Final answer:
Using Newton's version of Kepler's third law and substituting the necessary values, the orbital velocity of our solar system about the center of the Milky Way is calculated to be approximately 220 km/s, which is closer to option (a) approx 250 km/s.
Step-by-step explanation:
To calculate the orbital velocity of our solar system around the center of the Milky Way, we can use Newton's version of Kepler's third law, which relates the velocity of an orbiting object to the mass of the object (or collection of objects such as a galaxy) it is orbiting around. The mass of the Milky Way within the Sun's orbit is given as 1.5 × 10¹¹ times the mass of the Sun. We can convert the orbital radius from light years to meters and then apply the orbital velocity formula v = √(GM/r).
The distance from the center of the Milky Way to our solar system is 27,000 light-years, which is about 2.56 × 10¹¹ meters. Using this information, along with the gravitational constant G (6.674× 10⁻¹¹ m³¹kg⁻¹¹s⁻²) and the mass of the Milky Way within our orbit (which we can calculate based on the mass of the Sun, 1.989 × 10³° kg), we can find the orbital speed.
Substituting the given values, we get v = √((6.674× 10⁻¹¹ m³¹kg⁻¹¹s⁻² × (1.5 × 10¹¹ × 1.989 × 10³° kg)) / (2.56 × 10¹¹ m)). Calculating this gives us an orbital speed of V ≈ 220 km/s, which is closer to option (a) than to the other options provided, although it is not exactly 250 km/s.