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Lecture 2
Stars: Color and Spectrum
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2.1 - Solar spectrum
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2.1 - Solar spectrum (as detected on Earth)
Wavelength[m]
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Light spectrum from atomic transitions
Inahotgas,atomscollideandatomictransi-ons
occur,withelectronsbeingpromotedtohigherorbits.
Theexcitedatomseventuallyemitphotons
andtheelectronsreturntolowerenergy
orbitals.Inhydrogen,transi)onstotheground
state(n=1)yielddiscretelightenergies(lines)
namedLymantransi-ons.Transi)onstothe
firstexcitedstate(n=2)yieldBalmerlines.
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Emission and absorption lines
LookingatthelightemiRedbystarsasa
func)onofthewavelength(emission
spectrum),onecaniden)fyspecific
transi)onsincertainatoms,suchasthe
n=3ton=2transi)oninhydrogen(alpha
line).
ButsincelightisemiRedbyseveral
atomsinnumerouselectronic
transi)ons,itiseasiertodetect
absorp-onlines.Aslightpropagates
throughthestellaratmosphere,itis
absorbedbyhydrogenatomsandthe
intensityisseenreducedatthose
wavelengths.
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Color-magnitude diagrams
Measuring accurate Te for ~102 or 103 stars is intensive task – spectra are
needed and also model of atmospheres.
Magnitudes of stars are measured at different wavelengths: standard system is
UBVRI
Band
U
B
V
R
I
λ[nm]
365
445
551
658
806
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Magnitudes and Temperatures
One has to model stellar spectra at
different temperature, e.g., Te = 40,000,
30,000, 20,000 K, to obtain a function
f(Te)) so that B - V = f(Te)
It amounts in separating the flux into
different wavelength bands, finding the
wavelength for maximum strength and
finding temperature which fits that.
Various calibrations can be used to provide
the color relation
B - V = f(Te)
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Magnitudes and Temperatures
Calibration of spectral types.
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Color of stars
Colorsofstarsarecomplextodefine.Starcolorindicesweredefinedbyusingthe
responseofphotographicplateswithbandwidthsspanningtheUltraviolet,Blue
andVisual(UBV)spectra.ThecolorindexistheBluemagnitudeminustheVisual
magnitude,wherethemagnitudeisgivenbyEq.(1.10).Hence,hotstarsare
characterizedbysmall,infactnega)ve,colorindexwhilecoldstarshavelarge
colorindex.
Astronomerscorrelatethe
colorindexwiththe
effec)vesurface
temperatureofastar.The
HRdiagram(next)isaplot
oftheluminosityofastar
orthebolometric
magnitude(totalenergy
emiRedbyastar)versusits
surfacetemperature,orits
colorindex.
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Color of stars
Ingeneral,aspectralclassifica)on
O,B,A,F,G,K,M
with1–10subgroupsisused(Sun:G2),whichisactuallypreRywellcorrelatedtothe
temperature.OstarsarethehoRestandtheleRersequenceindicatessuccessively
coolerstarsuptothecoolestMclass.Ausefulmnemonicforrememberingthe
spectraltypeleRersis“OhBeAFineGirl/GuyKissMe”.Informally,Ostarsarecalled
“blue”,B“blue–white”,Astars“white”,Fstars“yellow–white”,Gstars“yellow”,K
stars“orange”,andMstars“red”,eventhoughtheactualstarcolorsperceivedbyan
observermaydeviatefromthesecolorsdependingonvisualcondi)onsand
individualstarsobserved.
B
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F G
K
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2.2 - The Hertzsprung-Russell diagram
Thisdiagramshowstypicalmethods
usedbyastronomerstoinferstellar
proper)essuchassurface
temperature,distance,luminosity
andradii.
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The Hertzsprung-Russell diagram
M, R, L and T do not vary independently. There are two major relationships
– L with T
– L with M
The first is known as the Hertzsprung-Russell (HR) diagram or the colormagnitude diagram.
From the Stefan-Boltzmann law
2
4
eff
L = 4πR σ T
(2.1)
Astarcanincreaseluminositybyeitheruppingtheradiusorthetemperature.
Withtheradiusconstant,theluminosityversustemperatureinalog–logdiagramis
astraightline(mainsequence):log(L)=constant.log(Teff).
