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Retain the preceding isothermal assumption, so the gas temperature remains $T_0$. When two neutral hydrogen atoms become one molecular hydrogen molecule, the mean particle mass doubles. At fixed mass density and temperature the ideal gas law would halve the pressure, but the flowing gas is free to compress and change speed. It cannot in general keep density, pressure, and velocity all unchanged.

To describe a thin steady transition, label the atomic and molecular sides by $1$ and $2$ and neglect the integrated external force and radiative recoil across its small thickness. Local mass conservation and momentum conservation requireThe isothermal sound speed changes from $a_1^2=k_B{T}_0/m_H$ to $a_2^2=a_1^2/2$. Dividing the momentum flux by $j$ gives $u_2+a_2^2/u_2=J$, with $J=u_1+a_1^2/u_1$. The subsonic continuation is thereforeThus the gas becomes denser and slower. In the well-subsonic limit, where ram stresses are small compared with thermal pressure, the useful leading description of isothermal flow through molecular hydrogen formation isThe exact momentum condition gives a small pressure increase at low Mach number, rather than an imposed discontinuous factor-of-two pressure drop. The full expressions above apply when the subsonic speed is not asymptotically small.

Radiation of the molecular binding energy prevents that released energy from automatically heating the gas. It does not, by itself, determine the temperature: the constant-temperature result uses the inherited isothermal closure. If that assumption is relaxed, an energy equation including cooling and molecular degrees of freedom is needed; the wording about emitted binding energy alone does not uniquely determine all four changes.

Write the initial mass density as $\rho =mn$. The ideal gas law gives pressure $p=n{k}_{B}T$, and the sphere has $M=4\pi \rho R^3/3$. Here $k_B$ is the Boltzmann constant and $G$ is the gravitational constant. First obtain the gravitational potential energy by adding concentric shells. The mass inside radius $r$ is $M(r)=M{r}^3/R^3$, soIn a small contraction $dR<0$ at fixed total mass, the gravitational energy released is $-dW=3G{M}^2|dR|/(5R^2)$. The positive work required against the gas pressure is $-pdV=4\pi R^{2}p|dR|$. Equating these givesUse $R=(3M/(4\pi mn){)}^{1/3}$ to eliminate the radius. The uniform-sphere pressure-work collapse criterion isThe units are kilograms: $(k_{B}T/(Gm){)}^{3/2}$ has units $\mathrm{k}{\mathrm{g}}^{3/2}{\mathrm{m}}^{-3/2}$, and ${\rho }^{-1/2}$ supplies $\mathrm{k}{\mathrm{g}}^{-1/2}{\mathrm{m}}^{3/2}$. At fixed initial $n$ and $T$, $M>M_{\mathrm{crit}}$ makes gravity stronger than the pressure-work requirement in this model, allowing contraction.

This is a Jeans mass estimate for homologous contraction of an initially uniform sphere. It is not an exact stability limit for a hydrostatic gas sphere: uniform mass density and uniform temperature would give uniform pressure, rather than the gradient required by hydrostatic equilibrium. The numerical coefficient therefore differs from conventions based on a Jeans instability wavelength or a pressure-confined isothermal sphere. Subsequent contraction also depends on heat exchange; isothermal pressure support during gravitational collapse weakens relative to gravity as the radius decreases.

The exponential factor removes the first spatial derivative. Apply the chain rule explicitly:while ${\partial }_t^2\Delta \rho =e^{-z/(2H)}{\Psi }_{tt}$. Substituting into the supplied density equation, the terms proportional to ${\Psi }_z$ cancel, leavingThis is the atmospheric acoustic cutoff transformation. There is a sign error in the target equation in the original PDF, as well as in the transcription: the last term must be $+{\omega }_0^2\Psi$ for real positive ${\omega }_0$. If the displayed minus sign is retained literally, coefficient matching would require ${\omega }_0^2=-c^2/(4H^2)$, so it would not define a real characteristic oscillation period.

For the corrected equation, a plane wave $\Psi \propto e^{i(kz-\omega t)}$ has dispersion relationThus ${\omega }_0$ is the acoustic cutoff frequency: vertically propagating acoustic waves have $|\omega |\ge {\omega }_0$. The scale height and sound speed have units metres and metres per second, respectively, so ${\omega }_0$ has units ${\mathrm{s}}^{-1}$.

For an order-of-magnitude estimate in the lower terrestrial atmosphere, take $H\simeq 8.0\mathrm{km}$ and $c\simeq 340\mathrm{m}{\mathrm{s}}^{-1}$. These are representative atmospheric values, not additional data specified by the paper. They giveThe isothermal atmosphere approximation is local: a real atmospheric temperature profile and the thermodynamic response of the disturbance modify the precise cutoff and period.

