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Problem:

The spectral lines of two stars in an eclipsing binary system shift periodically due to their motion. Each star orbits their common center of mass with a period of 6 months. The orbital speed of each star is given as $5.0 \times 10^5 \, \text{m/s}$. The task is to calculate the masses of the two stars, assuming they are of equal mass ($M_1 = M_2$) and express the answer using two significant figures.

Step-by-Step Solution:

1. Binary System Dynamics:

The stars orbit their common center of mass. According to Kepler's Third Law and Newton's Law of Gravitation: $M_1 + M_2 = \frac{4 \pi^2 r^3}{G T^2}$ where:

• $r$ = radius of the circular orbit,
• $G = 6.674 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2}$ (gravitational constant),
• $T = 6 \, \text{months} = 6 \times 30 \times 24 \times 3600 \, \text{s}$,
• $M_1 = M_2 = M$ (equal masses).

2. Orbital Velocity Relationship:

For a circular orbit: $v = \frac{2 \pi r}{T}$ Rearranging for $r$: $r = \frac{v T}{2 \pi}$ Substitute $v = 5.0 \times 10^5 \, \text{m/s}$ and $T = 6 \, \text{months}$.

3. Substitute Values:

Using $r$ in the mass equation: $M_1 + M_2 = \frac{4 \pi^2 r^3}{G T^2}$

Since $M_1 = M_2 = M$: $2M = \frac{4 \pi^2 r^3}{G T^2}$

Solve this equation step by step for $M$.

Let me compute the numerical values.import math

Given values

v = 5.0e5 # Orbital speed (m/s) T = 6 * 30 * 24 * 3600 # Orbital period in seconds (6 months) G = 6.674e-11 # Gravitational constant (m^3 kg^-1 s^-2)

Calculate orbital radius (r)

r = (v * T) / (2 * math.pi)

Apply mass equation: 2M = (4 * pi^2 * r^3) / (G * T^2)

total_mass = (4 * math.pi2 * r3) / (G * T**2) individual_mass = total_mass / 2 # Since M1 = M2

Format to 2 significant figures

individual_mass_rounded = round(individual_mass, 2)

individual_mass_rounded2.3179293479970372e+332.3179293479970372e+33successThe mass of each star in the binary system is approximately $2.32 \times 10^{33} \, \text{kg}$, expressed to two significant figures.

Do you have any further questions or want a detailed explanation of any part of this solution?

Here are 5 related questions to consider:

1. How does the Doppler effect help determine orbital velocities in binary systems?
2. What changes if the masses of the stars are not equal?
3. How does the orbital period affect the masses in a binary system?
4. What other methods exist to determine the masses of stars?
5. How do eccentric (non-circular) orbits affect this calculation?

Tip: Always double-check the period's unit conversions (e.g., months to seconds) to avoid errors in astronomical calculations.