Stellar Volume 푉= (Area) (length) Collision = cross section 휎 ⋅ ℓ = 휎 퓋푡 Relative speed 퓋 Number of 푁 Number 푛 = 푁Τ푉 # of collisions = # of (other) particles in the volume = 푁 = 푛 휎 퓋 t # of collisions per unit = 푁 Τ푡 = 푛 휎 퓋 Time between 2 consecutive collisions (푁=1) (mean-free time), 푡col = 1 Τ (푛 휎 퓋) Distance between 2 consecutive collisions (푁=1) (mean-free path), ℓcol = 퓋푡col = 1 Τ (푛 휎 ) In general “encounters” between particles, or between a

and a . The “cross section” is the key. 2 Opacity −1 ′ 2 −1 −3 휅 cm = 휅 휌 cm g ⋅ g cm = ෍ 푛푖 휎푖 푖 Recall that ׬ 휅 푑푠 is the optical depth

If 휅휈 is frequency independent  휅 gray atmosphere

Usually an average opacity is used ׬ 휅휈 퐵휈 푑휈 Planck opacity  average of frequency 휅P = ׬ 퐵 푑휈 dependent opacity weighted by Planck function 휈

Svein Rosseland −1 Rosseland opacity  weighted ׬ 휅휈 푢휈 푑휈 by 푇 derivative, averaging 1Τ휅휈 1Τ휅R = ׬ 푢휈 푑휈 푢 휈, 푇 = 휕퐵휈 푇 Τ휕푇 3 푎+푏+푐 Arithmetic average, , e.g., 1 + 4 + 4 Τ3 = 3 3 Geometric average, 3 푎 ∙ 푏 ∙ 푐, e.g., 3 1 × 4 × 4 ≈ 2.52

1Τ푎 + 1Τ푏 + 1Τ푐 −1 Harmonic average, one of the Pythagorean means, , 3 1Τ1 + 1Τ4 + 1Τ4 −1 e.g., = 2 3

4 Bound-bound absorption Excitation of an electron of an to a higher state by the absorption of a photon. The excited atom then will be de-excited spontaneously, emitting a photon, or by collision with another particle. Bound-free absorption Photoionization of an electron from an atom () by the absorption of a photon. The inverse process is radiative recombination.

5 • Free-free absorption Transition of a free electron to a higher energy state, via interaction of a nucleus or ion, by the absorption of a photon. The inverse process is bremsstrahlung. • Electron Scattering of a photon by a free electron, also known as Thomson (common in stellar interior) or Compton (if relativistic) scattering. • H− absorption Important when < 104 K, i.e., dominant in the outer layer of low-mass (such as the )

Hydrogen anion (or protide)

Opposite is H cation (or or just ) 6 • Bound-bound, bound-free, and free-free opacities are collectively called Kramers opacity, named after the Dutch physicist H. A. Kramers (1894-1952). • All have similar dependence 휅 ∝ 휌 푇−3.5. • Kramers opacity is the main source of opacity in of 4 6 10 ~10 K, i.e., in the interior of stars up to ~ 1 M⊙. • In a much more massive, the electron scattering process dominates the opacity, and the Kramers opacity is important only in the surface layer. Kramers opacity 휅 ≈ 4 × 1025 1 + 푋 푍 + 0.001 휌 푇−3.5 [cm2g−1] 퐾푟 7 X=0.70 Z=0.02

Rosseland mean opacity Data from Iglesias & Rogers (1996) 8 For Thomson scattering, 2 8휋 푟푒 2 −1 휅휈 = = 0.20 1 + 푋 [cm g ] 3 휇푒푚e is frequency independent, so is the Rosseland mean. 2 −1 휅푒푠 = 0.20 1 + 푋 [cm g ]

Here 푟푒 is the electron classical (charge; Lorentz) radius, 푋 is the H mass fraction, and 휇푒 = 2Τ(1 + 푋) 푒2 푟 = = 2.82 × 10−15 [m]; experimentally 푟 < 10−18 [m] 푒 푚푐2 푒 −25 2 Classical electron cross section, 휎T = 6.65 × 10 [cm ]

