Binary systems

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Problem 1: Symmetric binary system

Find the strain tensor $\bar{h}_{\alpha\beta}$ of the metric perturbation far from a binary system with components of equal masses $M$. The radius of the orbit is $R$, orbital frequency $\omega$.
Hint: Calculating the quadrupole moment of the system gives \[\bar{h}^{\alpha\beta} =-\frac{8\omega^2 MR^{2}}{r} \left( \begin{matrix} \cos \left[ 2\omega (t-r) \right] & \sin \left[ 2\omega (t-r) \right] & 0 \\ \sin \left[ 2\omega (t-r) \right] & -\cos \left[ 2\omega (t-r) \right] & 0 \\ 0 & 0 & 0 \\ \end{matrix} \right)\]


Problem 2: Gravitational luminosity

Calculate the gravitational luminosity for the binary system with equal masses of the previous problem.
Hint: Calculating the quadrupole tensor's square and averaging gives \[L_{GW}=\frac{128}{5}M^{2}R^{4}\omega^{6}.\]


Problem 3: Asymmetric binary system

Calculate the gravitational luminosity for a binary system with components of arbitrary masses $M_1$ and $M_2$.
Hint: If $R$ is the distance between the components, then \[L_{GW}=\frac{32}{5}\frac{G^4}{c^5} \frac{M_1^2 M_2^2 (M_1 +M_2)}{R^5}.\]


Problem 4: Binary system evolution

Find the rate of the two components of a binary with masses $M_1$ and $M_2$ approaching due to emission of gravitational waves. Obtain the distance between them $R$ as a function of time.
Hint: Equating $dE(R)/dt$ to losses due to gravitational radiation, we obtain differential equation for $R(t)$. Integration yields \[ R(t)=4\;\Big[ \frac{G^{3}M_{1}M_{2}(M_1 +M_2)}{5 c^5} (t_{coal}-t)\Big]^{1/4}.\]


Problem 5: Binary system lifetime

Calculate the lifetime of a binary system of mass $M$ and orbital radius $R$ due to emission of gravitational waves alone. Make estimates for the Hulse-Taylor binary pulsar PSR1913+16 with the following characteristics: $M\approx 2.8 M_\odot$, $R\approx 4.5 R_\odot$, $R_\odot \approx 7\times 10^8$m.


Problem 6: Mass through GW frequency

Show that the mass of a binary can be determined through the frequency of gravitational waves it generates.
Hint: It is possible thanks to the relation \[f_{GW}=(1.9Hz)\Big(\frac{1.4M_{\odot}}{M}\Big)^{5/8} \Big(\frac{days}{t_{coal}-t}\Big)^{3/8}.\]