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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.

The Schönberg-Chandrasekhar limit is the largest mass fraction that an approximately isothermal inert core can have while remaining matched in hydrostatic and thermal equilibrium to a hydrogen-rich envelope. For a simple isothermal core and polytropic envelope,Because a helium core has larger mean molecular weight than its hydrogen-rich envelope, the limiting fraction is typically near ten per cent. Once shell burning grows the core beyond this limit, no neighboring equilibrium with an isothermal core exists: the core contracts and heats by Kelvin-Helmholtz contraction, the hydrogen-burning shell brightens, and the envelope expands rapidly toward a red giant.

Letbe the scalar moment of inertia and total kinetic energy. Sincesubstitution of the Euler momentum equation reduces the stress contribution, by the divergence theorem and isotropic pressure $\mathbf{P}=P1$, toFor self-gravity, $\int \rho \mathbf{r}\cdot \mathbf{F}dV=\Omega$, where $\Omega <0$ is the gravitational potential energy. Thus the stellar virial theorem is

Apply hydrostatic equilibrium to the isothermal core, taking its boundary pressure to be $P_c$. Its ideal-gas pressure integral scales as $\int P,dV\propto M_c{T}_c$, its volume as $R_c^3$, and its gravitational energy as $-G{M}_c^2/R_c$. The virial theorem therefore givesor, after absorbing fixed dimensional and structural factors into positive constants,Hydrogen-burning reactions are extremely temperature-sensitive, so expansion cools and suppresses burning while contraction heats and enhances it. This stellar thermostat keeps the shell and adjoining isothermal core near an approximately fixed $T_c$.

At fixed $M_c,T_c$, differentiating $P_c(R_c)$ gives

For a homologous envelope, hydrostatic balance gives $P_c\propto G{M}^2/R^4$ and the ideal-gas temperature scale gives $T_c\propto GM/R$. Eliminating $R$ yieldsup to composition and gravitational constants common to the sequence. A matching core exists only if this required pressure does not exceed $P_{c,\max }\propto T_c^4/M_c^2$. ThereforeThis maximum fractional isothermal-core mass is the mechanism behind the Schönberg-Chandrasekhar limit.