# Astronomy 217

## Prof. Andrew W. Steiner

Oct. 6, 2021

TA James Ternullo

## Last Time

• Light and Atoms

## 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 $S \Delta x$ $$\frac{\Delta I}{I} = - \left( n S \delta x \right) \frac{\sigma}{S} = - n \sigma \Delta x$$

• 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 $$L = 4 \pi R^2 \sigma_{\mathrm{SB}} T^4$$