Content1. BlackbodyRadiation2. PlanckCurveandApproximations3. EffectiveTemperature4. Brightnesstemperature5. Lambert6. FluxandIntensity7. Magnitudes8. PhotometricSystems9. ColorIndices10.SurfaceBrightness
Astronomical Observing Techniques 2018, Lecture 1: Black Bodies in Space [email protected]
Kirchhoff(1860):blackbodycompletelyabsorbsallincidentrays:noreflection,notransmissionforallwavelengthsandforallanglesofincidence.
CavityatfixedT,thermalequilibrium
Incomingradiationiscontinuouslyabsorbedandre-emittedbycavitywall
Smallholeà escapingradiationwillapproximateblack-bodyradiationindependentofpropertiesofcavityorhole
BlackbodyRadiation:Background
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Conservationofpowerrequires:
α +ρ +τ =1
α =absorptivityρ =reflectivityτ =transmissivity
BlackbodyRadiation
Astronomical Observing Techniques 2018, Lecture 1: Black Bodies in Space [email protected]
Kirchhoff’sLaw
• Blackbodycavityinthermalequilibriumwithcompletelyopaquesides
• Opaque->transmissivity τ =0• emissivity ε = amount of emitted radiation• Thermalequilibrium->ε +⍴ =1
εα
τ
τρα
ρε
=⎪⎭
⎪⎬
⎫
=
=++
−=
0
11
Kirchhoff’slaw appliestoperfectblackbodyatallwavelengths
Blackbodyabsorbsallradiation:⍺=ε=1
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BlackbodyRadiation:TelescopeDomes
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Credit NOAO/AURA/NSF: www.noao.edu/image_gallery/telescopes.html
SolidAnglesteradian• SolidAngleΩ:2Danglein3Dspace• Measureshowlargetheobjectappearstoanobserver
• Solidangleisexpressedinadimensionlessunitcalledasteradian (sr)
• Fullsphereis4𝜋 sr
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𝐴 = 𝜋𝑟&
PlanckCurve:EquationSpecificintensityIν ofblackbodygivenbyPlanck’slaw:
inunitsof[Wm-2 sr-1 Hz-1]
Inwavelengthunits:
inunitsof[Wm-3 sr-1]
Conversion betweenfrequencyówavelengthunits:
νν
λλλ
dcddcdv 22 or ==
( )1exp
122
3
−⎟⎠
⎞⎜⎝
⎛=
kThc
hTIν
νν
( )1exp
125
2
−⎟⎠
⎞⎜⎝
⎛=
kThc
hcTI
λλλ
Astronomical Observing Techniques 2018, Lecture 1: Black Bodies in Space [email protected]
• Blackbody(BB)isidealizedobjectthatabsorbsallEMradiation• Cold(T~0K)BBsareblack(noemittedorreflectedlight)• AtT>0KBBsabsorbandre-emitcharacteristicEMspectrum
Manyastronomicalsourcesemitclosetoablackbody.
Example:COBEmeasurementofthecosmicmicrowavebackground
BlackbodyRadiation:BlackBody
Astronomical Observing Techniques 2018, Lecture 1: Black Bodies in Space [email protected]
PlanckCurve:Emission,Power&Temperature
TotalradiatedpowerperunitsurfaceproportionaltofourthpoweroftemperatureT:
σ =5.67·10-8 Wm-2 K-4 (Stefan-Boltzmannconstant)
( ) 4TMddTI σνν
ν ==Ω∫ ∫Ω
AssumingBBradiation,astronomersoftenspecifytheemissionfromobjectsviatheireffectivetemperature.
