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Lecture 1. Course Overview. 5 Sep 2001

- Fill out cards: Name, major, year, Math, Phy, Ast courses taken- Hand out syllabus: grading, texts

Course outline and motivation. Theoretically oriented: models of stars:

- Physical model, isolated system, gas. Basic model very successful. (Solar model good at thefew percent level)

- Important for humans to understand sun.- Provides "lab" with conditions unattainable on Earth: large gravitational, magnetic fields, large

pressures, temperatures; isolated development over astronomical periods- Most important source/sink of matter in evolution of galaxies- Most of (visible) mass in universe in stars- Basic information about evolution of universe: age, origin, composition as function of time

I. ObservationsMass (composition) => Radiation, observed

1 EM radiation:a) I(<) (spectrum) radiated from top layers ("atmosphere")

(=> surface temp, pressure, comp). Now observed from ( ray to radio.b) Position (=> distance)c) Velocity (=> dynamics, mass)d) Variability (=> radius, internal modes)e) Polarization (=> geometry)

2 Other radiation:a) Neutrinos (=> nuclear reactions)

observed for Sun, SN1987Ab) Gravitational Radiation (=> compact object dynamics)

not yet observed (LIGO, yr 2002?)

Entire first section of coures devoted to seeing how we deduce fundamental properties of stars:M MassL Luminosity (total energy output)R RadiusChemical composition

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II. TheoryWhy are there stars? Combination of all four basic forces:1) Gravitational (attractive)2) Electromagnetism (atoms - attractive, pressure - repulsive, as long as there is energy)3,4) Strong, weak nuclear (energy source)

Mass range: 0.08 Mu - -50 Mu (1 Mu = 2×1033 gm)(below, "brown dwarfs", planets; above, not stable)

In the long view, stars are a temporary arrangement:Collapsing H,He cloud =>Mass -> Energy convertor (H -> C-Fe) =>Readjustment and remnant cooloff

But the time scalesare so long that we can (usually) use static models:

A. Atmospheres - visible layers (arbitrary, but important since this is what you can observe!)

Radiation transport, pressure balance, atomic processes=> observed T as function M,R,L,Comp of interior

B. Interiors - Almost entirely inferred from1) pressure balance2) thermal equilibrium3) radiative equilibrium4) equation of state5) energy generation

Hopefully, get L,R as fn M,comp. Just add atmosphere => observations

C. Evolution - as nuclear reactions change composition, L, R change.

Try to match with observed stars of different age. Gets pretty complicated, requiring muchcomputer time. Amazingly successful (e.g. prediction of neutrinos from SN1987A). Sofinal stellar parameter we would like to deduce from observation/theory:

Age

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Lecture 2. Stellar Distance; Flux and magnitude 7 Sep 2001

The biggest problem in observational astronomy. The fundamental method is based onastrometry (measurement of precise angular positions of stars) and geometry (surveyingtechniques):

Heliocentric parallax: nearby stars exhibit change of angular position with respect to moredistant ones, ellipse with period 1 year as earth goes around sun.

Define parallax p of star as semimajor axis of parallactic ellipse. (same as angular semimajoraxis of earth's orbit as seen from star). Then

tan p = 1 AU / D – p radians

Usually quoted in arcsec: p(") = (57.29 °/rad × 3600 "/°) (1 AU / D)

= (206265 AU)/D206265 AU = 3.086 × 1013 km / 1 parsec (pc) = 3.26 light yearsD(pc) = 1/p(")

" Cen: B = 0.76" D = 1.31 pc the closest star" CMa (Sirius): B = 0.38" D = 2.65 pc

Ground-based parallax ± 0.005 " (less than 1% of the image size!)no star with p > 1-50 stars with p > 0.2 (D < 5 pc)at 20 pc (p=0.05), can get distance to 10%. -1000 starsat 100 pc, it's uselessAlmost all parallax stars are low mass dwarfs (the most common stars)ground-based parallax done for 8000 stars.