Hydrostatic equilibrium with self-gravity. A resting fluid obeys $\nabla p=-\rho \nabla \Psi$. Spherical symmetry and the Poisson equation for Newtonian gravity giveEliminate the Newtonian gravitational potential derivative to obtainEquivalently, the shell theorem gives ${\Psi }^{\prime }=GM(r)/r^2$ and $M^{\prime }(r)=4\pi r^2\rho$, which leads to the same hydrostatic equilibrium equation.

A uniform-density liquid. With constant $\rho$, $M(r)=4\pi \rho r^3/3$. There is no central point mass, so the interior gravitational acceleration magnitude is $4\pi G\rho r/3$. ThusUse $p(R)=0$ at a vacuum boundary, neglecting surface tension. The uniform-density planetary pressure profile isThe central value can also be writtenIt has units pascals and decreases smoothly from its maximum at the centre to zero at the surface.

An isothermal gas. Here $c$ is the constant isothermal sound speed, with $p=c^2\rho$. Try $p=A/r^2$, so $\rho =A/(c^2{r}^2)$. The two sides of the equilibrium equation becomeThus $A=c^4/(2\pi G)$, giving the singular isothermal sphere,This does not describe a finite gaseous sphere in vacuum. Both pressure and mass density diverge at the centre, and $p(R)>0$ for every finite outer radius $R$, so it cannot meet $p=0$ there. Extending it to infinity removes the finite-radius pressure jump but makes the total mass infinite. It is useful as an idealized profile over an intermediate radial range, or as a truncated model with nonzero external pressure. The vacuum boundary obstruction for an isothermal sphere also applies to the regular, nonsingular isothermal solution; removing the central singularity does not produce a finite vacuum surface.

Steady outflow. In the wind calculation, take the central mass $M$ to be fixed and neglect the wind's own self-gravity. Constant mass flow rate and the isothermal equation of state giveThe steady radial Euler equation isSubstituting the density derivative givesThis is the isothermal Parker wind equation. A smooth sonic point requires both sides to vanish when $u=c$, soMaintaining constant temperature during expansion generally requires heat input; it is an isothermal model, not the adiabatic blast-wave model of Question 5.

The escaping branch at large radius. Integrate the wind equation, using dimensionless logarithm arguments:Set $x=r/r_s$ and $w=u^2/c^2$. ThenFor the accelerating transonic branch, $w(1)=1$, which fixes $C_0=-3$. At $x\gg 1$ on its supersonic branch, $w\to \infty$ and $\ln w$ is smaller than $w$. ConsequentlyA first correction makes the ordering explicit:This is the large-radius asymptotics of an isothermal Parker wind. The leading expression also holds for other supersonic branches with finite $C_0$.

The source implicitly selects an escaping wind. Positive steady outward flow alone does not determine that branch: a subsonic isothermal breeze also satisfies the differential equation. Its $w\ll 1$ limit gives $w\sim e^{-C_0}{x}^{-4}$, hence $u\propto r^{-2}$ and $\rho \to {\rho }_{\infty }>0$. Such a breeze needs nonzero pressure at infinity. Selecting a regular outflow from a subsonic base into vanishing ambient pressure instead gives the accelerating transonic wind used above.

The asymptotic speed has the same units as $c$; the logarithm is dimensionless. Its unlimited growth is a formal feature of indefinite isothermal heating and a prescribed point-mass potential. Real heating, surrounding pressure, and eventually the neglected wind mass limit that extrapolation.

Write $R=C{E}^a{\rho }_0^b{t}^d$ for a uniform ambient mass density. Equating mass, length and time dimensions gives $a+b=0$, $2a-3b=1$ and $-2a+d=0$. Thus the Sedov-Taylor blast wave hasThe dimensionless coefficient depends on the gas's adiabatic index and on whether $R$ denotes the shock radius. Solving the similarity profile for a monatomic ideal gas gives $\xi \simeq 1.15$, so taking it as unity is an order-of-magnitude approximation rather than an extra dimensional result.

Timed photographs of the 1945 atomic explosion provide a direct scale test: the measured fireball radius follows $R\propto t^{2/5}$ over the relevant blast-wave interval, and inserting the independently established explosion energy and atmospheric density gives a coefficient of order unity. The published photographic analysis tested the radius-time law and used its normalization to infer an explosion energy of the correct nuclear-test scale. This illustrates both the success and the limits of dimensional reasoning: the power law is fixed dimensionally, whereas the numerical coefficient requires a profile calculation or measured normalization.^[ref]

Neglecting the exterior temperature requires a strong shock, ${\mathcal{M}}_s=V_s/c_{s,u}\gg 1$, or ${\rho }_0{V}_s^2\gg p_u$. The swept-up mass must dominate the initial ejecta for this energy-conserving similarity stage, while radiative losses, gravity, ambient gradients and continuing energy input must be unimportant. A very hot exterior or a late weak disturbance introduces a sound-speed scale and invalidates the cold-upstream similarity assumptions.

