Final answer:
In a binary star system, the stars orbit a common center of mass and thus have the same orbital period. However, without additional information like distance or mass, the orbital period cannot be directly calculated from the speeds alone.
Step-by-step explanation:
When considering a binary star system where the two stars are moving in circular orbits, we utilize Newton's version of Kepler's third law to understand their motion. If one star, alpha, has an orbital speed of 36.0 km/s and the other, beta, has an orbital speed of 12.0 km/s, and assuming they're gravitationally bound with circular orbits, their orbital periods will be the same. This is because the stars orbit their common center of mass, thus completing one orbit simultaneously.
To find the orbital period without specific mass details, one must have more information such as the distance between the two stars or the sum of their masses. Given the information provided, we cannot calculate the orbital period of the system directly. Additional data must be provided such as the distance to the binary system or the mass of the stars.