Binary systems

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 $\omega$.

Problem 2: Gravitational luminosity

Calculate the gravitational luminosity for the binary system with equal masses of the previous problem.

Problem 3: Asymmetric binary system

Calculate the gravitational luminosity for a binary system with components of arbitrary masses $M_1$ and $M_2$.

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.

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.