A540 Review - Chapters 1, 5-10 Basic physics Boltzman equation Saha equation Ideal gas law Thermal velocity distributions Definitions Specific/mean intensities Flux Source Function Optical depth Black bodies Planck’s Law Wien’s Law Rayleigh Jeans Approx. Gray atmosphere Eddington Approx. Convection Opacities Stellar models Flux calibration Bolometric Corrections

Basic Assumptions in Stellar Atmospheres Local Thermodynamic Equilibrium Ionization and excitation correctly described by the Saha and Boltzman equations, and photon distribution is black body Hydrostatic Equilibrium No dynamically significant mass loss The photosphere is not undergoing large scale accelerations comparable to surface gravity No pulsations or large scale flows Plane Parallel Atmosphere Only one spatial coordinate (depth) Departure from plane parallel much larger than photon mean free path Fine structure is negligible (but see the Sun!)

Basic Physics – the Boltzman Equation Nn = (gn/u(T))e-Xn/kT Where u(T) is the partition function, gn is the statistical weight, and Xn is the excitation potential. For back-of-the-envelope calculations, this equation is written as: Nn/N = (gn/u(T)) x 10 –QXn Note here also the definition of Q = 5040/T = (loge)/kT with k in units of electron volts per degree, since X is in electron volts. Partition functions can be found in an appendix in the text.

Basic Physics – The Saha Equation The Saha equation describes the ionization of atoms (see the text for the full equation). For hand calculation purposes, a shortened form of the equation can be written as follows N1/ N0 = (1/Pe) x 1.202 x 109 (u1/u0) x T5/2 x 10–QI Pe is the electron pressure and I is the ionization potential in ev. Again, u0 and u1 are the partition functions for the ground and first excited states. Note that the amount of ionization depends inversely on the electron pressure – the more loose electrons there are, the less ionization there will be.

Basic Physics – Ideal Gas Law PV=nRT or P=NkT where N=r/m P= pressure (dynes cm-2) V = volume (cm3) N = number of particles per unit volume r = density of gm cm-3 n = number of moles of gas R = Rydberg constant (8.314 x 107 erg/mole/K) T = temperature in Kelvin k = Boltzman’s constant (1.38 x 10–16 erg/K) m = mean molecular weight in AMU (1 AMU = 1.66 x 10-24 gm)

Basic Physics – Thermal Velocity Distributions RMS Velocity = (3kT/m)1/2 Velocities typically measured in a few km/sec Mean kinetic energy per particle = 3/2 kT

Specific Intensity/Mean Intensity Intensity is a measure of brightness – the amount of energy coming per second from a small area of surface towards a particular direction erg hz-1 s-1 cm-2 sterad-1 Jn is the mean intensity averaged over 4p steradians

Flux Flux is the rate at which energy at frequency n flows through (or from) a unit surface area either into a given hemisphere or in all directions. Units are ergs cm-2 s-1 Luminosity is the total energy radiated from the star, integrated over a full sphere. F=sTeff4 and L=4pR2sTeff4

Black Bodies Planck’s Law Wien’s Law – Il is maximum at l=2.9 x 107/Teff A Rayleigh-Jeans Approx. (at long wavelength) Il = 2kTc/ l 4 Wien Approximation – (at short wavelength) In = 2hc2l-5 e (-hc/lkT)

Using Planck’s Law Computational form: For cgs units with wavelength in Angstroms

The Solar Numbers F = L/4pR2 = 6.3 x 1010 ergs s-1 cm-2 I = F/p = 2 x 1010 ergs s-1 cm-2 steradian-1 J = ½I= 1 x 1010 ergs s-1 cm-2 steradian-1 (note – these are BOLOMETRIC – integrated over wavelength!)

