HIA Summer SchoolMolecular Line ObservationsPage 1 Molecular Line Observations or “What are molecules good for anyways?” René Plume Univ. of Calgary Department of Physics & Astronomy more detailed notes can be found at

HIA Summer SchoolMolecular Line ObservationsPage 2 Molecules in the ISM molecules as of 2005 Molecular line observations are not just stamp collecting. Can be used to determine: - gas density - gas temperature - molecular abundances - cloud kinematics HOW??? - for that we need to look at properties of molecules & radiation

HIA Summer SchoolMolecular Line ObservationsPage 3 Molecular Excitation  Molecules can rotate Can also vibrate but we will focus on rotation Kinetic energy of rotation is: since Angular momentum  But this is quantum mechanics so rotation is quantized  Energies of rotational levels are low enough that collisions between particles can cause excitation & de-excitation Thus molecular rotation lines can be used to find physical conditions (densities & temps) in interstellar gas

HIA Summer SchoolMolecular Line ObservationsPage 4 A Perfectly Rigid Rotor  Solution of Schroedinger’s equation results in eigenvalues for the rotational energy where  J in the QN for the total ang mom J = 0, 1, 2, …… = rotation constant (often expressed in Hz)

HIA Summer SchoolMolecular Line ObservationsPage 5 NON-linear Molecules  In general All 3 MoI are different called an ASYMMETRIC ROTOR or ASYMMETRIC TOP MoI are labeled as I A, I B, I C I A < I B < I C  In some cases 2 MoI are equal called a SYMMETRIC ROTOR or SYMMETRIC TOP If 2 largest MoI are equal (I B = I C )  Have a prolate symmetric top  Foot ball shaped molecule  Linear molecule is a special case of a prolate symmetric top  CH 3 CN N H HH H HH C N C If 2 smallest MoI are equal (I A = I B )  Have an oblate symmetric top  Frisbee shaped molecule  NH 3

HIA Summer SchoolMolecular Line ObservationsPage 6 Symmetric Top Molecules  Much of the behavior can be deduced from Classical mechanics Molecule rotates about molecular axis with ang mom J Z And there is a precession of this axis about the total ang mom (J) Energy of rotation is: JZJZ J  For a PROLATE symmetric top I A < I B = I C Using the fact that J 2 = J A 2 + J B 2 + J C 2

HIA Summer SchoolMolecular Line ObservationsPage 7 Prolate Symmetric Top Molecules  Now we make the transition to quantum mechanics J 2 and J A 2 both have eigenvalues JZJZ J Energies = Where  Since I A B Second term is positive So each J value corresponds to a series of 2J+1 levels lying progressively higher in energy  J A is the projection of the total J onto the molecular axis K is the quantum number associated with this projection 2J + 1 values of K  K = 0, ±1, ±2,…,±J

HIA Summer SchoolMolecular Line ObservationsPage 8 Oblate Symmetric Top Molecules  For Oblate symmetric tops: I A = I B < I C And J 2 = J A 2 + J B 2 + J C 2 Energies = Where  Since I B < I C C < B Second term is negative So each J value corresponds to a series of 2J+1 levels lying progressively lower in energy  K ladders are RADIATIVELY DECOUPLED So populations across K-ladders are controlled by collisions More on this later….. See “Townes & Schaalow”

HIA Summer SchoolMolecular Line ObservationsPage 9 So What? What does this give us?  Consider an Interstellar cloud…….   0 SSoSo 0 I or B (T B ) I (bg) or B (T bg )  T K

Basics of Radiative Transfer  For a photon traveling in a straight line….  0 SSoSo 0  I = Specific Intensity (erg s -1 cm -2 sr -1 Hz -1 )  K = Absorption coefficient  j = Emission coefficient  d  = –K ds = Opacity   = 0 at observer and increases toward source (if K > 0) A measure of how far we see into the source

Basics of Radiative Transfer  0 SSoSo 0  The radiative transfer equation can be solved and, in LTE, can be re-written as: Planck function Where:

