Calculate the mean expectation value of the factor

Hint: In spherical coordinates,

Orbital period

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$\mathrm{sin}}^{3$ 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 $\mathrm{sin}}^{3$ i has its mean value.

Hint: In spherical coordinates,$(\theta ,\varphi )$ , 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 $\mathrm{sin}\left\{3\right\}\left(\theta \right)$ .

$v}_{1}=100k\frac{m}{s$

$v}_{2}=200k\frac{m}{s$

Orbital period$=2$ days

${M}_{1}=5.74e33g$

${M}_{2}=2.87e33g$

Calculate the mean expectation value of the factor

Hint: In spherical coordinates,

Orbital period

You can still ask an expert for help

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,

The Kepler equation for a binary star system is,

Step 3

The mean expectation value of

Step 4

From equation (1),

Hence,

Similarly,

Here

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