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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 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 $G{M}^2/(RL)$ 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 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.

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. 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.

Use stellar homology: dimensionless density, temperature and luminosity profiles are taken to have the same shape, so the dimensional scale relations follow from the structure equations. Mass conservation and hydrostatic support giveThe ideal gas equation then gives $T_c\propto \mu M/R$, with the dimensional constants held fixed. The specific stellar energy-generation rate has the prescribed hydrogen factor and temperature exponent, soIndependently, radiative diffusion in a star gives $T_c/R\propto \kappa {\rho }_{c}L/(R^2{T}_c^3)$, henceEquating the nuclear supply and radiative transport establishes composition-dependent CNO homology:At one fixed composition, these reduce to $R\propto M^{3/4}$ and $L\propto M^3$ as required.

The Stefan–Boltzmann law gives $T_e^4\propto L/R^2\propto M^{3/2}$, so $T_e\propto M^{3/8}$. Thus the logarithmic Hertzsprung-Russell diagram slope along this zero-age main sequence isIncreasing mass moves upward and leftward on the conventional diagram, whose temperature axis is reversed.

For composition changes, neglect the small heavy-element contribution to the mean molecular weight and set $Y\simeq 1-X$. Then ${\mu }^{-1}=(3+5X)/4$. The fixed trace catalyst abundance is absorbed into ${ϵ}_0$; this is not an assertion that an exactly metal-free star can undergo CNO burning. Substitution into the homology laws givesFor a retained, nonnegligible fixed metallicity $Z$, the corresponding mean-molecular-weight factor is $3+5X-Z$ instead. The displayed requested formula uses the trace-metal approximation.

At fixed mass, differentiating with respect to $X$ at the formal reference $X=1$ givesConsequentlyand the initial fully mixed hydrogen-burning evolutionary track has slopeHydrogen depletion has $dX<0$, so both luminosity and effective temperature initially increase. The star therefore moves upward and leftward, on a shallower logarithmic track than the mass sequence. The figure uses dimensionless luminosity and temperature relative to its initial reference star; the track is computed for declining $X$ at constant mass.

Ordinary cluster main-sequence turn-offs instead evolve toward brighter, cooler, more extended configurations and eventually the giant region. Normal stars are therefore not mixed throughout their entire mass during hydrogen burning: the core composition changes while the envelope retains hydrogen, and composition and entropy gradients develop. Core exhaustion and subsequent shell burning then produce very different structural evolution. Complete mixing is a special evolutionary scenario, not the standard explanation of ordinary giant formation.

These hydrogen-burning scalings must not be extrapolated literally to $X=0$. Their vanishing nuclear coefficient would make the inferred radius tend to zero and the effective temperature singular; in reality hydrogen exhaustion requires a different structural and energy-generation model. A homogeneous remnant may be helium-rich, but its later structure is not described by the same CNO law. Also, the cover-sheet radiation-pressure expression has a temperature-power typo: the physical term is $a{T}^4/3$. Radiation pressure was neglected in this derivation, so that typo does not enter the calculation.

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.

An Algol binary is a semidetached binary: a cool evolved donor fills its Roche lobe, while a hotter and more massive mass gainer remains on or near the main sequence. Roche-lobe overflow feeds the companion through a stream, sometimes forming an accretion disk. A suitably inclined system is an eclipsing binary. The prototype's periodic dimming led John Goodricke to propose an occulting companion in 1783; his original observations record an early eclipse interpretation.

The Algol paradox arises if the present masses are assumed to have been constant: the lower-mass star is the more evolved one, even though coeval isolated stars of higher mass normally exhaust core fuel sooner. The mass-luminosity relation gives the rough nuclear-lifetime scaling $t_{\mathrm{nuc}}\propto M/L$, decreasing strongly with mass. The resolution is binary mass-ratio reversal. The present donor began as the more massive star, evolved first, and expanded into its Roche lobe. Transferring much of its envelope made it less massive and made its initially less massive companion the present mass gainer. Their current masses therefore do not reveal their original evolutionary ordering.

On a Hertzsprung-Russell diagram, both components start on the zero-age main sequence. The initially more massive donor leaves the main sequence first, moving toward lower effective temperature and higher luminosity as a subgiant or red giant. During envelope stripping, it remains oversized and overluminous for its decreasing mass because its evolved core continues to supply energy. The accretor gains mass, moves to higher effective temperature and luminosity, and can undergo stellar rejuvenation if fresh hydrogen mixes into its core. After substantial envelope removal, the donor contracts to a hot stripped star; a sufficiently low-mass helium core ultimately becomes a helium white dwarf. The schematic below separates the two identities through the transfer episode rather than relabelling them when their masses cross.