•  Starsthathavethesameluminosityasdimmermainsequencestars,butareto
thelegofthem(hoRer)ontheHRdiagram,havesmallersurfaceareas(smaller
radii).
•  Bright,coolstarsareverylarge(RedGiants)andlieabovethemainsequence
line.
•  Starsthatareveryhotandyets)lldimmusthavesmallsurfaceareas(white
dwarfs)andliebelowthemainsequence.
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The Hertzsprung-Russell diagram
Stefan-Boltzmann law
2
4
e
L∝ R T
shows that L correlates with T
à  Hertzprung-Russell’s idea of plotting L vs. T and find a path in the diagram
where some information about R could be found à discovery of main sequence
stars (large majority of stars along the shaded band).
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The Hertzsprung-Russell Diagram (HRD)
Color Index (B-V)
Spectral type
–0.6
0
+0.6
OB A F G
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+2.0
K
M
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The HRD catalogue
TheHRDhasbeenpopulated
withobserva)onsof22,000stars
obtainedwiththeHipparcos
satelliteand1,000fromthe
Gliesecatalogueofnearbystars.
TheastronomerWilhelmGliese
publishedin1957hisfirststar
catalogueofnearlyonethousand
starslocatedwithin20parsecs
(65ly)ofEarth.
Hipparcos,waslaunchedin1989
bytheEuropeanSpaceAgency
(ESA),whichoperatedun)l1993.
wikipedia
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Mass-luminosity relation
Forthefewmain-sequencestarsfor
whichmassesareknown,thereisa
Mass-luminosityrela6on.
n
L ∝M
(2.2)
wheren=3-4.Theslopechangesat
extremes,lesssteepforlowandhigh
massstars.
Thisimpliesthatthemain-sequence(MS)
ontheHRDisafunc)onofmassi.e.from
boTomtotopofmain-sequence,stars
WemustunderstandtheM-Lrela)on
increaseinmass
andL-Terela)ontheore)cally.Models
mustreproduceobserva)ons.
Equa)on(2.2)onlyappliestoMSstars
with
Forstarsbiggerthan20M¤,onefinds
2<M<20M¤
anddoesnotapplytoredgiantsorwhite L~M.
dwarfs.
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Lifetime-Mass relation
Ifaconsiderablemassfrac)onofastarisconsumedinstellarevolu)on,then
thelife)meofastarisgivenby
(2.3)
τ ~M /L
τ ~ M −2 − M −3
for M < 20 M ⊗
τ ~ const
for M >> 20 M ⊗
Mass(M¤) Life-me
(years)
Spectral
type
60
3million
O3
30
11million
O7
10
32million
B4
3
370million
A5
1.5
3billion
F5
1
10billion
G2(Sun)
0.1
1000sbillion
M7
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(2.4)
M¤=Sun’smass
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Age and Metallicity relation
There are two other fundamental properties of stars that we can measure –
age (time t) and chemical composition (X, Y, Z).
Composition parameterized with the notation:
X = mass fraction of hydrogen H
Y = mass fraction of helium He
Z = mass fraction of all other elements
e.g., for the Sun: X¤ = 0.747 ; Y¤ = 0.236 ; Z¤ = 0.017
Note: Z is often referred to as metallicity
We would like to study stars of same age and same chemical composition – to
keep these parameters constant and determine how models reproduce the
other observables.
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Star clusters
Weobservestarclusters
• Starsallatsamedistance
• Dynamicallybound
• Sameage
• Samechemicalcomposi)on
Cancontain103–106stars
Openclustersarelooselyboundbymutual
gravita)onalaRrac)onanddisruptbyclose
encounterswithotherclustersandcloudsof
gas.
Openclusterssurviveforafewhundred
millionyears.
Themoremassiveglobularclustersare
boundbyastrongergravita)onalaRrac)on
andcansurviveformanybillionsofyears.
