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The Cepheid instability strip is the region of the Hertzsprung-Russell diagram where Cepheid variables are unstable to pulsation driven by the opacity mechanism.

A pre-main-sequence star with a developing radiative core can contract along a relatively horizontal path toward higher effective temperature in the Hertzsprung-Russell diagram. This differs from the nearly vertical fully convective Hayashi track.

The Hertzsprung gap is a sparsely populated region of the Hertzsprung-Russell diagram traversed rapidly by many stars evolving away from the main sequence toward the red giant region.

For a stellar homology sequence with $R\propto M^X$ and $L\propto M^Y$, the Stefan–Boltzmann law gives $T_{\mathrm{eff}}\propto M^{(Y-2X)/4}$. Eliminating mass gives the displayed Hertzsprung-Russell diagram slope when $Y\ne 2X$. The usual horizontal axis decreases in temperature toward the right, reversing the visual left-to-right sign. The relation is between logarithmic variables and depends on the specified structural assumptions.

The main-sequence turnoff is the location in a coeval population's Hertzsprung-Russell diagram where stars are leaving the main sequence after exhausting their core hydrogen. More massive stars generally have larger luminosity per unit mass and shorter nuclear lifetimes, so the most massive stars still on the main sequence constrain the population's age. In a simple model, equating the stellar nuclear timescale $t\sim qM/L(M)$ to the common age determines a turnoff mass; detailed calibration requires composition and stellar models.

Take homologous profiles and hold composition, hence mean molecular weight, fixed. The stellar mass conservation equation and stellar hydrostatic equation give ${\rho }_c\propto M/R^3$ and $P_c\propto G{M}^2/R^4$. With ideal gas pressure support, $T_c\propto M/R$. Integrating the specific nuclear rate through homologous profiles then givesThe radiative diffusion in a star equation gives $L_{\mathrm{rad}}\propto R{T}_c^4/({\kappa }_c{\rho }_c)$. Here ${\kappa }_c\propto {\rho }_c^{1/2}{T}_c^{-5/2}\propto M^{-2}R$, soEquality of generated and transported luminosity implies $M^7{R}^{-8}\propto M^5{R}^{-1}$, whenceThis is the radiative homology with density-half inverse-five-halves opacity scaling.

The Stefan–Boltzmann law defines the effective temperature by $L=4\pi R^2{\sigma }_{\mathrm{SB}}{T}_{\mathrm{eff}}^4$. Therefore $T_{\mathrm{eff}}\propto M^{29/28}$. In theoretical Hertzsprung-Russell diagram coordinates, the slope isThe usual graphical convention puts increasing temperature to the left, so the line descends as one moves right. If the horizontal coordinate is $-\log T_{\mathrm{eff}}$, its plotted slope is $-132/29$.

The specific intensity of a blackbody is direction-independent over its outward hemisphere. Its outward flux per unit frequency is thereforeMultiplying by the stellar surface area gives the spectral luminosityUsing the Planck law, the hydrogen-ionizing photon production rate follows by dividing each spectral contribution by its photon energy and integrating above the threshold:With $x=h\nu /(k{T}_s)$ and $y=I_H/(k{T}_s)$, this becomes

For the initial hydrogen-only photon balance, assume an ionization-bounded nebula at steady photoionization equilibrium, negligible dust absorption and ionizing-photon leakage, negligible collisional ionization, and local reabsorption of photons from direct ground-state recombinations. A ground-state recombination then causes another photoionization and produces no net removal of a proton. Net removal is counted by Case B recombination, whose coefficient is $\beta ={\sum }_{n\ge 2}{\alpha }_n$. Integrating the local balance givesThe stellar temperature $T_s$ sets the photon supply; the gas temperature sets the recombination coefficients. They are not generally equal. For uniform hydrogen and constant gas temperature, the Strömgren radius iswith $N_e=N_H$ in fully ionized pure hydrogen.

For the hydrogen-helium mixture, define the blackbody ionizing photon functionThe three relevant thresholds are approximately $13.6$, $24.6$ and $54.4\mathrm{eV}$, for ${\mathrm{H}}^0$, ${\mathrm{He}}^0$ and ${\mathrm{He}}^{+}$ respectively. Higher stellar temperature increases the relative supply of the harder photons. These cumulative photon rates cannot be assigned independently to the three species: a photon above a helium threshold can instead ionize hydrogen, and helium recombination radiation can also ionize hydrogen. The absorption cross-sections and the diffuse radiation determine the effective allocation.

