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To derive the nonrelativistic electron-degeneracy pressure in a hydrogen-helium mixture, take fully ionized hydrogen and helium with hydrogen mass fraction $X$, the electron number density isHere helium nuclei have mass $4m_p$ in the approximation used in the question. Filling the two spin states of each momentum mode up to the Fermi momentum $p_0$ givesThe isotropic momentum flux gives pressure $\frac{1}{3}\int pvdn$. For nonrelativistic electrons $v=p/m_e$, so the electron degeneracy pressure isSubstitute the composition-dependent $n_e$ and simplify powers of two:Comparing $5/3=1+1/n$ gives the polytropic index $n=3/2$. The result assumes complete electron degeneracy, $p_0\ll m_{e}c$, and negligible nuclear mass corrections.

Now adopt the stated approximate partially degenerate equation of state, keeping composition and hence $K,\mu$ fixed during contraction. Set $q=m/M$ and define the mass-averaged temperature ${\overline{T}}_M={\int }_0^{1}Tdq$. The stellar virial theorem from part (a) givesFor an index-$3/2$ stellar polytrope, the permitted gravitational-energy formula is $\Omega =-6G{M}^2/(7R)$. Also, using $\bar{\rho }=3M/(4\pi R^3)$,Thus, with $A=3K{\int }_0^1[3\rho /(4\pi \bar{\rho }){]}^{2/3}dq$,The normalized density $\rho /\bar{\rho }$ is a fixed function of $q$ for this homologous polytropic structure. Therefore its integral and $K$ fix $A$, which is independent of $M$ and $R$ for a common composition. The mean here is mass-weighted, rather than a volume average.

For the mass-averaged temperature maximum in a partially degenerate polytrope, put $\alpha =2\mu GM/(7\mathcal{R})$ and $\delta =\mu A{M}^{2/3}/(3\mathcal{R})$, both positive. As a function of $s=1/R$ the temperature is the concave quadraticIt reaches its maximum when $R=2\delta /\alpha =7A{M}^{-1/3}/(3G)$, withAt large radius, contraction increases the temperature as in a thermally supported star. Once electron degeneracy provides a sufficiently large fraction of the support, further contraction can lower the thermal temperature. The formal zero-temperature radius is $R_{\mathrm{deg}}=\delta /\alpha =R_{\max }/2$, with $R_{\mathrm{deg}}\propto M^{-1/3}$. The continuation to negative temperatures would be unphysical; the adopted sequence only describes its positive-temperature part.

The contraction-limited central temperature has the same mass scaling. Indeed, if the total polytropic pressure is $P=K_{\mathrm{tot}}{\rho }^{5/3}$, the approximate equation of state gives $T=(\mu /\mathcal{R})(K_{\mathrm{tot}}-K){\rho }^{2/3}$. The fixed density profile therefore fixes $T_c/{\overline{T}}_M$. A sufficiently small mass never reaches the central temperature required for sustained hydrogen burning, producing a hydrogen-burning minimum mass. Such an object becomes a cooling, degeneracy-supported brown dwarf instead of settling onto the main sequence. More massive contracting objects can ignite hydrogen before reaching this degeneracy-limited maximum. This conclusion uses the nonrelativistic, fixed-composition model; it does not require extrapolation into the relativistic regime.

Write $n_{56}$ and $n_{54}$ for the number densities of nickel-56 and iron-54. Both nuclei contain 28 neutrons, while their proton numbers differ by two. Dividing the positive-exponential nuclear statistical equilibrium formulas printed in the PDF therefore gives the nickel–iron nuclear equilibrium ratioThe neutron-density factors cancel. $K$ contains the isotope-dependent statistical and translational mass factors suppressed in the supplied proportionality; it is constant in this simplified calculation. The TeX conversion incorrectly reverses the exponential sign, so the PDF sign is essential here.

In the specified restricted composition, equality of total proton and neutron numbers meansConsequentlyDifferentiation, with the suppressed prefactors held constant, givesUsing the paper's supplied signed energy difference producesWith that negative value, $F$ tends to zero at both temperature endpoints, increases below this temperature and decreases above it. At the stationary point $d^2\log F/d{T}^2=-3/T^2<0$, so it is indeed a maximum for the printed toy data.

There is a genuine physical-data qualification: the stated negative energy difference is inconsistent with interpreting $Q$ as the conventional positive total nuclear binding energy of these actual isotopes. Using atomic masses, whose electron masses cancel in this reaction,The masses used are $1.00782503223\mathrm{u}$, $53.93960899\mathrm{u}$ and $55.94212855\mathrm{u}$, from NIST hydrogen masses, NIST iron masses and NIST nickel masses. Small electronic binding corrections do not change the sign. With a positive $\Delta B$, the simplified $F(T)$ above is strictly decreasing for $T>0$ and has no finite maximum. Thus the numerical maximum follows from the supplied negative parameter, not from consistent physical binding energies; changing the TeX sign to recover a maximum would not repair the PDF's data.

Nuclear statistical equilibrium is appropriate when the relevant forward and inverse strong or electromagnetic reactions are fast compared with changes of temperature, density and composition. It is useful in sufficiently hot late silicon-burning material and in the hottest explosive burning regions of massive stars. Some burning zones reach only quasi-equilibrium, and cooling or expansion eventually freezes the abundances out; an equilibrium expression cannot by itself predict the full ashes when these timescale conditions fail. Temperature, density, conserved baryon number and the electron fraction in stellar matter must all be specified. Negligible free-particle contributions in bulk bookkeeping are not the same as exactly zero free densities in the equilibrium formula.