HIA Summer SchoolMolecular Line ObservationsPage 12 Special cases  If  << 1 A Taylor expansion gives: e -   1; therefore: Absorption of background radiation by foreground cloud Emission of foreground cloud at a temperature T into the beam All background radiation is absorbed by the intervening cloud and there is only emission of foreground cloud at a temperature T into the beam Virtually no foreground emission and no absorption of background radiation  If  >> 1 e -   0; therefore:

HIA Summer SchoolMolecular Line ObservationsPage 13 Radiative Transfer using Einstein A Coefficients  As before, pass radiation through a slab of thickness ds Intensity changes  3 different processes to consider: Total amount of energy emitted spontaneously over the full solid angle 4 . Total amount of energy absorbedTotal amount of stimulated emission  Total amount of energy emitted is:

HIA Summer SchoolMolecular Line ObservationsPage 14 Radiative Transfer using Einstein A Coefficients  So the radiative transfer equation is:  Remember:  Therefore: After some algebra….

HIA Summer SchoolMolecular Line ObservationsPage 15 Column Densities from Observations  So, if  << 1 becomes  N 2 is the column density (cm -2 ) of particles in the upper state If B(T bg ) = 0 of course we can always subtract B(T bg ) from the observations by chopping on the sky

HIA Summer SchoolMolecular Line ObservationsPage 16 Column Densities from Observations  Now to incorporate the line profile, we measure the frequency or velocity integrated intensity:  So,  And therefore, Normalized to unity  If h << kT

HIA Summer SchoolMolecular Line ObservationsPage 17 Column Densities from Observations  For optically thin emission (all photons created escape cloud) for a single transition: ∫ T B dv = integrated intensity of the line A ul = spontaneous emission coefficient = frequency of the given transition  N u = f u N tot f u = fraction in upper state N tot = total column density directly from observables! want this

HIA Summer SchoolMolecular Line ObservationsPage 18 Column Densities from Observations  Fraction in the upper state given by the partition function. But…. So….  So to calculate the total column density of say 13 CO J = 2-1: See the file: example_column_den.doc for a worked example

HIA Summer SchoolMolecular Line ObservationsPage 19 Why do we give a #$&^%!? Dickens et al. 2000, ApJ 542, 870 L134N  So for every molecule we observe we can make a map of the column density (abundance)  But we often find that the distribution of different chemical species is different…  WHY? Not all related to excitation Many species/line have similar excitation requirements  T ex and n crit Only way to understand the complex distribution of molecular species is to apply both physics & chemistry to understand why the abundances vary with position  Age, temp, dust properties, etc.  So let’s look at interstellar chemistry for a moment….

HIA Summer SchoolMolecular Line ObservationsPage 20 Formation of Molecules  In any gas, atoms can chemically interact to form molecules  Given table above and the fact that He & Ne are chemically inert Expect molecules with H, C, N & O to be most abundant Most common is H 2 ElementAtomic NumberAbundance H11 He20.1 C64x10 -4 N O89x10 -4 Ne Cosmic elemental abundances

HIA Summer SchoolMolecular Line ObservationsPage 21 Gas-Phase Chemical Reactions Formation of molecule M Destruction of molecule M Reaction rate = k f (cm 3 s -1 ) Reaction rate = k d (cm 3 s -1 ) CR-destruction of molecule M Reaction rate =  cr (s -1 ) (related to CR flux) Photo-destruction of molecule M Reaction rate =  uv (s -1 ) (related to UV flux)

HIA Summer SchoolMolecular Line ObservationsPage 22 Gas-Phase Chemical Reactions So the rate of change of the Abundance of molecule M is given by: In reality there will be a reaction network of 1000’s of reactions and you need to solve for the abundance change in each simultaneously via a series of stiff differential equations I.e. UMIST data base reactions coupling 400 species

HIA Summer SchoolMolecular Line ObservationsPage 23 Time dependent chemical evolution Bergin, Langer & Goldsmith 1995, ApJ, 441, 222 Log time So you can begin to see why different species have different abundance distributions. It depends on the chemical history of the core

HIA Summer SchoolMolecular Line ObservationsPage 24 Gas - Grain Interactions  Dust grains are also important in chemistry Primarily responsible for the formation of H 2 In addition, all species can interact with grains  gas-phase species can accrete (adsorb or freeze-out) onto dust grains And thus be removed from gas phase  species attached to grain surfaces may react with one another Forming new species Which can later be ejected back into the gas phase (desorption or evaporation)

