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9.7 Astrophysics
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NOTE: The notes and worksheets for this topic are divided into four separate pages in order to keep download times within acceptable limits.  To access the other pages on this topic click on the links at the left marked 9.7 Astrophysics Page 2, Page 3 and Page 4.







Astrophysics is one of the most exciting fields of scientific research.  It draws on knowledge, understanding and skills from almost every other branch of Physics.  The wonders of the universe are revealed through technological advances based on tested principles of Physics.  Our understanding of the cosmos draws upon models, theories and laws in our endeavour to seek explanations for the myriad of observations made by various instruments at many different wavelengths.  Techniques such as imaging, photometry, astrometry and spectroscopy allow us to determine many of the properties and characteristics of celestial objects.  Continual technical advancement has resulted in a range of devices extending from optical and radio telescopes on Earth to orbiting telescopes, such as Hipparcos, Chandra and the Hubble Space Telescope (HST).

Explanations for events in our spectacular universe based on our understandings of the electromagnetic spectrum, allow for insights into the relationships between star formation and evolution (supernovae), and extreme events such as the high gravity environments of a neutron star or black hole.

This module increases students’ understanding of the nature and practice of Physics and the implications of Physics for society and the environment.

Note: Some internet browsers (eg Firefox) do not accurately display text symbols such as Greek letters used to represent quantities in Physics.  For example, capital delta is displayed as D and lower case phi as f in Firefox.  This is just something to be aware of in case you do come across such issues.  The square root sign is another one not displayed properly by some browsers.  Any symbols used in equations produced by equation editors will of course display properly.

NOTE: Numbers appearing in parentheses at the end of sentences or paragraphs refer to the references provided in the Bibliography at the end of these notes.







From time immemorial human beings have looked to the heavens in awe and tried to explain what they have seen.  Observational astronomy or at least some rudimentary form of it, has played a part in many of our ancient cultures – Mesopotamia, Egypt, India, China, the Celts, the Mayans and the Aztecs to name a few.  Modern astronomy is generally considered to have had its roots in the ancient Greek tradition of natural philosophy.  It was the Greeks, through Pythagoras (550 BC) and others, who developed the mathematical approach to the study of the universe that has continued through to the present day.  Socrates, Plato and Aristotle, the many great scholars of the Alexandrian period (300BC-200AD), the many great Islamic scholars of the 8th to 13th Centuries and people such as Nicolaus Copernicus (1474-1543AD), Tycho Brahe (1546-1601AD), Johannes Kepler (1571-1630AD), Galileo Galilei (1564-1642AD), Isaac Newton (1642-1717) and Albert Einstein (1879-1955) have been some of the huge number of people who have made great contributions to science and astronomy and thereby to our knowledge and understanding of the universe.  (1 & 2)





In 1609, Galileo Galilei, an Italian natural philosopher, changed the world of observational astronomy forever.  After hearing of the basic principle of the telescope, Galileo built a telescope of his own that had a magnification of about 10.  The potential of this instrument for military and commercial purposes so impressed the Venetian Senate that they funded the building of another larger telescope.  This time Galileo constructed a telescope with an aperture of about 5 centimetres and a magnification of about 20.  Galileo then used this telescope to make a series of astronomical observations that stunned the scientific world. (2)

By his own account, Galileo first observed the Moon on November 30 1609.  He observed the large dark patches that can be observed with the naked eye.  He also observed several smaller dark patches that could not be seen with the naked eye.  Over several weeks of observations, he noted that in these smaller spots, the width of the dark lines defining the spots varied with the angle of solar illumination.  He watched the dark lines change and he saw lighter spots in the unilluminated part of the Moon that gradually merged with the illuminated part as this part grew. The conclusion he drew was that the changing dark lines were shadows and that the lunar surface has mountains and valleys.  Galileo also observed that the moon was not perfectly spherical in shape.  (4)

Also check out the following link:


Galileo’s use of the telescope to identify features of the moon was ground-braking science in several ways.  Firstly, although Galileo was not the first person to study the heavens with a telescope, he was the first to do so in a systematic way and to record and interpret his observations and publish them for others to read.  Secondly, Galileo demonstrated the usefulness of the telescope as an astronomical instrument that enhanced observation beyond what was possible with the unaided eye.  In both of these ways he set an excellent example for other scientists to follow and earned the title of “the father of modern observational astronomy” (1).