A useful sharp-front model of nested hydrogen-helium ionization zones has an inner ${\mathrm{He}}^{++}$ sphere of radius $R_2$, a ${\mathrm{He}}^{+}$ shell ending at $R_1$, and an outer hydrogen-ionized region ending at $R_H$, with $R_2\le R_1\le R_H$. Let $n_H,n_{He}$ be uniform nuclear number densities. The electron number densities in these three regions are $n_H+2n_{He}$, $n_H+n_{He}$ and $n_H$. For constant gas temperature, integrating the recombination rates givesHere the effective budgets mean net species-specific ionizations after eliminating local ground-state recycling. Their calculation must include photon competition and any interspecies diffuse contribution; the helium coefficients label the recombining ion's emitted spectrum. Once these budgets are specified, solve the third equation for $R_2^3$, the second for $R_1^3-R_2^3$, and the first for $R_H^3$.

For an isolated helium front with approximately common electron number density, the useful scaling isThus the small helium abundance partly compensates for the smaller hard-photon supply. A cooler central star produces little ${\mathrm{He}}^{++}$ and can have a substantially smaller helium-ionized zone. A sufficiently hot star can ionize helium throughout most of the hydrogen region. If independent radius estimates place a helium front beyond the hydrogen front, the independent photon allocations are inconsistent: the fronts and transfer must be treated together. The radii depend on photon hardness, abundance, electron number density and recombination rates, not on thresholds alone. The sharp-front equations are an approximation; they do not assert three exact independently conserved photon budgets for a real mixed nebula.

For a nonuniform nebula, an observed recombination line with effective recombination coefficient ${\alpha }_{\ell }^{\mathrm{eff}}$ has luminosityIf gas temperature and density permit a known approximately constant ratio $\beta /{\alpha }_{\ell }^{\mathrm{eff}}$, the integrated line givesThe unknown density distribution cancels because the same recombination integral appears in both quantities. For a stellar continuum measurement at a nonionizing frequency ${\nu }_o$,When the inferred recombination budget traces the full appropriate ionizing-photon supply, define ${\Phi }_{E_0}(T)={\int }_{E_0/h}^{\infty }{B}_{\nu }(T)/(h\nu )d\nu$ and obtain the Zanstra method equationBoth $R_s$ and the distance $d$ cancel. Hydrogen and helium recombination lines probe different hardness thresholds, so their ratios and the stellar continuum constrain $T_s$ and test the assumed spectrum and trapping. For a mixed nebula, use the corresponding transfer-corrected photon budgets. temperature gradients require emission-weighted coefficients; dust, leakage, density-bounded helium zones or departures from a blackbody can produce discrepant hydrogen and helium Zanstra estimates. Line fluxes alone without an appropriate budget or continuum normalization need not uniquely determine $T_s$.

Finally, locate the derived $T_s$ and luminosity on the Hertzsprung-Russell diagram. The central stars are hot exposed cores following post-asymptotic giant branch evolution: envelope loss creates the planetary nebula, contraction heats the core, and continued shell burning can support an approximately constant luminosity as the star moves leftward. Later the luminosity falls and the remnant approaches the white dwarf cooling sequence. Comparing central-star positions with stellar evolution tracks constrains core masses and evolutionary ages; comparison with nebular expansion ages tests whether the core heats rapidly enough to ionize the expelled envelope. The central-star locus traces the transition from an envelope-losing giant to a compact stellar remnant.

In stellar homology, write $r=Rx$, $m=Mf(x)$ and $\rho ={\rho }_{c}h(x)$, with common dimensionless profiles. Integrating hydrostatic equilibrium from the surface, where the pressure is negligible, givesThe profile integral is common to the sequence. Similarly, mass conservation gives $M=4\pi {\rho }_c{R}^3{\int }_0^1{x}^{2}h(x)dx$. These prove the homology scaling of central stellar pressure:The ideal gas law consequently gives $T_c\propto \mu M/R$, with the gravitational and gas constants absorbed into the coefficient.

