Past exam of the astrophysics course of the University of Oxford / 2015 / 1

A close binary star can produce two compact stellar objects if both initial stellar masses are sufficiently large. The initially more massive star evolves first, transfers material or loses its envelope, and leaves a neutron star or black hole after core collapse. The system must remain bound despite mass loss and any natal kick. Subsequent evolution of the companion can produce a common envelope: orbital energy transferred to the envelope can expel it and greatly reduce the separation. A second collapse must again leave a bound pair. A sufficiently close surviving binary star then loses orbital energy and angular momentum through gravitational waves and may merge. Stable transfer, winds and kicks make this a possible channel rather than an inevitable outcome; dynamical encounters in dense stellar systems provide another channel.

A neutron star merger, or a neutron star tidally disrupted by a black hole outside its capture region, can leave a hot accretion disk around the remnant. Rapid accretion can power a relativistic jet and a short gamma-ray burst. Beaming means that an observer must lie sufficiently near the jet axis, and not every merger produces an observable burst. A clean binary black hole merger has no substantial baryonic fuel and is not ordinarily expected to power this mechanism. Disruption in a neutron-star–black-hole pair depends on binary mass ratio, black-hole spin and the neutron star's stellar radius.

The changing mass quadrupole moment produces an inspiral gravitational wave signal whose frequency and amplitude increase toward merger. The chirp mass controls the leading frequency evolution, while higher-order phasing and the merger carry information about binary mass ratio and spins. Neutron stars also introduce tidal effects. These signals motivate a search with an interferometer such as LIGO; the supplied ranges quantify the paper's proposed detection prospects, without assuming any particular later observing-run performance.

Neutron-rich material ejected dynamically or in remnant-disk outflows can undergo the r-process: successive neutron captures, followed by beta decays, make heavy nuclei. The yield depends on the amount ejected, its entropy, expansion and electron fraction; not all outflow is equally neutron rich. Radioactive products can power a kilonova. A neutron-star–black-hole event contributes if matter escapes capture. An isolated binary black hole merger has no analogous reservoir of neutron-rich baryonic material, so it contributes essentially no such nucleosynthesis.

Use constant component masses and a circular Keplerian orbit. Then $J=\mu \sqrt{GMA}$ gives $\dot{J}/J=\dot{A}/(2A)$, while $P=2\pi A^{3/2}/\sqrt{GM}$ gives $\dot{P}/P=3\dot{A}/(2A)=3\dot{J}/J$. Dividing the supplied loss law by $J$ therefore yieldsThe positive instantaneous orbital decay timescale is ${\tau }_P=P/|\dot{P}|$, since $\dot{P}<0$. For $t_G=10^{10}$ yr, the requested prescription givesIt is important to distinguish this prescription from the integrated gravitational-wave inspiral time. Integrating $\dot{A}=-C/A^3$ givesFinite contact radii supply a small lower-limit correction when the initial separation is much larger than the compact objects. Thus the periods below satisfy the printed instantaneous-timescale criterion and have a formal merger time $3.75$ Gyr. If instead the requirement is literally $t_{\mathrm{merge}}=10$ Gyr, every period limit is larger by $(8/3{)}^{3/8}=1.445$. Eccentric systems require a different decay calculation.

The binary of neutron stars has $M=2.8M_{\odot }$ and $\mu =0.7M_{\odot }$. Inserting these into the circular orbital decay timescale givesUsing the integrated ten-billion-year merger criterion instead would give $15.2$ h.

For the mixed pair, $M=11.4M_{\odot }$ and $\mu =(14/11.4)M_{\odot }=1.228M_{\odot }$. The prescribed instantaneous orbital decay timescale givesThe integrated ten-billion-year merger criterion instead gives $26.6$ h. Both are pre-disruption circular-orbit estimates; the final contact/capture process differs from that for two neutron stars.

For two black holes of equal mass, $M=20M_{\odot }$ and $\mu =5M_{\odot }$. The prescribed instantaneous orbital decay timescale givesThe integrated ten-billion-year criterion gives $51.9$ h. These large initial periods compared with a neutron-star pair reflect the stronger radiation reaction of the more massive system.

Treat the quoted distances as sharp isotropic detection radii in a homogeneous Euclidean population. With galaxy number density $n_g$, the number surveyed is $N_g=(4\pi /3)n_g{D}^3$. One event per year requires a per-galaxy merger rate $\mathcal{R}=1/N_g$ per year. ThusThe larger gravitational wave range gives enormous volume leverage: even a rare merger in a typical galaxy can yield frequent detections across the surveyed population. This supports promising detection prospects if astrophysical rates exceed these thresholds. The estimate is not a prediction of the rates themselves. Real sensitivity depends on sky direction, binary inclination, detector duty cycle and a signal-to-noise selection function; a quoted horizon distance is not automatically a volume-averaged range. Cosmological volume and source-time corrections also matter toward the largest quoted distance. The calculation follows the simple assumptions specified by the paper.

Infer the component masses and spins from the gravitational wave phasing, using the chirp mass together with binary mass ratio information. The paper's representative pairs have total masses $2.8$, $11.4$ and $20M_{\odot }$, and the more massive mergers terminate their inspirals at lower frequencies. A black-hole ringdown and its quasi-normal modes identify the final compact remnant, although both a neutron-star–black-hole merger and a sufficiently massive neutron-star pair can also leave a black hole.

Tidal deformability and any disruption cutoff provide a more direct signature of a neutron star than total mass alone. A disrupted neutron star may supply a gamma-ray burst, kilonova and afterglow; these help distinguish matter-containing mergers from a clean black-hole pair. Their absence does not establish a black-hole pair, because jets are beamed and ejecta may be faint or swallowed. Measured gravitational wave masses are redshifted detector-frame masses; association with a host can supply the redshift needed for source-frame masses. Classification should combine the waveform and any electromagnetic evidence, allowing for parameter degeneracies.