1 Components of Galaxies Gas The Importance of Gas Fuel for star formation (H 2 ) Tracer of galaxy kinematics/mass (HI) Tracer of dynamical history of interaction between galaxies (HI)
2 The Two-Level Atom Consider an atom comprised of only two levels, 1 & 2 n 2 atoms m -3 n 1 atoms m -3 When the atoms jumps from 2 to 1, it emits a photon of frequency ν 21 & energy hν 21, where h is the Planck constant (6.63x10-34 J. s).
3 Spontaneous Transition Probability from 2 to 1 is expressed in terms of the Einstein coefficient A 21 (s -1 per atom). The energy released per second per cubic meter through spontaneous decay is thus Stimulated Transition Probability is expressed in terms of the Einstein coefficient B 21 & the energy density U 21 (= I ν / c) such that the energy released per second per cubic meter per atom is, Similarly, B 12 represents the stimulated up transition probability.
4 The net number of absorptions per unit volume per second is then And thus the energy absorbed per unit volume per second is, Where κ ν is the absorption coefficient. Note that the source Function, S, is simply the ratio of the emission coefficient & the absorption coefficient j ν,
5 Given, I ν / c can be substituted for U 21 to get Balancing the absorption rate with the emission rate, In terms of Einstein coefficients, where the factor 4π comes from integrating over 4π steradian solid angle.
6 Rearranging the terms in A 21 And solving for I ν, In Thermodynamic equilibrium, Thus, given that the levels are populated as where g 1 & g 2 are statistical weights & T is the excitation temperature,
7 The Einstein coefficients are equal to.
9 Neutral Hydrogen A source of strong emission in galaxies is the 21 cm neutral hydrogen emission line. The line arises from the flip in the spin state of an electron in the ground state of neutral hydrogen.
10 Neutral Hydrogen, cont. Forbidden transition, with A 10 = 2.9x10-15 s -1. Transition rate lifetime is t 10 = 1.1x10 7 years Why is the signal strong? High HI column density (i.e., N HI = cm -2 ). I.e., if n 1 l is large, where l is the length along the line of sight containing HI, then h ν 10 n 1 l is large.
11 In galaxies, collisions dominate 21 cm transitions I.e., emitting clouds are in thermodynamic equilibrium. Determination of collision time: 1) collisional cross section σ = π (2x10-8 cm) 2 = 1x10-15 cm 2 2) atomic density N = 1 cm -3 The mean free path is Atomic gas in galaxies has T = 100K, thus the velocity v is, Thus, the time between collisions is
12 Most of the electrons in neutral hydrogen are in the F=1 state. This can be shown by comparing E = h ν / k T ex = 100 K (Typical HI T) Thus, where g F = 2F + 1. Since n 1 ~ 3n 0, then n HI = n 1 + n 0 = 4n 0
13 Optical Depth For HI at T = 100 K, h ν << k T. Thus, the spin states are populated where g 1 / g 0 = 3. Similarly, HI absorption occuring from transitions are proportional to Making the substitution n 0 = ¼ n H
14 The optical depth, τ, is thus, Where 1) N H = H column density 2) f(v) is a function that accounts for the contribution of 21 cm emission occuring at velocity v. f(v) is normalized such that, A useful form is where
