Note for ASTR 518 with Professor Rieke at UA
Note for ASTR 518 with Professor Rieke at UA
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Imaging and Astrometry 1 Optical imager desigi 11 Design Small field optical imagers tend to be very simple basically they can consist ofa liquid nitrogen LN dewar holding a CCD that looks out through a Window A Wheel places the desired filter in front of the Window and the telescope is focused directly onto the CCD see Figure 1 Although this imager is conceptually simple good performance requires attention to detail For example if the filters are too close to the CCD and telescope focus then any imperfections or dust on them will produce largeamplitude artifacts in the image A few other components can improve the quality of the data A shutter provides for accurate and uniform exposures A suitable type because it can provide exactly the same exposure over the entire CCD is based on a slide or curtain that opens across the CCD to initiate the exposure and then another CED Dewar window LN can lighl iighi baffles Filler Wheel Shielded cable in CCD controller Dcwar preamp electronic Shuucr Z r Field Corrector ADC V Figure 1 Optical camera schematic From Michael Bolte curtain that goes across in the same direction and speed to close off the light and end the exposure An atmospheric dispersion corrector ADC counters the nonachromatic refraction of the atmosphere of the earth Without the ADC images taken at small elevation angles will look like tiny spectra with the blue end pointing to the zenith because the atmospheric refraction bends blue light more than red light More L ambitious instruments use field correcting optical systems to feed a large field onto a mosaic of detectors achieving a good match of the projected pixel scale to L2 5 Filter FPA Figure 2 Field correcting optics for the 90 Prime Camera Figure 3 The 90 Prime Camera on the Telescope the expected seeing limited image size Figure 2 shows the optical layout for the 90P1ime camera which feeds a 4CCD mosaic at 045 arcsec per pixel and with a eld of 116 degrees square Figure 3 shows the instrument on the 90Inch The detector is cooled in the dewar at the top toward us Neither type of optical camera provides a pupil where lters can be placed Therefore the accuracy of their photometry depends on a high degree of lter uniformity and to some extent the lter pass band will vary over the eld anyway due to varying angles of incidence of the light from the telescope 12 Nyquist Sampling A basic question in imager design is the equivalent angular size of the array pixels If they are made too small then the eld of view of the instrument is unnecessarily made small while if they are too large the images will lose information on small angular scales There are many competing considerations for a given instrument but at least there is a rule that says how small is small enough that essentially no details are lost in an image The MTF of an ideal detector array with no gaps between pixels can be shown to be MTF s1n7z39ufs 1 7z39u f5 where u is the pixel width and is the spatial frequency or lpS where pS is the spatial period of the signal Figure 4 shows the MTFs for the cases with 05 l and 2 pixels per spatial period it shows that the loss of information with 2 pixels is modest but with 1 it is already severe Now if we identify fS l with 12 08 MTF o a 04 02 N2 N1 N05 02 04 06 08 1 f5 Figure 4 MTFs for sampling at 05 l and 2 pixels per spatial period the natural cutoff frequency of a telescope Dk we have shown that basically two pixels across the image diameter FWHIVI ND is a good goal for the pixel size that does not lose any spatial information In practice if we take multiple exposures offset in pointing by a fraction of a pixel size it might be thought that we could sample finely enough to recover the information lost with large pixels However Figure 4 shows that this strategy does not work once the pixels are as large as MD the highest spatial frequencies are lost completely from the images This result is often Col liindlor Fuhl Minors ppmm Army Wuulmv 3mi I lanr Vgt 3 Biwmymm Mum Figure 5 Layout of a nearinfrared camera After entering through the dewar window the light is focused by the field lens plus collimator on to the pupil where the cold stop is placed The remaining optics relay the focal plane to the detector array described as Nyquist sampling after the Nyquist Theorem which states that a bandwidthlimited signal with maximum frequency F and period P lF can be completely reconstructed from time samples at a time interval of PZ The situation with finite pixels is analogous but not completely identical with the assumptions in proving this mathematical result In addition there are situations in which sampling more nely than implied by the Nyquist Theorem is beneficial One cause is that real arrays fall slightly short of the ideal see Section 3 below and some of their aws can be overcome by ner sampling 2 Infrared imagers Figures 5 and 6 are the optical layout for the near infrared camera for Gemini from Hodapp et al possible Figure 6 The infrared camera optical train is folded to make it compact so the cooled volume can be as small as 2003 PASP To avoid being ooded with thermal background from the telescope and other surroundings the entire camera is cooled in a vacuum enclosure In addition to minimize the view of the telescope the optical train forms a pupil around which is placed a tight cold stop These two design considerations result in an instrument con guration changed substantially from that for the CCD camera although the two instruments