Prospects for measurements of the longitudinal proton structure function $F_L$ at the Electron Ion Collider

We explore the potential for extracting the longitudinal proton structure function $F_L$ at the future Electron-Ion Collider (EIC) through a Rosenbluth separation method. The impacts of differing assumptions on sample sizes, systematic uncertainties and beam energy scenarios are investigated. With a sufficiently large number of centre of mass energy configurations and well-controlled systematics, the EIC will measure $F_L$ to an unprecedented precision, even with relatively modest luminosities. The accessible kinematic range complements both fixed target and HERA data. In the most optimistic scenarios, the EIC data will be a highly competitive direct probe of the proton gluon density.