Heating and Cooling Processes - KROME...

Heating and Cooling Processes

Inga Kamp, KROME summer school November 26 – November 28, 2018, Concepcion, Chile

Inga Kamp

Heating and Cooling Processes 1.  Introduction 2.  Dust heating/cooling 3.  Line heating/cooling

I.  LTE II.  non-LTE

4.  Other Processes I.  Photoelectric and PAH heating II.  CR and X-ray heating III.  Ionisation heating IV.  H2: a special case V.  Dust thermal accommodation VI.  Bremsstrahlung VII.  Viscous heating VIII. Chemical heating


5.  Exercise 6.  Examples

I.  Planet forming disk

Literature


B.T. Draine, Physics of the Interstellar and Intergalactic Medium, Princeton University Press

Tielens & Hollenbach (1985), Photodissociation Regions: Basic Model, ApJ 291, 722

Hollenbach & Tielens (1997), Dense Photodissociation Regions, ARAA 35, 179

Summer School: Protoplanetary Disks: Theory and Modelling Meet Observations, EPJ Web of Conferences Vol 102, 2015 (Open Access)

From observations to interpretation 1. Introduction


a quantitative interpretation of line emission often requires dynamical/radiation/thermal/chemical models

of the studied astrophysical environment

Heating and cooling of a gas


dedt= −P dV

dt+ ρΓi −

i∑ ρΛk

k∑

e – internal energy of the gas t – time V – specific volume P – gas pressure ρ – gas density Γι – heating rate of process i Λκ – cooling rate of process k ρΓi =

i∑ ρΛk

k∑

change in internal energy If the cooling timescale (τcool) is much faster than the dynamical timescale (τdyn): If collisional coupling between gas and dust is inefficient:

⇒ Tgas

Tgas ≠Tdust

1. Introduction

How it all ties together 1. Introduction

Physical structure of the object (+ element

abundances, dust properties)

Dust opacity, dust temperature

Gas (+dust surface) chemistry

Gas energy balance (gas temperature)

Radiation field (e.g. photons/s/cm2/Hz)

Ray tracing (observables e.g. spectrum, line profile)

possibly update physical model (e.g. hydrostatic equilibrium, pressure equilibrium, thermal stability of dust/clouds)


Physical structure of the object (+ element

abundances, dust properties)

Dust opacity, dust temperature

Gas (+dust surface) chemistry

Gas energy balance (gas temperature)

Radiation field (e.g. photons/s/cm2/Hz)

Ray tracing (observables e.g. spectrum, line profile)

possibly update physical model (e.g. hydrostatic equilibrium, pressure equilibrium, thermal stability of dust/clouds)


1. Introduction

How it all ties together







Dust opacities Input data for models: •  grain size distribution •  grain composition (e.g. volume

fraction silicate, amorphous carbon, vacuum, ice, …)

κνabs =

Qabs (ν )πa2

43πa3ρgrain

absorption efficiency of a grain

mass of a grain

for single grain size a => can be generalized to grain size distribution using moments of the distribution <a2>, <a3>

[Woitke et al. 2016]


2. Dust heating and cooling

Dust radiative equilibrium Energy can be added or removed from the grain by absorption or emission of photons, or by inelastic collisions with atoms or molecules from the gas (grain heating).

stellar photons

isotropic dust emission

Tdust Tdust

uνhν∫ ⋅c ⋅hν ⋅Qabs (ν )πa

2dν = 4πa2Bν (Tdust )Qabs (ν )πa2 dν∫

photon number density

absorption efficiency of a grain

speed and energy of photon

Blackbody emission from a grain with size a



solving for Tdust requires continuum radiative transfer

often neglected, because small – but not in accretion disks

Dust temperature

A day in the life of four carbonaceous grains, heated by the local inter-stellar radiation field. τabs is the mean time between photon absorptions.

1 hour = 3600 s

Definition of temperature for very small grains: instantaneous vibrational temperature = temperature T(E) at which the expectation value of the energy would be equal to the actual grain energy

[Draine 2003]



Continuum radiative transfer Radiative transfer equation with dIνds

= −ανdust Sν − Iν( )

Iν – intensity [erg cm-2 s-1 Hz-1] ν – frequency s – physical path length Sν – source function aνdust – dust extinction coefficient [cm-1] aνdust,abs – dust absorption coefficient aνdust,sca – dust scattering coefficient jνdust – continuum emission coefficient

Sν =jνdust

ανdustν

jνdust =αν

dust,absB Tdust( )+ανdust,scaJν

ανdust =αν

dust,abs +ανdust,sca

I0 - dI I0

jν , aν

ds



Gas Temperature the gas has many possibilities to heat and cool due to the presence of a large variety of atoms/molecules (forest of line transitions, ionization, dissociation processes etc.)

