sign in sign up
<<
Home , Photometry (astronomy)

Intro to and Goals'of'Lab'#3:'Astrometry''

• Learn more about imaging proper/es of a CCD & & &&& & & & &&& & and coordinate systems &&& • Further applica/ons of least-square fi=ng &&&&& &&&&& • Query astronomical databases and catalogs && & & • Data reduc/on techniques for imaging data &&& & & & & &&& • Learn about proper mo/ons and astrometry & &&& & & & & The Needs of What are the physical laws that govern these objects? Does it move? Astrometry Measure gravitational interactions between Detection objects Do we see it? Initial discovery of astronomical Photometry objects What kinds of How bright is it? light Does it vary with time? does it emit? Measure rudimentary Look at astrophysical properties of processes that govern astronomical objects astronomical objects I.'Stellar'Photometry' What'are'the'essen/al'factors'that' define'what'an'“image”'looks'like?'

• DiffracHonIlimit:& – &and/or&imaging&system& (e.g.,&your&eye’s&pupil),&f(λ)& • SensiHvity:& – total&collecHng&area&of&the&telescope,& f(λ)& – throughput&of&opHcs&and&atmosphere,& f(λ)& – QE&of&detector,&f(λ)& – Frame&rate&of&detector& – Detector&spaHal&sampling&(pixel&size& and&plate&scale)& • Seeing:&& – Atmospheric&turbulence,&f(λ)& Point'Spread'Func/on'' of'DI'Telescope'Star'Field' NGC&6397,&Hubble& Why'do'bright'stars'“look”'bigger' than'faint'stars?' • The&FWHM&of&the&stars&are&the&same,&but&the&intensity&of&the&stars& are&different&with&respect&to&the&background&(or&image&stretch)&

SensiHvity&limit&& (or&image&stretch)& Principles'of'photometry'

• The&light&from&a&star&is&spread&over&several& pixels&nonIuniformly& • How&do&we&sum&the&light&to&get&a&measure&of& the&total&flux&from&the&star?& – IdenHfy&the&locaHon&of&the&star& – Select&the&associated&pixels&that&contain&the&stellar& flux&by&generaHng&a&masking®ion& – Sum&up&the&light& • Ensure&that&the&noise&and&background&is¬&included& Determining'the'center'

• For&each&star&we&can&find&its¢roid&by& determining&the&first&moments&along&a&2d& array,& y I ∑xiIi ∑ j j x = i y = j I ∑Ii ∑ j i j

where&I&is&the&intensity&at&each&pixel&locaHon&(x,y)& 'photometry'

• How&would&you&define&the&aperture&mask&on&the&star?& – What&radius&of&an&aperture?& – Where&would&you&define&the&sky®ion?&

Sky&annulus& aperture& Sky&annulus&

aperture& 'Photometric'Model'

• Star& – Brightness&

– Center&(x0,&y0)& – Width&(σ)& • Sky&background&in& annulus& – B& • Detector& r1& – QE,&readnoise,&dark¤t& r2& • Aperture&sizes& r3&

– r1,&r2,&r3& Photometric'Model'

• Write&down&an&expression&for&the&signal,&Si&,&in&units&of& photoelectrons& – In&an&individual&pixel&

Si = Fi + Bi + Qi + Ei

I& – Fi&is&the&stellar&signal&=&fi&t&&at&pixel&i&[e ]& • Different&for&every&pixel& I – Qi&is&the&dark&charge&=&ii&t&[e ]&in&a&given&pixel& • The&dark¤t&iivaries&from&pixel&to&pixel& • For&SNR&model&assume&constant& I& – Bi&&is&the&sky&background&=&bit&assumed&uniform&[e ]& • Varies&from&pixel&to&pixel,&for&SNR&model&assume&constant& I& – Ei&&is&the&readout&electronic&offset&or&bias&[e ]& • Varies&from&pixel&to&pixel,&for&SNR&model&assume&constant& & The'Stellar'Signal'

• The&stellar&signal&is&found&by&subtracHng&the&background&from&Si&and& summing&over&the&N&pixels&that&contain&the&star&

;&Flux&per&pixel& Fi = Si − (Bi + Qi + Ei )

N1 N1 ;&Total&flux&of&star&in&& FN = ∑ Fi = ∑Si − (Bi + Qi + Ei ) i=1 i=1 circular&aperture& 2 N1 = πr1

• Error&in&FN&is&due&to&noise&in&the&signal&itself,&FN& • Noise&due&to&dark&charge&per&pixel,&Qi& • Noise&from&the&background&per&pixel&(e.g.,&from&atmosphere),&Bi& 2 • The&read&out&noise&σRO &can&define&the&Error,&Ei& Noise'in'the'stellar'aperture'

• The&total&noise&is&a&funcHon&of&the&Poisson&noise&of&the&stellar&signal& and&noise&(background,&dark¤t,&readnoise)&in&the&stellar& aperture&(N1)&

N1 N1 # & N1& % ( FN = ∑Fi = ∑ Si −(Bi +Qi + Ei ) i 1 i 1 % ( = = $ Background '

