Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 63 / 4

Since the donor fills its Roche lobe, use $R_2=R_L$ and substitute its equilibrium mass-radius relation into the lobe approximation:Kepler's third law then givesso the Roche-lobe period-mass relation for a linear donor radius law isThe total binary mass cancels, rather than being assumed equal to the white-dwarf mass.

The orbital radii about the center of mass are $a_1=a{M}_2/M$ and $a_2=a{M}_1/M$. Summing the two orbital angular momenta givesWith transferred matter retained by the accretor, total mass is fixed. Logarithmic differentiation givesFor $\dot{J}=0$, the separation response is $\dot{a}/a=2(q-1){\dot{M}}_2/M_2$. Since $R_L\propto a{M}_2^{1/3}$ at fixed total mass,The second relation uses the supplied equilibrium donor sequence. As the donor loses mass, ${\dot{M}}_2<0$. If $q>4/3$, its Roche lobe shrinks fractionally faster than this donor radius, so the excess filling increases and drives more transfer: it is positive feedback. If $q<4/3$, infinitesimal loss instead reduces overfill on that assumed sequence, so some continuing driver is needed to retain contact.

There are two limits to the literal dynamical interpretation. The given initial range is $q<1$, so $q>4/3$ is an extrapolated instability test, not a state inside that range. Also, a true dynamical stability of binary mass transfer test needs the donor's adiabatic stellar radius response exponent ${\zeta }_{\mathrm{ad}}$, rather than its thermal-equilibrium exponent. In general,Thus $4/3$ is the result for the assumed ${\zeta }_{\ast }=1$ response. A fully convective idealized donor has ${\zeta }_{\mathrm{ad}}=-1/3$, giving a dynamical threshold $2/3$, already below $4/3$. The equilibrium radius law alone should not be used to assert a universal dynamical threshold.

One continuing driver is magnetic braking of a binary star. A magnetized wind carries away the donor's spin angular momentum; tides that maintain near-synchronous rotation replenish that spin from the orbit, removing orbital angular momentum. With an external $\dot{J}<0$, the lobe response becomesEquating this to the donor response proves the binary mass-transfer contact equation in this model:A negative torque therefore drives mass loss while maintaining contact for $q<4/3$, subject to the chosen donor response and braking assumptions.

To obtain the specified expansion, use the usual nova-ejection prescription: the escaping material is expelled isotropically from the white dwarf and carries its specific orbital angular momentum, with no additional torque. This is isotropic re-emission from a binary star. Circularity alone does not determine the angular momentum carried by the ejecta; the prescription is an additional physical assumption.

The accretor's orbital radius is $a_1=a{M}_2/M$, so its specific angular momentum satisfiesDuring ejection, take $\delta M_1=-\delta m$, $\delta M_2=0$ and $\delta M=-\delta m$, where $\delta m>0$. ConsequentlyDifferentiate $J=M_1{M}_2\sqrt{Ga/M}$ to first order:Using $M/M_1=1+q$ yieldsThe same lobe law now has changing total mass but unchanged donor mass, soThe donor radius is unchanged on the short eruption timescale, whereas its lobe expands. It therefore falls inside the lobe and Roche-lobe overflow stops in this idealized model. This is nova-induced binary detachment.

The need for the ejection prescription can also be seen algebraically. If the ejecta instead have mean specific angular momentum $j_{\mathrm{ej}}={\eta }_{J}J/M$, the same circular-orbit differentiation givesThe requested coefficient is obtained only when ${\eta }_J=q$. Other prescriptions can produce a different expansion or a contraction; the stated circularity condition does not rule them out. The ejection may be short compared with the transfer time while remaining slow compared with an orbit, consistent with circularity. If truly impulsive, an eccentricity is generally excited unless an additional circularization assumption is made.

Now put $\Gamma =-\dot{J}>0$ for the constant external torque. Keep the masses, mass ratio and torque coefficients fixed to leading order over one small-$\delta m$ cycle; corrections in those coefficients affect the calculated times only at the next order. During detachment there is no transfer, so fixed component masses giveThe fractional lobe gap $4\delta m/(3M)$ closes in timeDuring the semi-detached accumulation phase, use the same external torque and the equilibrium donor contact equation derived in (a):Accumulating the next layer therefore takesDivision cancels the layer mass, torque and angular momentum:The result uses $q<4/3$ for a positive steady accumulation rate, unchanged donor structure during the eruption and the same external torque in the two phases. Here the given $q<1$ ensures a positive denominator. It describes an idealized nova cycle; extra frictional loss, irradiation-driven donor changes or a different ejection angular momentum would change the ratio.