- #1
rcatalang
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- Homework Statement
- We have two massive particles of mass ##m_1## and ##m_2## that approach each other from infinity up to a certain point, ##x_f##. One has to calculate the energy radiated through gravitational waves.
- Relevant Equations
- Equation of motion: ##\ddot{x}=\frac{-GM}{x^2}##.
Energy lost (disregarding constants): ##E\propto \int_{t=0}^{t_f} \frac{(\dot{x})^2}{x^4}##
I have tried simplifying the integral (turning into an integral in terms of position variables) using the equation of motion. It's easy to show:
$$\frac{1}{x^4}=\frac{(\ddot{x})^2}{(GM)^2} $$
And therefore one can write:
$$\frac{1}{(GM)^2}\int_{t=0}^{t_f} (\dot{x}\ddot{x})^2 dt = \frac{1}{(GM)^2}\int_{\infty}^{x_f} \dot{x}(\ddot{x})^2 dx $$
But I have no idea about how to continue further with this integral. Any ideas on how to solve this integral?
$$\frac{1}{x^4}=\frac{(\ddot{x})^2}{(GM)^2} $$
And therefore one can write:
$$\frac{1}{(GM)^2}\int_{t=0}^{t_f} (\dot{x}\ddot{x})^2 dt = \frac{1}{(GM)^2}\int_{\infty}^{x_f} \dot{x}(\ddot{x})^2 dx $$
But I have no idea about how to continue further with this integral. Any ideas on how to solve this integral?