Let the mean particle mass be $\bar{m}=\mu m_H$, with $n_u$ counting total particles so that ${\rho }_u=n_u\bar{m}$. The ideal-gas law and the strong-shock pressure give the immediate post-shock temperatureThe robust prediction is $T_d\propto t^{-6/5}$. For a transparent numerical convention take $\mu =1$, $n_u=10^6{\mathrm{m}}^{-3}$ and $\xi =1$. ThenFor an ionized mixture with $\mu =0.6$ under the same total-particle-density convention, these normalizations become $5.13$ pc and $5.49\times 10^7$ K. The paper does not specify composition or whether its quoted density instead counts hydrogen nuclei; that convention must be stated, not hidden in a precise age.

The ambient-temperature limit of an adiabatic blast wave gives a formal matching estimate, $T_d\sim 10^4$ K, givesWith $\mu =0.6$ the age is about $1.3\times 10^6$ yr and the radius stays about 90 pc. This matching criterion already lies outside a very strong shock: $T_d=T_u$ in the strong-shock formula corresponds to ${\mathcal{M}}_s=\sqrt{16/(3\gamma )}\simeq 1.79$. A separate sonic estimate $V_s\sim c_{s,u}$ gives about $4.4\times 10^6$ yr and 133 pc for $\mu =1$; neither extrapolation is a precise transition solution. Using $\xi =1.15$ increases these inferred matching ages and radii by factors ${\xi }^{5/3}$.

In a real supernova remnant, substantial radiation losses commonly end the energy-conserving phase before these formal pressure-merging limits. The relevant comparison is $t_{\mathrm{cool}}\sim (3/2)n{k}_{B}T/(n_e{n}_i\Lambda (T))$ against the expansion time. A cooling function, composition and ionization state are needed to calculate that earlier age and radius; they are not supplied. The numerical values above answer the ambient-temperature breakdown estimate under explicit assumptions and must not be presented as a uniquely determined radiative shell-formation age.

After cooling, an interior-pressure-driven shell can approximately follow $R\propto t^{2/7}$ while pressure remains important. In the limiting momentum-conserving stage, $M_s\dot{R}\simeq \mathrm{const}$ and $M_s\propto {\rho }_u{R}^3$ give $\boxed{R\propto t^{1/4}}$. Ultimately ambient pressure and turbulent motions make the disturbance merge with the ISM. These slower stages respect the reduced energy budget rather than continuing an adiabatic strong-shock law indefinitely.

Stellar radiation and nebular ionization. The planetary-nebula spectral energy distribution is constrained by a shared radiation and thermal balance. The central star's effective temperature, luminosity and evolutionary state set its continuum and ionizing-photon supply. A hotter star shifts the ionizing spectrum and allows higher ionization stages; the total recombination output also depends on how many ionizing photons the gas actually absorbs. In an ionization-bounded nebula, approximate photoionization balance iswhereas a density-bounded nebula can leak photons. Dust absorption of ionizing photons changes this relation further. Stellar-atmosphere and wind effects make a blackbody an approximation, especially in the extreme ultraviolet. A changing core temperature and expanding envelope explain why nebular spectra evolve.

Nebular continua and recombination lines. Their strength depends on the emission measure, roughly $\int n_e{n}_{i}dV$, electron temperature, ionic abundances and radiative branching probabilities. For a recombination line,Case-B line ratios assume appropriate trapping of Lyman-series photons and escape of the lower-series radiation. Density, optical-depth effects and extinction can alter observed ratios. The free-free emissivity has the schematic dependence $j_{\nu }^{\mathrm{ff}}\propto n_e{n}_i{T}_e^{-1/2}\exp [-h\nu /(k_B{T}_e)]g_{\mathrm{ff}}$; the nebular free-bound continuum also carries ionization thresholds and recombination cross-sections. Two-photon emission depends on the population of the hydrogen $2s$ state and can be reduced by collisional mixing into $2p$. Density clumping matters because these rates are weighted approximately by density squared, not simply by total gas mass.

Collisionally excited lines. Below a line's critical density of an atomic transition,Elemental abundance, ionic fraction and electron temperature therefore all enter. The ionizing spectrum and local ionization parameter help determine which ion is present. Above the critical density of an atomic transition, collisional de-excitation competes with radiative decay and can suppress a forbidden line. Low-energy infrared fine-structure lines are generally less exponentially temperature sensitive than high-excitation optical/ultraviolet lines. Metal-line cooling also helps set the temperature, so changing an abundance changes both the number of emitters and the thermal balance; line luminosity need not be proportional to abundance at fixed everything else.