take very similarappearing data There are some serendipitous bene ts from this design For example the lters can be placed at a pupil Since a pupil is optically equivalent to the primary or secondary mirror small aws in a lter result in uniform loss of light but do not introduce artifacts into the images However all those optics impose their own issues 1 CosNe effects In general optical systems are most ef cient on axis The signal on an array is likely to fall off with distance from the center of the eld and due to effects such as vignetting this fall off may be different for point sources than for sky this type of problem exists with the wide eld correctors used in the optical also 2 Ghost images light re ected from refractive optics can get back to the array and provide weird extended images or even pointlike ones depending on the geometry Again even simple optical imagers can also produce ghosts due to re ections from the dewar window andor lter 3 Imager Data Reductions 31 The Issues lmagers have huge advantages over single detectors for nearly all astronomical observations They permit very accurate position determination and enable astrometry although an altemative approach was used for Hipparcos as discussed in Section 4 For photometry they 17 Allow centering on the source and setting other parameters of extraction of photometry after the fact 17 Let you use small apertures in crowded elds and to reject backgrounds for improved sensitivity 17 Allow differential photometry relative to other sources in the eld for accurate measurements under nonoptimum conditions 17 Provide much exibility for measurements of extended sources 17 Let you remove foreground stars 17 Let you use custom extraction apertures 17 Allow exibility in sky reference For extended sources they let you construct multiplecolor images and compare the behavior of a source in all the colors precisely However to gain these advantages there are number of steps that are required F v gt H Calibration must take into account the differing properties of the detectors in the array to be discussed below including 17 pixeltopixel variations in ampli er offset 17 pixeltopixel variations in dark current 17 pixeltopixel variations in responsivity Figure 7 Interpixel response gaps from John Hutchings Arrays also have a variety of their own issues l39l Optical 1 lnterpixel gaps and intrapixel response Since array pixels are discrete the sensitivity may have minima between pixels Figure 7 image from JWST HngTe array These quotgapsquot can have big effects ifthe pixels do not sample the PSF well It is also possible that the response varies over the face of a pixel Figure 8 shows the dependence of the signal on M the centering ofa source on a NICMOS Camera 3 pixel for this example 1 pixel N 15 ND where I D is the telescope aperture These effects will not 02 be detected through normal at elding image processing iiii u F160W ls 139 Amag 2 Fringingchannel spectra Arrays are based on thin parallel plate components When the 702 39 39 I 39 39 39 39 l 39 39 absorption in the detectors is low interference 0 0392 0 4 0 5 within the material causes fringing The nature of DP1X91 GaulET the fringing is a sensitive function of the spectral content ofthe illumination Figure 9 shows Figure 8 Signal vs centering from ffngng in a GMOSS CCD at N 09511111 Stiavelli et a WFC3 199901 l39l Electrical 1 Hot and dead pixels 2 Cosmic ray hits other transients 3 Nonlinearity and so saturation it may not be obvious when your signal is too strong for accurate data39 as the wells of typical readouts ll the detectors tend to be debiased resulting in lower signals 4 Latent images many electronic arrays retain an image at the 01 1 level on the next readout a er a bright source has been Figure 9 Fringing in a ID at 095 observed The images are usually from charged trapped at surface or interface layers in the detector These images Lalenl Images persistence usually decay over about 10 Figure 10 Latent images The source is removed after the first frame minutes Figure but its image persists while slowly fadingimage from Figer et a 10 Much 2002 SPIE 4850 longer decay times can result however if the array has been saturated by an extremely strong signal 5 MUX glow the readout transistors are sources of light through electroluminescence This can be picked up by the array contributing to a lack of atness in the images and also contributing noise Figure 11 from Figer 6 Electrical crosstalkghosting Various effects eg inadequate drive power on the output ampli ers produce secondary images see Figure 12 7 Pedestal effects readouts sometimes signal Ampli er MUX Glow resulting noise Figure 11 The raw readout is to the left Even after the signal has been corrected excess noise from the photons emitted by the MUX is visible to the right have electrical offsets that appear as structure in images Figure 13 32 Taking Good Data To be able to x these issues in the nal reduced images requires care in taking the data The first rule of array imaging is repetition L Multiple images allow systematic identi cation of outlier signals due to cosmic rays and other transients L They also allow replacing areas compromised by cosmic rays latent images hot or dead pixels ghosting etc with real data L By dithering the pointing on the sky between exposures moving the telescope slightly so the images fall on different parts of the array the sky signal itself can be used to atten the image as discussed below If the sky dominates the signal then fringing effects are removed to rst order along with many other potential contributors to non atness L Properly