Thirdly, Galileo’s assertion that there were features on the moon was a good example of the power of deductive reasoning from careful observation.  Galileo could not see the mountains and valleys – his telescope was not that good.  He deduced their presence from careful observation of the borders between the light and dark patches on the surface, eventually deciding that the dark lines were shadows and therefore that there had to be mountains and valleys in order for the shadows to be cast the way they were. 

Fourthly, Galileo’s telescopic observations provided clear evidence that the Aristotelian view of the universe was inaccurate.  Aristotelian doctrine stipulated that celestial bodies were perfectly smooth and spherical.  Clearly, this was not true for the moon and so the Aristotelian doctrine needed some amendment.  Further, since the moon had features, clearly the Earth was not unique in this respect and perhaps other heavenly bodies would also be found to have features.  Further still, if heavenly bodies could have features and therefore be imperfect, perhaps the Earth is a heavenly body too.

Galileo made many other observations with the aid of telescopes.  References 1, 2 & 3 give good accounts of these.






Telescopes are devices that help astronomers overcome the limitations of the human eye.  Since Galileo’s day, telescopes have become essential instruments in the study of astronomy.  Large telescopes can make images that are far brighter, sharper and more detailed than the images made by our eyes.  Telescopes have also been developed that can observe the universe at wavelengths outside the visible range.  Our present comments, however, will be restricted to optical telescopes, the most commonly used of all telescopes.

There are two basic types of optical telescope – the refracting and the reflecting.  Although the current Syllabus does not require you to know specific details about these telescopes, it is essential for any student of astronomy to have at least a rudimentary understanding of these very important instruments.  Therefore, we shall examine each very briefly here.





A convex lens is one that is fatter in the middle than at the ends.  When light rays pass through the lens, refraction causes the rays to converge to a point called the focus.  If the light rays entering the lens are all parallel, the focus occurs at a special point called the focal point of the lens.  The distance from the lens to the focal point is called the focal length of the lens.  Since the light coming from astronomical objects is coming from so far away, the light rays are essentially parallel.  So when light from an astronomical object is allowed to enter a convex lens, it is brought to a focus at the focal point.  Objects with a small angular size (eg a star) produce an image that is just a single bright dot.  Objects of large angular size (eg the moon) produce an extended image that lies in the focal plane of the lens.



A refracting astronomical telescope consists of a large diameter, long focal length, convex objective lens at the front of the telescope and a small, short focal length, convex eyepiece lens at the rear of the telescope.   The objective lens forms the image and the eyepiece lens magnifies this image for the observer.  See the diagram below.


Refractors are considered ideal for observing the fine, low-contrast details of the moon and planets.  They are really not appropriate for observing stars due to their susceptibility to chromatic aberration – a lens refracts different wavelengths by different amounts and so each colour ends up with a different focal point.  The result is that stars appear surrounded by fuzzy rainbow-coloured halos.  This aberration can be corrected but it is expensive to do so.  The main use for refractors today is by amateur astronomers.  For numerous reasons, professional astronomers today prefer reflecting telescopes.  (1 & 3)




A concave mirror has a shape as shown in the following diagram.  It causes parallel light rays to converge to a focus.  The distance between the reflecting surface of the mirror and the focal plane is the focal length of the mirror.



A reflecting astronomical telescope uses either a parabolic or spherical concave mirror as the objective or primary mirror.  This produces the image of the object being viewed.  How the observer then views this image depends on the exact design of the reflecting telescope. 