Strong and electromagnetic reactions preserve total charge and baryon number, so they conserve the proton-to-neutron ratio. Weak reactions need not: electron capture and beta-plus decay convert protons into neutrons and lower it. Such conversions occur in hydrogen-burning networks and in the nitrogen-to-neon chain during helium burning. At high density in advanced stellar cores and core collapse, electron capture becomes particularly important; neutrino emission also affects energy and lepton loss. Beta-minus decay changes the ratio in the opposite direction and can compete during later relaxation. Nuclear equilibrium for strong reactions does not automatically imply weak equilibrium or a fixed proton-to-neutron ratio throughout all stellar evolution.

For a nonrelativistic particle, $E(p)=m{c}^2+p^2/(2m)$. The Gaussian integral yieldsIndeed ${\int }_0^{\infty }{p}^2{e}^{-p^2/(2m{k}_{B}T)}dp=\sqrt{\pi }(2m{k}_{B}T{)}^{3/2}/4$, which combines with $4\pi /h^3$ to give the prefactor. For the two species this givesChemical equilibrium imposes ${\mu }_a+{\mu }_{\alpha }={\mu }_b+{\mu }_{\beta }$. If the two massless bath species have zero chemical potentials, then ${\mu }_a={\mu }_b$; their masses alone do not imply this chemical-potential condition.

Weak reactions such as $n+{\nu }_e\leftrightarrow p+e^{-}$, with leptons treated as relativistic, interconvert neutrons and protons. Negligible lepton chemical potentials give approximately $n_n/n_p=\exp [-(m_n-m_p)c^2/(k_{B}T)]$, since their degeneracies and masses are nearly equal. Equilibrium favors protons increasingly as temperature falls. But the weak reaction rate decreases faster than the expansion rate and eventually falls below it: cosmological weak freeze-out prevents the ratio from following its equilibrium exponential to zero. Free neutrons subsequently decay, reducing the frozen ratio further.

In Big Bang nucleosynthesis, cooling eventually makes deuterium survive photodissociation. Nuclear reactions then build helium through deuterium and the light intermediate nuclei, locking almost all surviving neutrons into stable helium-four. Neglecting binding-energy corrections and taking $r=n_n/n_p\le 1$, each helium nucleus uses two neutrons and two protons, so $n_{\mathrm{He}}=n_n/2$ andFor example $r=1/7$ gives a helium mass fraction $1/4$, with the remaining baryons predominantly hydrogen.

For an optically thin low-density gas in collisional ionization equilibrium, the astrophysical cooling function collects losses per pair of hydrogen nuclei into $\mathcal{C}=n_H^2\Lambda (T)$. Its units are $\mathrm{erg}\mathrm{c}{\mathrm{m}}^3{\mathrm{s}}^{-1}$; the time unit is missing from the printed guide to the vertical-axis labels. For the primordial atomic cooling curve, only hydrogen and helium provide bound-state coolants.

Near $10^4\mathrm{K}$, collisional excitation of hydrogen becomes effective and subsequent photon emission removes thermal energy. The excitation rate is exponentially suppressed below the atomic energy threshold. Atomic line cooling and collisional ionization generate a strong hydrogen feature around a few $10^4\mathrm{K}$; helium excitation and ionization produce further structure around $10^5\mathrm{K}$. Radiative recombination also contributes. Once the gas is highly ionized, these bound-state losses weaken. At $T\gtrsim 10^6$–$10^7\mathrm{K}$, thermal bremsstrahlung dominates, with approximately $\Lambda \propto T^{1/2}$ apart from slowly varying factors. The figure is an original qualitative sketch anchored to the two supplied values, not an atomic-rate calculation.

Increasing galactic metallicity adds many ionic transitions, producing metal-line cooling and substantially enhancing cooling, particularly in the $10^5$–$10^7\mathrm{K}$ range. Metals also provide low-energy fine-structure transitions below $10^4\mathrm{K}$. The change is not a uniform vertical shift: the positions and strengths of features depend on the elemental abundances and ionization state.

Photoionization removes bound electrons even where collisions alone would leave atoms neutral, often suppressing the hydrogen and helium line peaks and changing the metal-ion population. It also adds photoionization heating. The net thermal loss is then $\mathcal{C}-\mathcal{H}$, with possible heating-cooling equilibrium near $10^4\mathrm{K}$. A single density-independent $\Lambda (T)$ no longer describes every irradiated cloud: the answer also depends on radiation intensity, spectrum, shielding and density. The illustrative photoionized curve denotes altered cooling alone, not the net loss including heating.

Below $10^4\mathrm{K}$, primordial gas can use molecular line cooling, especially rotational and vibrational transitions of molecular hydrogen and, in suitable chemical conditions, hydrogen deuteride. Molecule formation and protection against photodissociation are essential. Enriched gas can additionally use metal-line cooling, other molecular species and dust thermal emission, with gas-dust energy exchange important at high density. At high cosmological redshift, residual free electrons can transfer energy to the cosmic microwave background through Compton cooling by the cosmic microwave background if the gas is hotter than the radiation. Expansion can also cool gas adiabatically, but neither that process nor Compton cooling by the cosmic microwave background is an $n_H^2$ atomic loss law. The atomic threshold is a cooling-channel limitation, not a universal minimum gas temperature.