sampled images are another form of repetition more than one pixel contributes to the signal Accurate photometry bene ts from spreading the light over multiple pixels which can also be done with dithering a lot L Modern arrays often include nonactive pixels that are electrically identical to those that detect photons For CCDs the same bene t is obtained by overscanning the array while for infrared arrays they are physical outputs called reference pixels They can be used to correct the r Electronic Ghost Images Figure 12 Electronic ghosts Figure 13 Pedestal effect on a NICMOS arrav images for slow drifts in the electronics and other such effects The second rule of imaging photometry is don t change anything D Artifacts like MUX glow pedestals and many others will disappear from your reduced data virtually completely if you are careful to take all your data science and calibration frames in identical ways for example the identical exposure times and readout cadences D Detector arrays also perform better when they reach equilibrium ie constant exposure times and readout cadences plus constant temperatures backgrounds and etc D Suppose you have done everything correctly Then you can preprocess your images to get rid of artifacts e g cosmic rays Of course archival data may not have been taken with good procedures and in fact often is not so you will need to be more ingenious to use it 33 Calibration Then you are ready to calibrate the data This step must take into account the differing properties of the detectors in the array D pixeltopixel variations in amplifier offset D pixeltopixel variations in dark current D pixeltopixel variations in responsivity Three unknowns require three sets of data D Offset frame sometimes called bias frame very short exposure no signals D Dark current frame long exposure no signals D Response frame sometimes called at field uniform illumination Image data reduction consists of D Subtract offset from data dark and response frames to obtain data dark and response D Scale dark to exposure time of data and response and subtract from them to get data and response D Divide data by response The result if the data frame has a uniform exposure then the product will be a uniform image at a level corresponding to the ratio of the exposure on the data frame to the exposure on the response frame exposure level of illumination multiplied by the exposure time Sources will appear on top of this uniform background For best results D Dark current and response frames may need to be obtained close in time to the data frames D It may be necessary to use identical integration times for dark response and data frames D Response and data frames should be taken with illumination of identical spectral character U Need a minimum of 3 frames on source 7 5 or more is better 7 to be sure there are no transient bad pixels e g cosmic ray hits D Permanent bad pixels can be masked out by replacing values with the average of those from surrounding pixels However doing so can give bad data that looks good It is better to take multiple exposures and move the source on the array between them to fill in the source structure with all good data 7 bad pixels then just reduce the integration time at some points in the image A good strategy for imaging is D Take repeated exposures of the field moving the source on the array between exposures D Generate the response frame by a median average of these frames 7 sources will disappear because they do not appear at the same place on any two frames D Obtain dark frames with the same exposure time as used for the data and response frames D Subtract dark from data and response also takes out offset divide corrected data by corrected response D Shift frames to correct for frametoframe image motions D Median average again to eliminate bad pixels and cosmic rays while gaining signal to noise on the source image In general the image reduction software will include standard or recommended procedures to generate the necessary calibration frames from your data and to shift and add all your science frames into one highquality image B Is thejob done Recall the list of possible array problems we discussed earlier Some of them should be taken care of at this stage although they might have required some extra processing 1 fringing 2 hot and dead pixels 3 cosmic ray hits 4 latent images and 5 MUX glow The next to last item might require generating a special at field frame designed to just capture the latents You might also have to identify electrical and optical ghost images and other such effects and fix them by hand or with custom routines 34 How to carry the measurements around In the late 197039s astronomers developed the Flexible Image Transport System FITS as an archive and interchange format for astronomical data files In the past decade FITS has also become the standard format for online data that can be directly read and written by data analysis software FITS is much more than just an image format such as JPG or GIF and is primarily designed to store scientific data sets consisting of multidimensional arrays and 2dimensional tables containing rows and columns of data A FITS file consists of one or more Header Data UnitsHDUs where the first HDU is called the quotPrimary Arrayquot The primary array contains an Ndimensional array of pixels This array can be a 1D spectrum a 2D image or a 3D data cube Any number of additional HDUs called quotextensionsquot may follow the primary array Every HDU consists of a ACSII formatted quotHeader Unitquot followed by an optional quotData Unitquot Each header unit consists of any