Problem — Vibrations of the Sun

Problem

Estimate the fundamental frequency of radial oscillation of the Sun, modelling it as a spherical star with mass $M_\odot \approx 2\cdot 10^{30}\;\text{kg}$ and radius $R_\odot \approx 7\cdot 10^8\;\text{m}$, oscillating under its own gravity. Compare with the observed value of $\sim 5\;\text{minutes}$ for solar oscillation modes.

Solution

The gravitational dynamical time of a self-gravitating star is $\omega \approx \sqrt{GM/R^3}$.

$GM = 6{,}67\cdot 10^{-11} \cdot 2\cdot 10^{30} = 1{,}33\cdot 10^{20}, \quad R^3 = (7\cdot 10^8)^3 = 3{,}43\cdot 10^{26}$

$\frac{GM}{R^3} = 3{,}88\cdot 10^{-7} \;\Rightarrow\; \omega = 6{,}2\cdot 10^{-4}\;\text{rad/s}, \quad T = \frac{2\pi}{\omega}$

$\ev{T_\text{grav} \sim 10^{4}\;\text{s} \ (\sim 3\text{ hours})}$

Comparison: the observed modes at $\sim 5\;\text{min}$ ($\sim 300\;\text{s}$) are much faster. They are therefore not the fundamental gravitational oscillation, but high-order acoustic modes (p-modes): sound waves trapped in the Sun’s outer layers.

Links

Topics: Oscillations and simple harmonic motion Concepts: Simple harmonic motion · Period and frequency Skills: Fermi estimates