Hipparcos satellite (mapping done 1997)(http://astro.estec.esa.nl/SA-general/Projects/Hipparcos/) )p = 0.001" => 20% at 200 pc120,000 stars

compare size of galaxy 10000 pc (10 kpc)

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Photometry

(Radiant) Flux. Observed brightness of star:

Total ("bolometric") Flux: erg/s cm2 F Integrated over 8Monochromatic Flux: erg/s cm2 (Å or Hz) F8 or F< SpectrumFilter Flux: erg/s cm2 FU, FB, etc

e.g. FU = I F8 SU(8) d8

Typical bandpasses:photographic: pg (bluish)photovisual: pv (human eye, yellow)photoelectric:broadband U (8max = .36: = 3600Å, )8 = 700Å, "near" ultraviolet)

B (4200 Å, )8 = 1000Å, blue)V (5200 Å, )8 = 900Å, yellow)

infrared R, I, J, K, L, M, N (7000Å - 10.2:)intermediate u,v,b,y (3500 -5500Å, )8 - 200 Å)

Often quoted as a ratio in logarithmic units, "magnitudes"

m - m1 = -2.5 log10( F/F1 )

F1 is a flux reference.

Rules of thumb1 mag ("1m") = -4 decibels2.5 mag = -1 dex (factor of ten)for small mag differences (< 0.5),

)m - relative difference )F/F

(Proof )m = -2.5 log10 ((F + )F)/F)

= (-2.5/ln 10) ln (1 + )F/F)

– -2.5/2.3 )F/F)

Apparent magnitude: flux at the Earth (corrected for atmospheric absorption): m (lower case): Zero points F0(m0 = 0) for various filter "photometric system" are not defined consistently, butyou can look them up in, eg, Allen Astrophysical Quantities":

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for mU, mB, mV, F0 – flux of " Lyrae (Vega) above atmos (for short, write mU=U, mB=B, mV=V) U B V

Vega: 0.02 0.03 0.03Sun: -25.96 -26.09 -26.74Full Moon: -14Max Venus: -3Sirius: -1Unaided eye <6ground tel <24 (atmos background - 21m/arcsec2)Hubble <29

Total range - 55m => 1022 !

Apparent magnitudes also defined formonochromatic flux F8 monochromatic magnitude m8

total flux F = I F8 d8 bolometric magnitude mbol

Warning #1: magnitudes are not linear:eg, what is total magnitude of binary system with two stars of equal magnitude?

Warning #2: published apparent magnitudes (and fluxes) always "corrected" to what would beseen above Earth atmos

Warning #3: there is also extinction from gas & dust between stars ("interstellar extinction"). Photometric catalogues do not remove this (process called "dereddening").

V0 = V - AV (V = observed mV; V0 = V in absence of extinction, Av = extinctionat V in magnitudes for this star)

B0 = B - AB (AB > AV, so that interstellar extinction causes "interstellarreddening")

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Lecture 3. Luminosity and Color: Temperature; 10 Sep 2001

Luminosity and Absolute magnitude

Combining distance and flux, have info to calculate another basic property of stars,

L (luminosity) = energy output in erg/s= Flux received at earth (erg/s cm2) × (area of sphere with radius = distance)= 4BD2 F

Can also define luminosity in filter bands

Lfil = 4BD2 Ffil

Often given in magnitude logarithmic units, referenced to D=10 pc (!). Absolute magnitude M:

F(D=10pc) = L/(4B102) = F(D)×(10/D2)

So m - M = -2.5 log(F(10)/F(D))= 5 log D - 5

m-M called distance modulus. Distances sometimes quoted this way!m-M D (pc) mV MV

Sun -31.6 1/206265 -26.7 4.83Vega -0.62 1/0.133=7.5 0.03 0.65 LV(Vega)/LV(Sun) = 47

0 10 5 10010 1000 = kiloparsec (kpc)25 106 = Megaparsec (Mpc)

Absolute mag for total luminosity called Mbol "absolute bolometric magnitude"

Mbol(star) - Mbol(sun) = -2.5 log L(star)/L(sun)

L(sun) = Lu = 3.826×1033 erg/s

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Colors, Color Index.

Observed Colors

Overall distribution of flux described by ratios of flux or differences of magnitudes in filters(color index):

U - B = -2.5 log [(FU//FU0)/(FB/FB0)] slope between 3600 and 4200ÅB - V = -2.5 log {(FB/FB0)/(FV/FV0)] slope between 4200 and 5500Å

Signs defined so that color is "redder" for larger color indexeg Vega, B-V = 0.00 U-B = -0.01 Sun B-V = 0.65 U-B = 0.13 the Sun is redder than Vega in both colors

U,B,V has been measured for - 100,000 stars, available in paper catalogs, or via internet atvarious archive sites.