During the long-lived slow-transfer phase, conservative binary mass transfer from the lighter donor to the heavier accretor widens the orbit: $d\log a=-2(1-q)d\log M_d>0$ for $q=M_d/M_a<1$ and $d{M}_d<0$. Transfer eventually stops when the donor's shrinking envelope can no longer maintain contact. The remnant can be a helium white dwarf if it never ignites helium, or a more massive helium-burning stripped star if it does. Later the mass gainer also leaves the main sequence. Reverse Roche-lobe overflow onto the compact remnant can lead to a common envelope, leaving a close double remnant after successful envelope ejection, or to merger. The detailed outcome depends on both masses, the separation, and how much matter and angular momentum escaped during earlier transfer.

The approximate mass-ratio boundary in the question has a stability interpretation. For an ideal fully convective donor star with adiabatic stellar radius response exponent ${\zeta }_{\mathrm{ad}}=-1/3$, the conservative Roche-radius approximation gives ${\zeta }_L=2q-5/3$. Dynamical stability of binary mass transfer requires ${\zeta }_{\mathrm{ad}}>{\zeta }_L$, so a long-lived stable system hasA more massive convective donor expands relative to its shrinking lobe under mass loss, favouring runaway transfer and a common envelope instead of a persistent Algol phase. This explains the approximate Algol mass-ratio stability limit in that model. It is not a universal observational boundary: a donor with a radiative envelope or a substantial evolved core has a different adiabatic response, and nonconservative loss changes the Roche-lobe response. Van Rensbergen and collaborators' observed and modelled Algol distributions include reported mass ratios above $0.7$ and discuss uncertainties in their determination. The literal claim that all Algols obey the same cutoff is therefore too strong.

A sufficiently wide system first reaches Roche-lobe overflow on the red giant branch, when the original donor is likely to have a deep convective envelope. Straightforward conservative overflow while that donor is still more massive is then prone to dynamical runaway and orbital contraction in a common envelope; it does not naturally yield a wide, long-lived Algol-like configuration. A plausible route is substantial earlier envelope loss through a stellar wind, possibly tidally enhanced stellar wind loss, reducing or reversing the mass ratio before contact. Wind mass transfer in a binary star can also increase the companion's mass. The lower donor mass, reduced envelope and larger core fraction make later transfer easier to stabilize. Alternatively, a detached pair with the same reversed evolutionary appearance may be interacting only through a wind and need never have undergone overflow. Its current width alone does not uniquely determine the initial orbit, but it indicates that prior mass loss or a more general nonconservative history must be considered, rather than applying the simple conservative convective-donor picture unchanged.

At fixed initial zero-metal composition, equating parts (i) and (ii) gives $R\propto M^{3/4}$ and $L\propto M^3$. Therefore $T_{\mathrm{eff}}\propto (L/R^2{)}^{1/4}\propto M^{3/8}$, and the fully radiative zero-age line on the Hertzsprung-Russell diagram has slope

For the convective sequence, equating parts (i) and (iii) at fixed composition givesIts slope is consequently

For a fully ionized zero-metal hydrogen-helium mixture with $Y=1-X$, the mean molecular weight obeysFor a radiative star of fixed mass,At the pure-hydrogen point $X=1$,andHenceAs hydrogen burns, $X$ decreases, so both $L$ and $T_{\mathrm{eff}}$ increase. The track moves upward and toward higher temperature from the zero-age main sequence, with initial slope $d\log L/d\log T_{\mathrm{eff}}=256/53\simeq 4.83$.

For a homologous star, hydrostatic equilibrium, mass conservation, and the ideal gas equation give the central scalingswhere $\mu$ is the mean molecular weight. Integrating the nuclear energy-generation law over a fixed homologous profile givesRadiative stellar structure gives independentlyEquating the two luminosities yieldsAt fixed zero-age composition, the stars are therefore homologous with

The effective temperature satisfies $L=4\pi R^2\sigma T_e^4$, so $T_e\propto M^{3/8}$. The zero-age main sequence consequently hasIt is a steep line rising toward high luminosity and high temperature on a Hertzsprung-Russell diagram.

For fully ionized hydrogen and helium with $Y=1-X$,while $\kappa \propto 1+X$. At fixed mass,and the corresponding radius relation is

At the pure-hydrogen zero-age point $X=1$,whereasHenceandAs hydrogen is consumed, $X$ falls, so both $L$ and $T_e$ rise. In the usual diagram with temperature increasing leftward, the evolutionary track initially moves upward and leftward from the zero-age main sequence.