Stellar parallax

Stellar parallax

the annual displacement

The mural quadrant at Uraniborg, from Tycho's own Astronomiae Instauratae Mechanica (1598). With this he looked for the annual shift of the stars and did not find it, and concluded that the Earth stands still. He was right about the looking and the not finding.

Cause	The Earth's position in its orbit
Largest known	0.77″ (Proxima Centauri)
Varies with distance	Yes, as 1/d
Demanded as proof	1543
First measured	1838, by Bessel[1]
Interval	295 years
The objection meanwhile	Sound
Defeated by	An artefact of the eye

Stellar parallax is the small annual displacement of a nearby star against the more distant ones, caused by the observer being carried from one side of the Earth's orbit to the other.^[1] It is the direct geometrical consequence of the Earth having a position that changes, and its size falls off as one over the distance to the star.

It is also the observation that heliocentrism was required to produce, failed to produce for two hundred and ninety-five years, and was disbelieved for failing to produce. The objection was not a bad one. It was a good one, and it was answered in the only way a good objection can be, which is by measurement.

The effect

A star observed in June and again in December is seen from two points separated by the diameter of the Earth's orbit, and should appear to shift against the background by an angle set by that baseline and its own distance. The parallax quoted is the half-angle, corresponding to the radius:

The relation defines the unit. A star whose parallax is one second of arc lies at one parsec, about 3.26 light years. No star is that close: the nearest, Proxima Centauri, has a parallax of 0.7687″ and everything else has less.^[2]

This is what makes the measurement hard, and it is also what makes it decisive. The shift is not merely small; it is graded, being twice as large for a star half as far away. Any account which produces parallax by some other means must produce it scaled to distances it has no independent way of knowing.

The objection

Tycho Brahe was the finest observer of the age before the telescope, and he looked. He found nothing, and he reasoned from the nothing.^[3]

If the Earth ran a great circle about the Sun, the near stars must shift, and they did not. Either the Earth stood still, or the stars lay at distances so great that the shift fell below his instruments. That much is simply correct, and it is where most accounts stop. Tycho did not stop there.

He had measured the stars. They were not points to him: a first-magnitude star showed a disc of a minute or two of arc, and a body of that apparent size, placed at the distance the Copernican scheme required, would have to be physically enormous, comparable to the whole orbit of the Earth. The heavens would be full of objects each larger than the Sun's entire circuit. Tycho found this absurd, said so, and kept the Earth where it was.^[4]

Why the objection was sound

It should be said plainly that the argument was valid. The geometry was right, the measurements were carefully made with the best instruments then existing, and the conclusion followed from the premises.

The premises were wrong in one particular. The stars are not discs. What Tycho measured was not a star but the response of his own eye and, later for others, of the telescope: a bright point source spreads into a small disc with rings, and the disc's size is set by the aperture and the wavelength rather than by the object. As telescopes improved the discs shrank, which is not what a real diameter does.

The effect now carries the name of Airy, who described it in 1835, and its consequence for resolution is the Rayleigh criterion. Neither man was thinking about Tycho. The strongest empirical argument ever raised against the motion of the Earth rested on a measurement of the instrument, which is a thing that has happened in this encyclopedia before.^[5]

Two hundred and ninety-five years

Everyone looked. Bradley looked in 1725 and found aberration instead. The measurement wanted an angle of a third of a second of arc, held steady against a background over a year, and no instrument could hold it.

Then three arrived at once.

Struve is the hard case. He published before either of the others and was not believed, partly because he had few observations and partly because he later revised the figure to about twice its first value – away from the truth, the modern parallax of Vega being 0.130″. His first answer was very nearly correct and his second was not, and he is remembered for neither.^[8]

Henderson had the answer in his desk for six years. He had made the observations at the Cape, come home, and doubted them; by the time he was persuaded, Bessel had published. He is the only man to have measured the distance to a star first and been told about it afterwards.^[6]

Bessel chose 61 Cygni not because it was bright but because it moved: a large proper motion suggests a near star. He was right, and the reasoning is better than the result.

What it shows, and what it does not

Parallax tracks the Earth's position. Aberration tracks its velocity. The two run a quarter of a year apart, and the difference between them is not a matter of interpretation but of calculus.

The distinction matters because the two are often run together in argument. An account which supposes the stars themselves to move must give them one set of motions scaled to their distances, to produce parallax, and a second set identical at every distance, to produce aberration, and must then keep the two three months out of step for ever.^[7]

Since

The quantity that could not be measured at all is now measured industrially. Hipparcos catalogued a hundred thousand parallaxes in the 1990s; Gaia has published them for something over a billion stars, at a precision of a few tens of microarcseconds, which is a shift of about the width of a human hair seen from a thousand kilometres.

Tycho's objection has therefore been answered about a billion times. It remains the best argument the other side ever had.

Parallax is still only the first rung. It reaches nearby stars and then runs out, and everything beyond it is measured against something calibrated on it – beginning with the period–luminosity relation that Henrietta Swan Leavitt found in 1912, which is how the distance to another galaxy was first obtained. Edwin Hubble got that distance in 1924, and every error in the rungs below him arrived in his answer with it.

See also

• Stellar aberration – the other annual displacement, a quarter-year out of phase
• Edwin Hubble – at the far end of the same ladder, inheriting every error in it
• Friedrich Bessel – who measured it first, having spent his life on the errors of observers
• Tycho Brahe – whose objection it answered
• Heliocentrism – the arrangement it was demanded of
• James Bradley – who went looking for it and found something else
• George Biddell Airy – whose disc explains the discs Tycho saw
• The CMB dipole – the third annual effect, and the newest
• Vulcan – a body looked for just as patiently, by the same methods, that was not there
• Peter van de Kamp – what the same technique does when the instrument is not accounted for

References

1. ^ Bessel's result for 61 Cygni was communicated in a letter to Herschel dated 23 October 1838 and set out in full in a paper dated at Altona on 13 December.
2. ^ The parsec is defined from the relation and not the other way about: it is the distance at which one astronomical unit subtends one second of arc. That no star is nearer than 1.3 parsecs is a fact about the sky and not about the unit.
3. ^ Tycho's instruments were accurate to something under a minute of arc, which is a remarkable figure for the naked eye and about two hundred times too coarse for the task.
4. ^ The argument is set out in his correspondence and was taken up after him by Riccioli, who repeated it with a telescope, measured the discs at about a sixth of Tycho's figure, and reached the same conclusion by the same route.
5. ^ See the Bedford Level experiment and celestial refraction for the pattern in its more usual direction, where a real property of the air is measured and reported as a property of the Earth.
6. ^ Henderson's caution was not foolish. His value was large, and a large parallax was exactly what an over-eager observer would report; he waited for confirmation from a second instrument, and the delay cost him the discovery and nothing else.
7. ^ See kinematic and dynamic equivalence, where the three annual effects are set against what each account must supply to produce them.
8. ^ Modern values, for the three: 61 Cygni 0.286″, α Centauri 0.75″, Vega 0.130″. Bessel's stands up best by a wide margin, which is the reason the priority is not merely a matter of dates.

External links

• Bessel, "On the parallax of 61 Cygni" – Monthly Notices of the Royal Astronomical Society 4 (1838)
• Tycho Brahe, Astronomiae Instauratae Mechanica (1598) – the mural quadrant plate, Wikimedia Commons
• Gaia – European Space Agency
• Stellar parallax – general reference