For a small number of stars, the total ("bolometric") magnitude has been measured. For these wehave calibrated the relationship between V and mbol:

mbol = V - BC (BC the bolometric correction, allows for flux outside V band)

BC is not defined rationally! Set=0 at B-V=0.3.

apparent and absolute magnitude colors are the same:

B - V = 2.5 log10 FV / FB = 2.5 log10 LV / LB = MB - MVso BC applies to absolute magnitudes, also

eg

BC(Sun) = 0.07, so Mbol (sun) = 4.75.

Procedure: to get luminosity in solra luminosities or in cgs for a star

Look up MV, BC -> Mbol , use ratios with sun:

L/L* = 10-(Mbol - 4.75)/2.5

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Blackbodies

Find that most stars have intrinsic UV, visible, and IR colors roughly similar to theoretical"blackbody"

Blackbody: entity that is in thermal equilibrium at some uniform temperature and absorbs all thelight that falls on it. In practice, light from an oven with a small pinhole in it is close to ablackbody.

Depends only on surface temperature, not composition.

Blackbody Colors

STELLAR COLOR DEPENDS MAINLY ON SURFACE TEMPERATURE

Monochromatic Flux observed at surface (erg/s-cm2-D)

F8 = (2Bhc2/85) / (ehc/k8T - 1)

Wavelength 8max of maximum of f8

8max T = 2900 (: °K) ("Wien's Law")

(eg for Sun, 8max = 5200 D = 0.52 : => T – 5600 °K)

Summarize in color-color plot: U-B color correlated with B-V color same way as blackbody

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Lecture 4. Blackbody Luminosity; Line spectra; Doppler Effect 12 Sep 2001

Blackbody Luminosity

Can also apply blackbody approximation to luminosity

Total energy emitted (luminosity) (erg/s)

L = (Surface Area) × F T4 (F = Stefan-Boltzmann const) = 4BR2 FT4 (Stefan-Boltzmann Law)

LUMINOSITY DEPENDS MAINLY ON SIZE AND SURFACE TEMPERATURE

=> knowing F(app mag), distance => L (abs mag) knowing T roughly from color => R

We have L, T, and R (only two are independent: any two -> third through Stefan=BoltzmannLaw)

Line Spectra; Doppler Effect

So can "classify" stars by U-B, B-V colors, corresponding roughly to surface temperature. Thisuses "photometric" (calibrated) data with R = 8/)8 - 5.

Actually, original classification scheme based on presence of small spectral features seen at R >100. Is because strength of features can be estimated by eye without calibration.

Reason for features understood by Kirkhoff (1859):1) Opaque, incandescent object emits continuum (e.g. blackbody)2) Transparent hot gas emits emission lines (certain 8's identified by chemical element)3) Opaque hot object observed through cooler gas shows continuum with absorptionlines at the same wavelengths

Most stars show absorption line spectra (3) due to transparent atmospheres with lessertemperatures than underlying opaque gas.

(Show abs lines)

Can get much information from line spectra. Easiest is info on motion:

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Stellar Motion

We know stars move relative to each other through angular motion:

Proper motion steady angular motion (relative to distant stars) due to "tangential" speed vT(perpendicular to line of sight).

denote tan : (rad/sec) – : = vT(km/s) / D (km): ("/yr) = (3.16×107 s/yr)/3.086×1013 km/pc) (vT(km/s)/D(pc))

= 4.74 vT(km/s)/D(pc)vT(km/s) = :("/yr) D(pc)/4.74

Typical velocities: 10-20 km/sec (cf earth orbit 18 km/sec)Barnard's star: : = 10.3"/yr (D=1.9 pc) largest known" Cen" : = 3.68Typical errors ): = ±0.002 "/yr (based on photos spanning decades)

=> observable to 10000 pcIs an angular vector, components :" and :* or vT" and vT* (RA and dec dir'ns)Usually combined with parallax. Must observe for years to split them apart.

