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# OBSER. ASTROPHYSICS PHYS 134

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This 121 page Class Notes was uploaded by Hailey Halvorson on Thursday October 22, 2015. The Class Notes belongs to PHYS 134 at University of California Santa Barbara taught by Staff in Fall. Since its upload, it has received 68 views. For similar materials see /class/227128/phys-134-university-of-california-santa-barbara in Physics 2 at University of California Santa Barbara.

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Date Created: 10/22/15

Physics 134 Observational Astrophysics Lecture 239 April 3 2008 Detectors and Image Displays mmmlul 4i 4 pk Review Last Lecture CCDs are low noise high quantum efficiency photon detectors made of silicon Charge is generated by the photoelectric effect transferred along columns by applying voltages and read out as a digital number The array of numbers can be displayed as an image or used as input to an arithmetic function Outline Announcements Observing opfianai af 830pm fonighf in BRDA3402 And Final Presenfa fions on June 2 3 6pm and June 3 1 35pm Observing af LCOGT at 9pm on April 15 What are we going to measure with a CCD Properties of CCDs that affect astronomical measurements Displaying Images from a CCD Camera the LUT L Project 1 Supernova Light Curves Magmtude o n 300 350 so 1 150 Days after Maximum Magmtude Considerations in CCD Operation 1 Gain eDU Gain is the number of es the AD requires to register an additional x gym 15ml count High gain produces digitization noise example High gain produces a larger dynamic range examplez 16bit ADU outputs numbers from 0 xiii quot CCD Binning xl J Vb IIIIIII g CCD Window 3ln1l ILFast Slarl Avl Slam Row Numher m cm llmnuar or Rnws Number n1 readout amuli ers l 2 Heaumu ampli ers Id wmdnw Delecwr lemperalum Frame 1 5620 CCD subsystem Elnseu B 4173340211 lzlnn lnnnzmauas l1 1 U91 7 ear 9th 13s NunImage Pixels l low DESIIECI 39n n 3 2m 4095 l l Emse cccc m 1 l Tn disk m we neannut mm mry W ss ata7 7zsieng n annutm am Fi esl SI ulmrnp I n quotJuggle cco Sum l can Gun Ampunm Dismiss to 65536 Lower gain is generally better as large as the full well of a pixel is sampled Considerations in CCD Operation 11 Quantum Ef ciency Quantum efficiency is the fraction of indicent photons detected It is a function of wavelength Depends on the type of CCD backfront illuminated thinned coatings etc ls closely related to the system throughput which can be measured using calibration stars and determines how long you need to observe HlRES LRlSrR suu l suu l lUUU Considerations in CCD Operation III Read Noise All electrons in the bucket look alike Whether from a cosmic source or thermal noise Distinguish signal from noise statistically Conversion of the analog signal to a digital number is not perfectly repeatable The electronics may also introduce spurious e s They produce an additive uncertainty in the final output value of each pixe Measure it With zero sec exposures This Read noise feW e piX todaygt but 10 years ago CCDs had Rn 50 e piX At xed array sizegt faster readout times lead to higher read noise swag segg em 9sz p293 Considerations in CCD Operation IV Dark Current v ma a EEthamaica Zuzuznzmzmzanzajsm 15D 13 m E 1 2m 2 2 W Every material above absolute zero is subject to thermal noise Dark current consists of e s freed from the valence band of the Silicon by thermal motion Measure it using long expsoures With the shutter closed 0 It is sensitive to temperature Operate at 100 C inside devvar filled With liquid air or N2 Considerations in CCD Operation V Pixel Size Pixel size affects spatial resolution Nyquist sampling Pixel size affects how much charge one bucket can hold full well capaa39ty blooming Onchip pixel binning allowed AD Saturation maximum DU 2Nblts 2 x 2 Pixel Binning ReadOut Stages Serial slim summed Regishar Ptxei E ii Parallel shin Register ml Hi gt II idi Considerations in CCD Operation VI Linearity Does the signal generated increase linearly with the number of incident photons How could you tell Most CCDs used at observatories respond linearly to photons up to N 40000 DU The full well capacity is usually several times higher than the point at which the reponse becomes nonlinear Charge Cnupled Device CCD Linearity Plot Llnear LeastSquares Regression Analysls 2000 Signal Level 1600 Data Point 1000 Figure 1 Mean Signal Analog Dlg al Units 2 00 LD 60 20 30 40 Sr Exposure Tlme Seconds Considerations in CCD Operation VII Charge Transfer Ef ciency CTE Charge Iransfer Efficiency describes how many electrons are lost each time charge is transferred High CTE required for photometric accuracy Charge Transfer Ef ciency in CCDs WW l i a Farthest Pixel i 7 I Homework solution was 99995 to get 90 of the photons after 2048 transfers 1 Nearest Pixel Figure 8 Considerations in CCD Operation VII Array Size XMosaicrOverview QLrTool Array size affects how much sky you monitor at once 2k X 4k is common Mosaic to 8k X 8k or more Array size affects readout time homework solution Nickel Telescope CCDImager Thinned Loral 2048 by 2048 CCD 033 5111 quot 1 39 31115101 um xl slow 20 86 116 IOOO 39 39 p 39 x1 fast 20 178 017 1000 o arcsecondSplxel x2 slow 17 803 021 1000 x2 fast 8 140 005 1000 39 What IS the eld Of VIEW x4 slow 17 100 006 1000 e x4 fast 8 148 002 1000 Keep countspix below 50000 to ensure linearity Read out with a single amplifier Gain 154 eADU 39 Dark Current 125 C 02 epiXhour CTE very good Scaling Algori rhm S rar r up ds9 How are 65536 dara values fH in ro 256 shades of gray Announcement It is now possible for each student to create hisher own account for storing data relating to your projects Each computer now has 2 new accounts Login name astronomer1 and astronomer2 Password physics To personalize one of these for yourself 10 Find a computer and account astronomer1 or astronomer2 that nobody else is using if you can log in using the physics password then nobody else got there first 2 Change your password to something else using the unix passwd command just type passwd and you will be prompted for old and new passwords 3 Change to your home directory by typing cd and ltentergt 4 Type mkiraf This will among other things create an lFlAF data directory tagged with your new user name such as irafimdirsastronomer2 5 Type chmod 755 irafimdirsastronomer1 or astronomer2 This changes the permissions on your new data directory so that it can be read by anybody but it can be written only by you Physics Probability expectation value mean median quartiles standard deviation normal amp Poisson distributions CCD noise sources capabilities limitations optics telescopes imaging Earth s atmosphere Leastsquares fitting Q 6qu Fitting in your projects light curves galaxy ot brightness profiles Pymquot Terminology amp nomenclature properties of stars clusters galaxies SNe pulsating stars many other tidbits Statistics Least Squares Fitting AGAIN See Taylor Ch 8 o Line Fitting by Least Squares Suppose we have a bunch of paired measurements 271141 and also that we think a good model of these should be 1 A3 B Also suppose that the errors in the x1 are negligible but that the ya have signi cant errors How may we choose A and B so that the model agrees with the observations as well as possible One answer is to choose A and B to minimize the summed squared deference between the model and the observations min Edyquot model2 wrt A B 0139 IZHE auwsesamx jasamm min Eggquot Axu B2 wrt A B Fitting a Constant 10 Let s nd the constant value A that best represents a bunch of observations in a leastisquares sense min 211 42 5 Equation to solve is A2111 21 or 7 2 1 7 2 A i 2111 i N 0 It s the average Fitting a Constant with Uncertainties 10 E uppose each individual measurement yi has an uncertainty E i I Now 01 associated With it E E E E yrAf min 2 03 y 7736 Equation to solve is A27 2 or 2 2 7 11 01 A 2003 This is a weighted Menage With the Weights proportional to lUf 0 Fitting a Straight Line Suppose We have a bunch of paired measurements 951111 x and also that We think a good model of these should be O Q y A BI 0 Choose A and B to minimize the 2x 0 s mmed squared difference between the model and the k observations or y91123X 5 min 2111 7 A 7 Baci2 Wl t A B so 7221117 A 7 Ba 0 3 72211 7 A 7 Bac aci U or see Taylor s notation Ch 8 p 184 0 A21 3295 By 0 A295 32x2 21y Uncertainty in the Fined Parameters The solution to these linear equations is