Astronomy 217

 
 

Prof. Andrew W. Steiner

 
 
Oct. 6, 2021
 
 

TA James Ternullo

Last Time

  • Light and Atoms

Today

  • Blackbody radiation

Bohr's Model, Revisited

  • With Bohr’s model for the Hydrogen atom we have a physical basis to understand the emission and absorption of light by atoms.
  • An electron \( L=n \hbar \) has energy $$ \begin{eqnarray} E_n &=& - \frac{m_e c^2 \alpha^2}{2} \frac{Z^2}{n^2} \\ &=& -13.6 \frac{Z^2}{n^2}~\mathrm{eV} \end{eqnarray} $$
  • An atomic transition from the \( n_2 \) orbital to the \( n_1 \) orbital emits a photon with wavelength given by $$ \frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2}-\frac{1}{n_2^2}\right) $$ where $$ R_H \equiv \frac{m_e c \alpha^2}{2 \hbar} = \left( 91.16~\mathrm{nm} \right)^{-1} $$

Kirchoff's Laws

  • Kirchhoff presented Three Laws of Spectroscopy to codify the observed behavior of spectral lines.
  • A hot solid or a hot, dense gas produces a continuous spectrum.
  • A hot, low-density gas produces an emission- line spectrum.
  • A continuous spectrum source viewed through a cool, low-density gas produces an absorption-line spectrum.
  • From Bohr's model, it is easy to understand #2 and #3. What about the continuous spectrum?

Intensity

  • To understand light in a dense medium, we need to follow the strength of the light as it progresses through that medium.
  • We describe the intensity, I, as the number of photons passing through a unit area, S (m2), per unit time, t (s).
  • The absorber has \( n \) atoms per \( \mathrm{cm}^{3} \) and a volume of \( S \Delta x \) $$ \frac{\Delta I}{I} = - \left( n S \delta x \right) \frac{\sigma}{S} = - n \sigma \Delta x $$

Radiative transfer

  • In the limit of small \( \Delta x \), this becomes a differential equation $$ \frac{dI}{I} = - n \sigma~dx $$
  • To find I as a function of x, if \( \sigma \) and \( n \) do not depend on \( x \), we can integrate $$ \int \frac{dI}{I} = \int - n \sigma~dx \Rightarrow \ln~I = - n \sigma x + C $$ where \( C \) is a constant.
  • This can be rearranged into the form $$ I(x) = I_0 e^{- n \sigma x} \qquad \mathrm{where}~I_0 = I(0) $$
  • Thus the intensity falls of exponentially with \( x \)
  • For larger \( n \) or larger \( \sigma \), \( I \) falls off more rapidly

Optical Depth

  • The form of the equation of radiative transfer $$ I(x) = I_0 e^{-n \sigma x} $$ suggests two important quantities
  • First, the column density $$ N(x) = \int_0^{x} n(x^{\prime})~dx^{\prime} $$ is the total number of particles in a column of length x and 1 square meter in cross section. \( N(x) = n x \) if \( n \) is constant
  • Second, the optical depth $$ \tau(x) = \int_0^{x} \sigma n(x^{\prime})~dx^{\prime} $$ is the measure of the attenuation when traveling through a column of length \( x \) \( \tau = \sigma N(x) \) if \( \sigma \) is constant

Optically Thick

  • If \( \tau = 1 \approx \sigma n x\), the intensity is reduced by almost a factor of 3 $$ I(x) = I_0 e^{-n \sigma x} = I_0/e $$
  • Thus \tau serves a similar role for the intensity as the decay constant, \( \lambda \), serves for radioactive decays
  • We can use \( \tau \) to characterize the "opacity" of an object
  • If \( \tau \gg 1 \), the object is opaque, or "optically thick"
  • If \( \tau \ll 1 \), the object is transparent, or "optically thin"
  • A related quantity is the "mean free path", \( \left< x \right> \), the average distance traveled before absorption $$ \left< x \right> = \left( \int_0^{\infty} x e^{-n \sigma x}~dx \right) \left( \int_0^{\infty} e^{-n \sigma x}~dx \right)^{-1} $$
  • \( \left< x \right> \) is the distance for \( \tau \) to increase by 1

Frequency Dependent

  • We defined the intensity, I, as the number of photons passing through a unit area, \( S \) (measured in m2), per unit time, t (measured in s).
  • But we have shown that the intensity depends on the atomic cross-sections and Bohr’s model makes it clear that \( \sigma \) depends on frequency (or wavelength), \( \sigma(\nu) \) or \( \sigma_{\nu} \).
  • Thus intensity will generally depend on frequency, \( I_{\nu} \), the "specific intensity".
  • The assumption that photons move in perfect alignment down a column is also simplistic. Thus, we need also consider the direction from which the photons come.
  • For this purpose, we introduce the angular equivalent of an area, the solid angle, \( \Omega \), with units of square degrees or steradians.