= 0.665 barns 9 10 11 Bowers & Deeming12 16 17 Absorption and Emission by

Hydrogen as an example …

18 QM versus classical  Action [energy time] or (momentum distance] is quantized in unit of ℏ

 Heisenberg’s uncertainty principle, 휎푥 휎푝 ≥ ℏΤ2  Pauli exclusion principle

Harwit

19 Lowest state of H, 푝2 푟2 ≈ ∆푝2 ∆푟2 ≈ ℏ2

Virial theorem, 2ℰ퐾 + ℰ푝 = 0 Lowest (ground state) energy 1 1 푍푒2 1 푝2 1 ℏ2 ℰ = − 퐸 = − = − ≈ − 1 2 푝 2 푟 2 휇 2휇 푟2 휇: reduced mass 푍푒2 ℏ2 ℏ2 = ⇒ 푟 = (퐁퐨퐡퐫’퐬 퐫퐚퐝퐢퐮퐬) 푟 휇푟2 휇푍푒2 1 푍푒2휇푍푒2 1 푍2휇푒4 ℰ = − = − 1 2 ℏ2 2 ℏ2 −9 For H, 푍 = 1, ℰ1 = −13.6 eV, 푟 ≈ 5.3 × 10 [cm] = 0.53 Å 20 ℎ ℎ de Broglie , 휆 = = 푝 푚푣 푍푒2 Virial theorem (classical uniform circular motion), 푚푣2 = 푟 Standing waves, 2휋 푟 = 푛휆

ℏ2 For the ground state, 푛 = 1, 푟 = 푚푍푒2

21 Virial theorem relation between (the time average of) the total kinetic energy and the total potential energy of a system in equilibrium Equation of motion (in the Lagragian form)

푑2푟Ԧ 휚 = 푓Ԧ − 훻푃 …… (1) 푑푡2 푑2푟Ԧ In , = 0, so 푓Ԧ = 훻P, and assuming 푑푡2 spherical symmetry and the force being self-gravitation 푑푃 퐺 푚 푟 휚 푟 = − (Hydrostatic equilibrium) 푑푟 푟2 푟 2 (and 푚 푟 = ׬0 4휋푟 휚 푑푟 (mass continuity/distribution

22 푑2푟Ԧ 휚 = 푓Ԧ − 훻푃 푑푡2 Take the vector dot of 푟Ԧ of (1), divide by 휚, define 푭 = 풇Τ휚 (force per unit mass, and then integrate, using the boldface for vectors

풅ퟐ풓 푑푚 (׬ 푑푚 풓 ⋅ = ׬ 풓 ⋅ 푭 푑푚 − ׬ 풓 ⋅ 훻푃 …… (2 풅풕ퟐ 휚

푑 푑풓 푑2풓 푑풓 2 1 푑2 Given 풓 ⋅ = 풓 ⋅ + = 풓2 푑푡 푑푡 푑푡2 푑푡 2 푑푡2

풅ퟐ풓 1 푑2 푑풓 2 So, ׬ 푑푚 풓 ⋅ = ׬ 풓2 푑푚 − ׬ 푑푚 풅풕ퟐ 2 푑푡2 푑푡 1 푑2퐼 퐼: moment of inertia = − 2ℰ 2 푑푡2 kin ℰkin: kinetic energy

23 Because 푑푚 = 휚 푑푉, the last term in (2), 푑푚 ׬ 풓 ⋅ 훻푃 = ׬ 풓 ⋅ 훻푃 푑푉 = ׬ 훻 ∙ 풓 ∙ 푷 푑푉 − 3 ׬ 푃 푑푉 휚 Note 풓 ∙ 푷 ∙ 푑푺 − 3 ׬ 푃 푑푉 = න 훻 풓푃 = 훻 ⋅ 풓 푃 + 풓 ⋅ 훻푃

훻 ⋅ 풓 = 3 Assuming spherical symmetry, Gauss’s theorem  volume 3 integral of the divergence to 4휋푅 푃푠 − 3 ׬ 푃 푑푉 surface integral =

24 Putting together, we have 1 푑2퐼 = 2ℰ + 3 න 푃 푑푉 + න 풓 ⋅ 푭 푑푚 − ර 푃풓 ⋅ 푑푺 2 푑푡2 kin where 풓 ⋅ 푭 (work) is virial; or 1 푑2퐼 = 2 ℰ + 3 න 푃 푑푉 + ℰ − 4휋푅3푃 2 푑푡2 kinetic potential external