Taylor p 184 7 21224 7 21211 A i f 7 N211 7 2121 B i A Where A NEI2 7 Eat2 Uncertainties If the observed yi all have uncertainties of 01 the uncertainties in the estimates for A and B are Fitting a Polynomial 10 Let s t a polynomial with more terms yABacCac2 Differentiate Wrt each parameter ABC Set all derivatives to zero yielding equations that look like y91 123x42x 6 A21 1329 02x2 A295 32952 02953 A2952 32953 02954 zoosgy I Smag 25 log Ltl L0 25 log2 t lh 06 tlh Days Since Explosion 5ma90Bt mag mag0 Bt A Bt Than h 06 I B Radial Scale Length of Spiral Disks Suppose we model the radial brightness distribution in a spiral galaxy as Rl0expth R 2 73 22 39 Howtoestimateh R g 24 Converttomagnitudesietakethelog E 26 J j 28 7 1 25 ogRl0 25ln10 th R 3 magRmagR 0109th R g 00 mgmw w vvgg azy So fit the magnitude vs radius to 05 w I IE v I 0 10 20 30 4 magR A B R r urcsec And get hR109lB Effective Radius Power Law of Elliptical Galaxies IR l0 exp FtR 1 we Vaucouleurs formula How do we estimate R 0 NGC 1399 Convert to magnitudes 1 3 I Y 4395 microns 6 mag 25 log IR mag mag0109 R In 2quot Change variables to z R quot4 E And then fit to E E mag ABz So B 109m 0 And R 5 1093 4 Scientific Method Hypothesis I Revision H Experiment I Measurement Cycle I Choose a measurement strategy Iquot 1 I Understand sources of error Design measurement to minimize these I Y Result acceptable N I I Means measuring stuff I Physics Physics Probability expectation value mean median quartiles standard deviation normal amp Poisson distributions CCD noise sources capabilities limitations optics telescopes imaging Earth s a mos39 here Leastsquares fIttIng lI 39J ds9 IFlAF DAOphot python 6qu Terminology amp nomenclature properties of stars clusters galaxies SNe pulsating stars many other tidbits In ClassExercise Confined to Quarters1 Statistics o Probability Y Rielative frequency de nition Suppose an experiment has 0 N possible outcomes and these are Pquot WWIUI iliigx xnm o Essential properties of probabilities P 2 0 an 1 Other de nitions of probability are possible and use describing nut relative frequency but rather a state of incomplete knowledge 0 Probability de ned on a continuous variable If the set of outcomes is continuous not discrete then we talk about probability density de ned so that the probability of nding an outcome between a and x d3 is given by fail M Pmdv Notice that in this common usage Pm is really the derivative of a probability you must integrate it over an interval to get a real probability I Probability of A 85 B two dj eren experiments The probability that the rst yields a ran A and the second yields B Is PA l B PAPBA If the two experiments are independent of each other this reduou to PA amp B PAPB This formula is very dangeroual endenne inappropriately leads to ludicrous mistake o Expectation Value Suppose there is a probability density a distribution P over a variable 2 so that the probability of observing a value between a and a dm is P dcc Now consider any function f For example 3 might be the velocity of a particle and f mmz might be its energy Then we use lt f gt to denote the expectation value of f namely the weighted average of f over all possible a weighted according to Par That is lt f gt If fm Padx 0 Mean Variance RM S Suppose we have N numbers 13 perhaps different measurements of some quantity The mean value is just the average 372 f is a measure of the typical value of the xi We also desire a measure of the central condensation of the 921 ie how closely the individual numbers cluster around the mean One such measure is the variance 02 de ned as 02 N11 55 239 A more intuitively useful quantity is the root mean square or RM S usually denoted by a and equal to the square root of the variance RMS E a 0212 This is a measure of the typical distance between each m and 1quot 0 Mean and Variance Given Probability Densities If we have a probability density PL39 we can compute expectation values lt a gt memda lt 23 lt 1 gt2 gt m lt ac gt2Pcrda This last quantity is also usually denoted by 0392 To distinguish this property of a theoretical probability distribution from the similar property of a set of measurements for the moment we will call the former thing 0 If we draw a number N of samples from the distribution Px we expect that as N grows large r gt lt x gt The law of large numbers RMS gt 012 a Gaussian Normal Distribution The Gaussian distribution is the familiar bell curve it is called norm because it is the most common one encountered in statistics It is usually written as 2 1 a G mm m This is de ned in such a way that lt35 gt 0 1 l l and ltm2gtltx f2gt02o 087 7 e 0 05 m a 06 r a E a 04 r 4 8 E 4710 02 7 7 020 05i0 O l l l 6 4 2 4 2 O DEV ATION Gaussians turn up in many context We will counter them most often as describing the distribution of the average of many small noise terms each randamly distri buted with mean value equal to zero and a nite a fact the central limit theorem assures that the distribution of such an average is always Gaussian no matter what the shape of the individual distri butions so long as the RMS is nite Moreover when all of the terms have the same distribution terms the RMS of the average is 0 Full Width at Half Maximum FWH M In addition to the RM 5 another useful measure of the width of a probability distribution is the Full Width at Half Maximum or FWH M This is the width of a distribution Px measured across the points where the value of P is duh onehalf of its maximum It is a handy thing because from a graph of Pc one can simply read it off For a Gaussian distribution FWHM 2x21n2a 2350 04 39 39 39 0 11 I 03 0 i4 39 o 110 013 f 0039 A o Multivariate Distributions and Moments Probability densities are often de ned in terms of more than one independent variable Thus just as Pacdr expresses the probability of nding a value between 10 and L39 dx Pm ydmdy is the probability that the rst value in an ordered pair lies between an and w dos and that the second value lies between 3 and y dy This notion can be extended in the obvious way to as many dimensions as you like The notion of expectation values may then be extended lt fa1aN gt ffx1mNPm1mNda1dmN In particular we will encounter various moments of such distributions equal to expectation values of products of the coordinate values viz lt 1111302 gt lt m m3m 2 gt or whatever The secondorder moments lt gt lt wimj gt will appear prominently soon P1 y PwPy a product distribution if and only if the variables a and y are independent of one another see the discussion of Probability of A amp B above 0 Errors Adding in Quadrature Suppose we measure a quantity that is the sum of contributions several sources each with its own average value and each af icted with independent noise having zero mean and a specified RM S Thus Q 2 qi where qz qio 65 and we have lt 6 gt 0 lt 62 gt2 of and lt EiEj gt 0 What then is lt C2 gt and what is lt 62 lt Q gt2 gt The expectation of a sum is the sum of the individual expectations so ltQgt Ziqw The expected variance is 2 022 lt 21 a gt 22 lt zzgt l Zi2j ilt6i j gt Zi 7i2 Thus the expectation values and the variances add linearly but the RM S values add in quadrature VERY IMPORTANT This only follows if all the 6 are independent Starting up lFiAF Image Reduction and Analysis Facility In an X window illhis gels you lo your home director you should see several files including oginc which sels environmenl variables and loads packages on iraf slal1 p xglerm amp slans up an xglerm window Many of iraf39s funclions do nol work properly unless cl is run from an xg erm Is In lhe xglerm window dw slans dw in a separale window cl command languagequot lhis slaI1s IRAF Vou should see El stud Ilt NERDIRRFNET PErlRRF ReVlSlan 216 rpl Nav 30 152705 MST 2 1px ls the RELERSED vepSlap a anr v21a suPPDrclHQ PE 55s melcame La lRRF detalled lnFurmaLlan ahaut a eammand have pelp ltcammandgt pw cammand ap laad a pae age have lts name T Pe Package ar lagaul ca get But a the EL rape nems La what s new ln the vepSlan a the system gag are uslng vlSlc httplrafnet 1 gal have questlans Dr La repart prahl rpe Fallamlug cammands ar paelages are currently deflned adccdram clcsucll language naaa saftaals Ella gemlnl l sts ahsalete stsdas datala gmlsc mlaals plat s st dhms lmages nnlsc p La eclgt I Ta llst the avallahle eammanas have 7 ar 77 m7 ems in get in run a La exit a Fwd wt m nmamp Each entry is a package containing a number of iraf tasks each of which allows you to do