This question has to do with binary star systems, where 'i' is the angle of incl

Nann 2021-09-15 Answered
This question has to do with binary star systems, where i is the angle of inclination of the system.
Calculate the mean expectation value of the factor sin3 i, i.e., the mean value it would have among an ensemble of binaries with random inclinations. Find the masses of the two stars, if sin3 i has its mean value.
Hint: In spherical coordinates, (θ,ϕ), integrate over the solid angle of a sphere where the observer is in the direction of the z-axis, with each solid angle element weighted by sin{3}(θ).
v1=100kms
v2=200kms
Orbital period =2 days
M1=5.74e33g
M2=2.87e33g
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Expert Answer

Nathalie Redfern
Answered 2021-09-16 Author has 99 answers

Step 1
For a binary star system, the ratio of their velocities and their masses are related as,
m1m2=v2v1
=200kms100kms
=2

The Kepler equation for a binary star system is,
m1+m2=P2πG(v1+v2)3sin3(i)
=(2×24×3600s)2π(6.67×1011Nm2kg2} (105ms1+2×105ms1)3sin3(i)
=(1.11×1031kg)sin3(i)(Md2×1030kg)
=5.556Mdsin3(i)
Step 3
The mean expectation value of sin3 i be given as,
sin3(i)=0π/2sin3(i){sinidi}
0π2sin4(i)di
=[3i8sin(2i)4+sin(4i)32]0π2
=3π16
=0.59
Step 4
From equation (1), m1=2m2
Hence,
3m2=5.556Mdsin3(i)
m2=5.556Md3(0.59)
=3.1389Md
Similarly,
m1=6.3Md
Here Md defined as the solar mass. Solar mass equal to approximately 2×10

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