Igor at 13:52, 15 January 2013

2013-01-15T13:52:30Z

Cosmo All at 03:08, 8 January 2013

2013-01-08T03:08:56Z

Cosmo All at 02:53, 8 January 2013

2013-01-08T02:53:25Z

Cosmo All: Created page with "9"

2012-12-26T12:07:12Z

Created page with "9"

New page

[[Category:Weak field limit and gravitational waves|9]]

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 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
 <br/>
-Hint: \[h\sim MR^{2}\omega^{2}/r\sim 10^{-38}\]
+<div class="NavFrame collapsed">
-<!--<div class="NavFrame collapsed">
    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <p style="text-align: left;"></p>
+     <p style="text-align: left;">\[h\sim MR^{2}\omega^{2}/r\sim 10^{-38}\]</p>
    </div>
-</div>--></div>
+</div></div>

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	[[Category:Weak field limit and gravitational waves\|9]]		[[Category:Weak field limit and gravitational waves\|9]]
		+
		+
		+
		+
		+
		+
		+	<div id=""></div>
		+	<div style="border: 1px solid #AAA; padding:5px;">
		+	=== 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.
		+	<div class="NavFrame collapsed">
		+	<div class="NavHead">solution</div>
		+	<div style="width:100%;" class="NavContent">
		+	<p style="text-align: left;">\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}</p>
		+	</div>
		+	</div></div>
		+
		+
		+	<div id=""></div>
		+	<div style="border: 1px solid #AAA; padding:5px;">
		+	=== 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
		+	<br/>
		+	Hint: \[h\sim MR^{2}\omega^{2}/r\sim 10^{-38}\]
		+	<!--<div class="NavFrame collapsed">
		+	<div class="NavHead">solution</div>
		+	<div style="width:100%;" class="NavContent">
		+	<p style="text-align: left;"></p>
		+	</div>
		+	</div>--></div>
		+
		+
		+	<div id=""></div>
		+	<div style="border: 1px solid #AAA; padding:5px;">
		+	=== Problem 3: Supernova explosion===
		+	Estimate the amplitude of the gravitational wave produced by a supernova explosion in our Galaxy.
		+	<div class="NavFrame collapsed">
		+	<div class="NavHead">solution</div>
		+	<div style="width:100%;" class="NavContent">
		+	<p style="text-align: left;">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.</p>
		+	</div>
		+	</div></div>
		+
		+
		+	<div id=""></div>
		+	<div style="border: 1px solid #AAA; padding:5px;">
		+	=== 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.
		+	<div class="NavFrame collapsed">
		+	<div class="NavHead">solution</div>
		+	<div style="width:100%;" class="NavContent">
		+	<p style="text-align: left;">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
		+	\[h\sim \| h_{\alpha\beta}\|
		+	\approx \frac{10^{-21}}{r/15Mpc}.\]</p>
		+	</div>
		+	</div></div>
		+
		+
		+	<div id=""></div>
		+	<div style="border: 1px solid #AAA; padding:5px;">
		+	=== 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.
		+	<div class="NavFrame collapsed">
		+	<div class="NavHead">solution</div>
		+	<div style="width:100%;" class="NavContent">
		+	<p style="text-align: left;">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,
		+	\[\frac{F_{GR}}{F_{\odot}}\sim {10}^{-6}.\]</p>
		+	</div>
		+	</div></div>
		+
		+
		+	<div id=""></div>
		+	<div style="border: 1px solid #AAA; padding:5px;">
		+	=== 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.
		+	<div class="NavFrame collapsed">
		+	<div class="NavHead">solution</div>
		+	<div style="width:100%;" class="NavContent">
		+	<p style="text-align: left;">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
		+	\[L_{GW}^{\odot +Jup}\simeq 5kW.\]</p>
		+	</div>
		+	</div></div>
		+
		+
		+	<div id=""></div>
		+	<div style="border: 1px solid #AAA; padding:5px;">
		+	=== 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.
		+	<div class="NavFrame collapsed">
		+	<div class="NavHead">solution</div>
		+	<div style="width:100%;" class="NavContent">
		+	<p style="text-align: left;">\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.)</p>
		+	</div>
		+	</div></div>

Gravitational Waves: scale of the phenomenon - Revision history

Igor at 13:52, 15 January 2013

Cosmo All at 03:08, 8 January 2013

Cosmo All at 02:53, 8 January 2013

Cosmo All: Created page with "9"