Dust and molecules. Nebular dust spectral features identify the carriers as well as contributing to the integrated luminosity. In an optically thin dust component,The grain mass sets its normalization, while the radiation field, stellar distance, grain size and absorption efficiency determine its temperature distribution. For approximately grey heating, radiative equilibrium suggests $T_d\propto L_{\ast }^{1/4}{r}^{-1/2}$; non-grey grains change this scaling. Long-wavelength opacity is often approximated as ${\kappa }_{\nu }\propto {\nu }^{{\beta }_d}$. Small aromatic grains undergo transient heating, so one equilibrium temperature is not adequate for every infrared feature. Carbon-to-oxygen chemistry, grain processing and shielding affect aromatic versus silicate bands. Molecular line emission further depends on molecule abundance, collisional density, kinetic temperature, ultraviolet pumping and shocks, while shielding and dissociation set whether molecules survive at all.

Finally, interstellar extinction preferentially suppresses ultraviolet and optical light, and dust reradiates some absorbed energy in the infrared. Aperture, angular resolution and distance must be treated consistently before comparing stellar, nebular and dust components. These dependencies make the combined spectrum an energy and abundance diagnostic, but also prevent a unique reconstruction from a single broad-band colour.

A planetary nebula combines a hot post-asymptotic-giant-branch stellar core, its ionized expelled envelope, neutral/molecular material and interstellar dust. Its ultraviolet-to-infrared spectral energy distribution is therefore not a single-temperature blackbody. The principal components are:

The observed mixture also contains attenuation by internal and foreground interstellar extinction. A spectrum containing all these components need not spatially coincide: stellar light, ionized gas, dusty clumps and the molecular envelope sample different regions.

Cold molecular hydrogen is difficult to observe directly: its homonuclear symmetry forbids electric-dipole rotational transitions, and its lowest useful quadrupole rotational excitation is high compared with a cold cloud's temperature. Carbon monoxide is relatively abundant, has a permanent dipole moment and has low-lying rotational transitions that can be collisionally excited in cold gas. The low-$J$ lines lie in accessible millimetre bands and suffer much less dust extinction than visible light. They therefore reveal molecular-gas velocities and spatial structure efficiently.

The advantages do not turn CO into a direct mass counter without a model. Common-isotope lines can be optically thick; excitation can be subthermal or have multiple temperatures. Abundance changes with metallicity, chemistry, shielding and photodissociation, and CO can freeze onto grains. Molecular hydrogen can also survive where CO is weak, giving CO-dark gas. A calibrated carbon-monoxide tracer of molecular hydrogen conversion may be useful for appropriate cloud populations but should not be transplanted uncritically to newly formed supernova ejecta. The line calculations here infer CO mass, not total molecular mass unless a CO-to-hydrogen abundance is separately known.

For optically thin CO rotational thermometry of the same emitting population, the upper-level abundance ratio isThe difference $E_3-E_1$ can be obtained directly from the two successive transition frequencies, $h({\nu }_{32}+{\nu }_{21})$, or approximately from $10k_B{T}_r$. Using the supplied rounded molecular data gives $(E_3-E_1)/k_B=27.66$ K. ThereforeThe observed ratio is $2.46\times 10^{26}/(1.33\times 10^{24})=184.96$, henceAt 20 K the predicted ratio would be only about 61, so the two detections reject that single-temperature optically thin 20-K model. The inferred value is an excitation temperature under the stated assumptions, not a model-independent measurement of the gas's kinetic temperature.

Raising the temperature populates higher rotational states and reduces the fraction in $J=1$. At $T\simeq 101$ K, the rotational partition function gives $f_1\simeq 0.0768$, so the same low-$J$ luminosity requiresabout $3.9$ times the 20-K estimate. Applying the independently computed $J=3$ fraction to the measured high-$J$ line gives the same mass within the model, providing a consistency check rather than an independent proof of its assumptions.

Optical depth, density-dependent excitation, multiple temperature components, beam dilution, aperture mismatch, line blending and flux calibration all need investigation. For one optically thick isothermal source with equal emitting area and velocity width, the Rayleigh-Jeans limit instead gives line-energy luminosity approximately proportional to ${\nu }^3$, so the 3–2/1–0 ratio is of order 27, far below 185. That illustrates why optical-depth and emitting-area assumptions cannot be ignored; different source areas or excitation structures can change this simple comparison. Optically thick clumps can hide mass, and a high-temperature component may dominate the 3–2 line while colder material contributes much of the 1–0 emission.