Radial Velocitymotion vR along line of sight relative to sun. Observed in Doppler shift of spectral

absorption lines:

vR/c = )8/8 - 5 × 10-5

Real tiny, but does not depend on distance. Can now get vR to m/s for bright objects. Alsomeasure vR for most distant objects in universe (QSO's, to lesser accuracy)

Total space velocity of star v = (vT2 + vR

2)1/2

vector components vR, vT", vT*

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Lecture 5. Stellar Mass; Binary Stars 17 Sep 2001

Stellar Mass; Binary Stars

Mass the most fundamental stellar property. Measure gravitational acceleration at somedistance. Only accurate way: observable test orbit.

Fortunately, large proportion of stars (-50%) are in binary star systems.Unfortunately, orbits of most of them not observable.

Visual binaries:observe periodic mutual proper motion of stars

resolve two stars in orbit: nearby, large orbit.period < few hundred years.

Spectroscopic binaries: observe periodic orbital radial velocity of stars

motion > 3 km/s: small orbit

Almost disjoint set. A few binaries (eg " Cen) are both.

AnalysisGeneralization of Kepler's IIIrd Law to two objects of comparable mass (each object haselliptical orbit of period P, semimajor axis a1, a2 around common center of mass):

G(M1 + M2)P2 = 4 B2 a3

a = a1 + a2

Also, from definition of center of mass:

M1 a1 = M2 a2 {Henceforth, convenient units: notice for 1=sun, 2=earth, M1 + M2 – Mu, P = 1 yr, a = 1 AU:

(M1 + M2)/Mu = (a/1AU)3 / (P/1yr)2

where Mu = 2 × 1033 gm

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Visual binaries

Observability: if 0.1 < M1 + M2 < 10 Mu

P < 100 yr=> a3 < 1002 × 10 = 105; a < 50 AUa (arcsec) > 0.5 => D < 100 pc ! Pretty restrictive (future: interferometry)

Procedure: to get M1, M2 need 3 pieces of data:

1) a in arcsec, from relative positions of stars (relative orbit of one star around other is ellipse with same eccentricity asindividual orbits, semimajor axis = sum of two, one star at focus)

2) a1/a2 from absolute astrometry of center of mass3) distance D to get a1, a2 in AU

The major complication: orbital plane is not face on, is inclined. Standard definition:

Inclination i = angle between line of sight and normal to orbital planei = 0: face on, no projection effectsi = 90: orbit projects into straight line

How to do (1) with inclination:a) Trivial for circular orbits (rare for wide orbits): circle => ellipse; a = semimajor axisb) For elliptical orbits, ellipse projects into another ellipse.

How do you tell difference between (a) and (b)? one star is at focus of actual ellipse, and inclination displaces apparent focus of projected

ellipse:

1) Draw line through center and actual focus (projection of actual major axis): e = relativedistance

2) Projection of actual minor axis = bisector of line parallel to major axis3) Construct projection of circumscribed circle by expanding along projected minor axis by

factor 1/(1 - e2)1/2 4) a = semi-major axis of projected circumscribed circle

Nowadays, actual results obtained by computer fit(Typical results: Show Allen pp 236-237)

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Lecture 6. Stellar Mass Continued; Spectroscopic Binary Stars 19 Sep 2001

Spectroscopic Binaries

Observability (assume circular orbits)v = 2B a/Pa3/P2 = M1 + M2 => (v/2B) = [(M1 + M2)/a]1/2 large v => small a

v > 3 km/sec (10% accuracy with 300 m/s errors)=> a < 1.6 AU (only close binaries)

These are so close that they are almost never resolved objects. Even at D = 1 pc, 2 < 1.6 arcsec. Very few visual biniaries are also spectroscopic binaries (exception: " Cen)

Procedure:Observe vrad/c = )8/8 of spectral lines as function of time: P = repetition time

(Nomenclature: phase N = (t - t0) / P repeats mod 1.0)Form of vrad(N) => eccentricity, orientation

For elliptical orbit it can be shown:

vrad(t) = vCM + 2Ba sin i/(P(1-e2)1/2) (e cos T + cos (2(t)+T))

a = semi-major axis of elliptical orbit around center of massP = periodi = inclinatione = eccentricity2(t) = angle of star from periastron (a complicated fn if e > 0)T = longitude of periastron (angle between periastron and line of nodes)

Astronomical terminology:vCM = ( (the "gamma velocity")

K = 2B a sin i /(P(1-e2)1/2) (curve amplitude, also has units of velocity)

Since stars 1 and 2 are opposite the CM, T = T1 = T2 + 180, and 2 = 21 = 22 so

v1,2(t) = ( ± K1,2(e cos T + cos (2(t) + T))

