Final answer:
Using the orbital speed and period of an eclipsing binary system and applying Kepler's third law, we can calculate the sum of the masses of two stars. The individual masses are then determined by the fact that they are equal due to their equal orbital speeds and Doppler shifts, which can be divided by two to find each mass.
Step-by-step explanation:
To determine the masses of two stars in an eclipsing binary system based on the Doppler shifts of their spectral lines, we use Kepler's laws and the Doppler effect. Given that both stars have the same orbital speed of 64,000 m/s and an orbital period of 5 months, we first convert the period into seconds:
5 months * 30 days/month * 24 hours/day * 3600 seconds/hour = 12,960,000 seconds.
The total speed of the stars in their orbit is twice the orbital speed of one star (since they move in opposite directions), which is 2 * 64,000 m/s = 128,000 m/s. Using the circumference of their orbit, we find their separation distance:
Circumference = Velocity * Period, hence Separation (d) = Circumference / (2π) = (128,000 m/s * 12,960,000 s) / (2π).
Now, using Kepler's third law, which states that:
P2 = (4π² / G(M1 + M2))d3,
we can solve for the sum of the masses (M1 + M2) of the two stars.
Substituting the known values and rearranging, we obtain:
M1 + M2 = (4π² / G) * (d3 / P2).
Since the stars have equal mass (as indicated by equal Doppler shifts and orbital speed), we can find the individual masses by dividing the total mass by two:
M1 = M2 = (M1 + M2) / 2.
Therefore, we can conclude the masses of the stars from the given orbital speed and period using Kepler's laws and the Doppler effect.