= Solution
For a slowly evolving spherical star, use a Lagrangian <enclosed mass> coordinate $m$. Let $u$ be specific thermal internal energy and $D/Dt$ follow a mass element. The <stellar energy balance equation> is
$$
\boxed{\frac{\partial L}{\partial m}
=\epsilon_{\rm nuc}-\epsilon_\nu-\frac{Du}{Dt}
-P\frac{D}{Dt}\left(\frac1\rho\right)
=\epsilon_{\rm nuc}-\epsilon_\nu+\epsilon_{\rm grav}.}
$$
The outward <luminosity> is energy transported through the mass shell, $\epsilon_{\rm nuc}$ is nuclear rest-mass energy released per unit mass and time, and $\epsilon_\nu$ is escaping <neutrino> power per unit mass. Define the <gravothermal stellar energy generation> by $\epsilon_{\rm grav}=-Du/Dt-PD(1/\rho)/Dt$. At fixed composition it is $-TDS/Dt$ by the <first law of thermodynamics>. During composition changes the full internal-energy derivative must use the appropriate equation of state, with nuclear rest-mass changes counted only once. If a tabulated nuclear rate already subtracts reaction-neutrino energy, that loss must not be subtracted a second time in $\epsilon_\nu$.
In a nearly steady <main sequence> star, <stellar nuclear fusion> supplies most of the <luminosity>. Before stable hydrogen ignition, <Kelvin-Helmholtz contraction> releases gravitational energy. In a gas-supported hydrostatic star the <stellar virial theorem> gives $U\simeq-\Omega/2$, so about half the change in gravitational binding energy heats the star and half is available for radiation. The associated timescale is $GM^2/(RL)$. After nuclear burning ends, a <white dwarf> can shine by losing stored thermal energy; crystallization releases latent heat, and composition separation can add gravitational energy. These are heat and gravitational reservoirs rather than sustained hydrogen fusion.
<Stellar nuclear fusion> is possible because lighter nuclei can combine into products with larger <nuclear binding energy>. The energy per reaction is $Q=(m_{\rm initial}-m_{\rm final})c^2$, including the relevant particles consistently. Although thermal energies are below the <Coulomb barrier>, <quantum tunnelling> permits fusion. The <thermonuclear reaction rate> averages a cross-section over the thermal relative-speed distribution:
$$
r_{12}=\frac{n_1n_2}{1+\delta_{12}}\langle\sigma v\rangle,\qquad
\epsilon_{12}=\frac{r_{12}Q_{\rm deposited}}\rho.
$$
For nonresonant charged-particle reactions the energy weighting contains $\exp[-E/(kT)-\sqrt{E_G/E}]$; the compromise between the thermal tail and penetration produces the <Gamow peak>. A resonance can greatly enhance the rate. Density, abundances and <temperature> therefore all affect the <stellar energy-generation rate>, and power laws such as $T^4$ or $T^{16}$ are local approximations.
For the <proton–proton chain>, first produce deuterium and helium-3:
$$
p+p\longrightarrow {}^2\mathrm H+e^++\nu_e,\qquad
{}^2\mathrm H+p\longrightarrow{}^3\mathrm{He}+\gamma.
$$
The initial reaction is slow because it converts a proton to a neutron through the weak interaction. It is the bottleneck that allows long hydrogen-burning lifetimes. The alternative pep reaction $p+e^-+p\to{}^2\mathrm H+\nu_e$ also feeds the same chain. There are three principal <proton-proton chain branches>. In pp I, two helium-3 nuclei terminate the chain:
$$
{}^3\mathrm{He}+{}^3\mathrm{He}\longrightarrow{}^4\mathrm{He}+2p.
$$
In pp II, helium-3 first captures an existing helium-4 nucleus, then <electron capture> and proton capture finish the branch:
$$
{}^3\mathrm{He}+{}^4\mathrm{He}\longrightarrow{}^7\mathrm{Be}+\gamma,\qquad
{}^7\mathrm{Be}+e^-\longrightarrow{}^7\mathrm{Li}+\nu_e,\qquad
{}^7\mathrm{Li}+p\longrightarrow2\,{}^4\mathrm{He}.
$$
In pp III, the beryllium-7 instead captures a proton:
$$
{}^7\mathrm{Be}+p\longrightarrow{}^8\mathrm B+\gamma,\qquad
{}^8\mathrm B\longrightarrow{}^8\mathrm{Be}^{*}+e^++\nu_e,\qquad
{}^8\mathrm{Be}^{*}\longrightarrow2\,{}^4\mathrm{He}.
$$
Relative branch weights change with <temperature> and composition. All branches convert four protons into a net helium-4 nucleus, with two weak conversions and two <neutrinos>, allowing for the electron consumed in pp II. After including positron annihilation, the atomic-mass energy budget is approximately \b[$26.7\,\mathrm{MeV}$ per net helium-4 nucleus], but the deposited heat is smaller by the branch-dependent escaping-neutrino energy. Near ordinary low-mass main-sequence conditions, $\epsilon_{pp}\propto\rho X_H^2T^4$ is a useful local approximation.
The <CNO cycle> provides an alternative catalyzed <hydrogen burning> route in hotter cores. The <CNO-I cycle>, also called the <CN cycle>, consists of
$$
\begin{aligned}
{}^{12}\mathrm C+p&\longrightarrow{}^{13}\mathrm N+\gamma,\\
{}^{13}\mathrm N&\longrightarrow{}^{13}\mathrm C+e^++\nu_e,\\
{}^{13}\mathrm C+p&\longrightarrow{}^{14}\mathrm N+\gamma,\\
{}^{14}\mathrm N+p&\longrightarrow{}^{15}\mathrm O+\gamma,\\
{}^{15}\mathrm O&\longrightarrow{}^{15}\mathrm N+e^++\nu_e,\\
{}^{15}\mathrm N+p&\longrightarrow{}^{12}\mathrm C+{}^4\mathrm{He}.