number of 80character records which have the general form KEYNAME value comment string The keyword names may be up to 8 characters long and can only contain uppercaseletters the digits 09 the hyphen and the underscore character The value of the keyword my be an integer a oating point number a character string or a Boolean value the letter T or F There are many rules governing the exact format of keyword records so it usually best to rely on a standard interface software like CFITSIO IRAF or the IDL astro library to correctly construct or parse the keyword records rather than directly reading or writing the raw FITS file 4 Astrometry Coordinates Wellreduced images invite us to think of the positions of astronomical objects a direction that leads us to the topic of astrometry 7 the procedures used to set up systems of positions and to measure individual objects accurately within these systems 4 1 Coordinate Systems An astrometric coordinate system can be envisioned as coordinate grid projected up into the sky upon which the positions of celestial objects are measured Such a grid has a fundamental great circle and a secondary one a great circle is the intersection ofa plane running through the center of a sphere with the surface of that sphere There are four such grids in common use be c nadir Y0 Figure 14 Horizon Coordinates 1 Horizon the fundamental circle runs around the horizon and the secondary one runs from it over the zenith This system is the basis of the altitudeazimuth altaz coordinates used to point most large telescopes As shown in Figure 14 the great circle passing through the zenith and north and south celestial poles de nes the zero point of azimuth where it intersects the horizon circle to the north Any object on the celestial sphere lies on a great circle perpendicular to the horizon circle and the azimuth A for this object is the angular distance measured eastwards from the zero point to the rst intersection of its great circle with the horizon one The altitude a of the object is measured along this circle from the horizon circle ifit is above the horizon and 7 ifit is below 2 Equatorial Although the horizon Figure 15 Equatorial Coordinates 3 system is convenient for telescopes it has the disadvantage that the coordinates of any object depend on the place and time of the observation Another system is needed in which the position of an object remains xed in the coordinate system Equatorial coordinates ful l this role Figure 15 the fundamental circle is the celestial equator and declination is measured along a secondary great circle that runs through the object and is perpendicular to the celestial equator for north and 7 for south The zero point is the position of the Vernal Equinox and right ascension is measured from there eastward to where the declination circle for the object intersects the celestial equator The Vernal Equinox is de ned as the point where the celestial equator and ecliptic the apparent path of the sun across the sky intersect in March ie the placement of the sun the moment in March when it is directly overhead as viewed from the equator Thus the zero is roughly at midnight within the vagaries of civil time at the Autumnal Equinox The units of right ascension 0c are hours minutes and seconds while declination 5 is measured in degrees and minutes and seconds of arc The celestial meridian or local meridian is the great circle along which lie the north and south celestial poles and the zenith point directly overhead The meridian of a source is the great circle along which lie the north and south celestial poles and the source in question The hour angle of the source is the angular distance from the celestial meridian to the meridian of the source measured to the west and in hours minutes and seconds It is equivalently the time until the source transits the celestial meridian negative hour angle or the time since it transited positive Ecliptic This coordinate system is the natural one to use when dealing with members of the solar system The fundamental great circle is the ecliptic 7 the apparent path of the sun across the sky The zero point is the Vernal Equinox and the ecliptic longitude 7 is measured from there eastward The ecliptic latitude 5 is measured along a great circle perpendicular to the ecliptic and passing through the north and south ecliptic poles positive if north of the ecliptic and negative if south Galactic For problems centered on places in the Milky Way the Galactic system is preferred Its fundamental great circle is the plane of the Galaxy the Galactic equator and the zero point since 1958 is close to the Galactic Center it was originally intended to be the Galactic Center but we have learned since where this region is more accurately and it is about 5 arcmin away from the coordinate system zero The prel958 system is designated by I and the newer one by 11 Roman numerals To place an object in Galactic coordinates we rst determine a second great circle passing through it that is perpendicular to the Galactic Equator and passes through the north and south Galactic Poles The Galactic longitude l of the source is measured from the zero point eastward to where this circle intersects the equator The Galactic latitude b is measured along the second great circle for north and 7 for south 42 Sidereal Time To make use of these coordinate systems we need to synchronize watches 7 that is to place the celestial objects in the sky as a function of some time system A way to do so is to determine when an object is on the local meridian Consider the Earth at position E1 on the diagram The star shown is