Stellar parallax: The measurement of stellar parallax

After the publication of the Copernican theory of the Universe, which stated that the observed behaviour of planets could be as easily explained if it was assumed that the Earth revolved about the Sun, repeated attempts were made to measure the parallaxes of the brighter, and presumably nearby, stars. Almost 300 years were to elapse, however, before success was achieved, first by Bessel at Konigsberg in 1838, then by Henderson at the Cape of Good Hope and by F Struve at Dorpat soon
after. The values they obtained showed why it took three centuries to detect the parallactic movements of stars. For the star 61Cygni, Bessel found a parallax of 0''·314; Henderson measured the parallax of α Centauri to be almost three-quarters of one second of arc while Struve showed that of Vega to be about one-tenth of 1 second of arc. These are very small angles. In fact, only 23 stars are known with parallaxes of 0''·24 or greater, Proxima Centauri having a parallax close to 0''·75.
The modern method of measuring a star’s parallax involves the use of photographic plates or CCD detectors. In principle, records are taken six months apart of the area of the sky surrounding the star. If the star is near enough, the shift of the Earth from one side of its orbit to the other should produce a corresponding apparent shift of the star against the very faint stellar background. This shift will change the right ascension and declination of the star and it is essentially
these changes in coordinates that are measured. Because the shifts are very small, they are measured using faint reference stars, so faint that they are presumably far enough away for their own parallactic displacements to be negligible. To fix our ideas, let us consider one such reference star only, with right ascension α_R and declination δ_R. Let the heliocentric right ascension and declination of the parallax star be α and δ and let its apparent coordinates be α₁, δ₁ and α₂, δ₂ at the times the first and second records are taken.
Now the change in a star’s right ascension due to parallax will be given by an expression of the
form
α' − α = P × F
where α', α are the star’s apparent and heliocentric right ascensions, P is its parallax and F is a function of the star’s equatorial coordinates, the Sun’s longitude and the obliquity of the ecliptic. This function will have a particular value at any given date and this value, from a knowledge of the form of the function, can be calculated.
Let its values be F₁ and F₂ when the two records were made. Then
α₁ − α = P × F₁ α₂ − α = P × F₂.
Subtracting, we obtain
α₁ − α₂ = P(F₁ − F₂)
or, introducing the reference star’s right ascension,
(α₁ − α_R) − (α₂ − α_R) = P(F₁ − F₂).
The quantities (α₁ − α_R) and (α₂ − α_R) are the differences between the right ascensions of the parallax star and the reference star and can be measured on a suitable measuring engine or with reference to the pixel grid of the detector. Hence,
P = (α₁ − α_R) − (α₂ − α_R)/F₁ − F₂.
In practice, several plates or frames are taken at each epoch and more than one reference star is used, the two epochs (separated by six months) being chosen so that the most advantageous value of F₁ − F₂ is obtained. The practical limit to this method from Earth-based telescopes is quickly reached. Only parallaxes greater than 0''·01 can be measured at all reliably and only a few thousand stars have had their parallaxes measured in this way. A major step forward in accuracy was the launching of the artificial Earth satellite Hipparcos by the European Space Agency in August 1989. Its 0·30 m telescope measured the positions, proper motions1 and parallaxes of about 120 000 stars to an accuracy of better than 0''·002. It also measured the brightnesses and colours of more than one million stars.