Absorption Coefficient and Optical Depth Gas absorbs photons passing through it Photons are converted to thermal energy or Re-radiated isotropically Radiation lost is proportional to Absorption coefficient (per gram) Density Intensity Pathlength Optical depth is the integral of the absorption coefficient times the density along the path

Radiative Equilibrium To satisfy conservation of energy, the total flux must be constant at all depths of the photosphere Two other radiative equibrium equations are obtained by integrating the transfer equation over solid angle and over frequency

Convection If the temperature gradient then the gas is stable against convection. For levels of the atmosphere at which ionization fractions are changing, there is also a dlogm/dlogP term in the equation which lowers the temperature gradient at which the atmosphere becomes unstable to convection. Complex molecules in the atmosphere have the same effect of making the atmosphere more likely to be convective.

The Transfer Equation For radiation passing through gas, the change in intensity In is equal to: dIn = intensity emitted – intensity absorbed dIn = jnrdx – knrIn dx dIn /dtn = -In + jn/kn = -In + Sn This is the basic radiation transfer equation which must be solved to compute the spectrum emerging from or passing through a gas.

Solving the Gray Atmosphere Integrating the transfer equation over frequency: The radiative equilibrium equations give us: F=F0, J=S, and dK/dt = F0/4p LTE says S = B (the Planck function) Eddington Approximation (I independent of direction)

Monochromatic Absorption Coefficient Recall dtn = knrdx. We need to calculate kn, the absorption coefficient per gram of material First calculate the atomic absorption coefficient an (per absorbing atom or ion) Multiply by number of absorbing atoms or ions per gram of stellar material (this depends on temperature and pressure)

Physical Processes Bound-Bound Transitions – absorption or emission of radiation from electrons moving between bound energy levels. Bound-Free Transitions – the energy of the higher level electron state lies in the continuum or is unbound. Free-Free Transitions – change the motion of an electron from one free state to another. Scattering – deflection of a photon from its original path by a particle, without changing wavelength Rayleigh scattering if the photon’s wavelength is greater than the particle’s resonant wavelength. (Varies as l-4) Thomson scattering if the photon’s wavelength is much less than the particle’s resonant wavelength. (Independent of wavelength) Electron scattering is Thomson scattering off an electron Photodissociation may occur for molecules

Neutral hydrogen (bf and ff) is the dominant Source of opacity in stars of B, A, and F spectral type

Opacity from the H- Ion Only one known bound state for bound-free absorption 0.754 eV binding energy So l < hc/hn = 16,500A Requires a source of free electrons (ionized metals) Major source of opacity in the Sun’s photosphere Not a source of opacity at higher temperatures because H- becomes too ionized (average e- energy too high)

Dominant Opacity vs. Spectra Type Low Electron scattering (H and He are too highly ionized) Low pressure – less H- Electron Pressure He+ He Neutral H H- H- High (high pressure forces more H-) O B A F G K M

The T(t) Relation In the Sun, we can get the T(t) relation from Limb darkening or The variation of In with wavelength Use a gray atmosphere and the Eddington approximation In other stars, use a scaled solar model: Or scale from published grid models Comparison to T(t) relations iterated through the equation of radiative equilibrium for flux constancy suggests scaled models are close

Hydrostatic Equilibrium Since dtn=kn rdx dP/dx=kn r dP/dtn=gr or dP/dt n = g/kn

The Paschen Continuum vs. Temperature 50,000 K 4000 K

Calculating Fl from V Best estimate for Fl at V=0 at 5556A is Fl = 3.54 x 10-9 erg s-1 cm-2 A-1 Fl = 990 photon s-1 cm-2 A-1 Fl = 3.54 x 10-12 W m-2 A-1 We can convert V magnitude to Fl: Log Fl = -0.400V – 8.451 (erg s-1 cm-2 A-1) Log Fn = -0.400V – 19.438 (erg s-1 cm-2 A-1) With color correction for 5556 > 5480 A: Log Fl =-0.400V –8.451 – 0.018(B-V) (erg s-1 cm-2 A-1)

Bolometric Corrections Can’t always measure Fbol Compute bolometric corrections (BC) to correct measured flux (usually in the V band) to the total flux BC is usually defined in magnitude units: BC = mV – mbol = Mv - Mbol