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31 March, 06:35

We discover a nearby star with two planets. The first planet has an orbit period of 10 years and is in a circular orbit with radius 106 km. The second planet has an orbit period of 15 years. What is its orbit radius? You may assume it is also in a circular orbit.

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  1. 31 March, 10:19
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    Answer: 139 Km.

    Explanation:

    The question tells us that a planet A has an orbit period of 10 years and its circular orbit has a radius of 106 Km, whilst a planet B has an orbit period of 15 years (also assuming a circular orbit), both orbiting a nearby star.

    This information allow us to use the Kepler's 3rd law, for the special case in which the orbit is circular.

    Kepler's 3rd law, tells that there exist a direct proportionality between the square of the orbit period, and the cube of the orbit radius (in the more general case, with the cube of the semi-major axis of the elipse), for celestial bodies orbiting a same star.

    (like Earth and Mars orbiting Sun).

    So, for planet A and planet B (orbiting a same star), we can write the following:

    (TA) ² / (TB) ² = (rA) ³ / (rB) ³

    Replacing by TA = 10 years, TB = 15 years, rA = 106 Km, and solving for Rb, we get RB = 139 Km.
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