The radiative diffusion in a star equation has a characteristic gradient $T_c/R$. Hence $L\propto R{T}_c^4/({\kappa }_c{\rho }_c)$. Using Kramers' opacity law, and comparing fixed-composition homologous models, this becomesThe volume integral of the stellar energy-generation rate $ϵ={ϵ}_0\rho T^{\nu }$ instead givesThe dimensionless integrals behind these relations do not depend on $M$ along a homologous sequence. In stellar thermal equilibrium the two luminosities agree, proving Kramers homology with a variable nuclear exponent:For the specified proton–proton chain exponent $\nu =7/2$, the mass factor cancels. ThusThe model captures two important features of the lower main sequence: hydrogen burning primarily uses the proton–proton chain, and gas rather than radiation pressure supplies the principal support. A radiative interior and a Kramers' opacity law approximation are useful near the Sun. It is a simplified local model: the Sun has a convective envelope, the coolest main sequence stars become fully convective, and real opacities and burning exponents vary. In particular a literally constant radius is not a realistic law over the entire lower main sequence.

On this proton–proton chain branch $T_c\propto M$, because $R$ and mean molecular weight are fixed. Normalizing to the Sun gives the upper mass estimateFor the specified CNO cycle exponent $\nu =23/2$, the homology relation instead reads $R^{14}\propto M^8$. Substituting into the radiative luminosity relation givesThis branch retains the deliberately idealized fully radiative assumptions, even though real CNO cycle cores are often convective.

For a Hertzsprung-Russell diagram, use the Stefan–Boltzmann law, $L=4\pi R^2\sigma T_{\mathrm{eff}}^4$. On the proton–proton chain branch $T_{\mathrm{eff}}\propto M^{11/8}$, so $L\propto T_{\mathrm{eff}}^4$. On the CNO cycle branch $T_{\mathrm{eff}}\propto M^{57/56}$, giving $L\propto T_{\mathrm{eff}}^{292/57}$. Thus the upper branch rises slightly more steeply toward the hot, luminous upper left in a logarithmic Hertzsprung-Russell diagram. The sketch below matches the approximate branches continuously at their respective burning crossovers; the true crossover has both heating terms and is smooth.

The proton–proton chain begins with reactions between hydrogen nuclei, so its specific heating scales as $X_H^2\rho T^{7/2}$ in this approximation. The catalytic CNO cycle involves hydrogen and CNO nuclei, and scales as $X_H{Z}_{\mathrm{CNO}}\rho T^{23/2}$. Here $X_H$ is the hydrogen mass fraction and $Z_{\mathrm{CNO}}$ the CNO mass fraction, not the total abundance of every metal. ConsequentlyKeeping $X_H$ fixed while dividing $Z_{\mathrm{CNO}}$ by 256 gives the CNO crossover temperatureThe mass estimate uses the unchanged proton–proton chain branch below crossover, together with the stipulated unchanged opacity and mean molecular weight.

The low-CNO proton–proton chain sequence therefore lies on the same locus, but extends to twice the transition mass. Beyond its later crossover the low-CNO CNO cycle sequence has the same logarithmic slope $292/57$ but a different normalization. At a fixed mass where both sequences are CNO-dominated, lowering the nuclear coefficient by 256 givesThus those low-CNO models are smaller, hotter and modestly more luminous at the same mass. At the same effective temperature, their CNO branch is lower in luminosity than the solar-composition CNO branch; these comparisons hold different quantities fixed.

For fully ionized hydrogen and helium with mass fractions $X$ and $Y=1-X$, the inverse mean molecular weight counts nuclei plus free electrons:Thus ${\mu }_1=8/13$ for $X_1=0.7$, ${\mu }_2=4/7$ for $X_2=0.8$, and ${\mu }_1/{\mu }_2=14/13$. Keep the same $M,{\kappa }_0$ and dimensionless structure constants when comparing these toy sequences. From part (a),Eliminating $R$ gives the composition shift of a convective pre-main-sequence track:Consequently the tracks in a Hertzsprung-Russell diagram are almost vertical, with $d\log L/d\log T_e=20$. At equal luminosity, $T_{e,1}/T_{e,2}=(14/13{)}^{2/5}\simeq 1.030$: the $X=0.7$ track lies to the hotter, left-hand side of the $X=0.8$ track. Since contraction decreases $R$, both $L$ and $T_e$ decrease in this particular opacity model. Both arrows therefore point downward and slightly rightward on the conventional hot-left diagram.

The plotted reference scales are arbitrary; the horizontal separation is the predicted $\Delta {\log }_{10}{T}_e=(2/5){\log }_{10}(14/13)$. They are schematic Hayashi tracks under the specified atmospheric opacity law, rather than calibrated stellar models.