on the meridian at midnight by the clock But three months later when the Earth reaches position E2 the same star is on the meridian at 6 pm by the clock Our clocks are set to run approximately on solar time sun time But for astronomical observations we need to use sidereal time star time Consider the rotation of the Earth relative to the stars We de ne one rotation of Earth as one sidereal day measured as the time between two successive meridian passages ofthe same star Because of the Earth39s orbital motion this is a little shorter than a solar day In one year the Earth rotates 365 times relative to the Sun but 366 times relative to the stars So the sidereal day is about 4 minutes shorter than the solar day t i The local sidereal time LST is the sidereal time at a particular location It is zero hours when the Vernal Equinox is on the observer s local meridian and by de nition of the hour angle the LST is thus the hour angle of the Vernal Equinox ithat is if the Vernal Equinox is on the local meridian in two hours it will be Figure 16 Time and local meridian tvvo hours 30 degrees on the celestial equator west of the meridian and the hour angle will be 2 hours By the de nition of right ascension the LST is also the right ascension of any source that is on the local meridian Equivalently the hour angle ofa source is the LST minus its right ascension 43 Coordinate Transformations Although each of the coordinate systems has its use they do pose the problem of transforming from one to another Here are the formulae for that purpose First to transform from azimuth A and altitude a to hour angle h and declination 5 for an observer at latitude on the earth of d cos5 sinh cos a sinA sin5 sin sina cos cosa cosA cos5 cosh cos sina sin cosa cosA The inverse goes from hour angle and declination to azimuth and altitude cosa sinA cos5 sinh sina sin sin5 cos sin5 cos cos5 cosh cosa cosA 7cos sin5sin cos5 cosh From equatorial to ecliptic coordinates where the obliquity inclination of the equator of the earth against the ecliptic is 823 26 21448 the transformation is cos cosl cos5cosa cos sinl cos5sina cosgsin5sing sin cos5sinasin8sin5cosg where 7 and i are the ecliptic longitude and latitude respectively The inverse transformation from ecliptic to equatorial is cos5 cosa cos cosl cos5sina cos sinl cosgsin5sing sin5cos sinlsingsin cosg The previous two sets of transformations are relatively simple mathematically because all the systems are centered on the earth The conversion to Galactic coordinates does not have this attribute and is more complex There are a number of webbased coordinate transformation calculators that can be used eg httpnedwwwipaccatecheduformscalculatorhtm or httpheasarcgsfcnasagovcgibinToolsconvcoordconvcoordpl or one can nd details in Lang 2006 or Kattunen et al 2007 44 De nitions The most accurate celestial positions are obtained through very long baseline interferometry VLBI in the radio accurate to a milliarcsec or better Therefore in 1997 the IAU adopted the International Coordinate Reference System ICRS based on VLBI coordinates for 212 radio sources Because these objects are extragalactic and indeed very distant they should have no proper motions and the definition should remain in place indefinitely We will discuss VLBI position determination in Chapter Z The ICRS is transferred in the optical to 118218 stars all with accurate measurements of positions and proper motions based on the Hipparcos satellite data 45 World Coordinate System As astronomy becomes more and more panchromatic it has become a necessity to have an efficient method to place an image of a field accurately on the sky and in the appropriate equatorial coordinates so it can be matched with identifications at other wavelengths To implement this capability suitable information is now placed in the FITS header of many types of astronomical data A common example is to link each pixel in an astronomical image to a specific direction on the sky such as right ascension and declination In general the FITS world coordinate system WCS of an image is defined by keywords in the FITS header The basic idea is that each axis of the image has a coordinate type a reference point given by a pixel value a coordinate value and an increment A rotation parameter may also exist for each axis A common of set of keywords used to de ne the WCS of an image are CRVAL n coordinate value at reference point CRPIX n array location of the reference point in pixels CDELT n coordinate increment at reference point CTYPE n axis type 8 characters CROTA n rotation from stated coordinate type The FITS WCS standard defined 25 different projections which are specified by the CTYPE keyword For a complete description of the FITSWCS projections and definitions see Greisen and Calabretta 2002 Calabretta and Greisen 2002 and Greisen et al 2006 There are a number of software packages that aid the astronomer in accessing the astrometric information using the WCS of the image or to write the WCS of an image to the header A few of the most commonly used packages are WCStools WCSLIB IRAF and packages in the astronomy IDL library If an adequate WCS does not exist for an image the basic steps are to 1 Read in the FITS image and its header 2 Find all the stars in the image 3 Final all stars in a reference catalog in a region of the sky where the image header says the telescope is pointing 4 Match the reference stars to the image stars 5 Using one of the above WCS software packages perform a fit between the matched star s pixel and reference positions Write the resulting WCS information to the header Unless very