The Sun seen from Barnard's Star
- Round
- Theory
- Points
- 10
- Level
- Bronze
- Taught in
- Section S3
Adapted from IOAA 2012 (Rio de Janeiro), theory problem 11. We have restated it in our own words. The original gives no data, so we have added the values that the official solution uses. The solution was last checked on 16 October 2026.
Jupiter pulls the Sun round a small orbit, so that seen from Barnard's Star the Sun would swing to and fro across the sky. What is the angular amplitude of this motion, the largest angle by which the Sun moves from its mean position, and what is its period? Take Jupiter's orbit as a circle of radius 5.204 au, the Sun as 1047 times as massive as Jupiter, and the distance of Barnard's Star as 1.834 pc.
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The centre of mass lies on the line from the Sun to Jupiter, at a distance
and the Sun moves on a circle of this radius, once in each orbit of Jupiter.
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Seen from 1 pc, 1 au covers 1″, so by the small-angle approximation the largest angle by which the Sun moves from its mean position, seen from 1.834 pc, is
From Barnard's Star the circle is seen at a slant, as an ellipse, and this angle is its long half-axis.
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The period is the period of Jupiter, which in years and astronomical units is
AnswerThe Sun swings up to 0.0027″ from its mean position, less than three thousandths of an arcsecond, once every 11.87 years. From Barnard's Star its circle is seen at a slant, as an ellipse whose long half-axis is this angle. Measuring motions as small as this is one way to find giant planets around nearby stars (Chapter 9).
This problem also appears among the olympiad problems of Section S3, Maths toolkit, which teaches the ideas it uses. Equation and figure numbers refer to the course.