and many other molecules



and many other atomic lines







Local Thermodynamic Equilibrium if collisions dominate, level populations for an atom/molecule follow from the Boltzmann equation rotational level populations are often in LTE since their energies Erot are often «1 eV ⇒ collisions can easily thermalize them The critical density of a line is a measure for the density at which LTE roughly holds


3. Line heating and cooling

Local Thermodynamic Equilibrium the line emission (cooling) can be derived from the level populations


Λk (ν ij ) = niLTEAijhν ijβesc (τ ij )

νij – frequency of the line τij – optical depth of the line Aij – spontaneous emission probability βesc – escape probability ni – population number of the upper level

1D slab geometry complete re-distribution Gaussian profile

θ

cosθ=µ

with the escape probability

[Avrett & Hummer 1965]


different in cases of large velocity gradients

Velocity gradients

Why would velocity gradients impact line RT?

radially expanding velocity field v = dv

dr!

"#

$

%&r

r

v



Velocity gradients

Why would velocity gradients impact line RT?

Large Velocity Gradient (LVG) approximation if Δν from one cell to the next is larger than the line width, the photon is “shifted out of the line” and can escape τLVG is the total optical depth along the path for any frequency

[Sobolev 1957] radially expanding velocity field v = dv

dr!

"#

$

%&r

r

v Δν ij =ν ijv(rn )− v(rn−1)( )

c

βesc =1− e−τ LVG

τ LVG



Velocity gradients

[Beckwith & Sargent 1993, Pontoppidan et al. 2009]

disks have a Keplerian velocity field

LVG approximation does not work since a line can interact with itself at various locations along a ray (e.g. top and bottom of the disk, near- and far-side)

45o inclination



Statistical Equilibrium if LTE does not hold, we need to solve the detailed equations of statistical equilibrium (SE) for each energy level i



addition from higher levels due to sponatenous + stimulated emission

addition from lower levels due to absorption

collisions ending in level i

loss into lower levels due to sponatenous + stimulated emission

loss into higher levels due to absorption

collisions leaving level i

ni - population of level i νij - frequency of the line P(νij) – radiation field at frequency νij

dnidt

= njj>i∑ Aji +BjiP(ν ji )( )+ nj

j<i∑ BjiP(ν ji )+ njCji

j≠i∑

−ni Aij +BijP(ν ij )( )j<i∑ − ni BijP(ν ij )

j>i∑ − ni Cij

j≠i∑

Aij - Einstein A coefficient (spontaneous emission) Bji – Einstein B coefficient (absorption) Bij – Einstein B coefficient (stimulated emission)

Statistical Equilibrium if LTE does not hold, we need to solve the detailed equations of statistical equilibrium (SE) for each energy level i with the stimulated emission coefficient and the relation between stimulated emission and absorption coefficient



Statistical Equilibrium

Solve numerically using e.g. Newton-Raphson => level population numbers for rotational, vibrational levels of all electronic states => for some purposes (cold low density environments), only ground electronic, vibrational state populated, hence only rotational level populations needed



The two-level atom E1, n1, g1

E0, n0 g0

E10 n1n0=

A10c2

2hν 3g1g0P(ν10 )+C01

A10 + A10c2

2hν 3P(ν10 )+C10

n1n0=

C01A10 +C10

without background radiation: What happens if collisions are negligible?

n1n0=g1g0

c2

2hν 3P(ν10 )

1+ c2

2hν 3P(ν10 )

if P(ν10) is a blackbody radiation field with Tgas

n1n0=g1g0

c2

2hν 32hν 3

c21

ehν10 /kTgas −1"

#$

%

&'

1+ c2

2hν 32hν 3

c21

ehν10 /kTgas −1"

#$

%

&'

=g1g0e−hν10 /kTgas

radiation can also produce LTE !



The two-level atom E1, n1, g1

E0, n0 g0

E10= 92 K n1n0=

A10c2

2hν 3g1g0P(ν10 )+C01

A10 + A10c2

2hν 3P(ν10 )+C10

n1n0=

C01A10 +C10

without background radiation: What happens if collisions are negligible?

n1n0=g1g0

c2

2hν 3P(ν10 )

1+ c2

2hν 3P(ν10 )

if P(ν10) is a blackbody radiation field with Tgas

n1n0=g1g0

c2

2hν 32hν 3

c21

ehν10 /kTgas −1"

#$

%

&'

1+ c2

2hν 32hν 3

c21

ehν10 /kTgas −1"

#$

%

&'

=g1g0e−hν10 /kTgas

radiation can also produce LTE !