A protostar leaves this contraction sequence when its assumptions fail. Increasing central stellar temperature makes stellar nuclear fusion significant: burning of deuterium can temporarily slow contraction, and sustained hydrogen burning brings the star to the main sequence. In a sufficiently massive pre-main-sequence star, a radiative core develops, the interior ceases to share one adiabat, and evolution can proceed along a hotter Henyey track. Changing atmospheric opacity, partial ionization, or substantial continuing accretion also changes the predicted path. For sufficiently low mass, electron degeneracy pressure instead limits the increase of central temperature, as in part (c).

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.

Introduce the characteristic scalesThen $\rho ={\rho }_{\ast }b$, $P=P_{\ast }p$ and $T=T_{\ast }t$. The gas-only equation of state gives the important dimensionless identity $p=bt$. The PDF's CNO cycle approximation has temperature power 13; the converted TeX's power 1.3 is an OCR error.

Since $d/dr=R^{-1}d/dx$, the mass and hydrostatic pressure support equation immediately becomeThus $dq/dx=x^{2}b$ and $dp/dx=-bq/x^2$. The stellar structure equations for nuclear heating givesoFor radiative diffusion in a star, $\kappa \rho ={\kappa }_0{\rho }^2{T}^{-3}$, and thereforeThis gives the radiative gradient coefficientIn an efficiently convective region, the monatomic adiabatic temperature gradient is $2/5$. Hence $dt/dx=(2/5)(t/p)(dp/dx)=-2q/(5x^2)$. If the radiative gradient is shallower, radiation suffices; if it is steeper than the adiabatic gradient, the Schwarzschild criterion selects convection and the shallower adiabatic profile. Thus

The regular central conditions are $q(0)=l(0)=0$ with finite positive $p(0),t(0)$. In particular $q\sim b(0)x^3/3$ and $l\sim Eb(0{)}^{2}t(0{)}^{13}{x}^3/3$, so the pressure and temperature derivatives tend to zero at the centre. The normalized surface has $q(1)=l(1)=1$. In the idealized zero-pressure, zero-temperature surface used for stellar homology, $p(1)=t(1)=0$; a physical photosphere supplies small nonzero surface values that are neglected in these bulk homology relations.

The common dimensionless profiles and boundary conditions keep $D,E$ constant. With fixed ${\kappa }_0,{ϵ}_0$, $D$ constant yields $L{\mu }^{-7}{M}^{-5}=\mathrm{constant}$. Substitution into the nuclear coefficient then gives $R^{16}\propto {\mu }^6{M}^{10}$. Consequently the CNO homology with density-dependent inverse-cubic opacity isThe Stefan–Boltzmann law gives $T_e^4\propto L/R^2\propto {\mu }^{25/4}{M}^{15/4}$. Eliminating mass, rather than dropping composition dependence, gives the Hertzsprung-Russell diagram relationThe additive constant depends on the fixed physical and structural coefficients; logarithms refer to fixed choices of units.

For physical evolution time $s$, distinct from the dimensionless temperature $t(x)$, hold stellar mass fixed and put $u=3+5X$, $u_0=3+5X_0$. The mean molecular weight approximation and the homology relation give $L=L_0(u_0/u{)}^7$. Converting hydrogen mass to helium releases energy at rate $L$, so fuel conservation is $M{E}_{H}dX/ds=-L$. ThereforeIntegration with the initial composition proves the homogeneous fuel-depletion luminosity feedback lawIt is valid while hydrogen remains and the homology assumptions hold. Fuel exhaustion occurs when $u=3$, before the formal denominator vanishes; the apparent infinite luminosity is not a physical endpoint of this stellar model.

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.

The helium mass fraction is $Y=1-X-Z$. For the two compositions, $Y_1=0.28$ and $Y_2=0.18$. The mean molecular weight is thereforeAt fixed mass and with the same prescribed opacity coefficient, part (a) gives $T_e\propto {\mu }^{1/2}{R}^{1/8}$ and $L\propto {\mu }^2{R}^{5/2}$. Eliminating radius givesThis is the composition shift of a convective pre-main-sequence track. The higher-helium mass fraction case $X_1=0.7$ is hotter at a given luminosity, lying to the left of the $X_2=0.8$ case in a Hertzsprung-Russell diagram. Its fixed-luminosity temperature ratio is $(349/324{)}^{2/5}\simeq 1.030$.

Both stars contract, so both $L$ and $T_e$ decrease in this particular opacity model. Their nearly vertical Hayashi tracks therefore run downward and slightly to the right, with arrows in that direction:

The plotted luminosity and effective temperature are normalized to arbitrary common references; no opacity normalization or absolute age was specified. The comparison holds the given opacity coefficient fixed.

