The solar system is 25,000 light years from the center of our Milky Way galaxy. One light year is the distance light travels in one year at a speed of 3.0 x 10⁸ m/s . Astronomers have determined that the solar system is orbiting the center of the galaxy at a speed of 230 km/s . (c) The gravitational force on the solar system is the net force due to all the matter inside our orbit. Most of that matter is concentrated near the center of the galaxy. Assume that the matter has a spherical distribution, like a giant star. What is the approximate mass of the galactic center?

Verified step by step guidance

1. Convert the distance of the solar system from the center of the Milky Way from light years to meters. Recall that 1 light year is the distance light travels in one year. Use the speed of light (3.0 x 10^8 m/s) and the number of seconds in a year (approximately 3.15 x 10^7 seconds) to perform this conversion.
2. Convert the orbital speed of the solar system from kilometers per second to meters per second to maintain consistency in units.
3. Use the formula for centripetal force, which is the net force acting on the solar system due to its circular motion around the center of the galaxy. The formula is F = (m*v^2)/r, where m is the mass of the solar system, v is the orbital speed, and r is the radius of the orbit (distance from the center).
4. Assume the mass of the solar system is negligible compared to the mass of the galaxy, and rearrange the centripetal force formula to solve for the mass of the galactic center (M). Use the gravitational force formula F = (G*M*m)/r^2, where G is the gravitational constant (6.674 x 10^-11 N(m^2)/(kg^2)), and equate it to the centripetal force formula to solve for M.
5. Substitute the known values (G, r, and v) into the rearranged formula to find the approximate mass of the galactic center.