There are many different reflector designs.  The one shown below is called a Newtonian Reflector after Isaac Newton who designed it.  This is in common use by amateur astronomers.  A small flat mirror is placed at a 45o angle in front of the focal point.  This secondary mirror deflects the light rays into an eyepiece lens at the side of the telescope, where the image of the object can be viewed. Other popular designs include: the prime-focus, where an observer or detector is placed at the focal point inside the barrel of the telescope; the Cassegrain focus, where a hole is made in the centre of the primary mirror and a convex secondary mirror is placed in front of the original focal point to reflect light back through the hole; and the coude focus, in which a series of mirrors reflects the light rays away from the telescope to a remote focal point in a coude room (special laboratory) located below the telescope.  Both amateur and professional astronomers use the Cassegrain design while almost exclusively it is professional astronomers who use the prime-focus and coude designs. (1 & 3)


There are many good websites that give details of the different kinds of telescope designs that are available.  One I can suggest is:


See my Useful Links page for other telescope links.





Many people believe that the main purpose of a telescope is to magnify the object being viewed.  In fact, the two main purposes of any kind of telescope are to gather light from faint sources and to resolve those sources clearly.  Let us now examine the meaning of the two terms “sensitivity” and “resolution”.

The sensitivity of a detecting system is a measure of the weakest signal discernable by the system (5).  So, for an optical telescope, the sensitivity is defined as the light-gathering power of the telescope.  The light-gathering power is dependant upon the light-collecting area of the lens or mirror used as the objective.  Mathematically, then, the sensitivity of an optical telescope is directly proportional to the square of the lens or mirror diameter. (3)

For example, a human eye that is fully adapted to the dark has a pupil diameter of about 5 mm.  By comparison, each of the two Keck telescopes on Mauna Kea, Hawaii, uses a concave mirror of 10 m diameter to collect light.  Thus, the ratio of the sensitivity of the Keck telescopes to that of the human eye is (10 000 mm)2/ (5 mm)2.  That is, the Keck telescopes are 4 million times more sensitive than the human eye.

Clearly, the bigger the telescope, the better the sensitivity (all other things being equal).  Astronomers often refer to the light bucket of a telescope.  The bigger the bucket, the more light it can hold and the more sensitive the telescope.

The angular (or optical) resolution of a telescope gauges how well fine details can be seen.  By definition the angular resolution of a telescope is the minimum angular separation between two equal point sources such that they can be just barely distinguished as separate sources.  In simpler language, the angular resolution of a telescope is an angle that indicates the sharpness of the telescope’s image.  The smaller the angle, the finer the details that can be seen and the sharper the image. (3)

Note that the term “resolving power” can be used interchangeably with the term “resolution”

It is worth considering for a moment why there is a limit to the angular resolution we can achieve, even in perfect viewing conditions.  When a beam of light passes through a circular aperture such as a telescope it tends to spread out, blurring the image.  This phenomenon is called diffraction, as you should remember from the Preliminary Course.  The diffraction pattern of a point source of light as seen through a circular aperture is as shown below.




The central bright spot is known as the Airy disk.  The maxima (bright bands) become fainter very quickly as you move outward from the centre.  If we view two point sources of light (two stars) whose angular separation is greater than the angular resolution of the telescope, the sources can easily be distinguished. (1)




If we view two point sources of light whose angular separation is equal to the angular resolution of the telescope, the two sources can only just be distinguished as separate.  If the sources were any closer together, the telescope image would show them as a single source.  By definition, two images are said to be unresolved when the central maximum of one pattern falls inside the location of the first minimum of the other. (1)



Mathematically, the angular resolution of a telescope can be expressed as:



where qmin = the diffraction-limited angular resolution in arcseconds, l = wavelength of light in metres and D = diameter of telescope objective in metres.  Remember that 1o = 60= 60 arcminutes and 1= 60 = 60 arcseconds. (3)

The angular resolution can also be expressed in terms of radian measure as:


where the only difference is that qmin is expressed in radians. (1)

Clearly, the larger the diameter, D, of the objective, the more sensitive the telescope (ie the larger D2) & the better the resolution (ie the smaller qmin).  Resolution is also better when observing shorter rather than longer wavelengths.


