Finding The Masses Of Main Sequence Stars
Consider main sequence binary stars with circular orbits in planes parallel to the line of sight. For any such pair, define m1, m2, d1, d2, v1, v2 and T to be the masses, center of mass distances, velocities and period of the stars respectively. G is the gravitational constant:
v1 = 2πd1 / T, v2 = 2πd2 / T, m1d1 = m2d2 and Gm1m2 / (d1 + d2)2 = m1(2π / T)2d1.
Since d1 = m2(d1 + d2) / (m1 + m2), m1 + m2 = (4π2 / G) (d1 + d2)3 / T2.
Since (d1 + d2) / T = (v1 + v2) / (2π), m1 + m2 = (v1 + v2)3 T / (2πG).
Since m1 = (m1 + m2) / (1 + d1 / d2) and m2 = (m1 + m2) / (1 + d2 / d1):
m1 = (1 / (2πG)) (v1 + v2)3 T / (1 + d1 / d2)
m2 = (1 / (2πG)) (v1 + v2)3 T / (1 + d2 / d1).
v1 and v2 can be found by measuring Doppler shifts. T, d1 / d2 and d2 / d1 can be found from visual inspection. Lastly, distances and luminosities can be found by measuring parallaxes.
Doing all these measurements for several main sequence binary stars will empirically confirm the mass luminosity relationship. The mass luminosity relationship can then be used to find the mass of any main sequence star by measuring its luminosity.
v1 = 2πd1 / T, v2 = 2πd2 / T, m1d1 = m2d2 and Gm1m2 / (d1 + d2)2 = m1(2π / T)2d1.
Since d1 = m2(d1 + d2) / (m1 + m2), m1 + m2 = (4π2 / G) (d1 + d2)3 / T2.
Since (d1 + d2) / T = (v1 + v2) / (2π), m1 + m2 = (v1 + v2)3 T / (2πG).
Since m1 = (m1 + m2) / (1 + d1 / d2) and m2 = (m1 + m2) / (1 + d2 / d1):
m1 = (1 / (2πG)) (v1 + v2)3 T / (1 + d1 / d2)
m2 = (1 / (2πG)) (v1 + v2)3 T / (1 + d2 / d1).
v1 and v2 can be found by measuring Doppler shifts. T, d1 / d2 and d2 / d1 can be found from visual inspection. Lastly, distances and luminosities can be found by measuring parallaxes.
Doing all these measurements for several main sequence binary stars will empirically confirm the mass luminosity relationship. The mass luminosity relationship can then be used to find the mass of any main sequence star by measuring its luminosity.
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