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The parallax formula is an essential tool in astronomy for measuring the distance to stars. This formula helps astronomers determine how far a star is from Earth by using the angle of its apparent shift in position as observed from different points in Earth's orbit. The formula is written as: \( d = \frac{1}{p} \) where \( d \) represents the distance in parsecs and \( p \) is the parallax angle measured in arcseconds. By understanding this formula, we can translate small angular measurements into vast cosmic distances. In our exercise, we start by recognizing that the star's parallax angle is 0.1 arcseconds.

Parsecs are the standard unit of measurement used by astronomers to express distances between astronomical objects. A parsec is defined as the distance at which one astronomical unit subtends an angle of one arcsecond. This makes parsecs very useful for describing stellar distances. In our exercise, we use the given parallax angle of 0.1 arcseconds and apply it to the parallax formula. Substituting in the formula \( d = \frac{1}{p} \) with \( p = 0.1 \), we calculate the distance to be: \[ d = \frac{1}{0.1} = 10 \text{ parsecs} \]. Therefore, the given star is 10 parsecs away from Earth.

An arcsecond is a unit used to measure very small angles in the sky. One arcsecond is \(1/3600\) of a degree, making it a crucial unit for precise astronomical observations. When we talk about a star's parallax, we're referring to its apparent movement due to Earth's orbit around the Sun. Even though these shifts are tiny, measuring them in arcseconds allows us to employ the parallax formula accurately. For example, in our exercise, the parallax of 0.1 arcseconds means the star appears to shift by this very tiny angle due to our perspective from Earth. Using arcseconds in the parallax formula provides astronomers with the power to convert these minuscule angles into a comprehensible distance in parsecs.