Past exam of the astrophysics course of the University of Oxford / 2016 / 3 / Solution

A gravitational wave is a propagating tidal perturbation of spacetime. In the weak-field limit its transverse polarizations alternately stretch and compress the separation of freely falling test bodies. A laser interferometer such as LIGO measures the differential optical phase of orthogonal arms. The mirrors approximate free test masses over the detection band; arm cavities enhance the phase response, while suspension, seismic isolation and optical control reduce instrumental disturbances. A strain $h$ produces relative arm-length changes of order $hL$. Coincidence and waveform consistency between detectors distinguish candidate astrophysical signals from local noise.

Potential sources include black-hole and neutron-star binary inspirals and mergers, asymmetric core-collapse supernovae, nonaxisymmetric spinning neutron stars, and a stochastic background from many unresolved sources or early-Universe processes. Their detectability depends on frequency, strain, duration and detector noise; a source producing gravitational radiation need not be observable in a terrestrial interferometer's band.

For slowly varying approximately isotropic stellar wind loss, decreasing the binding mass widens the orbit. The prescribed Jeans-mode mass loss equation givesHot massive stars have metal-line-driven winds. Higher metallicity generally strengthens the radiative coupling and mass loss, increasing orbital expansion and leaving less massive black holes. The supplied initial-contact formula also assigns larger zero-age radii and initial minimum separations at higher metallicity.

The initial total mass is $2M^0$, while the final total mass is $2M_{\mathrm{BH}}$. Neglecting supernova mass loss as requested, their ratio has no additional factor of two. The wind-limited black-hole binary merger window therefore has lower boundaryA nonempty allowed initial-separation interval requires $a_{\min }^V<a_{\max }^{\mathrm{merger}}$. Substituting the supplied final masses gives the following separation bounds, all in units of $R_{\odot }$:The arguments 30 and 100 indicate the initial mass of each star in solar units.

Under these particular approximate formulae, only the $30M_{\odot }$ initial systems at $Z_{\odot }/10$ and $Z_{\odot }/20$ have overlapping bounds. The $100M_{\odot }$, $Z_{\odot }/20$ system is a near miss, not formally allowed: $75.43>71.13$. Given approximate evolutionary scalings, that small mismatch should not be treated as a precise astrophysical exclusion.

Lower metallicity favors merging remnants here through weaker winds, smaller initial stellar radii and larger final black-hole masses. More massive binaries can also produce stronger detectable signals over a suitable frequency range, although the detection volume and band limitations must be considered. The observed rate further depends on formation efficiency, metallicity history, delay times and binary survival. Common-envelope evolution, stellar rotation, supernova mass loss and natal kicks could alter this simplified channel; the wind-only inequalities do not establish that such mergers are impossible through other channels.