Using a telescope to see more detail in an image

5.4 Using a telescope to see more detail in an image

There is a fundamental limit to the detail in the image produced by a telescope which is caused by the effects of diffraction. Assuming that the telescope has a circular aperture, and in the absence of an atmosphere, then the image formed by a point source is a central disc surrounded by a number of concentric rings rapidly decreasing in brightness. In a perfect telescope the central disc contains 84% of the light with the remainder in the surrounding rings – most in the fi rst ring which has about twice the diameter of the disc. This pattern, called the Airy disc, shown in Figure 5.8, is named after the Astronomer Royal, George Airy, who gained some notoriety for not pursuing the discovery of the planet Neptune.

The angular size of this pattern, ∆θ, is a function of both the wavelength of light, λ, and the diameter of the telescope objective, D:

∆θ ⫽ 1.22 λ/D

The angular size of the Airy disc is related to what is termed the resolution of a telescope. For a given wavelength the resolution increases with telescope aperture and is inversely proportional to D.

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The calculation of this formula requires calculus but an approximate derivation can be gained by using a little quantum mechanics. It is based on the Heisenberg Uncertainty Principle which states, in one of its forms, that the more accurately we know the position of an object, the less accurately we can know its momentum. The principle states that the product of the uncertainty in x, ∆ x , times the uncertainty in p, ∆ p , must be less that Planck’s constant, h. When a photon passes through an aperture (like a lens) its position is determined to a certain extent so we cannot be so sure about its momentum along the axis or axes in which it is constrained. This is at right angles to the direction of the incoming photon and the result is to give the photon some uncertainty in its future direction. A beam of individual photons passing through the aperture thus ‘spreads out’ giving rise to the Airy disc. You can immediately see that the smaller the aperture through which the photons pass, the greater the knowledge about their position and hence the less well determined is their momentum and the Airy disc gets larger – agreeing with the formula given above.

Consider a photon passing through a slot aperture along the x axis which has

a width D in the y axis as shown in Figure 5.9. It has come from infi nity along the x axis and has a momentum p. Quantum mechanics states that the momentum, p x , of the photon is given by:

p x ⫽ h/λ

where h is Planck’s constant and λ is the wavelength.

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The fact that it has passed through the slit will give the photon an uncertainty in its momentum in the y direction of ∆ p y , where as ∆x ⫽ D:

∆p y D ⫽h

From the momentum equation, h ⫽ p x λ, So,

∆p y D ⫽p x λ ,

Giving

∆p y /p x ⫽ λ/D.

The uncertainty in ∆p y gives an uncertainty in the direction in which the photon continues of ∆θ ⫽ ∆p y /p x so, substituting from above gives:

∆θ ⫽ λ/D.

This is the angular uncertainty when the photon’s position has only been constrained along one axis. A lens of aperture D will constrain the photons along two axes, and one would expect the uncertainty to become greater. Hence the factor 1.22 derived by the formal calculation.

5.4.1 An interesting worked example of the eff ects of diff raction

In Section 3.5.4 it was pointed out that the lunar laser refl ectors were made up of 100 small refl ectors rather than one large one. Let us see why. The laser pulse sent from a telescope on Earth is refl ected by 100, 3.8 cm square, corner-cube

refl ectors forming a square array whose sides are 46 cm in length.

Consider fi rst a single 46 cm square refl ector. The light pulse refl ected by the refl ec- tor could be thought of as a beam of light coming from beyond the Moon’s surface that has passed through a square aperture of this size on its way to Earth. Assuming that a green laser is used, the beam will thus be spread over an angle given by:

∆θ ⫽ ∼1.22 λ/D

⫽ ∼1.22 ⫻ 5.5 ⫻ 10 ⫺7 /0.46 ⫽ ∼1.5 ⫻ 10 ⫺6 rad

The beam will thus be spread over this angle and could thus be seen over an area of the Earth whose diameter, d, is given by R ⫻ ∆θ, where R is the distance of the Moon’s surface from a typical point on the surface of the Earth.

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d ⫽ ∼1.5 ⫻ 10 ⫺6 ⫻ 378 000 ⫽ ∼0.57 km

The radius of this area will thus be ∼0.28 km. However during the 2.5 s the pulse takes to reach and return from the Moon, the telescope will have moved as the Earth rotates around its axis. If the telescope that sent the pulse was located at a latitude of 40° then the distance moved, d t , would be given by the circumference of the Earth at latitude 40 times the ratio of

2.5 s to 1 day: d t ⫽ 2 ⫻ π ⫻ [6780 ⫻ cos (40)] ⫻ 2.5/24 ⫻ 3600

d t ⫽ 0.94 km

So the laser ranging telescope will have moved out of the region where it could receive the returned echo!

By using smaller individual refl ectors, in this case 3.8 cm, the refl ected beam will have an angular size 46/3.8 times larger and, on the Earth, will cover an area of radius 3.4 km which is comfortably greater than the telescope’s change of position.

5.4.2 The eff ect of diff raction on the resolution of a telescope

If one considers a 150 mm telescope observing in green light of 5.5 ⫻ 10 ⫺7 m wavelength, one gets a size for the Airy disc of 4.4 ⫻ 10 ⫺6 rad which is 0.9 arcsec:

∆θ ⫽ 1.22 λ/D

⫽ 1.22 ⫻ 5.5 ⫻ 10 ⫺7 /0.15 rad ⫽ 4.4 ⫻ 10 ⫺6 rad

⫽ 4.4 ⫻ 10 ⫺6 ⫻ 57.3 ⫻ 3600 arcsec ⫽ 0.9 arcsec

Larger aperture telescopes will theoretically give higher resolutions but, in practice the resolution is usually limited by what is called the ‘seeing’ – a func- tion of turbulence in the atmosphere. The atmosphere contains cells of gas with slightly differing refractive indices which are carried high above the telescope by the wind and act rather like the glass used for screens which blur what is seen beyond. A star is effectively a point source and should theoretically have an image size given by the Airy disc of the telescope aperture. In practice a stellar image

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size as seen from the UK will probably be or order 2–3 arcsec across and is highly unsteady. This is one reason why professional telescopes are located on high mountains on islands such as La Palma in the Atlantic Ocean and Hawaii in the Pacifi c Ocean. At such locations, there is far less atmosphere above the telescope and the air tends to be less turbulent as it has been fl owing over the sea. Under the best conditions the seeing might limit the resolution to half an arcsecond, so larger aperture telescopes will see more detail but not signifi cantly more than

a telescope whose aperture is ∼400 cm across. The best location for an optical telescope is in space, as in the case of the Hubble Space Telescope, where its full resolution of 1/20th of an arcsecond at visible wavelengths may be realised.

The turbulence of the atmosphere and hence seeing varies from night to night. In bad seeing the image of a star will appear bloated and the Moon can appear to be boiling! On such nights the image quality will be totally determined by the atmosphere. But, rarely, the atmosphere can be still and then the aperture, type of telescope and the quality of the optics will determine what you can see. The seeing tends to have more effect on large aperture telescopes, so when the seeing is not good a smaller telescope may actually give better views of the planets.