Astronomical Observing Techniques 2018, Lecture 1: Black Bodies in Space [email protected]
PlanckCurve:Approximations
Planck:
Highfrequencies(hv >>kT)èWienapproximation:
Lowfrequencies(hv <<kT)è Rayleigh-Jeans approximation:
( ) ⎟⎠
⎞⎜⎝
⎛−=kTh
chTI νν
ν exp22
3
( ) 22
2 22λ
νν
kTkTc
TI =≈
( )1exp
122
3
−⎟⎠
⎞⎜⎝
⎛=
kThc
hTIν
νν
Astronomical Observing Techniques 2018, Lecture 1: Black Bodies in Space [email protected]
PlanckCurve:PlanetRadiation
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EffectiveTemperature:Wien’sLawTemperaturecorrespondingtomaximumspecificintensitygivenbyWien’sdisplacementlaw:
mK 1098.2or mK 10096.5 3max
3
max
−− ⋅=⋅= TTvc
λ
CoolerBBshavepeakemission(effectivetemperatures)atlongerwavelengthsandatlowerintensities:
1
10
100
1000
10000
0.100 1.000 10.000 100.000 1000.000
wavelength [μm]
effe
ctiv
e te
mpe
ratu
re [K
] T=293K
Astronomical Observing Techniques 2018, Lecture 1: Black Bodies in Space [email protected]
EffectiveTemperature:SolarSpectrum
Astronomical Observing Techniques 2018, Lecture 1: Black Bodies in Space 14http://en.wikipedia.org/wiki/Sunlight#mediaviewer/File:Solar_Spectrum.png
BrightnessTemperature:GreyBodiesManyemittersclosetobutnotperfectblackbodies.Withwavelength-dependentemissivityε<1:
Example:theSun(likemanystars)
( ) ( )1exp
125
2
−⎟⎠
⎞⎜⎝
⎛⋅=
kThc
hcTI
λλ
λελ
Astronomical Observing Techniques 2018, Lecture 1: Black Bodies in Space [email protected]
Brightnesstemperature:temperatureofaperfectblackbodythatreproducestheobservedintensityofagreybodyobjectatfrequencyν.
Forlowfrequencies(hv <<kT):
OnlyforperfectBBsisTb thesameforallfrequencies.
( ) ( ) vb IkvcTT 2
2-Rayleigh
Jeans 2⋅=⋅= νενε
BrightnessTemperature:Definition
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Lambert:CosineLaw
Lambert’scosinelaw:radiantintensityfromanideal,diffusivelyreflectingsurfaceisdirectlyproportionaltothecosineoftheangleθ betweenthesurfacenormalandtheobserver.
Johann Heinrich Lambert (1728 – 1777)
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Lambert:Lambertian EmittersRadianceofLambertian emittersisindependentofdirectionθ ofobservation(i.e.,isotropic).
PerfectblackbodiesareLambertian emitters!
Twoeffectsthatcanceleachother:1. Lambert’scosinelawà radiant
intensityanddΩ arereducedbycos(θ)
2. EmittingsurfaceareadA foragivendΩ isincreasedbycos-1(θ)
Astronomical Observing Techniques 2018, Lecture 1: Black Bodies in Space [email protected]
Lambert:TheSunaLambertian Emitter?
http://www.pa.msu.edu/people/frenchj/moon/moon-5day-1807.jpghttp://sdo.gsfc.nasa.gov/data/Astronomical Observing Techniques 2018, Lecture 1: Black Bodies in Space [email protected]
FluxandIntensity• EnergyfluxF ofstar= π× intensityI averagedoverdisk• Stellardiskaverageinpolarcoordinatesr,φ
• Substituter withRsinθ,μ=cosθ
• Fluxintegratedoverhemisphere
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I = 1πR2
I(r)r dr dϕ0
R
∫0
2π
∫
I = 2 I(θ )0
π /2
∫ sinθ cosθdθ = 2 Iµdµ0
1
∫
F = I θ( )0
π /2
∫0
2π
∫ cosθ sinθdθdφ = 2π Iµdµ0
1
∫
R
r
θ
SummaryofRadiometricQuantities
*10-26 Wm-2Hz-1 =10-23 ergs-1cm-2Hz-1 iscalled1Jansky
*
*
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OriginsinGreekclassificationofstarsaccordingtotheirvisualbrightness.Brighteststarswerem=1,faintestdetectedwithbareeyewerem=6.
FormalizedbyPogson (1856): 1st mag~100× 6th mag
Magnitude Example #stars brighter
-27 Sun
-13 Fullmoon
-5 Venus
0 Vega 4
2 Polaris 48
3.4 Andromeda 250
6 Limitofnakedeye 4800
10 Limitofgoodbinoculars
14 Pluto
27 Visible light limitof8mtelescopes
Magnitudes:Origin
Astronomical Observing Techniques 2018, Lecture 1: Black Bodies in Space [email protected]
Apparentmagnitudeisrelative measureofmonochromaticfluxdensityFλ ofasource:
M0 definesreferencepoint(usuallymagnitudezero).
