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Week8
• MidterminclassonThursday– Reviewdocumentposted– Prepara;onemphasizedinTue/Wedsessionsthisweek– Expect½;mespentonconceptualques;ons,½spentoncalcula;on
• DayAssignment4wasposted– WatchoutforthatMoon
• Lab4pending,butnothinghappensun;laJermidterm– (howdoesApril1or2lookforyou?)
• Topics– PhotometryandFilters– Magnitudes– Imagecalibra;on– Sourceextrac;on(aperturevs.PSFphotometry)
StellarPhotometry
• Why?Precisionstellarphotometryisthegatewaytoobserva;onalastrophysics.
HR Diagram in Omega Centauri
StellarPhotometry
• Why?Precisionstellarphotometryisthegatewaytoobserva;onalastrophysics.
HR Diagram in Omega Centauri
StellarPhotometry
• Why?Precisionstellarphotometryisthegatewaytoobserva;onalastrophysics.
• Time variability in pulsating stars (top) and eclipsing binaries (bottom).
• Variable stars are key “standard candles” calibrated by amplitude and period measurements.
• Eclipsing binaries permit direct measurement of stellar dimensions and masses, putting stellar models to the test.
StellarPhotometry
• Why?Precisionstellarphotometryisthegatewaytoobserva;onalastrophysics.
Planetary Transits
StellarPhotometry
• Why?Precisionstellarphotometryisthegatewaytoobserva;onalastrophysics.
Planetary Transits…. with precision
Mul;-PlanetSystemsandPowerSpectralAnalysis
StellarPhotometry
• Why?Precisionstellarphotometryisthegatewaytoobserva;onalastrophysics.
Asteroid Light Curves
MagnitudesforReal
• FromASTR2110,2120,etc…considertwotargetswithmeasuredfluxesf1andf2
• Afluxra;oof10isadifferenceof2.5magnitudes,afluxra;oof100
is5magnitudes.
• But,exactlywhatiswhatisf???
m1 − m2 = − 2.5 log f1f2
⎛
⎝⎜
⎞
⎠⎟
Fluxvs.FluxDensity
• Fluxistheamountofenergyfromasourcethatpassesthroughasquarecen;meteradistance,d,awayintegratedoverallwavelengths.– Alsoreferredtoas“bolometricflux”.Unitsareerg/cm2/sorwa^s/m2
• Fluxdensityaccountsforthefactthattheenergypassingthroughthatsquarecen;meteriswavelengthdependentandwemaycare,forexample,abouttheenergytraversingthatsquarecen;meterinapar;cularfilterbandpass.– Unitsareerg/cm2/s/µmorwa^s/m2/Hz
Flux = Luminosity4πd 2
Integra;ngFluxAcrossaFilterBandpass• Measuringafluxrequireslookingthroughafilteratan
astronomicalsource.– Thatfilterhasabandpass(variabletransmissionwithwavelength)– Thetargethasaspectrum(structurewithinthefilterbandpass)
Integra;ngFluxAcrossaFilterBandpass• Measuringafluxrequireslookingthroughafilteratan
astronomicalsource.– Thatfilterhasabandpass(variabletransmissionwithwavelength)– Thetargethasaspectrum(structurewithinthefilterbandpass)
filter_fluxBand = filter_transmission(λ)*0
∞
∫ target_flux_density(λ) dλ
Integra;ngFluxAcrossaFilterBandpass• Measuringafluxrequireslookingthroughafilteratan
astronomicalsource.– Thatfilterhasabandpass(variabletransmissionwithwavelength)– Thetargethasaspectrum(structurewithinthefilterbandpass)
filter_fluxBand = filter_transmission(λ)*0
∞
∫ target_flux_density(λ) dλ
Integra;ngFluxAcrossaFilterBandpass• Measuringafluxrequireslookingthroughafilteratan
astronomicalsource.– Thatfilterhasabandpass(variabletransmissionwithwavelength)– Thetargethasaspectrum(structurewithinthefilterbandpass)– Tobestrictlycorrectonemustaccountforatmosphericandtelescope/
instrumenttransmissionasafunc;onofwavelength.
• Observingtwostarssimultaneously(inthesameimageforexample)calibratesoutthe;mevariableterms,atm_transinpar;cular.
filter_fluxBand = filter_transmission(λ)*0
∞
∫ target_flux_density(λ)
* atm_trans(λ) * inst_trans(λ) dλ
FilterCharacteris;cs/Terminology
• Cutoffandcut-onwavelengths– Definedby½powerpointsatfilteredges
• Centerwavelength:½(cutoff+cut-on)• Bandpass:(cutoff–cut-on)• Effec;vewavelength:Weightedaverageofproductoftransmission
andwavelength(centroidwavelengthaccoun;ngfortransmission)
• Refertoon-boarddiscussion
Hertzvs.Nanometers
• Conver;ngabandpassfromwavelengthunitstoHertzisnotassimpleasdividingthewavelengthintothespeedoflight…Onemustdifferen;atethewavelength/frequencyequa;ontogettheproperconversion:
λν = c ν =cλ
Δν =cλ 2
Δλ
FilterWheels
FilterTechnologies
• Filterscansimplybecoloredglasswiththedye/glasstransmissionenforcingatransmissionspectrumvs.wavelength.
