(c) the equation d = 1/p, where p is the parallax in seconds of arc and d…

The idea

The equation d = 1/p gives the distance d in parsecs to a star when its parallax p is measured in seconds of arc. The relationship is inverse: a larger parallax means a nearer star, and a smaller parallax means a more distant star. For example, a star with p = 0.1″ lies at d = 1/0.1 = 10 pc, while Proxima Centauri with p ≈ 0.77″ lies at about 1.3 pc. In multiple-choice questions you may be asked to calculate d from p, to find p from d, to convert the result to metres or light-years, or to recognise that the equation only works when p is in seconds of arc.

What earns the marks

• States the equation d = 1/p with d in parsecs and p in seconds of arc.
• Applies the inverse relationship correctly, so a smaller parallax gives a larger distance.
• Substitutes a parallax value in seconds of arc and evaluates d without a unit error.
• Converts a distance in parsecs to metres or light-years when required.
• Recognises that the equation assumes the parallax angle is measured with a baseline of 1 au.

Common mistakes and how to put them right

• Substituting the parallax in degrees or arcminutes instead of seconds of arc.

  Put it right: The equation only gives parsecs when p is in seconds of arc.

• Inverting the relationship and writing d = p.

  Put it right: The correct form is d = 1/p, so a parallax of 0.5″ gives 2 pc, not 0.5 pc.

• Forgetting to convert a distance in parsecs to metres when the question asks for an SI answer.

  Put it right: Multiply by about 3.09 × 10¹⁶ m pc⁻¹.

• Using the total angular shift over six months as p.

  Put it right: The parallax angle is half that shift, so the calculated distance would be too small by a factor of two.

Exam tips

• Write down d = 1/p and label the units before substituting, so you can see immediately whether p is in seconds of arc.
• If the question gives p in milliarcseconds, convert to seconds of arc by dividing by 1000 before using the equation.
• Check the reasonableness of your answer: nearby stars have distances of a few parsecs, so a result of thousands of parsecs should prompt a unit check.
• When converting to metres, use 1 pc ≈ 3.09 × 10¹⁶ m and keep the calculation in standard form to avoid power-of-ten slips.