The calculated mass of each star in the Alpha Centauri binary system is approximately 4.735 × 10^29 kilograms, based on Kepler's third law of planetary motion.
To determine the mass of the stars in the Alpha Centauri binary system, you can use Kepler's third law of planetary motion, which can be extended to binary star systems:
T^2 = (4π^2 / G * (m1 + m2)) * a^3
where:
T is the orbital period,
G is the gravitational constant (6.674 × 10^(-11) m^3 kg^(-1) s^(-2)),
m1 and m2 are the masses of the two stars,
a is the semi-major axis of the binary orbit.
In the case of equal-mass stars, m1 = m2 = m, and the semi-major axis a is half of the separation (a = 1/2 * separation).
Given that T = 2.52 × 10^9 s and a = 3.45 × 10^12 m, we can substitute these values into the equation and solve for m:
(2.52 × 10^9)^2 = (4π^2 / G * (2m)) * (3.45 × 10^12)^3
Now, solve for m:
m = (4π^2 * (3.45 × 10^12)^3) / (G * (2.52 × 10^9)^2)
Let's calculate this:
m = (4 * π^2 * (3.45 × 10^12)^3) / (G * (2.52 × 10^9)^2)
Now, plug in the values and solve for m. Note that π is approximately 3.14159.
m ≈ (4 * (3.14159)^2 * (3.45 × 10^12)^3) / (6.674 × 10^(-11) * (2.52 × 10^9)^2)
Calculating this will give you the mass m of each star in the binary system.
m ≈ (4 * 9.8696 * (1.26425 × 10^38)) / (6.674 × 10^(-11) * 6.3504 × 10^18)
m ≈ (4 * 9.8696 * 1.26425 × 10^38) / (6.674 * 6.3504 × 10^7)
m ≈ (4 * 9.8696 * 1.26425 × 10^38) / (42.4407 × 10^7)
m ≈ (4 * 9.8696 * 1.26425 × 10^38) / (4.24407 × 10^8)
m ≈ (4 * 9.8696 * 1.26425 × 10^38) / (424.407 × 10^6)
m ≈ (4 * 9.8696 * 1.26425 × 10^38) / 424407000
m ≈ (201.18816 × 10^38) / 424407000
m ≈ 4.735 × 10^29 kg
Therefore, the mass of each star in the Alpha Centauri binary system is approximately 4.735 × 10^29 kilograms.