accurate pointing data can be associated with the data obtaining this information often requires conducting an automated search to match objects detected in the image with a catalog of objects on the sky If this search is to be fast it cannot proceed by brute force One strategy is to sort the objects in the image in order of decreasing brightness and then to match them with a list similarly sorted of catalog objects in the same region of sky Once a match has been achieved it is usually necessary to correct the image data for distortions and other effects that might make the coordinates less accurate away from the specific region of the match 46 Changes in Celestial Coordinates Unfortunately we are not done The linking of the equatorial coordinate system to the celestial equator and poles means that the grid of the system shifts due to a number of motions of the earth In addition to use the system accurately there are additional effects to be taken into account Fortunately all of the items listed below are well understood and with care can be compensated sufficiently well that they do not interfere with obtaining accurate positions for any objects we wish to observe 461 Precession Because the earth is not perfectly spherical the gravitational fields of the moon and sun exert a torque on it The result is that it precesses like a spinning top its axis describing cones with a half angle of about 2350 centered on the north and south ecliptic poles A precessional cycle takes about 26000 years Similar torques exerted by the other planets add an additional precession term with a period of about 41000 years The planets also result in a change in the obliquity tilt of the poles ofthe earth over a range of about 215 to 2450 Of course these effects also change the direction of the celestial equator and parameters that depend on it drift For example the zero of the equatorial system is set by the intersection of the celestial equator and the ecliptic which is currently drifting at about 50 arcsec per year Therefore coordinate systems defined by the Vernal Equinox must be specified for a certain date The specified year is called the Equinox not epoch as is commonly assumed We currently usually use coordinates for Equinox J20000 but one will find coordinates for equinoxes of 1900 1950 and so forth Calculators such as httpnedwwwipaccatecheduformscalculatorhtm or httpheasarcgsfcnasagovcgibinToolsconvcoordconvcoordpl are convenient for converting from one equinox to another 462 Nutation On top of precession the gravity of the sun and moon cause a number of smaller shortperiod motions the largest is 186 years long These terms are called nutation after Latin for nodding Like precession they can be determined and compensated accurately 463 Parallax Of course one of the primary goals of astrometry is to measure parallax and determine stellar distances For nearby stars this effect must be accounted for in any accurate position determination The Hipparcos satellite has measured accurate parallaxes for virtually all nearby stars that are reasonably bright 464 Proper Motion Nearby stars also move measurably across the star 7 in about 500 cases at a rate of l arcsec per year or more In these cases the coordinates need to be updated to the current date to have an accurate position for the star The Hipparcos satellite data have been used with earlier astrometry to provde measurements of proper motions Where there is a long time baseline or under circumstances permitting very accurate positional measurements they can be measured by other means also including to much fainter levels than are reached by Hipparcos 465 Refraction The index of refraction of air under standard conditions is about 10003 and diminishes with reduced pressure Therefore light from outside the atmosphere that enters obliquely is bent slightly Objects that are really 35 arcmin below the horizon will appear in visible light to be right on the horizon if we could see that clearly The refractive index is significantly smaller in the near infrared and at longer wavelengths reducing this effect 466 Aberration Because of the finite speed of light the apparent position of an object is displaced in proportion to the transverse velocity of the earth moving through space relative to the vector of the beam of light Much of this effect is periodic over a year and can have an amplitude as large as 20 arcsec It should not be confused with parallax which becomes larger the closer the object Aberration occurs with the same amplitude for all objects in the same direction 5 Astrometric Instrumentation and Surveys Astrometry is an important branch of science in its own right besides providing coordinates for us all It is the foundation of our distance scales a basic way to identify members of populations of stars has provided fundamental evidence for the existence of a supermassive black hole in the Galactic Center and is a promising approach to search for planets around other stars just to name a few examples The most straightforward method for measuring stellar positions is by means of images of the sky The accuracy of the position measurement of an image can be estimated roughly as the full width at half maximum divided by the signal to noise This guideline clearly breaks down at low signal to noise if you do not detect a source you cannot locate it at all It also fails at very high signal to noise the underlying reason is that the further one pushes the position below the FWHM the better it is necessary to understand the structure of the image Fortunately it is not required that the image be perfect aberrationfree