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Star cluster known as the Pleiades
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Globular clusters
•  Inclusters,tandZmustbesamefor
allstars
•  HencedifferencesmustbeduetoM
•  ClusterHRD(orcolor-magnitude)
diagramsarequitesimilar–age
determinesoverallappearance
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Globular vs. Open clusters
Globular
Open
• MSturn-offpointsinsimilarposi)on.Giant
branchjoiningMS
• Horizontalbranchfromgiantbranchtoabove
theMSturn-offpoint
• Horizontalbranchogenpopulatedonlywith
variableRRLyraestars(periodicvariablestars-
theprototypeofsuchastarisinthe
constella)onLyra)
• MSturnoffpointvaries
massively,faintestis
consistentwithglobulars
• Maximumluminosityof
starscangettoMv≈-10
• Verymassivestarsfound
intheseclusters.
Thedifferencesareinterpreted
duetoage–openclustersliein
thediskoftheMilkyWayand
havelargerangeofages.
Globularclustersareallancient,
withtheoldesttracingtheearliest
stagesoftheforma)onofMilky
Way(≈12×109yrs).
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Doppler Shift in Sound
If the source of sound is moving, the pitch changes.
Doppler Shift in Light
Shift in wavelength is
Δλ = λ - λ = λ0 v / c
Δλ = λ – λo = λov/c
0
(2.5)
λ  is the observed (shifted)
wavelength
λo is the emitted wavelength
v is the source velocity
c is the speed of light
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Redshift and Blueshift
• 
Observed increase in
wavelength is called a
redshift
• 
Decrease in observed
wavelength is called a
blueshift
• 
Doppler shift is used to
determine an object’s
velocity
•  Edwin Hubble (1889-1953) and colleagues
§  measured the spectra (light) of many galaxies
§  found nearly all galaxies are red-shifted
•  Redshift (z)
λobserved - λrest v
z=
=
λrest
c
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(2.6)
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Hubble’s Law
Recessional velocity
Hubble found the amount of redshift depends upon the distance
•  the farther away (d), the greater the redshift (v)
v = H0 d
(2.7)
H0 ~ 70 km/s/Mpc
Hubble’s data
distance to galaxy
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The expansion of the Universe
•  Distances between galaxies are
increasing uniformly.
•  There is no need for a center of the
universe.
Cosmological Redshift
Universe expands à redshift. The wavelengths get more stretched.
Size of the
Universe when the
light was emitted.
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Size of the Universe
now, when we observe
the light.
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Looking Back in Time
• 
It takes time for light to reach us: (a) c = 300,000 km/s, (b) We see things
“as they were” some time ago.
• 
The farther away, the further back in time we are looking
–  1 billion ly means looking 1 billion years back in time.
• 
The greater the redshift, the further back in time
–  redshift of 0.1 is 1.4 billion ly which means we are looking 1.4 billion years
into the past.
All galaxies are moving away from each other à in the past all galaxies were
closer to each other.
All the way back in time, it would mean that everything started out at the same
point - then began expanding.
This starting time is called the Big Bang.
The age of the Universe can be calculated using Hubble’s Law
v = H0 d
à
d = v / H0
But distance = velocity x time. The time is how long the expansion has been
going on à The Age of the Universe)
(2.8)
à
Universe
0
t
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=1/ H
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Cosmic Microwave Background (CMB)
As Universe expanded its temperature decreased and so did the
temperature of the radiation.
This radiation should be cosmologically redshifted
- mostly into microwave region – about 2.75 K
Twenty years after its prediction, it was found by
Penzias and Wilson in 1964, for which they got
Nobel prize. It is incredibly uniform across sky and
the spectrum follows incredibly close to Planck’s
blackbody radiation spectrum.
Above: how the sky looks at T=2.7 K.
Right: distribution of radiation as a function
of wavelength measure by the COBE satellite
compared to blackbody radiation for T=2.7 K.
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CMB Anisotropies
ShortlyagertheCMBwasdiscoveredone
realizedthatthereshouldbeangular
varia)onsintemperature,asaresultof
densityinhomogenei)esintheUniverse.
ThedenserregionscausetheCMBphotonsto
begravita)onallyredshigedcomparedto
photonsarisinginlessdenseregions.
TheamplitudeoftheTfluctua)onsisroughly
1/3ofthedensityfluctua)ons.
As)mepassed,overdenseregionsbecame
gravita)onallyunstableandcollapsedtoform
galaxies,clustersofgalaxiesandallother
structuresweseeintheUniversetoday.
