Lecture 2 - Faculty Web Sitesfaculty.tamuc.edu/cbertulani/ast/lectures/Lec2.pdf · introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 15 The first is known as the Hertzsprung-Russell

introduc)ontoAstrophysics,C.Bertulani,TexasA&M-Commerce 11

Lecture 2

Stars: Color and Spectrum






2.1 - Solar spectrum



2.1 - Solar spectrum (as detected on Earth)


Wavelength[m]


Light spectrum from atomic transitions


Theexcitedatomseventuallyemitphotonsandtheelectronsreturntolowerenergyorbitals.Inhydrogen,transi)onstothegroundstate(n=1)yielddiscretelightenergies(lines)namedLymantransi-ons.Transi)onstothefirstexcitedstate(n=2)yieldBalmerlines.

Inahotgas,atomscollideandatomictransi-onsoccur,withelectronsbeingpromotedtohigherorbits.


Emission and absorption lines


LookingatthelightemiRedbystarsasafunc)onofthewavelength(emissionspectrum),onecaniden)fyspecifictransi)onsincertainatoms,suchasthen=3ton=2transi)oninhydrogen(alphaline).

ButsincelightisemiRedbyseveralatomsinnumerouselectronictransi)ons,itiseasiertodetectabsorp-onlines.Aslightpropagatesthroughthestellaratmosphere,itisabsorbedbyhydrogenatomsandtheintensityisseenreducedatthosewavelengths.

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

Color-magnitude diagrams


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)


Magnitudes and Temperatures


Calibration of spectral types.

Magnitudes and Temperatures


Color of stars


Astronomerscorrelatethecolorindexwiththeeffec)vesurfacetemperatureofastar.TheHRdiagram(next)isaplotoftheluminosityofastarorthebolometricmagnitude(totalenergyemiRedbyastar)versusitssurfacetemperature,oritscolorindex.

Colorsofstarsarecomplextodefine.StarcolorindicesweredefinedbyusingtheresponseofphotographicplateswithbandwidthsspanningtheUltraviolet,BlueandVisual(UBV)spectra.ThecolorindexistheBluemagnitudeminustheVisualmagnitude,wherethemagnitudeisgivenbyEq.(1.10).Hence,hotstarsarecharacterizedbysmall,infactnega)ve,colorindexwhilecoldstarshavelargecolorindex.


Color of stars


Ingeneral,aspectralclassifica)on O,B,A,F,G,K,M

with1–10subgroupsisused(Sun:G2),whichisactuallypreRywellcorrelatedtothetemperature.OstarsarethehoRestandtheleRersequenceindicatessuccessivelycoolerstarsuptothecoolestMclass.AusefulmnemonicforrememberingthespectraltypeleRersis“OhBeAFineGirl/GuyKissMe”.Informally,Ostarsarecalled“blue”,B“blue–white”,Astars“white”,Fstars“yellow–white”,Gstars“yellow”,Kstars“orange”,andMstars“red”,eventhoughtheactualstarcolorsperceivedbyanobservermaydeviatefromthesecolorsdependingonvisualcondi)onsandindividualstarsobserved.

B F G K


2.2 - The Hertzsprung-Russell diagram

Thisdiagramshowstypicalmethodsusedbyastronomerstoinferstellarproper)essuchassurfacetemperature,distance,luminosityandradii.

The Hertzsprung-Russell diagram


The first is known as the Hertzsprung-Russell (HR) diagram or the color-magnitude diagram.

M, R, L and T do not vary independently. There are two major relationships – L with T – L with M

L = 4πR2σTeff4 (2.1)From the Stefan-Boltzmann law

Astarcanincreaseluminositybyeitheruppingtheradiusorthetemperature.Withtheradiusconstant,theluminosityversustemperatureinalog–logdiagramisastraightline(mainsequence):log(L)=constant.log(Teff).

•  Starsthathavethesameluminosityasdimmermainsequencestars,butaretothelegofthem(hoRer)ontheHRdiagram,havesmallersurfaceareas(smallerradii).