Stellar nucleosynthesis and the subsequent return of matter explain most elements heavier than the primordial hydrogen and helium. Big Bang nucleosynthesis supplied predominantly those two elements, with small light-isotope abundances; stars do not make every element from an initially metal-free mixture. Different stellar masses reach different central temperatures and leave different remnants. The mass comparisons below describe representative isolated, near-solar-composition stars; mass loss and binary interaction can alter the final core. A nucleus contributes to later generations only if it is eventually expelled into the interstellar medium, rather than remaining locked in a remnant.

A $1M_{\odot }$ star spends its main sequence life converting hydrogen to helium, predominantly through the proton–proton chain. After core hydrogen burning ends, a hydrogen-burning shell supplies an expanding red giant envelope and builds an electron-degenerate helium core. First dredge-up exposes material whose CNO cycle has redistributed the original carbon and nitrogen abundances. The helium flash starts core helium burning: the Triple-alpha process makes carbon, and carbon-12 alpha capture makes oxygen. Once core helium is exhausted, the Asymptotic giant branch has a carbon-oxygen core with helium-burning shell and hydrogen-burning shell. Shell instability causes AGB thermal pulses; subsequent third dredge-up, where efficient enough, can bring carbon and neutron-capture products into the envelope. Its efficiency at such low mass depends on composition and mixing. The star does not ordinarily ignite core carbon burning. Stellar winds remove the envelope, a hot remnant may illuminate a planetary nebula, and a carbon-oxygen white dwarf retains much of the core's synthesized matter.

A $5M_{\odot }$ star generally uses the more temperature-sensitive CNO cycle for main-sequence hydrogen burning, with a substantial convective core. Its hotter helium core ignites core helium burning without the low-mass degeneracy-driven helium flash, producing carbon and oxygen. After core helium exhaustion it develops double-shell Asymptotic giant branch burning and a degenerate carbon-oxygen core. Second dredge-up and repeated third dredge-up alter the envelope composition. In sufficiently hot models, hot-bottom burning at the base of the convective envelope converts dredged-up carbon into nitrogen through the CNO cycle; consequently a carbon-producing shell need not produce a carbon-rich final envelope. Hot helium-shell pulses can activate the ${}^{22}\mathrm{Ne}(\alpha ,n){}^{25}\mathrm{Mg}$ neutron source. Strong stellar winds expel these processed layers. A conventional isolated five-solar-mass model ends as a carbon-oxygen white dwarf, below the mass range that normally develops sustained core carbon burning; mass loss and mixing govern the exact boundaries between outcomes.

A $32M_{\odot }$ star has core CNO cycle hydrogen burning, followed by core helium burning. The core then contracts through carbon burning, neon burning, oxygen burning and silicon burning, with active shells outside the successive exhausted cores. Carbon burning produces chiefly nuclei such as neon and magnesium. Neon burning combines photodisintegration and alpha capture; oxygen burning makes silicon-group nuclei. Silicon burning is a network of captures and photodisintegration approaching nuclear statistical equilibrium, rather than simply fusion of pairs of silicon nuclei. It produces an iron-group mixture governed by temperature, density and weak reactions. The maximum in nuclear binding energy per nucleon near the iron group prevents continued exothermic fusion from supplying ordinary hydrostatic support. Stellar neutrino energy loss accelerates the advanced stages; electron capture and photodisintegration help remove support during iron-core collapse.

The massive star can return products both before and during collapse. Radiation-driven stellar winds may expose and expel layers enriched in helium, nitrogen, carbon and oxygen. A successful core-collapse supernova ejects additional hydrostatic products and creates further nuclei by shock-driven explosive burning, including alpha-rich and iron-group products. The nickel-56 decay chain converts radioactive nickel into iron through cobalt. Initial mass alone does not fix the explosion or remnant: substantial fallback or a failed supernova can leave a black hole and retain much of the core material. Thus the possible core-collapse supernova contribution is not a guarantee that a thirty-two-solar-mass star ejects every element it made.

For nuclei well beyond the iron group, neutron capture is a principal assembly mechanism, avoiding the large Coulomb barrier for charged-particle fusion. Beta-minus decay converts captured neutrons into protons and raises the atomic number. In the s-process, captures are slower than relevant beta decays, so the sequence usually follows nuclei near stability. The main site is the Asymptotic giant branch: ${}^{13}\mathrm{C}(\alpha ,n){}^{16}\mathrm{O}$ is an important neutron source in low-mass intershell material, while ${}^{22}\mathrm{Ne}(\alpha ,n){}^{25}\mathrm{Mg}$ becomes important in hotter pulses and in massive-star helium/carbon burning. Third dredge-up and stellar winds make these products available to the interstellar medium. Barium and lead illustrate heavy products; the detailed yield depends on seed abundance, neutron exposure and branching at unstable nuclei.

In the r-process, neutron captures outrun beta decays, building very neutron-rich nuclei before they decay toward stable heavy isotopes. This needs much larger neutron densities than the s-process and can produce very heavy nuclei, including uranium. Neutron-rich ejecta from neutron star mergers provide an established site: observed kilonova emission and spectroscopy support heavy-element synthesis in such ejecta. Rare magnetically driven stellar-collapse outflows are additional candidate sites. An ordinary core-collapse supernova cannot automatically be assumed to supply a complete heavy r-process pattern.

Not every heavy isotope belongs to the two capture sequences. Gamma-process nucleosynthesis in hot supernova material uses photodisintegration on pre-existing heavy seed nuclei through neutron, proton and alpha-emission channels, helping produce proton-rich isotopes bypassed by the main capture paths. This is distinct from simply using charged-particle fusion to build every element beyond the iron group.

