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)). A blackbody curve of higher T always is greater than a blackbody curve of a lower T. Lambda_max*T = .29 cm*deg. Sigma and a can be related to fudamental constants. sigma = 2Pi^5*k^4/(15c^2*h^3) and a = 8Pi^5*k^4/(15c^3*h^3). There are characteristic temperatures that B_v can describe. The brightness temperature is the temperature a blackbody of the same brightness at the same frequency. I_v = B_v(T_b). Color temperature is the temperature of a blackbody that emits the same frequency (color) as the source. It can be used to fit data of a source of unknown distance, and it can correctly define the temperature of an optically thin source. Effective temperature is the temperature a blackbody of the same flux would have. F = integral of cos(theta)*I_v*dv*d(omega) = sigma*T_eff^4.

Sec 1.6

Because j_v = alpha_v*B_v, there is necessarily a relation between emission and absorption. In an atom, this corresponds to a higher and lower energy level, by which an electron can change by absorbing or emitting a photon. A_21 is the transition probability/time that a photon will be spontaneously emitted. B_12*J is the probability/time that a photon will be absorbed. B_21 is the probability/time that a passing photon will perturb an electron and cause it to emit a photon of equal phase. The Einstein relations are: g_1*B_12 = g_2*B_21, and A_21 = (2hv^3/c^2)*B_21. Rewriting the absorption/emission coefficients of the RT equation in these terms yields j_v = (hv_0/4*Pi)*n_2*A_21*phi(v) and alpha_v = (hv/4*Pi)*phi(v)*(n_1*B_12 - n_2*B_21). The RT equation is now dI_v/ds = (hv/4*Pi)*phi(v)*(n_1*B_12 - n_2*B_21)*I_v + (hv_0/4*Pi)*n_2*A_21*phi(v), and the Source Function = (n_2*A_21)/(n_1*B_12 - n_2*B_21). For LTE, n_1/n_2 = (g_1/g_2)*e^(hv/kT). For non LTE, this equation is not satisfied.

Chp. 2

Sec. 1

F = q(E + v/c X B). Del dot D = 4*Pi*rho, where D = e*E. Del X E = -1/c partialB/partialt. Del dot B = 0. Del X H = 4Pi/c*j + 1/c partialD/partialt, where H = B/mu. The Poynting vector S = c/4Pi*E X H.

Sec. 2

In a vacuum, Maxwell's equations become Del dot E = 0. Del X E = -1/c partialB/partialt. Del dot B = 0. Del X B = 1/c partialE/partialt. The solutions of these equations are E = E_0*e^(i*(k dot r - omega*t)) and B = B_0*e^(i*(k dot r - omega*t)). but with E perpendicular to B. Solving yields that E_0 = B_0, and the phase velocity is c.

Sec. 3

A frequency*time uncertainty principle exists: delta*omega * delta*t >1. Using this and Fourier transforms, detailed information can be found about the Energy based on the frequency, or based on the time.

Sec. 4

Polarization affects the direction of propagation of the E and B waves. Waves can be linear, elliptically, or circularly polarized. In reality, light is not monochromatic, so data comes in superpositions of multiple monochromatic polarized waves. These waves are called quasi-monochromatic.

Chp. 3

Sec. 1

The Leinard- Wiechart potentials are Phi = [q/kR] and A = [qu/ckR]. The k factor concentrates potentials into a cone when v approaches c.

Sec. 2

The radiation fields of moving charges are: E_rad(r,t) = q/c(n/(k^3*R) X ((n-Beta) X dBeta/dt)). B_rad(r,t) = [n X E_rad]. |E_rad| = |B_rad|. dW/domega*dsolid angle = (q^2*omega^2/(4*Pi*c))|Integral of n X (n X Beta)*e^(i*omega(t' - n dot r_0(t')/c) dt' |^2.

Sec. 3

When Beta << 1, E_rad = [q/(Rc^2)n X (n X dv/dt)], and B_rad = [n X E_rad]. |E_rad| = |B_rad| = q/(Rc^2)*dv/dt*sin(Theta). The Poynting vector becomes c/(4*Pi)*q^2/(R^2*c^4) (dv/dt)^2*sin^2(Theta). P = 2q^2/(3*c^3)*(dv/dt)^2 (Larmor's formula). The dipole approximation is that P = 2/(3*c^3)*(d^2d/dt^2)^2. Applying this, E(t) = (d^2d/dt^2)*sin(Theta).(c^2*R_0).