nor even that images being compared for example in a given field of view of an imager 7 be identical just very well understood Standard accuracy limits for photographic astrometry based on combining the results of multiple observations of the same field are about 10 milliarcsec mas To reach this level of accuracy requires that the telescope be very stable In addition from our discussion of issues like interpixel gaps and intrapixel response variations and their analogs for photographic plates it is very desirable that the image scale be large enough f of the telescope large enough that the image of a star is spread over many pixels A century ago the best solution was longfocus refracting telescopes but as the engineering of re ectors improved they proved more than competitive Although specialized astrometric telescopes are often used for large programs eg the 155 cm telescope with a at secondary ofthe US Naval Observatory with care good results can also be obtained with ones of more conventional design In the l9Lh Century visual astrometry was a central topic in astronomy A number of specialized telescopes and instruments were developed to allow accurate measurements such as transit telescopes sometimes called meridian circles to mark the passage of a star over the meridian heliometers telescopes with split lenses to measure angular distances from their neighbors and various micrometer adjustable sighting devices integrated with eyepieces An ambitious program was initiated in the late l9Lh Century the Carte du Ciel to obtain allsky astrometry using the newly available highsensitivity photographic plates and eighteen identical refracting telescopes each with 30cm aperture In fact the project proved too ambitious and observations dragged out for more than 50 years by which time the product was becoming obsolete 254 printed volumes in various formats This effort was replaced by Hipparcos and Tycho the latter of which has a similar limiting magnitude about llth and number of stars to the Carte du Ciel The work invested in the photographic effort has assumed new importance however because it provides a long baseline for determining accurate proper motions Astrometry based on the Palomar Optical Sky Survey POSS has been pursued to provide good guide stars for HST Guide Star Catalog Version 22 provides allsky measurements to accuracies of 200250 mas and down to about 193911 magnitude More recent astrometric data have been obtained with electronic imagers One noteworthy examples is the Sloan Digital Sky Survey SDSS in the optical The SDSS uses a 25m Ritchey Cretien telescope with a wellcorrected 3 diameter field Toward the edges of this field there are 22 400X2048 pixel CCDs optimized for astrometry These detectors avoid saturation on bright stars through faster readout and neutral density filters up to SDSS r N 8 and can detect stars down to r N 17 well Therefore they include a huge number of stars from the Tycho2 catalog described below and other astrometric catalogs to establish the overall reference frame and then extend this frame to their detection limit The SDSS photometry CCDs saturate at r N 14 so the astrometric reference can be transferred to them using stars between l4Lh and l7Lh magnitude and the photometry CCDs extend the astrometry to about r N 22 We describe the reduction steps since they are typical of position determination with digital imaging data To start the CCD data are run through a standard reduction computer pipeline which carries out the steps we described earlier in this chapter to obtain highquality images Positions are then measured off these fully reduced images First the images are smoothed to minimize noise artifacts to avoid degrading the resolution the smoothing length is adjusted according to the image sizes on the data frame The pipeline divides each physical pixel into 3 X 3 subpixels and quartic interpolation is used to estimate the peak position of the image within the subpixels of the peak physical pixel This result is compared with a first moment position calculation the sum of the signal times the distance from some fiducial point A number of additional steps correct these estimates for possible biases The resulting astrometric accuracy is 50 7 100 mas the latter at the sensitivity limit ofr N 22 Pier et al 2003 Another survey that provides accurate positions is 2MASS The 2MASS survey uses 2 arcsec pixels but with multiple sightings of each source It has proven possible to use these sightings to obtain accurate astrometry errors of N 80 mas relative to Tycho2 from the composite images down to K magnitude of about 14 This accuracy is achieved by modeling the positions of the Tycho2 stars as detected by the 2MASS cameras to identify and correct a number of error sources such as wandering of the telescope pointing and drifts in the image distortion These methods work because 2MASS has a large instantaneous field of view 85 arcmin on a side and was scanned rapidly 7 1 arcmin per second so very large areas were covered quickly compared with the time over which the potential error sources change Thus a large number of astrometric calibration stars could be tted together to obtain an accurate astrometric solution and determine the necessary corrections to the true positions Hipparcos is currently the ultimate astrometric reference source for the optical range The instrument concept was very different from the historical imaging approach It had a Schmidt telescope that included a beamcombining mirror that superimposed two fields of view about 10 in size and 580 apart on the sky The detector was an