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Friday, 09 May 2014 00:00

Stellar Parallax

By  Gideon
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Consider two stars that appear close together in the night sky. Suppose that one star is relatively close to our solar system (what this means exactly we will come to shortly) while the second star is extremely distant.

As the Earth revolves around the sun, the apparent positions of the two stars will shift. The far distant star will not suffer any noticeable change in position, but the nearer star will be seen to move around the distant star in an ellipse.

Let \(\alpha\) be the maximum angular separation between the two stars expressed in radians.

If \(d_{o}\) is one A. U., i.e. the distance from the Earth to the Sun, and \(d_{\star},\) is the distance from the Sun to the star, then \( d_{\star} \alpha \approx d_{0}.\)

Because in practice the distances to stars are so great, and the angles so small, this approximation is an excellent one and we can write:

\[ d_{\star} = \frac{ d_{0}}{\alpha} = \frac{1 (A.U.)}{\alpha} \]

In this way, if we can measure the angle of parallax, \(\alpha\) we can find the distance to a star.

Parallax1d-600

In the above figure, the distance from the Earth to the Sun, one Astronomical Unit, or A.U. is equal to \(d_{\star} \sin{\alpha}.\)

But because \(\alpha\) is so small, and measured in radians, we can use the approximation:

\[ \sin{\alpha} \approx \alpha.\]

Using this approximation we find again:

\[ d_{\star} = \frac{ d_{0}}{\alpha} =  \frac{ 1 A.U. }{\alpha}\]

The yellow star is the nearer star, and the blue star is considered to be so far away that its position remains fixed throughout the year.

Paralallax2d-600

The first figure shows the situation from the position of an observer who is at rest with respect to the sun.

The second figure shows the situation from the perspective of an observer from a definite position on the surface of the Earth.

The two views of the star are separated by six months of time.

In practice, \(\alpha\) is tiny for all stars, and cannot be measured at all with the naked eye, and is even extremely difficult to measure with a telescope.


The star that is closest to our solar system, Proxima Centauri, has a parallax of almost three-quarters of one second of arc, all other stars have a smaller angle of parallax.

Of al the stars that can be seen from Earth, only twenty-three have an angle of parallax greater than \(.24\) seconds of arc.

 

image 1514e-Proxima-Centauri

 

A distance that corresponds to one second of arc is called a parsec. A parsec is equal to about 206 thousand astronomical units, or about 3.26 lightyears, or about \(2 \times 10^{13}\) miles, or about \(3.1 \times 10^{18}\) cm.

Proxima Centauri is about 4.243 light years, or about 1.3 parsecs, from our solar system. In other words, our sun's nearest neighbor is more than a parsec away so that all stars have a parallax of less than one second of arc.

The absence of any evidence for parallax may have contributed to Copernicus' reluctance to publish his theory; the best physical evidence seemed to be against it.

Parallax was never discovered with naked eye astronomy.

Copernicus published De Revolutionibus in 1543. Sixty-five years later, in 1608, Hans Lipperhey invented the telescope.

And yet it took another two hundred and thirty years before parallax was finally verified by Bessel in 1838.

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