\end{aligned}
$$
The carbon seed is regenerated: the net reaction again consumes four protons and produces one helium-4 nucleus, two positrons and two <neutrinos>. The slow ${}^{14}\mathrm N(p,\gamma){}^{15}\mathrm O$ reaction governs the ordinary cycle and causes nitrogen-14 to accumulate. The <CNO-II cycle> branches through ${}^{15}\mathrm N(p,\gamma){}^{16}\mathrm O$, ${}^{16}\mathrm O(p,\gamma){}^{17}\mathrm F$, ${}^{17}\mathrm F\to{}^{17}\mathrm O+e^++\nu_e$, and ${}^{17}\mathrm O(p,\alpha){}^{14}\mathrm N$, after which the main cycle continues. Thus CNO nuclei are catalytic in the closed cycles, although their relative abundances change. A local rate is $\epsilon_{\rm CNO}\propto\rho X_HX_{\rm CNO}T^\eta$ with a much steeper exponent than the <PP chain>; the preceding homology problem specifies $\eta=16$. The concentration of this heating near the centre favors a <convective core> in hotter, more massive hydrogen-burning stars.
Once hydrogen is exhausted in a core, contraction can raise its <temperature> enough for <core helium burning>. The <Triple-alpha process> overcomes the absence of stable mass-5 and mass-8 nuclei by maintaining a tiny transient beryllium-8 population:
$$
{}^4\mathrm{He}+{}^4\mathrm{He}\rightleftharpoons{}^8\mathrm{Be},\qquad
{}^8\mathrm{Be}+{}^4\mathrm{He}\longrightarrow{}^{12}\mathrm C^{*}
\longrightarrow{}^{12}\mathrm C+\gamma\text{ rays}.
$$
The <Hoyle state>, an excited carbon-12 resonance near the three-alpha threshold, makes the second capture efficient enough despite beryllium-8's very short lifetime. The net conversion is $3\,{}^4\mathrm{He}\to{}^{12}\mathrm C$, releasing about \b[$7.27\,\mathrm{MeV}$]. Because three alpha particles are required, the specific rate scales as $\rho^2Y^3$; a schematic resonant rate is
$$
\epsilon_{3\alpha}\propto\rho^2Y^3T_8^{-3}e^{-44/T_8},\qquad
T_8=T/(10^8\,\mathrm K).
$$
Its local <temperature> exponent is $-3+44/T_8$, of order forty near $T_8=1$. A degenerate core cannot expand promptly to regulate this increase, giving a <helium flash>; in a nondegenerate core the <stellar thermostat> permits stable helium burning. The competing ${}^{12}\mathrm C(\alpha,\gamma){}^{16}\mathrm O$ reaction determines much of the eventual carbon-oxygen mixture.
If the core becomes hot enough, <carbon burning> follows, for example through ${}^{12}\mathrm C+{}^{12}\mathrm C\to{}^{20}\mathrm{Ne}+\alpha$ and ${}^{12}\mathrm C+{}^{12}\mathrm C\to{}^{23}\mathrm{Na}+p$. <Neon burning> begins with <photodisintegration>, ${}^{20}\mathrm{Ne}+\gamma\to{}^{16}\mathrm O+\alpha$, followed by alpha capture such as ${}^{20}\mathrm{Ne}+\alpha\to{}^{24}\mathrm{Mg}+\gamma$. The first step consumes heat, but the combined rearrangement can release net energy. <Oxygen burning> includes ${}^{16}\mathrm O+{}^{16}\mathrm O\to{}^{28}\mathrm{Si}+\alpha$ and channels producing phosphorus and other nearby nuclei. At still higher temperatures, <silicon burning> proceeds through <photodisintegration> and particle captures in a reaction network approaching quasi-statistical equilibrium, producing iron-group nuclei. It is not simply direct silicon-plus-silicon fusion; the products depend on the electron fraction and weak-interaction timescale.
The increasing <nuclear binding energy> per nucleon supplies progressively less energy as products approach the iron group. Further fusion past that region cannot provide sustained net heat to support a core. Massive stars can therefore end with an unstable iron-group core; lower-mass stars do not attain all these burning stages and instead leave remnants such as <carbon-oxygen white dwarfs>. Heavy-element neutron captures can synthesize nuclei beyond the iron group without being the principal hydrostatic power source.
Finally, <stellar neutrino energy loss> competes with every heat source. Reaction <neutrinos> accompany the proton–proton and CNO chains. Hot or dense material also emits <neutrino> pairs through $e^-+e^+\to\nu+\bar\nu$, plasmon decay, $e^-+\gamma\to e^-+\nu+\bar\nu$, and electron-ion bremsstrahlung. These <neutrinos> usually escape, unlike the <photons> whose transport is diffusive; <neutrino> trapping requires much more extreme collapse conditions. Late burning has smaller fuel-energy reservoirs and strong <neutrino> cooling, so its duration is much shorter than core hydrogen burning. The surface <luminosity> is the escaping photon power, $L=4\pi R^2\sigma T_{\rm eff}^4$, not the sum of <photons> and unobserved <neutrino> power. <Radiative diffusion in a star>, convection and electron conduction redistribute energy, while the energy-balance equation distinguishes true sources, losses and storage. Accretion or mass loss adds boundary energy and mechanical work when present, rather than changing the nuclear reaction budget.
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