Binary systems - Revision history

Igor at 13:50, 15 January 2013

2013-01-15T13:50:56Z

Cosmo All at 03:08, 8 January 2013

2013-01-08T03:08:55Z

Cosmo All at 02:47, 8 January 2013

2013-01-08T02:47:19Z

Cosmo All at 02:41, 8 January 2013

2013-01-08T02:41:38Z

Cosmo All: Created page with "8"

2012-12-26T12:07:01Z

Created page with "8"

New page

[[Category:Weak field limit and gravitational waves|8]]

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 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 $\omega$.
 <br/>
-Hint: Calculating the quadrupole moment of the system gives
+<!--Hint: Calculating the quadrupole moment of the system gives
 \[\bar{h}^{\alpha\beta}
 	=-\frac{8\omega^2 MR^{2}}{r}
@@ Line 19: / Line 19: @@
 	& -\cos \left[ 2\omega (t-r) \right] & 0  \\
 & 0 & 0  \\
-\end{matrix} \right)\]
+\end{matrix} \right)\]-->
 <div class="NavFrame collapsed">
    <div class="NavHead">solution</div>
@@ Line 60: / Line 60: @@
 Calculate the gravitational luminosity for the binary system with equal masses of the previous problem.
 <br/>
-Hint: Calculating the quadrupole tensor's square and averaging gives
+<!--Hint: Calculating the quadrupole tensor's square and averaging gives
-\[L_{GW}=\frac{128}{5}M^{2}R^{4}\omega^{6}.\]
+\[L_{GW}=\frac{128}{5}M^{2}R^{4}\omega^{6}.\]-->
 <div class="NavFrame collapsed">
    <div class="NavHead">solution</div>
@@ Line 76: / Line 76: @@
 Calculate the gravitational luminosity for a binary system with components of arbitrary masses $M_1$ and $M_2$.
 <br/>
-Hint: If $R$ is the distance between the components, then
+<!--Hint: If $R$ is the distance between the components, then
 \[L_{GW}=\frac{32}{5}\frac{G^4}{c^5}
-	\frac{M_1^2 M_2^2 (M_1 +M_2)}{R^5}.\]
+	\frac{M_1^2 M_2^2 (M_1 +M_2)}{R^5}.\]-->
 <div class="NavFrame collapsed">
    <div class="NavHead">solution</div>
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 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.
 <br/>
-Hint: Equating $dE(R)/dt$ to losses due to gravitational radiation, we obtain differential equation for $R(t)$. Integration yields
+<!--Hint: Equating $dE(R)/dt$ to losses due to gravitational radiation, we obtain differential equation for $R(t)$. Integration yields
 \[ R(t)=4\;\Big[
 	\frac{G^{3}M_{1}M_{2}(M_1 +M_2)}{5 c^5}
-	(t_{coal}-t)\Big]^{1/4}.\]
+	(t_{coal}-t)\Big]^{1/4}.\]-->
 <div class="NavFrame collapsed">
    <div class="NavHead">solution</div>
@@ Line 144: / Line 144: @@
 Show that the mass of a binary can be determined through the frequency of gravitational waves it generates.
 <br/>
-Hint: It is possible thanks to the relation
+<!--Hint: It is possible thanks to the relation
 \[f_{GW}=(1.9Hz)\Big(\frac{1.4M_{\odot}}{M}\Big)^{5/8}
-	\Big(\frac{days}{t_{coal}-t}\Big)^{3/8}.\]
+	\Big(\frac{days}{t_{coal}-t}\Big)^{3/8}.\]-->
 <div class="NavFrame collapsed">
    <div class="NavHead">solution</div>

← Older revision	Revision as of 03:08, 8 January 2013
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−	+	__TOC__

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 <div id=""></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1: Gravitational luminosity===
+=== Problem 2: Gravitational luminosity===
 Calculate the gravitational luminosity for the binary system with equal masses of the previous problem.
 <br/>
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 <div id=""></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1: Asymmetric binary system===
+=== Problem 3: Asymmetric binary system===
 Calculate the gravitational luminosity for a binary system with components of arbitrary masses $M_1$ and $M_2$.
 <br/>
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 <div id=""></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1: Binary system evolution===
+=== 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.
 <br/>
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 <div id=""></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1: Binary system lifetime===
+=== Problem 5: Binary system lifetime===
 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.
 <div class="NavFrame collapsed">
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 <div id=""></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1: Mass through GW frequency===
+=== Problem 6: Mass through GW frequency===
 Show that the mass of a binary can be determined through the frequency of gravitational waves it generates.
 <br/>