For stars, under hydrostatic equilibrium and if 푃ext = 0, 2 ℰk + ℰp = 0 25 1 푑2퐼 LHS = 0  stable = 2 ℰ + ℰ 2 푑푡2 k p LHS < 0  collapsing LHS > 0  expanding

퓔퐤: a variety of kinetic 퓔퐩: a variety of potential energies  Kinetic energy of  Gravitation  Bulk motion of  Magnetic  Rotation  Electrical field  …  …

Note ℰtotal = ℰ푘 + ℰ푝, governs if the system is bound (ℰtotal < 0)

For stars, mostly ℰ푝 = Ω (gravitational energy; negative) 26 For higher energy states, 푝푛 푟푛 = 푛ℏ

2 2 2 2 4 푝푛 푛 ℏ 푍 휇 푒 ℰ푛 = − ≈ − 2 = − 2 2 2휇 2휇푟푛 2푛 ℏ For the 푛-th radial state, the phase volume is 2 2 4휋푝푛 Δ푝푛 4휋푟푛 Δ푟푛 , # of possible states with principle quantum number 푛

Total phase space volume 16휋2푛2ℏ3 = = ∝ 푛2 volume of unit cell ℏ3 The electron spin is either parallel or anti-parallel to that of the nucleus, so the n-th state has 2푛2 different substates, all having the same energy. 27 28 푍2휇 푒4 Note that ℰ ∝ 휇 ℰ = − 푛 푛 2푛2ℏ2 푚푒푚푝 푚푒 For normal H, 휇퐻 = = ≈ 푚푒 1 − 푚푒Τ푚푝 푚푒+푚푝 1+푚푒Τ푚푝

푚푒푚퐷 2푚푒푚푝 For deuteron, 휇퐷 = = ≈ 푚푒 1 − 푚푒Τ2푚푝 > 휇퐻 푚푒+푚퐷 푚푒+2푚푝

So the D lines are 1.5 Å shorter in

2 Note also that ℰ푛 ∝ 푍 , so for He II (푍 = 2, with 1 e−) , 푍2 is 4 larger, and with a different 휇. https://archive.stsci.edu/fuse/scisumm/sci_d2h.html29 푛

Grotrian diagram

Lang Gray & Corbally30 31 • For the ground state, the orbital angular momentum is ℓ = 0. The total spin angular momentum is 퐹 = 0 (spin opposite) or 퐹 = 1 (spin parallel) Hyperfine splitting

• For 푛 = 2, ℓ =1, and with spin, a total angular momentum of ℓ ℓ + 1 ℏ2 =2ℏ2 3 substates, ℏ, 0, −ℏ, 푚 = 1, 0, −1 (magnetic quantum number) Fine structure, ∆ℰ very small, ~10−5 Ev But if there is an external B field  Zeeman splitting 32 Free-free 퐇+  Free-free or free-bound to any level  Cascading down  emission of of different energies

− 퐇 Stars: ample supplies of free 푒− from Na, 퐻 + 푒− ⟶ 퐻− + ℎ휈 Ca, Mg, … with low-ionization potentials  He atom similar, with the second 푒− weakly bound, shielded by the first 푒− −  ℰbinding H = 0.75 eV, only 1 bound state; transitions  continuum  Absorption by H− immediately followed by reemission − The we see mostly is due to continuum transitions by 퐻 33 퐇ퟐ • Main constituent of cold clouds, not important in stars, except in the coolest substellar objects (brown dwarfs or planetary- mass objects) • Lacking a permanent electric dipole moment, so very difficult to detect. A rotationally excited would radiate through a relatively slow electric quadrupole transition. • Only detected in a hot medium, where stellar or stellar excites vibrational and electronic states which then decay relatively quickly.

