Advantages of large diameter telescopes
1. Angular resolution of telescope:
- When light passes through a circular aperture of diameter D, the first minimum occurs at angle given by
- [math] sinθ = \frac{1.22λ}{D} [/math]
- To demonstrate the diffraction of light in a laboratory, it is necessary to direct a beam of light through a very narrow slit – then we can see the light spread out.
- However, the effect of diffraction is apparent when light enters a telescope aperture, even though the telescope has a diameter of many centimeters or even meters.
- However, we shall work with the approximation that the minimum occurs for small angles at
- [math] θ = \frac{λ}{D} [/math]
- So, when light from a star passes through a telescope, the image of the star has a measurable width due to diffraction as the light passes through the lens or mirror aperture.
Figure 1 Sine wave function in telescope- Diffraction affects how well a telescope can resolve fine detail. Figure 2 shows the idea.
Figure 2 Diffraction pattern- Figure 2 shows the diffraction pattern due to two small sources of light, after passing through a narrow aperture.
2. Rayleigh’s criterion:
- Figure 3 shows how the intensity will appear for different values of y.
- In Figure 3 (a) the lamps are close to the slit, so their angular separation is relatively large and we see two separate patterns of intensity.
- In Figure 3 (b) the lamps are further away, so that they are just resolved, and in Figure 3 (c) the lamps are so far away that the eye cannot see any small dip in intensity between the lamps – so they cannot be resolved.
- Figure 3 (b) shows the Rayleigh criterion for resolution.
- When the first minimum of one of the sources coincides with the maximum of the second source, we can just see (resolve) the two separate sources. This rule is only a guide because some people’s eyes are better than others.
Figure 3 The Rayleigh criterion waves (a) (b) (c) - Rayleigh’s criterion for resolution can be written as follows – when two sources, emitting light of wavelength , have an angular separation and are viewed through an aperture of diameter D:
- If [math] θ > \frac{λ}{D} [/math] the sources can be resolved.
- If [math] θ = \frac{λ}{D} [/math] the sources can just be resolved.
- If [math] θ < \frac{λ}{D} [/math] the sources cannot be resolved
3. Collecting power:
- Collecting power is proportional to the square of the diameter (d) of the primary mirror or antenna. This means that as the diameter increases, the collecting power increases exponentially.
- [math] \text{Collecting Power (CP)} ∝ d^2 [/math]
- This relationship shows that:
– Doubling the diameter will increase the collecting power by a factor of [math] 4 (2^2) [/math]
-Tripling the diameter will increase the collecting power by a factor of [math] 9 (3^2) [/math] - The increased collecting power allows for:
– Deeper observations
– Higher sensitivity
– Better resolution - This proportionality is a fundamental principle in telescope design and astronomy, highlighting the importance of diameter in determining a telescope’s capabilities.
4. Students should be familiar with the rad as the unit of angle.
- Angular separation of two lamps
- Two lamps are separated by a distance of 1.2 cm, and they are placed 4 m away from a narrow slit of width [math] 2 \times 10^{-4} m [/math]. They are viewed through a blue filter, which allows light of wavelength [math] 4.8 \times 10^{-7} m[/math] to pass. Will an observer be able to resolve the lamps?
- Given data:
- separation of the lamps [math] = x =1.2 cm [/math]
- distance from the slit [math] = y = 400 cm [/math]
- Width of narrow slit [math] = D = 2 \times 10^{-4} m [/math]
- Wavelength of light [math]= λ = 4.8 \times 10^{-7} m[/math]
- Find data:
- Angle between the lamps [math] = \theta = ? [/math]
- Formula:
- [math] sin θ = θ = \frac{x}{y} [/math]
- Solution:
- [math] sin θ = θ = \frac{x}{y} \\ θ = \frac{1.2}{400} \\
θ = 3 \times 10^{-3} \text{rad} [/math] - The smallest angle that the observer will be able to
- [math] θ = \frac{λ}{D} \\
θ = \frac{4.8 \times 10^{-7}}{2 \times 10^{-4}}\\
θ = 2.4 \times 10^{-3} \text{rad} [/math] - Because [math]θ > \frac{λ}{D} [/math] the lamps may be resolved.
5. Comparison of the eye and CCD as detectors in terms of quantum efficiency, resolution, and convenience of use.
- a comparison of the eye and CCD (Charge-Coupled Device) as detectors in terms of quantum efficiency, resolution, and convenience of use:
- Quantum Efficiency:
- optical instruments give us fresh insight into our surroundings, and this is particularly so in the field of astronomy.
- The first way we look at stars is to use our eyes, but we see more when we use binoculars or a small telescope.
- A telescope gathers lighter than our eyes, so we see fainter objects, and the larger aperture of the telescope allows us to resolve more detail.
- However, astronomers realized, around the start of the twentieth century, that even more information could be gathered by using a camera together with a telescope.
- By ‘driving’ a telescope so that it rotates at the same rate as the Earth, it is possible to track stars exactly over a long period of time. Then a very long exposure photograph can be taken, and the film developed later.
- Now, all telescopes used by professional astronomers use cameras with charge-coupled devices (CCDs) to detect the light from stars and galaxies. A CCD is a slice of silicon that stores electrons freed by the energy of incoming photons.
- The charge on the electrons builds up an image as a pattern of pixels. CCDs are much more sensitive to light than photographic film, and they have the advantage that information can be stored in digital form and processed by computers.
- Now cameras using CCDs are readily available to us all, and astronomers use high-quality CCDs with hundreds of megapixels to take long-exposure photographs of deep space.
- A CCD has a very high quantum efficiency. What this means is that a very high percentage of photons that strike the CCD produce charge carriers, which are then detected. Quantum efficiency is defined:
- [math] QE = \frac{\text{number of electrons produced per second}}{\text{number of photons absorbed per second}} [/math]
- Eye: 1-5% (depending on wavelength and individual vision)
CCD: 50-90% (depending on wavelength and CCD type) 
Figure 4 comparison of the eye and CCD (charge-coupled device)- The CCD has a much higher quantum efficiency, meaning it can detect a larger percentage of incoming photons.
- Resolution:
Eye: 200-300 microradians (approximately 1-2 arcminutes)
CCD: 1-10 microradians (approximately 0.05-0.5 arcseconds) - The CCD has a much higher resolution, allowing for more detailed images.
- Convenience of Use:
Eye: Easy to use, no special training required
CCD: Requires specialized equipment and training to operate - The eye is much more convenient to use, as it’s always available and doesn’t require any special equipment or training. However, the CCD offers much higher performance and is essential for many scientific and astronomical applications.
- The eye’s quantum efficiency and resolution can vary greatly depending on individual vision and conditions. The CCD’s performance can also vary depending on the specific type and quality of the device.