something more or less useful A few iraf general features To quit iraf type logout To load a package of tasks type the package name Unixstyle redirection gt lt and pipes I work inside the cl Old commands can be retrieved edited and rerun using the uparrow key or the historyfeature history or A cd and Is commands work inside the cl The wildcard character usually works to help select files There is a lot of online help Type packagename to get a short synopsis of the tasks it contains Type phelp taskname to learn about a task and allow scrolling around in the help file A useful alias which is automatically defined is imdir which is set to imdir lirafimdirsyourloginname Most tasks have hidden parameters initially set to plausible defaults Examine these using lpar taskname edit them using epar taskname Tutorials Jeannette Barnes IRAF for Beginners link on the web site UW Astro 480 lFlAF series esp 1 links also on the web site So Let s explore one of the more useful tasks imexamine Type the following cd imdir change to your image directory data0 subdir cd data0 subdir data0 imexamine M16R 1 put the M16Rfits image into frame 1 of the ds9 display Notice that the fits suffix is optional invoke the imexamine task Click on the title bar of the ds9 display to highlight it Now place your cursor on the medium bright star at 524616 The cursor should now be a small circle Center the cursor on the star and type c Stands for column plot A new graphics window should appear showing a plot with a big spike our star at row 616 Type I ell Stands for line plot You get the same thing but spike at 524 So far kind of hohum But now for something really cool Type r Stands for radial plot This finds the center of the star nearest the cursor plots brightness vs radial distance fits a Gaussian plus pedestal to the result and prints a bunch of useful info to the graphics display BTW by clicking on the ds9 display you put yourself in image cursor mode To get back into command mode leave the cursor on the ds9 display and type q Now if you highlight the xgterm further commands will behave properly Doing stuff in the wrong mode causes many problems SAOImage dss QEJE M m 2m a m mm Ml ms R mm mumquot was ha M at w W WW The radial plot you just did should look like this 5 mum 38 Stellar magnitude peak height of 39 G 39 ADU arbitran zero aussuan several estimates total counts under backgroun f FWHM d Gaussian ADU Intensity ADU imexamine can do a lot of other things To get a list in the xgterm type phelp imexam Task names do not have to be complete merely unique Scroll down with d uP with u D P quot MWP39mm full page down with f up with b prrt re aperture ara erame m aux Far tua easmars e e1 e2 11 12 umquot you nd thIs Flt Sausslan ta Image lines Fit in Sausslan ta Image calumns Lme ht Statistics tmaaeseettar quotPixels mean metlan staueu mlquot max Next Frame ar Image nuerpiat Prevlaus Frame ar Image mu Radial Praflle plat uttr Flt and aperture sum values Surface plat nutput Image centered an eursar parameters autput reautput riautp entered ueetar plat Fram tua eurser pastttars Vectar plat aetueer tua eursar pastttars laaflle a line lxval xarlan uartatr dx nu r theta Buick BausslanMaffat radial Praftle plat and Flt Try some of these out Get familiar with how they work Quite a few of the upcoming nomeworK 39 39 39 39 39 anv 50 rY lpar imexam Iook at imexamine39s many hidden parameters Physics Probability expectation value mean median quartiles standard deviation normal amp Poisson distributions lt3 lies 39 39 escopes imaging Earth satmos here Leastsquares fIttIng p CCD calibration lt IRAF using imarith or python 6qu Stellar magnitudes maybe rminology amp nomenclature pro 39 v e of stars I r stars In ny othertidbits Telescope filters CCD wdefects gt Rxy7t lt 1 dirt moths etc Si sensitivity etc AmplifierADC Bxy Gain G ADUe AXY ADU Expose for N pe NDe D k I trons Light from sky atimet I a 391 Ixy9 Photoelectrons n plxe pilotons per in 1 pixel Dxyt N S mes projected pixel 1XY7 RXy t 0Ise ou per sec Alxv lev G Dixv t Ixv7 Rlxwv t Rxv7 1x5y5quot Axy Dxy I RxyO ltRgt The process of getting a good flat image is a little bit involved Here we go imarith flat1 bias1 t2 subtract the bias result into a temp file imhead t2 lo look for exposure time It s 33412 sec imarith darkf b 02227 t3 This gives a biascorrected dark scaled by the exposure time 02227 33412150 which we can subtract from t2 imarith t2 t3 flat1bd This gives a flat that is corrected for bias and dark Now ideally our flat image would have an absolute calibration That is we would know how many photons per pixel went into the telescope and how many ADU were measured Then with a calibrated value for the gain we could determine the absolute efficiency of the system But we don t have all that information So for now we will settle for a calibration image that measures the xy dependence of the efficiency and that leaves the total ADU counts pretty much unchanged Thus we should normalize the corrected flatfield image so that its typical values are close to 1 imstat flat1bd Shows statistics of the corrected flat Mean 38663 imarith flat1bd 38663 flat1bdn bias and darkcorrected normalized flat Now finally imarith dat1bd flat1bdn dat1 cal the longsought calibrated image This is better than dat1bd but apparently not by a whole lot The lowresponse spots are less obvious but not gone The most important differences are hard to see they are fewpercent changes in the brightness of some things relative to others because of big blotches in the flat Now we know how to estimate the signal from a single star starting from a CCD Image of the star and a lot of calibration images How reliable is this estimate That is what number should we put in the statement Star Signal 12292 ADU Recall that we identified 3 sources of noise in slide 2 Noise associated with the photoelectrons Star Sky 039star osky Noise associated with the dark current odark Noise associated with the amplifierADC oread Notice that all of these noise sources are independent of one another The Read noise is a function of the electronics and knows nothing of fluctuations in The star signal The dark noise is a function of the CCD and knows nothing Of the read noise And so on Therefore to estimate the total noise we can Add these 4 terms in quadrature 0393 tot c tar osky odark mead2 So how do we estimate these individual noise terms Read noise is the easiest The noise we measure is the sum of the noise values for each of the Npix pixels inside our photometry circle Also the noise on one pixel is pretty nearly independent of the noise on its neighbors But we know Stats I the sum of a lot of identical independent noise terms results in a Gaussian noise distribution with or ad E opixel Npix opixel 2 For this camera opixel is about 2 ADU so oread 24 349 1400 ADU For later convenience let s convert this to units of e assuming the reciprocal gain is 6 eADU to get 6read2 36 elADU 1400 2 50000 e The terms involving photoelectrons and dark current are a different matter These are examples of what Taylor calls counting experiments in which countable objects photoelectrons or dark current electrons arrive in the detector at some possibly unknown but definite average rate Such processes are modeled using the Poisson distribution They result in noise that is called Poisson noise or sometimes shot noise o Poisson Distribution The other distribution that we will encounter frequently is the Poisson distribution Pv A This describes processes in which events photons arriving at our detector for instance occur at some rate A per unit time Let us count events for a time 739 The probability that we will count exactly 19 events in that time is given by the Poisson distribution Pk A7quot exp 739 This distribution has the properties that ie t e expected number of arrivals equals 739 times the mean rate A and also ltk T2gt AT ltkgt Thus the RM S is the square root of the number of expected events The Poisson distribution is asymmetrical because it must be zero for negative 6 For values of AT that are even moderately large however it becomes indistinguishable from a Gaussian that is peaked at k AT with 7 ATVA 04 I I 0 L21 03 o k4 39 ltgt x210 3 02 012 00 3 0 5 10 15 Now back to our star brightness measurement Since the star signal the sky signal and the dark signal are all things that accumulate at definite average rates elsec per pixel they can all be modeled as Poisson processes This means 68tal39 Nstar 74000 