Lecture 2. Stars: Color and Spectrum. introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 1

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1 Lecture 2 Stars: Color and Spectrum introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 1

2 introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 2

3 introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 3

4 introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 4

5 2.1 - Solar spectrum introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 5

6 2.1 - Solar spectrum (as detected on Earth) Wavelength [m] introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 6

7 Light spectrum from atomic transitions In a hot gas, atoms collide and atomic transi-ons occur, with electrons being promoted to higher orbits. The excited atoms eventually emit photons and the electrons return to lower energy orbitals. In hydrogen, transi)ons to the ground state (n=1) yield discrete light energies (lines) named Lyman transi-ons. Transi)ons to the first excited state (n=2) yield Balmer lines. introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 7

8 Emission and absorption lines Looking at the light emired by stars as a func)on of the wavelength (emission spectrum), one can iden)fy specific transi)ons in certain atoms, such as the n=3 to n=2 transi)on in hydrogen (alpha line). But since light is emired by several atoms in numerous electronic transi)ons, it is easier to detect absorp-on lines. As light propagates through the stellar atmosphere, it is absorbed by hydrogen atoms and the intensity is seen reduced at those wavelengths. introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 8

9 Color-magnitude diagrams Measuring accurate T e for ~10 2 or 10 3 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] introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 9

10 Magnitudes and Temperatures One has to model stellar spectra at different temperature, e.g., T e = 40,000, 30,000, 20,000 K, to obtain a function f(t e )) 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(t e ) introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 10

11 Magnitudes and Temperatures Calibration of spectral types. introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 11

12 Color of stars Colors of stars are complex to define. Star color indices were defined by using the response of photographic plates with band widths spanning the Ultraviolet, Blue and Visual (UBV) spectra. The color index is the Blue magnitude minus the Visual magnitude, where the magnitude is given by Eq. (1.10). Hence, hot stars are characterized by small, in fact nega)ve, color index while cold stars have large color index. Astronomers correlate the color index with the effec)ve surface temperature of a star. The HR diagram(next) is a plot of the luminosity of a star or the bolometric magnitude (total energy emired by a star) versus its surface temperature, or its color index. introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 12

13 Color of stars In general, a spectral classifica)on O, B, A, F, G, K, M with 1 10 subgroups is used (Sun: G2), which is actually prery well correlated to the temperature. O stars are the horest and the lerer sequence indicates successively cooler stars up to the coolest M class. A useful mnemonic for remembering the spectral type lerers is Oh Be A Fine Girl/Guy Kiss Me. Informally, O stars are called blue, B blue white, A stars white, F stars yellow white, G stars yellow, K stars orange, and M stars red, even though the actual star colors perceived by an observer may deviate from these colors depending on visual condi)ons and individual stars observed. B F G K introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 13

14 2.2 - The Hertzsprung-Russell diagram This diagram shows typical methods used by astronomers to infer stellar proper)es such as surface temperature, distance, luminosity and radii. introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 14

15 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 L = 4πR 2 4 σt (2.1) eff A star can increase luminosity by either upping the radius or the temperature. With the radius constant, the luminosity versus temperature in a log log diagram is a straight line (main sequence): log(l) = constant. log(t eff ). Stars that have the same luminosity as dimmer main sequence stars, but are to the leg of them (horer) on the HR diagram, have smaller surface areas (smaller radii). Bright, cool stars are very large (Red Giants) and lie above the main sequence line. Stars that are very hot and yet s)ll dim must have small surface areas (white dwarfs) and lie below the main sequence. introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 15

16 The Hertzsprung-Russell diagram Stefan-Boltzmann law L R 2 T 4 shows that L correlates with T e 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). introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 16

17 The Hertzsprung-Russell Diagram (HRD) Color Index (B-V) Spectral type O B A F G K M introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 17

18 The HRD catalogue The HRD has been populated with observa)ons of 22,000 stars obtained with the Hipparcos satellite and 1,000 from the Gliese catalogue of nearby stars. The astronomer Wilhelm Gliese published in 1957 his first star catalogue of nearly one thousand stars located within 20 parsecs (65 ly) of Earth. Hipparcos, was launched in 1989 by the European Space Agency (ESA), which operated un)l wikipedia introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 18

19 Mass-luminosity relation For the few main-sequence stars for which masses are known, there is a Mass-luminosity rela6on. L M n (2.2) where n = 3-4. The slope changes at extremes, less steep for low and high mass stars. This implies that the main-sequence (MS) on the HRD is a func)on of mass i.e. from botom to top of main-sequence, stars increase in mass Equa)on (2.2) only applies to MS stars with 2 < M < 20 M and does not apply to red giants or white dwarfs. We must understand the M-L rela)on and L-T e rela)on theore)cally. Models must reproduce observa)ons. For stars bigger than 20 M, one finds L ~ M. introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 19

20 Lifetime-Mass relation If a considerable mass frac)on of a star is consumed in stellar evolu)on, then the life)me of a star is given by τ ~ M / L (2.3) τ ~ M 2 M 3 for M < 20 M τ ~ const for M >> 20 M (2.4) Mass (M ) Life-me (years) 60 3 million O million O million B million A billion F5 Spectral type 1 10 billion G2 (Sun) s billion M7 M = Sun s mass introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 20