15 Hydrogen Column Density The above expression can be rewritten & solved for hydrogen column density Or for τ << 1,
16 In terms of HI Mass, where, 1) d is in units of Mpc 2) S(v) dv is in units of Jy km s -1.
17 HI in Emission & Absorption HI is seen in both emission & absorption, depending on the optical depth & presence of background continuum source. Consider the equation of radiative transfer, Multiplying through by e -τ ν & rearranging the terms,
18 For an isothermal cloud in LTE, Sν = Bν = constant. Integrating yields, For this equation, 1) I ν (0) is the intensity of the background source 2) S ν is the intensity generated within the cloud. Typically, radio astronomers use temperatures instead of fluxes, where temperature is simply that given by the Planck function in the Rayleigh-Jean regime. Thus, for I ν & S ν, where T B = brightness temperature & T cloud = cloud temperature
19 Thus, How does the temperature measured change with τ? Consider a cloud observed in front of a background source of temperature T B (0) telescope Cloud Background Source
20 Molecular Gas Molecular hydrogen clouds are the sites of star formation Dust grain surfaces are the catalyst for H 2 formation The ratio of the number densities of dust grains to hydrogen atoms is n g / n H = The formation rate of H 2 is thus, where πa 2 is the grain cross section (a = 1x10-7 m) and For gas at a temperature of 100 K, v H = 1000 m s -1. Thus,
21 The destruction rate of H 2 is s -1. Specifically, 1) s -1 for N H2 < m -2 2) & s -1 for N H2 = m -2. Thus, the destruction rate is equal to the formation rate when, The cross section for photodissociation is σ PD = m 2. Thus, where l is the path length.
22 Thus, the condition required for significant shielding of dissociating radiation is, Thus, for a cloud with moderate hydrogen number density of n H ~ 10 8 m -3, the core can be molecular if the shell thickness is on the order of l ~ 1 pc.
23 CO as a Tracer of Molecular Hydrogen H 2 has no easily observed transitions 28 µm from space 2 & 12 µm only warm (T=100K) H 2 CO collisionally excited by H 2.
24 Collisional Excitation of CO The collisional coefficient is given by where 1) v H2 is the velocity of H 2 molecules 2) σ CO is the collisional cross section of CO 3) N CO is the number density of CO For significant emission from collisions, C = A J,J-1, where A J,J-1 is the Einstein coefficient for spontaneous emission. Thus,
25 The CO lines are optically thick, τ >> 1, thus N crit must be divided by the number of steps, N esc, it takes for a photon to random walk out of the molecular cloud, Thus, for CO(1 0), N crit > 3000 cm -3 (for τ << 1), and few 100 cm -3 (for τ >> 1).
26 The Mass of Molecular Hydrogen Clouds CO luminosity can be used to calculate the H 2 mass. The CO luminosity, L CO, of a molecular cloud is, where, 1) R = radius of the cloud 2) T CO v = flux of the CO emission line For a cloud in virial equilibrium, twice the time averaged kinetic energy KE is equal to the negative of the time-averaged potential energy W
27 Substituting in for v in the luminosity equation, Because, then, ½ Solving for M cloud yields,
28 It turns out that i.e., high density clouds are hot, low density clouds are cold. Thus, The proportionality constant is α.
29 What is α? (Scoville & Sanders, in Inter- Stellar Processes) L CO vs. dynamical mass in molecular clouds So, α = 4 M solar (K km s -1 pc 2 ) -1 Note the scatter!
30 Dynamical Mass Determinations In external galaxies, the virial theorem can be used to calculate the dynamical mass of a CO disk, & thus determine an upper limit on α. Dynamical mass is where, 1) v los = line of sight velocity 2) R is the radius of the disk 3) i is the inclination angle. Note that v los v true i
31 Properties of HI & H 2 in Galaxies The HI extent > optical extent of galaxies (2 3 R H at n = cm -2, where R H is the Holmberg Radius the radius at which Σ pg = 26.5 mags arcsec 2 ) Optical HI extent has a sharp cutoff (at n HI ~ cm -2 ) HI Many galaxies have HI holes in the central regions which correspond to the location of the stellar bulge. (Gilmore et al., pg 203)
32 Distribution vs. R HI Holes Some (like the Milky Way) have Molecular Rings (Scoville & Sanders 1987, in Interstellar Processes, pg 21)
33 HI, H 2, & Dust in the Milky Way (Gilmore et al., pg 68)
34 Counter-Example No HI hole No molecular ring (Young & Scoville 1991, ARAA, 29, 608)
35 Properties of HI & H 2 in Galaxies, cont. H 2, & not HI follows the distribution of dust & Hα emission in galaxies This is because dust is a catalyst for H 2 formation and stars form in molecular clouds
37 HI Profile HI Disk Inner part thin (half height ~ 500 pc) Outer part thick & warped (i.e., beyond the optical light) (B&M pg. 504)
38 HI Rotation Curves Useful tracer of mass in the galaxy halo (Galaxies & Cosmology, Combes et al., pg 71)
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