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Quasineutrality gives $n_e=4\times 10^{26}{\mathrm{m}}^{-3}$. Using the supplied $90^{\circ }$ collision-time expression and $\ln \Lambda =10$ givesThe printed absorption coefficient then yields $\kappa =1.62\times 10^2{\mathrm{m}}^{-1}$. Since the unabsorbed fraction is $e^{-\kappa L}$,This uses exactly the coefficient specified by the paper and neglects reflection and changing plasma conditions. With $\nu =1/{\tau }_{ei}$ instead, the corresponding thickness is $2.14\mathrm{mm}$, illustrating the convention dependence discussed in part (c).

Downward-from-space optical depth at wavelength $\lambda$ iswhere ${\kappa }_{\lambda }$ is the mass absorption coefficient. A nearly isothermal, hydrostatic equilibrium, compositionally well-mixed atmosphere with height-independent ${\kappa }_{\lambda }$ has $\rho \propto e^{-z/H}$ and $\tau \propto e^{-z/H_{\tau }}$ with $H_{\tau }=H=R_{a}T/g$. Thus temperature, mean molecular mass and gravity set the basic scale height. Changes in the absorber mixing ratio, pressure broadening, temperature dependence of line strengths, collision-induced absorption and departures from hydrostatic equilibrium modify it. For example if local absorption scales as ${\rho }^2$ in an isothermal atmosphere, $H_{\tau }=H/2$.

Use a steady planar sonic ablation model with absorbed intensity $I_a$, critical mass density ${\rho }_c=(A_i/Z)m_u{n}_c$, and $n_c={ϵ}_0{m}_e(2\pi c/\lambda {)}^2/e^2$. In the usual isothermal-electron sonic balance, $I_a\simeq 4{\rho }_c{c}_s^3$ and the ablation pressure is $P_A\simeq 2{\rho }_c{c}_s^2$. Eliminating $c_s$ givesFor $A_i/Z\simeq 2$ and complete absorption, the supplied laser parameters give about $1.15\times 10^{13}$ Pa, or $115$ Mbar. Mach one alone fixes neither the material mass per electron nor the enthalpy/absorption coefficient. Different sonic-flow closures change the order-one constant; incomplete absorption multiplies the estimate by its fraction to power $2/3$. The robust result is the two-thirds scaling, rather than a material-independent numerical pressure.

The principal non-water visible-light absorbers can be grouped into phytoplankton pigments, coloured dissolved organic matter (CDOM), and non-algal particles (NAP), including organic detritus and absorbing mineral material. A useful additive decomposition of the absorption coefficient isThe supplied sketch separates the three non-water contributions; its amplitudes are arbitrary and are not concentrations or measured spectra.

Phytoplankton pigments give prominent absorption maxima around $440$ and $675\mathrm{nm}$, associated with chlorophyll a, and a relative minimum in the green. Accessory pigments, including carotenoids and phycobilins, broaden or add absorption bands, so an assemblage is not just a pure chlorophyll-a solution. Intracellular packaging and cell size also modify the shape and absorption per unit pigment.

Coloured dissolved organic matter consists of dissolved light-absorbing organic compounds, often including humic material from terrestrial runoff and marine degradation. Its absorption is strongest at the blue/ultraviolet end and declines approximately exponentially toward red, without the narrow chlorophyll red peak. A common local parameterization is $a_g(\lambda )=a_g({\lambda }_0)e^{-S_g(\lambda -{\lambda }_0)}$ with positive spectral slope $S_g$; here $S_g$ is an optical slope, not gas saturation.

Non-algal particles include nonliving organic detritus, pigment degradation products and absorbing suspended mineral particles. Their aggregate absorption often also decreases approximately exponentially from blue to red, with variable slope and mineral-dependent departures. Suspended sediments can strongly scatter as well as absorb; bright sediment-laden water is not evidence of a red absorption peak. Chlorophyll-containing living cells belong in $a_{\mathrm{ph}}$, so these categories should not be double counted.^[ref]

For a closed two-level system with total population $N$, excited-state lifetime $\tau$ and statistical sublevel populations, let $W_a={\sigma }_{12}I/(\hslash \omega )$. The steady rate equation $0=W_a[N_1-(g_1/g_2)N_2]-N_2/\tau$ gives absorption coefficient $\kappa =N{\sigma }_{12}/(1+I/I_{s,a})$. For a uniform slab of thickness $L$ and transmission $T=I_{\mathrm{out}}/I_{\mathrm{in}}$, integrating $dI/dz=-\kappa I$ proves $\ln T+I_{\mathrm{in}}(T-1)/I_{s,a}=-N{\sigma }_{12}L$. Resolved sublevel optical pumping requires a more detailed model.

