Past exam of the astrophysics course of the University of Oxford / 2018 / 3

GW170817 supplied the first direct gravitational wave observation of a binary neutron star inspiral. The chirping waveform tests compact-binary general relativity, constrains masses and tidal effects, and links a class of known compact binaries to observable sources of gravitational waves. The electromagnetic counterpart identifies the host and distance, enabling complementary source and cosmological inference. This is distinct from the earlier binary-black-hole detections; it established a system involving matter that could also be observed electromagnetically.^[ref]

The short gamma-ray burst GRB 170817A followed the merger signal by about $1.7\mathrm{s}$, providing direct evidence that at least some short gamma-ray bursts arise from neutron star mergers. The association connects their high-energy emission and relativistic outflows to compact-binary coalescence. It does not establish that every short burst has this progenitor, nor that the observed gamma-ray luminosity is independent of viewing angle.^[ref]

The evolving optical and infrared kilonova was consistent with radioactive heating of neutron-rich merger ejecta. Rapid neutron capture, the r-process, can synthesize nuclei heavier than iron; decay of the newly formed unstable nuclei powers the transient. Its redward evolution and large inferred opacity supported production of heavy-element-rich ejecta. Thus mergers are a demonstrated important site for heavy-element production, while the fractions of individual elements and their total cosmic contribution remain model-dependent. The transient was not a direct identification of every very heavy element.^[ref]

Treat the stated $120\mathrm{Mpc}$ as the radius of a complete, volume-limited search, with uniform galaxy density $n_g$. Its surveyed volume and number of galaxies areThe observing time is $\Delta t=117/365.25=0.3203\mathrm{yr}$. One detection estimatesor one merger per approximately $2.3\times 10^4$ years per Milky-Way-type galaxy. The equivalent volume rate is about $431\mathrm{Gp}{\mathrm{c}}^{-3}\mathrm{y}{\mathrm{r}}^{-1}$. The event's actual $40\mathrm{Mpc}$ distance is not the survey radius and must not replace $120\mathrm{Mpc}$ in the exposure.

A representative Galactic core-collapse supernova rate is about two per century, roughly $4.6\times 10^2$ times larger than this merger-rate estimate. The historical ${}^{26}\mathrm{Al}$ estimate was $1.9\pm 1.1$ per century.^[ref] The one-event Poisson uncertainty is large. Detector orientation, duty cycle, completeness and whether the quoted radius is an optimal horizon or an effective range also change the exposure; the boxed value is the specified simple-volume estimate.

Keep both component masses constant during a circular, gravitational-radiation-driven inspiral. Since $J=\mu \sqrt{GMA}$, differentiation gives $\dot{J}/J=\dot{A}/(2A)$. Combining this with the supplied angular-momentum loss yieldsThe expression $A/\dot{A}$ printed in the source is negative during inspiral. A positive characteristic shrinking timescale isThe actual remaining coalescence time follows by integrating $A^{3}dA=-Kdt$:This constant factor does not change the requested separation scaling.

For an old, approximately coeval population, let $dN/d{A}_0=C$ over the relevant separation interval, and write the positive delay as $t_m=B{A}_0^4$. Conservation of event counts under this change of variable givesThereforeThe scaling applies between the minimum and maximum delays of the assumed separation distribution. It is a rate per delay interval, not the number per logarithmic delay interval.

For the Double-degenerate Type Ia supernova scenario, measure the Type Ia supernova rate per formed stellar mass in old galaxies, grouped by the time since their main star-formation episode. Correct the event counts for observing time and detection efficiency. In an ideal burst population, the normalized rate measures the delay-time distribution directly. If gravitational waves inspiral dominates the delay and the initial white-dwarf separations are uniformly distributed in separation, c predicts a slope $-3/4$.

More generally, if $dN/dA\propto A^{\beta }$ and $t_m\propto A^4$, the predicted rate scales as $t_m^{(\beta -3)/4}$. A distribution uniform in $\log A$ has $\beta =-1$ and gives $t_m^{-1}$. Thus a slope differing from $-3/4$ does not alone rule out mergers: the separation distribution must also be tested.

An extended star formation history gives a convolution, $R(t)=\int d\tau \mathrm{SFR}(t-\tau )\Psi (\tau )$, rather than the single-burst expression. The time since star formation stopped need not equal each progenitor's age. Stellar lifetimes before formation of the second white dwarf, common-envelope evolution, masses and mass ratios, metallicity, delay-dependent explosion efficiencies, galaxy-age estimates and survey selection all affect the comparison. Some CO-white-dwarf mergers need not make Type Ia supernovae, and other channels can contribute to the observed rate. Agreement in slope and normalization therefore supports the hypothesis under stated assumptions but is not a unique identification of the channel.