Finally, binary stars change both production and delivery. Roche-lobe overflow and wind mass transfer in a binary star strip envelopes, change subsequent core growth, and expose processed layers. Mass transfer from an Asymptotic giant branch donor can enrich a surviving companion in carbon and s-process products, as in a barium star; this redistributes existing yields rather than creating them in the recipient. Accretion onto a white dwarf can trigger a classical nova with explosive hydrogen-burning products. Thermonuclear disruption of a carbon-oxygen white dwarf in a Type Ia supernova, through suitable accretion or the Double-degenerate Type Ia supernova scenario, supplies a major delayed iron-group contribution, including iron; it is not the same event as collapse of a massive iron core. Compact binaries containing two neutron stars can later merge and eject r-process material. Thus binaries affect stellar mass histories, enrich companions, open thermonuclear explosion channels, and create heavy-element-producing compact-remnant mergers.

The key distinction is between making nuclei and ejecting them: ordinary low/intermediate-mass stars mainly return processed envelopes, massive stars add advanced and explosive burning products when those layers escape, and neutron-capture environments supply much of the material beyond iron.

The continuum exoplanet transit photometry depth is approximately the opaque projected area ratio, assuming a corrected or negligible limb darkening contribution:For a Sun-sized host, $R_p\simeq 6.96\times 10^7\mathrm{m}\simeq R_J\simeq 11R_{\oplus }$. The natural size analogue is Jupiter, not Earth.

The expected bulk constituent is molecular hydrogen, with helium next most abundant for a retained primary planetary atmosphere. The water band detects a strong trace absorber; it does not imply that water is the dominant gas. This interpretation assumes a conventional H/He gas giant; the measured radius by itself is not a measurement of mass or composition.

For an optically thin, low-density plasma, the astrophysical cooling function packages collisional radiation losses and temperature-dependent ion fractions into the coefficient multiplying $n_H^2$. Because the volume loss rate has units $\mathrm{erg}\mathrm{c}{\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$, the coefficient has units $\mathrm{erg}\mathrm{c}{\mathrm{m}}^3{\mathrm{s}}^{-1}$. The energy-density unit printed in the numerical hint cannot be the unit of this coefficient. Interpret the quoted logarithmic values in the dimensionally consistent cooling-coefficient unit.

Assume collisional ionization equilibrium, a primordial hydrogen-helium mixture and no external photoheating. In the specified range, the primordial atomic cooling curve has the following features. Just above $10^4\mathrm{K}$, thermal electrons begin to excite neutral hydrogen efficiently; subsequent line emission, especially Lyman-alpha emission, causes a steep rise. The excitation rate contains a threshold factor of order $e^{-10.2\mathrm{eV}/(k_{B}T)}$. Hydrogen line cooling is strong near a few times $10^4\mathrm{K}$; the supplied value at $1.5\times 10^4\mathrm{K}$ provides a useful low-temperature label.

As hydrogen becomes ionized, neutral-hydrogen line cooling declines. Helium excitation and ionization produce a further shoulder or peak around $10^5\mathrm{K}$. Collisional excitation, collisional ionization and radiative recombination all contribute: excitation photons remove electron kinetic energy, ionization consumes it, and recombination produces free-bound radiation. Once hydrogen and helium are almost fully stripped, their bound-state cooling disappears and the curve falls into a relatively inefficient interval. At high temperature, thermal bremsstrahlung dominates, with an approximate $\Lambda \propto T^{1/2}$ tail and a weak Gaunt-factor correction.

The sketch uses the two supplied numerical labels and a qualitative hydrogen-helium interpolation; it is not a tabulated atomic-rate calculation.

Metal-line cooling raises the cooling coefficient markedly over much of $10^5$–$10^7\mathrm{K}$, because heavier elements supply many ions and excitation transitions after hydrogen and helium have lost their bound electrons. It also broadens and reshapes the line-cooling peaks. Fine-structure lines can permit cooling below the hydrogen atomic threshold; molecular hydrogen can likewise cool metal-free gas below that threshold, but lies outside the requested temperature range. At sufficiently high temperatures thermal free-free emission again dominates the continuum. The enhanced curve in the figure is a schematic comparison, not a numerical claim about a specified metallicity. Thus metals generally shorten the cooling time and extend the temperature range of efficient cooling; ionization state, abundance and radiation field determine the actual curve.

Assume a hydrogen-rich gas with approximately solar elemental ratios and local thermochemical equilibrium when identifying the limiting compositions. At $T\lesssim 500\mathrm{K}$, a representative set of dominant molecules is ${\mathrm{H}}_2,{\mathrm{H}}_{2}O,{\mathrm{CH}}_4,{\mathrm{NH}}_3$. In the deeper $T\gtrsim 1300\mathrm{K}$ regime, it is ${\mathrm{H}}_2,{\mathrm{H}}_{2}O,\mathrm{CO},{\mathrm{N}}_2$. Helium is an abundant atom, not a molecule. The carbon and nitrogen switches, carbon monoxide–methane quenching and nitrogen–ammonia quenching when frozen by mixing, are represented byLow temperature and high pressure favor the right-hand sides, while hotter gas favors carbon monoxide and ${\mathrm{N}}_2$. Exact boundaries depend on pressure and composition; ${\mathrm{CO}}_2$ can become important at high metallicity. If mixing is strong, the cool upper atmosphere need not retain its local-equilibrium four-species ordering.

