Gravitational Waves: scale of the phenomenon

Problem 1: Athletic challenge

A world champion in sprint starts the race. Estimate the portion of energy he spends that goes into production of gravitational waves.

solution

\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}

Problem 2: Spinning dumb-bell

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

solution

\[h\sim MR^{2}\omega^{2}/r\sim 10^{-38}\]

Problem 3: Supernova explosion

Estimate the amplitude of the gravitational wave produced by a supernova explosion in our Galaxy.

solution

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.

Problem 4: Pulsar binary

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.

solution

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 \[h\sim | h_{\alpha\beta}| \approx \frac{10^{-21}}{r/15Mpc}.\]

Problem 5: Neutron star mergers

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.

solution

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, \[\frac{F_{GR}}{F_{\odot}}\sim {10}^{-6}.\]

Problem 6: The Jovian generator

In the Solar system the most considerable source of gravitational waves is the subsystem of Sun and Jupiter. Estimate its power.

solution

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 \[L_{GW}^{\odot +Jup}\simeq 5kW.\]

Problem 7: Gravity in atom

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.

solution

\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.)