Final answer:
To derive the star's mass in terms of b, t, and g, we equate the universal gravitational force equation with the gravitational force acting as the centripetal force in an orbital system. We solve for the mass of the star (M) and express it using the given variables, which results in the expression ms = b / (Gt).
Step-by-step explanation:
To derive an expression for the mass ms of a star in terms of the given variables b, t, and the universal gravitational constant g, we can refer to the provided equation for gravitational force, F = GmM, where G is the universal gravitational constant, m is the mass of a smaller object, and M is the mass of a larger object such as Earth or a star.
In a scenario where we are considering a satellite or an object orbiting a star, the gravitational force acting on the object can also be expressed as the centripetal force required to keep it in orbit, which is given by F = mv²/r, where v is the orbital speed of the object, m is its mass, and r is the radius of the orbit.
Now, assuming b is the orbital speed squared (v²), t is the radius of the orbit, and G is the universal gravitational constant, we can use the equation F = GmM and equate it to F = mv²/r to solve for the mass of the star M. After canceling the mass m from both sides and solving for M, we get
M = bv / (Gt)
Therefore, using the provided variables, the correct expression for the mass of the star is:
m₀ = b / (Gt)