To preserve the hot-region reactant $A_1$ aloft, require transport to beat its conversion at the quench region. With a mixing length $\ell$ and vertical eddy diffusivity $K_{zz}$,For the simplest use of the supplied atmospheric scale height reference, assume the same temperature and mean molecular weight as the reference atmosphere and take $\ell =H$. Since $H\propto 1/g$, $H\simeq 30/0.4=75\mathrm{km}$, giving $K_{zz}\gtrsim 5.6\times 10^4{\mathrm{m}}^2{\mathrm{s}}^{-1}=5.6\times 10^8\mathrm{c}{\mathrm{m}}^2{\mathrm{s}}^{-1}$.

The hotter quench layer requires a temperature correction if the reference is ordinary Jupiter; the temperature of that scale-height reference was not specified. An explicit estimate using $T_q=1300\mathrm{K}$, dimensionless mean molecular weight $\mu =2.3$, and $g\simeq 0.4\times 25=10\mathrm{m}{\mathrm{s}}^{-2}$ givesThus a scale-height-based estimate using the hot layer is of order $10^{10}\mathrm{c}{\mathrm{m}}^2{\mathrm{s}}^{-1}$. Both estimates state their assumptions: the reference scaling alone does not include the temperature ratio. Choosing $\ell =0.1H$ lowers the threshold by a factor of $100$, and a full quench calculation needs the reaction timescale along the profile. The dependence of a quench diffusivity on mixing length shows why the stated reaction timescale supplies a mixing constraint, not a unique measured coefficient.

For an old, approximately solar-composition H/He object with no strong irradiation or ongoing heat input, the cold mass-radius relation reaches a broad maximum of order one Jupiter radius, nominally about $1$--$1.2R_J$, at a few Jupiter masses. This is a property of that composition and low-entropy sequence, not a composition-independent theorem from hydrostatic equilibrium alone. Young isolated substellar objects can be larger while cooling; strict thermal steady state of a non-burning isolated body would require its internal luminosity to vanish.

At low mass, an incompressible fixed-composition approximation gives $M\simeq 4\pi \bar{\rho }{R}^3/3$, so $R\propto M^{1/3}$. Ice giants contain dense heavy material and modest H/He envelopes, so real radii depend strongly on composition and are not obtained by extending a pure-hydrogen curve through them.

Gas giants approach a regime in which Coulomb interactions, compression and partial electron degeneracy pressure compete. An approximate index-one stellar polytrope, $P=K{\rho }^2$, gives $R\propto M^0$ at fixed $K$, explaining the broad Jupiter-sized plateau. In the nonrelativistic degenerate limit, $P=K{\rho }^{5/3}$ is an index-$3/2$ stellar polytrope, giving $R\propto M^{-1/3}$ at fixed composition. This is a limiting slope; finite-temperature brown dwarfs do not follow that exact power law over their entire observed mass range.

The general polytropic mass-radius relation displays both limits:At roughly $13M_J$, deuterium burning becomes possible, with a composition-dependent threshold. This is a conventional planet/brown-dwarf division rather than a unique formation definition. At roughly $0.075$--$0.08M_{\odot }$, about $75$--$85M_J$, sustained hydrogen burning can support a low-mass star. Its nuclear energy supply maintains a specific entropy that prevents indefinite cooling onto the cold substellar sequence; its approximate main-sequence relation is $R\propto M$ over a limited low-mass interval.

Ice giants chiefly lose formation heat and some gravitational/radiogenic energy, with convection in fluid regions and thermal conduction or inhibited convection where composition stratifies the interior. Gas giants and non-burning brown dwarfs radiate stored heat and contraction energy; their deep interiors are largely convective, with radiative photospheres. Helium separation can supply extra gravitational energy in mature giant planets, while brown dwarfs above the deuterium threshold have a temporary nuclear contribution. Low-mass stars are powered primarily by hydrogen fusion and are largely or fully convective at the lowest stellar masses. The transition to sustained fusion is the essential change from a cooling body to a long-lived thermal balance.

At fixed mass, three broad controls of a super-Earth-sized body's observed radius are:

Bulk composition, envelope fraction/composition, and thermal/irradiation history are three independent controls. If mass is not fixed, the mass itself is an additional major variable. The label super-Earth does not ensure a rocky composition or an Earth-like atmosphere, and a radius alone does not determine which of these effects dominates.

For solar elemental abundances, take a molecular hydrogen- and helium-dominated atmosphere with atmospheric carbon-to-oxygen ratio near $0.55$. The principal oxygen-bearing molecule is generally water; the carbon and nitrogen carriers depend on both temperature and pressure.

At the cool $500\mathrm{K}$ observable upper layers, methane is the expected main carbon reservoir under chemical equilibrium, with abundant water and much less carbon monoxide. At the hotter layers approaching $1400\mathrm{K}$, carbon monoxide becomes the main carbon carrier and water contains much of the oxygen not bound in it. The relevant law of mass action follows fromLower temperature favors the exothermic methane-forming direction, while increasing pressure favors the side with fewer molecules. This explains why the change of dominant carrier cannot be specified by temperature alone.

Nitrogen is distributed between molecular nitrogen and ammonia. Cooling favors ammonia, but low pressure favors molecular nitrogen; it is therefore unsafe to call ammonia dominant throughout the $500\mathrm{K}$ low-pressure region. The exact partition requires the equilibrium constant for ${\mathrm{N}}_2+3{\mathrm{H}}_2\rightleftharpoons 2{\mathrm{NH}}_3$.

Carbon dioxide is usually a minor constituent at solar composition; hydrogen cyanide and acetylene are much less abundant than the principal carbon carriers in this oxygen-rich equilibrium case. Condensation can remove refractory species where a condensation curve is crossed. The robust cool-atmosphere expectation is a molecular hydrogen–helium background with methane and water, changing toward carbon monoxide in hotter layers. Precise mixing ratios require thermodynamic data and an explicitly specified elemental inventory after rainout.

