Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 37 (Formation, Structure and Evolution of Stars)

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 $-132/29$.

A fully convective, monatomic ideal gas is approximately isentropic through its deep interior. Its adiabatic index is $5/3$, so $P=K{\rho }^{5/3}$ and $T\propto K{\rho }^{2/3}$. The $n=3/2$ stellar polytrope scaling, or direct substitution of the central homology scalings into $K=P_c/{\rho }_c^{5/3}$, gives $K\propto R{M}^{1/3}$.

Near the photosphere, hydrostatic balance expressed in optical depth gives $P_{\mathrm{ph}}\sim g/{\kappa }_{\mathrm{ph}}$, with $g\propto M/R^2$. Treat $T_{\mathrm{ph}}$ as proportional to $T_{\mathrm{eff}}$ and match to the convective adiabat, neglecting a varying surface entropy jump in this homology approximation. Since $\rho \propto P/T$ and ${\kappa }_{\mathrm{ph}}\propto \rho T^{10}\propto P{T}^9$,The adiabat gives $P\propto K^{-3/2}{T}^{5/2}$, so the surface condition yieldsThe Stefan–Boltzmann law consequently gives $L_{\mathrm{surf}}\propto M^{4/7}{R}^{16/7}$. The same specific nuclear law as before still gives $L_{\mathrm{nuc}}\propto M^7{R}^{-8}$. Equating these scalings produces $R^{72/7}\propto M^{45/7}$. Thus the fully convective homology with tenth-power surface opacity isThen $T_{\mathrm{eff}}\propto M^{3/16}$, and the theoretical HR slope isIts steepness expresses the relatively weak change of effective temperature with mass. The graph using $-\log T_{\mathrm{eff}}$ has the negative of this slope.

The radiative family is a stylized approximation to gas-pressure-supported main sequence stars with mostly radiative interiors and moderately temperature-sensitive hydrogen burning, such as the roughly solar-mass regime. Its opacity exponents are local approximations, not a universal composition-independent opacity law. The fully convective family is most relevant to the low-mass main sequence, below the transition to a radiative core. Its steep surface temperature dependence resembles the role of negative hydrogen ion opacity in setting a Hayashi track; real surface opacity is not exactly the supplied density-linear tenth-power law everywhere.

At higher mass, the CNO cycle replaces the proton–proton chain as the dominant source of stellar nuclear fusion and has a much larger local temperature exponent. This promotes a convective core. Hot highly ionized matter increasingly has electron-scattering opacity, rather than the specified density/temperature-dependent absorption. Radiation pressure becomes important in the equation of state, changing $T_c\propto M/R$ and eventually making the luminosity relation approach an Eddington luminosity scaling. Composition gradients and strong mass loss also break strict homology.

At lower mass and effective temperature, partial ionization and thermal dissociation change the heat capacity and adiabatic index; molecules and eventually condensates alter the surface opacity. Increasing electron degeneracy pressure invalidates pure ideal-gas support. Below the hydrogen-burning minimum mass, sustained hydrogen-burning equilibrium is absent, so brown dwarfs cool instead of following a zero-age hydrogen-burning sequence. These changes explain why neither derived slope should be extrapolated indefinitely.

The envelope is light and geometrically thin, so neglect its self-gravity and replace $m(r)$ and $r$ by the core's fixed $M,R$. The stellar hydrostatic equation becomes $dP/dz=-\rho g$, where $g=GM/R^2$. Since $d\Sigma /dz=-\rho$, it gives $dP/d\Sigma =g$. Neglecting the surface pressure yields $\boxed{P=g\Sigma }$.

With $F=L_r/(4\pi R^2)$, the luminosity equation gives $dF/dz=\rho ϵ$, and radiative diffusion in a star gives $dT/dz=-3\kappa \rho F/(4ac{T}^3)$. Dividing each by $d\Sigma /dz$ givesThe signs reflect that column mass increases inward: the temperature rises inward while outward flux decreases toward its source-free inner boundary.

