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PHYS362 set8b

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Published on November 14, 2007

Author: Boyce

Source: authorstream.com

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PHYS362 – Advanced Observational Astronomy:  PHYS362 – Advanced Observational Astronomy Professor David Carter Slide Set 8 Interferometers:  Interferometers Angular resolution is given by: θ = (1.22 λ) / D D is the diameter of the (assumed circular) aperture of the telescope. There are limits in terms of cost and mechanical stability to the sizes of telescopes. Interferometry aims to obtain the resolution appropriate to the separation rather than the size of the telescopes. Two aperture interferometer:  Two aperture interferometer Place a mask in front of the aperture of a telescope with two small holes in it. Look at a close pair of stars, see if we can separate their images. Two aperture interferometer:  Two aperture interferometer If we look at a single star with the whole aperture we get the diffraction pattern from the circular aperture (as we saw before). Two aperture interferometer:  Two aperture interferometer Rayleigh Criterion – Images can be resolved when the separation is given by 1.22λ / D (D is aperture diameter) Two aperture interferometer:  Two aperture interferometer View a star through one small hole only in the mask. Image is now much broader. Two aperture interferometer:  Two aperture interferometer Now view a star through two infinitely small apertures with separation d in the screen. This is the same as Youngs’s slits. Gives a series of fringes with separation λ/D. Two aperture interferometer:  Two aperture interferometer For pair of small but not infinitely small apertures this is multiplied by the aperture diffraction envelope Two aperture interferometer:  Two aperture interferometer Now consider two stars imaged through this pair of holes, still with separation d: If the stars are superimposed then the fringes will just add. As the stars move apart the fringe systems move apart, until the angle of separation is α = λ/2d. Then the fringe systems cancel if the stars are equal brightness. Then if the stars are 2α apart the fringe systems add again, if they are 3α apart they cancel etc. Two aperture interferometer:  Two aperture interferometer Resolution of the two aperture interferometer is actually better than that of the whole aperture, by a factor of 2.44 d/D. However there is an ambiguity in the separation, and it is inflexible in the sense that d cannot be adjusted. The Michelson Stellar Interferometer:  The Michelson Stellar Interferometer Developed for optical observations of stars to determine their angular diameters. This is a development of the two aperture interferometer where the separation d can be varied. Used by Michelson to measure stellar angular diameters in the early 20th century. Apertures replaced by movable mirrors mounted on a rigid bar on top of the aperture of the 100 inch telescope at Mount Wilson. The Michelson Stellar Interferometer:  The Michelson Stellar Interferometer Mirrors are moved through various separations, and from the separations which give maximum and minimum fringe visibility the stellar diameter can be calculated. The Intensity Interferometer:  The Intensity Interferometer As the baseline goes up phase errors introduced by the atmosphere become important Not so true at longer (radio) wavelengths where signals remain coherent over long baselines. Intensity interferometer, devised by R.Hanbury-Brown, solves this by interfering only low beat frequencies on longer baselines Used again for measuring stellar diameters. The Intensity Interferometer:  The Intensity Interferometer The Intensity Interferometer:  The Intensity Interferometer Each receiver receives optical radiation through a narrow band filter. Radiation from the two sources produces a beat frequency which is passed through an electronic filter Signals from the two receivers are multiplied in a correlator. The Intensity Interferometer:  The Intensity Interferometer Consider two spectral components at slightly different frequencies, one from each source. At aerial 1 we receive the components: Aν1,s1 = E1 sin(2πν1t + φ1) Aν2,s2 = E2 sin(2πν2t + φ2) A is amplitude, φ is phase. Intensity received by the photomultiplier is: IA = CA[E1 sin(2πν1t + φ1) + E2 sin(2πν2t + φ2)]2 The Intensity Interferometer:  The Intensity Interferometer We analyse this in the same way we analysed the output from the square law mixer. IA = CA[E12 sin2(2πν1t + φ1) + E22 sin2(2πν2t + φ2) + 2E1E2 sin(2πν1t + φ1) sin(2πν2t + φ2) ] The Intensity Interferometer:  The Intensity Interferometer