Zero electric dipole moment 34 Stahler & Palla 35 CO molecules • Simple and abundant, in gaseous or solid form

• Strong ℰbinding =11.1 eV  self-shielding against UV field • with a permanent electric dipole moment; radiating strongly at radio frequencies. • 12C16O easiest to detect; isotopes 13C16O, 12C18O, 12C17O, 13C18O also useful • Low critical density for excitation  CO used to study large- scale distribution of molecules, as a tracer of H2 in dense clouds • 12C16O almost always optically thick; same line from other rare isotopes usually not  estimate of column density (total mass) 6 of molecular gas 푁퐻= 10 푁13 퐶푂 36 2.6 mm = 115 GHz

Only 5 K above the ground level … can be excited by collisions with ambient molecules or CMB photons

Stahler & Palla 37 CO bandheads in the Becklin-Neugebauer (BN) object, an IR-emitting, embedded, massive (~7 M⊙)

Gaballe & Persson (1987) Stahler & Palla38 Stahler & Palla 39 Stellar matter largely gas or . Molecules form primarily below 6000 K, only OB stars do not contain molecules. Absorption band spectra, e.g., due to MgH, CaH, FeH, CrH, NaH, OH, SiH, VO, and TiO, etc. Late-type stars exhibit TiO

NH3 and collision-induced absorption by H2 in brown dwarfs

40 Gray & Corbally 41 Effect of the new H2–H2 and H2–He CIA opacity on synthetic spectra of brown dwarfs. The spectra shown are cloudless models with Teff = 1500, 1000, and 500 K, with log g = 5 (cgs) and solar . The spectra computed with the new CIA opacities are shown in blue. The red lines show spectra computed with the older CIA opacity and the same (T, P) structures. The are calculated for d = 10 pc and are displayed at a resolving power of R = 500. Didier Saumon et al. 201242 Plane-Parallel Atmosphere

43 Solar Atmosphere • Lowest layer of the atmosphere; the solar “disk” we see; thickness ~300 km (cf. 2 R⊙ ∼ 1.4 million km) • Darker and density thinner than the photosphere; pinkish (hence the name); thickness ~500 km • Corona Outermost layer; hot (1 to 2 million K); 1 millionth of photosphere; visible during a total or with a coronograph; extending millions of km 44 Solar photosphere ≈ 300 km thickness ≤ 0.1% R⊙ ⟶ plane parallel approximation OK Recall the transfer equation, 푑퐼휈 = 퐼휈 − 푆휈 푑휏휈 and the vertical optical depth, 0 .휏휈 푧 = ׬푧 휅휈 푑푧 For a ray at an angle, 푑푧 = 푑푠 cos 휃, so in general,

푑퐼휈 휏휈,휃 cos 휃 = 퐼휈 휏휈, 휃 − 푆휈 휏휈 푑휏휈 45 The solution 휏 −휏휈 휈 − 휏휈−푡휈 퐼휈 휏휈 = 퐼휈 0 푒 + ׬0 푆휈 푡휈 푒 푑 푡휈

In the atmosphere  no incident radiation, with infinite optical depth ∞ −푡휈 sec 휃 퐼휈 0, 휃 = න 푆휈 푡휈 푒 푑 푡휈 ⋅ sec 휃 0

46 ∞ −푡휈 sec 휃 퐼휈 0, 휃 = න 푆휈 푡휈 푒 푑 푡휈 ⋅ sec 휃 This gives the intensity from the “disk” 0 of the star At the edge, 휃 → 휋Τ2, sec 휃 → ∞,

퐼휈 0, 휋Τ2 → 0 At the center, 휃 = 0, sec 휃 = 1, ∞ −푡휈 퐼휈 0, 0 = න 푆휈 푡휈 푒 푑 푡휈 0  The limb of a stellar disk is dimmer than to the center (on the average hotter seen to the same optical depth) .

For Sun, 퐼limb ≈ 80% 퐼disk center @550 nm; dimmer in the blue 47 [Rsun]

[nm]

Unsold, p.169 48 ∞ −푡휈 sec 휃 퐼휈 0, 휃 = න 푆휈 푡휈 푒 푑 푡휈 ⋅ sec 휃 0 Approximate the source function by Taylor expansion, 푆휈 ≈ 푎휈 + 푏휈 휏휈 ⟶ 퐼휈 0, 휃 = 푎휈 + 푏휈 cos 휃

So 퐼휈 = 푆휈 휏휈 = cos 휃 . (Eddington-Barbier relation) The specific intensity on the surface at position 휃 is the source function at the optical depth 휃. The effect of limb darkening observable in details for the Sun  measuring 퐼휈 across the solar disk  mapping the depth dependence of 푆휈  to probe the structure in the atmosphere Seen also in some eclipsing binaries, in large stars by interferometry, or in exoplanet transits. de Boer & Seggewiss49 Recall that 퐹휈 = ׬ 퐼휈 cos 휃 푑휈 푑휔 1 2 ׬ 푎 + 푏 cos 휃 cos 휃d cos 휃 = 푎 + 푏 = 0 휈 휈 휈 3 휈