Let $B=\mu g/\mathcal{R}$ for constant mean molecular weight. The gas equation of state and $P=g\Sigma$ give $\rho =B\Sigma /T$. With the supplied opacity and heating laws, put $y=T^7$ and $x={\Sigma }^2/2$. ThenDefine $A=21{\kappa }_{0}B/(4ac)>0$. Since $dx/d\Sigma =\Sigma$, the equations become $dy/dx=AF$ and $dF/dx=-{ϵ}_{0}B{y}^2$. Thus the inverse-square-opacity burning envelope satisfies

At the outer boundary $x=0$, the radiative-zero approximation neglects photospheric temperature relative to the burning layer, so $y=0$ and $y^{\prime }=A{F}_s=AL/(4\pi R^2)$. This is an approximation, not an exactly zero physical photospheric temperature. At the base $x_b={\Sigma }_0^2/2$, $y=y_b=T_0^7$. There is no luminosity entering from the helium core because all the luminosity is generated in the envelope, so $F(x_b)=0$ and $y^{\prime }(x_b)=0$.

Multiply the second-order equation by $y^{\prime }$ and integrate using the base conditions:Therefore the base temperature obeysThe physical branch has $y^{\prime }\ge 0$. A second integration givesThe integral is finite because its endpoint singularity is proportional to $(1-t{)}^{-1/2}$. Hence $y_b\propto x_b^{-2}\propto {\Sigma }_0^{-4}$ and $F_s\propto y_b^{3/2}\propto {\Sigma }_0^{-6}$. More explicitly,This envelope-mass comparison holds the core and the composition/opacity/heating normalizations fixed; the coefficient depends on $g$.

The steady solutions are susceptible to thin-shell instability. At fixed overlying column, base pressure is constrained by $g\Sigma$, so $\rho \propto T^{-1}$. The nuclear heating per unit mass scales as $T^{14}$, whereas the radiative cooling through a fixed-column profile scales as $T^7$: the opacity decreases as $T^{-3}$ and diffusion adds the fourth-power temperature dependence. A positive temperature perturbation therefore increases heating faster than cooling; thin-layer expansion does not supply the strong pressure relief of a whole-star expansion. The inverse relation $F_s\propto {\Sigma }_0^{-6}$ also means consuming the envelope raises the corresponding steady luminosity rather than damping the burning. This idealized model predicts runaway or flashes, despite nondegenerate gas. It is not a claim that all accreting white dwarfs are unstable: sufficiently thick burning layers, altered reaction regimes or other transport/accretion conditions can stabilize burning outside these assumptions.

For a circular binary with negligible spin, the two orbital radii about the center of mass are $a_1=a{M}_2/M$ and $a_2=a{M}_1/M$. Summing their orbital angular momenta givesUsing Kepler's third law, ${\Omega }^2=GM/a^3$, also gives $J=M_1{M}_2\sqrt{Ga/M}$. Under conservative binary mass transfer, $M$ and $J$ stay fixed, so $a\propto (M_1{M}_2{)}^{-2}$. Since $d\log M_2/d\log M_1=-q$, the Roche-lobe radius response exponent isFor the supplied giant response, luminosity stays fixed during the mass-loss perturbation and ${\zeta }_{\ast }=d\log R/d\log M_1=-0.27$. The change of the logarithmic overfill is $({\zeta }_{\ast }-{\zeta }_L)d\log M_1$. Because the donor loses mass, this reduces overfill and stabilizes transfer precisely when ${\zeta }_{\ast }>{\zeta }_L$. Thus the stated model requiresThis is a response criterion within the given radius law; a different adiabatic donor response would change the dynamical threshold.

The initially more massive star evolves first. If it waits until it has the supplied giant structure while still having $q>1$, conservative overflow is unstable by this criterion. If Roche-lobe overflow instead begins while the donor is on the main sequence or shortly after central hydrogen exhaustion, its largely radiative-envelope response can allow transfer before a deep convective giant envelope develops. Mass loss and accretion can then reverse the mass ratio before it reaches the giant state. This is the binary mass-ratio reversal route to an Algol binary, through Case A mass transfer or early case B mass transfer.

