What is the period of a planet with mass 1.3 × 10²⁸ kg orbiting 70 Virginis at a radius of 7.1 × 10¹⁰ m, in terms of Earth-days?

To determine the period of a planet orbiting a star, we can use Kepler's Third Law of planetary motion, which states that the square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit. The general formula is given by:

[ T^2 = \frac{4\pi^2}{G(M+m)} r^3 ]

where ( T ) is the orbital period, ( r ) is the radius of the orbit (in this case, the distance from the planet to the star), ( G ) is the gravitational constant (( 6.674 \times 10^{-11} , \text{m}^3/\text{kg/s}^2 )), and ( M ) is the mass of the star. The mass of the planet itself can generally be neglected when compared to the mass of the star.

For the star 70 Virginis, it has a mass of approximately ( 1.25 \times 10^{30} ) kg. Plugging the values into the formula, we can rearrange it to solve for ( T ):

1. Calculate the semi-major axis ( r^3 ):

(