항성 대기의 정의 Basic Definition: 별의 안과 밖의 경계 영역 지구대기의 경계 ? 목성형 대기의 경우 ? 두 계수로 정의 –Effective temperature – NOT a real temperature, but rather the “ temperature ” needed in 4  R 2 T 4 to match the observed flux at a given radius –Surface gravity – log g (note that g is not a dimensionless number!) Log g for the Earth is 3.0 (10 3 cm/s 2 ) Log g for the Sun is 4.4 Log g for a white dwarf is 8 Log g for a supergiant is ~0

항성 대기의 기본 가정 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!)

Solar granulation

Basic Physics – Ideal Gas Law PV=nRT or P=NkT where N=  /  P= pressure (dynes cm -2 ) V = volume (cm 3 ) N = number of particles per unit volume  = density of gm cm -3 n = number of moles of gas R = Rydberg constant(8.314 x10 7 erg/mole/K) T = temperature in Kelvin k = Boltzman ’ s constant (1.38 x 10 – 16 erg/K)  = mean molecular weight in AMU (1 AMU = 1.66 x gm)

Basic Physics – Thermal Velocity Distributions Most probable velocity =(2kT/m) 1/2 Average velocity = (8kT/p m) 1/2 RMS Velocity = (3kT/m) 1/2

Basic Physics – the Boltzman Equation N n = (g n /u(T))e -X n /kT Where u(T) is the partition function, g n is the statistical weight, and X n is the excitation potential. For back-of-the-envelope calculations, this equation is written as: N n /N = (g n /u(T)) x 10 –  X n Note here also the definition of  = 5040/T = (log e)/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. For hand calculation purposes, a shortened form of the equation can be written as follows N 1 / N 0 = (1/P e ) x x 10 9 (u 1 /u 0 ) x T 5/2 x 10 –  I P e is the electron pressure and I is the ionization potential in ev. Again, u 0 and u 1 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 Definitions Specific intensity/mean intensity Flux The K integral and radiation pressure Absorption coefficient/optical depth Emission coefficient The source function The transfer equation & examples Einstein coefficients

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 J is the mean intensity averaged over 4  steradians

Flux Flux is the rate at which energy at frequency 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.

Sun From the luminosity and radius of the Sun, compute the bolometric flux, the specific intensity, and the mean intensity at the Sun ’ s surface. L = 3.91 x ergs sec -1 R = 6.96 x cm

Solution F=  T 4 L = 4  R 2  T 4 or L = 4  R 2 F, F = L/4  R 2 Eddington Approximation – Assume I  is independent of direction within the outgoing hemisphere. Then … F =  I J = ½ I  (radiation flows out, but not in)

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

The K Integral and Radiation Pressure

흡수 계수와 광학 깊이 Gas absorbs photons passing through it –Photons are converted to thermal energy or –Re-radiated isotropically( 산란 ) Radiation lost is proportional to –Absorption coefficient (per gram) –Density –Intensity –Path length Optical depth is the integral of the absorption coefficient times the density along the path

There are two sources of radiation within a volume of gas – real emission, as in the creation of new photons from collisionally excited gas, and scattering of photons into the direction being considered. We can define an emission coefficient for which the change in the intensity of the radiation is just the product of the emission coefficient times the density times the distance considered. Emission Coefficient

The Source Function The source function S is just the ratio of the emission coefficient to the absorption coefficient The source function is useful in computing the changes to radiation passing through a gas

The Transfer Equation For radiation passing through gas, the change in intensity I is equal to: dI = intensity emitted – intensity absorbed dI = j  dx –   I  dx dI /d  = -I + j /  = -I + S This is the basic radiation transfer equation which must be solved to compute the spectrum emerging from or passing through a gas.

Pure Isotropic Scattering The gas itself is not radiating – photons only arise from absorption and isotropic re-radiation Contribution of photons proportional to solid angle and energy absorbed: J is the mean intensity dI/d  = -I + J v The source function depends only on the radiation field

Pure Absorption No scattering – photons come only from gas radiating as a black body Source function given by Planck radiation law

Einstein Coefficients Spontaneous emission proportional to N n x Einstein probability coefficient j  = N n A ul h Induced emission proportional to intensity   = N l B lu h – N u B ul h

숙제 4-1 During the course of its evolution, the Sun will pass from the main sequence to become a red giant, and then a white dwarf. Estimate the radius of the Sun in both phases, assuming log g = 1.0 when the Sun is a red giant, and log g=8 when the Sun is a white dwarf. Assume no mass loss. Give the answer in both units of the current solar radius and in cgs or MKS units.

숙제 4-2 Using the ideal gas law, estimate the number density of atoms in the Sun ’ s photosphere and in the Earth ’ s atmosphere at sea level. For the Sun, assume T=5800K, P=10 5 dyne cm -2. How do the densities compare?

숙제 4-3 What are the RMS velocities of 7 Li, 16 O, 56 Fe, and 137 Ba in the solar photosphere (assume T=5800K). How would you expect the width of the Li resonance line to compare to a Ba line?

숙제 4-4 At (approximately) what Teff is Fe 50% ionized in a main sequence star? In a supergiant? What is the dominant ionization state of Li in a K giant at 4000K? In the Sun? In an A star at 8000K?

숙제 4-5 Compare the contribution of radiation pressure to total pressure in the Sun and in other stars. For which kinds of stars is radiation pressure important in a stellar atmosphere?

숙제 4-6 Consider radiation with intensity I (0) passing through a layer with optical depth  = 2. What is the intensity of the radiation that emerges?

숙제 4-7 A star has magnitude +12 measured above the Earth ’ s atmosphere and magnitude +13 measured from the surface of the Earth. What is the optical depth of the Earth ’ s atmosphere?