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AstronomicalObservingTechniques
Lecture1:BlackBodiesinSpace
ChristophU.Keller
[email protected]
Outline
1.  BlackBodyRadia<on
2.  AstronomicalMagnitudes
3.  PointSourcesandExtendedSources
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BlackbodyRadia@on
Kirchhoff(1860):blackbodycompletelyabsorbsallincident
rays:noreflec<on,notransmissionforallwavelengthsand
forallanglesofincidence.
CavityatfixedT,thermalequilibrium
Incomingradia<onis"thermalized“by
con<nuousabsorp<onandre-emissionof
radia<onbycavitywall
Smallholeàescapingradia<onwill
approximateblack-bodyradia<on
independentofproper<esofcavityor
hole.
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Kirchhoff’sLaw
Conserva<onofpowerrequires:
α + ρ +τ = 1
α
ρ
withα=absorp<vity,ρ=reflec<vity,τ=transmissivity
cavityinthermal
equilibriumwith
completelyopaque
sides:
ε = 1− ρ
α + ρ +τ = 1
τ =0
⎫
⎪
⎬ α =ε
⎪
⎭
τ
ε=emissivity
Kirchhoff’slawappliestoperfectblackbodyatallwavelengths
Radiatorwithε=ε(λ)<1obencalledgreybody
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TheColorofTelescopeDomes
Credit NOAO/AURA/NSF: www.noao.edu/image_gallery/telescopes.html
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Defini@onofaBlackBody
• Blackbody(BB)isidealizedobjectthatabsorbsallEMradia<on
• Cold(T~0K)BBsareblack(noemieedorreflectedlight)
• AtT>0KBBsabsorbandre-emitcharacteris<cEMspectrum
Manyastronomicalsources
emitclosetoablackbody.
Example:COBEmeasurement
ofthecosmicmicrowave
background
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BlackBodyEmission
SpecificintensityIνofblackbodygivenbyPlanck’slaw:
2hν 3
1
Iν (T ) = 2
inunitsof[Wm-2sr-1Hz-1]
c ⎛ hν ⎞ exp⎜
⎟ −1
⎝ kT ⎠
Inwavelengthunits:
2hc 2
1
-3sr-1]
I λ (T ) = 5
inunitsof[Wm
λ
⎛ hc ⎞
exp⎜
⎟ −1
⎝ λkT ⎠
Conversionoffrequencyówavelengthunits:
c
c
dv = 2 dλ or dλ = 2 dν
λ
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ν
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EmissionóPoweróTemperature
Totalradiatedpowerperunitsurfacepropor<onaltofourthpower
oftemperatureT:
4
I
T
d
ν
d
Ω
=
M
=
σ
T
ν
Ων
σ=5.67·10-8Wm-2K-4(Stefan-Boltzmannconstant)
∫∫ ( )
AssumingBBradia?on,astronomersoAenspecifytheemissionfrom
objectsviatheireffec?vetemperature.
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Effec@veTemperatures
Temperaturecorrespondingtomaximumspecificintensitygiven
byWien’sdisplacementlaw:
c
T = 5.096 ⋅10 −3 mK or λmaxT = 2.98 ⋅10 −3 mK
vmax
CoolerBBshavepeakemission
(effec<vetemperatures)atlonger
wavelengthsandatlower
intensi<es:
effective temperature [K]
10000
T=293K
1000
100
10
1
0.100
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1.000
10.000
100.000
1000.000
wavelength [µm]
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UsefulApproxima@ons
2hν 3
1
Planck:
Iν (T ) = 2
c
⎛ hν ⎞
exp⎜
⎟ −1
⎝ kT ⎠
Highfrequencies(hv>>kT)èWienapproxima<on:
3
2
h
ν
⎛ hν ⎞
Iν (T ) = 2 exp⎜ −
⎟
c
⎝ kT ⎠
Lowfrequencies(hv<<kT)èRayleigh-Jeansapproxima<on:
2ν 2
2kT
Iν (T ) ≈ 2 kT = 2
c
λ
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SolarSpectrum
http://en.wikipedia.org/wiki/Sunlight#mediaviewer/File:Solar_Spectrum.png
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GreyBodies
Manyemieersclosetobutnotperfectblackbodies.