σ 2 = F + N B + Q +σ 2 F N 1 ( RO ) Poisson signalnoise  Poisson noisewithin r1

• Where&&and&&is&the&mean&background&and&dark¤t&per& 2& pixel&and&N1&=&πr1 Including'all'noise'sources'

• You&must&also&include&the&noise& N1& from&the&background&annulus&(N23),&

2 2 σ Sky = ( B + Q +σ RO ) / N23

r1& 2 2 2 N1 = πr1 , N23 = πr3 − πr2   r2& Star Sky r3& N23& • Then&you&get&the&total&noise&source& from&stellar&aperture&and&sky&annulus,&

σ 2 = F + N B + Q +σ 2 + N B + Q +σ 2 / N F N 1 ( d RO ) 1 ( d RO ) 23 Poisson signalnoise   Poisson noisewithinr1 Poisson noisewithinr2

Signal F SNR = N F + N B + Q + σ 2 + N B + Q + σ 2 / N N 1 (       RO ) 1 (       RO )  23 Signal Noise SkyDark& RON SkyDark& RON instar aperture inskyaperture

• How&do&we&choose&r1&,&r2&,&r3?& – Signal&increases&with&N1& – Noise&increases&with&N1&and&decreases&with&N23&

For&more&generalized&with&N&number&of&frames&see&Mclean,&“Electronic&Imaging&in&”& How'does'flux'change'' with'aperture'radius?'

• Suppose&the&stellar&signal&has&a&2Id& Gaussian&shape&

2 F0 ⎡ 1 ⎛ ri ⎞ ⎤ 2 2 2 Fi = 2 exp⎢ − ⎥ , ri = (x − x0 ) + (y − y0 ) 2πσ 2 ⎝ σ ⎠ ⎣ ⎦ i

r1 F = 2π rFdr N ∫0 i

– This&tells&us&how&FN&changes&with&aperture& radius& Stellar'profile'and'integral'

• You&can&determine& what&percentage&of& light&is&found&per& aperture&radius& – Useful&for&determining& aperture&radii& – Useful&when&you&need& to&use&“aperture& correcHon”&& • When&the&total&flux&is& outside&the&aperture,& like&for&crowded&field& photometry& – Useful&for&defining&SNR& per&radius& II.'Astrometry' Astrometry' • Astrometry&is&the&measurement&of& posiHon&and&moHon&of&celesHal& objects& • Oldest&branch&of&astronomy&& – In&129&BC&Hipparcos&generated&the& 1st&catalog&of&stars&with&apparent& brightness&and&posiHonal&accuracy&to& 1°& – RevoluHonized&by&Tycho&Brahe&in& 16th¢ury&and&measured&posiHons& of&stars&to&1&arcminute& – PosiHons&were&measured&to&1& arcsecond&precision&by&the&18th& century&with&new&instruments&and& the&telescope& – In&1989&Hipparcos&satellite&launched& Courtesy&ESA& and&had&an&astrometric&precision&of& 0.001”& & Importance'of'astrometry'

• Astrometry&is&used&to&determine:& – Distances&to&nearby&stars&using¶llax& – Proper&moHons&of&stars&& • KinemaHc&structure&of&the&& – Distance&scale&of&the&Milky&Way&and&beyond& • Stellar&formaHon&and&evoluHon&& – Orbits&of&solar&system&objects,&binary&stars,&exoplanets,&stellar& populaHons& • Direct&way&of&determining&the&mass&of&celesHal&objects& • Determine&the&mass&of&stellar&black&holes&and&the&supermassive&black& hole&at&the&GalacHc&Center& • Determine&the&orbit&of&asteroids&and&& – Probability&of&Earth&gePng&slammed!& – Solar&System&formaHon&& – Astrometric microlensing • Determine the mass of the lens source accurately • Number density compact objects Proper Mo(on (from Lecture #7) ''''' • Objects&in&our&solar&system,&galaxy,&and&extragalacHc& sources&have&their&own&velociHes& – Closer&the&source&and&higher&the&velociHes&the&greater& moHon&on&the&projecHon&of&the&sky,&which&is&known&as& “proper&moHon”& • Typical&proper&moHon&of&nearby&stars&is&0.1”&per&year& • Typical&proper&moHon&of&main&belt&asteroids&is&~0.5I1’&per&hour& • NEO’s&can&move&have&as&fast&as&2.5°&per&day& Arcturus&radial&velocity&

Δt& Radial&velocity&

μ& Star’s&velocity& &

Proper&moHon& Barnard’s&star,&10.3”&per&year& Astrometry'discovered'the'dwarf' 'Eris'(the'“Pluto'killer”!)' • Eris&moves&at&1.75”& per&hour&and&was& discovered&by&1.2m& telescope&(Brown&et& al.&2005,&ApJL)& – 27%&more&massive& than&Pluto&

Eris&

Eris’s&moon&Dysnomia& Palomar&(1.2m&telescope)&Discovery&Image!& Hubble& Astrometry'determined'the'mass'of' the'MW’s'supermassive'black'hole''' • Using&adapHve& opHcs&and&8I10m& groundIbased& telescope&the& relaFve&astrometric& precision&is&100&μas& (0.0001”)& • Monitor&the&stellar& orbits&over&15+& years& UCLA&GalacHc&Center&Group&

