Exploring Telescopes and the Diffraction Limit

1. Lecture 2 : Telescopes and the Diffraction Limit • Large astronomical telescopes are used as cameras. • Can apply simple optics: • all targets so distant, light enters as parallel rays • parallel rays focussed distance f (= focal length) from lens/mirror • rays passing through centre of thin lens are undeviated Objectives: Describe simple telescopes Diffraction limit and its importance for telescopes Importance of collecting area PHYS1005 – 2003/4

2. Basic camera: • Can use small angle approximation  • 2 stars separated by α produce image of size s on detector • s = f α N.B. Telescopes are described according to the diameter D of their objective lens or primary mirror e.g. WHT 4.2m on La Palma PHYS1005 – 2003/4

3. To look through a telescope, need an eyepiece: • Image now magnified (see increased angle between ray bundles) • Easy to show that magnification factor M = f1 / f2 (try it!) N.B. irrelevant for stars (why?) cf. optical specification of binoculars e.g. 10 X 50 means M = 10, D = 50 mm PHYS1005 – 2003/4

4. Resolution and the Diffraction Limit • Resolution of telescopeangular resolution i.e. smallest angle at which 2 point sources can be separated • Set by wave nature of light and is the ultimate limit • Passage of light through a lens (or any aperture)  spreading • e.g. spreading of waves through narrow entrance in harbour wall  wavefronts become curved ≡ spread in direction of waves  blurring of image • Blurring (or diffraction) is a standard result in optics PHYS1005 – 2003/4

5. For light of wavelength λ, telescope aperture D, angular resolution δθ is • δθ = 1.22 λ / D (in radians)  the Diffraction Limit • Applies to optical and radio telescopes, the eye, binoculars, i.e. any optical instrument! • e.g. in 1985, laser beam (diam. 4.7cm) of visible (λ = 500 nm) directed towards Space Shuttle, orbiting at altitude of 350 km. What was the diameter of the beam when it reached the Shuttle? • Answer: δθ = 1.22 λ / D = 1.22 x 5 x 10-7 / 0.047 = 1.3 x 10-5 and beam diameter = 1.3 x 10-5 x 350,000 = 4.5 m! PHYS1005 – 2003/4

6. Diffraction limit is important practical design limit: • but optical instruments can be designed that are diffraction-limited • however, large ground-based optical telescopes are limited by atmospheric “seeing” which blurs images • best seeing on Earth? Typically 0.3 – 1” •  large optical telescopes limited by atmosphere, not optics •  radio and space-based telescopes are limited by diffraction • e.g. observing at λ = 500 nm, at what telescope aperture D does the diffraction limit = 1”? • Answer: δθ = 1/206265 = 1.22λ / D • i.e. D = 206265 x 1.22 x 5 x 10-7 = 0.12 m ! •  large telescopes gain little in resolution (due to atmosphere) • So why are they built? • Answer: to gather more light! PHYS1005 – 2003/4

7. VLT, Chile ING, La Palma Gemini, Hawaii/Chile PHYS1005 – 2003/4

8. Gain in sensitivity with telescope D: • e.g. the VLT 8m telescopes are equipped with CCD detectors capable of detecting signals as low as Pmin = 10-15 W. How far away could it detect the light from a 60W light bulb? • Answer: assume light bulb emits power L equally in all directions. Then at distance d, there will be a flux F (power/unit area) of • F = L / 4 π d2 • Telescope collecting area = π D2 / 4 and so power P gathered is • P = F x π D2 / 4 = L D2 / 16 d2 • Set this = Pmin (assumes 100% efficient telescope/detector) and rearrange  • d = √ (LD2 / 16 Pmin) = 4.9 x 108 m ( ≈ 300,000 mls, > Moon distance!) • N.B. d α D PHYS1005 – 2003/4