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tokens: exact derivative of P ) and (the derivative when just GWs are considered) because this difference will be important for subsequent discussions. In this case , so thenGiven that will be difficult to measure, I focus on the simpler scenario where η = 1 with . To find the second derivative including tides, it is helpful to first rewrite Equation ( 6 ) asTaking the derivative of both sides of this expression, Rewriting all the first derivatives in terms of , dividing by J orb , and then collecting similar terms, the final result isWhen combined with Equation ( 6 ) for (using for consistency), this expression provides an analytic relations for . Comparing Equations ( 9 ) and ( 12 ) shows that tidal interactions make the second derivative even more negative than the GW only case. 4. Braking Index Beyond the values of the first and second derivative, another way to think about tides is in terms of a braking index. The basic idea is analogous to pulsars (Shapiro Teukolsky 1983 ) where one wishes to measure n such thatThe main difference is that here Ω refers to the orbital frequency of the binary rather than the pulsar spin frequency (and perhaps it should be referred to as an “acceleration index” here), although also see discussions of the braking index by Nelemans et al. ( 2004 ) and Stroeer et al. ( 2005 ) in the context of AM CVn systems. It is straightforward to showUsing the relations from Sections 2 and 3 , for GWs only (Webbink Han 1998 ),while when tides are includedThus fitting for the power law in real systems could be another way to infer the presence of tides. Although the exact value of n depends on the specific parameters of the binary and the degree of tidal locking, simply showing that n > 11/3 would demonstrate tides are occurring without having to know these parameters. 5. The Rate of Change of Tidal Locking An additional factor in that is often not considered is , the rate of change of the tidal locking. This will depend in detail on the model for the tidal interaction. In Piro ( 2011 ), the tide is treated using a parameterization with a standard quality factor Q . When this tidal interaction is integrated forward in time, it is found that the binary reaches an equilibrium spin at any given point where the ratio of the tidal forcing frequency to the orbital frequency is roughly the ratio of the tidal synchronization time τ tide to the GW inspiral time , i.e.,Furthermore, this work finds that for constant Q , . This ratio thus gets smaller at shorter orbital period, resulting in a more tidally locked binary. In reality, this will depend on the details of how the tides act, and so I consider a general form of with β > 0. Assuming a model of this form and taking the derivative results inThis particular model has a couple of important consequences for . First, it makes even more negative. Second, it adds corrections to even when η ≈ 0 because it implies a rapidly changing η , while for η ≈ 1 this model gives . In the future, it would