Withwavelength-dependentemissivityε<1:
2hc 2
1
I λ (T ) = ε (λ )⋅ 5
λ
⎛ hc ⎞
exp⎜
⎟ −1
⎝ λkT ⎠
Example:theSun
(likemanystars)
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BrightnessTemperature
Brightnesstemperatureistemperatureaperfectblackbodywould
havetoreproducetheobservedintensityofagreybodyobjectat
frequencyν.
Forlowfrequencies(hv<<kT):
2
Rayleighc
Tb = ε ν ⋅ T = ε ν ⋅
Iv
2
Jeans
2kv
OnlyforperfectBBsisTbthesameforallfrequencies.
( )
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Lambert’sCosineLaw
Lambert’scosinelaw:radiantintensityfromanideal,
diffusivelyreflec<ngsurfaceisdirectlypropor<onaltothe
cosineoftheangleθbetweenthesurfacenormalandthe
observer.
Johann Heinrich Lambert
(1728 – 1777)
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Lamber@anEmiNers
RadianceofLamber<anemieersisindependentofdirec<onθof
observa<on(i.e.,isotropic).
Twoeffectsthatcanceleachother:
1.  Lambert’scosinelawàradiant
intensityanddΩarereducedby
cos(θ)
2.  EmivngsurfaceareadAfora
givendΩisincreasedbycos-1(θ)
Perfectblackbodiesare
Lamber?anemiJers!
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TheSun:Lamber@anEmiNer?
http://www.pa.msu.edu/people/frenchj/moon/moon-5day-1807.jpg
http://sdo.gsfc.nasa.gov/data/
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FluxandIntensity
•  EnergyfluxFofstar=π×intensityIaveragedoverdisk
•  Stellardiskaverageinpolarcoordinatesr,φ
2π R
r
1
I =
I(r)r dr dϕ
2 ∫ ∫
πR 0 0
•  Subs<tuterwithRsinθ,μ=cosθ
R
θ
π /2
1
I = 2 ∫ I(θ ) sin θ cosθ dθ = 2 ∫ I µ d µ
0
0
•  Fluxintegratedoverhemisphere
F=∫
2π
0
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∫
π /2
0
I (θ ) cosθ sin θ dθ dφ = 2π ∫ I µ d µ
1
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SummaryofRadiometricQuan@@es
*
*
*10-26Wm-2Hz-1=10-23ergs-1cm-2Hz-1iscalled1Jansky
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Op@calAstronomersuse‘Magnitudes’
OriginsinGreekclassifica?onofstarsaccordingtotheirvisualbrightness.
Brighteststarswerem=1,faintestdetectedwithbareeyewerem=6.
FormalizedbyPogson(1856):1stmag~100×6thmag
Magnitude
#starsbrighter
-27
Sun
-13
Fullmoon
-5
Venus
0
Vega
4
2
Polaris
48
Andromeda
250
6
Limitofnakedeye
4800
10
Limitofgoodbinoculars
14
Pluto
27
Visiblelightlimitof8mtelescopes
3.4
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Example
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ApparentMagnitude
Apparentmagnitudeisrela?vemeasureof
monochroma<cfluxdensityFλofasource:
⎛ Fλ ⎞
mλ − M 0 = −2.5 ⋅ log⎜⎜ ⎟⎟
F0 ⎠
⎝
M0definesreferencepoint(usuallymagnitudezero).