Specific Intensity

  • If we ask how much energy is carried by photons with frequencies between \( \nu \) and \( \nu+d \nu \), over time \( dt \), originating in a small solid angle \( d\Omega \). $$ d E = I_{\nu}~dt~dA~d\Omega~d\nu $$ thus intensity has units of \( \mathrm{J}~\mathrm{s}^{-1}~\mathrm{m}^{-2}~ \mathrm{sr}^{-1}\mathrm{Hz}^{-1} \)
  • The total intensity includes all frequencies $$ I = \int_0^{\infty} I_{\nu}~d\nu $$
  • In situations when the specific intensity is the same in all directions (isotropic), it is useful to define the mean specific intensity $$ J_{\nu} = \frac{1}{4 \pi} \int I_{\nu}~d\Omega $$

Two-level Atom

  • Consider a simple atom which has two states, residing in an isotropic radiation field, \( J_{\nu} \).
  • Only photons with energy \( h \nu = E_2 - E_1 \) can interact with this atom.
  • The level populations change at rates of $$ \frac{d n_1}{dt} = - n_1 B_{12} J_{\nu} \quad \mathrm{and} \quad \frac{d n_2}{dt} = - n_2 B_{21} J_{\nu} - n_2 A_{21} $$ where \( B_{12} \) is the Einstein absorption coefficient (proportional to the cross section), \( B_{21} \) is the Einstein stimulated emission coefficient, and \( A_{21} \) is the Einstein A coefficient (for spontaneous emission)
  • We also have $$ B_{12} = \frac{g_2}{g_1} B_{21} \quad \mathrm{and} \quad A_{21} = \frac{2 h \nu^3}{c^2} B_{21} $$

Statistical Equilibrium

  • Eventually, the two-level atom will reach steady state, which is termed statistical equilibrium. Then $$ \frac{d n_1}{dt} = \frac{d n_2}{dt} \quad \mathrm{or} \quad n_1 B_{12} J_{\nu} = n_2 B_{21} J_{\nu} + n_2 A_{21} $$
  • Rearranging terms and using the relations that relate \( B_{12} \), \( B_{21} \), and \( A_{21} \), one can calculate ratio of level populations as a function of the mean intensity \( J_{\nu} \) $$ \frac{n_2}{n_1} = \frac{J_{\nu} B_{12}} {J_{\nu} B_{21} + A_{21}} = \frac{(g_2/g_1) J_{\nu} B_{21}} {J_{\nu} B_{21} + (2 h \nu^3/c^2) B_{21}} $$
  • This simplifies to $$ \frac{n_2}{n_1} = \frac{g_2}{g_1} \frac{J_{\nu}}{J_{\nu} + (2 h \nu^3/c^2)} $$

Planck Function

  • If the mix of atoms and photons is in LTE (a stronger assumption than statistical equilibrium), the Boltzmann equation tells us that, $$ \frac{n_2}{n_1} = \frac{g_2}{g_1} \exp \left( \frac{E_1-E_2}{k T} \right) = \frac{g_2}{g_1} \exp \left( - \frac{h \nu}{kT} \right) $$
  • Equating this relation with the population relation for \( J_{\nu} \), $$ \frac{n_2}{n_1} = \frac{g_2}{g_1} \frac{J_{\nu}}{J_{\nu} + (2 h \nu^3/c^2)} $$ gives $$ \frac{J_{\nu}}{J_{\nu} + (2 h \nu^3/c^2)} = \exp \left( - \frac{h \nu}{kT} \right) $$ or, the "Planck function" $$ J_{\nu} = \frac{2 h \nu^3}{c^2} \frac{1}{e^{h \nu/k/T}-1} $$

Blackbody Shape

  • Planck’s function was able to explain the observed shape of the Blackbody. $$ J_{\nu} = \frac{2 h \nu^3}{c^2} \frac{1}{e^{h \nu/k/T}-1} $$
  • There are two useful limits. For low-energy photons, \( h \nu \ll kT \) we have the "Rayleigh-Jeans" limit $$ J_{\nu}(T) = \frac{2 h \nu^3}{c^2} \frac{k T}{h \nu} = \frac{2 k T}{c^2} \nu^2 $$
  • For high-energy photons, \( h \nu \gg kT \), we have the Wien limit $$ J_{\nu}(T) = \frac{2 h \nu^3}{c^2} \exp \left( -\frac{h \nu}{k T} \right) $$

Emitted Flux

  • The flux of energy emitted per unit area, per unit time in a interval of frequency, can be calculated by integrating the intensity over the solid angle.
  • The result, \( F_{\nu} = \pi I_{\nu} \) has units of \( \mathrm{s}^{-1}~\mathrm{m}^{-2}~ \mathrm{Hz}^{-1} \).
  • The total flux is calculated by integrating over frequency $$ F = \frac{2 \pi h}{c^2} \int_0^{\infty} \frac{\nu^3~d \nu}{e^{h \nu/k/T}-1} = \frac{2 \pi^5 }{15 h^3 c^2} (kT)^4 = \sigma_{\mathrm{SB}} T^4 $$ where \( \sigma_{\mathrm{SB}} = 5.67 \times 10^{-8} ~\mathrm{J}~\mathrm{s}^{-1}~\mathrm{m}^{-2} ~\mathrm{K}^{-4} \) is the "Stefan-Boltzmann" constant.
  • The total luminosity is the flux times the area of the emitting surface $$ L = 4 \pi R^2 \sigma_{\mathrm{SB}} T^4 $$