ࡱ> [ ״bjbj zjjl>>>>>>>RZGZGZG8G|JRlNN"NNNQQQO W w>QP^QQQ/>>NN///Qj0>N>N/Q//Ǻ >>NN `s>REZG1 0}+/+/RR>>>>PHYS 228 Astronomy & Astrophysics Lecture Notes from Zeilik et al. Chap. 1 Celestial Mechanics & the Solar System Heliocentric Model of Copernicus General Terms: Zodiac evening star morning star Inferior Planet - orbits closer to Sun than Earth (see Fig. 1-2) elongation superior conjunction inferior conj Superior Planet - orbits farther from Sun than Earth (see Fig. 1-2) conjunction quadrature opposition Orbital Periods of Planets Synodic period -time taken by planet to return to same position in sky, relative to Sun, as seen by Earth Sidereal period- time taken by planet to complete one orbit w/ respect to stars Equations relating synodic and sidereal periods of a planet: Let S = synodic period of planet, P = sidereal period of planet, E = sidereal period of Earth = 365.26 d Then 1/S = 1/P - 1/E Inferior Planet 1/S = 1/E - 1/P Superior Planet Defn: Astronomical Unit: Average orbital radius of Earth Keplers Three Laws of Planetary Motion Elliptical Orbits Radius vector of planet sweeps out equal periods in equal times P2 = a3 Geometrical Properties of Elliptical Orbits r + r = 2a = constant (1-1) by defn of ellipse b2 = a2 - a2e2 = a2(1-e2) (1-2) from Pythagorean theorem where a = semimajor axis of ellipse b = semiminor axis e = eccentricity r = a(1-e2)/(1 + e cos () (1-3) equation of ellipse A = (ab (1-5) area of ellipse e = (ra - rp)/(ra + rp) eccentricity of ellipse ra/rp = (1 + e)/(1 - e) ratio of aphelion distance to perihelion distance ra + rp = 2a relation between aphelion/perihelion distances and major axis rp = a(1 - e) relations between aphelion/perihelion distances, semimajor ra = a(1 + e) axis, and orbital eccentricity Law of Areas and Angular Momentum dA/dt = rvt/2 = r2(d(/dt)/2 = H/2 = A/P = (ab/P Keplers 2nd law - of areas v2 = G(m1 + m2) [2/r - 1/a] vis viva equation (velocity at any point r in orbit) vper = 2(a/P [(1+e)/(1-e)]1/2 velocity at perihelion vaph= 2(a/P [(1+e)/(1-e)]-1/2 velocity at aphelion  Newtonian Mechanics Law of inertia F = ma Equal but opposite forces (action & reaction) Newtons Law of Universal Gravitation Fcent = mv2/r (1-14) v = 2(r/P Fgrav = G Mm/r2 (1-15) where G = 6.67 x 10-11 m3/kg . s2 Fgrav = (GM(/R(2)m = gm = mg W = mg Weight where g = 9.81 m/s2 Newtons Form of Keplers 3rd Law P2 = 4(2/G(m1+m2) a3 PHYS 228 Astronomy & Astrophysics Lecture Notes from Zeilik et al. Chap. 2 Solar System in Perspective Planets A. Motions: ecliptic prograde (direct) orbits - CCW seen from above Earths orbital plane synchronous rotation - sidereal rotation equals sidereal orbital period oblateness of planet ( = (re - rp)/re Two Major Categories of Planets Terrestrial Planets - Mercury, Venus, Earth, Mars relatively small, low mass similar or smaller than Earths solid composition ( high density orbit relatively close to Sun Jovian Planets - Jupiter, Saturn, Uranus, Neptune large, high mass (15 to 318 M( ) liquid & gaseous comp ( low density orbit far from Sun B. Planetary Interiors Differentiated - heavy elements sink to center, light elements rise to surface Explains Earths Ni-Fe core, but light silicate rocks in crust Expect Jovian planets to have rocky cores, surrounded by light H & He Average density ((( = M/(4(R3/3) C. Surfaces Albedo A = amount reflected/amount incident Blackbody radiation & Planck curve for warm objects Wiens law (max = 2898/T ((m) Ex. Sun T( 6000 K ( (max = 0.5 (m Stefans Law E = (T4 W/m Subsolar Temperature (approx. equilibrium noontime temp near equator) Tss = (Rsun/rp)1/2 (1-A)1/4 Tsun ( 394 (1-A)1/4 rp-1/2 Planetary Equilibrium Temperature (derived from energy balance averaged over entire planet) Te = (1-A)1/4 (Rsun/2rp)1/2 Tsun ( 279 (1-A)1/4 rp-1/2 D. Atmospheres Mercury, Moon - no atmospheres Venus, Mars - CO2 atmosphere Earth - N2 , O2 Jovian Planets - H, He Law for Perfect Gas P = nRT Maxwellian Distribution - most probable speed vp = (2kT/m)1/2 Average KE per particle (KE( = m (v2( Solve for root mean square speed vrms = (v2(1/2 = (3kT/m)1/2 Escape speed from planet w/ mass M, radius R: vesc = (2GM/R)1/2 Note: for vesc ( vrms, atmosphere escapes into space in few