2P1/2

2P3/2

[CII] 158 µm


IR pumping UV pumping

LTE often a good assumption

CO molecule



Heating and cooling of a gas


[Peter Woitke]

For the net cooling rate, one can either calculate the net creation rate of photon energy (radiative approach), or one can calculate the total destruction rate of thermal energy (collisional approach)

with



[diffuse cloud, no molecules: Draine 2011]



At different temperatures, different main coolants will dominate the energy balance: at low temperatures (few 100 K) the fine structure lines, at high temperatures (few 1000 K) atomic lines


[Hollenbach & Tielens 1997 ]


dense PDR with n = 2.3×105 cm-3, G0 = 105

C+/C/CO C+/C/CO H/H2








W – work function of bulk dust material Y – electron yield εGRAIN – efficiency of heating ΦC – Coulomb potential of the dust grain σabs – dust absorption cross section

4. Other processes

Photoelectric heating

10 ≤ hν ≤13.6eVPhotons with cannot ionize hydrogen, but can ionize dust grains => ejection of a photoelectron.

ΓPE = εGRAINndustσ dustabs χ

photoelectric heating rate

with the integrated FUV (912-2050 Å) radiation field

χ =λuλ

912

2050

∫ dλ

λuλDraine

912

2050

∫ dλ

photon energy density [erg/cm3]

[Hollenbach & Tielens 1997]


W – work function of bulk dust material Y – electron yield εGRAIN – efficiency of heating ΦC – Coulomb potential of the dust grain σabs – dust absorption cross section

Photoelectric heating

10 ≤ hν ≤13.6eVPhotons with cannot ionize hydrogen, but can ionize dust grains => ejection of a photoelectron.

ΓPE = εGRAINndustσ dustabs χ

photoelectric heating rate

with the integrated FUV (912-2050 Å) radiation field

χ =λuλ

912

2050

∫ dλ

λuλDraine

912

2050

∫ dλ

photon energy density [erg/cm3]



need to know the charge on dust grains which depends on grain sizes, radiation filed, density (recombination)

4. Other processes

4. Other processes

Other important heating processes


•  PAH heating: depends on the size of the PAH, the radiation field and density

• Cosmic Ray and X-ray heating: high energy radiation produces super-thermal electrons through ionisation (e.g. K-shell) that heat the gas through Coulomb interactions with thermal electrons

•  Ionisation heating: FUV radiation ionises metals and the electrons heat the gas (compared to CR and X-rays, FUV radiative transfer is more intertwined with dust and H2 – shielding)

all depend on the radiation field (solving full RT) and CRs (attenuation into the medium)

PAH heating PAHs get charged according to the local radiation field, densities, PAH abundance

IPk – Ionisation Potential of PAHk

nkPAH – PAH density

ne – electron density σk

PAH – PAH absorption cross section kk

PAH – PAH recombination coefficient Tgas – gas temperature Jν – radiation field Yk

ν – photoelectron yield sν – self-shielding factor

PAH ionisation is heating

PAH recombination with free e- is cooling



4. Other processes

Cosmic Ray and X-ray heating


4. Other processes

High energy radiation produces super-thermal electrons through ionisation (e.g. K-shell) that heat the gas through Coulomb interactions with thermal electrons

ΓCR =ζCR QHCRnH +QH2

CRnH2 +...( )ΓCR ≈ ζCR 5.5 ⋅10

−12nH + 2.5 ⋅10−11nH2( )

ΓXray =ζXray QHCounH +QH2

CounH2 +...( )

for H/H2 mixture

ζXray – Xray primary ionisation rate QCou – energy thermalized via Coulomb interactions ζCR – primary CR ionisation rate QCR – energy thermalized via Coulomb interactions

[Dalgarno et al. 1999]

Ionisation heating


4. Other processes

FUV radiation ionises metals and the electrons heat the gas compared to higher energy ionisation, the FUV radiative transfer is more complicated due to dust (τUV), H2 (sC,H2) and C (sC,C) self-shielding, leading to an ionisation rate

ΓC =1.602 ⋅10−12RC

phnC

RCph = sC,CsC,H2χ0αCe

−τUV αC – carbon ionisation rate χ0 – strength of FUV radiation field

H2: a special case


4. Other processes

H2 is the most abundant molecule and its excitation and chemistry couple strongly to the radiation/thermal balance:

• H2 can self-shield against photodissociation

• H2 formation on dust leads to “hot” excited H2 – H2exc

• H2 absorbs FUV radiation (pumping) – H2exc

• H2exc de-excites through radiation or collisions or

reacts chemically

H2exc

H2 heating


4. Other processes

Γdiss = 6.4 ⋅10−13Rph

H2nH2

Dissociation heating: kinetic energy of the H-atoms is ~0.4 eV

RH2ph – photodissociation rate of H2 including

dust and self-shielding RH2exc

coll – collisional de-excitation rate ΔE – pseudo vibration level ~2.6 eV (v=6)