At fixed temperature and pressure, the atmospheric carbon-to-oxygen ratio controls the division of the elemental inventory among carbon monoxide, methane, water and other molecules. For an oxygen-rich mixture, water can remain after carbon monoxide consumes its share of oxygen. Increasing the atmospheric carbon-to-oxygen ratio reduces the oxygen available for water, particularly in the hotter layers where carbon monoxide is stable. Near or above unity there, water can become strongly depleted while excess carbon feeds methane, hydrogen cyanide and acetylene. This familiar hot carbon-rich behavior should not be applied unchanged to every cool layer: at $500\mathrm{K}$, methane formation can leave substantial water even at high atmospheric carbon-to-oxygen ratio.

Increasing the atmospheric metallicity of a giant planet at fixed elemental ratios increases the heavy-element inventory relative to molecular hydrogen and helium. Abundances of water and the major carbon-bearing molecules generally rise while molecular hydrogen remains dominant. In that regime, the law of mass action for $\mathrm{CO}+{\mathrm{H}}_2\mathrm{O}\rightleftharpoons {\mathrm{CO}}_2+{\mathrm{H}}_2$ givesWhere carbon monoxide and water each scale roughly linearly with enrichment $Z$, carbon dioxide consequently scales approximately as $Z^2$. This scaling changes when molecular hydrogen ceases to dominate or chemical partitioning changes. Higher enrichment can also favor carbon monoxide over methane at fixed conditions, increase mean molecular weight, and reduce the atmospheric scale height and transmission-feature amplitudes. Cloud formation and atmospheric condensate rainout alter the observable elemental ratios relative to the bulk ones.

A mass-radius curve of solar-composition substellar objects must specify age, irradiation and composition. For a mature, weakly irradiated hydrogen–helium sequence, the qualitative mass-radius relation has three main trends:

The conventional deuterium-burning mass near $13$ Jupiter masses marks a commonly used giant-planet/brown dwarf classification scale, not a sharp discontinuity in radius or a universal formation boundary. Young brown dwarfs are larger and cool by contraction; irradiation, a heavy-element core and hot-Jupiter radius inflation shift planets away from a single solar-composition curve.

The drawing is a qualitative mature sequence, not an interpolation of a specified evolutionary model grid. Three relevant observations, using representative historical measurements, are:

Assume an isolated star of initial mass five solar masses of approximately solar stellar metallicity. Rotation, convective overshooting and mass loss change numerical ages and the extent of a blue loop, so the ages below are estimates. Take zero age at the zero-age main sequence; the pre-main-sequence star phase adds a much shorter contraction time.

The original schematic Hertzsprung-Russell diagram marks the stages discussed below. The strip denotes the approximate Cepheid instability strip; the track is illustrative, rather than a computed stellar model.

The main sequence lasts roughly $8\times 10^7$–$10^8$ years. CNO cycle hydrogen burning is concentrated in a mixed convective core. Its declining hydrogen mass fraction and rising helium mass fraction are nearly uniform there, while the retreating core leaves a composition gradient. The unprocessed envelope remains hydrogen rich. At the terminal-age main sequence, core hydrogen is exhausted and a hydrogen-burning shell takes over.

The inert helium core grows through shell burning. The Schönberg-Chandrasekhar limit is approximatelyBeyond this limit an isothermal nondegenerate core cannot remain in thermal equilibrium with its envelope. Core contraction and envelope expansion carry the star across the Hertzsprung gap. An initial slower shell-burning interval can precede the rapid crossing; the crossing itself is governed by Kelvin-Helmholtz contraction, with a representative $G{M}^2/(RL)\sim 10^5$–$10^6$ years. First dredge-up then mixes hydrogen-processed material into the convective envelope, lowering its hydrogen mass fraction and increasing helium and nitrogen, while leaving a composition discontinuity where the envelope later retreats.

At an age still of order $10^8$ years, core helium burning begins quietly: the core is nondegenerate, so there is no helium flash. The Triple-alpha process and subsequent alpha capture build a carbon-oxygen core. Core helium burning lasts roughly $1.6\times 10^7$–$2.2\times 10^7$ years in representative solar-composition models. Pols's stellar-evolution notes illustrate the substantial dependence of these lifetimes on convective overshooting.

During a blue loop the envelope contracts and effective temperature increases, before the star returns redward. The loop depends on core and envelope structure and the hydrogen discontinuity left by first dredge-up. A loop reaching the Cepheid instability strip produces two further crossings, blueward and redward, in addition to the earlier rapid crossing of the Hertzsprung gap. Cepheid variables pulsate through the opacity mechanism, involving helium ionization. Neither loop extent nor all three strip crossings are guaranteed for every composition or mixing prescription. The hydrogen-burning shell continues moving outward in enclosed mass while central helium is depleted. Lattanzio's five-solar-mass tutorial shows these composition changes.

After helium exhaustion, at an age roughly $1.0\times 10^8$–$1.2\times 10^8$ years, the star ascends the Asymptotic giant branch. An inert carbon-oxygen core is surrounded by helium-rich material and a hydrogen-rich envelope. Second dredge-up mixes helium and hydrogen burning products into the envelope and reduces the hydrogen-exhausted core mass. Subsequently a helium-burning shell and hydrogen-burning shell alternate in importance. A thermal pulse of an asymptotic-giant-branch star causes intershell convection, expansion and temporary quenching of hydrogen burning; third dredge-up can carry carbon and slow neutron-capture process products outward. Between pulses hydrogen burning rebuilds the helium layer. These mechanisms are illustrated in Lattanzio's AGB tutorial.

