Trigonometric Parallax

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TRIGONOMETRIC PARALLAX Dr. Himadri Sekhar Das Department of Physics Assam university, Silchar Lecture Note: 1

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Lecture on Trigonometric parallax

Transcript of Trigonometric Parallax

Page 1: Trigonometric Parallax

TRIGONOMETRIC PARALLAX

Dr. Himadri Sekhar Das

Department of Physics

Assam university, Silchar

Lecture Note: 1

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What is parallax ?

Parallax is an apparent displacement or difference of orientation of an object viewed along two different lines of sight, and is measured by the angle or semi-angle of inclination between those two lines.

Source: Wikipedia

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What is Stellar Parallax?

Stellar parallax is the apparent change in the position of a star that is caused only by the motion of the Earth as it orbits the Sun.

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Thus the trigonometric parallax of a star may be defined as the angle p (in arc-second) subtended at the star A by the mean radius a of the Earth’s orbit around the Sun. This is also called heliocentric or annual parallax.

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A measurement of the parallax angle p is given by tan p = a/d d = a / tan p ≈ a / p [tan p ≈ p for small angle] d = 1 / p AU (1 AU = 1.495978 x 1011 m)

Here p is in radian. πc = 1800

1c = 1800/π = [(180 × 60 × 60)/3.1428]’’ = 206265’’

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Thus, d = 206265 / p’’ AU

Defining a new unit of distance, the parsec (pc) (parallax-second), as

1 pc = 206265 AU = 3.085678 x 1016 m which leads to

d = 1 / p ’’ pc

By definition, when the parallax angle p = 1 ’’, the distance to the star is 1 pc.

Thus, 1 parsec is the distance from which the radius of Earth’s orbit (= 1 AU) subtends an angle of 1 ’’.

Also, 1 pc = 3.261633 lyr (light years)

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Source: http://www.astronomy.ohio-state.edu/~pogge/Ast162/Movies/parallax.html

The top half of each frame shows the appearance of the sky as seen from the Earth (ignoring the Sun), and the bottom half shows a fixed view looking down from above onto the plane of the Earth's orbit around the Sun (the ecliptic).

A red star is shown located some distance to the right .

When viewed from the moving Earth (top panel), the red star appears to move first west (towards the right) then east (towards the left) with respect to the distant background stars which are so far away that their parallax motions are too small to be seen at this scale.

This movie demonstrates Trigonometric Parallax.

In the second half, we move the star 2x farther away and run through another year. Now the annual the trigonometric parallax motions are 2x smaller because the distance to the star is 2x greater. This fact, that the trigonometric parallax of a star is inversely proportional to its distance from the Sun gives us a direct measurement of the star's distance.

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Our nearest star alpha Centauri has a parallax of p=0.742-arcsec:

•The star alpha Centauri has a parallax of p=0.742-arcsec:

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Limitations:

If the stars are too far away, the parallax can be too small to measure accurately. In general, the greater the distance, the smaller the parallax, and so the less precise the distance measurement will be.

The smallest parallax measurable from the ground is about 0.01-arcsec. This means that from the ground, the method of Trigonometric Parallaxes has the following limitations:

good out to 100 pc

only get 10% distances out to a few parsecs. only a few hundred stars are this close

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Hipparcos:

The Hipparcos satellite (launched by the European Space Agency in 1989) measured precision parallaxes to an accuracy of about 0.001-arcsec. Hipparcos measured parallaxes for about 100,000 stars Got 10% accuracy distances out to about 100 pc Good distances for bright stars out to 1000 pc. Hipparcos represented a great leap in our knowledge of the distances (and motions) of nearby stars. The catalog was just released in late 1997, and is already having an impact on many areas of astronomy that rely in accurate distances.

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Thank you!!