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Parallax, Distance and Parsecs

Using parallax to calculate distance to nearby stars

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Parallax, Distance and Parsecs

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In Astronomy, parallax is a method used to determine the distance to the closest stars. This technique for measuring astronomical distances is very important because it is a geometric method and independent of the object being observed.
 
Introduction to Astronomy Series
  1. Introduction to Astronomy
  2. The Celestial Sphere - Right Ascension and Declination
  3. What is Angular Size?
  4. What is the Milky Way?
  5. The Magnitude Scale
  6. Sidereal Time, Civil Time and Solar Time
  7. Equinoxes and Solstices
  8. Parallax, Distance and Parsecs
  9. Flux
  10. Luminosity of Stars
  11. Apparent Magnitude, Absolute Magnitude and Distance
  12. Variable Stars
  13. Spectroscopy and Spectrometry
  14. Redshift and Blueshift
  15. Spectral Classification of Stars
  16. Hertzsprung-Russell Diagram
  17. Kepler's Laws of Planetary Motion
  18. The Lagrange Points
  19. What is an Exoplanet?
  20. Glossary of Astronomy & Photographic Terms

Astronomers use an effect called parallax to measure distances to nearby stars. The principal of Parallax can easily be demonstrated by holding your finger up at arm's length. Close one eye, then the other and notice how your finger appears to move in relation to the background. This occurs because each eye sees a slightly different view because they are separated by a few inches.

If you measure the distance between your eyes and the distance your finger appears to move, then you can calculate the length of your arm.

This same principle can be used on a larger scale to calculate the distance to an object in the sky, only we use different points on the Earth's orbit instead of looking through alternate eyes. This is a fantastic way of measuring distance as it relies solely on geometry. Parallax calculations are based on measuring two angles and the included side of a triangle formed by the star, Earth on one side of its orbit and Earth six months later on the other side of its orbit.

Calculating parallax requires that the objects Right Ascension and Declination be recorded accurately so that we know the object's precise location on the celestial sphere.

Using parallax to calculate distance to a star
Using parallax to calculate distance to a star

We take a measurement of the position of an object relative to the other background stars during the winter months, and then again 6 months later, in the summer, when the Earth has moved 180° around its orbit around the Sun to give maximum separation distance.

In this diagram (not to scale) during the summer the position of the object appears to be at point A in the sky. Six months later, during the winter, it appears to be at point B. The imaginary line between the two opposite positions in the Earth's orbit is called the baseline. The half baseline is the Earth's orbit radius.

We know the radius of the Earth's orbit radius (r), and we can calculate the angle, θ from the observed apparent motion, measured in radians. Finally, we just need a little trigonometry to calculate the distance, d.

tan theta=R/d
Equation 8 - Pythagoras Triangle Trig

Since the value of theta measured is going to be very small, we can approximate tan θ = θ. Rearranging to solve for d gives us:

d=R/theta
Equation 9 - Pythagoras Triangle Trig

This equation forms the basis for a new unit of length called the parsec (PC). A parsec is defined as the distance at which 1 AU subtends 1 arcsecond. So an object located at 1pc would, by definition, have a parallax of 1 arcsecond.

Worked Example

The parallax measured for α Centauri is 0.74 arcseconds Calculate the distance in light years to α Centauri.

d=1/p
Equation 10 - Distance Calculation using Parallax
d={1.5*10^11}/0.74
d=1.351pc
Equation 11 - Distance Parallax Calculation

1 AU is equal to 1.4960x1011 meters and 1 parsec is equal to 3.26 light years, which makes α Centauri 4.405 light years away.

Last updated on: Thursday 20th July 2017

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