C Two stars have same form, different amplitudes K2/K1 = a2/a1 = M1/M2 / q, and opposite sign

C a only appears as a sin i. inclination does not affect form so is not determinable

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Notice:For e = 0 (circular). Can set T=0 wlog; 2 = 20 + 2BN

v1,2 = ( ± K1,2 cos 2K1,2 = 2B a1,2 sin i/P same as above, reduced by sin i

For e > 0, T = 0v1,2 = ( ± K1,2 (e + cos 2(t))highest velocities, symmetric about apastron, periastron

For e > 0, T = 90, 270v1,2 = ( ± K1,2 cos (2(t) + T)zero-crossings at apastron, periastron; equally placed

If observe both velocity curves (stars must be of comparable flux), have double-linespectroscopic binary:

a sin i = a1 sin i + a2 sin igives

(M1 + M2) sin3 i = (a sin i)3 / P2

and q = K2/K1 = M1/M2

so get M1,2 sin3 i separately

Realities:! Difficult to get i. If see mutual eclipse, i – 90.! If only observe one, single-line spectroscopic binary. Need some other estimate for q to getmasses separately.! For close binaries, stars may be exchanging mass, hence not "normal"

But notice also:� YOU DON'T NEED TO KNOW THE DISTANCE! Since radial velocities can be done onobjects of arbitrary distance, the potential sample of masses is much larger, not confined to thesolar neighborhood

Neat fact for future: if binary both spectroscopic and visual, visual orbit => i; spectroscopic =>masses, visual => distance. "dynamic distance"

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Lecture 7. Stellar Mass; Radius; Spectral Type 21 Sep 2001

Bottom-line: Mass-Luminosity relation

For ~ 90% of stars (Main Sequence Stars)Find L/Lu – (M/Mu)3.45

=> most luminous stars will use up fuel first:

amount of fuel = const × Mrate of use = const × Llifetime - M/L - M-2.45 - L-0.7

Stellar Radii

Usually, we can only get this indirectly from L, T(surf), and the Stefal Boltzmann Law,

L = 4BR2 F T4

There are two direct methods that work on a small subset of stars

! Eclipsing Binaries! Interferometry (large, nearby stars like Betelgeuse)

Eclipsing Binaries

For close binaries, i – 90°, get mutual eclipses => total flux is periodically variable

(Note: nomenclature for variable stars (believe it or not!:Constellation (genitive)R - Z; RR -RZ SS -SZ .. ZZ; AA -AZ BB -BZ .. QZ; (omitting J) V335 V336 ...

so "UW Badgeris" would be OK if there were a constellation "badger")

Same stellar area is blocked out in two eclipses/ period. Deepest one ("primary eclipse") is whenhottest star is behind:

)F(primary) = (Area of smaller star) × (Luminosity/Area of hotter star) / 4BD2

)F(secondary) = (Area of smaller star) × (Luminosity/Area of cooler star) / 4BD2

Luminosity/Area = 4BR2 F T4 / 4BR2 = FT4 (called surface brightness)

)F(primary eclipse)/)F(secondary eclipse) = (T(hotter)/T(cooler))4

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Eclipse length should be same for both:t1: time of first contactt2: beginning of totalityt3: end of totalityt4: time of last contact

For i=90°, circular orbit: diameter of larger star relative to size of orbit:d1/2Ba = ((t3-t1) + (t4-t2))/2P

for the smaller star:d2/2Ba = ((t2-t1) + (t4-t3))/2P

If this is also a double-line spectroscopic binary, can get a in cm, so we have R1 and R2 (don'tneed distance!)

Complications (easily modelled): ! Non-uniform light over stellar disk ("limb darkening")! Inclination not 90°! Eccentricity > 0 (rare for these close binaries)

Problems (not so easily modelled):! Reflection of light from one star off another. cos 2 effect.! Tidal distortion of stars. cos 22 effect.! Light from mass transfer stream (Eeek!)

