From the Sun's orbital speed of \(200 \mathrm{~km} \mathrm{~s}^{-1}\), find the mass within its orbit at \(r=8 \mathrm{kpc}\). Show that the average density inside a sphere of this radius around the Galactic center is \(\sim 0.03 \mathcal{M}_{\odot} \mathrm{pc}^{-3}\), so that \(t_{\mathrm{fr}} \sim 100 \mathrm{Myr}\). This is a typical density for the inner parts of a galaxy. Processes such as bursts of star formation that involve large parts of the galaxy happen on roughly this timescale, because gravitational forces cannot move material any faster through the galaxy.

To understand the mass within the Sun's orbit around the Galactic center, we use its orbital velocity and distance. The Sun orbits the center of the Milky Way at a speed of 200 km/s, at a distance of 8 kiloparsecs (kpc). This data helps us estimate the mass contained within this radius. The formula for circular orbital velocity is:
\( v = \sqrt{\frac{G M}{r}} \).
Here, \( v \) is the orbital speed, \( G \) is the gravitational constant, \( M \) is the mass, and \( r \) is the orbital radius. Rearranging the formula to solve for mass (\( M \)) gives:
\[ M = \frac{v^2 r}{G} \].
By inserting the known values, we can calculate the Galactic center mass inside the Sun's orbit.

The orbital velocity formula helps determine the speed at which objects orbit a galactic center due to gravitational forces. For any object in circular orbit, its velocity is influenced by the mass it orbits and its distance from the center. The formula is:
\( v = \sqrt{\frac{G M}{r}} \).
Here:

• \( v \) = orbital speed
• \( G \) = gravitational constant
• \( M \) = mass inside the orbit
• \( r \) = radius of the orbit

Using this formula helps understand how motion within galaxies is dependent on gravitational interplay and mass distribution.

In astronomy, calculating the density of an area within a galaxy helps us understand the matter distribution. To find the average density inside the Sun’s orbit, we use the mass calculated earlier and the volume of a sphere with radius equal to the Sun's orbit. The density formula is:
\[ \rho = \frac{M}{\frac{4}{3} \pi r^3} \].
Here:

• \( \rho \) = density
• \( M \) = mass
• \( r \) = radius
• \( \pi \) = 3.14159

This calculation helps confirm the mass density within the Sun's orbital range, which is typically around 0.03 solar masses per cubic parsec (\( \mathcal{M}_{\odot} \text{pc}^{-3} \)).

Gravitational forces are the backbone of orbital motion within galaxies. They ensure stars and other celestial bodies stay in their curved paths. Newton's law of gravitation states that every particle of matter in the universe attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers:
\[ F = G \frac{m_1 \cdot m_2}{r^2} \].
This equation is critical to understanding orbital motion and density within galaxies as it ties mass and distance to the gravitational attraction felt. For large structures like galaxies, gravitational forces dictate how stars orbit around the galactic center.

Free-fall time is the time it takes for an object to collapse under its gravity if no other forces act to support it. For galaxies, it’s an important factor in understanding how quickly processes like star formation can occur. It’s related to density by the formula:
\( t_{\text{fr}} = \frac{1}{\sqrt{G \rho}} \).
Here:

• \( t_{\text{fr}} \) = free-fall time
• \( \rho \) = density
• \( G \) = gravitational constant

By plugging in the calculated density, the free-fall time can be simplified to show that, typically, it’s around 100 million years (\( 100 \text{Myr} \)). This helps explain why large galactic processes, like star formation bursts, occur over such timescales.