measured in e with 1G 6 eADU csky Nsky 900000 The indivisible units we are counting are electrons not ADU odark Ndark 8700 Also Nsky and Ndark are summed over all the pixels in our photometry circle So at last we can finish the noise estimate getting otot osmr 0sky2 odark oread 2 2 Nstar Nsky Ndark oread 2 2 Nstar Npixsky dark O39rpix 2 12 where 6I9pix is the variance in e from reading a single pixel This is essentially Howell s CCD equation from his p 73 What about the numbers For this star we get Signal 12300 ADU 6 elADU 74000 e otot 74000 900000 8700 50000 102015 170 ADU star sky dark read The SN ratio is thus about 73 Notice that in this case the noise is completely dominated by the sky contribution It would be highly advantageous to make this smaller and the read noise as well by reducing the number of pixels used in estimating the total starsky signal The notion ofa magnitude scale to gauge the brightness of stars dates back to Hipparchus 150 BC who graded the visible stars into magnitudes according to their apparent brightness to the unaided eye The brightest stars in the sky were termed 15 magnitude and the faintest ones visible were 6 quot magnitude It turns out that the eye perceives equal multiplicative factors of energy ux as being equal steps in brightness and also that a differences of5 magnitudes on Hipparchus s scale corresponds pretty nearly to a factor of 100 The modern magnitude scale therefore de nes the apparent magnitude difference between two obiects as m1 m2 25 log uxflfluxz note log base 10 not In The zero point of the magnitude scale is defined in various ways one commonlyused de nition sets the magnitude of the bright star Vega equal to 00 in all wavelength bands On this scale one gets the following approximate Magnitudes for various astronomical obiects Sun Full Moon Sirius Vega Visual Limit Pluto FaintestHST 266 126 14 00 60 14 31 Magnitudes of stars are wavelengthdependent and are defined for many different wavelength bands Calibrated magnitudes using some standard filter and detector are usually indicated by the name of the filter eg UBVRl ultraviolet blue visual red infrared Johnson or Bessell systems bvyHbeta Stromgren system etc A color indexor just color is the difference between magnitudes measured with different filters hence a measure of the ratio of fluxes at different wavelengths They are denoted with an obvious notation The most commonly seen are BV and UB For both of these cooler redder stars generally have more positive color indices The magnitudes we measure are apparent they indicate a combination of the intrinsic brightness of the star and its distance The absolute magnitude is the magnitude that would be observed if the star were at a standard distance of 10 pc Absolute magnitudes are usually denoted by eg M where the subscript indicates the filter Distance Modulus Light intensity falls off as the inverse square of the distance Suppose a star has absolute magnitude M and lies at a distance of D pc The flux we receive from the star is then F F p10 392 Convert both sides to magnitudes take the log and multipy by 25 to get m M 5 ogD 5 where m 25 log F o The quantity 5 ogD 5 is called the distance modulus It is measured in Magnitudes and is a logarithmic measure of distance Distance Modulus Distance pc 0 10 5 100 10 1000 20 10 5 etc Telescopes are the most essential tool of the astronomer but we are not going to say very much about them For our purposes a telescope is any optical system that images the incoming plane wave from an unresolved obiect such as a star into a small point located on the telescope s focal surface One can then use a magnifier usually called an eyepiece to examine the image visually or place a piece of photographic film in the focal surface to obtain a picture or as we do these days replace the film with a CCD or other electronic detector and obtain an image that way Incoming 50quot5 Starlight 09th l focal length For now there are only two numbers describing a given telescope that we care about the focal length f and the aperture A The focal length determines the scale of the image If 2 stars are separated by an angle 9 radians on The sky then their images are separated by f0 in The focal surface J I r 9f e V 139 4 I I I I A f quot1 A V V Since 1 radian 206265 arcsec 2 objects separated by s mm in the focal surface Are separated by 0 206265 If arcsec in the sky For a fairly typical small telescope f 8m The size of a typical CCD pixel is about 10 micron 001 mm The projected size of a pixel on the sky is then s 001 mm 206265 8000 mm 026 arcsec The aperture A of a telescope is important for two reasons First the collecting area of the telescope is roughly depending on the design equal to TCA and the amount of light energy captured per unit time is proportional to this area So larger apertures allow one to see fainter objects faster As we have seen SN often scales like the square root of the number of collected photons So Other things being equal SNA And in the same observing time with similar Instruments etc a 25m telescope can achieve the same SN as a 1m telescope on objects that are about 1 stellar magnitude fainter Notice that we have assumed the same angular size for the images Formed by the two telescopes Aperture also matters because it influences the Angular resolving power of the telescope Even With perfect optics and no atmospheric disturbance The wave nature of light prevents telescopes from Making perfect pointlike images This effect is Called diffraction of light The angular size in radians of a diffractionlimited image is roughly Equal to the reciprocal of the telescope s aperture Measured in units ofthe wavelength of light Thus The biggerthe aperture the betterthe resolution More accurately in ideal circumstances one sees A point source as a socalled Airy disk with the LIN I I IALI Diameter of the 15 dark ring equal to 122 NA radians Most often groundbased telescopes are limited by smallscale variations in the Refractive index of the atmosphere a phenomenon called seeing The spatial Scale of the refracting elements is typically 10 cm or so yielding distorted Images with angular sizes of typically 1 to a few arcsec at 7 500 nm To get the most information out of a fuzzy image it is desirable to use a pixel size That provides 2 or more samples across the width of the image To locate objects in the sky it is useful to define cele ial coordinates which are the astronomical equivalents of longitude and latitude on Earth longitude iightAscension measured in hours Norm Sou glatitude ieclination measured in degrees The celestial poles are the extensions of the Earth s rotation axis the star Polaris happens to lie near the North Celestial Pole The celestial equator results from expanding the Earth s equator to infinity The zero point of RA is defined as one of the two places where the Earth s equatorial Plane intersects the Earth s orbital plane ie the plane of the ecliptic It is almost Fixed relative to the stars so it rotates around the Earth once per day skyquot quot m yuu are I I w quota L 39 39 ulluugllyuul uni HourAngle the difference in RA between the position of a particular object and the RA that corresponds to your meridian at the moment This is defined so that an object s HA is negative when it is east of the meridian and positive to the west Sidereal Time the RA that coincides with your local meridian at the moment Other important terms 1w u r39w he 4 A telescope is no good unless it can be accurately pointed at a target in the sky and held in alignment in spite of the Earth s rotation The hardware that does this is the telescope mount These come in a bewildering variety of forms we will be concerned only with two broad distinctions Equatorial mounts have one axis of motion aligned with the rotation axis of the Earth so they can follow the stars using motion only along this axis They are convenient to use but difficult to build in large sizes The orientation of the field of view remains fixed as the telescope tracks the stars AltitudeAzimuth AltAz mounts rotate around one vertical axis and one horizontal axis In general motions along both axes are required to follow the stars They are however simple to build and can be made very large The orientation of the field of view rotates in a complicated way as the telescope moves to follow a star Ghaervng Tonight at LCQGT What Introduction to a variety of small telescopes up to 16inch In the raw celestial