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Second Year Problem sheet 3 7th March 2002 Course ASTR2B15: Techniques in modern astronomy (Answers to be handed in on Friday 14th March) 1. A telescope is equipped with an adaptive optics system that includes a deformable mirror with effective actuator spacing, mapped back onto the primary mirror, of 20cm. If the seeing coherence length at 500nm is 0.20m and the fitting constant K for the deformable mirror is 0.15, what is the maximum Strehl ratio that the system can achieve at a wavelength of 1.6 microns? [10] 2. Explain, including a sketch, how a stellar coronograph works. Include in your explanation what the effect the focal plane mask has on the distribution in the conjugate pupil plane of light from the central star. What are the benefits that adaptive optics bring to coronography? [5] 3. The Very Large Telescope Array is made up of many circular 26m diameter dish radio telescopes. The maximum separation that can be achieved between telescopes in the array is 27 kilometres. At a wavelength of 10cm, what is the achievable angular resolution of a single telescope in the array and what is the smallest achievable angular resolution of the whole array? A radio source is emitting radiation at a frequency of 1 GHz. If heterodyne detection of the signal is used with a local oscillator of 0.97 GHz, what will be the frequencies of the various components of the signal coming out of the receiver? (before filtering and rectifying). [5] 4. A quarter wave plate is made of quartz that has a degree of birefringence J of 0.009. How thick must the quartz be to act as a quarter wave plate at a wavelength of 550nm? After an observation of light with a rotating polarizer no variation in intensity is seen. On the addition of a quarter wave plate before the polarizer again no variation is seen. What is the polarization state of the incident light? (Explain how you arrived at your answer). If linearily polarised light is incident on a half wave plate with the direction of polarisation at angle of to the optic axis of the plate, what is the state of polarisation of the emergent light? [5] 1. A telescope is equipped with an adaptive optics system that includes a deformable mirror with effective actuator spacing, mapped back onto the primary mirror, of 20cm. If the seeing coherence length at 500nm is 0.20m and the fitting constant K for the deformable mirror is 0.15, what is the maximum Strehl ratio that the system can achieve at a wavelength of 1.6 microns? Coherence length at 500nm =0.2m Coherence length at 1.6microns r1600 1600 r500 500 d 6 5 =0.808m 5 3 2 K r If d=0.2. k=0.15 and r=0.808m, then 5 3 0.2 0.0146 0.808 2 0.15 Since S.R. exp 2 therefore maximum S.R. S.R. exp(0.0146) 0.985 [10] 2. Explain, including a sketch, how a stellar coronograph works. Include in your explanation what the effect the focal plane mask has on the distribution in the conjugate pupil plane of light from the central star. What are the benefits that adaptive optics bring to coronography? A stellar coronograph is a device that suppresses the light from a central bright star preferentially to light from any object close to the star. This enables low contrast objects close to the star to be studied. Simple coronograph design The coronograph consists of a small focal plane mask that sits over the image of the central star. This focal plane mask conditions the remaining light from the central star that passes through to be at the edge of the pupil. This light is then removed by a Lyot stop, which consists of an undersized aperture at a pupil conjugate that masks off the light at the edge of the pupil. The size of the Lyot stop is also important. If the stop is small then it will suppress the central star well but the overall throughput of the system will be reduced. Typically stop sizes with diameters of 80 percent of the full aperture are used. The central secondary obstruction also needs to be masked out in the pupil plane as light from the central star appears around its edge. Advantages of adaptive optics and coronography The image size is smaller and fixed and therefore a smaller focal plane mask can be used and objects closer to the central star can be observed. The faint objects will also be sharpened and hence brighter. Therefore the combination of AO with coronagraphy will enable higher sensitivity and spatial resolution imaging of regions close to the central star. 3. The Very Large Telescope Array is made up of many circular 26m diameter dish radio telescopes. The maximum separation that can be achieved between telescopes in the array is 27 kilometres. At a wavelength of 10cm, what is the achievable angular resolution of a single telescope in the array and what is the smallest achievable angular resolution of the whole array? A radio source is emitting radiation at a frequency of 1 GHz. If heterodyne detection of the signal is used with a local oscillator of 0.97 GHz, what will be the frequencies of the various components of the signal coming out of the receiver? (before filtering and rectifying). Resolution of individual telescope is given by Rayleigh Criteria as θ =1.22 λ /d where λ is the wavelength and d the telescope diameter. Therefore at a wavelength of 10cm and with a telescope diameter of 26m. The resolution is θ =1.22 x0.1/26=4.69x10-3rad = 968 arcsec. Resolution of whole array is given by θ =λ /d where λ is the wavelength and d the greatest telescope separation diameter. Therefore at a wavelength of 10cm and with a greatest separation of 27km. The resolution is θ =0.1/27000=3.7x10-6rad = 0.76 arcsec. Frequencies coming out of heterodyne receiver are 2nd harmonic of signal 2 GHz 2nd harmonic of oscillator 1.94GHz Sum frequency 1.97GHz Intermediate frequency 0.03GHz - 30MHz DC signal 4. A quarter wave plate is made of quartz that has a degree of birefringence J of 0.009. How thick must the quartz be to act as a quarter wave plate at a wavelength of 550nm? After an observation of light with a rotating polarizer no variation in intensity is seen. On the addition of a quarter wave plate before the polarizer again no variation is seen. What is the polarization state of the incident light? (Explain how you arrived at your answer). If linearily polarised light is incident on a half wave plate with the direction of polarisation at angle of to the optic axis of the plate, what is the state of polarisation of the emergent light? For the 1/4 wave plate Optical path difference o. p. ( ne no ) d where d is the depth of the plate Phase difference If (ne no ) 0.009 , 2 (ne no )d 550nm , phase difference = 2 550x10 9 d 1.53x10 5 m 4(ne no ) 4 * 0.009 If no variation is seen with rotating polariser then light must either be circularly polarised, unpolarised or a mixture of both. If a quarter wave plate is added and no variation is seen again in the intensity then the light must be unpolarised, sine if there was an circularly polarised component this would be converted to linearly polarised light which would give an intensity maximum. If linearly polarised light is incident on a 1/2 wave plate at an angle of to the optics axis the light will stay linearly polarised but the plane of polarisation will be rotated by 2.