Escape velocity for Schwarzschild metric

I can’t fill in the gaps in my solution to this and assistance or a reference would be appreciated.

The question begins with the straightforward derivation of the EoM for a massive particle orbiting in the equatorial plane, as

$$ left( frac{du}{dphi}right)^2 = frac{c^2 k^2}{h^2} – alpha left( frac{c^2}{h^2} + u^2 right) $$ where $u = frac{1}{r}$, $ h, k$ are constants arising as $ alpha dot{t} = k$ and $dot{phi} = h u^2$, and $alpha = 1-frac{r_s}{r} $ where $r_s$ is the Schwarzschild radius.

It then says a stationary experimenter at radius $a > r_s$ projects a massive particle with speed $v$ normal to the radial direction, and asks me to show that in the case $h^2 > 3 r_s^2 c^2$ the particle will be ejected if $v$ exceeds an escape velocity similar in form to the Newtonian one.

Clearly the above condition restricts to the case of three real roots, and I think that the condition I want is that the smallest root of the above cubic (there’s an extra $u$ in the $alpha$) is $leqslant 0$, though I’m not entirely sure why that’s necessary/sufficient. Given that, I obtain the result $ v geqslant sqrt{frac{2GM}{a}} $.

Is this result correct? And could someone explain why that condition is the right one?