At lobe filling, the lobe approximation and Kepler's law give the pre-giant Roche-lobe-filling period limitIt increases with the donor's radius. Requiring $R_L<R_{\mathrm{BGB}}=1.68R_{\odot }(M_1/M_{\odot }{)}^{5/3}$ therefore yieldsThe companion mass cancels in this lobe approximation.

Since $P\propto a^{3/2}$ at fixed total mass, the separation scaling gives $\boxed{P(M_1{M}_2{)}^3=\text{constant}}$. Use solar-unit masses $m_i=M_i/M_{\odot }$ and $m=m_1+m_2$, and let $K=P(m_1{m}_2{)}^3$. ThenThe derivative vanishes at $m_1=5m/8$, and its derivative is $5/m_1^2+3/(m-m_1{)}^2>0$, while the ratio diverges at both endpoints. Hence this is the global minimum, the conservative pre-giant period-ratio minimum.

For the stated Algol parameters, $m=4$ and $K=3(1\cdot 3{)}^3=81$ days. At the minimum $m_1=2.5,m_2=1.5$, the period is $81/(2.5\cdot 1.5{)}^3=1.536$ days and $P/m_1^2=0.24576$ days, well below $P_0=0.8$ days. This is a possible initially more-massive donor configuration with overflow before the giant branch. The current $q=1/3$ is also below the giant stability threshold. Thus Algol is compatible with the proposed conservative early-overflow route, although this necessary geometric test alone does not prove its full evolutionary history.

For RT Lac, $m=2.4$ and $K=5.1(0.9\cdot 1.5{)}^3$. Its minimum occurs at $m_1=1.5,m_2=0.9$, where the mass product equals its current value, so $P=5.1$ days andNo conservative antecedent on this orbit can begin overflow before the giant branch. Beginning with an initially more massive giant instead encounters the instability just derived. RT Lac cannot follow the stated coeval, conservative Algol-formation model. A plausible alternative is rapid initial transfer with substantial mass and angular-momentum loss, via an outflow, strong wind or envelope-ejection episode. Such nonconservative binary mass transfer changes both total mass and $J$ and removes the period-product constraint. Its present mass ratio $0.9/1.5=0.6$ permits stable current giant transfer. The given data do not determine a unique mass-loss history or establish a specific ejection mechanism.

The Rosseland mean opacity is the transport opacity appropriate to an optically thick region close to local thermodynamic equilibrium. If ${\kappa }_{\nu }$ is monochromatic transport opacity and $B_{\nu }(T)$ the Planck function, thenIt is a weighted harmonic mean: low-opacity frequency windows carry much of the diffusive flux. One must combine the monochromatic processes before averaging, rather than adding their separately averaged opacities.

In hot fully ionized material, electron-scattering opacity supplies an almost density- and temperature-independent contribution at fixed composition. In the nonrelativistic Thomson scattering limit, ${\kappa }_{\mathrm{es}}={\sigma }_T{n}_e/\rho \simeq 0.2(1+X)\mathrm{c}{\mathrm{m}}^2{\mathrm{g}}^{-1}$. Free-free opacity, the absorption counterpart of thermal bremsstrahlung, comes from photon absorption during encounters between free electrons and ions. Its approximate mean obeys Kramers' opacity law, ${\kappa }_{\mathrm{ff}}\propto \rho T^{-7/2}$, with composition and quantum correction factors.

When ions retain bound electrons, bound-free absorption is photoionization; edges, occupations and ionization fractions strongly affect its frequency dependence. Bound-bound absorption produces absorption lines. Numerous metal lines can close low-opacity windows and produce important opacity features in partially ionized layers. Line populations depend on excitation as well as ionization, while Doppler broadening, pressure broadening and overlapping lines change the transport windows. Stimulated inverse processes must be included consistently with detailed balance and the local thermal radiation field.

In cooler hydrogen-rich atmospheres, negative hydrogen ion opacity from both bound-free and free-free processes is often important. Its strong temperature dependence helps determine convective surface relations. In still cooler layers, molecular bands, and at sufficiently low temperatures dust absorption/scattering, may be important. These processes cannot be represented by one Kramers power law over the whole density-temperature plane.