Additional rotational lines, especially the available 2–1 transition and higher-$J$ lines, test a one-temperature excitation diagram. Less abundant isotopologues help constrain opacity; resolved imaging and line profiles constrain emitting volume and homologous expansion, using the remnant age. Density information permits non-LTE statistical-equilibrium/radiative-transfer modelling. Dust emission can provide a complementary mass constraint with its own opacity and dust-to-gas uncertainties. A total gas mass additionally needs a justified CO abundance, accounting for molecular formation, destruction and unobserved material.

Inverse Compton scattering transfers energy from fast electrons to lower-energy photons. In the Thomson scattering limit the scattering is approximately elastic in the electron's instantaneous rest frame, but the transformations into and out of that frame can greatly increase the photon energy in the original frame.

In a blazar, relativistic jet electrons can scatter their own synchrotron radiation, producing synchrotron self-Compton radiation, or scatter photons from an accretion disk and its surrounding reprocessing material. The two spectral humps constrain electron Lorentz factors, the electron distribution and, with assumptions about geometry and relativistic beaming, the magnetic field. For the same electron population in the Thomson regime, the ratio of inverse-Compton to synchrotron power scales as $u_{\mathrm{ph}}/u_B$ in the plasma rest frame. It does not determine the field without information about the photon energy density and beaming.

In a galaxy cluster, hot thermal electrons scatter cosmic microwave background photons and slightly raise their mean energy, producing the thermal Sunyaev-Zeldovich effect. The spectral distortion measures the line-of-sight integral of electron pressure, approximately $y={\sigma }_T\int n_e{k}_B{T}_{e}d\ell /(m_e{c}^2)$ for nonrelativistic thermal electrons. Combined with X-ray emission, which weights density approximately quadratically, it constrains the gas density, temperature and mass distribution. These are distinct source populations and seed photon fields from the blazar example.

For the power derivation, use an isotropic incident radiation field of total photon energy density $u_{\mathrm{ph}}$ in the laboratory frame. Its stress-energy tensor has $T^{00}=u_{\mathrm{ph}}$ and $T^{zz}=u_{\mathrm{ph}}/3$, with zero mean flux. Boosting to an electron moving at $\beta c$ along $z$ givesThe radiation energy flux in this rest frame is $S_z^{\prime }=c{T}^{\prime 0z}$. Elastic Thomson scattering exerts the drag $F_z^{\prime }={\sigma }_T{S}_z^{\prime }/c$. The parallel three-force has the same value in the laboratory frame, so the electron's positive energy loss rate isThis isotropic inverse-Compton drag assumes photon energies small compared with $m_e{c}^2$ in the electron rest frame, negligible recoil and an isotropic seed field in the specified frame. An anisotropic photon field requires its directional moments; the recoil-dependent differential scattering cross-section of Compton scattering becomes necessary at larger rest-frame energies.

There are two conventions to qualify in the printed expression. First, the density multiplying $4/3$ must be $u_{\mathrm{ph}}$, not literally only the magnetic part of a propagating electromagnetic field. That field has equal electric and magnetic energy densities, so $u_{\mathrm{ph}}=2u_{\mathrm{mag}}$ and the corresponding formula is $P_{\mathrm{IC}}=(8/3){\sigma }_{T}c{u}_{\mathrm{mag}}{\beta }^2{\gamma }^2$. Second, this is net electron energy loss, rather than all incident plus scattered photon energy. At $\beta =0$ an electron still scatters power ${\sigma }_{T}c{u}_{\mathrm{ph}}$, but transfers no kinetic energy to the photons, so the inverse-Compton gain is zero.

For an individual incident photon of energy $\epsilon$, the electron-frame energy is ${\epsilon }^{\prime }=\gamma \epsilon (1-\beta \cos \theta )$. For a head-on photon followed by forward emission in the laboratory,Averaging an isotropic seed field gives $⟨{\epsilon }_{\mathrm{out}}⟩\simeq (4/3){\gamma }^2\epsilon$ for ultrarelativistic electrons. The original plot has an inverse-Compton maximum of order $10^8\mathrm{eV}$, a synchrotron maximum around $10^{-1}\mathrm{eV}$ and a disk component around $1\mathrm{eV}$. ThusThe head-on maximum gives the same order of magnitude, about $5\times 10^3$ to $1.6\times 10^4$. These are approximate readings of a broad spectrum, not precise fitted parameters. Disk and synchrotron seed photons need not be scattered by the same part of the electron distribution, and bulk jet Doppler factors also enter the observed peak relation. The indicated rest-frame seed energies remain well below the electron rest energy in this estimate.

