We prove that the size of the $e$-core of a partition taken under the Poissonised Plancherel measure converges in distribution to, as the Poisson parameter goes to $+\infty$ and after a suitable renormalisation, a sum of $e-1$ mutually independent Gamma distributions with explicit parameters. Such a result already exists for the uniform measure on the set of partitions of $n$ as $n$ goes to $+\infty$, the parameters of the Gamma distributions being all equal. We rely on the fact that the descent set of a partition is a determinantal point process under the Poissonised Plancherel measure and on a central limit theorem for such processes.