Matthew Trimble

Radiative Processes in Astrophysics (Rybicki and Lightman) Reading Assignment Summary

Chp. 1

Sec. 1.1

EM radiation is a spectrum of varying categories. c = wavelength*frequency. Energy = h*frequency. Temperature = Energy/k.

Sec. 1.2

When dealing with a macroscopic system, light can be viewed as rays, instead of individual photons. A source of light is isotropic if it emits an equal amount of energy in every direction, like a star. Flux is defined as energy per area per time.

Sec. 1.3

The specific intensity is the energy per area per time per solid angle per frequency. The net flux is the integral of the specific intensity*cos(angle away from the direction being measured along) d(solid angle). The momentum flux is 1/c * integral of specific intensity*cos^2(angle) d(solid angle). Net flux and momenrum flux are moments of the intensity. Flux = integral of net flux d(frequency). Momentum = integral of momentum flux d(frequency). Intensity = integral of specific intensity d(frequency). Specific energy density is the energy per volume per frequency range. dE = energy density as a function of solid angle*d(volume)*d(solid angle)*d(frequency). Specific energy density = (4*pi/c)*mean intensity. Mean intensity = 1/(4*Pi)* integral of specific intensity d(solid angle). Total radiation density = integral specific energy density d(frequency) = (4*Pi/c)* integral of the mean intensity d(frequency). The radiation pressure is 1/3 the energy density in an isotropic radiation field. The specific intensity of a ray is constant, meaning d(specific intensity)/d(length along ray) = 0. The flux at a uniformly bright surface = Pi*Brightness.

Sec. 1.4

Light interacts with matter by exchanging energy. Photons can either be absorbed or emitted by matter, and the specific intensity of the radiation will not be constant. Photons can also scatter off particles in matter, affecting the intensity of the light. dE = j*dV*d(omega)*dt, where j is the spontaneous emission coefficient. Monochromatically, dE = j_v*dV*d(omega)*dt*dv. j_v for an isotropic source is defined as 1/(4*Pi)*P_v, where P_v is the power per volume per frequency. Defining emissivity as energy per frequency per time per mass, dE = e_v*rho*dV*dt*d_nu*d(omega)/(4*Pi), where rho is the density of the matter. Relating the expressions fo dE, j_v = e_v*rho/(4*Pi). Because dV = dA*ds, dI_v = j_v ds. In the case of absorption, dI_v = -(alpha_v)*I_v*ds, where alpha is defined as n(sigma_v) or as rho*kappa, kappa being the opacity coefficient. Combining the changes in the specific intensity by absorption and emission, the radative transfer equation becomes dI_v/ds = -(alpha_v)I_v + j_v. When there is no apsorbtion, the alpha_v term is 0, and when there is no emission, j_v is 0. To simplify the radiative transfer equation by changing the variable s to tau_v: the optical depth. t_v(s)= integral of (alpha_v(s')) ds' from s_0 to s. The radiative transfer equation becomes dI_v/dtau_v = -I_v + S_v, where S_v is the source function, defined as j_v/(alpha_v). Solving the radiation equation leads to a function of I_v with respect to tau_v. I_v(tau_v) = I_v(0)e^-tau_v + integral of e^-(tau_v-tau'_v)*S_v(tau'_v) dtau'_v. The mean free path of radiation is the average optical depth a photon can travel before being absorbed. <tau_v> = integral of tau_v*e^-tau_v dtau_v from 0 to infinity = 1. The mean physical distance, l_v = <tau_v>/alpha_v = 1/alpha_v, is the average distance a photon can travel in homogeneous matter before being absorbed. Radiation carries momentum, so it exerts force on matter when it interacts with it. The radiation flux vector, F_v = integral of I_v*n*d(omega). The force per mass on the matter, f= 1/c * integral of kappa_v*F_v*dv.

Sec 1.5

Blackbody radiation is radiation in thermal equilibrium (I_v = B_v). Specific intensity is only dependant on T, and v=B_v(T). Kirchoff's law states that j_v = alpha_v*B_v(T). This makes the RT equation dI_v/dtau_v = -I_v + B_v(T). Because blackbody radiation is isotropic and homogeneous, p=1/3*u. Thermal radiation is radiation where S_v = B_v. The first law of thermodynamics states that dQ = dU + p*dV, with Q = heat and U is total energy. The second law of thermodynamics states that dS = dQ/T, where S is the entropy. The Stefan- Boltzmann law states that u(T) = aT^4. Relating that with the Planck function and emergent flux, the S-B equation can be rewritten as F= sigma*T^4. Sigma and a are derived constants. Entropy, S = 4/3*a*T^3*V. The adiabatic expansion for blackbody radiation is defined as T*V^1/3 = constant or pV^4/3 = constant. B_v(T)= (2*h*v^3/c^2)(e^(h*v/(k*T))-1)^-1. This law is extremely important because of different approximations of it. When h*v<<kT, the Rayleigh- Jeans law states that I_v = 2v^2/c^2*kT. When h*v>>k*T, the Wein law states that I_v = 2h*v^3/c^2*e^(-h*v/(k*T)).