For an optically thin collisionally excited line, the volume emissivity is approximately $j_{ul}=n_{e}n(X^i)q_{lu}(T_e)h{\nu }_{ul}$ below the transition's critical density. The excitation coefficient includes a factor $T_e^{-1/2}\exp [-\Delta E/(k_B{T}_e)]$, so temperature is needed to convert luminosity into an ionic abundance. Density is needed both to assess collisional quenching of a metastable line and to convert an emission measure into a gas mass. For a uniform recombining ion, for example, $M_i=m_i{L}_{\mathrm{line}}/[n_e{\alpha }_{\mathrm{eff}}h\nu ]$. Temperature structure, density clumping, extinction and unobserved ionization stages introduce further corrections.

A useful density doublet has closely spaced upper energies but different spontaneous decay rates and hence different critical densities. Nearly equal excitation energies reduce temperature dependence of the ratio; similar wavelengths reduce differential extinction and relative calibration errors. Metastable upper levels have sufficiently small Einstein coefficients for collisional de-excitation to compete at nebular densities.

Use statistical weights $g_2=6$, $g_3=4$. The source's O II ground-state term has a typographical error: it is $2s^{2}2p^3{}^4{S}_{3/2}^{\mathrm{o}}$, not ${}^2{S}_{3/2}$. A ${}^{2}S$ term has $J=1/2$, so the printed term is internally inconsistent. The tabulated $J$ values, wavelengths and rates can still be used.^[ref]

At high density, collisions establish a Boltzmann distribution between the upper states. Their energy splitting is small compared with $k_B{T}_e$, so $n_2/n_3\simeq g_2/g_3$. If both lines escape without absorption,The exact ratio includes $\exp [-(E_2-E_3)/(k_B{T}_e)]$, an approximately $0.2\%$ correction at $10^4\mathrm{K}$.

At low density, assume excitation from the common ground state and radiative decay back to it dominate, with no substantial collisional transfer between upper states. Statistical equilibrium gives $n_j{A}_{j1}=n_1{n}_e{q}_{1j}$. Detailed balance relates excitation to the supplied de-excitation coefficient, yieldingEach excitation then produces one line photon, so the spontaneous rate cancels:At high density the upper populations are collisionally thermalized and the ratio retains the radiative lifetimes; at low density radiative escape is nearly certain and the ratio is set by the excitation probabilities instead.

At $T_e=10^4\mathrm{K}$, the two de-excitation coefficients are $C_{21}=1.161\times 10^{-14}$ and $C_{31}=1.1825\times 10^{-14}{\mathrm{m}}^3{\mathrm{s}}^{-1}$. ThusThe useful density range lies between and around these scales. In the independent-upper-level approximation,The density sensitivity of a metastable doublet depends on the error measure. The fractional response $|d\ln R/d\ln n_e|$ is maximal atIf absolute ratio errors are constant, maximizing $|dR/d\ln n_e|$ instead gives $n_e=n_{c,2}$. An unspecified experimental noise model therefore does not define a unique sharper optimum than this transition-density range.

For Ne IV, the same calculation gives critical densities $4.23\times 10^{10}$ and $3.93\times 10^{11}{\mathrm{m}}^{-3}$, with fractional sensitivity peaking near $1.29\times 10^{11}{\mathrm{m}}^{-3}$. This is a useful higher-density diagnostic when O II is saturated. However, Ne${}^{3+}$ and O${}^{+}$ need not occupy the same ionization zone, so their inferred densities need not describe the same gas. The ultraviolet Ne IV pair also needs suitable observations and extinction correction. The proposal is useful with these qualifications, not a guaranteed replacement for the density of the O II emitting zone.

The density contrast is $\delta (\mathbf{x},t)=[\rho (\mathbf{x},t)-{\rho }_0(t)]/{\rho }_0(t)$, and ${\delta }_k(t)$ is one spatial Fourier mode with comoving wavevector $\mathbf{k}$. The scale factor is $a(t)$, $H=\dot{a}/a$ is the Hubble parameter, $c_s$ is the physical sound speed, ${\rho }_0$ the mean matter density, and $G$ the gravitational constant. For each fixed $k$ the displayed equation is an ordinary differential equation in cosmic time. It compares pressure restoration, self-gravity and expansion-induced damping:The pressure term resists compression; self-gravity enhances an overdensity; the $2H\dot{\delta }$ term slows the motion relative to a static background. The Newtonian treatment is appropriate to small perturbations on subhorizon scales within the specified fluid model, rather than an arbitrary gauge-independent treatment of superhorizon perturbations.

The physical Jeans length follows by equating pressure and gravity:The comoving Jeans wavenumber is $a{k}_{J,\mathrm{phys}}$. Perturbations well below the physical Jeans length are pressure-supported acoustic oscillations whose amplitude is modified by expansion; those above it admit gravitational growth when the background evolution permits. Pressureless matter in a matter-dominated universe is the example below. Merely having a large wavelength is not sufficient to apply this Newtonian equation outside the horizon.

