same as the temperature of the shocked gas defined before... ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University the halo bias function The Virial Temperature For a truncated, singular isothermal sphere of gas (no dark matter), the virial theorem implies a virial temperature µm V 2 T = p V 2 3.6 105 K vir vir 2 k vir × 100 km/s B where we have assumed that µ =0 . 59 , appropriate for a primordial gas (X,Y,Z) = (¾,¼,0) This is the definition of virial temperature most often adopted in the literature, and is identical to that defined above under the `assumption’ that ζ =3/2 CAUTION: in general, gas inside a (virialized) dark matter halo will have a temperature profile, and cannot be described merely by a single temperature. Nevertheless, the concept of `virial temperature’ is useful for order of magnitude estimates in galaxy formation theory, and is frequently used. ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University the halo bias function Radiative Cooling Thus far we have ignored radiative processes, which can cause both heating and cooling of the gas. In what follows we investigate how these radiative processes impact the gas in a virialized dark matter halo, focussing first on cooling. Let and be the volumetric heating and cooling rates, respectively. H C 1 3 [ ]=[ ]=ergs− cm− H C In what follows we ignore heating ( =0 ), and we assume that the gas is optically H thin, so that any photon that it emits escapes the system. It is useful to define the cooling function: Λ(T,Z) C2 ≡ nH which depends on the temperature, T, and composition (metallicity Z) of the gas, but not on its density. 1 +3 [Λ] = erg s− cm ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University the halo bias function Cooling Time The cooling time, the time it takes the gas to radiate away its internal energy, is given by ρε ρε t = cool ≡ n2 Λ(T ) C H where for the sake of brevity we don’t explicitely write down the metallicity dependence of the cooling function. Assuming an ideal, monoatomic gas, for which 1 k B T with this yields ε = γ 1 µm γ =5/3 − p 1 1 3nkBT 9 T n − Λ(T ) − tcool = 2 3.3 10 yr 6 3 3 23 1 3 2 nH Λ(T ) × 10 K 10− cm− 10− erg s− cm where we have assumed a completely ionized gas of primordial composition, for which n H =4 / 9 n , with n = ρ / ( µm p ) the number density of gas particles, which can be written as 2 5 3 fgas 1+δ Ωm,0h 3 n 9 10− cm− (1 + z) × 0.15 200 0.15 ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University Cooling Time 3nk T B 1 1 denser gas cools faster... tcool = 2 tcool n− ρ− 2 nH Λ(T ) ∝ ∝ In order to assess the impact of cooling on a system, we compare the cooling time to two other timescales: the age of the Universe, which is roughly the Hubble time 1 1 t = ρ¯ =Ω ρ H H(z) ∝ (Gρ¯)1/2 m crit the dynamical time (or `free-fall time’) of the system 3π 1/2 1 t = ρ¯ =¯ρ +¯ρ ff 32Gρ¯ ∝ (G ρ¯ )1/2 sys gas DM sys sys NOTE: the free-fall time is timescale on which gas cloud collapses in absence of pressure, and timescale on which system restores hydrodynamic equilibrium if disturbed.