FromtheobservedCMBangularanisotropiesin
temperature,itisstraight-forwardtoderive
whatdensityfluctua)onscreatedthem.
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CMB Anisotropy
Figure: temperature fluctuations
as measured by the satellite
Wilkinson Microwave Anisotropy
Probe (WMAP).
The fluctuations in temperature are at a
level of 10-5 T, and difficult to measure –
first detection was in 1992.
The angular distribution of the temperature fluctuations are expanded in terms
of spherical harmonics (any regular function of θ and φ can be expanded in
spherical harmonics)
∞
l
ΔT
(θ , ϕ ) = ∑
T
l=0
∑
almYlm (θ , ϕ )
(2.10)
m=−l
where the sum runs over l = 1, 2, . . .∞ and m = − 1, . . . , 1, giving 2l +1 values of
m for each l.
The spherical harmonics are orthonormal functions on the sphere, so that
∫ Y lm (θ ,ϕ )Y
*
l 'm '
(θ ,ϕ )d Ω = δll 'δmm '
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CMB Anisotropy
This allows us to calculate the multipole coefficients alm from
a lm = ∫ Y
*
lm
ΔT
θ ,ϕ
θ ,ϕ d Ω
T
( )
( )
Summing over the m corresponding to the same multipole number l we have the
closure relation
l
∑
m =− l
2
2l +1
Y lm (θ ,ϕ ) =
4π
Since alm represent a deviation from the average temperature, their expectation
value is zero, < alm > = 0 , and the quantity we want to calculate is the variance
< |alm|2 > to get a prediction for the typical size of the alm. The isotropic nature
of the random process shows up in the alm so that these expectation values
depend only on l not m. (The l are related to the angular size of the anisotropy
pattern, whereas the m are related to “orientation” or “pattern”.)
The brackets < > mean an average over all observers in the Universe. The
absence of a preferred direction in the Universe implies that the coefficients
alm
2
are independent of m.
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CMB Anisotropy
Since < |alm|2 > is independent of m, we can define
C l ≡ a lm
2
1
=
a lm
∑
2l +1 m
2
(2.11)
The different alm are independent random variables, so that
a lm a *lm = δlmδl ' m 'C l
The function Cl (of integers l ≥ 1) is called the angular power spectrum.
Inserting Eq. (2.11) in Eq. (2.10), one gets
" ΔT
%
θ ,φ '
$
# T
&
( )
2
=
∑a
lm
Y θ , φ ) ∑ a *l ' m ' Y
lm lm (
l 'm '
( )
= ∑ ∑Y lm θ , φ Y
ll ' mm '
m
*
l 'm '
( )
= ∑C l ∑ Y lm θ , φ
l
*
l 'm '
2
(θ ,φ ) a
(θ ,φ )
*
a
lm l ' m '
2l +1
=∑
Cl ≈
4π
l
∫
l (l +1)
C l d ln l
2π
(2.12)
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CMB Anisotropy
In the last step the approximation (valid for large values of l) is the reason why
instead of Cl one often uses
l(l +1)
Cl
2π
(2.13)
Thus, if we plot (2l + 1)Cl /4π on a linear l scale, or l(2l + 1)Cl /4π on a logarithmic
l scale, the area under the curve gives the temperature variance, i.e., the
expectation value for the squared deviation from the average temperature. It has
become customary to plot the angular power spectrum as l(l + 1)Cl /2π, which is
neither of these, but for large l approximates the second case.
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CMB Anisotropy
The different multipole numbers l correspond to different angular scales, low l to
large scales and high l to small scales. Examination of the functions Ylm(θ, φ)
reveals that they have an oscillatory pattern on the sphere, so that there are
typically l “wavelengths” of oscillation around a full great circle of the sphere.
2π 3600
Thus the angle corresponding to this wavelength is ϑ λ =
=
l
l
The angle corresponding to a “half-wavelength”, i.e., the separation between a
neighboring minimum and maximum is then
0
ϑ res =
π 180
=
l
l
This is the angular resolution required of the microwave detector for it to be able
to resolve the angular power spectrum up to this l.
For example, COBE had an angular resolution of 70 allowing a measurement up
To l = 180/7 = 26, WMAP had resolution 0.230 reaching to l = 180/0.23 = 783.
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