Question 44.1: Our Galaxy's mass. Estimate the total mass of our Galaxy usi......
Our Galaxy’s mass. Estimate the total mass of our Galaxy using the orbital data above for the Sun about the center of the Galaxy. Assume that most of the mass of the Galaxy is concentrated near the center of the Galaxy.
APPROACH We assume that the Sun (including our solar system) has total mass m and moves in a circular orbit about the center of the Galaxy (total mass M ), and that the mass M can be considered as being located at the center of the Galaxy. We then apply Newton’s second law, F=m a, with a being the centripetal acceleration, a=v^{2} / r, and F being the universal law of gravitation (Chapter 6).
Our Sun and solar system orbit the center of the Galaxy, according to the best measurements as mentioned above, with a speed of about v=200 \mathrm{~km} / \mathrm{s} at a distance from the Galaxy center of about r=26,000 \,\mathrm{ly}. We use Newton’s second law:
\begin{aligned}F & =m a \\G \frac{M m}{r^{2}} & =m \frac{v^{2}}{r}\end{aligned}
where M is the mass of the Galaxy and m is the mass of our Sun and solar system. Solving this, we find
M=\frac{r v^{2}}{G} \approx \frac{(26,000 \mathrm{ly})\left(10^{16} \mathrm{~m} / \mathrm{ly}\right)\left(2 \times 10^{5} \mathrm{~m} / \mathrm{s}\right)^{2}}{6.67 \times 10^{-11} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2}} \approx 2 \times 10^{41} \mathrm{~kg}
NOTE In terms of numbers of stars, if they are like our Sun \left(m=2.0 \times 10^{30} \mathrm{~kg}\right), there would be about \left(2 \times 10^{41} \mathrm{~kg}\right) /\left(2 \times 10^{30} \mathrm{~kg}\right) \approx 10^{11} or on the order of 100 billion stars.