F휈 = 푆휈 휏휈 = 2Τ3

4 Assuming LTE, so 푆휈 = 퐵휈, and a gray atmosphere (퐹 0 = 휎푇eff), 4 then 퐹휈 0 = 휋퐵휈 푇 휏 = 2Τ3 = 휎푇eff

This means 푇eff = 푇 휏 = 2Τ3 So the of the stellar surface is the temperature at optical depth 2/3.

50 by the ’s atmosphere with a zenith distance 푧 that needs to be accounted for in photometric measurements, i.e., the “atmospheric correction”.

−휏휈 sec 푧 퐼휈Τ퐼휈,0 = 푒

퐼휈,0, intensity outside the atmosphere, is estimated by extrapolation of the “extinction curve”.

51 Gray Atmosphere Conservation of energy, if energy is transported by radiation only  Radiation integrated over all directions and all frequencies = total energy

If 휅휈 (hence 휏휈) is frequency independent  gray (unphysical, except for Thomson scattering, but useful to get some first insights)

Rosseland (1924): a flux-weighted mean opacity (Rosseland mean), so that 휏휈 = 휅ഥds

52 Solar Atmosphere  Photosphere --- the visible “disk”; thickness ~300 km Temperature drops to a minimum of ~4200 K  Chromosphere --- a region extending ~2500 km above the limb; emission spectrum  “Transition region”  Corona --- tenuous outer envelope seen at eclipses or with a coronograph  Wind --- supersonically expanding

54 Athay (1976)55 Solar Structure

Thermonuclear fusion ≲ 0.25 푅⨀ Radiative core up to ≈ 0.80 푅⨀ Convective envelope Outer radiative layer

56 Model stellar by R. L. Kurucz (1979) for Unsold solar abundance, and for different 푇eff and 푔 57 Kurucz (1970) SAO Special Report #309 58 59 Kurucz (1979) 60 Kurucz (1979)61 Kurucz (1979) 62 Kurucz (1979)63 The energy distribution of Unsold 65 Line Broadening

66 Natural Broadening

QM Heisenberg energy-time uncertainty principle ∆퐸 Δ푡 ≥ ℎ That is, the energy of a given state cannot be specified more accurately than this  ∆휈 ≈ 1/ Δ푡. Typically Δ푡 ≈ 10−8 s (recall Einstein’s 퐴 coefficients), so the natural width of a line ≈ 5 × 10−5 nm . Meta-stable states have even much narrow lines.

67 Thermal Doppler Broadening Particle motion along line of sight  Doppler shift 푚푣2Τ2 = 3 푘푇/2 At a given temperature, a line due to a heavier element is narrower. At 6000 K, H moves at 푣 ≈ 12 km s−1, leading to a fractional Doppler broadening ∆휆Τ휆 ≈ 푣Τ푐 ≈ 4 × 10−5, so the H훼 line (656.3 nm) is broadened by 0.025 nm. The broadening is temperature and composition dependent.

68 Zeeman Broadening Energy levels spilt to 3 or more sublevels in a  Zeeman effect (Pieter Zeeman) Spectral lines closed spaced (∝ 퐁 strength), so difficult to resolve  line broadened Collisional Broadening Energy levels shifted by nearby particles, especially and electrons (“Stark Effect” due to E field); also called broadening. Density dependent

Additional broadening mechanisms: rotation, expanding, turbulence, …, etc. 69 Line Profile

 The details of a line profile: absorption coefficient as a function of frequency within the line  Superimposed on the Doppler profile (macroscopic motion of particles) are the radiative and collisional damping effects. An atom  a dipole; the electron oscillates when interacting with an incident EM wave In general, 푚푟ሷ = −푚푟휔2 − 푚훾푟ሶ − 푒퐸 푒푖휔푡 0 0 Novotny, p.199

