Answer:
The combined mass of the two stars is 2.9417 solar masses.
Explanation:
The mathematical expression for Kepler's third law is;
[tex]P^{2}[/tex] = [tex]\frac{4\pi ^{2} }{k^{2} (M_{1} + M_{2} }a^{3}[/tex]
Where: P is the period in days, a is the semimajor axis in AU, [tex]M_{1}[/tex] is the mass of the first star, [tex]M_{2}[/tex] is the mass of the second star and k is the Gaussian gravitational constant.
Given that;
P = 10 years = 3670 days (including two leap years)
a = 6.67 AU
k = 0.01720209895 rad
[tex]\pi[/tex] = [tex]\frac{22}{7}[/tex]
The sum of the masses of the two star can be determined by;
[tex](M_{1} + M_{2})[/tex] = [tex]\frac{4\pi ^{2} }{P^{2}k^{2} } a^{3}[/tex]
= [tex]\frac{4*(\frac{22}{7}) ^{2} } {(3650)^{2} * (0.01720209895)^{2} } (6.67)^{3}[/tex]
= [tex]\frac{11724.29601}{3942.2904}[/tex]
= 2.9417 solar masses
Thus the combined mass of the two star is 2.9417 solar masses.
