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Binary systems emit gravitational radiation. This causes the system to lose energy, which results in a shrinking of the semi-major axis. I have read on countless occasions that this 'inspiral' is adiabatic (here for example). What does this mean that the shrinking is adiabatic?

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The $Q=0$ is just a convenient definition for thermodynamics. The key to adiabacity is that the process is approximately in an equilibrium state at every step. In thermodynamics, this means that you want the process to be reversible, which means that you want the entropy change during the process to be zero, and since $dS = \frac{dQ}{T}$, this means that the heat added is necessarily zero.

But for other processes, the notion of heat added doesn't necessarily make sense, but an equliibrium process does. Typicallly, this is when you change some parameter on a process slowly enough that you don't change the basic physical situation. In addition to weak gravitational radiation causing inspiral where the orbiting objects are still orbiting in closed orbits, you get "adiabatic approximations" in quantum mechanics--for instance, when you consider a infinite square well where the walls are expanding at a small enough rate that the particle remains in what is approximately a ground state the entire time. Even more simply than that, you can consider a ball on a string problem, where the ball is rotating in two dimensions on a plane perpendicular to the string. If you change the length of the string at a rate slow enough that the ball stays rotating in what is approximately a circle and the string remains taut, you are adiabatically changing the system.

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I believe it's an approximation applied to that phase of the inspiral, stating that the relative change in orbital frequency over time is small with respect to the orbital frequency itself. This defines what is known as the adiabatic parameter:

$$ \xi = \frac{\dot\omega}{\omega^2} $$

Why adiabatic? No loss or gain of heat energy. Much like the heat death of the universe.

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Thanks for the response! I knew that adiabatic meant $dQ=0$, however, I do not see the relevance here. I mean, what does heat have to do with anything? –  user12345 Mar 21 '13 at 23:54
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Heat obviously isn't relevant here, and "adiabatic" doesn't just refer to heat. It's used more generally to refer to processes in which a change is carried out slowly enough so that the system always gets to settle down at every point along the way. For example, you can talk about a nucleus undergoing fission and whether or not the evolution of the shape is adiabatic. –  Ben Crowell Apr 24 '13 at 15:24
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