Collisional de-excitation: kinetic energy of H2 dissipated into the gas

Γcoll = ΔE ⋅RcollH2exc→H2 nH2exc − nH2e

−ΔE /kT( )( )

~10%

[Tielens & Hollenbach 1985]

~90%

cooling correction (collisional excitation)

[Stephens & Dalgarno 1973]

H2 heating


4. Other processes

H2exc

Formation heating: the surface reaction is exothermic and the assumption of equipartition of energy leads to Ekin~1/3 Ebind

RH2 – formation rate of H2 on dust grains Ebind – H2 binding energy 4.48 eV nH – number density of atomic hydrogen

Γform = 2.39 ⋅10−12RH2nH

[Black & Dalgarno 1976]

Dust thermal accommodation


4. Other processes

inelastic collisions between gas and dust thermalize the two (can heat or cool the gas) the influence of this process on the dust energy balance is usually neglected (however, see viscous heating later ...)

Γacc −Λacc = 4 ⋅10

−12π a2 ndustn H αacc (Tgas ) Tgas (Tdust −Tgas )

αacc(Tgas) – thermal accommodatiuon coefficient, ~0.1…0.5 Tgas – gas temperature Tdust – dust temperature n<H> – total hydrogen number density (nH+2nH2) <a2> – second moment of the dust size distribution n(a)~a-3.5

dust surface area

Processes relevant in special cases

4. Other processes

• Bremsstrahlung: in a plasma electrons and ions scatter off one another producing a continuum from radio wavelengths up to ≈ kT

Γchem = R(r)γ rchem

r∑ ΔHr


[Woitke et al. 2011]

Γvis dz∫ =3GM*

M8πr3

1− R*r

$

%&

'

()

•  Viscous heating: accretion causes friction

• Chemical heating/cooling: exothermic chemical reactions convert chemical potential energies into heat and endothermic reactions consume internal kinetic energy (cooling)

[D’Alessio et al. 1998] affects also Tdust







Exercise The typical hydrogen number density in the diffuse ISM is nH=1 cm-3, the radi-ation field is G0=1, and the density of C+ is n(C+)=5 10-4 cm-3, n(e-)=10-2 cm-3. Assume that photoelectric heating is the main heating process and fine structure emission by the [CII] 158 µm is the dominant cooling process.

Derive an estimate for the gas temperature using the two-level approximation for [CII].

k01(e− ) ≈1.5 ⋅10−6 (T )−0.5cm3s−1

ΓPE / nH =1.4 ⋅10−26G0erg / s

k01

Photoelectric heating rate per hydrogen atom


5. Exercise







Disks are layered structures: ionized/atomic, ion-molecules, molecules, ices

10 AU rcond 100 AU

ices

hot flaring surface

rich molecular

chemistry

hot gas: CO rich, no H2

[Aikawa et al. 2002, Kamp & Dullemond 2004, PPV chapters: Dullemond et al 2007, Bergin et al. 2007]

UV scattering by dust absorption by dust & gas

[PD database: van Dishoeck et al. 2006; Ly α up to 70-90% of LFUV: Schindhelm et al. 2012]

Ly α

Planet Forming Disk 5. Examples


Disks are layered structures: ionized/atomic, ion-molecules, molecules, ices

Ly α

dust properties in disks differ vastly from ISM

values at AV=1 χ=103 G0

n=1012 cm-3 χ=102 G0

n=1010 cm-3 χ=10-1 G0

n=109 cm-3


[Aikawa et al. 2002, Kamp & Dullemond 2004, PPV chapters: Dullemond et al 2007, Bergin et al. 2007]

[PD database: van Dishoeck et al. 2006; Ly α up to 70-90% of LFUV: Schindhelm et al. 2012]


These are not

classic PDR

s, but live

in a differen

t parameter sp

ace!!!



shown is only the “dominant” process [DIANA standard T Tauri disk: Woitke et al. 2016, Kamp et al. 2017]



[DIANA standard T Tauri disk: Woitke et al. 2016, Kamp et al. 2017]

Gas and dust couple efficiently below AV~1

as soon as molecules form, gas cooling becomes very efficient

C+/C/CO

different from ISM

different from ISM σdust (disk) << σdust (ISM)

Take away •  There is an overwhelming number of physical processes contributing to

gas heating and cooling •  In many astrophysical regions (PDR, warm/cold neutral medium, hot

ionized gas), only a subset of them dominate the energy balance – except planet forming disks that span a very wide range of conditions •  solving the energy balance of the dust requires knowledge of dust

properties (composition, sizes), optical constants etc. •  solving the energy balance of the gas requires knowledge of atomic/

molecular abundances, energy levels, line transitions (λ, Aij), collisional cross sections for all relevant collision partners (often e-, H, H2) and knowledge of the dust (see above)


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