21 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 = ; Y = ; Z = 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. introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 21

22 Star clusters We observe star clusters Stars all at same distance Dynamically bound Same age Same chemical composi)on Can contain stars Open clusters are loosely bound by mutual gravita)onal arrac)on and disrupt by close encounters with other clusters and clouds of gas. Open clusters survive for a few hundred million years. The more massive globular clusters are bound by a stronger gravita)onal arrac)on and can survive for many billions of years. Star cluster known as the Pleiades introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 22

23 Globular clusters In clusters, t and Z must be same for all stars Hence differences must be due to M Cluster HRD (or color-magnitude) diagrams are quite similar age determines overall appearance introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 23

24 Globular vs. Open clusters Globular MS turn-off points in similar posi)on. Giant branch joining MS Horizontal branch from giant branch to above the MS turn-off point Horizontal branch ogen populated only with variable RR Lyrae stars (periodic variable stars - the prototype of such a star is in the constella)on Lyra) Open MS turn off point varies massively, faintest is consistent with globulars Maximum luminosity of stars can get to M v -10 Very massive stars found in these clusters. The differences are interpreted due to age open clusters lie in the disk of the Milky Way and have large range of ages. Globular clusters are all ancient, with the oldest tracing the earliest stages of the forma)on of Milky Way ( yrs). introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 24

25 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 0 = λ 0 v / c Δλ = λ λ o = λ o v/c λ is the observed (shifted) wavelength λ o is the emitted wavelength v is the source velocity c is the speed of light (2.5) introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 25

26 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 ( ) and colleagues measured the spectra (light) of many galaxies found nearly all galaxies are red-shifted Redshift (z) z = λ observed - λ rest λ rest = v c (2.6) introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 26

27 Hubble s Law Hubble found the amount of redshift depends upon the distance the farther away (d), the greater the redshift (v) Recessional velocity Hubble s data distance to galaxy v = H 0 d H 0 ~ 70 km/s/mpc (2.7) introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 27

28 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. Size of the Universe now, when we observe the light. introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 28

29 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 = H 0 d d = v / H à 0 But distance = velocity x time. The time is how long the expansion has been going on à The Age of the Universe) à t Universe =1/ H 0 (2.8) introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 29

30 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. introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 30

31 CMB Anisotropies Shortly ager the CMB was discovered one realized that there should be angular varia)ons in temperature, as a result of density inhomogenei)es in the Universe. The denser regions cause the CMB photons to be gravita)onally redshiged compared to photons arising in less dense regions. The amplitude of the T fluctua)ons is roughly 1/3 of the density fluctua)ons. As )me passed, overdense regions became gravita)onally unstable and collapsed to form galaxies, clusters of galaxies and all other structures we see in the Universe today. From the observed CMB angular anisotropies in temperature, it is straight-forward to derive what density fluctua)ons created them. introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 31

32 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 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) ΔT T (θ,ϕ) = l=0 l a lm Y lm (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 m= l Y lm (θ,ϕ) Y * (θ,ϕ)d Ω = δ δ l 'm ' ll ' mm ' introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 32

33 CMB Anisotropy This allows us to calculate the multipole coefficients a lm from Summing over the m corresponding to the same multipole number l we have the closure relation l m= l Y lm (θ,ϕ) 2 = 2l +1 4π Since a lm represent a deviation from the average temperature, their expectation value is zero, < a lm > = 0, and the quantity we want to calculate is the variance < a lm 2 > to get a prediction for the typical size of the a lm. The isotropic nature of the random process shows up in the a lm 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 lm 2 a lm = * Y lm a are independent of m. ( θ,ϕ) ΔT T ( θ,ϕ )d Ω introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 33

34 CMB Anisotropy Since < a lm 2 > is independent of m, we can define 2 C l a lm = 1 2 a 2l +1 lm (2.11) The different a lm are independent random variables, so that m a lm a * lm = δ lm δ l 'm ' C l The function C l (of integers l 1) is called the angular power spectrum. Inserting Eq. (2.11) in Eq. (2.10), one gets " ΔT ( T θ,φ % $ )' # & 2 lm = a lm Y lm θ,φ ( ) a * l 'm ' Y * ( θ,φ) l 'm ' ( ) l 'm ' = Y lm θ,φ Y * θ,φ l 'm ' ll ' mm ' = C l Y lm θ,φ l m ( ) ( ) a lm a * l 'm ' 2 2l +1 = 4π C l introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 34 l l (l +1) C 2π l d ln l (2.12)

35 CMB Anisotropy In the last step the approximation (valid for large values of l) is the reason why instead of C l one often uses l(l +1) 2π (2.13) C l Thus, if we plot (2l + 1)C l /4π on a linear l scale, or l(2l + 1)C l /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)C l /2π, which is neither of these, but for large l approximates the second case. introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 35

36 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 Y lm (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 ϑ λ = 2π = 3600 l l The angle corresponding to a half-wavelength, i.e., the separation between a neighboring minimum and maximum is then ϑ res = π l = 1800 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 7 0 allowing a measurement up To l = 180/7 = 26, WMAP had resolution reaching to l = 180/0.23 = 783. introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 36