Exercise: Calculate the optical resolution in arcseconds of the 3.9 m Anglo-Australian Telescope at Siding Springs when observing starlight of wavelength 540 nm.  (Answer: 0.035)




There are many problems associated with ground-based astronomy.  The main problems concern atmospheric distortion and the resolution and absorption of radiation.



It may appear from the equations for angular resolution given above that the resolution can be improved without limit by simply making bigger and bigger telescopes.  Unfortunately, this is not true.  In practice, the turbulent nature of the atmosphere places a limit on an optical telescope’s resolving power.  Local changes in atmospheric temperature and density over small distances create regions where light is refracted in nearly random directions, causing the image of a point source to become blurred.  The image appears to undergo rapid changes in brightness and position, a phenomenon known as scintillation.   Since almost all stars appear as point sources, even through the largest telescopes, atmospheric turbulence produces the well-known “twinkling” of stars. (1 & 5)

A measure of the limit that atmospheric turbulence places on the resolution of a telescope is called the “seeing disk”.  This disk is the angular diameter of the star’s image broadened by turbulence.  Astronomers refer to the seeing conditions at a particular observatory on a particular night, since seeing conditions depend on the existing atmospheric conditions.  Some of the very best seeing conditions in the world are found at the observatories on top of Mauna Kea in Hawaii, where the seeing disk is often as small as 0.5 arcseconds. (3)  Kitt Peak National Observatory near Tucson, Arizona, USA and Cerro-Tololo Inter-American Observatory in Chile are also well known for their excellent seeing conditions (1).  Many optical telescopes have been built at both locations (1).  In general, most earth-based optical telescopes are limited by seeing to a resolution of no better than 1, regardless of their theoretical “diffraction limited” resolution (1).

As an aside, it is interesting to note that since the angular size of most planets is actually larger than the scale of atmospheric turbulence, distortions tend to be averaged out over the size of the image and the twinkling effect is removed (1).  So, stars twinkle and most planets do not.




Electromagnetic (EM) radiation of all kinds reaches Earth’s upper atmosphere from the universe beyond.  Astronomers are keenly interested in examining all this EM radiation, since every bit of it contains information that may help answer some of our questions about the universe.  Clearly, then, we have a problem.  As you should remember from “The World Communicates” topic in the Preliminary Course, the ability of EM radiation to penetrate Earth’s atmosphere is related to the wavelength of the radiation.  EM radiation of different wavelengths is absorbed by different amounts in the atmosphere.

Oxygen and nitrogen completely absorb all radiation with wavelengths shorter than 290 nm.  Ozone (O3) for instance absorbs most of the ultraviolet.  EM radiation beyond the near-ultraviolet (300-400 nm) never makes it to the ground.  Water vapour and carbon dioxide effectively block out all radiation with wavelengths from about 10 mm to 1 cm.  This makes ground observation of infrared radiation impossible with the exception of the near-infrared wavelengths from 1 to 10 mm.

Thus, of all the EM radiation that falls on earth from space, only the visible and radio (& microwave) bands, the near-infrared bands and the near-ultraviolet bands make it all the way to the ground without much absorption taking place on the way down.  For all intents and purposes the far-infrared, far-UV, X-ray and gamma-ray wavebands of the EM spectrum are effectively filtered out by absorption in the atmosphere well before they reach the ground.

These wavebands then, are only detectable from space.  To this end a number of telescopes have been placed in Earth orbit.  The Infrared Astronomical Telescope (IRAS) launched in 1983 and the Infrared Space Observatory (ISO) launched in 1995 have both made valuable discoveries.  IRAS for example found dust bands in our Solar System and around nearby stars and discovered distant galaxies, none of which was observable by ground-based optical telescopes.  The Space Infrared Telescope Facility (SIRTF) is due for launch in August 2003 (SIRTF Website).  During its 2.5-year mission, SIRTF will obtain images and spectra by detecting the infrared energy radiated by objects in space between wavelengths of 3 and 180 mm.  Most of this infrared radiation is blocked by the Earth's atmosphere and cannot be observed from the ground.  SIRTF will allow us to peer into regions of star formation, the centres of galaxies, and into newly forming planetary systems.  Also, many molecules in space, including organic molecules, have their unique signatures in the infrared.  Telescopes that observe in the far-ultraviolet, X-ray and gamma ray bands are also currently in operation. (3)



