# Resolution and Sensitivity

Relevant syllabus point:

• define the terms ‘resolution’ and ‘sensitivity’ of telescopes

## Function of Telescope

What does a telescope do? Contrary to popular opinion it does not 'magnify' light. Instead a telescope and instrument combination is generally used to gather light for one of two functions:

1. Imaging, in which pictures of celestial objects are clearly resolved, requiring optics that produce a sharp image, or
2. Photometry, where the incoming radiation is measured either for brightness or split to obtain a spectrum.

Nowadays, astronomers use telescopes each designed for a different part of the electromagnetic spectrum. Some telescopes are specialised for only one of the above functions but most are used for both. In order to fulfil these functions, a telescope should have high sensitivity and high resolution.

## Sensitivity

Sensitivity is a measure of the minimum signal that a telescope can distinguish above the random background noise. All other things being equal, a telescope of larger primary mirror or lens is more sensitive than one with a smaller primary.

The more sensitive a telescope, the more light it can gather from faint objects. The more light gathered, the fainter the object (or the more distant for a given class of object) that can be studied photometrically or imaged.

The size of a primary mirror or lens is normally expressed in terms of its diameter. A simple phrase often used by astronomers is that of the light bucket. The bigger the bucket, the more light can be poured into it.

The images below images of the same region of sky. The left hand image is simulates the image from a telescope of lower sensitivity than that on the right.

Low sensitivity
Higher sensitivity
Images adapted from SDSS field 756 from SkyServer

The right hand image from the more sensitive telescope reveals more and fainter stars and galaxies. The image on the right has a fainter limiting magnitude.

## Resolution

Have you ever tried to pick out a friend's face from a crowd? As you approach a crowd you can make out enough detail so that you know they are people and not cars. Walking closer you may be able to distinguish features such as the colour of a jacket or hair or the different height of individuals. At what distance can you clearly see the features of someone's face? What would happen if you kept walking closer. You may be able to see if they had shaved that morning. Eventually you could see individual skin pores - a scary thought. By observing someone from a closer distance, you have been able to resolve more detail, that is, see them more clearly.

Astronomers, unfortunately, are not able to move closer to stars and galaxies beyond our Solar System. So how can they see these distant objects more clearly? This is one of the key functions of a telescope - to resolve celestial objects. The higher the resolution of a telescope, the more details we can see from the images obtained on it. Technically we are referring here to the spatial or angular resolution of a telescope.

The three images below simulate the effect of differing resolution for the galaxy NGC 3521. The left hand image has low resolution, the middle image better resolution and the right hand image high resolution so that detail can be clearly seen.

Images adapted from image of NGC 3521 on SkyServer

The ability of a telescope to distinguish between, that is, resolve, close objects. For circular apertures, such as in telescopes, where the light rays from a source are parallel, as is the case for distant point sources of light such as stars, the light will be diffracted so as to form an Airy disc. The circular diffraction pattern formed contains 84% of the light in the central bright spot with decreasing percentages in the surrounding bright rings. The first diffraction ring should have less than 2% of the light in the central Airy disc.

Image provided by Brian Burton, Boston University

It is the size of the Airy disc that imposes a limit on resolution. Two objects are said to be resolved if their Airy discs are sufficiently separated to be seen as distinct. Rayleigh proposed the criterion that two point objects are just resolved if their angular separation is such that the central maximum from one point source lies on the first minimum of the other as shown in the image below:

Image: S. Karl

The theoretical resolving power of a telescope can be determined by the expression:

(Equation 1)

where θ = angular separation (in radians), λ = wavelength of light being collected and D = diameter of the primary mirror or lens. D and λ must both be in the same unit and this only applies where the size of the primary, D is >> λ. The image below shows

A more practical version of this equation expresses the theoretical value of the resolution in units of arcseconds. This is given by equation 2:

(Equation 2)

Note that this equation is not specified in the Board of Studies Physics syllabus or formulae sheet but understanding it will help you discuss the concept of resolution for telescopes.