days. vesc ( 10 vrms Condition for atmosphere to be retained for billions yrs or T ( GMm/150kR Moons, Rings, & Debris (read on own, will be discussed in Chap. 7) Moons Rings Asteroids Comets Meteoroids Interplanetary Dust Newtonian Mechanics Applied to Solar System Applications of Keplers 3rd Law Modern form of Keplers 3rd Law: P2 = 4(2a3/G(m1+m2) Launching Rockets (read equations on own) Height of rocket found by equating total energy at ground w/ total energy at maximum height h E = (KE + PE)ground = (KE + PE)h = const mv2 + 0 = 0 + mgh h ( v2/2g (for small h) h = v2/2g{R(/[R(-(v2/2g)]} (for large h) Orbits of Artificial Satellites Elliptical, parabolic, hyperbolic orbits PHYS 228 Astronomy & Astrophysics Lecture Notes from Zeilik et al. Chap. 3 Dynamics of the Earth Time and Seasons Terrestrial Time Systems celestial meridian upper transit celestial equator & poles vernal equinox sidereal time local sidereal time hour angle of the vernal equinox Solar time apparent solar time mean solar time Standard Time Zones Greenwich mean time or universal time (UT) Year sidereal year - period w/resp. to stars - tropical year - period w/ resp. to vernal equinox ( year of seasons ( Gregorian calendar anomalistic year - period between successive perihelion passages B. The Seasons Cause Eccentricity e = 0.017 to small to affect seasons significantly Seasons caused by tilt 23.50 of Earths axis w/ orbital axis, resulting in solar insolation varies due to angle of incidence: less in winter due to energy spread out more, more concentrated in summer fewer hours daylight in winter, more in summer radiation must penetrate more atmosphere in winter due to lower angle, more scattering Terms vernal & autumnal equinoxes summer & winter solstices Tropics of Cancer & Capricorn Arctic & Antarctic Circles - midnight sun Evidence of Earths Rotation (read on own) Coriolis Effect Cyclones, Anticyclones aCoriolis = 2 v x ( (constant acceleration) Foucaults Pendulum The Oblate Earth Evidence of the Earths Revolution about the Sun (read on own) Aberration of Starlight ( ( tan ( = v/c Stellar Parallax d = 206,265/(( (AU) Doppler Effect ((( = ((-(()/(( = vr/c Differential Gravitational Forces Tides Differential Tidal Forces: Non- point source masses attract near faces of each other stronger than far faces Results in tidal stretching ( bulges (see Fig. 3-15) Moons tidal stretching of Earth & Earths stretching of Moon (spring & neap tides) dF/dr = -2GM/R3 or dF = -(2GM/R3) dR where M, R = mass & radius of perturbing body (Moon here) Consequences of Tidal Friction synchronous rotation of Moon with its orbit slowing of Earths rotation recession of Moon Precession and Nutation Precession of equinoxes due to gravitational torque on rotating Earth by Sun & Moon - 50(/yr Nutation - wobbling of Earths rotation axis due to Moons orbit inclined 50 with ecliptic -sometimes above, sometimes below plane Roche Limit Limiting distance at which a body may approach another body without being tidally disrupted. d = 2.44 ((M/(m)1/3 R Roche limit for a fluid satellite d = 1.44 ((M/(m)1/3 R Roche limit for a rigid satellite PHYS 228 Astronomy & Astrophysics Lecture Notes from Zeilik et al. Chap. 4 The Earth-Moon System Dimensions (read on own) Eratosthenes - measured diameter of Earth c. 200 B.C. Dynamics Motions Months sidereal month - w/resp. to stars 27.322 d synodic month - w/resp to phases 29.531d draconic or nodical month - w/resp. to line of nodes (intersect ecliptic) 27.212 d anomalistic month - w/resp to consecutive perigee 27.555 d Synchronous rotation - tidal slowing libration in longitude & latitude Phases new (inf. Conj) waxing crescent first quarter (quadrature) waxing gibbous full (opposition) third quarter (quadrature) waning crescent Eclipses Solar Eclipses partial, total, & annular Lunar Eclipses partial & total Interiors (read on own) The Earth Average density ((( = 3M(/4(R(3 = 5520 kg/m3 Components Crust, Mantle, Core Determinations of interior from earthquakes longitudinal compression (P) waves transverse distortion (S) waves Hydrostatic Equilibrium downward force of gravity balanced by upward pressure otherwise object would expand or contract dP/dr = -( [GM/r2] Equation of hydrostatic equilibrium (EHE) States that internal pressure becomes smaller as one moves from center to surface. Ex. Use EHE to estimate central pressure inside planet or star: Consider region radius r w/ mass M and average density ((( inside a planet . Then must have M = 4/3 (r3 ((( Simplify by taking ( to be uniform, & that surface P << central P. Solving EHE for dP & subst for M above, get dP = -((( G (4/3)( ((( r dr = -(((2 (4/3)(G r dr Integrating over r from center (r=0) to surface (r=R), (0 ( ( dP = -(((2 (4/3)(G(r dr (Pc ( Pc = 2/3 (G (((2 R2 = 1.4 x 10-10 (((2 R2 (Pa) Subst. Values for Earth Pc = 1.7 x 1011 Pa = 1.7 x 106 atm B. The Moon Average density ((( = 3Mm/4(Rm3 = 3370 kg/m3 Solid, non- metallic core, indicated by density Thick, solid crust & mantle - geologically inactive Surface Features Earths Surface and Age Lithosphere, hydrosphere, & atmospheric layers Avg. density lithosphere 3300 kg/m3 Plate tectonics - crustal plates Age of Earth Obtained by radioactive dating techniques n/n0 = -exp(-(t) radioactive decay where n0 = original amt of radioactive substance n = current amt t = time elapsed since initial amt = decay constant Half -life Occurs when n/n0 = . Let ( = time when this occurs. ( = ln2 /( = 0.693/( Then, n/n0 = exp(-0.693t/() Example: #2 on worksheet The Lunar Surface Surface features terminator limb lowlands - maria (seas, basins) smooth, dark highlands - rugged, light impact craters primary craters secondary (satellite) craters rays ejecta blanket mountains mascons Composition regolith - thin layer of fine rocks & soil brecchias - conglomerate melted rocks anorthosite from highlands basaltic from marias Atmospheres The Moon Very slight atmosphere, since vesc/vrms < 10 for most molecules. B. The Earth Secondary atmosphere of N2, O2, Ar, CO2 result of vulcanism & outgassing Layers: trophosphere 0 - 15 km stratosphere mesosphere thermosphere 90 - 250 km exosphere Ideal Gas: P = nkT = (kT/m where n = (/m number density (in m-3) ( = mass density (in kg/m-3) m = mean molecular wt (in units of H mass or amu) Atmospheric pressure obeys exponential law: P = Po exp(-h/H) where P = pressure at height h Po = pressure at surface h=0 H = scale height = const Defn: Scale height H = height at which P falls to 1/e Po = 0.368 Po H ( kT/gm where g = GM/R2 = surface gravity Observed effect of atmosphere atmospheric transparency atmospheric seeing measured in seconds of arc Magnetic Fields Lunar Magnetism (omit) Earths Magnetosphere Van Allen Radiation Belts inner belt 1R( - 2R( outer belt 3R( - 4R( Lorentz force F = q(v x B) on charge F = qvB magnitude only Charged particle spiraling in B field Fcent = FB mv2/r = qvB so r = mv/qB gyroradius D. Aurorae Aurora borealis - named by Galileo Aurora australis See film later Evolution of Earth-Moon System (Read on own) Lunar History Earth History PHYS 228 Astronomy & Astrophysics Lecture Notes from Zeilik et al. Chap. 5 The Terrestrial Planets: Mercury, Venus, and Mars Mercury Motions Orbital radius a = 0.39 AU Orbital period Porb = 88d Rotation period Prot = 59d = 2/3 Porb from Doppler shift Physical Characteristics Mass M = 0.055 M( Radius R = 0.38 R( T = 700 K (sunlit side), T = 100 K (dark side) Density ( = 5420 kg/m3 Since small planet with high density, must contain large % metals Ni-Fe core (molten?) Central Pressure see Worksheet exercise Surface Features Surface similar to Moon: maria, highlands, craters, Caloris Basin (similar to Mare Orientale) Different than Moon: found scarps (cliffs), no mountain ranges, more very large craters Magnetic Field very weak due to slow rotation Venus Motions Orbital radius a = 0.72 Au Orbital period Porb = 225d Rotation period Prot = 243d (retrograde rotation) Physical Characteristics Mass & radius slightly smaller than Earths Density ( = 5200 kg/m3 Interior believed to resemble Earths interior Atmosphere 96% CO2 ( strong greenhouse effect Single cell model of global circulation due to slow rotation (Earth has 3-cell model) Sulfuric acid clouds Surface Features Surface features determined by radar, flatter than Earth upland plateaus cratered terrain shield volcanoes similar to Hawaii huge canyons Magnetic