Visible light is scattered in two different ways as it passes through the atmosphere.  In Mie scattering, suspended dust particles with sizes similar to the wavelength of the light scatter light by reflection.  In molecular or Rayleigh scattering, molecules of air (oxygen or nitrogen) with sizes much smaller than the wavelength of the light scatter light by absorption and re-radiation (5).

Both of these processes effectively decrease the intensity of the light coming from astronomical sources as it passes through the atmosphere.  The second process is also responsible for the blue colour of the sky during the day, which effectively blocks our view of stars, planets and other astronomical objects in daytime (3).

It is interesting to note that the saying “once in a blue moon” has an astronomical origin.  On rare occasions the moon does indeed appear blue.  This is due to Mie scattering in the upper atmosphere by dust particles with just the right size to scatter red light preferentially over blue, leaving the moon looking decidedly blue.  (Mie scattering is a complex function of wavelength and can make an object appear either redder or bluer depending on the size of the scattering particle.)  Blue moons were seen in 1883 after the eruption of Krakatoa and in 1950 after severe forest fires in Canada (5).




Radio telescopes have angular resolution problems.  These are not caused by atmospheric turbulence, as is the case for optical telescopes.  The problem for radio telescopes is that angular resolution is directly proportional to the wavelength being observed.  The longer the wavelength, the larger the angular resolution and the worse the image (3).

There is a practical limitation on the size of the objective lens for a ground-based refracting telescope.  Since light must pass through the objective lens, it can only be supported at its edges.  So, when the size and weight of the lens is increased, deformation of its shape occurs due to gravity (1).  This affects the resolution of the image.

Be aware that there are other factors that can affect the resolution of lens and mirror systems but all of these can affect space-telescopes just as much as ground-based telescopes.  Chromatic aberration in lenses was mentioned earlier.  Spherical aberration, coma, astigmatism, curvature of field and distortion of field can occur with both lenses and mirrors (1).  Lenses can suffer from defects in the material from which they are made and from deviations in the desired shape of their surfaces (1).  All of these effects can and are compensated for when constructing telescopes.





Clever techniques have been developed to improve the resolution and/or sensitivity of ground-based observational systems.  The techniques examined here are: active optics, adaptive optics and interferometry.




Deformations in the reflecting surface of a mirror reduce the quality of the image formed.  For this reason, before the 1980’s, large diameter primary mirrors in reflecting telescopes had to be made very rigid and very thick, usually about one-sixth the diameter of the mirror.  This prevented any change in shape of the mirror due to changes in the force of gravity acting on the mirror as it moved to different positions around the sky.  Unfortunately, the resulting mirror was very heavy and took a long time to reach thermal equilibrium each night, reducing the resolution achievable and producing extraneous seeing effects. (6)

Since that time, however, primary mirrors have been made much thinner.  The twin 8-metre diameter Gemini telescopes in Hawaii and Chile for example, have primary mirrors that are only 20 cm thick (6).  Although these mirrors do change shape as the telescope changes its orientation and experiences changes of temperature, a system of active optics ensures that the image is of very high resolution.  An “Active Optics” system is one that compensates for the deforming effects of gravity on a telescope’s mirrors, maintaining their surface accuracy and alignment (5).

As the telescope tracks across the sky, reference stars within the field of view are observed and analyzed by an image analysis system to determine any distortions in the observed light wavefronts due to deformations in the primary mirror.  A computer then calculates the necessary corrections in the shape of the mirror to eliminate these distortions.  If these corrections are determined to be statistically reliable by the telescope operator, they are sent to an array of electromechanical actuators on the back of the primary mirror, which push or pull on a section of the primary to change its shape in the required way.  Active optics systems correct the primary mirror shape about once per minute.  (5, 6 & 7)

Note that the rapid image distortions due to atmospheric turbulence are ignored by the image analysis system used in active optics systems.  Active optics systems are only employed to compensate for the various deformation effects in the telescope structure and the mirrors, and for effects due to inhomogeneities of the air temperature in the dome itself. (7)

The first telescope to use active optics was the 3.58 m New Technology Telescope (NTT) in Chile, which commenced operation in March 1988 (1).  The instrument employs 75 adjustable pressure pads on the back of the primary to modify automatically the shape of the mirror when it is in different positions (1).