•  Bright,coolstarsareverylarge(RedGiants)andlieabovethemainsequenceline.

•  Starsthatareveryhotandyets)lldimmusthavesmallsurfaceareas(whitedwarfs)andliebelowthemainsequence.


42eT RL∝Stefan-Boltzmann law 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).

The Hertzsprung-Russell diagram


The Hertzsprung-Russell Diagram (HRD)

Color Index (B-V) –0.6 0 +0.6 +2.0 Spectral type O B A F G K M

TheHRDhasbeenpopulatedwithobserva)onsof22,000starsobtainedwiththeHipparcossatelliteand1,000fromtheGliesecatalogueofnearbystars.

The HRD catalogue


TheastronomerWilhelmGliesepublishedin1957hisfirststarcatalogueofnearlyonethousandstarslocatedwithin20parsecs(65ly)ofEarth.

Hipparcos,waslaunchedin1989bytheEuropeanSpaceAgency(ESA),whichoperatedun)l1993.

wikipedia

Forthefewmain-sequencestarsforwhichmassesareknown,thereisaMass-luminosityrela6on. wheren=3-4.Theslopechangesatextremes,lesssteepforlowandhighmassstars.

Thisimpliesthatthemain-sequence(MS)ontheHRDisafunc)onofmassi.e.fromboTomtotopofmain-sequence,starsincreaseinmass WemustunderstandtheM-Lrela)on

andL-Terela)ontheore)cally.Modelsmustreproduceobserva)ons.

Mass-luminosity relation

nM L ∝

Equa)on(2.2)onlyappliestoMSstarswith2<M<20M¤anddoesnotapplytoredgiantsorwhitedwarfs.

Forstarsbiggerthan20M¤,onefindsL~M.


(2.2)


Lifetime-Mass relation

τ ~ M −2 −M −3 for M < 20 M⊗

τ ~ const for M >> 20 M⊗

Ifaconsiderablemassfrac)onofastarisconsumedinstellarevolu)on,thenthelife)meofastarisgivenby

Mass(M¤) Life-me(years)

Spectraltype

60 3million O3

30 11million O7

10 32million B4

3 370million A5

1.5 3billion F5

1 10billion G2(Sun)

0.1 1000sbillion M7

τ ~ M / L (2.3)

(2.4)

M¤=Sun’smass

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.

Age and Metallicity relation



Weobservestarclusters• Starsallatsamedistance• Dynamicallybound• Sameage• Samechemicalcomposi)onCancontain103–106stars

Star clusters

Star cluster known as the Pleiades

Openclustersarelooselyboundbymutualgravita)onalaRrac)onanddisruptbycloseencounterswithotherclustersandcloudsofgas.Openclusterssurviveforafewhundredmillionyears.Themoremassiveglobularclustersareboundbyastrongergravita)onalaRrac)onandcansurviveformanybillionsofyears.

•  Inclusters,tandZmustbesameforallstars

•  HencedifferencesmustbeduetoM

•  ClusterHRD(orcolor-magnitude)diagramsarequitesimilar–agedeterminesoverallappearance

Globular clusters


Thedifferencesareinterpretedduetoage–openclusterslieinthediskoftheMilkyWayandhavelargerangeofages.Globularclustersareallancient,withtheoldesttracingtheearlieststagesoftheforma)onofMilkyWay(≈12×109yrs).

Globular vs. Open clusters


Globular Open • MSturn-offpointsinsimilarposi)on.GiantbranchjoiningMS

• HorizontalbranchfromgiantbranchtoabovetheMSturn-offpoint

• HorizontalbranchogenpopulatedonlywithvariableRRLyraestars(periodicvariablestars-theprototypeofsuchastarisintheconstella)onLyra)

• MSturnoffpointvariesmassively,faintestisconsistentwithglobulars

• MaximumluminosityofstarscangettoMv≈-10

• Verymassivestarsfoundintheseclusters.


Doppler Shift in Sound

If the source of sound is moving, the pitch changes.