← Older revision		Revision as of 02:41, 8 January 2013
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	[[Category:Weak field limit and gravitational waves\|8]]		[[Category:Weak field limit and gravitational waves\|8]]
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		+
		+
		+
		+
		+	<div id=""></div>
		+	<div style="border: 1px solid #AAA; padding:5px;">
		+	=== 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 $\omega$.
		+	<br/>
		+	Hint: Calculating the quadrupole moment of the system gives
		+	\[\bar{h}^{\alpha\beta}
		+	=-\frac{8\omega^2 MR^{2}}{r}
		+	\left( \begin{matrix}
		+	\cos \left[ 2\omega (t-r) \right]
		+	& \sin \left[ 2\omega (t-r) \right] & 0 \\
		+	\sin \left[ 2\omega (t-r) \right]
		+	& -\cos \left[ 2\omega (t-r) \right] & 0 \\
		+	0 & 0 & 0 \\
		+	\end{matrix} \right)\]
		+	<div class="NavFrame collapsed">
		+	<div class="NavHead">solution</div>
		+	<div style="width:100%;" class="NavContent">
		+	<p style="text-align: left;">Let the orbit lie in the $xy$-plane. Then the masses have trajectories
		+	\[\left( R\cos \omega t,\ R\sin \omega t,0 \right),
		+	\quad
		+	\left( -R\cos \omega t,\ -R\sin \omega t,0 \right)\]
		+	From the definition of the second mass moment
		+	\[I^{\alpha\beta}(t)=\int d^{3} x\;
		+	\rho (t,\mathbf{x})x^{\alpha}x^{\beta},\]
		+	($\rho$ is the rest-mass density) we find for binary system
		+	\[I^{\alpha\beta}=MR^{2}
		+	\left( \begin{matrix}
		+	1+\cos 2\omega t & \sin 2\omega t & 0 \\
		+	\sin 2\omega t & 1-\cos 2\omega t & 0 \\
		+	0 & 0 & 0 \\
		+	\end{matrix} \right).\]
		+	As we have seen above in the long-wavelength approximation $\lambda \gg R$ and the distant-source approximation $r\gg R$ the strain tensor is
		+	\[ \bar{h}^{\alpha\beta}(t,\mathbf{x})
		+	=\frac{2}{r}\ddot{I}^{\alpha\beta}(t-r)\]
		+	Therefore,
		+	\[\bar{h}^{\alpha\beta}
		+	=-\frac{8\omega^2 MR^{2}}{r}
		+	\left( \begin{matrix}
		+	\cos \left[ 2\omega (t-r) \right]
		+	& \sin \left[ 2\omega (t-r) \right] & 0 \\
		+	\sin \left[ 2\omega (t-r) \right]
		+	& -\cos \left[ 2\omega (t-r) \right] & 0 \\
		+	0 & 0 & 0 \\
		+	\end{matrix} \right)\]
		+	Note that this representation of $\bar{h}^{\alpha\beta}$ is already in transverse traceless gauge for propagation in the $z$ direction. Also, the frequency of the emitted radiation is twice the orbital frequency by the symmetry of the problem.</p>
		+	</div>
		+	</div></div>
		+
		+
		+	<div id=""></div>
		+	<div style="border: 1px solid #AAA; padding:5px;">
		+	=== Problem 1: Gravitational luminosity===
		+	Calculate the gravitational luminosity for the binary system with equal masses of the previous problem.
		+	<br/>
		+	Hint: Calculating the quadrupole tensor's square and averaging gives
		+	\[L_{GW}=\frac{128}{5}M^{2}R^{4}\omega^{6}.\]
		+	<div class="NavFrame collapsed">
		+	<div class="NavHead">solution</div>
		+	<div style="width:100%;" class="NavContent">
		+	<p style="text-align: left;">The components of $Q^{\mu\nu}$ were calculated in the previous problem, so the quadrupole formula can be immediately applied to find the luminosity. The trace $Q_{\alpha}^{\alpha}=2MR^{2}$ is independent of time, so the time derivates of $I^{\alpha\beta}$ and $Q^{\alpha\beta}$ coincide. Averaging over the orbital period provides an additional factor of $1/2$ . The result for ${{L}_{GW}}$ is then
		+	\[L_{GW}=\frac{128}{5}M^{2}R^{4}\omega^{6}.\]</p>
		+	</div>
		+	</div></div>
		+
		+
		+	<div id=""></div>
		+	<div style="border: 1px solid #AAA; padding:5px;">
		+	=== Problem 1: Asymmetric binary system===
		+	Calculate the gravitational luminosity for a binary system with components of arbitrary masses $M_1$ and $M_2$.
		+	<br/>
		+	Hint: If $R$ is the distance between the components, then
