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?

Answer 1

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.