This study investigates how graduate students' sense of belonging (SB) influences their intent to persist (IP) in physics and astronomy programs, and how this relationship is shaped by the basic psychological needs that drive motivation--autonomy, competence, and relatedness--as well as gender. Grounded in self-determination theory, the analysis treats these three needs as mediators and gender as a moderator. A quantitative survey was administered to graduate students in the Department of Physics and Astronomy at a large public land-grant R1 Midwestern university in the USA. Using probit regressions, we found that SB significantly predicts IP. Autonomy may play a compensatory role when SB is high, competence amplifies the effect of SB on IP, and relatedness buffers against low SB. Gender moderates the relationship: women report lower IP at low levels of SB but exceed men when SB is strong. These findings underscore the importance of fostering a sense of belonging, academic confidence, and social connection--particularly for women in male-dominated STEM fields.