Math Problem Statement
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.
Solution
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 from orbital parameters is:
where:
- is the mass of the galaxy within the Sun's orbit,
- is the Sun's orbital radius around the galactic center,
- is the orbital period,
- is the gravitational constant, approximately .
Step 1: Convert Units
-
Orbital Period : The Sun's orbital period is given as 230 million years. We convert this to seconds:
-
Orbital Radius : The distance is given as 27,000 light-years. We convert this to meters:
Step 2: Substitute Values and Solve for Mass
Calculating this yields:
Final Answer
The estimated mass of the Milky Way Galaxy within the Sun's orbit is:
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
- What is the significance of calculating only the mass within the Sun's orbit?
- How would this calculation change if the Sun were at a different distance from the galactic center?
- What assumptions are made in using this method to estimate galactic mass?
- How does dark matter affect the accuracy of this estimate?
- 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.
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Math Problem Analysis
Mathematical Concepts
Orbital Mechanics
Gravitational Dynamics
Kepler's Laws
Formulas
M = (r^3 * (2π / T)^2) / G
Theorems
Kepler's Third Law
Suitable Grade Level
College Physics or Astronomy
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