If we assume the ocean is flat and extends indefinitely in all directions, there is a strategy that guarantees we can catch the pirates in at most 800,000 years.
Put our current location as the origin of a coordinate system. We will describe our position in polar coordinates, as a function of time: $(r(t),\theta(t))$ (where we have arbitrarily chosen a direction to be $\theta=0$, and $t=0$ is when we realized we had been robbed).
We begin by traveling in the $\theta=0$ direction for 20 hours, putting our position at $(420, 0)$. We are then the same distance from the origin as the pirates. Next, we will travel in a spiral, in a manner so that $r'(t)=20$ at all times. This guarantees we will always be the same distance from the origin as the pirates. For $t\geq 20$, we will have $r(t)=420+20(t-20)=20t+20$.
Our speed is
$$
\sqrt{(r')^2+r^2(\theta')^2} = 21\text{ mph},
$$
and $r'(t)=20$ for $t>20$, so
$$
\theta'(t)=\sqrt{\frac{41}{r^2}}=\frac{\sqrt{41}}{20+20t}.
$$
If there is a $t\geq 20$ for which $\theta(t)$ is the angle in which the pirates fled, we will catch them. This means we will certainly catch the pirates by the time $\theta$ has increased from $0$ to $2\pi$. If $t_0$ is the time this happens, we have
$$
2\pi=\int_{20}^{t_0}\theta'(t)\,dt=\int_{20}^{t_0}\frac{\sqrt{41}}{20+20t}dt.
$$
Solving for $t_0$ gives
$$
t_0=21\mathrm{exp}\left(\frac{40\pi}{\sqrt{41}}\right)-1\approx 7,005,043,026.
$$
This means we can catch the pirates in at most 7,005,043,026 hours, or about 800 millennia. Better later than never!
