When planets orbit a star, both orbit their common "center...

When planets orbit a star, both orbit their common "center of mass" (Unit 17), which lies at a point between them closer to the center of the more massive object in proportion to the ratio of their masses. (If the planet is $1 / 10^{\text {th }}$ as massive as the star, the point will be $10 \times$ farther from the planet than from the star.) a. By how many AU does the Sun shift in each of the cases in the previous problem? b. If viewed from 30 light-years away, how large will its angular shift be? (Refer to the angular size formula in Unit $10 .$ )

When planets orbit a star, both orbit their common "center of mass" (Unit 17), which lies at a point between them closer to the center of the more massive object in proportion to the ratio of their masses. (If the planet is $1 / 10^{\text {th }}$ as massive as the star, the point will be $10 \times$ farther from the planet than from the star.)
a. By how many AU does the Sun shift in each of the cases in the previous problem?
b. If viewed from 30 light-years away, how large will its angular shift be? (Refer to the angular size formula in Unit $10 .$ )

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Key Concepts

Astronomical Distance Units

In astronomy, distances are commonly expressed using units like astronomical units (AU) for intra-solar system distances and light years for interstellar distances. Understanding these units is critical for accurately computing and converting measurements such as the displacement of a star or the angular shift observed from a distant vantage point.

Angular Measurement and Angular Size Formula

Angular size, or the apparent size of an object in the sky, is determined by the physical size of the object and its distance from the observer via the angular size formula. This concept is used to convert a physical displacement (such as a solar shift measured in astronomical units) into an angular measurement as seen from a great distance, allowing for comparisons and observations from different reference points.

Center of Mass (Barycenter)

This concept deals with the point where the mass of two or more orbiting bodies effectively balances. In orbital dynamics, both objects (for example, a star and a planet) orbit around their common center of mass, whose location is influenced by the ratio of their masses. This principle is fundamental to understanding the motion and gravitational effects within a binary system.

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