• Alterna;velyfilterbandpassescanbecustomdesignedusingmul;-layerreflec;ve(!)coa;ngs.– Theconspiracyofreflectedandtransmi^edwavesinterferingwitheachother
canproducearemarkably“square”hightransmissionbandpass.
InterferenceFiltersvs.AngleofIncidence
• Thetransmissionofbulkcoloredglassisindependentofangle.• Becauseinterferencefiltersrelyonphasedelaybetweenmul;ply
reflectedwaves,anglesincreasepathlengthsanddecrease(!)theeffec;vewavelengthofthefilter.
PhotometryisUl;mately“Compara;ve”(eventhoughitstrivestobeabsolute!)
• Themagnitudeequa;onalwaysquan;fiesadifference(whichisactuallyafluxra;o)anditis“easy”tomeasurera;os.
• Asphysicists,however,wewouldliketoknowtheanswersinSIunits.– Fluxesinerg/cm2/sorfluxdensi;esinerg/cm2/s/µm…
• InrareinstancesaninstrumentcalibratedtoSIunitsgetspointedatastar.– ForthemostpartweareleJtomeasuremagnitudedifferencesbetweenstars
(viainstrumentalcounts–whicharepropor;onaltofluxes)andbootstrapourwaytoaphysicallycalibratedstandard.
– Modernskysurveysprovidebillionsofcalibratedsourcesatavarietyofwavelengths(theremightalwaysbeacalibratorinyourfieldofview).
– Imagetwostarsatthesame;me,extractthecounts,andyougetthefluxra;o/magnitudedifferenceindependentoftheatmosphere(tofirstorder).
m1 − m2 = − 2.5 log f1f2
⎛
⎝⎜
⎞
⎠⎟
MagnitudeZeropoints
• 0thmagnituderequiresadefini;on(acomparisonposter-child).• Sincemeasuringrela;vefluxesis“easy”thestarVegaisdefinedto
bemagnitude0.00inallbands.
• AtsomepointVegahasbeenquan;ta;velyassessedineveryfilter
mstar = − 2.5 log fstarfVega
⎛
⎝⎜⎜
⎞
⎠⎟⎟
Magnitudes,Percentages,andSNR
• Considerthefluxdifferenceoftwostarsthatdifferby0.1magnitude– Sincemagnitudesarelogarithmicamagnitudedifferencecorrespondstoa
mul;plica;ve/factordifferenceinbrightnesssoitdoesn’tma^erifwearetalkingaboutmagnitude10.0vs.10.1or4.0vs.4.1
• Itturnsoutan0.1magdifferenceisclosetoa10%differenceinfluxandan0.01magdifferenceisclosetoa1%differencebecause….
• Consideran0.1magdifference
• Taylorseriesexpandthissmalldifference:• Expandingaboutx=0
f (x +Δx) = f (x) + ʹf (x)*Δx + ...
m1 −m2 = 0.1 = − 2.5log f1f2
⎛
⎝⎜
⎞
⎠⎟
f1f2=10
−0.12.5⎛
⎝⎜
⎞
⎠⎟
10x = 1 + ln(10)*Δx for x = −0.1 / 2.5 10x = 1− 2.3* 0.12.5⎛
⎝⎜
⎞
⎠⎟
Magnitudes,Percentages,andSNR
• Expandingaroundx=0
• So,thankstothefactthatln(10)ispre^ycloseto2.5thefluxra;ofor0.1magdifferencecorrespondstoabouta10%differenceinflux.
• Similarly,ifamagnitudemeasurementisuncertainby+/-0.1magthatuncertaintytranslatesto10%orSNR=10.– 10.73+/-0.01magisabouta1%uncertaintyinmagnitudecorrespondingto
SNR=100
f1f2=10
−0.12.5⎛
⎝⎜
⎞
⎠⎟
10x = 1 + ln(10)*Δx for x = − 0.12.5⎛
⎝⎜
⎞
⎠⎟ 10x =1− 2.3* 0.1
2.5⎛
⎝⎜
⎞
⎠⎟ = 0.91
ABMagnitudes
• Justtomakethingsmoreconfusing,about20yearsagoaseparatesetofmagnitudezeropointswereestablishedtotaketheempiricalmeasurementofVega“outoftheequa;on”and;eittothefundamentalunitsoffluxdensity.
• fν hereisinunitsoferg/s/cm2/Hertz– One“Jansky”is10-26Wa^s/m2/Hz
• Theconstant,48.600,makesAB=Vforaflatspectrumsource• Seeh^p://adsabs.harvard.edu/doi/10.1086/160817
mAB = − 2.5log fν + 48.600 =− 2.5log fv3631 Jy⎛
⎝⎜
⎞
⎠⎟
RegisteringABandVegaMagnitudes
InstrumentalMagnitudes
• Sincemagnitudesarera;os,anyconstantofpropor;onalitythatscalesthemeasurement(forexampleconver;nginstrumentalcountstophysicalfluxunits)cancelsout.
• Thesimpleimplica;onisthatyoucanevaluatemagnitudedifferencessimplybyapplyingthemagnitudeequa;ontoextractedinstrumentalcounts.
m1 − m2 = − 2.5 log φ f1φ f2
⎛
⎝⎜
⎞
⎠⎟
minstrumental = − 2.5 log counts( )