The supermassive black hole at the center of our Milky Way Galaxy is believed to be 4.3 × 10⁶ solar masses.The nearest of the stars orbiting it has a period of 11.5 years.What is the semimajor axis of the orbit of this star (in AU)?

Question

Quizplus · Accepted Answer

We can use Kepler’s Third Law to relate the period and semimajor axis of the orbit:
(T/2π)² = (a³/GM)
where T is the period of the orbit, a is the semimajor axis, G is the gravitational constant, and M is the mass of the central object (the black hole in this case).

Rearranging for a, we get:
a = [(GM)(T²)/(4π²)]^(1/3)

Substituting the values given:
a = [(6.67 × 10^-11 m³ kg^-1 s^-2)(4.3 × 10^6 M☉)(11.5 years x 3.15 x 10^7 s/year)²/(4π²)]^(1/3)

a ≈ 1.2 x 10¹³ m

Converting to AU:
a ≈ 828 AU

The Supermassive Black Hole at the Center of Our Milky