Sec. 4

The force of a linearly polarized wave = e*epsilon*E_0*sin(w_0*t), and dipole moment d = -(e^2*E_0/(m*w_0^2))*epsilon*sin(w_0*t). dP/dsolid angle = *dsigma/dsolid angle = c*E_0^2/(8*Pi)*dsigma/dsolid angle. So dsigma/dsolid angle of polarized light = r_0^2sin^2(Theta), where r_0 = e^2/(mc^2). Integrating gives the cross section: sigma = 8/3*Pi*r_0^2. If this is an electron, it is called Thomson scattering. If the light is unpolarized, dsigma/dsolid angle = r_0^2/2*(1 + cos^2(Theta)).

Sec. 5

If the time a process occurs is on the scale larger than the time it takes for the radiation to cross an electron's radius, the radiation works like a perturbation of a particle's motion. F_rad = m*tau*d^2v/dt^2.

Chp. 5

Sec. 1

Bremsstrahlun is hen the Coulomb field of a charged particle accelerates another charged particle. Defining the collision time tau as b(the impact parameter)/v, the Fourier transform of the dipole moment's second derivative is approximately e/(2*Pi*omega^2)*deltav, if omega*tau is much less than 1, and 0 if omega*tau is much greater than 1. Estimating deltav to be 2*Z*e^2/(mbv), the expression for the energy/solid angle/frequency becomes 8*Z^2*e^6/(3*Pi*c^3*m^2*v^2*b^2) if b << v/omega, and 0 if b>> v/omega. Integrating to find the emission/time/volume/frequency yields dW/(domega*dV*dt)= 16*Pi*e^6/(27^(1/2)*c^3*m^2*v)*n_e*n_i*Z^2*g_ff(v,omega). G_ff is the Gaunt factor: a function of the emission frequency and of the particle's energy.

Sec. 2

Out of a velocity range d^3*v, the probability a particle is within that range is proportional to e^(-m*v^2/(2*k*t)) d^3*v. Treating d^3*v as isotropic, the probability becomes v^2*e^(-m*v^2/(2*k*T))dv. The photon energy must be less than the kinetic energy of the particle. v_min = (2*h*nu/m)^(1/2). The energy/frequency/volume/time becomes 6.8E-38*Z^2*n_e*n_i*T^(-1/2)e^(-h*nu/(k*T))*g_ff_bar, where g_ff_bar is the Gaunt factor averaged over the velocity. Integrating over the frequency gives the ower/volume/time (2*Pi*k*T/(3*m))^(1/2)*32*Pi*e^6/(3*h*m*c^3)*Z^2*n_e*n_i*g_B_bar, where g_B_bar is the frequency averaged velocity averaged Gaunt factor.

Sec. 3

Thermal bremsstrahlung can be absorption as well as emission. The emission coefficient j_vff= alpha_vff*B_v(T). j_vff = 1/(4*Pi) dW/(dt*dV*dnu), so alpha_vff becomes 3.7E8*T^(-1/2)*Z^2*n_e*n_i*nu^-3*(1-e^(-h*nu/(k*T)))g_ff_bar. In the R-J limit, h*nu <<k*T, alpha_vff = .-18*T^(-3/2)*Z^2*n_e*n_i*nu^-2*g_ff_bar.

Sec. 4

Relativistic bremsstrahlung can be treated like Compton scattering of "virtual quanta" of the scattering ion's electric field. In the electron's frame, dW'/*(dA'*domega')=(Ze)^2/(Pi^2*b'^2*c)*(b'*omega'/(gamma*c))^2K_1^2(b'*omega'/(gamma*c)). Becaus dA' in low frequencies is the Thomson cross section, dW'/domega' = sigma_T*dW'/(dA'*domega') But dW/domega = dW'/domega', so the observed emission in the lab frame is dW/domega = 8*Z^2*e^6/(3*Pi*b^2*c^5*m^2)*(b*omega/(gamma^2*c)^2*K_1(b*omega/(gamma^2*c)). Using b_min = h/(m*c), in a low frequency limit, dW/(dt*dV*domega) = 16*X^2*e^6*n_e*n_i/(c*c^4*m*2)*ln(.68*gamma^2*c/(omega*b_min)). Integrating over the frequency gives an expression for power in a thermal distribution of electrons. dW/(dV*dt) = 1.4E-27*T^(1/2)*Z^2*n_e*n_i*g_B_bar(1 + 4.4E-10*T).