Inpractice,measurementsthroughtransmissionfilterT(λ)thatdefinesbandwidth:
⎟⎟⎠
⎞⎜⎜⎝
⎛⋅−=−
00 log5.2
FFMm λ
λ
Magnitudes: ApparentMagnitude
( ) ( ) λλλλ λλ dTdFTMm ∫∫∞∞
+−=−00
log5.2log5.2
Astronomical Observing Techniques 2018, Lecture 1: Black Bodies in Space [email protected]
• absolutemagnitude=apparentmagnitudeofsourceifitwereatdistanceD=10parsecs
• m¤ ≈-27,M¤ =4.83atvisiblewavelengths• MMilky Way=−20.5àM¤- MMilky Way=25.3• LuminosityLMilkyWay =1.4×1010 L¤
DmM log55−+=
Magnitudes:AbsoluteMagnitude
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( )2
80
mW1052.2 ; log5.2 −
∞
⋅=⋅−=∫
bolbol
bol FF
dFM
λλ
Magnitudes:BolometricMagnitudeBolometricmagnitude isluminosityexpressedinmagnitudeunits= integralofmonochromaticfluxoverallwavelengths:
Ifsourceradiatesisotropically:
BolometricmagnitudecanalsobederivedfromvisualmagnitudeplusabolometriccorrectionBC:
BCislargeforstarsthathaveapeakemissionverydifferentfromtheSun’s.
W10827.3 ; log5.2log525.0 26⋅=⋅−⋅+−= ΘΘ
LLLDMbol
BCMM Vbol +=
Astronomical Observing Techniques 2018, Lecture 1: Black Bodies in Space [email protected]
PhotometricSystems
Filtersusuallymatchedtoatmospherictransmissionà differentobservatories=differentfiltersàmanyphotometricsystems:
Astronomical Observing Techniques 2018, Lecture 1: Black Bodies in Space 26
0
20
40
60
80
100
300 500 700 900 1100
trans
mis
sion
[%]
U B V R I
0
20
40
60
80
100
300 500 700 900 1100
trans
mis
sion
[%]
Thuan-GunnFilters
u v g r i z
0
50
100
300 500 700 900 1100
trans
mis
sion
[%]
SDSS Filters
u' g' r' i' z'
0
20
40
60
80
100
300 500 700 900 1100
trans
mis
sion
[%]
wavelength [nm]
Stromgren Filters
u v b y
PhotometricSystems:ABandSTMAG
( ) 60.48log5.2 −⋅−= νFABm
ForgivenfluxdensityFv,ABmagnitudeisdefinedas:
• objectwithconstantfluxperunitfrequency intervalhaszerocolor• zeropointdefinedtomatchzeropointsofJohnsonV-band• usedbySDSSandGALEX• Fv inunitsof[ergs-1 cm2 Hz-1]
STMAGsystemdefinedsuchthatobjectwithconstantfluxperunitwavelength intervalhaszerocolor.STMAGsareusedbytheHSTphotometrypackages.
Astronomical Observing Techniques 2018, Lecture 1: Black Bodies in Space [email protected]
Color IndicesColor index=differenceofmagnitudesatdifferentwavebands=ratiooffluxesatdifferentwavelengths
Color-magnitude diagram for a typical globular cluster, M15.
• Color indicesofA0Vstar(Vega)aboutzerolongward ofV
• Color indicesofblackbodyinRayleigh-Jeanstailare:B-V=-0.46U-B=-1.33V-R=V-I=...=V-N=0.0
Astronomical Observing Techniques 2018, Lecture 1: Black Bodies in Space [email protected]
• Absolutemagnitudedefinition:
• InterstellarextinctionE orabsorption A affectstheapparentmagnitudes
• Needtoincludeabsorptiontoobtaincorrectabsolutemagnitude:
DmM log55−+=
Color Index: InterstellarExtinction
ADmM −−+= log55
( ) ( ) ( ) ( ) ( )intrinsicobserved VBVBVABAVBE −−−=−=−
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PhotometricSystems:ConversionsFλ Fν
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SurfaceBrightness:Point&ExtendedSources
Surfacebrightness[mag/arcsec2]isconstantwithdistance!
Pointsources=spatiallyunresolved
Brightness ~1/distance2
Sizegivenbyobserving conditions
Extended sources=wellresolved
Surfacebrightness~const(distance)
Brightness ~1/d2 andareasize~1/d2
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SurfaceBrightness:CalculatingSurfacebrightnessofextendedobjectsinunitsofmag/sr ormag/arcsec2
SurfacebrightnessSofareaA inmagnitudes:
AmS 10log5.2 ⋅+=
Observedsurfacebrightness[mag/arcsec2]convertedintophysicalsurfacebrightnessunits:
with
[ ] [ ]2102 /pclog5.2572.21mag/arcsec ΘΘ ⋅−+= LSMS
-13326 s erg 10×3.839 W 10×3.839 ==ΘL
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