coordinates seeing wind shake magnitudes and more When 8001000 PM Tues 15 April hope your taxes are done Where Parking lot on N side of LCOGT building 6740 Cortona Dr Goleta Free advice DRESS WWRMLY and ER NILE FLASHUGHT You will be standing around In the wind burning very few calories for a long time A serious coat is a must Hat and gloves are not crazy If you are not too warm when you start you will be freezing when you are done 39 39 39 Here s a map Ignore the green arrow Take Los Carneros north from campus 1 block past Hollister turn L on Castillan After1 block turn L on Cortona Turn R into 3quot driveway Park in this lot and walk N to the back side of building to the telescopes The security lighting really stinks but the site is mighty convenient for fixing things Physics Probability expectation value mean median quartiles standard deviation normal amp Poisson distributions lt3 lies 39 39 escopes imaging Earth satmos here Leastsquares fIttIng p CCD calibration lt IRAF using imarith or python 6qu Stellar magnitudes maybe rminology amp nomenclature pro 39 v e of stars I r stars In ny othertidbits Telescope filters CCD wdefects gt Rxy7t lt 1 dirt moths etc Si sensitivity etc AmplifierADC Bxy Gain G ADUe AXY ADU Expose for N pe NDe D k I trons Light from sky atimet I a 391 Ixy9 Photoelectrons n plxe pilotons per in 1 pixel Dxyt N S mes projected pixel 1XY7 RXy t 0Ise ou per sec Alxv lev G Dixv t Ixv7 Rlxwv t 1x5y5quot Axy Dxy I Rxyquot To carry out this process we need Axy Observed ADU at each pixel on the detector Bias signal at each pixel Gain of electronic system in ADUe note reciprocal of usual gain t Exposure time in sec Dxy Dark current e per pixel per sec at each pixel Bxyt Response of the opticaldetector system to uniform illumination in the wavelength band actually used for the observation Bxv G t is specified by us when we take an image G is known with fair accuracy from laboratory tests of the camera and generally changes slowly with time timescale of years How can one measure Bxy Dxy Bxyt Bxy Bias doesn t depend on exposure to light or integration time on the CCD So this can be estimated by taking an image with the shutter closed no light and with an integration time of zero no dark current Dxy Doesn t depend on exposure to light but does depend on integration time dark current electrons are generated in the CCD silicon at a constant mean rate So this can be estimated by taking an image with the shutter closed but with a long integration time Note This image will be contaminated with the Bias pattern Bxy which must be subtracted in order to see the true Dark signal Bxy Does depend on exposure to light and moreover cares about the color This must be measured using a spatially uniform source of white light whatever that means observed through the same filters etc that are used for the actual observation Nonzero exposure times are required so the raw Flatfield images as such things are called must be corrected both for Dxyt and for Bxy in order to measure true values of Flxyt Calibration of R in absolute units photoelectrons produced per incident photon requires that the calibration light source be of known brightness in addition to being spatially uniform Now let s go through the process of CCD calibration using lRAF s imarith function Start xgterm ds9 and the iraf c as before cd imdir gets you to your image dir cd data1 and to the new data1 directory s so you can easily cut and paste big filenames Now we want to rename the images we are going to play with for typing convenience and so we don t accidentally overwrite something important imcopy bias080308 bias1 we will use this average master bias imcopy dark080308 darkf ditto for this flat imcopy flatR080308 flat1 This flat was taken with an R filter use only on R data imcopy sn2007uyR08030801 dat1 cut and paste this one Is See what you ve done bias1 is an honest bias frame so we don t have to mess with it So much for Bxy dark1 needs work because it has the bias signal in it Also it corresponds to an exposure time that may not be the one used in our dat1 image So do this imarith dark1 bias1 dark1 b subtract blast from dark1 call the result dark1b dark1b is actually t Dxy Now what is the exposure time for dark1 or dark1 b imhead dark1b long Show the fits headerof dark1 This contains a lot of info including the exposure time which happens to be 150s Now what is the exposure time for our data image dat1 lmhead dat1 lo long can be abbreviated gives exptime300s So lmarith dark1b 20 dark1bs We can subtract this from dat1 So let s do that imarith dat1 bias1 t1 Put result in a temporary file because imarith is dumb imarith t1 dark1bs dat1bd Result dat1bd is corrected for bias and dark This is a good time to see how we are doing display dat1 1 The display task just displays things scaled somehow display t1 2 biassubtracted data into frame 2 display dat1bd 3 bias and darkcorrected data into frame 3 Not bad The background level about 450 ADU is now probably real sky brightness not bias and most but not quite all of the isolated hot pixels have gone away To get a fairer comparison between dat1 and dat1bd try display dat1bd 3 contrast09 This forces the scaling to be similar for the two plots I found the value 09 by trial and error There are still some visible problems a few hot pixels and a few places eg near x380 y550 where the values are too small These are probably defects in the CCD where the sensitivity to light is smaller than usual To fix these we need a flat field The process of getting a good flat image is a little bit involved Here we go imarith flat1 bias1 t2 subtract the bias result into a temp file imhead t2 lo look for exposure time It s 33412 sec imarith darkf b 02227 t3 This gives a biascorrected dark scaled by the exposure time 02227 33412150 which we can subtract from t2 imarith t2 t3 flat1bd This gives a flat that is corrected for bias and dark Now ideally our flat image would have an absolute calibration That is we would know how many photons per pixel went into the telescope and how many ADU were measured Then with a calibrated value for the gain we could determine the absolute efficiency of the system But we don t have all that information So for now we will settle for a calibration image that measures the xy dependence of the efficiency and that leaves the total ADU counts pretty much unchanged Thus we should normalize the corrected flatfield image so that its typical values are close to 1 imstat flat1bd Shows statistics of the corrected flat Mean 38663 imarith flat1bd 38663 flat1bdn bias and darkcorrected normalized flat Now finally imarith datbd flat1bdn dat1cal the longsought calibrated image This is better than dat1bd but apparently not by a whole lot The lowresponse spots are less obvious but not gone The most important differences are hard to see they are fewpercent changes in the brightness of some things relative to others because of big blotches in the flat mn39ia lisn of the flux mmin ing an cuerall m To do this we draw a circle conceptually anyhow around 392 Hquot W W 39339 mquot the star of interest and add up all the ADU inside this circle This number is ADU T0 Star Npix avg sky Npix is pi r here r is the radius of our cirole To estimate avg sky we draw an annulus outside the circle and measure the average brightness of the pixels in that annulus in m wad a m 1 is v Now we know how to estimate the signal from a single star starting from a CCD Image of the star and a lot of calibration images How reliable is this estimate That is what number should we put in the statement Star Signal 12292 ADU Recall that we identified 3 sources of noise in slide 2 Noise associated with the photoelectrons Star Sky 039star osky Noise associated with the dark current odark Noise associated with the amplifierADC oread Notice that all of these noise sources are independent of one another The Read noise is a function of the electronics and knows nothing of fluctuations in The star signal The dark noise is a function of the CCD and knows nothing Of the read noise And so on Therefore to estimate the total noise we can Add these 4 terms in quadrature 0393 tot c tar osky odark mead2 So how do we estimate these individual noise terms Read noise is the easiest The noise we measure is the sum of the noise values for each of the Npix pixels inside our photometry circle Also the noise on one pixel is pretty nearly independent of the noise on its neighbors But we know Stats