ASTRONOMY FUNDAMENTALS - Chapter 1, Exercise 1 Solution
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Calculating Distance from Parallax Angle

PROBLEM
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A star is measured with a parallax angle of 0.5 arcseconds. Calculate its
distance in both parsecs and light-years.

SOLUTION
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By definition, distance in parsecs is the reciprocal of the parallax angle
in arcseconds:

    distance (parsecs) = 1 / parallax angle (arcseconds)
    distance (parsecs) = 1 / 0.5
    distance (parsecs) = 2 parsecs

Converting to light-years, using the real conversion 1 parsec ≈ 3.26
light-years:

    distance (light-years) = 2 parsecs x 3.26 ly/parsec
    distance (light-years) ≈ 6.52 light-years

ANSWER: The star is 2 parsecs away, or approximately 6.52 light-years.

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WHY THIS WORKS AS AN ANSWER

The relationship distance = 1/parallax falls directly out of how the parsec
is defined in the first place: a star with exactly 1 arcsecond of parallax
is, by definition, exactly 1 parsec away. Since parallax angle shrinks as
distance grows (a more distant star shifts less against the background
sky), the two quantities are inversely proportional - half the parallax
angle means twice the distance, which is exactly what this calculation
shows: a 0.5 arcsecond parallax (half of 1 arcsecond) corresponds to 2
parsecs (double the 1-parsec reference distance).

For context, this 2-parsec distance is actually closer than any real
star system other than the Sun - the nearest real star system, Alpha
Centauri, is about 1.33 parsecs away (4.34 light-years), with a real
measured parallax of about 0.751 arcseconds. A star at exactly 0.5
arcseconds parallax would be even closer than Alpha Centauri - a
genuinely nearby star by real astronomical standards, and not one
that's actually been found this close in reality.