Chapter 13 · Section 9 of 10
A world champion in sprint starts the race. Estimate the portion of energy he spends that goes into production of gravitational waves.
\begin{align} P_{GR}&=\frac{G}{5c^5}(\dddot{Q})^2 \sim \frac{GM^{2}L^{4}}{5c^5 t^6} \sim 10^{-45}W, \\ P&\sim \frac{1}{2}\frac{ML^2}{t^2}\frac{1}{t} \sim {{10}^{3}}W, \\ \eta&\equiv \frac{P_{GW}}{P}=\sim 10^{-48}. \end{align}
Imagine a dumb-bell consisting of two 1-ton compact masses with their centers separated by 2 meters and spinning at 1 kHz about a line bisecting and orthogonal to their symmetry axis.Estimate the amplitude of gravitational waves at the distance of $r=300km$ from this source
\[h\sim MR^{2}\omega^{2}/r\sim 10^{-38}\]
Estimate the amplitude of the gravitational wave produced by a supernova explosion in our Galaxy.
The energy flux the wave carries is quadratic in amplitude, so from energy conservation \[h^{2}R^{2}=const,\] where $h$ is the amplitude, $R$ distance to source. At If $R_*$ is the characteristic size of the supernova, and $h_0$ the characteristic amplitude at such distances from the source, then at Earth the amplitude will be \[h=h_{0}\frac{R_*}{R}.\] If the explosion leads to creation of a neutron star or a black hole, $R_*$ can be estimated by the gravitational radius \[R_{*}\sim \frac{2GM}{c^2} \approx 3km\frac{M}{M_\odot}.\] The amplitude $h_0$ is determined by the efficiency $\alpha $ of the conversion of the full energy into that of gravitational waves \[h_{0}=\alpha^{1/2},\] so \[h\sim \alpha^{1/2} \frac{M}{M_\odot}\frac{3km}{R(km)}.\] If the explosion happens at the center of our galaxy $R\sim 10kpc\approx 3\times 10^{7}km$, then assuming $M/M_{*}\approx 3$, we will obtain \[h\sim \alpha^{1/2}\times 10^{-17}.\] For $\alpha \sim 10^{-2}\div 10^{-6}$ the value is reasonably encouraging $h\sim {10}^{-18}\div {10}^{-21}$. But reality might be quite different.
Consider a pair of $1.4{M}_{\odot}$ neutron stars $15Mpc$ away (e.g., near the center of the Virgo galactic cluster) on a circular orbit of $20km$ radius and orbital frequency of $400Hz$. Estimate the amplitude of gravitational wave.
The frequency of gravitational waves is twice the orbital frequency, thus $800Hz$. For the case of the source being dominated by its rest-mass density $\mu $ (non-relativistic internal velocities) \[h^{\alpha\beta}(t,x) =\frac{2G}{c^4}\frac{d^2}{dt^2} Q^{\alpha\beta}( t-r/c).\] In the considered case \[h\sim | h_{\alpha\beta}| \approx \frac{10^{-21}}{r/15Mpc}.\]
Estimate the energy flux of gravitational waves, registered on Earth, from a pair of merging neutron stars in the Virgo cluster with the same parameters. Compare with the energy flux from the Sun.
For a gravitational wave with amplitude $h\sim {{10}^{-21}}$ and frequency $\omega \sim 800Hz$ the energy flux is \[F_{GW}=\frac{1}{32\pi }\frac{c^3}{G}h^{2} \omega^{2}\approx 3mW/m^{2}.\] The radiation energy flux from the Sun is about \[F_{\odot}\approx 1400W/m^{2}.\] Hence during the brief moment when the waves of a coalescing binary neutron star system in the Virgo cluster pass the Earth, \[\frac{F_{GR}}{F_{\odot}}\sim {10}^{-6}.\]
In the Solar system the most considerable source of gravitational waves is the subsystem of Sun and Jupiter. Estimate its power.
The gravitational luminosity for this binary system is \[L_{GW}^{\odot +Jup} =\frac{32}{5}\frac{G^4}{c^5} \frac{M_{\odot}^{2}M_{Jup}^{2} (M_{\odot}+M_{Jup})}{R^5}.\] For values \[M_{\odot}\simeq 1.9\times 10^{30}kg, \quad M_{Jup}\simeq 10^{-3}M_{\odot}, \quad R\simeq 8\times 10^{11}m\] this gives \[L_{GW}^{\odot +Jup}\simeq 5kW.\]
Estimate the lifetime of the hydrogen atom in the $3d$ state with respect to decay into $1s$ due to gravitational (as opposed to electromagnetic) interaction, and emission of a graviton.
\begin{align*} \Gamma \simeq& 0.36\;G m_{e}^{2} \omega \alpha^{4}, \\ &Gm_{e}^{2}\approx 1.75\times 10^{-45}, \\ &\omega =12eV, \\ &\quad\tau =\Gamma^{-1}\approx 2\times 10^{38}\sec. \end{align*} Compare this with the age of the Universe $T_{Universe}\sim 10^{17}\sec$ (for full solution see Lightman et al.)