The integrated absorption coefficient of an isolated spectral line is its line strength. For a mass coefficient write ${\kappa }_{\nu }=S{\varphi }_{\nu }$, where $\int {\varphi }_{\nu }d\nu =1$. The units of $S$ include the chosen spectral-coordinate unit; a strength integrated over wavenumber is not numerically interchangeable with one integrated over frequency. Changing normalized width at fixed strength redistributes opacity without changing its spectral integral.

Consider a plane-parallel atmosphere in hydrostatic equilibrium, with constant gravitational acceleration $g$ and a uniformly mixed absorber of mass fraction $q$. Let its mass absorption coefficient be the Lorentz line shape ${\kappa }_{\nu }=(S/\pi )\Delta /[(\nu -{\nu }_0{)}^2+{\Delta }^2]$, with pressure-independent integrated strength $S$ and half-width $\Delta (p)={\Delta }_{s}p/p_s$. This assumes a fixed-temperature pressure broadening model over the pressures used.

For vertical propagation from the top to pressure $p$, optical depth is ${\tau }_{\nu }=(q/g){\int }_0^p{\kappa }_{\nu }(p^{\prime })d{p}^{\prime }$. Integration givesThe strong-line limit is $C\gg 1$. At optical depth one, $|\nu -{\nu }_0|=\Delta (p)/\sqrt{e^{1/C}-1}\simeq \Delta (p)\sqrt{C}$, so the escaping frequencies lie in the far wings. Expanding the logarithm there recovers ${\tau }_{\nu }\simeq qS{\Delta }_s{p}^2/[2\pi g{p}_s(\nu -{\nu }_0{)}^2]$. This identifies both the dimensionless control parameter and the physical approximation; pressure broadening still requires a suitable collisional regime.

Choose a closed two-level system, equal stimulated absorption and emission strengths, a monochromatic pump, and no extra dephasing. Let $p_g+p_e=1$ be the lower and upper populations, and let spontaneous emission remove upper population at rate $\Gamma =1/{\tau }_e$. In this convention the natural linewidth is full width at half maximum $\Gamma$ in angular frequency, its half width at half maximum is $\Gamma /2$, and its full width at half maximum in hertz is $\Gamma /(2\pi )$. All $\Delta$ below are angular detunings.

Let $s=I/I_{\mathrm{sat}}$ be the on-resonance saturation parameter. An effective stimulated transition rate that reproduces the monochromatic steady response isHere $R$ describes each stimulated direction, while $\Gamma$ describes irreversible spontaneous emission. The rate equation gives the steady populations; it is not a description of transient Rabi oscillations. Setting ${\dot{p}}_e=0$ givesand the population difference relevant to net weak-probe photon absorption isThe pump depletes the lower population by $p_e$, but reduces the population difference by $q=2p_e$: stimulated emission contributes as well as missing ground-state absorbers. On resonance, $s=1$ gives $p_e=1/4$, $p_g=3/4$, $d=1/2$; $s=9$ gives $p_e=0.45$, $p_g=0.55$, $d=0.10$. As $s\to \infty$, the two populations approach $1/2$, rather than producing a population inversion. Far off resonance, $p_e\to 0$ and $d\to 1$.

For moving atoms use $\Delta =\delta -kv$ in these pump-prepared populations. A weak counterpropagating probe interrogates them near $\delta +kv=0$. A simple population-only approximation to its absorption coefficient averages the weak-probe Lorentzian line shape times $d(\delta -kv)$ over the Maxwell-Boltzmann velocity distribution. It neglects interference between the beams, changes of velocity during illumination, and multilevel optical pumping. Those restrictions matter for real alkali chemical atoms.