에너지 전달 복사 전달, 대류 전달

Black Bodies Wien’s Law – Peak intensity Stefan-Boltzman Law – Luminosity Planck’s Law – Energy Distribution Rayleigh-Jeans approximation Wien approximation

Wien’s Law – Peak Intensity Il is max at lmax = 2.90 x 10 7/T (Angstroms) (or l’max = 5.1 x 107/T where l’max is the wavelength at which In is max)

Luminosity – Stefan Boltzman Law F = sT4 or L = 4p R2 sT4

In = constant x n3 e (-constant x n /T) Planck’s Law Rayleigh-Jeans Approximation (at long wavelength) In = 2kTn2/c2 = 2kT/l2 Wien Approximation – (at short wavelength) In = constant x n3 e (-constant x n /T)

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

Energy Transport 복사 전달식의 기본 해 (Formal solution of the transfer equation) 복사 평형(Radiative equilibrium) 회색대기(The gray atmosphere) 주연감광(Limb darkening)

dIn = intensity emitted – intensity absorbed The Transfer Equation Recall: for radiation passing through a gas, the change in In is equal to: dIn = intensity emitted – intensity absorbed dIn = jnrdx – knrIndx or dIn/dtn = -In + Sn

The Integral Form A solution usually takes the form Where One must know the source function to solve the transfer equation For LTE, Sn(tn) is just the Planck function Bn(T) The solution is then just T(tn) or T(x)

Geometry In real life, we are interested in In from an arbitrary direction, not just looking radially into the star In plane parallel geometry we have azimuthal symmetry, so that

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

Integrating Over Solid Angle Assume knr and Sn are independent of direction, and substitute the definitions of flux and mean intensity: becomes: Then integrate over frequency:

(integrating over frequency…) LHS is zero in radiative equilibrium, so The third radiative equilibrium condition is also obtained by integrating over solid angle and frequency, but first multiply through by cos q. Then

3 Conditions of Radiative Equilibrium: In real stars, energy is created or lost from the radiation field through convection, magnetic fields, and/or acoustic waves, so the energy constraints are more complicated

Solving the Transfer Equation in Practice Generally, one starts with a first guess at T(tn) and then iterates to obtain a T(tn) relation that satisfies the transfer equation The first guess is often given by the “gray atmosphere” approximation: opacity is independent of wavelength

Solving the Gray Atmosphere Integrating the transfer equation over frequency: gives or The radiative equilibrium equations give us: F=F0, J=S, and dK/dt = F0/4p

Eddington’s Solution (1926) Using the Eddington Approximation, one gets Chandrasekhar didn’t provide a rigorous solution until 1957 Note: One doesn’t need to know k since this is a T(t) relation

Limb Darkening This white-light image of the Sun is from the NOAO Image Gallery. Note the darkening of the specific intensity near the limb.

Limb Darkening in a Gray Atmosphere Recall that so that as q increases the optical depth along the line of sight increases (i.e. to smaller tn and smaller depth and cooler temperature) In the case of the gray atmosphere, recall that we got:

Limb darkening in a gray atmospehre so that In(0) is of the form In(0) = a + bcos q One can derive that and

Comparing the Gray Atmosphere to the Real Sun

Review Forms of the radiative transfer equation Conditions of radiative equilibrium Gray atmospheres Eddington Approximation Limb darkening

Convective Energy Transport Stability criterion for convection Adiabatic temperature gradient When is convection important Convection in the Sun The Mixing Length Formalism

대류에 의한 에너지 F = r Cp v DT 태양 : r = 10-7 g /cm3 Cp = 2.5 R =8 * 107 erg/(g K) for H DT = ~ 100 v = few km/s  2* 108 erg/(s cm2) (복사 ~ 1011 )  asymmetric line profile Granulation  noise acoustic wave power ;chromosphere  motions in magnetic field lines  Alfven wave convection  mixing : chemical homogeneity

Criterion for Stability against Convection If we displace an element of gas, will it continue to move in the same direction? P2’ = P2 If r2’ < r2, the element will continue to rise. P2’=P2 r2’ T2’ P2, r2, T2 Displaced gas Initial gas: P1, r1, T1

Stability against Convection Since P2=P2’ (the gas will adjust to equalize the pressure), then r2T2=r2’T2’ To be stable against convection, r2’ must be greater than r2 (원소 밀도가 주변 밀도가 보다 높아야 원소가 못 올라감 ) Thus, T2 must be greater than T2’ (주변온도가 원소 온도보다 높아야 못 올라감) That is, the temperature in the moving element must decrease more rapidly than in the surrounding medium (주변온도의 변화가 원소 온도의 변화보다 적어야 대류가 발생 못함) dT/drelement > dT/drsurroundings

Stability Criterion in Terms of Pressure Since pressure falls upward in the atmosphere, the stability criterion can be rewritten as: Take the derivative and multiply by P/T to get: or

Adiabatic Equilibrium 원소 주변이 복사평형일 경우, 원소와 주변사이에 열교환이 없으므로 올라가는 가스는 아디아배틱(에너지 전달없음) 평형에 있다. adiabatic equilibrium (i.e. no energy transfer). For gas in adiabatic equilibrium, PVg = constant and where g = 5/3 for ionized gas and is less for neutral or incompletely ionized regions near the surface. (Recall that g is related to the polytropic index as g = n/(n+1) and is the ratio of the specific heat of the gas under constant pressure to the specific heat of the gas under constant volume.)

