The Sun moves in a near-circular orbit about the Galactic center at radius \(R_{0} \approx 8 \mathrm{kpc}\), with speed \(V_{0} \approx 200 \mathrm{~km} \mathrm{~s}^{-1}\). If all the mass of the Milky Way were concentrated at its center, show that its total mass would be about \(7 \times 10^{10} \mathcal{M}_{\odot}\), and that a nearby star would escape from the Galaxy if it moved faster than \(\sqrt{2} V_{0}\). In fact, we see local stars with speeds as large as \(500 \mathrm{~km} \mathrm{~s}^{-1}\); explain why this tells us that the Galaxy contains appreciable mass outside the Sun's orbit.

Circular orbital motion is the movement of an object along a circular path around a central point due to the balance of forces. In the Milky Way, the Sun orbits the Galactic center in a near-circular path. This circular motion is maintained because the gravitational force between the sun and the central mass acts as the necessary centripetal force.

The orbit requires the centripetal force to point towards the center, directed along the radius of the circular path. For the sun, this force is essential to keep it moving in a circle around the galaxy. The centripetal force needed for the circular motion is provided by the gravitational attraction of the mass at the Galactic center. Thus, the gravitational force matches the centripetal force required to maintain this circular orbit.

Gravitational force is the attractive force that acts between two masses. According to Newton's law of gravitation, this force can be given by the equation: \( F_g = \frac{G M m}{R_0^2} \). Here, \( G \) is the gravitational constant, \( M \) is the mass of the Milky Way, \( m \) is the mass of the Sun, and \( R_0 \) is the distance between the Sun and the Galactic center.

In our context, this gravitational pull keeps the Sun bound to its orbit around the Galactic center. It's essential to realize that the gravitational force not only depends on the masses involved but also on the distance separating them. The greater the distance from the center, the weaker the gravitational attraction. Hence, the Sun's gravitational interaction with the Galactic center results in its stable circular motion.

Centripetal force is the force that keeps an object moving in a circular path, directed towards the center of the rotation. For the Sun, this force must equal the gravitational force to maintain a circular orbit. The centripetal force can be expressed as \( F_c = \frac{m V_0^2}{R_0} \), where \( V_0 \) is the Sun's orbital speed.

By equating the centripetal force formula to the gravitational force: \( \frac{G M m}{R_0^2} = \frac{m V_0^2}{R_0} \), we can derive the mass within the Sun's orbit. Simplifying gives us \( \frac{G M}{R_0} = V_0^2 \). This relationship confirms that gravitational force provides the necessary centripetal force for the Sun's circular motion.

Escape velocity is the minimum speed an object needs to escape the gravitational pull of a celestial body. For the Sun's orbit in the Milky Way, the escape velocity can be determined using the formula: \( V_{esc} = \sqrt{2} V_0 \). Given that \( V_0 \approx 200 \) km/s, the escape velocity works out to \( V_{esc} \approx 283 \) km/s.

If a nearby star moves faster than this escape velocity, it can overcome the Milky Way's gravitational pull and escape the galaxy. Observing stars with speeds exceeding this implies other gravitational sources within the galaxy, hinting at a more distributed mass, rather than all mass being centrally located.

Mass distribution within galaxies is not uniform. Instead of being concentrated only at the center, substantial mass exists outside the Sun's orbit. This observation is supported by the high velocities of nearby stars. For example, some stars in the Milky Way move at speeds around 500 km/s, which is well above the escape velocity calculated for a central mass. This suggests additional mass spread throughout the galaxy.

Thus, considering mass distribution helps understand the observed stellar velocities. The presence of appreciable mass outside the Galactic center contributes significantly to the overall gravitational potential, keeping these high-speed stars bound within the galaxy, supporting the concept of dark matter. This distributed mass must be accounted for to explain the gravitational effects observed both in the Milky Way and other galaxies.