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Orbital Radius & Detection Techniques of Extrasolar Planets

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** Please see the attached file for the complete solution response **

Please show all calculations.

I only need help with part (a)i and ii of the question.

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Solution Summary

This solution is an illustrated description of how to calculate the orbital radius of extrasolar planets about their star systems, it also describes appropriate and inappropriate detection techniques for such planets given their distances and orbital orientations from us.

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I'll start by letting you know the key things you have to understand to address this question appropriately:

1. In a system like that of stars and planets, there is always a center of mass. This center of mass is always closer to the more massive body, and in (almost) all cases, the star in a system is more massive than the planets in the same system, the result being that the center of the mass for a star-planet system will be closer to the star than to the planet.

2. The star and planet both orbit their common center of mass in their different orbits; since the center of mass is always closer to the star than to the planet, the star's orbit is always shorter than the planet's, the result being that we hardly notice the star's motion as compared to the planet's. [Most people usually neglect the star's motion in a system especially if the star is much more massive than the planet, it will only appear the planet is the one orbiting the stationary star (like we always say that the earth and other planets orbit the sun), but this is not exactly true; the sun is also in orbital motion about the center of mass of the solar system just that this center of mass is very close to the sun due to its massiveness, and so the sun's orbit is also really too short compared to those of the planets, and we tend to ignore this motion]

3. Due to the brightness of the stars and their great distances away from us, it is impossible to directly image planets in their systems from earth with the technologies we have today, especially for really far systems like the 1000 pc mentioned in this question. [I have just indirectly said that Direct Imaging ...

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