# Why does the magnitude of the Gravitational wave reach maximum peak upon merger?

We all know that objects which are accelerating in space-time produce Gravitational waves. My question is why is there a burst of GW upon two black holes merging. Also, does that mean that let's say when two stars or planets collide does that also produce a sudden spike in GW?

## 1 Answer

The power emitted from a rotating quadrupole is given by:

$$P = \frac{32}{5} \frac{G}{c^5} I_{zz}^2 \epsilon^2 \omega^6 \tag{1}$$

where $\omega$ is the angular frequency of the rotation and $I_{zz}^2 \epsilon^2$ is related to the geometry (the quadrupole moment) of the rotating object.

The key point here is that the power is proportional to the sixth power of the angular velocity $\omega$. As the two black holes spiral in towards each other conservation of angular momentum means the orbital frequency increases, and the power radiated increases as the sixth power of this angular frequency. That's why the emitted gravitational wave intensity peaks at the moment of merger.

Strictly speaking the equation (1) is an approximation derived in the linear regime, and it will fail to accurately describe the gravitational wave emission in the highly non-linear spacetime near to the merger. However it gives the general idea that the frequency of rotation is by far the dominant factor.