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Spectral Classification

E.G. in sun 1000's of "Fraunhofer" absorption lines ID'd with:Symbol AbundanceH Hydrogen 90% by number (relative amounts worked out much later)He Helium 10%C,N,O,Ne 0.1% |Si,Mg,Fe,Al 0.01% | "heavies" or "metals"Most everything else |Some simple molecules (CH, CN)

Looking at stars, find repeatable pattern of line strengths:

< 1890 "objective prism" (R - 500) classes A,B ... Q based on strength of H lines

1890-1920 objective prism classes rearranged into continuous sequence based on all lines => Henry Draper ("HD") catalogue of - 250,000 stars (m < about 10). Newarrangement::

O B A F G K M "early" "late"

With practice, could interpolate types: B0 B1 ... B9 A0

This sequence was found to correlate well with colors (surface temperature): O B A F G K M hot -> cool

Change of line strengths for O - M now understood to be due not to change in abundances, but toeffects of temperature on relative population within an element of:

1) ionization (how many electrons still attached)2) electron energy levels (particular 8 only absorbed by particular energy level)

So spectral classification is complementary to photometric classification:advantage disadvantage

photometric can go very faint need photometric calibration; IS reddening correctionspectral no cal req'd brighter stars, longer exposures

+

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Lecture 8. Luminosity Class; HR Diagram 24 Sep 2001

Extra goodies from spectral classification:

Finally, slit spectrograph photographic spectra (360-470 nm, R=4000) show effects of Radius(actually, surface gravity => pressure). Morgan Keenan Type ("MK")

Luminosity class: I supergiant (subclass Ia, Iab, Ib)II bright giantIII giantIV subgiantV dwarf

These are named by size, since, by blackbody approximation, for constant surface temperature(spectral type), luminosity % radius2

(turns out luminosity class also affects photometric colors subtly, so this also complicatesphotometric classification)

So, end up with 2D spectral classification scheme which works on -90% visible stars:

, Ori B0Ia" Lyr A0VSun G2V" Ori M1.5Iab

But, there are "spectral peculiarities", e.g.emission lines: add "e"O star CNO emission lines: add "f"magnetic A star metal line strength: "p", "m"stars with low abundances of heavy elements "metal weak"

And others require whole new schemes, e.g.Carbon Stars: R, N, SWolf-Rayet Stars: WN, WC, WO very strong C N O emission linesWhite dwarfs: DO, DA, DB, DC extremely fuzzy or no linesBrown Dwarfs (New!): L (Methane bands), T (H2O bands!)

Now have enough data to see the most famous correlation among stellar properties:

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Classification Diagrams (Hertzsprung, Russell, ca 1912). Various forms, but all haveluminosity (up), redness (right). E.G.

Hertzsprung-Russell ("H-R") Diagram MV (down) vs Spectral type (right)Color-Magnitude ("CM") Diagram MV (down) vs B-V (right)Theoretical H-R L (up) vs T (left) (sometime called "theoretical H-R")

First done for nearby stars with heliocentric parallax. Found

Most stars on "Main Sequence" (lum,hot,large, massive) => (faint,cool,small, light)). (= "MK Dwarfs")

Some stars much brighter than MS, ("Giants")Later,

Some stars much fainter than MS, called "White Dwarfs"Most recently,

Substellar objects below and to the right of MS, called "Brown Dwarfs"

Then done for clusters of stars. Nearest clusters called "open" or "galactic" clusters (100-1000 stars, loosely grouped).

For cluster with known distances lower main sequences (below F) agree (more or less) withnearby stars.

Distance Methods based on C-M

Main Sequence fitting. For stars known to be at same (unknown) distance, can plot mV instead of MV on C-M. Thenwhole diagram shifted vertically by m-M, but shape not changed. Match mV vs B-V (correctedfor interstellar reddening) or spectral type with calibration of MV vs B-V or spectral type => m-M => distance.

E.G. M92, m-M = 15.3 => log D = (15.3 + 5) / 5 = 4.1, D=11,000 pc = 11 kpc)

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Lecture 9. Census; Radii; Composition 26 Sep 2001

Spectroscopic distance (bad astronomical jargon: spectroscopic "parallax")Using clusters of known distance, can find examples of most MK types and calibrate MV.If confident of your MK type, can look up best guess at MV (spectral type), get m-M. Risky,luminosity class at certain spectral types has considerable luminosity spread, and luminosityclassing not easy.