Using the trigonometric addition formulae this becomes: IA = ½ CA{[E12 + E22] - E12 cos(4πν1t + 2φ1) - E22 cos(4πν2t + 2φ2) - 2E1E2 cos(2π(ν1+ ν2) t + φ1 + φ2) + 2E1E2 cos(2π(ν1- ν2) t + φ1 - φ2)} We use an electronic filter to pass only this last beat component, which is much lower frequency than anything else IA = CA E1E2 cos(2π(ν1- ν2) t + φ1 - φ2) The Intensity Interferometer:  The Intensity Interferometer For aerial B we have phase differences caused by time delays (P1/c) and (P2/c) for sources 1 and 2. The intensity at receiver B is: IB= CB[E1 sin(2πν1(t + P1/c) + φ1) + E2 sin(2πν2(t + P2/c) + φ2)]2 and the same analysis gives the component passed by the electronic filters as: IB = CB E1E2 cos[2π(ν1- ν2) t + φ1 - φ2 + (2π/c)(ν1P1- ν2P2 )] The Intensity Interferometer:  The Intensity Interferometer We now form a correlation function k(s) by multiplying these two outputs for various values of the separation s between the aerials. k(s) = CACB E12 E22cos[2π(ν1- ν2) t + φ1 - φ2] cos[2π(ν1- ν2) t + φ1 - φ2 + (2π/c)(ν1P1- ν2P2 )] Using: cosA cosB = ½ [cos(A+B) – cos(A-B)] k(s) = ½ CACB E12 E22[cos[(2π/c)(ν1P1- ν2P2 )] +cos[4π (ν1- ν2)t + 2(φ1 - φ2) + (2π/c)(ν1P1- ν2P2)]] The second term is time variable, and if we integrate over time it disappears. The Intensity Interferometer:  The Intensity Interferometer k(s) = ½ CACB E12 E22cos[(2π/c)(ν1P1- ν2P2 )] Path differences are given by (referring back to the diagram): P1 = s sinφ P2 = s (sinφ + θ cosφ) (for small θ) P1 – P2 = -s θ cosφ We are using narrow band optical filters so ν1  ν2  ν k(s) = ½ CACB E12 E22cos[(2 π ν s θ cosφ)/c] The Intensity Interferometer:  The Intensity Interferometer This assumes so far that we have two monochromatic sources with frequency very close, but in practice we have to integrate the output for continuum sources over the bandpasses of the optical and electronic filters. This integration produces the result that we still get a fringe pattern, whose amplitude is the square of the amplitude of the visibility function of a Michelson interferometer of the same range of baselines. k(s) = V(s)2 Radio Interferometers:  Radio Interferometers Radio interferometers work on the principle of the Michelson interferometer. Simplest example consists of two aerials a fixed distance apart, which track a source across the sky: Correlation Interferometer:  Correlation Interferometer In a correlation interferometer the outputs from the two aerials are multiplied in a correlator rather than added. If the output voltages from the two elements of the interferometer are: V1 + N1 and V2 + N2 where the V values are signal voltage and N are the uncorrelated noise components. Correlation Interferometer:  Correlation Interferometer Product of the signals is: U = (V1 + N1) x (V2 + N2) = V1 V2 + N1 V2 + N2 V1 + N1 N2 when time averaged, the last three terms will average to zero (as the components are uncorrelated) <U> = < V1 V2 > the correlation interferometer is much more stable than the adding interferometer. Radio Interferometers:  Radio Interferometers Radio Interferometers:  Radio Interferometers Path difference is given by: p = s cosθ Suppose that the source is on the equator and the aerials are aligned east-west. As the source rises the path difference only changes slowly, in fact if θ  t: dp/dt  dp/dθ = s sinθ So the path difference changes slowly with time as the source rises, changes rapidly as the source passes transit, and slowly again as the source sets. Radio Interferometers:  Radio Interferometers We get a set of fringes as the path difference passes through integral and half integral numbers of wavelengths. Radio Interferometers:  Radio Interferometers Now consider two sources of equal brightness, separated by an angle β. When the objects are near the horizon, the path differences for the two objects are nearly equal, and the fringe patterns of the two sources add (they do not interfere with other because they are mutually incoherent). However near transit the difference between the two path differences is significant. Radio Interferometers:  Radio Interferometers Radio Interferometers:  Radio Interferometers Source 2 is straight overhead (actually on the meridian is enough), so the path difference from this source is zero. Source 1 has a path difference between the aerials of p = s sinβ  s β. If we move the aerials (e.g. slide them along a railway track) so that s = λ / 2β then this source will be at a fringe minimum at transit. The two fringe patterns will at transit be one half a fringe spacing out of phase. Radio Interferometers:  Radio Interferometers Radio Interferometers:  Radio Interferometers By adjusting the baselines we can get information