Doppler Shift in Light Shift in wavelength is Δλ = λ – λo = λov/c λ  is the observed (shifted)

wavelength λo is the emitted wavelength v is the source velocity c is the speed of light

Δλ = λ -λ0 = λ0v / c(2.5)


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)

z = λobserved -λrestλrest

=vc

(2.6)


Hubble’s Law

Hubble’s data

distance to galaxy

Rec

essi

onal

vel

ocity

v = H0 d

H0 ~ 70 km/s/Mpc

(2.7)

Hubble found the amount of redshift depends upon the distance

•  the farther away (d), the greater the redshift (v)


Cosmological Redshift Universe expands à redshift. The wavelengths get more stretched.

Size of the Universe when the light was emitted.

Size of the Universe now, when we observe the light.

•  Distances between galaxies are increasing uniformly.

•  There is no need for a center of the universe.

The expansion of the Universe


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.

But distance = velocity x time. The time is how long the expansion has been going on à The Age of the Universe) à

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 à v = H0 d d = v /H0

tUniverse =1/H0 (2.8)


Cosmic Microwave Background (CMB)

This radiation should be cosmologically redshifted - mostly into microwave region – about 2.75 K

As Universe expanded its temperature decreased and so did the temperature of the radiation.

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.


CMB Anisotropies ShortlyagertheCMBwasdiscoveredonerealizedthatthereshouldbeangularvaria)onsintemperature,asaresultofdensityinhomogenei)esintheUniverse.

ThedenserregionscausetheCMBphotonstobegravita)onallyredshigedcomparedtophotonsarisinginlessdenseregions.

TheamplitudeoftheTfluctua)onsisroughly1/3ofthedensityfluctua)ons.

As)mepassed,overdenseregionsbecamegravita)onallyunstableandcollapsedtoformgalaxies,clustersofgalaxiesandallotherstructuresweseeintheUniversetoday.

FromtheobservedCMBangularanisotropiesintemperature,itisstraight-forwardtoderivewhatdensityfluctua)onscreatedthem.


Figure: temperature fluctuations as measured by the satellite Wilkinson Microwave Anisotropy Probe (WMAP).

CMB Anisotropy

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)

ΔTT(θ,ϕ ) =

l=0

∞

∑m=−l

l

∑ almYlm (θ,ϕ ) (2.10)

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

Ylm (θ ,ϕ )∫ Y *l 'm ' (θ ,ϕ )dΩ = δll 'δmm '


CMB Anisotropy

Summing over the m corresponding to the same multipole number l we have the closure relation

Ylm (θ ,ϕ )m=−l

l

∑2

=2l +14π

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

2lma are independent of m.

This allows us to calculate the multipole coefficients alm from

alm = Ylm*∫ θ ,ϕ( )ΔTT θ ,ϕ( )dΩ


CMB Anisotropy

The different alm are independent random variables, so that

alma*lm = δlmδl 'm 'Cl

The function Cl (of integers l ≥ 1) is called the angular power spectrum.

Cl ≡ alm2=

12l +1

alm2

m∑ (2.11)

Since < |alm|2 > is independent of m, we can define

Inserting Eq. (2.11) in Eq. (2.10), one gets

ΔTT

θ ,φ( )"

#$

%

&'

2

= almYlm θ ,φ( ) a *l 'm ' Y*l 'm ' θ ,φ( )

l 'm '∑

lm∑

= Ylm θ ,φ( )mm '∑ Y *

l 'm ' θ ,φ( ) alma *l 'm 'll '∑

= Cll∑ Ylm θ ,φ( )

m∑

2=

2l +14πl

∑ Cl ≈l (l +1)2π∫ Cl d ln l

(2.12)


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)2π

Cl

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.

(2.13)


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. Thus the angle corresponding to this wavelength is The angle corresponding to a “half-wavelength”, i.e., the separation between a neighboring minimum and maximum is then 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.

ϑλ =2πl=3600

l

ϑ res =πl=1800

l

Documents

Lecture 2 - Faculty Web Sitesfaculty.tamuc.edu/cbertulani/ast/lectures/Lec2.pdf · introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 15 The first is known as the Hertzsprung-Russell