HIA Summer SchoolMolecular Line ObservationsPage 25 Bergin, Langer & Goldsmith 1995, ApJ, 441, 222 Log time

HIA Summer SchoolMolecular Line ObservationsPage 26 Example of grain freeze out - H 2 O & O 2 Steady state chemical models predict high H 2 O and O 2 abundances (since the 70’s) Maréchal et al. 1997,A&A,324,221 Observed Abundance ranges

HIA Summer SchoolMolecular Line ObservationsPage 27 Example of freeze out - O 2 & H 2 O Abundances  Why are O 2 and H 2 O abundances so low? Freeze out onto dust grains Followed by chemical reactions on the surface of grains Roberts & Herbst 2002, A&A, 395, 233

HIA Summer SchoolMolecular Line ObservationsPage 28 Surface Migration & surface reactions  Once adsorbed onto a grain surface the species does not just sit in one spot It can migrate from binding site to binding site  via quantum tunneling through the potential wells that separate each binding site for light species (H, D, etc.) Timescale for H tunneling is 1.5x s  via thermal hopping for heavier species In a typical grain, H will visit all 2x10 6 binding sites in ~ seconds  thus it will visit each binding site many times before evaporating  heavier atoms will visit each binding site in ~ 100 hours (again less than the evaporation timescale) So during this migration, species can encounter one another at various binding sites and react  Forming new species  which can eventually evaporate back into gas phase  the best way to produce observed abundances of certain species like H 2 CO and CH 3 OH

HIA Summer SchoolMolecular Line ObservationsPage 29 Hot Cores - an example of Grain surface reactions Rodgers & Charnley 2003, ApJ, 585, 355 Chemical abundances in a collapsing envelope as a function of distance from a protostar (after 10 5 years)

HIA Summer SchoolMolecular Line ObservationsPage 30 No Really…Why Do We Care??  Distribution and abundance of molecules critical to understand chemistry Understanding chemistry is important in understanding:  ionization fraction Controls magnetic support of cloud against collapse  thermal balance Controls thermal support of cloud against collapse

HIA Summer SchoolMolecular Line ObservationsPage 31 Molecular Cloud Cooling Goldsmith & Langer 1978, ApJ, 222, 881 Speaking of temperature…how do we measure the temperature in a molecular cloud?

HIA Summer SchoolMolecular Line ObservationsPage 32 Temperatures from Observations  One way is to take a transition from a linear molecule that is: Optically thick Low energy In LTE Apply radiative transfer and use the R-J limit  Since:  And if  >> 1:  So:

HIA Summer SchoolMolecular Line ObservationsPage 33 Temperatures from Observations (Better Way)  Remember, for a symmetric top molecule 2 principal moments of inertia are equal Rotational energy levels described by 2 QN  J - the total angular momentum  K - the projection of J along axis of symmetry We get K-ladders  Since there is no dipole moment perpendicular to symmetry axis There are no radiative transitions across K ladders K ladders connected ONLY through collisions Popn of one K ladder wrt another should reflect a thermal distribution at the kinetic temperature

HIA Summer SchoolMolecular Line ObservationsPage 34 Methyl Acetylene Can only decay radiatively within a given “K” ladder  populations across “K” ladder will reflect gas temperature

HIA Summer SchoolMolecular Line ObservationsPage 35 Temperature of Molecular Hydrogen Gas K = 0K = 1K = 2

HIA Summer SchoolMolecular Line ObservationsPage 36 Temperatures from Observations  Again assuming optically thin emission Integrated intensity is proportional to column density in upper state  In LTE  In general, T ex will be different for different pairs of levels But if populations of ALL levels are in LTE we get our old friend:

HIA Summer SchoolMolecular Line ObservationsPage 37 Temperatures from Observations mxy

HIA Summer SchoolMolecular Line ObservationsPage 38 Temperature of Molecular Gas Observations! See the file: example_temp.doc for a worked example

HIA Summer SchoolMolecular Line ObservationsPage 39 Temperature Distributions Observations! Temperature Map of the Orion Molecular Ridge