Question

The Mass of the Sun Calculate the mass of the Sun, noting that the period of the Earth's orbit around the Sun is 3.156 ✕ 107 s and its distance from the Sun is 1.496 ✕ 1011 m. SOLUTION Conceptualize Based on the mathematical representation of Kepler's third law expressed as T2 = 4π2 GMS a3 = KSa3, we realize that the mass of the central object in a gravitational system is related to the orbital size and Correct: Your answer is correct. of objects in orbit around the central object. Categorize This example is a relatively simple Correct: Your answer is correct. problem. Solve Kepler's third law for the mass of the Sun: MS = 4π2a3 GT2 Substitute the known values (Enter your answer in kg.): MS = 1990000000000000000000000000000 Correct: Your answer is correct. kg In the Example The Density of Earth, an understanding of gravitational forces enabled us to find out something about the density of the Earth's core, and now we have used this understanding to determine the mass of the Sun! EXERCISE If we find an unknown comet returning to the Sun every 296 years (for example, we know that Comet Halley returns every 75.6 years), determine the radius a (semi-major axis) of its orbit (in m).

Answer 1

According to Kepler's third law, mass of the sun can be calculated as follows:

$T^2=\frac{4\pi^2}{GM_s}a^3\\M_s=\frac{4\pi^2}{GT^2}a^3\\M_s=\frac{4\pi^2}{6.67\times 10^{-11}(3.156\times10^7)^2}(1.496\times10^{11})^3\\M_s=1.98\times 10^{30} kg$

Kepler's third law can also be written as:

$P^2=a^3$

where P is period in years and a is semi major axis in au

$1au=1.496\times10^{11}m$

Substitute the values to find semi-major axis of comet's orbit around the Sun:

$a^3=(296)^2\\a=44.41 au = 6.64\times10^{12} m$