I the sum of a lot of identical independent noise terms results in a Gaussian noise distribution with or ad E opixel Npix opixel 2 For this camera opixel is about 2 ADU so oread 24 349 1400 ADU For later convenience let s convert this to units of e assuming the reciprocal gain is 6 eADU to get 6read2 36 elADU 1400 2 50000 e The terms involving photoelectrons and dark current are a different matter These are examples of what Taylor calls counting experiments in which countable objects photoelectrons or dark current electrons arrive in the detector at some possibly unknown but definite average rate Such processes are modeled using the Poisson distribution They result in noise that is called Poisson noise or sometimes shot noise o Poisson Distribution The other distribution that we will encounter frequently is the Poisson distribution Pv A This describes processes in which events photons arriving at our detector for instance occur at some rate A per unit time Let us count events for a time 739 The probability that we will count exactly 19 events in that time is given by the Poisson distribution Pk A7quot exp 739 This distribution has the properties that ie t e expected number of arrivals equals 739 times the mean rate A and also ltk T2gt AT ltkgt Thus the RM S is the square root of the number of expected events The Poisson distribution is asymmetrical because it must be zero for negative 6 For values of AT that are even moderately large however it becomes indistinguishable from a Gaussian that is peaked at k AT with 7 ATVA 04 I u 0 k2 03 0 14 39 o 110 E 02 01 00 i 3 0 5 10 15 Now back to our star brightness measurement Since the star signal the sky signal and the dark signal are all things that accumulate at definite average rates elsec per pixel they can all be modeled as Poisson processes This means 68tal39 Nstar 74000 measured in e with 1G 6 eADU csky Nsky 900000 The indivisible units we are counting are electrons not ADU odark Ndark 8700 Also Nsky and Ndark are summed over all the pixels in our photometry circle So at last we can finish the noise estimate getting otot osmr 0sky2 odark oread 2 2 Nstar Nsky Ndark oread 2 2 Nstar Npixsky dark O39rpix 2 12 where 6I9pix is the variance in e from reading a single pixel This is essentially Howell s CCD equation from his p 73 What about the numbers For this star we get Signal 12300 ADU 6 elADU 74000 e otot 74000 900000 8700 50000 102015 170 ADU star sky dark read The SN ratio is thus about 73 Notice that in this case the noise is completely dominated by the sky contribution It would be highly advantageous to make this smaller and the read noise as well by reducing the number of pixels used in estimating the total starsky signal The notion ofa magnitude scale to gauge the brightness of stars dates back to Hipparchus 150 BC who graded the visible stars into magnitudes according to their apparent brightness to the unaided eye The brightest stars in the sky were termed 15 magnitude and the faintest ones visible were 6 quot magnitude It turns out that the eye perceives equal multiplicative factors of energy ux as being equal steps in brightness and also that a differences of5 magnitudes on Hipparchus s scale corresponds pretty nearly to a factor of 100 The modern magnitude scale therefore de nes the apparent magnitude difference between two obiects as m1 m2 25 log uxflfluxz note log base 10 not In The zero point of the magnitude scale is defined in various ways one commonlyused de nition sets the magnitude of the bright star Vega equal to 00 in all wavelength bands On this scale one gets the following approximate Magnitudes for various astronomical obiects Sun Full Moon Sirius Vega Visual Limit Pluto FaintestHST 266 126 14 00 60 14 31 Magnitudes of stars are wavelengthdependent and are defined for many different wavelength bands Calibrated magnitudes using some standard filter and detector are usually indicated by the name of the filter eg UBVRl ultraviolet blue visual red infrared Johnson or Bessell systems bvyHbeta Stromgren system etc A color indexor just color is the difference between magnitudes measured with different filters hence a measure of the ratio of fluxes at different wavelengths They are denoted with an obvious notation The most commonly seen are BV and UB For both of these cooler redder stars generally have more positive color indices The magnitudes we measure are apparent they indicate a combination of the intrinsic brightness of the star and its distance The absolute magnitude is the magnitude that would be observed if the star were at a standard distance of 10 pc Absolute magnitudes are usually denoted by eg M where the subscript indicates the filter Distance Modulus Light intensity falls off as the inverse square of the distance Suppose a star has absolute magnitude M and lies at a distance of D pc The flux we receive from the star is then F F p10 392 Convert both sides to magnitudes take the log and multipy by 25 to get m M 5 ogD 5 where m 25 log F o The quantity 5 ogD 5 is called the distance modulus It is measured in Magnitudes and is a logarithmic measure of distance Distance Modulus Distance pc 0 10 5 100 10 1000 20 10 5 etc Physics Probability expectation value mean median quartiles standard deviation normal amp Poisson distributions CCD noise sources capabilities limitations optics telescopes imaging Earth s atmosphere Leastsquares fitting 6qu CCD calibration lt IRAF using ccdproc ot python Celestial coordinates Telescope mounts rminology amp nomenclature pro 39 ofstars I y e stars In ny othertidbits To locate objects in the sky it is useful to define cele ial coordinates which are the astronomical equivalents of longitude and latitude on Earth longitude iightAscension measured in hours Norm Sou glatitude ieclination measured in degrees The celestial poles are the extensions of the Earth s rotation axis the star Polaris happens to lie near the North Celestial Pole The celestial equator results from expanding the Earth s equator to infinity The zero point of RA is defined as one of the two places where the Earth s equatorial Plane intersects the Earth s orbital plane ie the plane of the ecliptic It is almost Fixed relative to the stars so it rotates around the Earth once per day skyquot quot m yuu are I I w quota L 39 39 ulluugllyuul uni HourAngle the difference in RA between the position of a particular object and the RA that corresponds to your meridian at the moment This is defined so that an object s HA is negative when it is east of the meridian and positive to the west Sidereal Time the RA that coincides with your local meridian at the moment Other important terms 1w u r39w he 4 A telescope is no good unless it can be accurately pointed at a target in the sky and held in alignment in spite of the Earth s rotation The hardware that does this is the telescope mount These come in a bewildering variety of forms we will be concerned only with two broad distinctions Equatorial mounts have one axis of motion aligned with the rotation axis of the Earth so they can follow the stars using motion only along this axis They are convenient to use but difficult to build in large sizes The orientation of the field of view remains fixed as the telescope tracks the stars AltitudeAzimuth AltAz mounts rotate around one vertical axis and one horizontal axis In general motions along both axes are required to follow the stars They are however simple to build and can be made very large The orientation of the field of view rotates in a complicated way as the telescope moves to follow a star IRAF tasks have a lot of hidden parameters Sometimes it is desirable to set these to nonstandard values 1 Do the usual startup mantra cd xgterm amp ds9 amp cl cd imdir 2 Display an image cd data1 display dat1cal 1 3 Now look at the parameters of the display function lpar display You should see something like this These are the default values l aaiiaeneaaariaie QEE inaae a ae aiepialea nane a ae amen inia These for instance govern how r the ADUto grayscale conversion is done a pixel mask had pixel dlsPlay naneloverlayllnterPolate had pixel colors m we saw mm paraererase na erase unFllled area seleei rane a s dlsPlag Fro r repeat n rer i of 33 range c ear med n contrast adjustment eale Elaarlthm 33 Full inaae n range We can change these in a i 325 P2233 52 temporary way by entering Wm 335225 new values on the command 5 widow horlzontal naanicieaii ne aiepial al naanirieaiian spatial interpaiaiar