Difference between revisions of "Binary systems"

From Universe in Problems
Jump to: navigation, search
 
Line 10: Line 10:
 
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 $\omega$.
 
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/>
 
<br/>
Hint: Calculating the quadrupole moment of the system gives
+
<!--Hint: Calculating the quadrupole moment of the system gives
 
\[\bar{h}^{\alpha\beta}
 
\[\bar{h}^{\alpha\beta}
 
=-\frac{8\omega^2 MR^{2}}{r}
 
=-\frac{8\omega^2 MR^{2}}{r}
Line 19: Line 19:
 
& -\cos \left[ 2\omega (t-r) \right] & 0  \\
 
& -\cos \left[ 2\omega (t-r) \right] & 0  \\
 
   0 & 0 & 0  \\
 
   0 & 0 & 0  \\
\end{matrix} \right)\]
+
\end{matrix} \right)\]-->
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
 
   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
Line 60: Line 60:
 
Calculate the gravitational luminosity for the binary system with equal masses of the previous problem.
 
Calculate the gravitational luminosity for the binary system with equal masses of the previous problem.
 
<br/>
 
<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="NavFrame collapsed">
 
   <div class="NavHead">solution</div>
 
   <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 $M_2$.
 
Calculate the gravitational luminosity for a binary system with components of arbitrary masses $M_1$ and $M_2$.
 
<br/>
 
<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}
 
\[L_{GW}=\frac{32}{5}\frac{G^4}{c^5}
\frac{M_1^2 M_2^2 (M_1 +M_2)}{R^5}.\]
+
\frac{M_1^2 M_2^2 (M_1 +M_2)}{R^5}.\]-->
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
 
   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
Line 100: Line 100:
 
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.
 
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/>
 
<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[
 
\[ R(t)=4\;\Big[
 
\frac{G^{3}M_{1}M_{2}(M_1 +M_2)}{5 c^5}
 
\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="NavFrame collapsed">
 
   <div class="NavHead">solution</div>
 
   <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.
 
Show that the mass of a binary can be determined through the frequency of gravitational waves it generates.
 
<br/>
 
<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}
 
\[f_{GW}=(1.9Hz)\Big(\frac{1.4M_{\odot}}{M}\Big)^{5/8}
\Big(\frac{days}{t_{coal}-t}\Big)^{3/8}.\]
+
\Big(\frac{days}{t_{coal}-t}\Big)^{3/8}.\]-->
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
 
   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>

Latest revision as of 13:50, 15 January 2013



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 $\omega$.


Problem 2: Gravitational luminosity

Calculate the gravitational luminosity for the binary system with equal masses of the previous problem.


Problem 3: Asymmetric binary system

Calculate the gravitational luminosity for a binary system with components of arbitrary masses $M_1$ and $M_2$.


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.


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.


Problem 6: Mass through GW frequency

Show that the mass of a binary can be determined through the frequency of gravitational waves it generates.