Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 39 (Atomic Astrophysics)

The specific intensity of a blackbody is direction-independent over its outward hemisphere. Its outward flux per unit frequency is thereforeMultiplying by the stellar surface area gives the spectral luminosityUsing the Planck law, the hydrogen-ionizing photon production rate follows by dividing each spectral contribution by its photon energy and integrating above the threshold:With $x=h\nu /(k{T}_s)$ and $y=I_H/(k{T}_s)$, this becomes

For the initial hydrogen-only photon balance, assume an ionization-bounded nebula at steady photoionization equilibrium, negligible dust absorption and ionizing-photon leakage, negligible collisional ionization, and local reabsorption of photons from direct ground-state recombinations. A ground-state recombination then causes another photoionization and produces no net removal of a proton. Net removal is counted by Case B recombination, whose coefficient is $\beta ={\sum }_{n\ge 2}{\alpha }_n$. Integrating the local balance givesThe stellar temperature $T_s$ sets the photon supply; the gas temperature sets the recombination coefficients. They are not generally equal. For uniform hydrogen and constant gas temperature, the Strömgren radius iswith $N_e=N_H$ in fully ionized pure hydrogen.

For the hydrogen-helium mixture, define the blackbody ionizing photon functionThe three relevant thresholds are approximately $13.6$, $24.6$ and $54.4\mathrm{eV}$, for ${\mathrm{H}}^0$, ${\mathrm{He}}^0$ and ${\mathrm{He}}^{+}$ respectively. Higher stellar temperature increases the relative supply of the harder photons. These cumulative photon rates cannot be assigned independently to the three species: a photon above a helium threshold can instead ionize hydrogen, and helium recombination radiation can also ionize hydrogen. The absorption cross-sections and the diffuse radiation determine the effective allocation.

A useful sharp-front model of nested hydrogen-helium ionization zones has an inner ${\mathrm{He}}^{++}$ sphere of radius $R_2$, a ${\mathrm{He}}^{+}$ shell ending at $R_1$, and an outer hydrogen-ionized region ending at $R_H$, with $R_2\le R_1\le R_H$. Let $n_H,n_{He}$ be uniform nuclear number densities. The electron number densities in these three regions are $n_H+2n_{He}$, $n_H+n_{He}$ and $n_H$. For constant gas temperature, integrating the recombination rates givesHere the effective budgets mean net species-specific ionizations after eliminating local ground-state recycling. Their calculation must include photon competition and any interspecies diffuse contribution; the helium coefficients label the recombining ion's emitted spectrum. Once these budgets are specified, solve the third equation for $R_2^3$, the second for $R_1^3-R_2^3$, and the first for $R_H^3$.

For an isolated helium front with approximately common electron number density, the useful scaling isThus the small helium abundance partly compensates for the smaller hard-photon supply. A cooler central star produces little ${\mathrm{He}}^{++}$ and can have a substantially smaller helium-ionized zone. A sufficiently hot star can ionize helium throughout most of the hydrogen region. If independent radius estimates place a helium front beyond the hydrogen front, the independent photon allocations are inconsistent: the fronts and transfer must be treated together. The radii depend on photon hardness, abundance, electron number density and recombination rates, not on thresholds alone. The sharp-front equations are an approximation; they do not assert three exact independently conserved photon budgets for a real mixed nebula.

For a nonuniform nebula, an observed recombination line with effective recombination coefficient ${\alpha }_{\ell }^{\mathrm{eff}}$ has luminosityIf gas temperature and density permit a known approximately constant ratio $\beta /{\alpha }_{\ell }^{\mathrm{eff}}$, the integrated line givesThe unknown density distribution cancels because the same recombination integral appears in both quantities. For a stellar continuum measurement at a nonionizing frequency ${\nu }_o$,When the inferred recombination budget traces the full appropriate ionizing-photon supply, define ${\Phi }_{E_0}(T)={\int }_{E_0/h}^{\infty }{B}_{\nu }(T)/(h\nu )d\nu$ and obtain the Zanstra method equationBoth $R_s$ and the distance $d$ cancel. Hydrogen and helium recombination lines probe different hardness thresholds, so their ratios and the stellar continuum constrain $T_s$ and test the assumed spectrum and trapping. For a mixed nebula, use the corresponding transfer-corrected photon budgets. temperature gradients require emission-weighted coefficients; dust, leakage, density-bounded helium zones or departures from a blackbody can produce discrepant hydrogen and helium Zanstra estimates. Line fluxes alone without an appropriate budget or continuum normalization need not uniquely determine $T_s$.

Finally, locate the derived $T_s$ and luminosity on the Hertzsprung-Russell diagram. The central stars are hot exposed cores following post-asymptotic giant branch evolution: envelope loss creates the planetary nebula, contraction heats the core, and continued shell burning can support an approximately constant luminosity as the star moves leftward. Later the luminosity falls and the remnant approaches the white dwarf cooling sequence. Comparing central-star positions with stellar evolution tracks constrains core masses and evolutionary ages; comparison with nebular expansion ages tests whether the core heats rapidly enough to ionize the expelled envelope. The central-star locus traces the transition from an envelope-losing giant to a compact stellar remnant.

