A binary system has a period of 5 years and semi-major axis of 3 AU. The primary star is 2 times the mass of the secondary, what is the approximate ate mass of the secondary star in terms of solar masses?
To calculate the approximate mass of the secondary star in terms of solar masses, we can use Kepler's Third Law of Planetary Motion:
T^2 = (4π^2 / G * (M1+M2)) * a^3
Where:
T = Period of the binary system (in this case, 5 years)
a = Semi-major axis of the binary system (in this case, 3 AU)
M1 = Mass of the primary star
M2 = Mass of the secondary star
G = Gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2)
From the given information, we know that the primary star is 2 times the mass of the secondary star:
M1 = 2M2
Substituting this into the equation:
T^2 = (4π^2 / G * (2M2 + M2)) * a^3
T^2 = (4π^2 / G * 3M2) * a^3
T^2 = (4π^2 * a^3) / (G * 3M2)
Now, solving for M2:
M2 = (4π^2 * a^3) / (G * 3 * T^2)
Substituting the known values into the equation:
M2 = (4π^2 * (3 AU)^3) / (6.67430 × 10^-11 m^3 kg^-1 s^-2 * 3 * (5 years)^2)
Converting AU to meters:
1 AU = 1.496 × 10^11 meters
M2 = (4π^2 * (3 * 1.496 × 10^11 meters)^3) / (6.67430 × 10^-11 m^3 kg^-1 s^-2 * 3 * (5 years)^2)
M2 ≈ 1.57 × 10^30 kg
To convert this mass into solar masses, we divide it by the mass of the Sun:
1 solar mass ≈ 1.989 × 10^30 kg
M2 ≈ (1.57 × 10^30 kg) / (1.989 × 10^30 kg)
M2 ≈ 0.789 solar masses
Therefore, the approximate mass of the secondary star in terms of solar masses is approximately 0.789 solar masses.