Estimate of the Galaxy's mass

Problem

Our Galaxy completes one full rotation in the galactic plane in about $250$ million years. The Sun is about $2{,}6\cdot 10^{20}\ \text{m}$ from the galactic centre. Assuming a circular orbit and that most of the galactic mass is concentrated within the Sun’s orbit, estimate the order of magnitude of the Galaxy’s mass and comment on the comparison with the visible stellar mass ($\sim 10^{11}\,M_\odot$).

Solution

Setting up. We treat the Sun as a satellite in circular orbit around the mass $M$ enclosed within its orbital radius. Gravity provides the centripetal force; equivalently, we use Kepler’s third law: $\frac{GMm}{r^2} = m\,\frac{4\pi^2 r}{T^2} \quad\Longrightarrow\quad M = \frac{4\pi^2 r^3}{G\,T^2}$

Converting the period. With $1\ \text{year} \approx 3{,}15\cdot 10^7\ \text{s}$: $T = 250\cdot 10^6\ \text{years} \approx 2{,}5\cdot 10^8 \cdot 3{,}15\cdot 10^7\ \text{s} \approx 7{,}9\cdot 10^{15}\ \text{s}$

Numerical substitution. With $r = 2{,}6\cdot 10^{20}\ \text{m}$ and $G = 6{,}67\cdot 10^{-11}\ \text{N}\,\text{m}^2/\text{kg}^2$: $M = \frac{4\pi^2\,(2{,}6\cdot 10^{20})^3}{6{,}67\cdot 10^{-11}\cdot (7{,}9\cdot 10^{15})^2} = \frac{39{,}5\cdot 1{,}76\cdot 10^{61}}{6{,}67\cdot 10^{-11}\cdot 6{,}2\cdot 10^{31}} \approx \frac{6{,}9\cdot 10^{62}}{4{,}2\cdot 10^{21}}$

$\ev{M \sim 1{,}7\cdot 10^{41}\ \text{kg} \sim 10^{41}\ \text{kg}}$

Comment. The visible stellar mass is $\sim 10^{11}\,M_\odot \approx 10^{11}\cdot 2\cdot 10^{30}\ \text{kg} = 2\cdot 10^{41}\ \text{kg}$, of the same order of magnitude as the dynamical mass estimated here within the Sun’s orbit. However, extending the same reasoning to more distant stars, rotation curves remain flat instead of declining: this requires much more non-luminous mass (dark matter) in the galactic halo. Kepler’s estimate is therefore only a lower bound on the Galaxy’s total mass.

Links

Topics: Gravitazione Concepts: Leggi di Keplero · Moto circolare uniforme · Legge di gravitazione universale · Forza centripeta Skills: Stime di Fermi · Impostazione simbolica