about sources on different scales. If we have a whole line of N radio telescopes, and we have sufficient gain in the receivers to split the signal from each (N-1) ways, then we can treat each pair as a two element interferometer, and get N(N-1)/2 baselines at once. Radio Interferometers:  Radio Interferometers In general: The instantaneous output of a two element interferometer is a measure of one component of the two dimensional Fourier transform of the pattern of radiation from the objects in the field of view of one of the telescopes. Our aim is to measure all of these Fourier components, then use the inverse Fourier transform to reconstruct the image. Aperture Synthesis:  Aperture Synthesis Which Fourier component we are measuring depends upon the vector separation B of the two aerials as viewed from the source. If the linear separation is fixed the vector separation changes as the source is tracked over 24 hours due to the rotation of the earth. The vector separation is plotted in a plane perpendicular to the rotation axis of the Earth, called the (u,v) plane. u is in the local East direction, whilst v is North. Aperture Synthesis:  Aperture Synthesis We aim to sample the (u,v) plane as well as possible with a series of 12 hour observations with the aerials in a fixed configuration (we only need 12 hours because for the other 12 hours the signal is the complex conjugate of the first). Aperture Synthesis:  Aperture Synthesis Changing orientation of an antenna pair with time, for a source at the North celestial pole Aperture Synthesis:  Aperture Synthesis Track of a single aerial pair in the (u,v) plane for a 12 hour observation at a declination of 35 (hence teh track is an ellipse not a circle) Aperture Synthesis:  Aperture Synthesis Series of tracks in the (u,v) plane for a two element interferometer at a series of different spacings, or alternatively a linear array giving a range of spacings Aperture Synthesis:  Aperture Synthesis Early aperture synthesis telescopes (e.g. at Cambridge) are on an East-West baseline. This is fine for sources at Northern declinations, but on the equator the coverage of the (u,v) plane in the v direction collapses. As the Point Source Response Function is basically the Fourier transform of the coverage of the (u,v) plane, this means such telescopes have no resolution in a North-South direction Aperture Synthesis:  Aperture Synthesis Later aperture synthesis telescopes such as the Very Large Array (VLA) have a different configuration. The VLA has its aerials in a Y shaped configuration to give better sampling of the (u,v) plane at near equatorial declinations. Aperture Synthesis:  Aperture Synthesis The Very Large Array (VLA) aperture synthesis telescope in New Mexico Processing of aperture synthesis data:  Processing of aperture synthesis data Because the (u,v) plane is not completely sampled for instance because there is a limit to how close aerials are able to be, or because the time allowed to to short to allow 12 hours with all necessary spacings, the Point Source Response Function which is the Fourier Transform of the sampling in the (u,v) plane tends to be complicated. Deconvolution and Wiener filtering are inapplicable to the case where we do not completely sample the spatial frequency plane The Visibility map:  The Visibility map The Visibility map V(u,v) is the Fourier transform of the true Intensity map I(x,y) of the source. V(u,v) = F(I(x,y)) The Dirty beam and the Dirty map:  The Dirty beam and the Dirty map The Dirty beam is the Point Source Response Function of an aperture synthesis observation. It is the inverse Fourier transform of the sampling distribution of the (u,v) plane: b(x,y) = F-1(S(u,v)) The dirty map is the inverse Fourier transform of the measured signals from the interferometer I(x,y) = F-1(V(u,v)S(u,v)) = F-1(V(u,v)) F-1 (S(u,v)) I (x,y) = b(x,y) * I(x,y) A (very) dirty beam:  A (very) dirty beam The CLEAN algorithm:  The CLEAN algorithm Developed in the 1970s the CLEAN algorithm is a process carried out on the image which models it as a series of delta functions, each contributing a multiple of the dirty beam at some point in the map. Start with I0 = 0 then find the point of highest residual intensity in the dirty map. At this point create a delta function with amplitude μ times the value in the map at this point. μ is called the loop gain, and is chosen to give the best results (a common practice in astrophysics!) The CLEAN algorithm:  The CLEAN algorithm Convolve this with the dirty beam, and subtract it from the dirty map to create a new residual map. Repeat this process, can be tens of thousands of iterations In = I - b*In-1 Meanwhile store