= Solution
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.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-317-evolution.png]
{title=Schematic <stellar evolution> of an initially five-solar-mass <star>}
The <main sequence> lasts roughly $8\times10^7$–$10^8$ 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 approximately
$$
q_{\rm SC}\simeq0.37\left(\frac{\mu_{\rm env}}{\mu_{\rm core}}\right)^2\sim0.08\text{--}0.1.
$$
Beyond 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 $GM^2/(RL)\sim10^5$–$10^6$ 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 $1.6\times10^7$–$2.2\times10^7$ years in representative solar-composition models. https://www.astro.ru.nl/~onnop/education/stev_utrecht_notes/chapter9-11.pdf[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. https://users.monash.edu.au/~johnl/StellarEvolnV1/m5z02evoln.html[Lattanzio's five-solar-mass tutorial] shows these composition changes.
After helium exhaustion, at an age roughly $1.0\times10^8$–$1.2\times10^8$ 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 https://users.monash.edu.au/~johnl/StellarEvolnV1/AGBevoln.html[Lattanzio's AGB tutorial].
The following original <stellar composition profile> sketches distinguish hydrogen exhaustion from helium exhaustion; their mass boundaries are illustrative.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-317-composition.png]
{title=Schematic internal composition profiles during <stellar evolution>}
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 \b[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.
Back to article page
