Problem — Earth's orbital angular momentum

Problem

Estimate the magnitude of the Earth’s angular momentum due to its orbital motion around the Sun. Data: Earth’s mass $\approx 6\cdot 10^{24}\;\mathrm{kg}$, orbital radius $\approx 1{,}5\cdot 10^{11}\;\mathrm{m}$, period of one year.

Solution

Treat the Earth as a point mass in circular orbit: $L = m\,v\,r$, with orbital speed $v = \dfrac{2\pi r}{T}$. The period is $T = 1\;\text{year} \approx 3{,}15\cdot 10^7\;\mathrm{s}$. $v = \frac{2\pi\cdot 1{,}5\cdot 10^{11}}{3{,}15\cdot 10^7} \approx 3{,}0\cdot 10^4\;\mathrm{m/s}$ $L = m\,v\,r = 6\cdot 10^{24}\cdot 3{,}0\cdot 10^4\cdot 1{,}5\cdot 10^{11}$ $\ev{L \sim 3\cdot 10^{40}\;\mathrm{kg\,m^2/s}}$ This is an enormous angular momentum, which makes the Earth’s orbit extraordinarily stable: it would take a colossal torque to alter it appreciably.

Connections

Topics: Dinamica rotazionale Concepts: Momento angolare Skills: Stime di Fermi