The Boltzmann distribution for two atomic levels at thermodynamic equilibrium iswhere $\omega$ is the statistical weight of an atomic level. For a resolved level of total angular momentum $J$, $\omega =2J+1$ when magnetic substates are unresolved.

To extend this counting to the continuum, assume an ideal, nondegenerate plasma with classical free electrons and neglect the small difference between the translational masses of the two heavy ionic species. Let $z_e=e^{{\mu }_e/kT}$ be the electron fugacity in the Saha equation, with the electron chemical potential measured relative to its rest energy. The two electron spin states and the phase-space cell volume $h^3$ giveChemical equilibrium for capture equates the bound species' chemical potential to the sum for its parent ion and free electron. Thus applying Boltzmann factors to a bound level at energy $-I_n$ gives $N_n/N_{+}=({\omega }_n/{\omega }_{+})z_e{e}^{I_n/kT}$. Eliminating the electron fugacity proves the level-specific Saha ionization equation:The factor two is the electron spin multiplicity; the factor with power $3/2$ is the inverse free-electron translational state density. A total-ionization Saha equation sums the bound states into an internal partition function, whereas this expression refers to one specified level.

For dielectronic recombination, write the entrance ion as $X^{(z+1)+}$ in ground state $i$, and the intermediate state of $X^{z+}$ as $d=(j,n\ell )$. Capture excites the core from $i$ to $j$ while binding the incident electron. The resonance energy relative to the entrance continuum is $\bar{E}=E_{ij}-I_{n\ell }>0$. The doubly excited state can eject the electron by autoionization, or undergo radiative stabilization to a bound state.

Let $A_{di}^a$ be the partial autoionization rate back to the entrance ion, $A_a$ the total autoionization rate, $A_r$ the total radiative decay rate, and $A_s$ the rate of radiative decays that stabilize into bound levels. For an isolated narrow resonance, the Saha-Boltzmann equilibrium ratio has the bound-state energy replaced by the positive resonance energy:At equilibrium, inverse dielectronic capture and autoionization fluxes balance: $N_e{N}_i{C}_d=N_d^{\mathrm{eq}}{A}_{di}^a$. This establishes the capture coefficient even when the actual plasma is not in local thermodynamic equilibrium. In the low-density, weak-radiation recombining plasma, the intermediate level instead has the stationary balanceMultiplying its population by the successful stabilization rate gives the actual recombination coefficientHere the equilibrium population was used only to determine the capture rate by detailed balance; it was not assumed to equal the intermediate population when photons escape. This is precisely why the competing decay probabilities appear.

Use the supplied core-transition relation between the Einstein coefficients and atomic oscillator strength,Here $\alpha$ is the fine-structure constant and $a_0$ the Bohr radius, not a recombination coefficient. Factor out $A_{ji}$ and defineSubstitution yieldsThis definition accommodates several entrance and decay channels. In the common spectator-electron approximation with one autoionization channel and stabilization by the core transition, $A_s=A_r=A_{ji}$ and $A_{di}^a=A_a$. If all coupled substates of the captured $n\ell$ electron are summed, $g_d=2(2\ell +1)g_j$, givingFor an intermediate state resolved by fine structure, retain its actual $g_d$ instead of that summed weight. The stabilization probability itself is $A_s/(A_a+A_r)$; the requested ${\beta }_{j,n\ell }$ is not just this branching probability, because the prefactor has already factored out the core radiative rate and statistical weights of atomic levels. Radiative transitions to another still-autoionizing state require that state's eventual survival probability, rather than automatically counting every emitted photon as stabilization.

The total dielectronic recombination coefficient sums all accessible positive-energy resonances. In this isolated-resonance approximation it has the formwith atomic factors included in $c_d$. The resonance-temperature dependence of dielectronic recombination follows by differentiating a single term's logarithm:Thus low temperatures suppress a positive-energy resonance, although resonances very near threshold can still matter; at sufficiently high temperature a fixed set of resonances falls as $T^{-3/2}$. Several core-excitation series can produce several peaks. When autoionization is much faster than radiation, the capture-and-survival product is limited by the stabilizing radiative rate; when capture is weak, autoionization is the limiting inverse-capture rate. For a fixed Rydberg angular channel, autoionization typically decreases at high principal quantum number while the core radiative rate changes little. Capture into very high levels is vulnerable to subsequent collisional or field ionization, so finite-density survival can suppress the total rate. dielectronic recombination is resonant and strongly temperature-selective; its magnitude can be important in coronal ionization balance.

