What is the average separation distance period of a binary system with a period of 32 years?

In a binary star system, the average separation distance between the two stars can be determined using Kepler’s third law of planetary motion, which relates the orbital period of two objects to their average separation. Specifically, for a binary star system, the law states that the square of the orbital period (in years) is proportional to the cube of the semi-major axis of their orbit (in astronomical units, AU).

The mathematical relationship can be represented as:

( P^2 = a^3 )

where:

• ( P ) is the period of the orbit in years,

• ( a ) is the average separation distance in astronomical units.

In this case, the period ( P ) is given as 32 years. Plugging this into the equation, we get:

( 32^2 = a^3 )

Calculating ( 32^2 ) gives:

( 1024 = a^3 )

To find ( a ), we take the cube root of 1024:

( a = \sqrt[3]{1024} )

Calculating the cube root of 1024 yields approximately 10.079.

The closest value among the provided choices to 10.079 AU is