Estimate the mass of the Milky Way Galaxy from the fact that the Sun orbits the galactic center every 230 million years at a distance of 27,000 light-years. (As we'll discuss in Chapter 19, this calculation actually tells us only the mass of the galaxy within the Sun's orbit.)
Express your answer to three significant figures and Include the appropriate units.

To estimate the mass of the Milky Way Galaxy within the Sun's orbit, we can use Kepler’s Third Law adapted for orbital motion around a massive central object. The relevant formula for calculating mass \( M \) from orbital parameters is:

\[
M = \frac{r^3 \cdot (2 \pi / T)^2}{G}
\]

where:
- \( M \) is the mass of the galaxy within the Sun's orbit,
- \( r \) is the Sun's orbital radius around the galactic center,
- \( T \) is the orbital period,
- \( G \) is the gravitational constant, approximately \( 6.674 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \).

### Step 1: Convert Units

1. **Orbital Period \( T \)**: The Sun's orbital period is given as 230 million years. We convert this to seconds:
   \[
   T = 230 \times 10^6 \, \text{years} \times (365.25 \times 24 \times 3600) \, \text{s/year} = 7.255 \times 10^{15} \, \text{s}
   \]

2. **Orbital Radius \( r \)**: The distance is given as 27,000 light-years. We convert this to meters:
   \[
   r = 27,000 \, \text{light-years} \times (9.461 \times 10^{15} \, \text{m/light-year}) = 2.554 \times 10^{20} \, \text{m}
   \]

### Step 2: Substitute Values and Solve for Mass \( M \)

\[
M = \frac{(2.554 \times 10^{20})^3 \cdot (2 \pi / 7.255 \times 10^{15})^2}{6.674 \times 10^{-11}}
\]

Calculating this yields:

\[
M \approx 1.99 \times 10^{41} \, \text{kg}
\]

### Final Answer

The estimated mass of the Milky Way Galaxy within the Sun's orbit is:

\[
\boxed{1.99 \times 10^{41} \, \text{kg}}
\]

This result is expressed to three significant figures and in units of kilograms.

Would you like further details, or do you have any questions?

### Related Questions

1. What is the significance of calculating only the mass within the Sun's orbit?
2. How would this calculation change if the Sun were at a different distance from the galactic center?
3. What assumptions are made in using this method to estimate galactic mass?
4. How does dark matter affect the accuracy of this estimate?
5. What other methods can astronomers use to estimate the mass of galaxies?

### Tip

Remember, Kepler's Laws are only strictly accurate in systems with a central mass; in galaxies, the distributed mass means the actual mass could be higher.

Estimate the mass of the Milky Way Galaxy from the fact that the Sun orbits the galactic center every 230 million years at a distance of 27,000 light-years. (As we'll discuss in Chapter 19, this calculation actually tells us only the mass of the galaxy within the Sun's orbit.) Express your answer to three significant figures and include the appropriate units.

Discover how to estimate the mass of the Milky Way Galaxy within the Sun's orbit, given that the Sun completes an orbit every 230 million years at a distance of 27,000 light-years from the galactic center. This solution involves using Kepler's Laws and gravitational formulas, suitable for college-level physics or astronomy students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation