Given that the mass of the Milky Way galaxy is 10^11 times that of the Sun and that the sun is 2.6x10^20 meters from its center, what is the Sun's orbital speed around the center of the galaxy? How long does it take the Sun to orbit the Milky Way? (In this problem, we assume that the galaxy can be treated as a single, spherical blob of matter. Strictly speaking, this isnt correct, but the far more elaborate math needed to calculate the problem properly ends up giving almost the same answer)

To find the Sun's orbital speed around the center of the Milky Way galaxy, we can use the concept of gravitational force.

First, let's find the gravitational force between the Sun and the center of the galaxy. The force of gravity is given by Newton's law of universal gravitation:

F = G * (m1 * m2) / r^2

Where:
F is the gravitational force,
G is the gravitational constant (approximately 6.67430 x 10^-11 N(m/kg)^2),
m1 is the mass of the Sun,
m2 is the mass of the galaxy (in this case, the Milky Way),
and r is the distance between the center of the galaxy and the Sun.

Given:
Mass of the galaxy (m2) = 10^11 times the mass of the Sun (m1)
So, m2 = 10^11 * m1

Distance from the center of the galaxy to the Sun (r) = 2.6x10^20 meters

We can rewrite the gravitational force equation as:

F = G * (m1 * (10^11 * m1)) / r^2

Now, we can calculate the gravitational force between the Sun and the center of the galaxy.

Next, we will calculate the centripetal force, which is responsible for keeping the Sun in its orbit around the center of the galaxy. The centripetal force can be defined as:

Fc = m1 * v^2 / r

Where:
Fc is the centripetal force,
m1 is the mass of the Sun,
v is the orbital speed,
and r is the distance from the center of the galaxy to the Sun.

The gravitational force and the centripetal force are equal, so we can set them equal to each other and solve for v:

G * (m1 * (10^11 * m1)) / r^2 = m1 * v^2 / r

Simplifying the equation:

G * (10^11 * m1^2) / r = v^2

Now, we can solve for v, the orbital speed of the Sun.

Now, to calculate the time it takes for the Sun to orbit the Milky Way, we can use the formula for the period (T) of a circular orbit:

T = (2 * π * r) / v

Where:
T is the orbital period,
π is a mathematical constant (approximately 3.14159),
r is the distance from the center of the galaxy to the Sun, and
v is the orbital speed.

Using the values calculated above, we can find the time it takes for the Sun to complete one orbit around the Milky Way.