What is Energy conservation Rate in the Universe?

Size: px

Start display at page:

Download "What is Energy conservation Rate in the Universe?"

Oswald Andrews
3 years ago
Views:

1 Thermal Equilibrium Energy conservation equation Heating by photoionization Cooling by recombination Cooling by brehmsstralung Cooling by collisionally excited lines Collisional de-excitation Detailed Balancing Critical Densities Comparison of heating and cooling rates 1

2 Energy conservation Heating provided by photoionization of electrons Cooling provided by: 1) recombination lines (mostly H, He) 2) brehmsstralung (free-free radiation) 3) collisional excitation of heavy ions, and subsequent radiation Thermal equilibrium: G = L r + L ff + L c (G energy gained by photoionization, L energy lost by radiation due to the above processes) Ionization: creates an electron with energy ½ mv i 2 = h(ν- ν 0 ) Recombination: electron gives up energy = ½ mv f 2 Net energy that goes into heating: ½ mv i 2 ½ mv f 2 2

3 Heating Heating provided by photoionization of electrons For a pure hydrogen nebula: G(H ) = energy input/vol/sec (ergs s -1 cm -3 ) G(H ) = n H 0 ν 0 4π J ν hν h(ν - ν 0 ) a ν (H 0 ) dν = n e n p α A (H 0,T ) 3 2 kt i (using ionization equil.) T i is the initial electron temperature T i (electron temperature) is low close to the star, as hν 0 photons are absorbed in the inner nebula first However, as τ increases, T i increases due to absorption of photons with energies > hν 0 3

4 Energy Loss by Recombination L R (H) = n e n p kt β A (H 0,T) where β A = recomb. coeff. averaged over kinetic energy β nl (H 0,T) = 1 v σ kt nl ( H 0,v) f(v) 1 2 mv 2 dv 0 On the spot approximation: L R (H) = n e n p ktβ B (H 0,T) where β B summed over all states but ground For a pure H Nebula: G(H) L R (H) (free-free radiation not very important) For recombination of He: - same formulae as for heating and cooling of H Recombination of other elements: - usually not important, since heating and cooling proportional to ionic densities (and abundances are low) 4

5 Brehmsstralung (Free-Free Radiation) L ff = 1.42 x Z 2 T 1 2 g ff n e n + where g ff = Gaunt factor, weak function of n e and T - quantum mechanical correction for classical case - between 1.0 and 1.5 for H II regions - brehmsstralung usually not very important at nebular temperatures; recombination and collisional excitation dominates (but dominant cooling mechanism in T = K intracluster gas) 5

6 Collisionally Excited Radiation Dominated by collisional excitation of low-lying levels of heavy elements (e.g., O +,O ++,N + ) Excited levels are mostly metastable, which result in forbidden or semi-forbidden lines (low A values) ΔE kt, so very important coolants, despite lower abundance Consider two levels: lower (1) and upper (2) Collision cross-section: σ 12 (v) = πh2 m 2 v 2 Ω 12 ω 1 (for 1 2 mv 2 > χ where χ = hν 12 ) where Ω 12 = collision strength from levels 1 to 2 (essentially constant with temperature at these electron velocities) ω 1 = statistical weight for level 1 6

7 Collision Strengths (Osterbrock & Ferland, p. 53) calculated quantum-mechanically Ex) Collision strength for [O III] 3 P 1 D = 2.29 Radiative Transitions : 1 D 2 3 P 2 : λ D 2 3 P 1 : λ4959 J = 1 7

8 Ex) Energy-Level Diagram for [O III], [N II] (Osterbrock & Ferland, p. 59) 8

9 Partial Grotrian Diagram for [O III] O +2 (Carbon-like): - Ground: 1s 2 2s 2 2p 2-2 outer shell electrons - L-S coupling from Partial Grotrian Diagrams of Astrophysical Interest, Moore, C.E. & Merrill, P.W., NSRDS National Bureau of Standards, Vol. 23 (1968) L = S =

10 - detailed balancing: populations of levels remain constant in equilibrium - rate of population of a level = rate of depopulation - relation between cross sections for collisional excitation and de-excitation can be derived from thermodynamic equilibrium (Osterbrock & Ferland, p. 50): Consider a two-level transition (1- lower, 2 - upper): ω 1 v 1 2 σ 12 (v 1 ) = ω 2 v 2 2 σ 21 (v 2 ) Collisional De-Excitation where 1 2 mv 1 2 = 1 2 mv χ (where χ = hν 12 ) So : σ 21 (v 2 ) = π h2 m 2 v 2 2 Ω 12 ω 2 (similar to formula for σ 12 (v 1 ) 10

11 The collisional de - excitation rate is: # de-excitations/vol/sec = n e n 2 q 21 q 21 = v σ 21 f(v) dv (q 21 in cm 3 s 1 ) 0 (note similarity to recombination) The collisional excitation rate is : #collisions / vol / sec = n e n 1 q 12 where q 12 = ω 2 ω 1 q 21 e -χ kt Note: q ij is a function of (σ ij,v) which is a function of (Ω ij, v) or (Ω ij, T) Collision Rates 11

12 Energy Loss by Collisional Processes 1) Single excited level, low n e L C = n e n 1 q 12 hν 12 (every excitation followed by radiative transition) 2) Single excited level, higher n e n e n 1 q 12 = n e n 2 q 21 + n 2 A 21 # collisions/vol/sec = # de-excitations/vol/sec + # transitions/vol/sec - solve above eqn. for n 2 to get relative level populations n 1 Solve for population of level 2 : n 2 = n(x) n 1 n(x) = number density of element X L c = n 2 A 21 hν 12 12

13 3) For multiple levels, use detailed balancing: - multiple equations for each level, # in = # out For each level i of an ion X : n j n e q ji + n j A ji = j i j>i n i n e q ij + n i A ij j i j<i (transitions into i) = (transitions out of i) together with : i n i = n(x) can be solved for the population in each level n i. L c = i n i j<i A ij hν ij 13

14 Critical Density - For a given level i, n c is the density at which # radiative transitions/vol/sec = # de-excitations/vol/sec Let n e = n c when this occurs: n i A ij = n c n i q ij j<i n c (i) = j<i j i A ij q ij j i - At densities n e > n c, line emission from i!j is significantly suppressed. 14

15 Transition Probabilities ( A values) (Osterbrock & Ferland, p. 56) 15

16 Critical Densities for Some Important Levels (Osterbrock & Ferland, p. 60) 16

17 Heating and Cooling Rates for a Low-Density Gas effective heating = cooling G - L R = L ff + L c Per n e n p - G L R : dashed L ff + L c : solid (Osterbrock & Ferland, p. 62) 17

18 Heating and Cooling Rates for n e = 10 4 cm -3 Collisional de-excitation raises temperatures (Osterbrock & Ferland, p. 63) 18

Heating & Cooling in the Interstellar Medium

Section 7 Heating & Cooling in the Interstellar Medium 7.1 Heating In general terms, we can imagine two categories of heating processes in the diuse ISM: 1 large-scale (mechanical, e.g., cloud-cloud collisions),