An opacity calculation therefore needs the chemical abundances, the equation of state, electron density and bound-state populations. Saha ionization equation and thermal excitation populations give useful dilute LTE estimates; high-density pressure ionization, nonideal effects and altered level populations require corrections. Composition matters even when one writes only $\kappa (\rho ,T)$: hydrogen, helium and the metal abundance pattern are additional fixed inputs.

In dense degenerate cores, thermal conduction by electrons can dominate energy transport. An equivalent conductive opacity may be introduced so that $1/{\kappa }_{\mathrm{eff}}=1/{\kappa }_R+1/{\kappa }_{\mathrm{cond}}$, because the two diffusive fluxes add; conduction is not itself a photon absorption process. Very hot regimes also require departures from the simple nonrelativistic scattering approximation. Near the optically thin photosphere, a Rosseland diffusion description alone is inadequate, and frequency-dependent radiative transfer with appropriate boundary conditions is required. These regime changes are why realistic stellar opacity tables, rather than one global power law, enter stellar structure.

The most direct stellar mass determination uses orbital dynamics in a binary star. A resolved orbit and an independent distance give the relative semimajor axis $a$ from its angular size; measuring the period then gives the total mass by Kepler's third law:Individual barycentric orbital sizes, or the ratio of the two radial-velocity amplitudes, give $M_1/M_2=a_2/a_1=K_2/K_1$. A photocenter orbit alone needs the luminosity ratio to infer the barycentric orbit.

For a double-lined spectroscopic orbit, the line-of-sight amplitudes determine $a_i\sin i=K_{i}P\sqrt{1-e^2}/(2\pi )$. Combining them givesAn eclipsing binary light curve constrains the inclination and relative radii, turning these projected quantities into individual masses. Eclipse and orbit modeling must allow for eccentricity, surface brightness, limb darkening and possible tidal distortion.

If only one spectrum is measured, the binary mass function isWithout an inclination and an estimate of the visible star's mass it does not fix the unseen mass uniquely, but it provides useful constraints and lower limits.

Less direct estimates combine a spectroscopic surface gravity with a radius inferred from distance, bolometric flux and effective temperature: $R^2=L/(4\pi {\sigma }_{\mathrm{SB}}{T}_{\mathrm{eff}}^4)$ and $M=g{R}^2/G$. These depend on atmosphere modeling. Locating a star on an evolutionary track or using a calibrated main-sequence mass-luminosity relation also estimates mass, but assumes composition and evolutionary state; giants and mass-transfer products need not obey a single main-sequence relation. Dynamical binary masses provide the empirical calibration against which such model-dependent methods are tested.

Nuclear burning explains both the large available energy and its release under stellar-core conditions. Converting hydrogen into helium releases approximately $0.007$ of the fuel rest-mass energy. The net proton–proton chain reaction can be written $4p+2e^{-}\to {}^4\mathrm{He}+2{\nu }_e$, with about $26.7$ MeV available per helium nucleus, including the neutrino energy loss. The CNO cycle achieves the same net hydrogen-to-helium conversion with catalysts. Measured nuclear reaction rates and stellar-core temperatures give rates capable of supplying the observed luminosity.

The energy budget distinguishes stellar nuclear fusion from alternatives. Kelvin-Helmholtz contraction has a characteristic Kelvin-Helmholtz cooling time $t_{\mathrm{KH}}\sim G{M}^2/(RL)$, only tens of millions of years for a solar-type star. Geological ages and stellar-population ages are much longer. Chemical binding energy per unit mass is smaller still. Nuclear fuel in a substantial fraction of the core gives a stellar nuclear timescale of billions of years, explaining long-lived main-sequence stars.

Main sequence luminosities, the correlation of lifetime with mass, and cluster turnoffs are reproduced by models in which central hydrogen is consumed and helium accumulates. Spectroscopic abundances and the appearance of helium-, carbon- and heavier-element-rich evolved material agree with sequential nuclear processing and stellar dredge-up. These observations support the reaction network and evolutionary consequences, rather than independently measuring every reaction in every star.