These paths cease to apply when their assumptions fail. Deuterium-burning energy-generation rate can temporarily slow contraction; sustained hydrogen burning ends the purely gravitational phase. Formation of a radiative core breaks the fully convective adiabat and can lead to a Henyey track toward hotter effective temperature. Partial ionization, a changing atmospheric opacity, and electron degeneracy pressure alter the equation of state or boundary relation. Accretion luminosity, changing mass, and magnetic inhibition of convection can also modify a young star's path.

In stellar homology, the dimensionless radial profiles are the same after scaling radius, enclosed mass, pressure, temperature and luminosity. At corresponding radii let $r^{\prime }=\xi r$, $m_r^{\prime }=m{m}_r$, $T^{\prime }=tT$, $P^{\prime }=pP$ and $L_r^{\prime }=l{L}_r$. The printed $R^{\prime }=\xi r$ is understood as this local radial scaling, with surface radius $R^{\prime }=\xi R$.

Let $q={\rho }^{\prime }/\rho$ and keep composition, opacity coefficient and nuclear coefficient fixed first. Mass conservation gives $q=m/{\xi }^3$, and hydrostatic equilibrium gives $p=mq/\xi =m^2/{\xi }^4$. The ideal gas law then gives $t=p/q=m/\xi$. Scaling the stellar energy-generation rate equation and the stellar radiative temperature gradient gives respectivelyThe second follows from $dT/dr\propto -\kappa \rho L_r/(r^2{T}^3)$, not from energy production. Equating the two luminosity scalings givesThis assumes the denominator is nonzero and a consistent homologous family exists. If the mean molecular weight differs by $u={\mu }^{\prime }/\mu$, then $t=um/\xi$ andRatios of the opacity and energy-generation coefficients multiply the right-hand side. Thus fixed composition is a real restriction, not an automatic property of every stellar sequence.

Use the Stefan–Boltzmann law $L=4\pi R^2\sigma T_{\mathrm{eff}}^4$ to locate the family on a Hertzsprung-Russell diagram. It implies $T_{\mathrm{eff}}\propto M^{(Y-2X)/4}$ andFor proton–proton chain burning take the usual local approximation $\eta =4$, with Kramers' opacity law $\lambda =1$, $\nu =-7/2$. ThenFor the CNO cycle with electron-scattering opacity, $\lambda =\nu =0$, soA conventional local choice $\eta =16$ gives $R\propto M^{15/19}$ and slope $76/9\simeq 8.44$. Choosing $\eta =17$ instead gives slope $60/7\simeq 8.57$. The source specifies no numerical nuclear exponents, and the effective exponent changes with temperature; the general expression is the unambiguous answer.

The Hertzsprung-Russell diagram places hotter stars to the left. Its branches rise toward higher luminosity and mass, and the CNO cycle/electron-scattering opacity branch has the steeper logarithmic slope. Their illustrative joining point and normalization are arbitrary because the proportional opacity and reaction laws do not specify absolute stellar scales. These are the fully radiative ideal-gas homology predictions, not exact observed main sequence relations: convection and increasing radiation support limit those assumptions in real stars.

For PP chain burning and the Kramers' opacity law, insert $\eta =4$, $\lambda =1$, and $\nu =-7/2$ into the homology relations for radiative stars:Thus the radiative homology with proton-proton burning and Kramers opacity isTo obtain the Hertzsprung-Russell diagram slope, apply the Stefan–Boltzmann law $L=4\pi R^2\sigma T_{\mathrm{eff}}^4$:Equivalently $T_{\mathrm{eff}}\propto L^{69/284}$. The Hertzsprung-Russell slope of a stellar homology sequence refers here to logarithmic luminosity versus logarithmic effective temperature; a conventional diagram reverses its horizontal axis, so the line descends toward the right.

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 Stefan–Boltzmann law defines the effective temperature through $L=4\pi R^2{\sigma }_{\mathrm{SB}}{T}_{\mathrm{eff}}^4$. The stellar homology relations give $R\propto M^{1/13}\propto L^{1/71}$, soThusIn a logarithmic Hertzsprung-Russell diagram, with $\log L$ plotted vertically against $\log T_{\mathrm{eff}}$,The usual horizontal axis increases in effective temperature toward the left, so the hotter, more luminous end lies toward the upper left. The quoted algebraic slope uses increasing $\log T_{\mathrm{eff}}$ as the coordinate.

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.