For a gas of particle mass $m$ and temperature $T$, write $c_s^2=\Gamma k_{B}T/m$, with $\Gamma =1$ for the isothermal sound-speed convention or the appropriate adiabatic heat-capacity ratio otherwise. ThenAt fixed density it is larger for hotter or lighter particles, and it decreases with increasing density. Numerical coefficients differ between sound-crossing and self-gravitating-sphere conventions; the expression here uses the Fourier instability boundary of the stated equation.

In an Einstein-de Sitter universe, the spatially flat Friedmann-Lemaître-Robertson-Walker metric is $d{s}^2=-c^{2}d{t}^2+a^2(t)d{\mathbf{x}}^2$. Conservation of pressureless matter gives ${\rho }_0{a}^3=\mathrm{constant}$; the Friedmann equation then gives $a\propto t^{2/3}$, $H=2/(3t)$ and ${\rho }_0=1/(6\pi G{t}^2)$. With $c_s=0$,Trying $\delta =t^p$ gives $p^2+p/3-2/3=(p-2/3)(p+1)=0$. Hence the Einstein-de Sitter density-growth modes areThe growing mode increases only as a power law because the expansion dilutes and slows the collapsing matter.

Conformal time is defined by $d\eta =dt/a(t)$, so the metric becomes $d{s}^2=a^2(-c^{2}d{\eta }^2+d{\mathbf{x}}^2)$. Choosing its origin at the big bang gives $\eta \propto t^{1/3}$ and $a\propto {\eta }^2$. ThereforeEquivalently the equation is ${\delta }^{\prime \prime }+2{\delta }^{\prime }/\eta -6\delta /{\eta }^2=0$, where primes denote conformal-time derivatives.

Before and through cosmological recombination, photons diffuse between hotter and cooler parts of the tightly coupled photon–baryon fluid. Radiative viscosity and heat transport erase short-wavelength acoustic perturbations. As free electrons disappear, the photon mean free path grows, enhancing this diffusion before decoupling. Silk damping can be represented by a factor approximately $e^{-k^2/k_D^2}$ in the linear acoustic amplitude. The comoving Silk wavelength may be defined as ${\lambda }_D=2\pi /k_D$; the associated diffusion-length convention is $r_D=1/k_D$.

A random-walk estimate is $r_D^2\sim \int c{\lambda }_{\gamma }dt/a^2$ up to transport and baryon-loading factors, with physical ${\lambda }_{\gamma }=(n_e{\sigma }_T{)}^{-1}$. Scales much shorter than the damping wavelength are strongly suppressed. This is photon diffusion, not the Jeans boundary and not erasure of every cold-dark-matter perturbation. After decoupling, photon free streaming is a distinct part of the evolution.

An H II region is photoionized interstellar gas around young massive stars. Its optical spectrum contains hydrogen and helium recombination lines, collisionally excited forbidden metal lines, and weaker free-free, free-bound and two-photon continua. Typical hot main-sequence ionizing stars have effective temperatures of a few $10^4\mathrm{K}$; their ultraviolet output, age and metallicity determine the ionization balance. Gas and dust can also emit strongly outside the visible band.

A planetary nebula is expelled evolved-star material illuminated by a much hotter exposed stellar core. Its central star can reach temperatures of order $10^5\mathrm{K}$, allowing higher ionization stages and prominent He II emission. The gas may carry nucleosynthetic enrichment from its progenitor, while expansion lowers density and changes line strengths with age. Both object classes produce nebular spectra, but their ionizing sources and evolutionary contexts differ; neither spectrum is simply the blackbody spectrum of the exciting star.

For an optically thin recombination line under Case B recombination, the line emissivity isThe effective coefficient includes recombination cascades feeding the line. For approximately common conditions and matching sampled volumes,The wavelength factor is essential because the tabulated energy flux counts photons weighted by their energies. More generally the result is an emissivity-weighted ionic ratio, not an unweighted pointwise abundance.

Assume corrected extinction and underlying stellar absorption, suitable electron temperature and density, negligible additional collisional or optical-depth effects on the chosen lines, and essentially ionized hydrogen. The total helium abundance is thenprovided neutral helium is negligible or an ionization correction is applied. A line emitted by neutral He after recombination measures the recombining He${}^{+}$ population; He II $4686$ measures recombining He${}^{++}$.

Using the supplied line coefficients and fluxes givesThereforeIf hydrogen and helium dominate the mass, this corresponds to helium mass fraction $Y\simeq 4y/(1+4y)\simeq 0.379$, with $y=He/H$. It is an approximate elemental estimate under the ionization assumptions, not merely a ratio of two observed line fluxes.