Solar neutrinos provide direct information from the burning core because they escape without the long radiative diffusion delay of photons. Their measured energies and fluxes test weak reactions in the hydrogen-burning network. The historical deficit of electron neutrinos did not require abandoning nuclear power; flavor conversion changes the measured electron component. Later flavor-sensitive measurements confirmed the total flux expected from solar nuclear burning, as described in the SNO collaboration's first-results report. Thus the evidence combines an adequate energy source, quantitatively successful stellar models and a direct nuclear by-product. Cooling remnants and brief gravitationally powered evolutionary phases are exceptions to the statement that a star's instantaneous luminosity is nuclear.

An Asymptotic giant branch star has an approximately degenerate carbon-oxygen core, a helium-burning shell, a hydrogen-burning shell farther out, and a deep convective envelope. During an interpulse interval, the hydrogen-burning shell supplies most of the luminosity and deposits helium ash beneath it. The helium layer gradually grows and its temperature rises until Triple-alpha process heating becomes large.

The helium layer is geometrically thin. A thermal perturbation greatly increases its nuclear rate, but at first expansion of the layer does not provide sufficient cooling to offset the heating. This Härm–Schwarzschild instability causes a thermal pulse of an asymptotic-giant-branch star: a helium-shell flash with an enormous but brief increase in internal helium luminosity. It is a shell instability, not repeated ignition of the entire stellar core.

The flash creates an intershell convection zone, expands the layers above it and cools or extinguishes hydrogen burning temporarily. Much of the flash energy goes into expansion and internal rearrangement, so the huge peak shell luminosity need not immediately appear as a comparable surface luminosity. After the layer expands and cools, the helium flash subsides; hydrogen burning later resumes and replenishes the helium layer, repeating the cycle.

After a pulse, the convective envelope may penetrate into the intershell: third dredge-up transports newly formed carbon and slow neutron-capture process products to the surface. This helps explain carbon stars and heavy-element enrichment. Strong winds remove the envelope over successive pulses. When the envelope is too small to sustain the giant configuration, the star leaves the AGB and exposes a hot core that becomes a white dwarf. Pulse behavior, dredge-up efficiency and the number of cycles depend on core/envelope mass, composition and mass loss.

The endpoint is set chiefly by the final core mass and composition, with envelope loss and binary interaction modifying the relation to initial mass. A star that never sustains hydrogen fusion is a brown dwarf, which cools; it is not a post-main-sequence remnant of a hydrogen-burning star. Very low-mass hydrogen-burning stars have such long lifetimes that their predicted eventual helium-white-dwarf endpoints need not yet be represented by isolated evolved examples.

Low- and intermediate-mass stars generally lose their giant envelopes and leave white dwarfs supported by electron degeneracy pressure. Most have carbon-oxygen cores; more massive degenerate cores may be oxygen-neon, while a helium white dwarf commonly results from binary stripping before helium ignition. The exposed hot core can ionize the expelled gas as a planetary nebula. The remnant subsequently cools without sustained central fusion. The Chandrasekhar limit constrains a cold nonrotating electron-supported core; it is not a universal initial-star mass cutoff.

More massive stars can proceed through advanced burning to an iron-group core. Further fusion of such nuclei does not provide energy, while electron captures and photodisintegration reduce pressure support and drain energy. Collapse can produce a core-collapse supernova and a neutron star, or a black hole if a sufficiently massive core collapses directly or gains substantial fallback. The remnant outcome depends on final core structure and the success of ejecting the mantle, rather than only on the original mass. Some oxygen-neon cores instead undergo an electron-capture supernova.

Binary evolution introduces additional branches: an accreting white dwarf may undergo classical novae, a thermonuclear Type Ia supernova leaving no ordinary white-dwarf remnant, or collapse in an appropriate core-composition regime. Extremely massive pair-unstable cores can be disrupted entirely by a pair-instability supernova. These possibilities are distinct from simple isolated-star cooling or ordinary core collapse. The broad white-dwarf/neutron-star/black-hole distinction follows the remnant support mechanism and surviving core mass; NASA's stellar overview and stellar-type account describe the corresponding observational classes.