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9.7 Option - Astrophysics: 2. Parallax

Extract from Physics Stage 6 Syllabus (Amended October 2002). © Board of Studies, NSW.
[Edit: 30 June 09]

define the terms parallax, parsec, light-year

• Parallax is defined as the change in the apparent position of a nearby object with respect to distant objects as the observer’s position changes. The apparent change in position is usually quoted as an angle.
  
  
• In astronomy, parallax (sometimes called “annual parallax”) means specifically the apparent angular displacement of a star against the background of much more distant stars, as seen by an observer on Earth moving in orbit around the Sun. The trigonometric parallax is defined as half the annual parallax. Parallax of a star is measured in arc seconds (1 arc second = 1” =1/3600 degree).
  
  
• The parsec (pc)is a unit of distance commonly used in astronomy. It is defined as the distance away a star would have to be in order for its annual parallax to be 1”. (Note that no known star is close enough to Earth to have a parallax this big.) By definition of annual parallax, the parsec is also the distance at which the radius of Earth’s orbit (a distance of one astronomical unit, AU) subtends an angle of 1”. 1 parsec = 3.1 x 10¹⁶ m.
  
  
• Another unit of distance used in Astronomy is the light year (ly). It is defined as the distance light (or other electromagnetic radiation) travels in one year. Its value is approximately 9.46 x 10¹² km. 1 parsec=3.26 ly.

Watch a movie that illustrates parallax of a star: Trigonometric Parallax Movie , Richard W. Pogge, The Ohio State University, Ohio, USA.

gather and process information to determine the relative limits to trigonometric parallax distance determinations using recent ground-based and space-based telescopes

• Information about ground-based and space-based methods of astrometry is difficult to find in one location. Generally, the information is located separately and little comparison is made. Try to improve the efficiency of your data gathering techniques by scanning possible sources for information explicitly on this topic.
  
  
• Process the information by comparing the limits stated in a range of different secondary sources to assess their reliability. Error in the calculation of distance from parallax increases with diminishing parallax, and different authors accept different limits on distances over which parallax is useful in calculating distance with accuracy.

A website which provides a useful starting point for recent space-based distance measurement is From Hipparchus to Hipparcos Catherine Turon.

One site on future space telescopes is: Science and Technology: GAIA European Space Agency.

Astronomy magazines such as Sky & Telescope maybe useful, such as the article on Hipparcos in the June 1999 edition.

explain how trigonometric parallax can be used to determine the distance to stars

• Trigonometric parallax is defined as half the angular shift of a star (in arc seconds) as observed from Earth over a period of six months.
  
  
• Because even stars close to Earth are still at vast distances, to see any appreciable change in their position with respect to the distant background, a considerable change in the observer’s position is required. The greatest baseline achievable for a ground-based telescope is the diameter of Earth’s orbit. By observing stars at a six monthly interval, the change in the observer’s position becomes twice the Earth-Sun distance or 2 Astronomical Units (AU).
  
  
• Parallax data is collected by photographing the same star field twice, from opposite points of Earth’s orbit, then measuring the annual shift of stars that are relatively close to Earth against the background of much more distant stars. From knowing the angular size of the photograph, the annual parallax of any close star can be calculated. Trigonometric parallax is calculated as half the annual parallax. The angular shift in the nearby star is still very small and is usually measured as a fraction of an arc second Even Proxima Centauri, the closest visible star to Earth other than the Sun, has a parallax of only ~0.8 arc second.
  
  
• The concept of trigonometric parallax is an application of geometry. The distance to a star is calculated geometrically from its parallax. From the diagram below, the distance d is related to the parallax angle p and the Earth-Sun distance D by the relation tan p = D/d. Since we know D, d can be determined if we can measure p.
  

  For example, a star that has a parallax of say 1 arc second will be at a distance of:

  

• A large number of relatively near stars, whose distances can be calculated accurately from parallax measurements, are used as reference stars for a range of techniques to estimate distances to much more distant stars, including some in neighbouring galaxies.

discuss the limitations of trigonometric parallax measurements

• Trigonometric parallax is half the apparent annual angular displacement of a star against the background of much more distant stars, as seen by an observer on Earth moving in orbit around the Sun. Trigonometric parallax is used to calculate distance to nearby stars.
  
  
• The usefulness of parallax in measuring distance to stars is limited because the parallax angle of even nearby stars is extremely small. The largest trigonometric parallax, for the nearest star other than the Sun, is less than one arc second (0.772”).
  
  
• In addition, the atmosphere “blurs” stellar images, making measurement of small angles very difficult. In practice, for Earth based telescopes, the limit is about 0.01”, with the result that parallax is good only for the relatively small number of stars up to about 100 pc away. For stars more than 100 pc away the parallax angle becomes too small to measure accurately from the ground.
  
  
• Other less direct methods must then be used for determining the distances to celestial objects which are more than 100 pc away. However parallax measurements of nearby stars are still vital as they indirectly underpin many of the less direct techniques for measuring distance to remote celestial objects.
  
  
• The limitations of trigonometric parallax measurements are lessened by placing a telescope out in space away from the scattering effects of the Earth’s atmosphere. The sharpness of the image is then determined predominantly by the quality of the optics and size of the telescope objective, rather than the atmosphere. Additionally, a space telescope can observe stars at shorter wavelengths, which has the effect of increasing the resolving power. The Hipparcos orbiting telescope is capable of resolving parallax angles as small as one milli-arc second (0.001”), allowing distances to about 1000 pc to be measured with reasonable accuracy.
  
  
• The parallax technique could be extended by increasing the baseline from which the measurements are made, so that annual parallax for any object would larger. For example, observations could be made from the larger orbit of Mars, or a satellite could be placed in orbit around the sun at some distance outside Earth’s orbit, such as Gaia planned for 2012.
  
  
• As technology is improved, smaller angles may be measured more accurately. The planned 2012 launch of Gaia into a non-eclipsing orbit 1.5 million km further out than the Earth’s orbit, should achieve measurements of parallax as small as 50 micro-arc second, up to 20 times more accurate than Hipparcos.

solve problems and analyse information to calculate the distance to a star given its trigonometric parallax using:

• The identified strategy for calculating distance to a star is to substitute the measured parallax angle into the equation where:
  
  d represents the distance measured in parsecs (pc), and
  p represents the parallax angle measured in arc seconds (“)
  
  
• You may need to analyse information by comparing two photographs taken 6 months apart showing a nearby star that has apparently moved against the background pattern of stars, to measure the parallax angle of the nearby star.

A useful website: Virtual Experiment 3 – Distance to Alpha Centauri A using Virtual Parallax Measurements , Brian von Konsky, Curtin University of Technology, Western Australia.

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