Lecture 7: Molecular Transitions (2) Line radiation from molecular clouds to derive physical parameters H 2 CO (NH 3 ) See sections 5.1-5.3.1 and 6.1 of Stahler & Palla Column density Volume density (Gas temperature) Rotational levels: E rot = J(J +1) BhJ(J +1) 2I Rotational quantum number Rotational constant (Hz) The moment of inertia of H 2 is the smallest of any diatomic molecule! its energy levels are widely spaced. Rotational levels of H 2 decay through electric quadrupole transitions, in which J decreases by 2. J=2 0 has an associated energy change of 510 K (λ=28.2 µm).
H 2 total energy: E tot = E rot + E vib + E elect E vib = hν 0 (v +1/2) Vibrational quantum number In the hot environments where vibrational states are excited, H 2 relaxes through rovibrational transitions, in which J and v change (the change in v is unrestricted, while ΔJ =0 or ±2). Designation: v = initial vibrational state; v = final vibrational state: v -v O(J ) if J -J =2 v -v Q(J ) if J -J =0 v -v S(J ) if J -J =-2 Electronic transitions have an energy separation of the order 10 5 K. The ground electronic state contains 14 vibrational levels, plus a continuum of levels with E > ΔE diss = 4.48 ev, the molecule s binding energy. Indirect radiative dissociation: a photon with E > 11.2 ev excites H 2 to a higher electronic state. About 85% of the time, H 2 drops back to ground through electronic and rovibrational transitions (fluorescence). Emitted lines range from UV to IR. Lyman band: cluster of transitions from first excited electronic state to ground state. Werner band: from second excited electronic state to ground state. In order to dissociate, the molecule must decay from an excited electronic state to a vibrational continuum level lying above the v=14 level in the ground state.
E rot = BhJ(J +1) ΔJ = ±1 Carbon Monoxide (CO) The J=1 state is elevated above the ground by 4.8x10-4 ev or, equivalently, 5.5 K! easy to excite in a quiescent cloud. Within a molecular cloud, excitation of CO to the J=1 level occurs primarily through collisions with the ambient H 2. In a cloud with relatively low, each upward transition is followed by emission of a photon. Conversely, when is high, the excited CO usually transfers its excess energy to a colliding H 2 molecule, with no emission of a photon. The critical density separating these two regimes is 3x10 3 cm -3. The Two-Level System: see S&P Appendix B Collisional coefficient Einstein coefficient for absorption γ lu ΔE γ ul B ul A ul Collisional deexcitation coefficient Einstein coefficient for stimulated emission Einstein coefficient for spontaneous emission Problem: find the level populations and as a function of the ambient kinetic temperature T kin and density. In order for the populations of both levels to remain constant in time, we need: γ lu + J = γ ul + B ul J + A ul Total rate of collisional excitations per unit time and per unit volume Probability per unit time of being excited radiatively
In Local Thermodynamic Equilibrium at high densities (i.e. when radiative transitions are negligible): γ lu γ ul = = g $ u exp ΔE ' & ) g l % k B T kin ( When is so low that that radiative transitions dominate: J = A ul A = ul /B ul B ul (g l /g u B ul )exp(δe /k B T rad ) 1 Under the assumption of thermodynamic equilibrium (J ν =B ν, much broader than the profile function φ(ν)): J B ν 0 = 2hν 0 3 /c 2 exp(δe /k B T rad ) 1 A ul = 2hν 3 0 B c 2 ul g l = g u B ul γ lu + J = γ ul + B ul J + A ul $ exp ΔE ' $ & ) = f coll exp ΔE ' $ & ) + (1 f coll )exp ΔE ' & ) % k B T ex ( % k B T kin ( % k B T rad ( f coll + n crit (1+ c 2 J /2hν 0 3 ) fraction of all downward transitions due to collisions n crit A ul /γ ul Excitation Temperature k B T ex /ΔE 10 8 6 4 2 0 k B T kin /ΔE k B T rad /ΔE 0 1 2 Ambient Density log( /n crit )