Density Census: For nearby stars out to several hundred pc, can detect all types, so can do avolume-limited survey:%

Sp O B A F G K M TotI-IV 0.08 0.2 0.6 0.05 1% of stars giantsV 3×10-5 0.2 0.8 4 9 14 73 90% stars MS, mostly MWD 2 3 2 1 1.5 10% of stars WD's

Apparent Magnitude Census: Weighted towards brighter stars1% 10 22% 19 14% 31 3% randomly-chosen brt star

most likely K giant!

C-M systematicsFor open clusters, find upper main sequence, presence of supergiants, position of giants variessystematically.

1) Very top of main sequence (hottest, most luminous stars) always hooks to right.2) Rare to have O MS stars, but those that do have M supergiants (h,P Per), and are associated

with much interstellar dust, gas.3) Can form progression of clusters ordered by MS "turnoff" point, correlates with position of

giants:

Explanation: All these together => differences due to age (+ composition)

Most stars on MS: MS is where stars spend most of lifetime Only see most luminous stars in a cluster that has just formed: Most luminous stars use

up fuel fastest, not surprising they die first.Hook to right in MS: points in direction of post-MS evolution => supergiants, giants.GC's turnoff at G: Must be very old ($ age of Sun, a G2)

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Theoreticians HR Diagram: L-Teff-R

To compare MV - Color(spectrum) with fundamental properties of stars, must convert toluminosity-temperature:

mV, D => MV; +BC => Mbol; +Mbol(sun),Lu => L

How to get temperature? Have seen already can guesstimate T(surface) by assuming F8 -blackbody, and use Wien's Law (maybe after fitting a smooth curve through f8)

T (°K) - 2928 °K/8max(:)

This doesn't give reliable, reproducible values, but is approximately equal to:

The best definition, Effective Temperature, derived from integrated brightness:

Recall for perfect, spherical blackbody of radius R, temp T

L(blackbody) = 4BR2 FT4

For a non-blackbody, Teff is T that would give the observed L for a blackbody

L(star) = 4BR2 FTeff4

Notice, with this form of C-M (logL/LÀ vs log Teff : "Theoretician's C-M") can plot straight linesof: constant R: two fundamental quantities on same plot. Dividing through by values forSun

L/LÀ = (R/RÀ)2 (Teff/Teff(À))4

log (L/LÀ) = 2 log(R/RÀ) + 4 log(Teff/Teff(À))

Even without a good calibration Teff vs spectral type can use Tsurf from blackbody fit to get roughradii:

=> More luminous MS stars are slightly larger=> R varies from .01 RÀ (White Dwarfs) to 1000 RÀ (Supergiants)

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Stellar Composition, Rotation, Magnetic Field

Bottom line so far: stellar appearance (L, R => T => Spectrum) governed by Mass, Age.Several other effects can modify this (mostly observed by subtle effects on spectrum):! Composition! Rotation! Magnetic Fields! Binarity

CompositionEffect discovered through photometry of rare stars in solar neighborhood that appeared to fall

below main sequence. Called "Subdwarfs".

Then found (using MK spectral types), these not underluminous for their color, but too blue fortheir luminosity. Blue colors due to weakness of metal lines, which are primarily in Uand (less) B bands. Better term "metal-poor dwarfs".

Detailed analysis of strength of lines in metal-poor stars (later in course):H, He stay in same ratio (actually He very difficult in these cool stars)All heavier elements reduced pretty much in proportion (interesting exceptions later). Can quantify by proportion of Iron/H relative to Sun:

"metallicity" [Fe/H] = log (n(Fe)/N(H)) - log (n(Fe)/n(H))u

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Lecture 10. Composition; Rotation; Magnetic Field 28 Sep 2001

Metal-poor stars very important for galactic structure studies: all very old, apparently formedfrom gas before galaxy had much heavy elements. Called "Population II" stars (opposedto "Population I" stars with composition similar to sun).

In our galaxy, many globular clusters metal poor. (recall cool main-sequence turnoff of GC =>very old).

Since most of these stars very faint, methods developed to use broad-band color data to estimateand correct for "metallicity".

Just as in interstellar reddening, compute (or calibrate from stars with good MK types) "line-blanketing vector" in U-B vs B-V color-color plot.