the position and amplitude of each delta function in a CLEAN component table. Carry on until you think you are fitting just noise. The CLEAN algorithm:  The CLEAN algorithm Construct a CLEAN beam I, b(x,y), which is usually a two dimensional elliptical Gaussian fit to the dirty beam. Convolve each CLEAN component by the CLEAN beam, and add them all up to form the CLEAN map. Maximum Entropy:  Maximum Entropy Maximum Entropy methods produce the map which is consistent with the observations, but which is consistent with some prior knowledge. To do this define an entropy parameter: H = - Σpixels I [ln(I/M)-1] Where M is a “default” image (for instance a lower resolution map). Maximum Entropy:  Maximum Entropy Find the map I which has the maximum value of this parameter, subject to the additional constraint that the differences between the true and dirty visibility map have the correct noise distribution. χ2 = Σspatial frequencies[(V - V)2 / σ2] Maximum Entropy is faster than CLEAN for large images, but is much less popular. It is a mixture of a maximisation in the image plane and a minimisation in the (u,v) plane, and you need to know the noise properties in the (u,v) plane. Closure Phase:  Closure Phase The atmosphere and the communications between the aerials introduce phase errors between aerials, but these can be reduced by adding the phase differences around triangles of baselines, when they cancel out. This provides extra constraints which can be introduced into the Maximum entropy maximisation. Closure Phase:  Closure Phase Self Calibration:  Self Calibration Self calibration is effectively using a source in the field of view (primary beam) to define the dirty beam, rather than using a separate observation. Atmospheric phase errors contribute to the dirty beam and the advantage of self calibration is that the same atmospheric errors are present. More Detail (if you are interested):  More Detail (if you are interested) http://scienceworld.wolfram.com/physics/topics/SynthesisImaging.html (Introduction to a whole range of physics topics) http://wwwuser.oat.ts.astro.it/sedmak/scuolatecno2002/venturi/ (Lecture notes on radio astronomy form the University of Bologna, but the content is in English) N.B. You won’t be examined on anything thats in here but not in my notes, the extra stuff is for your interest if you want to know more. Very Long Baseline Interferometry:  Very Long Baseline Interferometry Telescopes are in a variety of countries, even a variety of continents. Too far apart to interfere the signals directly. Signals are recorded on magnetic tape with the telescopes pointing in the same direction, and an accurate time from calibrated atomic clocks is associated with each signal. Signals are brought together in a computer, and the signal multiplication is done digitally. MERLIN:  MERLIN Telescopes at Jodrell Bank and around the country (furthest is in Cambridge). Signals are brought back to Jodrell Bank by Microwave link, which has limited bandwidth. Signal multiplication is done in software. Upgrade planned which will replace the mircowave links with dedicated high capacity optical fibres, vastly increasing the range of frequencies (bandwidth) which can be transmitted and interfered. Special purpose hardware correlator is required to do the signal multiplication. MERLIN:  MERLIN Locations and images of the individual MERLIN radio telescopes Optical Aperture Interferometry:  Optical Aperture Interferometry Interferometry with telescope arrays is possible in the optical, but much more difficult. Because of the smaller number of photons you can’t amplify the signal as you do at radio wavelengths, you just split it up. Atmospheric phase errors are much more serious. The phase error changes more rapidly with angle on the sky, restricting you to much smaller fields of view. Optical Aperture Interferometry:  Optical Aperture Interferometry The development of closure phase techniques has changed optical aperture synthesis from impossible to very difficult, and various people, at Cambridge, the Keck telescope and the European Southern Observatory are working on optical aperture synthesis experiments. Involve very accurately calibrated delay paths to ensure that the phase differences are maintained. Technically very difficult. Cambridge project concept (on Tenerife):  Cambridge project concept (on Tenerife) ESO VLT Interferometer (Chile):  ESO VLT Interferometer (Chile) The tracks running from the large telescopes outwards are for placing smaller “outrigger” telescopes at various stations to give a range of baselines. The outriggers themselves are not built yet.

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