Astronomical Observing Techniques 2018 Lecture 1: Black …keller/Teaching/AOT... · 2018. 1. 22. · Content 1. Blackbody Radiation 2. Planck Curve and Approximations 3. Effective

AstronomicalObservingTechniques2018

Lecture1:BlackBodiesinSpace

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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


Conservationofpowerrequires:

α +ρ +τ =1

α =absorptivityρ =reflectivityτ =transmissivity

BlackbodyRadiation


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


BlackbodyRadiation:TelescopeDomes

Astronomical Observing Techniques 2018, Lecture 1: Black Bodies in Space 6

Credit NOAO/AURA/NSF: www.noao.edu/image_gallery/telescopes.html

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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


• Blackbody(BB)isidealizedobjectthatabsorbsallEMradiation• Cold(T~0K)BBsareblack(noemittedorreflectedlight)• AtT>0KBBsabsorbandre-emitcharacteristicEMspectrum

Manyastronomicalsourcesemitclosetoablackbody.

Example:COBEmeasurementofthecosmicmicrowavebackground

BlackbodyRadiation:BlackBody


PlanckCurve:Emission,Power&Temperature

TotalradiatedpowerperunitsurfaceproportionaltofourthpoweroftemperatureT:

σ =5.67·10-8 Wm-2 K-4 (Stefan-Boltzmannconstant)

( ) 4TMddTI σνν

ν ==Ω∫ ∫Ω

AssumingBBradiation,astronomersoftenspecifytheemissionfromobjectsviatheireffectivetemperature.


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


PlanckCurve:PlanetRadiation


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


EffectiveTemperature:SolarSpectrum

Astronomical Observing Techniques 2018, Lecture 1: Black Bodies in Space 14http://en.wikipedia.org/wiki/Sunlight#mediaviewer/File:Solar_Spectrum.png

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BrightnessTemperature:GreyBodiesManyemittersclosetobutnotperfectblackbodies.Withwavelength-dependentemissivityε<1:

Example:theSun(likemanystars)

( ) ( )1exp

125

2

−⎟⎠

⎞⎜⎝

⎛⋅=

kThc

hcTI

λλ

λελ


Brightnesstemperature:temperatureofaperfectblackbodythatreproducestheobservedintensityofagreybodyobjectatfrequencyν.

Forlowfrequencies(hv <<kT):

OnlyforperfectBBsisTb thesameforallfrequencies.

( ) ( ) vb IkvcTT 2

2-Rayleigh

Jeans 2⋅=⋅= νενε

BrightnessTemperature:Definition


Lambert:CosineLaw

Lambert’scosinelaw:radiantintensityfromanideal,diffusivelyreflectingsurfaceisdirectlyproportionaltothecosineoftheangleθ betweenthesurfacenormalandtheobserver.

Johann Heinrich Lambert (1728 – 1777)


Lambert:Lambertian EmittersRadianceofLambertian emittersisindependentofdirectionθ ofobservation(i.e.,isotropic).

PerfectblackbodiesareLambertian emitters!

Twoeffectsthatcanceleachother:1. Lambert’scosinelawà radiant

intensityanddΩ arereducedbycos(θ)

2. EmittingsurfaceareadA foragivendΩ isincreasedbycos-1(θ)


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


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

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SummaryofRadiometricQuantities

*10-26 Wm-2Hz-1 =10-23 ergs-1cm-2Hz-1 iscalled1Jansky

*

*


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


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


• 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


( )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 +=


PhotometricSystems

Filtersusuallymatchedtoatmospherictransmissionà differentobservatories=differentfiltersàmanyphotometricsystems:


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

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PhotometricSystems:ABandSTMAG

( ) 60.48log5.2 −⋅−= νFABm

ForgivenfluxdensityFv,ABmagnitudeisdefinedas:

• objectwithconstantfluxperunitfrequency intervalhaszerocolor• zeropointdefinedtomatchzeropointsofJohnsonV-band• usedbySDSSandGALEX• Fv inunitsof[ergs-1 cm2 Hz-1]

STMAGsystemdefinedsuchthatobjectwithconstantfluxperunitwavelength intervalhaszerocolor.STMAGsareusedbytheHSTphotometrypackages.


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


• Absolutemagnitudedefinition:

• InterstellarextinctionE orabsorption A affectstheapparentmagnitudes

• Needtoincludeabsorptiontoobtaincorrectabsolutemagnitude:

DmM log55−+=

Color Index: InterstellarExtinction

ADmM −−+= log55

( ) ( ) ( ) ( ) ( )intrinsicobserved VBVBVABAVBE −−−=−=−


PhotometricSystems:ConversionsFλ Fν


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


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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Astronomical Observing Techniques 2018 Lecture 1: Black …keller/Teaching/AOT... · 2018. 1. 22. · Content 1. Blackbody Radiation 2. Planck Curve and Approximations 3. Effective