**3.1 The value of** *η<sup>G</sup>*

In Appendix A.1, we prove that

$$
\eta\_G = \frac{r\_1}{r\_2} \ge .44 \quad \sqrt{\frac{K-1}{K}} \tag{36}
$$

for MBH traveling at subsonic speeds with M< 0*:*8. *K* is the ratio of average temperature in the hot gas trail and ambient temperature of stellar material. A possible route for estimating *K* is outlined in Appendix A.2.
