Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 322 (Binary Stars)

In a circular binary star, the distances from the centre of mass are $a_1=a{M}_2/M$ and $a_2=a{M}_1/M$. Summing the two orbital angular momenta givesEquivalently, $J=\mu \sqrt{GMa}$ by Kepler third law.

First consider a rapid conservative perturbation. Both $M=M_1+M_2$ and $J$ are fixed, while $d{M}_1=-d{M}_2$. With $q=M_2/M_1$,soThe Roche lobe formula then gives its mass-radius exponentAfter mass loss, dynamical stability of binary mass transfer requires the donor to shrink relative to its lobe. Since $d\log M_2<0$, this means ${\zeta }_{\mathrm{ad}}>{\zeta }_L$, or

For stable secular conservative binary mass transfer, put $m={\dot{M}}_2/M_2$. The angular-momentum and contact conditions areElimination of $\dot{a}/a$ givesHere magnetic braking of a binary star removes orbital angular momentum, while the donor's expansion maintains Roche-lobe overflow.

In a cataclysmic variable, hydrogen-rich material accumulates on a degenerate white dwarf. Degeneracy prevents initial expansion from regulating its temperature, so nuclear ignition produces the thin-shell instability and a classical nova. Nuclear burning of hydrogen to helium releases about $0.007m{c}^2$, whereas the binding energy at a white-dwarf surface is only of order $G{M}_{1}m/R_1$, typically a few $10^{-4}m{c}^2$. Even modest coupling can therefore eject all the newly accreted envelope without disrupting the white dwarf.

Finally suppose every transferred mass element is expelled by isotropic re-emission from a binary star. Then ${\dot{M}}_1=0$, $\dot{M}={\dot{M}}_2$, and expelled matter carries the white dwarf's specific angular momentum $j_1=(M_2/M{)}^2{a}^2\Omega$. HenceOn the other hand, logarithmic differentiation of $J=M_1{M}_2{M}^{-1/2}{G}^{1/2}{a}^{1/2}$ and of the Roche-lobe radius givesEliminating the separation produces

In centre-of-mass and relative coordinates, the kinetic energy separates into centre-of-mass motion plus $\mu v^2/2$, while the mutual potential is $-G{M}_1{M}_2/r=-GM\mu /r$. In the centre-of-mass frame the orbital energy is therefore

The force is central, so the specific relative angular momentum $h=r^2\dot{\theta }$ is conserved. With $u=1/r$, the radial equation becomes the Binet equationwhose solution isHere $\theta$ is the true anomaly measured from periapsis and $e$ is the orbital eccentricity. Substitution into the energy, or evaluation at an apsis, givesand hence

Immediately before the supernova the circular relative speed obeys $v^2=GM/a$. The impulsive supernova kick in a binary star changes it toApplying the energy formula just after the explosion at the unchanged separation $r=a$ givesUsing the original circular-speed relation,The post-explosion system is bound exactly when $1+2\alpha \cos \psi +{\alpha }^2<2M^{\prime }/M$. The left side ranges from $(1-\alpha {)}^2$ to $(1+\alpha {)}^2$, so every kick direction remains bound ifwhereas every direction unbinds the stars if

For an unbound orbit, conservation of relative specific energy at infinity gives $V^2/2=v^{\prime 2}/2-G{M}^{\prime }/a$. Therefore

A common envelope is invoked because many observed compact binaries have separations far smaller than the radii of their progenitor giants; ordinary conservative transfer cannot remove enough orbital energy and angular momentum. It begins when rapid or dynamically unstable Roche-lobe overflow engulfs the companion, or when a companion is swallowed by an expanding giant. Drag inside the envelope transfers orbital energy and angular momentum to gas, causing a fast inspiral. If the deposited energy ejects the envelope before the cores touch, a close exposed-core binary survives; otherwise the cores merge. Partial ejection or delayed thermal readjustment can complicate either outcome.

The common-envelope energy formalism writesHere $M_d,M_c,M_{\mathrm{env}}$ are donor, core, and envelope masses; $\lambda$ describes envelope structure; and ${\alpha }_{\mathrm{CE}}$ is the efficiency with which released orbital energy unbinds gas. Solving estimates $a_f$, while failure to supply $E_{\mathrm{bind}}$ predicts merger.

A carbon--oxygen white dwarf can make a Type Ia supernova when carbon ignites under degenerate conditions. Burning carbon and oxygen to iron-group and intermediate-mass nuclei releases of order $10^{44}$ J, comparable to and exceeding the white dwarf's gravitational binding energy, so the thermonuclear flame disrupts the star. Radioactive ${}^{56}\mathrm{Ni}$ and then ${}^{56}\mathrm{Co}$ decay power the optical light curve by depositing gamma-ray and positron energy. Possible triggers include near-Chandrasekhar mass central ignition and sub-Chandrasekhar detonations initiated by an accreted helium shell or a merger.

One route from zero-age masses $8M_{\odot }$ and $5M_{\odot }$ is as follows. The $8M_{\odot }$ primary evolves first, fills its Roche lobe, and transfers its envelope; if stable, the orbit usually widens after mass-ratio reversal, leaving a carbon--oxygen core that becomes the first white dwarf. The rejuvenated secondary later evolves and becomes the more massive giant. Transfer to the white dwarf then has an extreme mass ratio and is dynamically unstable, producing a common envelope. Inspiral ejects the giant envelope and leaves the first white dwarf close to the secondary's exposed helium or carbon--oxygen core, which becomes the second white dwarf. Depending on the earlier separation, a second unstable episode may be required. Once detached, gravitational-wave emission from a binary system removes energy and angular momentum, steadily shrinking the orbit until the white dwarfs contact. This is the Double-degenerate Type Ia supernova scenario.

If white-dwarf mass transfer is dynamically stable, the lighter donor is gradually disrupted or transferred at rates that may lead to quiescent burning, off-centre carbon ignition, and conversion to an oxygen--neon remnant followed by accretion-induced collapse, rather than prompt thermonuclear disruption. Dynamically unstable transfer can overcome this by producing a violent merger: rapid stream impact, tidal heating, shocks, and helium or carbon hotspots may detonate before the remnant settles and ignites carbon off centre. Whether this happens is the central ignition problem of the double-degenerate channel.