araer cereaiieaie ieiine re level la ae aie ia ea Sreiie ei a la aispi Or we can change them for as Sreylevel rile cantalnln v a ea u EFZQZS QZLSLEiEQ Zi CZEieww long as we like by edi ing quot 9 parameter le using the epar task E PREKRBE Lv TRSK displad unwrapmam l lnage Reductlan and analdSis raeilicd inane to be dlshlaued Frame La he driccen lnta BPM had pixel mask nane had pixel displad nanelaverlayllnlerPalaLe red had pixel calars des erase rane na erase dnrilled area a dindad lsplag Frame being laaded na repeat prediads displad Parameters na scale inage La pic displad dindad na displad rang up aredlevels near nedian 025 cantrast adjustment Far zscale algorithm n lsplag Full Image incenSicd range 1 display dindad vertlcal nagnicieacian o SPaLlal lnterpalatar arder Oirepllcate 111nea no ninindn aredlevel La he isp aded 1000 naxindn aredlevel La he dlsPlayed linear areglevel Lransfarmatlan linearllpglnaneldser rile ntalnlng user deflned look up table a o autoscaling autorange lack 100 ADU hite 1000 ADU Tn eparam dlsp ay and play with the parameters until you get this Notice the weird editing syntax Basically you scroll to the line you want to edit using up or down arrows and then type what you want When you do up or downarrow again what you typed replaces what was there before To get out type quotd means drld if you want to save your changes Do this now quot2 also exits saving changes quotc exits without saving changes Now do displa dat1cal 1 This should look much different from before You can override the changes you just made from the command line by turning on the zrange parameter displa dat1cal zscale when zscale is on 21 and 22 parms are ignored Or you can mess around with the scaling parameters explicitly leaving the autoscaling zrange parameter off displa dat1cal z1100 2210000 displa dat1cal contrast01 we have done this before Also VERY USEFUL you can set all parameters back to their default values thereby undoing any disastrous changes you may have made unlearn displa For safety it is good practice to do an unlearn on any complicated task before you start to fiddle with it Who knows where it s been A r ll5 genera Now to correct CCD images for bias dark and flatfields without so much fuss The task you want is called ccdproc To get at it you need to load packages imred to load a package type its name ccdred The ccdproc task is capable of figuring out which images it should use for bias dark and flat and which are to be calibrated using these But only if the right keywords appear in the headers of the respective images Our Tenagradata do not have these keywords so we will supply them by setting the approptiate hidden parameters of ccdproc We wish to calibratethe im age sn2007uyR08030801fit this is the same image we calibrated using imarith last time So do the following epar ccdproc There are a lot of parameters Edit them so the result looks Vou will have to set ccdtype it make this blank zero it These are the names dark if of the images to be flat if used for calibration trim if Setting this causes a if part of each image to it be trimmed off it because the flat has it bad values along 2 dges trimsec it This describes the if part of the images to it keep the rest is thrown it away El studentwpmait F image Reauetiam and Rnalysxs raeiiitg PREKRBE ccdred TRSK cchroc sn2007uyR08011601 List of run images ta earreet List of autput cn images n ima e t e ta earreet Maximum image eaemims memory im Mhytes i ma List praeessims steps onlyv m LFIldlm time Read aut axis column FIX had run times and columns7 RPPly overseaquot strip earreetiamv level Image ta reaaaut earreetiamv e e ta seam earreetiamv uma riie describing tme bad times and columns Uverscan strip im s a num m aata seetiam a level eaiipratiam image aarxoaozoacit mm x eaumt eaiipratiam image nnmmmant39 Frimse earreetiam images 1 Minimum Flat Field value E Far HELP Now do ccdproc sn2007uyR08030801 This will pop you into epar for one last look at your params Type quotd You should get a flock of identical and incomprehensible error messages And that s it By default ccdproc overwrites the original data file with the calibrated data There are several ways to see that this has happened For instance try imhead sn2007uyR08030801 lo page You should see a number of new keywords inserted by IRAF to tell you what processing was done and what ancillary files were used to do it Also notice that the size of the image is now 1016 x 1016 not 1024 x 1024 You can also look at the calibrated image and compare it with dat1cal displa dat1cal 1 displa sn2007uyR08030801 2 put the 2 images in different frames and blink them They should be virtually identical except for a shift If you don t like the scaling of the image fix it by changing parameters in The display task using epar Now that you know how If we had a long list of images taken with the same filter we could Process them all at once by putting a wildcard or an image list in place of sn2007uyR08030801 when we invoke ccdproc We don t have such a list but consider sn2007uyV08030801 This image was taken on the same night as the other one but with a different filter Thus zero and dark are the same as are trim etc but the flat image name must be changed to the obvious thing epar ccdproc now change the value of the flat keyword ccdproc sn2007uyV08030801 and you re done Cool huh So for good measure now calibrate sn2007uyR08011401 This is a different night so you will need new zero and dark as well as flat Again the naming convention should be obvious Let s look at our results I ve set the scaling to get something that looks good displa sn2007uyR080114701 1zscale zrangez10 221000 displa sn2007uyR080308701 22scale zrangez1100 221100 v t Thisd is a supernova which was discovered in Dec 2007 It was near maximum brightness on 080114 left image and a lot fainter 2 months later on 080308 right image How much fainter Let s use the radial plot function in imexamine to make a quick estimate imexam sn2007uy080308R701 2 is rimexam chooses parameters of the radial plot epar rimexam function in imexamine see help imexam SEE Make the parameters 539 ndEnt pirfi Look like this llaae Realaaam ram 5 You need to es Standard banner The radius buffer gtmle widt and iterati keywords Raalus X axxs label Plxel Value V axls label es UverPlaL prafxle claw Prafxle type a 1 enter abject 1n aperture7 lca s maffat a htract hackgraund7 r ah er pf radius adjustment Iteratlans mnawlmn Plat paints Instead pf lines7 Paint marker character7 nu lag scale y axls es draw hax araund Periphery pf mindaw E Far HELP Try again imexam sn2007uy080308R701 2 Center the cursor on the SN and do a radial plot r This looks much better u get Date Supernova Ref Star SN Ref 080114 1360 1526 166 080308 1486 1489 003 So if we assume the reference star was constant over this Time the SN changed in R magnitude by 163 MidTerm 2 Tues May 27 YOU SHOULD EXPECT 2 QUESTIONS One will be multiplechoice and will deal with various aspects of image reduction and IRAF procedures This might include correction for bias dark and flatfield as well as aspects of aperture photometry image registration and image combination You will not be asked to do anything on a computer The second question will deal with concepts of stellar magnitudes and with leastsquares fitting of data Review Taylor Chap 8 and the lecture notes from Thurs May 16 You will not be asked to remember any formulas nor to do anything with computers but you will need a calculator We will allow 45 minutes for the midterm and use the rest of the class time to work on projects ewu Physics Probability expectation value mean median quartiles standard deviation normal amp Poisson diStribUtions CCD noise sources capabilities limitations optics telescopes imaging Earth s atmosphere lt Leastsquares fitting gt Aperture photometry with imexamine radial plots Terminology amp nomenclature properties of stars clusters galaxies SNe pulsating stars many other tidbits Let s iaak at our iesuiis i ve 50 the scaling an get same hing that Banks good displa sn2007uyR08011401 1 zscale zrange z10 221000 displa sn2007uyR08030801 2 zscale zrange z1100 221100 i s is a supei39nmla which was discavei39ed in Dec 2337 it was near maximum hrigi nness an 080114 Ibe image and 21 im fainter 2 munihs Hater on 080308 right image Haw much fainter Let s use the radial psi function in imexamine to make a chk catimam imexam sn2007uy080308R01 2 epar rimexam rimexam chooses parameters of the radial