Find [Fe/H] - [Fe/H]Hyades - -5 *(U-B)(Done with respect to Hyades since Hyades has well-established main sequence. Detailedanalysis of individual stars finds Hyades slightly metal-rich compred to sun:

[Fe/H]Hyades = 0.025 )Range of metallicities [Fe/H] -2.5 to +0.5

Other effects of composition: form of giant branch in CM diagrams of globular clusters

Rotation

! Has effect on sharpness of lines in spectrum:

Difference in radial velocity between approaching and receding limb of star:

)v = 2 vrot sin i (i=inclination of polar axis to line of sight)

)8 = 8 ()v/c)

Sun: Prot = 30 days = 2.6×106 secR = 7×105 kmvrot = 2BR/P = 1.7 km/sec)8/8 = 6×10-6

)8(5000 Å) = 0.03 Å not observable

Largest observed rotations vrot = 250 km/sec, )8 - 4 Å very mushy lines.

Find most stars hotter than F rotate rapidly. B-V where rotation stops correlated with age,probably due to spindown caused by mass-loss with magnetic field.

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! Extremely large rotation starts to have effect on color: effective gravity at equator is reducedby centrifugal acceleration.

Magnetic Field

Actually usually very difficult to observe. Best use Zeeman effect, causes lines to be split:

)8/8(zeeman splitting) = C(atom) 8 eH/4Bmec - 8 × 5×10-5 H(gauss)

for H=1000 gauss )8/8 (4300 Å) = 2×10-6 (even smaller than rotation effect for sun)

With very clever differential methods, can measure this if rotation is not too fast.

Sun: mean field, few gauss; in sunsplots, 1000 gauss (kgauss)Magnetic A stars and peculiar B stars: (1-10 kgauss)

Exception: In magnetic white dwarfs, can have H = 109 gauss!! This large enough to manglespectrum and effect structure and atmosphere.

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Stars at non-visible Wavelengths

Photospheric radiation mainly confied to UV-vis-IR as blcakbody curve:

F8 = (2Bhc2/85) / (ehc/k8T - 1) = c1 8-5 / (exp(c2/8T) - 1)

-> c1 8-5 exp(-c2/8T) (8 << 8max, eg X-Ray, or UV for cool stars; "Wien Distribution")

-> (c1/c2) T 8-4 (8 >> 8max, eg far IR, radio; "Rayleigh-Jeans Distribution")

Radio folks prefer talking in F<

F< = F8 d8/d< = (2Bh <3/c2) / (eh</kT - 1)

-> 2Bc-2kT <2 (h< << kT, or 8 >> 8max)

UV; X-Ray.

For short wavelengths, since Wien Distribution drops exposnentially, photosphere does notcontribute: Observed light is non-photospheric:

1) Coronae: very hot, thin envelopes T = 105 - several × 106 °K

Emission lines from highly ionized Fe, etc

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Lecture 11. UV Accretion Disks; IR Dust shells 1 Oct 2001

2) Accretion: matter begin dumped onto star piles up in opaque "accretion disk", emits accordingto blackbody:

3/2 kTacc - 1/2 m vacc2 - 1/2 m vesc

2 (for accretion onto surface)

for m = mH (pure hydrogen), v = 16 km/s at T = 11,600 °K

Tacc - 11,600 (vesc/16 km/s)2

but vesc = (2GM/R)1/2 = 620 km/sec (M/MÀ / R/RÀ)1/2

So Tacc - 450,000 °K × (M/MÀ) / (R/RÀ) (!!)

This will give black-body peak @ 8 = 50 D or so for MS stars: soft X-ray , hard UVFor White Dwarf, R - .01 RÀ => T > 106 °K !

This is origin of light in "X-Ray Binaries", brightest stellar objects in X-Ray sky

IR; Radio

For long wavelengths, Rayleigh-Jeans Law drops off more slowly F< - <2

1) Rayleigh - Jeans "tail" of photosphere seen into microwave2) Dust envelopes

Dust of radius a at distance D from star absorbs stellar light and re-emits at "equilibriumblackbody temperature" Teq(D)

Energy absorbed - L* × Ba2 / 4BD2

Energy emitted - 4Ba2 F Teq4

Setting them equal for equilibrium

FTeq4 = 4B R*

2 FT*4 × (Ba2/4Ba2 )/ 4BD2

(Teq/T*)4 - 1/4 (R*/D)2

This is the same relation that gives equilibrium temperstures of planetsTeq - D-1/2

Temperatures 30 - 1000 °K: 8max = 2 : - 100 : in infrared:

(Lecture Continues - "Understanding Stellar Spectra")