Images of astronomical objects are blurred and degraded by atmospheric turbulence. "Adaptive optics" is a technology for sharpening turbulence-degraded images, by using fast-moving, flexible mirrors to "unscramble" the optical distortion and thereby improve the angular resolution.

The key elements of an adaptive optics system are a wave-front sensor, an adaptive mirror, and a control computer, as shown in the diagram below.  Let us talk through this diagram to explain how the system works.

An optical wavefront passing through air is distorted by turbulence.  The light is collected by the telescope, and fed to the adaptive optics system.

The wave-front sensor measures the distortion, the control computer calculates the mirror shape needed to remove the distortion, and this correcting shape is applied to the adaptive mirror by a series of fast-acting actuators, to reconstruct the undistorted image.  This procedure is repeated about 1000 times per second, to track the rapidly varying turbulence.  It is this speed that is the major difference between adaptive and active optical systems.

In correcting the distorted image, the system uses as a reference either a real guide star in the field of view or an artificial guide star created by laser light backscattered off air molecules in the field of view.  Images made with adaptive optics are almost as sharp as if the telescope were in the vacuum of space, where there is no atmospheric distortion and the only limit on angular resolution is diffraction.  (1, 3 & 5)

The 3.6 m Canada-France-Hawaii Telescope (CFHT) at Mauna Kea Observatory, Hawaii, is an example of a telescope using an adaptive optics system.

Confusion sometimes arises over the difference between active optics and adaptive optics. Adaptive optics can correct for turbulence in the atmosphere by means of very fast corrections to the optics, whereas active optics only corrects for much slower variations. Thus, whereas adaptive optics (as on the CFHT) can reach the diffraction limit of the telescope, active optics (as on the NTT) only allows the telescope to reach the ambient seeing. (7)



Try this link for some good information on Adaptive Optics: http://www.abc.net.au/science/articles/2010/08/05/2974623.htm

See my Useful Links page for more on Active & Adaptive Optics.





As mentioned previously, since angular resolution is directly proportional to the wavelength being observed, radio telescopes have inherently poor angular resolution.  Obviously, the resolution can be improved by increasing the diameter of the receiving dish but there is a practical limit to the size of an individual dish.  The largest single radio dish in existence is the 300 m diameter dish at the Arecibo Observatory in Puerto Rico (1).  The resolution of this radio telescope, when observing at a wavelength of say 21 cm is around 175, compared to the 1 resolution achievable by optical telescopes.

A technique called interferometry has been used to greatly improve the resolution of radio telescopes.  At its most simple, two radio telescopes separated by a large distance observe the same astronomical object.  The signals from each telescope are then combined to produce an interference pattern, which can be analysed by computers to reveal details of the object.  The effective angular resolution of two such radio telescopes is equivalent to that of one gigantic dish with a diameter equal to the baseline, or distance between the two telescopes.  Two telescopes used in this way are called a radio interferometer.  (1 & 3)


In the diagram above, q is the pointing angle to the radio source being observed, L is the difference in distance travelled by the radio waves to each of the telescopes and d is the distance between the two telescopes, called the baseline.  It can be shown mathematically (1) that:


This equation allows the position of the source to be accurately determined using the interference pattern produced by combining the signals from the two telescopes.  Obviously, increasing the distance between the telescopes improves the resolution.  It is also true that the resolution can be improved by increasing the number of telescopes comprising the interferometer. (1)

