Prospects for measurements of the longitudinal proton structure function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>F</mml:mi><mml:mi>L</mml:mi></mml:msub></mml:math> at the Electron Ion Collider

We explore the potential for extracting the longitudinal proton structure function <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>F</a:mi><a:mi>L</a:mi></a:msub></a:math> 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 center-of-mass energy configurations and well-controlled systematics, the EIC will measure <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>F</c:mi><c:mi>L</c:mi></c:msub></c:math> to an unprecedented precision, even with relatively modest luminosities. The accessible kinematic range complements both fixed-target and Hadron-Electron Ring Accelerator data. In the most optimistic scenarios, the EIC data will be a highly competitive direct probe of the proton gluon density.