plot function in imexamine see help imexamquot ake the parameters 3 Stu enteizarfalt QLEEJ Look like thIs Image Reductmn and anaiusis raeiiitu You will need to edit The radius buffer ues Standard banner Tnb X ex s V exls label UverPiat praflle ritv Prafile tape ta cit s enter uhieet in aperture7 it and subtract heckgraund7 Radius ii Pixel Value d iterati keywords n at radius Back aund buffer uiut aund uia a N er af radius adiustment iteretlans ackaraund x arder o hwy Magnit aund u arder ude zera paint Maffat beta 8 Parameter ues piut paints insteau af lines7 Pius paint marker cherecter7 drew hax eraund Periphery af mindaw E Far HELP Try again imexam sn2007uy080308R01 2 Center the cursor on the SN and do a radial plot r This looks much better a NEW get Date Supernova Ref Star S N Ref 080114 1360 1526 166 080308 1486 1489 003 So if we assume the reference star was constant over this Time the SN changed in R magnitude by 163 See Taylor Ch 8 o LineFitting by Least Squares Suppose we have a bunch of paired measurements mum and also that we think a good model of these should be 1 A9 B Also suppose that the errors in the 032 are negligible but that the 141 have signi cant errors How may we choose A and B so that the model agrees with the observations as well as possible One answer is to choose A and B to minimize the summed squared dz er ence between the model and the observations min 29yl 771063602 wrt A B 0139 4 g A g min 24yquot Arm B2 wrt A B irr t3 3f gt523919 Diameter arcmin 2 diff2 346 ltgt 9 10 7 O 7 O 7 o 7 3 7 2 diff2 223 6 7 4 7 2 7 0 w w w w w 7 5 10 15 25 30 o CurveFitting by Least Squares We can do the same thing with more exotic functions letting the model 3 f9A B G be a complicated function of many parameters ABC The idea remains the same however the best model in a least squares sense is the one that satis es min Egg model2 wrt 148 C 2500 7 2000 1500 ADU 1000 500 l l 200 205 210 215 220 X pixel o Curve Fitting by lVIinimum X2 Suppose we have plausible estimates of the errors 0 not necessarily all the same that are associated with each measurement yi Then we usually wish to minimize not the summed squared differences but rather these things normalized by the squares of their individual errors as mm X2 V mo 8 Q1 This has the advantage that veryr uncertain measurements those with large I contribute little to the result Even better if our estimates of the a are fairly accurate we expect each term to add about 1 to the sum Thus if there are N terms we expect X2 25 N If X2 32gt N then either we do not understand our errors or the chosen function is a bad t to the data 0 Methods Special Case of Linear Least Squares To know how leastisquares methods can fail it helps to know how they Work Take the simple but very common case in which the model function is the sum of several speci c functions f1ac each multiplied by an unknown factor a1 We Want to estimate the face tors given some data 9 We are then trying to minimize the cost function a ie min a2 2km iziaim To do this We differentiate this expression with respect to each of 2 the a1 in turn and set each derivative 33 equal to zero This leads to a set of linear equations that We must solve for the coefficients a1 X11141 l BJ Here the entries AU of the matrix A and those of the vector Bi are Aw Ek xk 39 f175k Bi BMW 110 So the matrix AU depends only on the functions f1 and on the pi sitions 951 Where We have data samples While the vector Bi depends on the functions and on the data va ues o The Most Common Cause of Failure If any two of the functions f ack are the same or indeed if any of the f1ack is a linear combination of some of the others then the matrix A is singular meaning that its determinant vanishes Since solving a set of linear equations requires in effect dividing by detA this means that one or more of the coe icients a Will become in nite and the rest Will become meaningless This is a Bad Thing If one of the flack is merely almost a linear combination of some of the others then A becomes almost singular and the a1 thou h not formally in nite are still meaningless Also a Bad Thing This result is very common in least squares problems for example When tting coefficients of powers of ac To avoid these numerical disasters keep the number of functions fl to a minimum Make sure there are more data points yack than there are functions preferably a lot more And most importantly choose your functions carefully so that they have signi cantly dif ferent shapes When sampled On the grid wk Ideally make them orthogonal on that grid ie choose them so ha EkfdkaJWk 0lt 6w 6qu Physics Probability expectation value mean median quartiles standard deviation normal amp Poisson distributions Leastsquares fitting Terminology amp nomenclature pr of stars clus n 39 N pulsating ny other tidbits operties CCD noise sources capabilities limitations optics telescopes imaging Earth s atmosphere Web tools for Planning observations ds9 IFlAF DAOphot Observing What Where Targets calibratiqn stars E EEIIE zuuabb CBET Tabb bysbbveyeb zonemummy by Lmk Obsen amw sugenybva Search babe CBET Tabb bysbbveyeb 20080424 25 by 0 m7 ay R A b n o Mag T73Typ MT 6 9 N c 5036 b ayRA Tamsm T75 Deb T5 57 0 wesy and u a sbmyy by me nuc eus by NGC 5036 Debbyem ymage o Mag Tbs Type Ta cyAspeamm wesy e Ta CHASE A ybuangsbz DebT 7mm y and a 7 sbmyy by me nuc eus by Eso mun Dysbbveyy ymage Reyeyenbes caETyasD nsyems Mag gt 2b 7 yReyeyenbes ATEL m7 CBET T355 2 nabz CBET T353 bysbbveyeb 20080422 by by ROTSE bbuabbrayybn u oFuu dm 0 Lu n ay eb an a 5 yybn was 0 Mag T77 Type Ta mbue aTax ay RA 2h and a a sbmyy by me benye asmakm DebT mums YUHhe has gaTaxy Dyscuve ymage ESPLOO ymage zuuaby CBET Tasu bysbbveyeb zuuauwg 2y by ROTSE bbuabbrayybn o Ebunb m an anunymuus gaTaxy ay RA 0 Lbbayeb y o Mag T72 abx CBET mmd m M06 77 9 wesy and a a sbu by me ben ey cyA s ebymm Reyeyenbes 9 easy b Type ya yaeyeyenbes 0 ma bysbbveyeb 20050422 35 byTym Pucks s an new 4 M06 y and T 2 n m by me bem BET T353 74 ay RA MhSAmSS o Mag T5 5 Type M p zuuabw 0 Few CBET ms bysbbveyeb 2a b m uec ymy am Lbbayeb 5 Mag ya a Type easy Ta mam2y yyzbms a 2nb5 ey by 29 by E Gmda 0545 DebT mzusga DebT MUDSS AS a me yybay gaTaxy Dysbbveg wage and R Gaghanu 2 m b 4 Dysbbveyyymage qaumsmuobsymage CBET T359 w MabDbnaTb and Tym Pucks 5y quot08 6 b yyy by me benye anb4 sbu yReyeyenbee cBETyasa ybyuec may Dyscuvem ymage SN 2nu5by 2ubaby CBET T345 bysbbveyeb ZuuBMVQ 2y by Lyby observaybry supembva Search o Faund m use 4553 ay RA 8h53m54 54 DebT 35 o Lbbayeb easy and a sbmyy by me nuc eus by use 4553 nysbbveyy ymage o Mag Tbs Type Ta Reyeyenbes CBET Tasa Eumment W 1 0 SNZOOSbl 1430 a Fulntlng 11m1ts are 12 HQ 7529 27320e 2000 tn 514 02 gt 73111 66666666666666 0 Fuanmom39mNmmbthll L46 030413 s 030424 030422 Ia Ia p In w Hm mHm HUH Hg 1 a II Eumment W 13 02 200301 143012 27 E911 uthers tun 01d 1 Ia Dr tun NNanbln39mFu39m39mbLnFuL 0 7 3205 2000 Falnt 1 mm a Fulntlna 11m1ts are HQ 7529 tn 514 DE gt 73111 E Stlng dJSE 030413 Type Ia x dJSE 030424 Type Ia tun suuth dJSE 030422 Type Ia dJSE 030325 Type 11 For How Long SIN Saturation Overheads Rough photon count rates for Nickel Telescope Star mag 200 B 50 els V 40 els R 25 els Star mag 175 To reach SIN 30 how many e What about sky noise Saturation 1 unbinned pix 50000 ADU 75000 e T39ime to saturate depends on how many pixels are covered by a steller image Let s guess 1 for unbinned pix Saturation charge 75Ke At V 15 rate is 4Kels Saturation reached in 18 0s Probably don t want to expose more than 3 min in any filter Setup tasks Ordering the targets AcceptableOptimal times to observe Calibration concerns Setup Naming convention Bias Dark Open Flats BVR Exposure times How many images Coordinate alignment Focus Shall Konsola D x Sessmn Em1 V9w Euuxmams Semngs Hem Eumment a 20000 9 9 030403 Type 19 20000 3 9 030419 Type Ia 20000 2 9 030413 Type Ia 20000 2 9 030419 Type Ia 20000 7 9 030422 Type Ia 20000 2 9 030329 Type IIn 20000 9 9 030413 Type Ia 20000 2 9 030329 Type I 20000 5 9 030422 Type I 20000 3 9 030417 Type I 54 4 4 21 20000 5 9 030401 Type 19 4 2 132950 510317 20000 3 0 9 030421 Type Ia 1240777 32 Lucal 99 919 1r Ia Dr 99 Falnt r 9 9 ma n 99 ntErEstlng mg 1191 are HQ 7529 9 514 099 gt 73111 0509 0515 0522 0529 2011 2019 2022 2027 0555 0550 0544 0541 2107 2114 2121 2127 0453 0452 0445 0440 Check these 2145 2153 2201 2209 0421 0413 0405 0359 1507 1531 1559 1929 9977129

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