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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 Galaxy?
  5. The Astronomical Magnitude Scale
  6. Sidereal Time, Civil Time and Solar Time
  7. Equinoxes and Solstices
  8. Parallax, Distance and Parsecs
  9. Luminosity and Flux of Stars
  10. Kepler's Laws of Planetary Motion
  11. What Are Lagrange Points?
  12. Glossary of Astronomy & Photographic Terms
  13. Astronomical Constants - Some Useful Constants for Astronomy

When we talk about distance in astronomy, we are usually talking very, very large numbers. Far too many to describe them in terms of miles or kilometres. When we realised just how big space was, we needed some new units. In modern astronomy, we often use the Astronomical Unit or the Lightyear.

Space is big. Really big. You just won't believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it's a long way down the road to the chemist, but that's just peanuts to space.Douglas Adams, The Hitchhiker's Guide to the Galaxy

The Astronomical Unit is the average distance from Earth to the Sun. We say average because the Earths orbit is elliptical, varying from a maximum (aphelion) to a minimum (perihelion) and back again once a year.

Elliptical Orbits - Aphelion, Perihelion, Perigee, Apogee

Elliptical Orbits - Aphelion, Perihelion, Perigee, Apogee

Due to this variation, the Astronomical Unit is now defined as exactly 149,597,870,700 metres (about 150 million kilometres, or 93 million miles). You can see why we don't express this as kilometres! For objects in the solar system, their orbits are typically given in terms of the Astronomical Unit (AU). Earth being 1AU, Venus at 0.72AU, Jupiter at 5.2AU. These values are much easier to work with. If you want to convert AU to KM or Miles, simply multiply the Earths orbital radius by the AU value.

For distances outside of the solar system, the light year distance is often used. A light year is defined as the distance light travels in a year. Since the speed of light is constant, the distance is also constant. Light travels at around 300,000 kilometres per second, so these numbers can get very big, very fast. In one year, light travels about 10 trillion km. One light-year is equal to 9,500,000,000,000 kilometers, or 63,241 AU.

M31 - The Andromeda Galaxy

M31 - The Andromeda Galaxy

For large numbers like this, we often use scientific notion. We write a light year as 9.5x1012 km. This is called scientific notation. We simply move the decimal place to the left until we get to the smallest significant figure and count the number of times we moved the decimal place. Even using light years as a measure of distance we still deal with very large numbers. The Andromeda galaxy is the nearest galaxy to the Milky Way, and at a distance of 2.5 million light years, it's quite a bit further than walking down the road to the chemist. The furthest observed galaxy is EGS8p7 which is more than 13.2 billion light years away. Because we know how far away it is, and that the speed of light is constant, we know that the light from that galaxy has travelled for 13.2 billion years to arrive here. We are effectively looking back in time, to a point only a few hundred million years after the big bang. How cool is that?

The Hubble Ultra Deep Field is an image of a small region of space in the constellation Fornax which includes 10,000 galaxies of various ages, sizes, shapes, and colours.

The Hubble Ultra Deep Field is an image of a small region of space in the constellation Fornax which includes 10,000 galaxies of various ages, sizes, shapes, and colours.

NASA

How Are Astronomical Distances Measured?

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.

Pythagoras Triangle Trig
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:

Pythagoras Triangle Trig
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.

Distance Calculation using Parallax
Equation 10 - Distance Calculation using Parallax
Distance Parallax Calculation
Distance Parallax Calculation
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.

Tutorial Series

This post is part of the series Introduction to Astronomy. Use the links below to advance to the next tutorial in the couse, or go back and see the previous in the tutorial series.

One thought on “Parallax, Distance and Parsecs
  • 24th September 2019 at 5:00 pm

    the calculation for the work example is confusing. the distance for parallax of 0.74 is in parsec and the calculation should be written as 1/0.74 instead of AU/0.74. Nevertheless, the answer is correct.

    Reply

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