Emission in the J=1 0 line of 12 C 16 O Increasing the density in a cloud can enhance the J=1-0 emission, but only for subcritical values of. Observations in a given transition are most sensitive to 9 gas with densities near the corresponding n crit. Transfer of radiation in spectral lines The propagation of the specific intensity I ν is governed by the radiative transfer equation: di ν ds = α ν I ν + j ν I ν (0) I ν (s) I ν (Δs) s=0 s=δs I ν (Δs) = I ν (0)exp( α ν Δs) + j ν α ν [1 exp( α ν Δs)]
We consider the case in which both absorption and emission are due to transitions between two discrete levels in an atom or molecule. The macroscopic emission and absorption coefficients can be written in terms of the microscopic Einstein coefficients: j ν = hν 0 4π A ulφ(ν) j ν α ν = α ν = hν 0 4π ( B ul )φ(ν) A ul B ul = 2hν 0 3 /c 2 exp(hν 0 /k B T ex ) 1 I ν 0 (Δτ 0 ) = I ν 0 (0)e Δτ 0 + 2hν 0 3 /c 2 ( ) exp(hν 0 /k B T ex ) 1 1 e Δτ 0 The quantity of interest to the observer is not I ν itself, but rather the difference between I ν and the background intensity. Accordingly, we define a brightness temperature T B by: T B c 2 [I 2ν 2 ν (Δτ ν ) I ν (0)] k B If we make the final assumption that the background radiation field is Planckian with an associated temperature T bg : T B0 = T 0 [ f (T ex ) f (T bg )] 1 e Δτ 0 [ ] T B0 T B (ν = ν 0 ), T 0 hν 0 /k B f (T) [exp(t 0 /T) 1] 1 T Bo is what we measure if we are observing the source with a perfect telescope above the atmosphere, and with an angular resolution much smaller than the source size. 12
Column Density α ν = hν 0 4π ( B ul )φ(ν) Δτ 0 = 2 ln2 c 3 A ul (g u /g l ) 8π 3 / 2 ν 0 3 Δv - ' 1 exp ΔE * 0 / ), 2 Δs. ( k B T ex + 1 For the two-level system, ~n and Δs~N, the column density. Such an approximation is however not adequate for a system in which the two levels are closely spaced rungs in the ladder (e.g. CO). In these cases: Rotational partition function n = Q # Q = (2J +1)exp% $ j= 0 BhJ(J +1) k B T ex & ( ' Relation between CO and the total hydrogen column densities
NH 3 (inversion) lines to derive gas temperature [Not examined]! [Not examined] Optical depth from hyperfine satellites N. tot Δτ ν = Δτ 0 α i exp ν ν Δν 2 ( + 1 0 i 0 * - 3 ) i= 0 / Δν, 2 Temperature from population of metastable levels See Walmsley & Ungerechts (1983), A&A, 122, 164: Ammonia as a molecular cloud thermometer
The strength of B (Zeeman splitting) Total angular momentum Recall B-field morphology could be inferred from polarization observations. Zeeman splitting provides a means to measure field strength. Spin angular momentum Orbital angular momentum The Zeeman effect can be interpreted in terms of the precession of the orbital angular momentum vector in the magnetic field. In the case of the 21-cm line of H, the splitting Δν between the highest and the lowest levels is 2.8 B tot Hz, where B tot is the total field strength in µg. Measurements in OH emissioines yield field strengths ranging from 10 µg in the more rarefied outer portions of molecular clouds to ~50 µg in compact central regions. Object Type Diagn. B(µG) Ursa Major L204 NGC 2024 B1 S106 Sgr A/ West Diffuse Cloud Dark Cloud GMC clump Dense core HII Region Molecul. Disk Observed field strengths HI 10 HI 4 OH 87 OH 27 OH 200 HI 3000 W75 N Disk OH 3000 The general tendency for B to rise with density is an indication that the field lines in a cloud can be compressed along with the gas. The physical basis is the phenomenon of flux freezing. But this is not a general rule, as we will see in future lectures.