# Physics 616

• Prof. Andrew W. Steiner
(or Andrew or "Dr. Steiner")
• Office hour: 103 South College, Thursday 11am
• Email: awsteiner@utk.edu
• Homework: Electronically as .pdf
• You may work with each other on the homework, but you must write the solution in your own words

Use down and up arrows to proceed to the next or previous slide.

## Binary star system

• Begin with $$\frac{m^2}{\ell_0^2} = m \omega^2 \left( \frac{\ell_0}{2} \right)$$ (why?) thus $$\omega=\left(\frac{2 m}{\ell_0^3}\right)^{1/2}$$
• Schutz shows $$\begin{eqnarray} {I\!\!\!-}_{xx} &=& -{I\!\!\!-}_{yy} = \frac{1}{4} m \ell_0^2 e^{-2 i \omega t} \\ {I\!\!\!-}_{zz} &=& \frac{i}{4} m \ell_0^2 e^{-2 i \omega t} \end{eqnarray}$$ and then that the amplitude of the radiation is $\sim m \ell_0^2 \omega^2/r$ and that this value is $10^{-23}$ at Earth for PSR B1913+16

## Binary star system II

• This result is often quoted $$h \sim \frac{G}{c^4 r} \frac{d^2 Q}{dt^2}$$ where $$Q \equiv \int~d^3 x~\rho \left[ x_i x_j - x^2 \delta_{ij}\right]$$ or $$h \sim 10^{-22} \left(\frac{M}{2.8~\mathrm{M}_{\odot}}\right)^{5/3} \left(\frac{0.01~\mathrm{s}}{P}\right)^{2/3} \left(\frac{100~\mathrm{Mpc}}{r}\right)$$
• Unfortunately all sources too low in frequency or too far away to be currently observed this way
• Gravitational waves have wavelength which is much larger than the size of the source: no imaging
• Note that the "strain" (also called 'h') is defined by the Fourier transform and has different units

## Energy and Flux

• Flux of a wave with frequency $\Omega$ and amplitude $A$ is $$F = \frac{1}{32 \pi} \Omega^2 A^2$$ and the amplitude is (for example considering a wave moving in the z direction) $$\left< \left(\bar{h}^{TT}_{xx} \right)^2 \right> = \frac{1}{2} A^2$$ thus $$F = \frac{\Omega^2}{32 \pi} \left< \bar{h}^{TT}_{\mu \nu} \bar{h}^{TT \mu \nu} \right>$$

## Energy Loss

• Consider our object emitting spherical gravitational waves from last time. One can show that, at a distance r on the z axis, $$F = \frac{\Omega^6}{16 \pi r^2} \left< 2 {I\!\!\!-}_{ij} {I\!\!\!-}^{ij} - 4 n^{j} n^{k} {I\!\!\!-}_{ji} {{I\!\!\!-}_k}^{i} + n^i n^j n^k n^{\ell} {I\!\!\!-}_{ij} {I\!\!\!-}_{k\ell} \right>$$
• To get the luminosity, we integrate this over the sphere of radius $r$, and the general result is $$L = \frac{1}{5} \left< \dddot{I\!\!\!-}_{ij} \dddot{I\!\!\!-}^{ij} \right>$$ with a factor of $c^5/G$ to get the usual units.
• In fact, $c^5/G=4 \times 10^{59}~\mathrm{erg}/\mathrm{s}$ is a maximum luminosity (compare with the Eddington luminosity)

## Hulse-Taylor Binary

• Two $1.4~\mathrm{M}_{\odot}$ neutron stars
• From the previous results we get $$L = \frac{8}{5} M^2 \ell_0^4 \omega^6 \approx 4 \left( M \omega\right)^{10/3}$$
• For a period of 7.75 hours we get $L=1.71 \times 10^{-29}$
• Gravitational waves carry energy away, thus the period must decrease. Presuming circular motion, the period changes at a rate of $$\frac{dP}{dt} = -6 \times 10^{-6} \mathrm{s}/\mathrm{yr}$$ but the real number is an order of magnitude larger.
• Lifetime is short (thus the chirp) $$\tau_{GW} = 2.43 \left(\frac{M}{\mathrm{M}_{\odot}}\right)^{-5/3} \left(\frac{f}{100~\mathrm{Hz}}\right)^{-8/3}~\mathrm{s}$$

## Spinning Neutron Stars and Core-Collapse

• If neutron stars have a quadrupole deformation, then spinning neutron stars generate gravitational waves $$h \sim 2 \varepsilon \Omega^2 I_{\mathrm{NS}}/r$$ where $\varepsilon$ is the asymmetry about the spin axis
• How do you break the axial symmetry?
• Such an asymmetry has been referred to as a neutron star "mountain"
• This has not yet been detected, but has not yet been ruled out by LIGO
• Core-collapse supernovae must be relatively close to emit GWs (a few kpc) that are currently detectable, unless they rotate rapidly (then maybe up to 50kpc)
Gossan et al. (2016)

## Pulsar Timing Arrays

From Lommen (2015) based on Sesana et al.
• Pulsars emit regular bursts of radio; accuracy 100 ns over 5 years
• Detect GWs from massive BH binaries (possible) and relic GWs from the big bang (unlikely) and also "cosmic strings", etc.
• Background from many sources detected before any single source

## Group Work

• Complete 9.49b and c in Schutz. Also, compare 9.170 with the similar result in Lattimer's 2012 Ann. Rev. Nucl. Part. Sci. article. What is the origin of the differences in these two equations?