The following original stellar composition profile sketches distinguish hydrogen exhaustion from helium exhaustion; their mass boundaries are illustrative.

The final giant phases add at most a few million years at this level of accuracy. Strong stellar winds remove the envelope; the hot remnant can illuminate a planetary nebula and then cool as a carbon-oxygen white dwarf. A typical remnant is of order one solar mass. The ordinary isolated $5M_{\odot }$ case does not reach sustained carbon burning and core collapse. Thus the usual endpoint is a carbon-oxygen white dwarf, at a total age of order $10^8$ years, with its subsequent cooling age added separately.

helium changes cavity stability and ductility; void swelling changes bulk volume and component dimensions. These effects interact, but a helium bubble and a vacancy-rich void are not interchangeable descriptions.

neutron reactions produce helium from alloy constituents and impurities. For example, capture by boron-10 can produce helium through ${}^{10}\mathrm{B}+n⟶{}^7\mathrm{Li}+{}^4\mathrm{He}$; the production per displacement dose depends on composition and neutron spectrum. Retained helium has low solubility and is trapped at defects or in bubbles. Bubble nucleation and growth at grain boundaries can assist intergranular cavitation, lowering tensile ductility and creep-rupture resistance, particularly when temperature permits the relevant transport. This is important even before a large bulk void swelling is measurable. A primary study linked high-temperature ductility loss to observed bubbles and grain-boundary weakening in irradiated stainless steel.^[ref]

displacement damage continuously creates vacancies and self-interstitials. Their different capture efficiencies at dislocations and other sinks can leave a vacancy supersaturation available to cavities. A stable cavity grows when its net vacancy uptake exceeds vacancy emission and interstitial absorption. helium can help stabilise small nuclei, but the later volume increase is not simply the volume of gas inserted into an unchanged matrix. Changing helium content can also change cavity number, size and competition for point defects; a universal proportionality between helium concentration and void swelling is not justified.

For nuclear fuel cladding, void swelling can enlarge the diameter, alter the fuel–nuclear fuel cladding gap, reduce clearances and distort reactor coolant channels or nearby structures. Nonuniform void swelling generates additional stresses. Bubble-assisted embrittlement and stress-dependent creep then affect the ability of the wall to contain the fuel and internal pressure. The relevant measurements combine density or volume change, dimensional strain, cavity microscopy and mechanical tests. Pressurised-tube experiments explicitly separated void swelling, creep, plastic strain and precipitation strain; void swelling itself was approximately isotropic in that study.^[ref]

void swelling has a temperature window, rather than increasing monotonically with irradiation temperature. At sufficiently low temperature, vacancy migration and cavity growth are slow. At intermediate temperature, mobile defects, stable nuclei and unequal sink capture can support rapid cavity growth. At higher temperature, increased vacancy emission, recombination and changing sink populations can reduce the vacancy supersaturation and net void swelling. The precise peak depends on alloy, spectrum, dose rate and starting microstructure; lower void swelling at high temperature does not mean that creep or helium embrittlement is harmless there.

Distinguish void swelling $S=\Delta V/V_0$ from its slope. With displacement dose $D$, a simple schematic description iswhere $D_0$ represents incubation and $k(T)$ an approximately constant post-transient slope over a limited range. Real nucleation and sink evolution make the transition smooth; neither $D_0$ nor $k$ is a universal material constant. At larger dose, microstructural or compositional evolution may change the slope again.

neutron fluence is an integrated exposure, $\Phi =\int \varphi dt$. Under an unchanged spectrum it can track displacement dose, but equal fluence does not necessarily imply equal damage or helium production. Even equal dpa can give different void swelling at different dose rates because recombination and transport compete in real time. Thus compare specimens at specified temperature, spectrum, dose rate and metallurgical condition; do not transfer one void swelling curve using fluence alone.

Neutron capture by boron-10 can generate helium, leading to helium embrittlement, especially at elevated temperature. The relevant nuclear reaction isThe emitted alpha particle eventually becomes a helium atom in the metal. The product branch can also involve gamma emission, without changing this helium-producing balance. The production depends on boron isotopic content and the neutron energy spectrum; a thermal component is particularly relevant to this reaction.

Helium has very low solubility in steel and can be trapped at vacancies, interfaces and grain boundaries. Boron already segregated near grain boundaries can therefore create a locally unfavorable distribution of helium. Stabilized helium–vacancy clusters develop into bubbles or cavities. Under tensile stress, boundary cavities can grow and link, reducing ductility, fracture toughness and creep-rupture life through intergranular fracture. Early thin-foil observations and mechanical tests associated high-temperature ductility loss with helium-containing boundary bubbles and their coalescence.^[ref]

Irradiation temperature controls the transport and clustering of both helium and vacancies. At lower temperature, reduced long-range transport can retain small trapped clusters in the matrix. These can contribute to irradiation hardening, but extensive boundary-cavity growth is kinetically less favorable. At higher temperature, migration, cavity growth and material creep become faster, making intergranular fracture and loss of high-temperature ductility more likely. Trap distribution and prior microstructure can redistribute the gas, so there is no composition-independent critical temperature or universal monotonic law.

The mechanical-test temperature also matters. Material irradiated cold can become embrittled during subsequent hot testing or service when trapped gas and defects become mobile. Distinguish gas production during irradiation from the later conditions under which its mechanical consequences are expressed.

