Orbit of a definable set in a monster model ($O(\mathcal{D})$)

If $\mathcal{D}=\varphi (\mathcal{U},b)$, its automorphism orbit consists of the sets $\varphi (\mathcal{U},f(b))$. If this orbit is infinite, saturation constructs $|\mathcal{U}|$ pairwise distinct conjugates, so $|O(\mathcal{D})|=|\mathcal{U}|$.