An ordinary H II region would typically have $He/H\sim 0.08$–$0.10$ by number, with metallicity-dependent enrichment, rather than necessarily the larger value inferred here for an evolved-star nebula. Hydrogen ionization requires $13.6\mathrm{eV}$, neutral helium $24.6\mathrm{eV}$ and He${}^{+}$ $54.4\mathrm{eV}$. Typical H II region stars often produce little He${}^{++}$, so He II $4686$ can be extremely weak or absent despite substantial helium. He I $4471$ relative to H$\beta$ can still be useful, but neutral helium inside the H${}^{+}$ zone may require an ionization correction. A very hot planetary-nebula core is more favorable for observing both ionic stages, although ionization structure must still be checked.

A shock wave is a thin, irreversible compression layer across which fluid density, pressure, temperature and velocity change rapidly, while conserved fluxes satisfy the Rankine-Hugoniot conditions. In the shock frame, a compressive hydrodynamic shock converts ordered upstream kinetic energy into downstream thermal energy and increases entropy. The upstream normal flow is supersonic; the downstream normal flow is subsonic for an ordinary gas-dynamic compression shock.

In an accreting binary star, a supersonic stellar wind can be captured by a compact companion and shocked. An accretion stream striking an accretion disk, an accretion-disk boundary layer, and a magnetically guided accretion column provide other sites where infall energy is dissipated and high-energy radiation is generated. Shocks heat and decelerate material; a disk also needs transport of angular momentum before its material can reach the compact star.

In diffusive shock acceleration, charged particles scatter from magnetic irregularities on both sides of a shock and repeatedly cross the converging flows. The average energy gain is first order in the velocity jump divided by the particle speed. Escape competes with acceleration, producing a nonthermal distribution rather than a single downstream temperature. Supernova remnants are important candidate sources of Galactic cosmic rays; shocks associated with active galactic nuclei, their jets and lobes, and gamma-ray bursts are candidate extragalactic accelerators. Ultrarelativistic energies require sufficient size, magnetic field and lifetime, and losses must not outpace acceleration. In particular a relativistic particle's Larmor radius must fit within the accelerator; the existence of a strong shock alone does not guarantee unlimited particle energy.

A massive main-sequence donor loses material in a stellar wind. Gravitational capture by the neutron star can produce a nearly radial, supersonic infall when the captured material has little specific angular momentum. A magnetic field may channel the final infall toward polar accretion columns, where a standing shock decelerates the flow. Large angular momentum instead favours an accretion disk; strictly radial accretion is an idealization.

For mean particle mass $\mu$, the ideal gas law is $p_{\mathrm{d}}={\rho }_{\mathrm{d}}{k}_{\mathrm{B}}{T}_{\mathrm{d}}/\mu$. Combining it with b givesFor Newtonian free fall from a distance large compared with the stellar radius $R_{\ast }$, $v_{\mathrm{u}}^2\simeq 2G{M}_{\ast }/R_{\ast }$. ThereforeTODO: a unique numerical temperature and characteristic photon band require source data not supplied here. The held question gives $M_{\ast }=1.4M_{\odot }$ but gives neither $R_{\ast }$ nor a composition fixing $\mu$. The mass alone does not fix the free-fall speed or the temperature. For example, doubling $R_{\ast }$ at the same mass and composition halves both quantities in the last box.

A normalization, rather than an additional claimed datum, makes the scale explicit:Here $k_{\mathrm{B}}T$ is the characteristic particle energy; the mean translational energy of a nonrelativistic thermal particle is $3k_{\mathrm{B}}T/2$. If one additionally chose $R_{\ast }=10\mathrm{km}$ and $\mu =m_p$, this ideal one-temperature estimate would correspond to gamma-ray-scale thermal photon energies. Those are conditional choices, not information recovered from the source. Real accreting neutron stars commonly emit X-rays: electron-ion disequilibrium, radiative cooling, radiation pressure, reprocessing and relativistic infall invalidate a direct identification of this ideal ion-heating estimate with the escaping photon spectrum.

A spontaneous emission transition has probability per unit time $A_{ji}$ and emits a photon of energy $h{\nu }_{ji}$. For an optically thin gas, its angle-integrated emissivity is $j_{ji}=n_j{A}_{ji}h{\nu }_{ji}$. Electron collisions populate excited levels at rates $n_e{n}_i{q}_{ij}$; they also remove excitation collisionally at rates $n_e{n}_j{q}_{ji}$, returning the energy to the gas rather than producing the line photon. A collisionally excited line is usually emitted by a subsequent spontaneous radiative decay; collisional de-excitation instead quenches it.