= Solution
A $5M_\odot$ star of roughly solar <stellar composition> begins on the <zero-age main sequence> with a hot <convective core>, a mostly <radiative envelope>, and <hydrogen burning> dominated by the <CNO cycle>. The strong temperature dependence of the <CNO cycle> concentrates the <stellar energy-generation rate> near the centre, helping produce the <convective core>. Core mixing replenishes <hydrogen> within that region; it does not homogenize the entire star. The nuclear fuel supply gives a <stellar nuclear timescale> of order $10^8$ years. For a numerical comparison, Table 2 of https://arxiv.org/abs/1110.5049[Ekström et al.'s solar-metallicity models] gives core <hydrogen burning> lifetimes of about $88$ and $109$ million years for nonrotating and rotating $5M_\odot$ models, respectively.
After central <hydrogen> exhaustion, an inert <helium> core contracts and heats through <Kelvin-Helmholtz contraction>, while a <hydrogen-burning shell> supplies the growing <luminosity>. The envelope expands and cools: the track crosses the <Hertzsprung gap> toward the <red giant> region. Its deepening <convective envelope> causes <first dredge-up>, bringing material altered by <hydrogen burning> to the surface. The rapid structural crossing has a thermal timescale, roughly $10^5$–$10^6$ years for a star of this <mass>, rather than the full <stellar nuclear timescale>; an estimate follows from the <Kelvin-Helmholtz cooling time> $GM^2/(RL)$ with the evolving <radius> and <luminosity>.
The <helium> core reaches ignition before becoming strongly supported by <electron degeneracy pressure>. Thus a normal $5M_\odot$ star starts <core helium burning> without the <helium flash> characteristic of lower-mass stars. The <Triple-alpha process> produces <carbon-12>, and <carbon-12 alpha capture> produces <oxygen-16>; these reactions build a <carbon-oxygen core>. There is a <convective core> and continued <hydrogen-burning shell> activity. Core <helium> burning lasts order $10^7$ years: the same https://arxiv.org/abs/1110.5049[model table] gives approximately $19$ and $18$ million years. The changing core and envelope structure can produce a <blue loop> in the <Hertzsprung-Russell diagram>; its extent depends on <stellar composition>, mixing and <stellar rotation>.
After central <helium> exhaustion, the <carbon-oxygen core> contracts and becomes increasingly supported by <electron degeneracy pressure>. A <helium-burning shell> forms above it, beneath the <hydrogen-burning shell>. The star returns to a cool, luminous <Asymptotic giant branch> configuration. On this early branch, <helium-burning shell> activity is an important energy source; the overlying <hydrogen-burning shell> can weaken during readjustment and resume later. At this <mass>, a deepening <convective envelope> can cause <second dredge-up>. This shell-supported evolution to the first <AGB thermal pulse> takes order $10^6$ years, with model-dependent duration. As the <helium-burning shell> becomes sufficiently thin, the <thin-shell instability> ends steady shell burning and marks the onset of <AGB thermal pulses>.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2004/iii/paper-63-stellar-track.png]
{title=Schematic evolution of a five-solar-mass star to the first AGB thermal pulse}
The original sketch shows the direction of <stellar evolution>, with higher <effective temperature> to the left. It illustrates a possible <blue loop>; its coordinates and loop shape are schematic, rather than a computed stellar model. The stages before the first <AGB thermal pulse> are the requested track; the later remnant evolution is described below.
In the usual final evolution, <stellar winds> and an <AGB superwind> remove the envelope before the <carbon-oxygen core> can ignite substantial <carbon burning>. Once shell fuel is exhausted, the exposed core contracts and heats, potentially ionizing the expelled gas as a <planetary nebula>, and finally becomes a <carbon-oxygen white dwarf> that loses its stored thermal <energy> by cooling. The relevant competition is between the <AGB envelope-loss and core-growth timescales>, not just the total initial <mass>.
Much higher late-stage <mass loss> strips the envelope sooner, shortens the <Asymptotic giant branch> phase, and allows less core growth, usually producing a smaller <carbon-oxygen white dwarf>. Much lower <mass loss> prolongs shell burning and permits greater core growth. In an extreme hypothetical case, retention of the envelope could let a degenerate <carbon-oxygen core> approach the <Chandrasekhar mass> and undergo explosive <carbon burning>: the proposed <type 1.5 supernova> channel. This is not an inevitable fate for a $5M_\odot$ star. https://arxiv.org/abs/1410.5431[Doherty et al.'s evolutionary calculations] emphasize the uncertainty of that channel and find that their adopted <mass loss> removes carbon-oxygen envelopes before such ignition. If <carbon burning> instead proceeds without disruption, the possible further evolution depends on the resulting core and continued envelope retention; lowering <mass loss> alone does not establish an <electron-capture supernova> outcome.
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