Suppose we have a binary star system with an orbital radius of 15 AU and a period of 30 yrs. What is the total mass of this binary system?

To find the total mass of a binary star system, we can apply **Kepler's Third Law** in the following form:

\[
\left( \frac{P^2}{a^3} \right) = \frac{4\pi^2}{G(M_1 + M_2)}
\]

where:
- \( P \) is the period of the orbit (in years),
- \( a \) is the semi-major axis or orbital radius (in astronomical units, AU),
- \( M_1 \) and \( M_2 \) are the masses of the two stars (in solar masses),
- \( G \) is the gravitational constant, but in astronomical units, years, and solar masses, \( G \) simplifies the equation to:

\[
\frac{P^2}{a^3} = \frac{1}{M_1 + M_2}
\]

### Rearranging for total mass:
\[
M_1 + M_2 = \frac{a^3}{P^2}
\]

### Given:
- \( a = 15 \, \text{AU} \),
- \( P = 30 \, \text{years} \).

Now, we substitute these values into the equation:

\[
M_1 + M_2 = \frac{15^3}{30^2} = \frac{3375}{900} = 3.75 \, \text{solar masses}.
\]

### Final Answer:
The total mass of the binary star system is **3.75 solar masses**.

Would you like more details on how this formula works, or have any questions?

Here are some related questions to expand this concept:

1. How does the mass of the binary system affect the orbital period?
2. What would happen to the orbital period if the mass were doubled?
3. How can we estimate the individual masses of the stars in the binary system?
4. How would this system's mass compare to the mass of our Sun?
5. How do astronomers measure the orbital radius in such systems?

**Tip:** When using Kepler's law, make sure your units are consistent—AU for distance, years for time, and solar masses for mass.

Explore how to calculate the total mass of a binary star system with an orbital radius of 15 AU and a period of 30 years using Kepler's Third Law. This example demonstrates key principles of orbital mechanics, suitable for students in Grades 11-12 or college-level physics.

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