Edelen, J.P. Harris, J.R. Lewellen, J.W. Sun, Y. JACoW Geneva, Switzerland 978-3-95450-180-9 10.18429/JACoW-NAPAC2016-WEPOB25 English 953-955 WEPOB25 ion gun cathode electron simulation Contribution to a conference proceedings 2017 2017-01 http://dx.doi.org/10.18429/JACoW-NAPAC2016-WEPOB25 https://jacow.org/napac2016/papers/wepob25.pdf In this paper we derive analytical expressions for the output current of an un-gated thermionic cathode RF gun in the presence of back-bombardment heating. We provide a brief overview of back-bombardment theory and discuss comparisons between the analytical back-bombardment predictions and simulation models. We then derive an expression for the output current as a function of the RF repetition rate and discuss relationships between back-bombardment, field-enhancement, and output current. We discuss in detail the relevant approximations and then provide predictions about how the output current should vary as a function of repetition rate for some given system configurations.