		+	\[L_{GW}=\frac{32}{5}\frac{G^4}{c^5}
		+	\frac{M_1^2 M_2^2 (M_1 +M_2)}{R^5}.\]
		+	<div class="NavFrame collapsed">
		+	<div class="NavHead">solution</div>
		+	<div style="width:100%;" class="NavContent">
		+	<p style="text-align: left;">Let $R$ be the distance between the components. Then the quadrupole formula gives
		+	\[L_{GW}=\frac{32G}{5c^5}
		+	\Big(\frac{M_{1}M_{2}}{M_{1}+M_{2}}\Big)^{2}
		+	R^{4}\omega^6.\]
		+	Frequency $\omega$ is related to $R$ through
		+	\[\omega^{2}R^{3}=G(M_{1}+M_{2}),\]
		+	so for the gravitational luminosity we finally obtain
		+	\[L_{GW}=\frac{32}{5}\frac{G^{4}}{c^5}
		+	\frac{M_{1}^{2}M_{2}^{2}(M_{1}+M_{2})}{R^5}.\]</p>
		+	</div>
		+	</div></div>
		+
		+
		+	<div id=""></div>
		+	<div style="border: 1px solid #AAA; padding:5px;">
		+	=== Problem 1: 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.
		+	<br/>
		+	Hint: Equating $dE(R)/dt$ to losses due to gravitational radiation, we obtain differential equation for $R(t)$. Integration yields
		+	\[ R(t)=4\;\Big[
		+	\frac{G^{3}M_{1}M_{2}(M_1 +M_2)}{5 c^5}
		+	(t_{coal}-t)\Big]^{1/4}.\]
		+	<div class="NavFrame collapsed">
		+	<div class="NavHead">solution</div>
		+	<div style="width:100%;" class="NavContent">
		+	<p style="text-align: left;">From the virial theorem $2\left\langle T \right\rangle =-\left\langle U \right\rangle$, the energy of the system is
		+	\[E=-G\frac{M_1 M_2}{2R},\]
		+	so
		+	\[\dot{R}=\frac{2R^{2}}{GM_{1}M_{2}}\frac{dE}{dt}.\]
		+	Then from $dE/dt=-L_{GW}$ and using the expression for the gravitational luminosity, we obtain
		+	\[\dot{R}=-\frac{64G^{3}}{5c^{2}}
		+	\frac{M_{1}M_{2}(M_{1}+M_{2})}{R^3}.\]
		+	Integrating from present time $t$ to the future time of coalescence ${{t}_{coal}}$, we have $R(t)$:
		+	\[ R(t)=2\;\Big[\frac{16}{5}
		+	\frac{G^{3}M_{1}M_{2}(M_1 +M_2)}{c^5}
		+	(t_{coal}-t)\Big]^{1/4}.\]</p>
		+	</div>
		+	</div></div>
		+
		+
		+	<div id=""></div>
		+	<div style="border: 1px solid #AAA; padding:5px;">
		+	=== Problem 1: Binary system lifetime===
		+	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.
		+	<div class="NavFrame collapsed">
		+	<div class="NavHead">solution</div>
		+	<div style="width:100%;" class="NavContent">
		+	<p style="text-align: left;">For a binary with $M_1 =M_2 =M/2$ a rough estimate would be
		+	\[t_{chirp}\sim\frac{MV^{2}}{L_{GW}}
		+	=\frac{5M}{2}\Big(\frac{R}{M}\Big)^{4}.\]
		+	More accurate evaluation would use the time evolution of $R(t)$: for $M_1 =M_2 =M/2$ we get
		+	\[t_{chirp}=\frac{5}{16}\frac{c^5 R^4}{G^3 M^3}.\]
		+	This is of the order of $45\times 10^9$ years.</p>
		+	</div>
		+	</div></div>
		+
		+
		+	<div id=""></div>
		+	<div style="border: 1px solid #AAA; padding:5px;">
		+	=== Problem 1: Mass through GW frequency===
		+	Show that the mass of a binary can be determined through the frequency of gravitational waves it generates.
		+	<br/>
		+	Hint: It is possible thanks to the relation
		+	\[f_{GW}=(1.9Hz)\Big(\frac{1.4M_{\odot}}{M}\Big)^{5/8}
		+	\Big(\frac{days}{t_{coal}-t}\Big)^{3/8}.\]
		+	<div class="NavFrame collapsed">
		+	<div class="NavHead">solution</div>
		+	<div style="width:100%;" class="NavContent">
		+	<p style="text-align: left;">From the previous problem for gravitational wave frequency for a binary with components of equal masses we get
		+	\begin{align*}
		+	f_{GW}&=\frac{\omega }{\pi }
		+	=\frac{1}{\pi }\Big(\frac{GM}{4R^3}\Big)^{1/2}
		+	=\frac{2^{1/8}5^{3/8}}{8\pi}
		+	\Big(\frac{c^3}{GM}\Big)^{5/8}(t_{coal}-t)^{-3/8}=\\
		+	&=(1.9Hz)\Big(\frac{1.4M_{\odot}}{M}\Big)^{5/8}
		+	\Big(\frac{days}{\tau}\Big)^{3/8},
		+	\quad \tau \equiv t_{coal}-t.
		+	\end{align*}
		+	Hence the mass of the compact binary system can be derived from the frequency evolution.</p>
		+	</div>
		+	</div></div>