The key imbalance is preferential loss of self-interstitials to biased defect sinks, leaving a vacancy supersaturation that can feed void swelling. The initial displacement cascade creates both vacancies and self-interstitials, often in equal net numbers as Frenkel pairs; it need not create more vacancies than interstitials at birth.

Many nearby Frenkel pairs recombine. Others separate, become trapped or form dislocation loops. Surviving self-interstitials are usually much more mobile than vacancies over the relevant range. Dislocations also have an interstitial bias: their elastic interaction with defects makes them more efficient sinks for self-interstitials. Preferential interstitial removal to the dislocation structure can leave the mobile vacancy population above its thermal equilibrium value.

A minimal radiation damage rate theory makes the distinction explicit:Here $c_i,c_v$ are defect number densities, $G$ is the surviving mobile-pair production rate per volume, $K$ the recombination coefficient, $D_i,D_v$ the diffusion coefficients, and $k_i^2,k_v^2$ the respective sink strengths. The vacancy loss term includes thermal emission in an effective form. Equal production terms do not require equal steady populations or equal partitioning among sinks. This simplified model omits an explicit cluster-size distribution.

A stable cavity receiving a greater vacancy influx than interstitial influx and vacancy emission grows. Vacancies therefore condense into voids, producing void swelling. Helium can stabilize small cavity nuclei, but a growing vacancy void need not be a high-pressure gas bubble. Dislocation loops, precipitates and changes in the dislocation density evolve alongside the voids and modify the competing sink strengths.

There is no violation of atom conservation: atoms represented by removed self-interstitials have joined extended defects or surfaces, while the corresponding unfilled lattice volume can accumulate in cavities. The classic electron-microscope observations of cavities in irradiated type316 steel provide direct evidence for this microscopic basis of swelling.^[ref]

A parent initially at rest emits an alpha particle and a recoiling heavy daughter. momentum conservation gives equal and opposite momenta, so their nonrelativistic kinetic energies satisfyHere $Q$ is the kinetic energy available to this branch; excitation of the daughter reduces it and may be followed by photon emission. For an illustrative $Q=5$ MeV, $m_{\alpha }=4\mathrm{u}$ and $M_d=236\mathrm{u}$, the energies are $4.917$ MeV in the alpha particle and $83.3$ keV in the daughter. The heavy daughter receives much less energy, but that does not imply much less atomic damage.

The fast alpha particle loses most of its energy through electronic stopping: ionization and electronic excitation along a relatively long path. It also undergoes nuclear collisions, particularly towards the end of its range, and can produce displacement damage. The slower, heavy, highly charged recoil couples strongly to surrounding nuclei and generates a compact displacement cascade. Successive knock-on atoms create vacancies, interstitial defects and clusters. Some Frenkel pairs recombine as the cascade relaxes; the initial collision count is therefore not the number of defects that remain.

A quantitative illustration comes from brannerite rather than an assumed universal ceramic. Simulations in the primary natural-mineral study used a $4.5$ MeV alpha particle and an $80$ keV daughter recoil. The corresponding ranges were about $13$ micrometres and $23$ nanometres. At the study's assumed $50$ eV threshold displacement energy, its tabulated results give roughly $560$–$571$ recoil atomic displacements and $91$–$100$ alpha particle atomic displacements per radioactive decay. The numbers depend strongly on the displacement threshold and material, but the shorter recoil path contains the dominant and much denser ballistic radiation damage.^[ref]

In a susceptible ceramic, cascade cores can accumulate into an amorphous fraction; defects and associated relaxation can also change volume, thermal conductivity and mechanical response. molecular dynamics simulations of zirconolite found a strongly disordered core surrounded by more isolated point defects, supporting this distinction between local cascade damage and an entirely transformed specimen.^[ref]

The alpha particle eventually becomes a helium atom after slowing and charge exchange. helium accumulation can have later microstructural effects distinct from the initial knock-on cascade. Material-dependent recombination and annealing limit surviving disorder; a change in diffraction or radiation-induced swelling is not automatically radiation-induced amorphization, and radiation-induced amorphization is not automatically dissolution or release of the contained radionuclide.

For a classical ideal monatomic gas, $C_{V,\mathrm{He}}=3N{k}_B/2$ and $C_{p,\mathrm{He}}=5N{k}_B/2$. Treat copper as having one conduction electron per atom. Its degenerate-electron contribution isConsequently the electron-only estimates at5K areThe constant-pressure gas versus electronic metal heat capacity convention makes the second numerator appropriate to a constant-pressure gas measurement. For the solid, the low-temperature difference between $C_p$ and $C_V$ is usually small, but its phonon contribution is not fixed by $E_F$. In a Debye model,The supplied $E_F$ alone therefore does not determine the ratio to copper's total heat capacity; the electronic-only result is conditional.^[ref] An estimate using additional material data is possible: the specific-heat fit in Material Properties: OFHC Copper by National Institute of Standards and Technology gives $c_{p,\mathrm{Cu}}(5\mathrm{K})\simeq 0.155\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$. Using copper's rounded molar mass $0.0635\mathrm{kg}\mathrm{mo}{\mathrm{l}}^{-1}$ gives $C_{p,\mathrm{Cu}}\simeq 9.85\times 10^{-3}\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$, and henceThis supplementary estimate includes copper's lattice contribution and retains the ideal-gas approximation for helium. The NIST fit reports a 10% deviation from its source data below 15 K; neither that fit nor the stated pressure alone supplies helium nonideality corrections.^[ref]