Reference'frames'

• The&concept&of&posiHon&in&space&is&relaHve& – You&need&to&define&a&coordinate&system&(e.g.,&CelesHal& coordinates,&standard&coordinates)& – You&need&to&define&posiHon&relaHve&to&other&celesHal& objects&for&given&Equinox& • Everything&in&the&Universe&is&moving!& • A&“reference&frame”&is&a&catalog&of&known& posiHons&and&proper&moHons&of&celesHal&objects& – For&astrometry&you&use&reference&frames&to&define&the& posiHon&and&moHon&of&the&object&of&study& Reference'star'catalogs'

• AllIsky&surveys&have&generated&catalogs&of&stars& with&posiHons&per&Equinox&and&proper&moHons&& – Hipparcos&contains&posiHons,&proper&moHon,& parallaxes&of&118,218&stars.&Complete&to&V=7.5&mag.& – TychoM2'contains&posiHons&and&proper&moHon&of&over& 2&million&stars.&Complete&to&V=11.0&mag.& – USNOMB1&contains&posiHons&and&proper&moHon&of& over&1&billion&objects.&AllIsky&coverage&to&V=21&mag.& – Sloan'Digital'Sky'Survey'mapped&¼&of&the&sky&in&5& opHcal&filters&and&contains&287&million&objects&

We&will&be&using&the&USNOIB1&catalog&in&Lab& Interna/onal'Celes/al'Reference' System'and'Frame'(ICRS)' • Fundamental&reference&system&adopted&by& InternaHonal&Astronomical&Union&(IAU)& – Reference&system&of&celesHal&sphere&defined&by& the&barycenter&of&the&solar&system&& – Reference&frame&is&with&respect&to&distant& extragalacHc&objects&that&are&considered&fixed& • Based&on&radio&posiHons&from&the&VLBI&of&Quasars& • PosiHonal&accuracy&is&0.5&mas&(0.0005”)&

ICRS&–&VLBI&Quasar&reference& Coordinate'transforma/ons'' to'the'image'plane' • CelesHal&coordinates&are&defined&on&spherical& coordinate&system,&but&a&CCD&image&is&a&Cartesian& coordinate&system&on&the&plane&of&the&sky&

Celestial pole The&projected&coordinates&(X,&Y)&of&a& Y posiHon&on&the&celesHal&sphere&(α,&δ)& X in&the&vicinity&of&a&field¢ered&at& (α0,&δ0)&are,&

(α, δ) δ0 cosδ sin(α −α 0 ) X = − α cosδ 0 cosδ cos(α −α 0 ) + sinδ sinδ 0 0 Celestial equator ! sinδ 0 cosδ cos(α −α 0 ) − cosδ 0 sinδ Y = − cosδ 0 cosδ cos(α −α 0 ) + sinδ sinδ 0

! DerivaHon&in&the&Lab&writeIup& Conver/ng'to'pixel'coordinates'(x,y)' of'the'CCD' • The&CCD&has&plate&scale&(pixel&per&arcseconds)& that&is&defined&by&the&focal&length&(f)&of&the& camera&and&the&size&of&the&CCD&pixels&(p)& • For&an&“ideal”&camera&and&detector&the&x,&y& posiHon&on&the&detector&can&be&found&by,&

x = f (X p) + x0

y = f (Y p) + y0

• Of&course&a&number&of&factors&make&this&nonI ideal&with&opHcal&distorHon,&anamorphic& magnificaHon&and&the&CCD&rotaHon&on&sky& Let’s think about Affine transformation. Affine Transformation (from MathWorld): An affine transformation is any transformation that preserves collinearity (i.e., all points lying on a line initially still lie on a line after transformation) and ratios of distances (e.g., the midpoint of a line segment remains the midpoint after transformation). In this sense, affine indicates a special class of projective transformations that do not move any objects from the affine space to the plane at infinity or conversely. Geometric contraction, expansion, dilation, reflection, rotation, shear, similarity transformations, spiral similarities, and translation are all affine transformations, as are their combinations. In general, an affine transformation is a composition of rotations, translations, dilations, and shears.

Examples of Affine Transformation in matrix form: Examples of Affine Transformation in matrix form: Telescope Plate Scale

Diameter = D Focal length = F F ratio = F/D

δ θ

Focal plane Pupil plane = aperture stop (usually)

• F determines the pixel scale at the focal plane (the ratio between the angle on the sky and the physical dimension on the detector) - δ = θ x F (Suppose at 8.4-meter with an f/15 focal ratio, what is F?) • The focal ratio gives the size of the diffraction limit on the focal plane/detector. - x = F/D x λ • Plate scale is defined as the angle per unit displacement. - What is the plate scale of a 5-meter f/16 telescope?

• f = 5 x 16 = 80 m = 80,000 mm

• Plate scale = (1/3600)*(π/180)*80,000 mm => 1” = 0.387 mm or 2.58”/mm