Estimating the mass of the Sun

Problem

Estimating the mass of the Sun. Knowing that the Earth has period $T \approx 1$ year and distance $1$ AU $\approx 1{,}5\cdot 10^{11}$ m, estimate the mass of the Sun using Kepler’s third law.

Solution

Kepler’s third law for the Earth’s (nearly circular) orbit is $T^2 = \dfrac{4\pi^2 a^3}{GM_S}$, from which $M_S = \frac{4\pi^2 a^3}{G\,T^2}$

Convert the period: $T = 1\ \text{year} \approx 365 \cdot 86400 \approx 3{,}15\cdot 10^7$ s. Substitute $a = 1{,}5\cdot 10^{11}$ m and $G = 6{,}67\cdot 10^{-11}$: $M_S = \frac{4\pi^2 (1{,}5\cdot 10^{11})^3}{6{,}67\cdot 10^{-11}\,(3{,}15\cdot 10^7)^2} = \frac{39{,}5 \cdot 3{,}38\cdot 10^{33}}{6{,}67\cdot 10^{-11}\cdot 9{,}92\cdot 10^{14}}$ $M_S = \frac{1{,}33\cdot 10^{35}}{6{,}62\cdot 10^{4}} \approx 2\cdot 10^{30}\ \text{kg}$

The accepted value is $1{,}99\cdot 10^{30}$ kg: the estimate from Kepler’s third law alone is excellent.

$\ev{M_S \approx 2\cdot 10^{30}\ \text{kg}}$

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Topics: Gravitation Concepts: Kepler’s laws Skills: Fermi estimates