The Very Large Array (VLA) located near Socorro, New Mexico, consists of 27 radio telescopes in a moveable “Y” configuration with a maximum configuration diameter of 36 km.  Each individual dish has a diameter of 25 m and uses receivers sensitive at a variety of frequencies.  The signal from each of the separate telescopes is combined with all of the others and analysed by computer to produce a high-resolution map of the sky.  The resolution is comparable to that of the very best optical telescopes.  The 27 telescopes combine to produce an effective collecting area that is 27 times greater than that of an individual telescope.  (1 & 3)

To produce even higher resolution maps, a technique called very-long-baseline interferometry (VLBI) is used.  The Very Long Baseline Array (VLBA) consists of ten 25 m dishes at different locations between Hawaii and the Caribbean.  With VLBA, features smaller than 0.001 arcsec can be distinguished at radio wavelengths.  This angular resolution is 100 times better than a large optical telescope with adaptive optics.  Even better angular resolution can be obtained by adding radio telescopes in space to the array.  (1 & 3)

See also ALMA which is now the world's most powerful millimetre/submillimetre-wavelength telescope.

Note that interferometry is also used with optical telescopes.  The details of this will not be discussed here.  Check out the following link if you like.  It details the Very Large Telescope array, which is currently the world’s most advanced optical telescope array.


Just as an aside, the 0.001 arcsec resolution mentioned above is equivalent to being able to distinguish the two headlights on a car located on the moon from the earth (3).


(a)    Calculate the required diameter for a single radio dish to achieve an angular resolution of 1 when observing radio waves of wavelength 21 cm.  (Answer: 52.5 km)

(b)   The VLA has an effective diameter of 36 km.  Calculate the angular resolution achieved when observing the shortest receivable wavelength of 7 mm.  (Answer: 0.05 arcsec.  Note that this compares well with optical telescopes in terms of theoretical resolution and in practical terms is actually better, since radio telescopes are not greatly affected by seeing.  An angular resolution of 0.05 arcsec is sufficient to see a golf ball held by someone at a distance of roughly 125 km.)

(c)    Determine the diameter of the single radio telescope dish required to achieve the same sensitivity as the VLA.  (Answer: 130 m)









There are several different units used in Astronomy to measure distance (3).

u    The astronomical unit (AU) is the average distance between the earth and the sun.  1 AU = 1.496 x 108 km.  This is used primarily for distances within the Solar System.

u    The light year (ly) is the distance travelled by light in one year.
1 ly = 9.46 x 1015 m = 9.46 x 1012 km = 63 240 AU.  This is used for distances to the stars.

u    The parsec (parallax-second, symbol pc) is defined as the distance at which 1 AU perpendicular to the observer’s line of sight subtends an angle of 1 arcsec (1 second of arc).  See the diagram below.  1 pc = 3.09 x 1013 km = 3.26 ly.  This unit is used for distances to the stars.


Astrometry is the science of the accurate measurement of the position and changes in position of celestial objects.  The change in position of a celestial object can be due to either the real motion of the object itself or the motion of the Earth around its orbit, effectively shifting the point of observation.

As the point of observation shifts, a relatively nearby object appears to move against a set of more distant background objects.  This apparent change in the position of a nearby object as seen against a distant background due to a change in position of the observer is called parallax.

The phenomenon of parallax gives rise to an effective method for measuring the distance to nearby astronomical objects.  This method is called trigonometric parallax and is based on the method of triangulation used by surveyors.  It works in the following way.

We know that the direction of a nearby star from the earth changes as the earth orbits the Sun.  The nearby star appears to move against the background of more distant stars.  This motion is called stellar parallax.  Astronomers measure the parallax shift of the star from opposite sides of the earth’s orbit by making observations of the star six months apart.  The parallax shift (or angle) p is half the angle through which the star’s apparent position shifts as the earth moves from one side of its orbit to the other (3).  Since this is the maximum possible parallax shift for the star when observed from Earth, this particular parallax shift is often called the star’s annual parallax.  See the diagram below.