The solar corona is a hot, dilute plasma in which collisional ionization mainly proceeds by direct electron impact and by excitation-autoionization. The principal inverse ion-changing processes are radiative recombination and dielectronic recombination. Three-body recombination is negligible at ordinary coronal densities. Photoionization, stimulated recombination and charge exchange are normally secondary in this fully ionized, collision-dominated regime, although they should be reconsidered near irradiated or neutral interfaces.

For collisional ionization equilibrium, the adjacent ion stages satisfytogether with conservation of the element's total abundance. Thus the low-density ion fraction depends primarily on $T_e$. This fixes the amount of Si X available for line emission, but not the populations of its excited levels: a coronal approximation does not impose a Boltzmann level distribution.

Within an ion, electrons collisionally excite, de-excite and redistribute levels. Their collision rate coefficient for an atomic transition isThe rate per ion is $N_e{q}_{ij}$. Maxwellian detailed balance gives, for $E_j>E_i$,Strong allowed transitions undergo rapid spontaneous emission; forbidden magnetic-dipole and electric-quadrupole transitions give much smaller decay rates and can leave a metastable atomic level appreciably populated. Radiative cascades feed levels from higher states. Absorption and stimulated emission add rates proportional to the local radiation field through the Einstein coefficients when significant. Proton collisions can also mix closely separated levels separated by fine structure and should be included when their rates matter.

Write $f_i=N_i/N(\mathrm{Si}\mathrm{X})$. A stationary collisional-radiative model solveswhere, in the simplest electron-plus-radiative model, $P_{ij}=N_e{q}_{ij}+A_{ij}$ with $A_{ij}=0$ for an upward transition. Add proton and radiation rates when appropriate. Include enough levels and radiative branches to capture population feeding. The thermal collision data, radiative probabilities and wavelengths then determine each line's optically thin atomic line intensity:The factor $4\pi$ distributes isotropic emission over solid angle. photon intensities omit $h\nu$. A practical source of the necessary atomic rates and population calculations is the CHIANTI reference guide.

Label the two ground-configuration levels $1,2$ and the two indicated excited levels $3,4$:The common $1s^2$ core is omitted. Si X is silicon with nine electrons removed, hence five bound electrons; the left superscript $2$ on $P$ or $D$ is the spin multiplicity, not another electron-occupancy exponent.

The observed lines correspond to decays $3\to 1$ and $4\to 2$. For approximately uniform plasma at the indicated $T_e\simeq 1.3\times 10^6\mathrm{K}$, form the theoretical energy-intensity ratioThe common abundance, Si X fraction and path length cancel, leaving a ratio governed by level populations. The frequency factor is ${\nu }_{42}/{\nu }_{31}=347/356$ for the rounded wavelengths. With observed photon counts, first apply the detector calibration and use the corresponding photon-ratio convention.

The electron-density diagnostic from metastable populations can be seen in a reduced rate model. The upper ground level $2$ has only a weak radiative decay to $1$. Ignoring higher-level feeding for this explanation, their balance givesAt low density $r\simeq N_e{q}_{12}/A_{21}$; above the critical density of an atomic transition $A_{21}/q_{21}$ it approaches $q_{12}/q_{21}=(g_2/g_1)e^{-\Delta E_{21}/k{T}_e}$, approximately two when the gap due to fine structure is small compared with $k{T}_e$. This saturation can occur while the strong upper-level transitions still remain radiatively dominated.

Let $A_3,A_4$ be the total radiative loss rates of levels $3,4$, and $b_{31}=A_{31}/A_3$, $b_{42}=A_{42}/A_4$ their line branching fractions. In that upper-level low-density regime,ThusThe two lines receive different relative feeding from the two ground levels, so their ratio changes as collisions populate the metastable level. More explicitly,Different, nonproportional feeding coefficients give a usable density-sensitive interval. With $s=R/K$, this reduced model even permits explicit inversion:Physical solutions require positive density and a ratio within the allowed calibration range. The actual Si X calibration should use the full multilevel population system, not discard cross-excitation, cascades or all competing branches simply to use this illustrative formula.

Operationally, evaluate the model ratio at the known temperature over a density grid and match the calibrated observed ratio to it, propagating line-flux and atomic-data uncertainties. No observed ratio or numerical atomic rate set is supplied here, so the result is this inference procedure rather than a numerical density. At low- or high-density saturation, the ratio constrains density poorly. Deconvolve blends or sum every unresolved transition in the theoretical feature; for example, the neighboring ${}^2{D}_{3/2}\to {}^2{P}_{3/2}$ branch can contribute near the 356-Angstrom feature when unresolved. Atomic line identifications and that nearby branch are tabulated in Del Zanna's coronal-spectroscopy thesis.

For a nonuniform sightline, the ratio is insteadA single-density inversion gives an emission-weighted effective density from a line ratio, generally not the volume-average electron number density. The density estimate is meaningful within its temperature, optical-depth, line-blending and spatial-uniformity assumptions. This is the precise interpretation of an average density obtained from the observed pair.