Force on the electron Restoring force Damping force Force by EM wave 70 Lorentz (damping) profile 훾 휙 Δ휈 = 2휋Δ휈 2 + 훾Τ2 2 Classical treatment Atom absorbing a photon  excited  푒− oscillates as a dipole 2 Equation of motion: 푚푟ሷ = −4휋푟휈0 2 푒2 Such a dipole radiates with power ℙ = 푟ሷ 2 3 푐3 Energy is radiated away  damping force to slow down the 푒− 2 푒2 The force is ℱ = 푟ഺ 2, and for a small damping 3 푐3  a simple harmonic motion (around 휈0)… 71 The velocity and position vector of a projectile with a resistance force proportional to its instantaneous speed −훽v, e.g., in a free-fall body in the air

Spiegel

72 Bower & Deeming

73 The effective number of oscillators  oscillator strength, relates the spectral line to harmonic electron-oscillators, and is related to the Einstein 퐵 coefficient ℎ휈 휋푒2 න 휎푖푗 휈 푑휈 = 퐵푖푗 = 푓 line 4휋 푚푒푐

The oscillator strength 푓, the ratio [QM transition rate]/[Classical rate], is dimensionless, and is also related to 퐴 coefficient 8 휋2푒2휈2 푔푗 퐴푗푖 = 3 푔푖 푓 푚푒푐

74 For Balmer lines, 푓 H훼 = 0.641, 푓 H훽 = 0.119, 푓 H훾 = 0.044. Kramers computed the analytic approximation for H,

6 −𝑔푖 2 1 1 𝑔푏푏 푓푗푖 = 푓푖푗 = 2 2 3 3 3 𝑔푗 3 3 휋 𝑔푖 1Τ푖 −1Τ푗 푗 푖

Where 푔푏푏 is the Kramers-Gaunt factor, or Gaunt factor, for the bound-bound transition to correct for the QM effect, and is on the order of unity.

Bower & Deeming75 Doppler profile

1 2 휙 Δ휈 = exp − Δ휈ΤΔ휈D 휋 Δ휈D

This has a FWHM of 2Δ휈D ln 2

Δ휈D or Δ휆D is defined by the most probable speed.

Voigt profile Doppler core + Damping wings (convolution) +∞ 휙 Δ휈 = න ℒ ∆휈 − Δ휈′ 𝒟 Δ휈′ 푑Δ휈′ −∞

76 2 2 휆0 Δ휆 휅휆 = 푟푒 푓 푛푖 휋 exp − Δ휆0 Δ휆퐷 Decays exponentially

휆4 훾 휅 = 푟 푓 푛 휋 0 휆 푒 푖 4휋푐 Δ휆 2

푟푒 is classical electron radius; Unsold 푛푖 is density at lower level 77 Photons of different wavelengths carry different Equivalent Width energies. Which line is “stronger” (how much energy is missing in a spectral line)?  Compare with local “continuum”, i.e., where there is no absorption

퐼푐 − 퐼휆 휙휈 = 퐼푐

∞ 퐼푐 − 퐼휆 −휏휆 푊휆 = න 푑휆 = න 1 − 푒 푑휆 푊 푊 푊 퐼 휆 = 휈 = v −∞ 푐 휆 휈 푐 is the equivalent width (W), which measures the absorption (strength) of a spectral line, where 퐼휆 is the line profile, and 퐼푐 is the continuum (at the same 휆). 푊 in unit of Å or mÅ ; traditionally negative for emission lines. 78 휋푒2 휎 = 푓휙 휎휈푑휈 = 휎휆푑휆 휈 푚푐 휈

휏휈 = 휅휈ds = 푛 휎휈푑푠 = 푁 휎휈, where 푁 is the column density 휋푒2 휏 = 푁 푓 휆2 휙 휆 푚푐2 0 휈

(1) For a weak line (휏휆 ≪ 1) 휋푒2 푊 = න 휏 푑휆 = 푁 푓휆2 ∝ 푁푓 휆 휆 푚푐2 0 Or 푊 휋푒2 휆 = 푁 푓휆 = 8.85 × 10−13 푁 [cm−2] 푓 휆 [cm] 푚푐2 0 푖 12 Spitzer79 (2) For a strong line (휏휆 ≫ 1), damping wings important 푊휆 ∝ 푁푓

(3) For an intermediate case

푊휆 ∝ ln 푁푓

Line strength (equivalent width)  abundance

Curve of growth

80 Square-root part

flat part

linear part

81