Clearly, then, by using a right-angled triangle, as shown above, the angle p is known, as is the length of the side opposite this angle – the radius of the Earth’s orbit.  Therefore the distance d to the nearby star can be calculated using trigonometry as follows:

                     sin p = radius of Earth’s orbit / d

Since, the angle p is very small, we can use the approximation sinq = tanq = q for q small, and hence we have that the distance, d, in parsecs, to the nearby star is given by: 


where p = parallax angle (annular parallax) of the star in arcseconds. 

Clearly, the larger a star’s annual parallax, the closer the star is to Earth. 

Note also that for objects within our own Solar System, it is possible to use trigonometric parallax, with the diameter of the Earth as the baseline to calculate distances to these objects.  For such an object, observations of the object are made 12 hours apart to obtain the diurnal (or geocentric) parallax angle and then a similar procedure to that described above is used to determine the distance to the object. 

EXERCISE: Barnard’s star has a parallax angle of 0.545 arcsec.  Determine the distance from Earth to the star. (1.83 pc)




Limitations of Trigonometric Parallax Measurements

Measuring parallax angles from the ground is very difficult due mainly to the atmospheric blurring discussed earlier.  Even with the very best optical telescopes in the world under excellent seeing conditions, parallaxes smaller than about 0.01 arcsec are extremely difficult to measure from the ground (3).  Therefore, trigonometric parallax measurements used with ground-based telescopes can give fairly reliable distances only for stars nearer than about 1/0.01 = 100 pc.

In 1989 the European Space Agency (ESA) launched the satellite Hipparcos, an acronym for High Precision Parallax Collecting Satellite, in order to collect much more precise parallax measurements from the perfect seeing environment of space.  In over four years of observations, Hipparcos measured the parallaxes of 118 000 stars with an accuracy of 0.001 arcsec.  From the data collected, astronomers have been able to determine stellar distances by trigonometric parallax out to several hundred parsecs, and with much greater precision than was possible with ground-based observations (3).

Check out the link to the Hipparcos Web Site on my Useful Links page.  There is also a link to the ESA’s GAIA project.  This satellite is due to be launched in 2010 and will measure the parallaxes of about 1 billion stars (1% of our Milky Way Galaxy) down to an accuracy of 10 microarcsec, which is about 100 times more accurate than the Hipparcos data.



Go to the next page of the Astrophysics Topic





1.      Carroll, B.W., & Ostlie, D.A. (1996).  "An Introduction To Modern Astrophysics", New York, Addison-Wesley Publishing Company Inc.

2.      Gondhalekar P. (2001). "The Grip of Gravity", Cambridge, Cambridge University Press

3.      Kaufmann, W.J. III, & Freedman, R.A. (1999).  "Universe", (5th Edition), New York, W.H. Freeman & Company

4.      "Sidereus Nuncius or the Sidereal Messenger". Translated with introduction, conclusion, and notes by Albert Van Helden. Chicago: University of Chicago Press, 1989

5.      Ridpath, I. (Ed.) (1997).  "Oxford Dictionary of Astronomy", Oxford, Oxford University Press

6.      Hollow, R.  "Why Build Big Telescopes?", paper presented at Science Teachers Workshop 2002

7.      http://www.eso.org/projects/aot/introduction.html and http://www.ls.eso.org/lasilla/sciops/ntt/telescope/esontt.html

8.      Eisberg, R. & Resnick, R. (1974).  "Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles", Canada, Wiley

9.      Playoust, D.F., & Shanny, G.R. (1991).  "An Introduction to Stellar Astronomy", Queensland, The Jacaranda Press

10. Bhathal R., (1993).  "Astronomy for the Higher School Certificate", Kenthurst, Kangaroo Press Pty Ltd

11.  Dawes, G., Northfield, P. & Wallace, K. (2003).  "Astronomy 2004 Australia – A Practical Guide to the Night Sky", Australia, Quasar Publishing

12.  Andriessen, M., Pentland, P., Gaut, R. & McKay, B. (2001). "Physics 2 HSC Course", Australia, Wiley

13. Schilling, G. (2004). "Evolving Cosmos", Cambridge, Cambridge University Press





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