Inprac<ce,measurements
throughtransmissionfilterT(λ)
thatdefinesbandwidth:
∞
∞
m − M = −2.5 log T (λ )F dλ + 2.5 log T (λ )dλ
λ
∫
0
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λ
∫
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PhotometricSystems
Filtersusuallymatchedtoatmospherictransmission
à differentobservatories=differentfilters
à manyphotometricsystems:
U
B
V
R
I
u
transmission [%]
60
40
20
0
300
500
u'
700
g'
900
transmission [%]
g
r
i
z
r'
i'
50
z'
SDSS Filters
500
700
900
1100
80
60
Thuan-Gunn
Filters
40
20
0
300
1100
100
0
300
v
100
80
500
700
u
100
transmission [%]
transmission [%]
100
v
900
b
1100
y
80
60
Stromgren Filters
40
20
0
300
500
700
900
1100
wavelength [nm]
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ABandSTMAGSystems
ForgivenfluxdensityFv,ABmagnitudedefinedas:
m( AB ) = −2.5 ⋅ log Fν − 48.60
• objectwithconstantfluxperunitfrequencyintervalhaszerocolor
• zeropointdefinedtomatchzeropointsofJohnsonV-band
• usedbySDSSandGALEX
• Fvinunitsof[ergs-1cm2Hz-1]
STMAGsystemdefinedsuchthatobjectwithconstantfluxperunit
wavelengthintervalhaszerocolor.STMAGsareusedbytheHST
photometrypackages.
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ColorIndices
Colorindex=differenceofmagnitudesatdifferentwavebands=
ra<ooffluxesatdifferentwavelengths
• ColorindicesofA0Vstar(Vega)
aboutzerolongwardofV
• Colorindicesofblackbodyin
Rayleigh-Jeanstailare:
B-V=-0.46
U-B=-1.33
V-R=V-I=...=V-N=0.0
Color-magnitude diagram for a typical globular
cluster, M15.
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AbsoluteMagnitude
Absolutemagnitude=apparentmagnitudeofsourceifitwereat
distanceD=10parsecs:
M = m + 5 − 5 log D
MSun=4.83(V);MMilkyWay=−20.5àΔmag=25.3àΔlumi=14billionLo
However,interstellarex<nc<onEorabsorp<onAaffectsthe
apparentmagnitudes
E (B − V ) = A(B ) − A(V ) = (B − V )observed − (B − V )intrinsic
Needtoincludeabsorp<ontoobtaincorrectabsolutemagnitude:
M = m + 5 − 5 log D − A
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BolometricMagnitude
Bolometricmagnitudeisluminosityexpressedinmagnitudeunits=
integralofmonochroma<cfluxoverallwavelengths:
∞
M bol = −2.5 ⋅ log
∫ F (λ )dλ
0
Fbol
; Fbol
W
= 2.52 ⋅10
m2
−8
Ifsourceradiatesisotropically:
L
M
; LΘ = 3.827 ⋅10 26 W
bol = −0.25 + 5 ⋅ log D − 2.5 ⋅ log
LΘ
Bolometricmagnitudecanalsobederivedfromvisualmagnitude
plusabolometriccorrec<onBC:
M bol = M V + BC
BCislargeforstarsthathaveapeakemissionverydifferentfromtheSun’s.
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PhotometricSystemsandConversions
Fλ
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F
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PointSourcesandExtendedSources
Pointsources=spa<allyunresolved
Extendedsources=wellresolved
Brightness~1/distance2
Surfacebrightness~const(distance)
Sizegivenbyobservingcondi<ons
Brightness~1/d2andareasize~1/d2
Surfacebrightness[mag/arcsec2]isconstantwithdistance!
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Calcula@ngSurfaceBrightness
Surfacebrightnessofextendedobjectsinunitsof
mag/srormag/arcsec2
SurfacebrightnessSofareaAinmagnitudes:
S = m + 2.5 ⋅ log10 A
Observedsurfacebrightness[mag/arcsec2]convertedinto
physicalsurfacebrightnessunits:
2
2
S mag/arcsec = M Θ + 21.572 − 2.5 ⋅ log10 S LΘ /pc
withLΘ = 3.839 ×10 26 W = 3.839 ×1033 erg s -1
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