The Temperature Gradient 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.

When Is Convection Important? 불투명도가 높아지면 온도 경사도가 높아진다.(i.e. 불투명도가 높아져 복사에 의한 에너지 전달이 비효율적으로 된다.) F 형보다 차거운 항성은 표면 대류층을 갖는다.( surface convection zones) 이 표면대류층은 만기형 항성으로 가면서 깊어져 cool M dwarfs에서 항성전체가 fully convective해진다. 표면대류는 채층을 형성하게 하고 광구 높은 위치에 온도 최저 지역 이후에 에너지 전달에 중요한 역할을 하는 acoustic or magnetic 전달을 유도한다. 대류는 항성내부구조에서는 중요한 에너지 전달이므로 더 자세히 취급됨.

결과적으로 Teff>9000K 에서는 일반적으로 대류는 존재하지 않음 For A stars, a thin convective zone is present (100-200 km) below the visible layers Giant stars are less likely to be convective at a given temperature than main sequence stars because the low density of the atmosphere makes convection inefficient for transferring energy

Convection in the Sun We observe convection in the Sun in the form of granulation. Each granule is the top of a rising column of hot gas, and the granules are surrounded by cooler falling gas The granules are typically 1000 km in size Note that the distance from t = 1 to t=25 in the Sun is less than about 100 km, just a fraction of the size of a convective cell

Solar Granulation

The Mixing Length Formalism 대류를 막는 안정 조건을 유도했으나 , 대기에서 대류에 의해 전달되는 에너지를 수학적으로 계산하는 방법을 개발해야 한다. Convection is a hard problem The mixing length formalism was developed in the 1950’s (e.g. Erika Bohm-Vitense) and is still the most widely used treatment of convection A proper theory of convection is beginning to come from 2D and 3D hydrodynamical calculations

Mixing length ½ rn2 = g Dr l (1/2 mv2 = mgh) Dr/r = DT/T v =(glDT/T)1/2 ==> F = r Cp(g l/T)1/2 DT3/2 = r Cp(g/T)1/2 l2[(dT/dx)cell -(dT/dx)ave]3/2

Definition of the Mixing Length The mixing length L is the distance traveled by a convective cell before merging into the surrounding medium The “mixing length to pressure scale height ratio” (a = L/H) just expresses the assumed mixing length in terms of a characteristic atmospheric length H (the distance over which the pressure is reduced by the factor e) In the case of no convection, a=0 When convection is present, a is typically assumed to be about 1.5, although values from zero up to 2-3 are used.

문제 5-1,2,3 문제 5-1: Calculate the wavelength at which In is maximum in the Sun and at which Il is maximum in the Sun 문제 5-2: What is the spectral type of a main sequence star in which Il is maximum at H_alpha? A giant star? 문제5-3: What is the peak wavelength of an 05 star at 35000K (if it were radiating as a black body!)?

문제 5-4 문제 5-4: What is the approximate absolute magnitude of a DA white dwarf with an effective temperature of 12,000, remembering that its radius is about the same as that of the Earth?

문제 5-5 문제5-5 The flux of M3’s IV-101 at the K-band is approximately 4.53 x 105 photons s–1 m–2 mm-1. What would you expect the flux to be at 18 mm? The star has a temperature of 4250K.

문제 5-6,7 문제 5-6 :You are studying a binary star comprised of an B8V star at Teff = 12,000 K and a K2III giant at Teff = 4500 K. The two stars are of nearly equal V magnitude. What is the ratio of their fluxes at 2 microns? 문제 5-7: In an eclipsing binary system, comprised of a B5V star at Teff = 16,000K and an F0III star at Teff = 7000K, the two stars are known to have nearly equal diameters. How deep will the primary and secondary eclipses be at 1.6 microns?

문제 5-8,9 문제 5-8: Calculate the radius of an M dwarf having a luminosity L=10-2LSun and an effective temperature Teff=3,200 K. What is the approximate density of this M dwarf?   문제 5-9 : Calculate the effective temperature of a proto-stellar object with a luminosity 50 times greater than the Sun and a diameter of 3” at a distance of 200 pc.

문제 5-10,11 문제 5-10 : You want to detect the faint star of an unresolved binary system comprising a B5V star and an M0V companion. What wavelength regime would you choose to try to detect the M0V star? What is the ratio of the flux from the B star to the flux from the M star at that wavelength? 문제 5-11: You want to detect the faint star of an an unresolved binary system comprising a K0III giant and a DA white dwarf with a temperature of 12,000 K (and MV=10.7). What wavelength regime would you choose to try to detect the white dwarf? What is the ratio of the flux from the white dwarf to the flux from the K giant at that wavelength?

문제 5-12 문제 5-12: The opacity, effective temperature, and gravity of a pure hydrogen gray atmosphere are k = 0.4 cm2 gm-1, 104K, and g=2GMSun/RSun2. Use the Eddington approximation to determine T and r at optical depths t = 0, ½, 2/3, 1, and 2. Note that density equals 0 at t = 0.

숙제 5-13 Using the Kurucz models, map out the effective temperatures and surface gravities at which significant flux is carried by convection at optical depth 1 for main sequence stars and for supergiants. Again, assume g = 5/3.