So what does it mean?

Firstly, resolution is inversely proportional to the size of the primary mirror. The larger the diameter of the mirror, the smaller the value of θ, the theoretical resolution. A large telescope therefore theoretically can resolve more detail than a small telescope at a given wavelength.

How does an 8m telescope compare with the human eye when it comes to resolving detail? If we assume that a fully-dilated pupil has a diameter of 7mm, (ie 7 x 10-3m) and we are observing in yellow light at a wavelength of 550nm (5.50 x 10-7m) then:

Theoretical resolution for a human eye is given by = 2.1x105 x 5.50x10-7 / 7x10-3 = 16.5 arcseconds.

For an 8-m telescope: = 2.1x105 x 5.50x10-7 / 8 = 0.014 arcseconds.

The second point is that the wavelength at which an astronomer wishes to observe also determines the detail that can be seen as resolution is proportional to wavelength, θ ∝ λ. The longer the wavelength, the lower the theoretical resolution for a telescope of given size. Hence an optical telescope such as Gemini that can also observe at near-infrared wavebands should theoretically obtain lower resolution observing an object in the IR than at shorter wavelength visible light. As we shall see below, however, other factors come into play that reduce the actual resolution obtained by telescopes.

## Difficulties

If an optical device such as an eye or telescope achieves its theoretical resolution in operation it is said to be diffraction limited. In practice, this is not always achieved. The human eye, for instance, has imperfections on the cornea which normally degrade its resolution to about 1 arcminute, compared with the 16.5 arcseconds or about 0.3 arcminutes that equation 2 above determines. Modern optical telescope mirrors generally approach their theoretical limits for smoothness so should not suffer from this problem.

Large mirrors traditionally were made very thick to avoid the problem of flexing which would distort any image. Glass is very heavy, necessitating heavy mounts and drives to support the telescope, and also retains heat quite well. This is a problem as it takes a long time to cool down at night. The warmth of the mirror can heat the air above it, causing turbulent convection cells that diminish the seeing of the telescope.

Photometry traditionally did not require the level of resolution necessary for effective imaging but modern multi-fibre spectroscopes such as 2dF on the Anglo-Australian Telescope are only effective if numerous objects in a dense field (such as a star cluster or deep galaxy cluster) can be individually resolved.

## Further Information

Cosmic Reference Guides - Sensitivity is a clear, short page with images that is part of NASA's Cool Cosmos site.

Cosmic Reference Guides - Spatial Reolution is another page form the site. It has a clear explanation and useful comparison images.

Resolution Page from S. Karl's microscopy site provides a concise and technical explanation of resolution. Discusses lenses in microscopy.

The Purpose of a Telescope is a simple short page from a set of course notes at Cornell University. Shows the 'light bucket' model and provides links to other pages.

The Resolution of a Telescope - Dawes, Rayleigh and Sparrow is a site from a manufacturer of optics for quality amateur telescopes. It is reasonably technical and clearly written with some useful diagrams.

What is Resolution is a short page with a number of images comparing resolution as Airy disks and as astronomical images.

## Questions

1. What is the relationship between the diameter of the primary mirror of a telescope and its sensitivity?

2. Assuming the human eye has a pupil diameter of 7mm, how many more times sensitive is a) a 10cm telescope, b) an 8.1m Gemini telescope?

3. What is the theoretical resolution at the 21 cm waveband for a) 22m Mopra, b) 64m Parkes and c) 303m Arecibo radio telescopes?

4. Complete the table below:

Telescope Diameter of Primary Mirror (m) Theoretical Resolution at 550nm (arcseconds) Sensitivity compared with amateur 20cm reflector
Amateur Newtonian reflector
0.20

1
HST
2.3

AAT
3.9

Gemini
8.1

planned Overwhelmingly Large Telescope (OWL)
100

5. Why does the HST achieve higher resolution in actual use than the larger AAT?
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