imagedissectorscanner basically a photomultiplier with the ability to place its sensitive field anywhere over the large sensitive area that filled the usable focal plane ofthe telescope A fine grid was placed over the field viewed by this detector with alternating opaque and transparent bands The satellite was put into a slow roll causing any star image to conduct a controlled drift across the grid The region around a known star was isolated with the image dissector scanner with a field diameter of 38 arcsec The resulting modulation of the star signal as the telescope scanned it over the grid produced an oscillating signal A similar signal was produced by a star in the second field of view differing by 580 in placement on the sky The phase difference between the two signals could be analyzed for an accurate determination of the apparent angle between the two stars These relative positions are ambiguous at the level of the period of the grid 12 arcsec but previous measurements of the stellar positions just to a modest fraction of an arcsec permit this ambiguity to be removed Positions of more than 100000 stars were measured complete to V 73 to an accuracy of 7 1 7 3 mas in this way with proper motions based on comparison with earlier astrometry such as the Carte du Ciel typically accurate to 1 7 2 mas Comparing these numbers it is clear that the currentepoch accuracy of the Hipparcos coordinates has degraded significantly due to the uncertainties in proper motion 7 the epoch of observation is 199125 so by 2011 typical errors will be about 40 mas The satellite carried a second instrument that gathered astrometric data to an accuracy about 25 times lower but on more than two million stars It also obtained homogeneous B and V photometry These measurements are contained in the Tycho2 catalog the current best reduction which is 95 complete to V 115 with positional accuracies of 10 7 100 mas better for brighter stars and proper motions typically accurate to 1 7 3 masyr The next step for space astrometry is the proposed European Space Agency GAIA mission GAIA goes back to image analysis but in a grand way As with Hipparcos a key element is for the spacecraft to roll slowly and for images from two widely separated areas on the sky to be brought to a single focal plane At the focal plane GAIA will have large number of CCDs aligned so the stellar images can be tracked across the detector through timedelayintegration TDI That is the signal charge is shifted by the CCD charge transfer structure so it just follows the star image and collects all the charge from it as it drifts across the detector array Because GAIA will use so many detectors and in a high performance mode it can achieve a huge gain over Hipparcos It is expected to reach 201h magnitude with positional errors of a few hundred microarcsec pas and to make measurements to about 4 pas at l2Lh magnitude the sensitivity limit of the Tycho catalog Perryman et al 2001 NASA has been evaluating another and complementary approach to highaccuracy astrometry the Space Interferometer Mission SIM SIM is a 6meter baseline interferometer An interferometer is very sensitive to pointing because a slight change in angle toward the target changes the path lengths to the inputs and therefore changes the phase of the pattern of interference between the two beams The orientation of SIM in inertial space will be measured to pas accuracy using interferometer observations of a bright star in the same direction as its science target For simplification we can then consider its pointing to be locked to pas accuracy in an inertial frame The field of its science interferometer is directed toward the target by a steerable mirror which is then locked in place Once inside the instrument the light from the target is taken to delay lines which compensate for the path difference in the interferometer as a result of the target acquisition Once compensation has been reached the delay lines are also locked The beams are then combined and the fringes and their phase measured to determine the relative direction to the target The final astrometric errors are expected to be about 04 pas per axis for a single measurement sequence Further Reading Howell Handbook of CCD Astronomy 2quotd edition 2006 Good general coverage of CCDs and their use in professional astronomy Kovalevsky J 2002 Modern Astrometry 2quotd Ed Springer Kovalevsky J and Seidelmann P K 2004 Fundamentals of Astrometry Cambridge University Press Lindegren L 2005 The Astrometric Instrument of GAIA Principles in Proceedings of the GAIA Symposium The ThreeDimensional Universe with GAIA ed C Turon K S O Flaherty M A C Perryman ESA SP576 Smart W M and Green R M 1977 Textbook on Spherical Astronomy 63911 edition Cambridge University Press a classic Starck and Murtagh Astronomical Image and Data Analysis 2006 References Calabretta MR amp Greisen EW 2002 quotRepresentations of celestial coordinates in FITSquot Astronomy amp Astrophysics 395 10771122 Greisen EW amp Calabretta MR 2002 quotRepresentations of world coordinates in FITSquot Astronomy amp Astrophysics 395 10611075 Greisen EW Calabretta MR Valdes FG amp Allen SL 2006 quotRepresentations of spectral coordinates in FITS Astronomy amp Astrophysics 446 747771 Kattunen H Kroger P Oja H Poutanen M amp Donner K J 2007 Fundamental Astronomy 5Lh edition Springer Lang K R 2006 Astrophysical Formulae 3rd Ed Springer Pier J R et a1 2003 AJ 125 1559 Perryman M A C de Boer K S Gilmore G Hog E Lattanzi M G Lindegren L Luri X Mignard F Pace 0 and de Zeeuw P T 2001 AampA 369 2001
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