Question

A new planet is discovered orbiting a distant star. Observations have confirmed that the planet has a circular orbit with a radius of 18 AU and takes 109 days to orbit the star. Determine the mass of the star.
AU stands for "astronomical unit". It is the average distance between Earth & the Sun. 1 AU ≈ 1.496 × 1011 m.

Answer 1

Answer:

Mass of star is $1.31\times10^{35}$ kg.

Explanation:

The cube of orbital radius is equal to the square of its orbital time period is known as Kepler's law.

$T^{2} = (\frac{4\pi^{2} }{GM})r^{3}$          .....(1)

Here T is time period, r is orbital radius, G is universal gravitational constant and M is the mass of the star.

According to the problem,

Time period, T = 109 days = 109 x 24 x 60 x 60 s = 9.41 x 10⁶ s

Orbital radius, r = 18 AU = 18 x 1.496 x 10¹¹ m = 2.70 x 10¹² m

Gravitational constant, G = 6.67 x 10⁻¹¹ m³ kg⁻¹ s⁻²

Substitute these values in equation (1).

$(9.41\times10^{6}) ^{2} = (\frac{4\pi^{2} }{6.67\times10^{-11}\times M})(2.70\times10^{12}) ^{3}$

M = $1.31\times10^{35}$ kg