Field none detected thus far (too slow rotation of core?) Mars Motions Orbital radius a = 1.52 AU Orbital period Porb = 780d Rotation period Prot = 24.6h Inclination axis I = 250 Physical Characteristics Mass M = 0.11 M( Radius R = 0.53 R( T = 225 K (Range 210 - 300K) Density ( = 3900 kg/m3 (slightly higher than Moon) Atmosphere 95% CO2 similar to Mars Surface Features From Earth: Canali fictitious Polar caps, seasonal changes From space probes Arroyo channel which water flows into only occasionally Valles Marinaris valley of the mariners Magnetic Field very weak Evolution of surface read on own PHYS 228 Astronomy & Astrophysics Lecture Notes from Zeilik et al. Chap. 6 The Jovian Planets Jupiter Motions Orbital radius a = 5.2 AU Orbital period Porb = 11.9 yr Rotation period Prot = 9h 50m (fastest planetary rotation) Note differential rotation due to fluid body, also distinct oblate shape. Physical Characteristics Mass M = 318 M( Radius R = 11.2 R( Mean density <(> = 1330 kg/m3 Composition 75% H, 24% He, 1% all heavier elements (by mass) also CH4 (methane), NH3 (ammonia), H2 (molecular H) Atmosphere 78% H, 20% He, 2% all heavier elements (by mass) from Voyager specroscopy (same as Suns) Features Visible in Atmosphere Belts (dark, deeper) and zones (light, higher) parallel to equator Great Red Spot (GRS) rotates CCW like vortex, extends several km above surrounding zone high pressure, ascending material Wind speeds fastest near equator, fall off rapidly away from equator Internal Structure Core of liquid metallic H Central Pc = 2/3 ( G<(>2 R2 = 1.4 x 10-10 <(>2 R2 = 1.2 x 1012 Pa = 1.2 x 107 atm Core Tc ( 40,000 K Thermal Emission from Jupiter & other Jovian Planets (see Table 6-1) Jupiter emits 1.7 x energy receives from Sun ( internal power source Possible sources of heat: internal energy left over from formation gravitational contraction conversion of grav PE ( thermal KE of gas Grav. PE spherical mass: U ( GM2/R Energy loss therefore found by taking time derivative: dU/dt ( (-GM2/R2) dR/dt = -(U/R) dR/dt We deduce from observations & models internal power source radiating at rate: dU/dt ( 4 x 1017 W Solving previous eqn. for dR/dt gives rate of planetary contraction necessary to sustain this rate of energy radiation: dR/dt = -(R2/GM2 ) dU/dt = -(R/U) dU/dt ( -8.5 x 10-12 m/s ( -0.3 mm/yr ! Thermal Conductivity of Heat from Interior Temperature Gradient (T/(x from center to surface: (T/(x = (T2 - T1)/(r2 - r1) = (Ts - Tc)/(R - 0) Thermal conductivity (: ( = - H/[A((T/(x)] = -(dU/dt)/[4(R2 (T/(x] Magnetic Field B ( 4 x 104 T at surface, very strong due to rapid rotation of metallic H core Produces powerful radiation belts - inner, middle, & outer magnetosphere inner surface to 6 RJ middle 6 RJ to 50 RJ outer beyond 50 RJ Charged particles (es & ps) generates synchrotron radiation in region 1.3 to 3 RJ B ( 4 x 10-4 T at 1 RJ bursts radio emission Can calculate gyroradius & frequency of spiraling particles r = mv/qB gyroradius f = v/2(r = v/2((mv/qB) = qB/2(m cyclotron frequency Aurorae visible Saturn Motions Orbital radius a = 9.5 AU Orbital period Porb = 29.5 yr Rotation period Prot = 10h 14m (slightly less rapid than Jupiter) Physical Characteristics Mass M = 95 M( Radius R = 9.0 R( Mean density <(> = 680 kg/m3 (much less than Jupiter, less than water) Composition similar to Jupiter & Sun Features Visible in Atmosphere Deeper atmosphere, less colorful than Jupiter Wind speeds at equator 4x faster than Jupiter Internal Structure Probably similar to Jupiter Magnetic Field B much weaker than Jupiter (1/34) Still strong enough to have radiation belts Uranus First planet to be discovered in historical times, by W. Herschel (1781) Voyager 2 flyby in 1986 No internal heat souce apparently Motions Orbital radius a = 19.2 AU Orbital period Porb = 84.0 yr Rotation period Prot = 17h 14m (retrograde) Physical Characteristics Mass M = 14.5 M( Radius R = 4.0 R( Mean density <(> = 1600 kg/m3 (slightly more than Jupiter) Composition similar to Jupiter & Sun T ( 59 K Magnetic Field Tilted ~59o from spin axis Neptune Predicted to exist & discovered by J.C. Adams & U. Leverrier (1846) Voyager 2 flyby in 1989 Motions Orbital radius a = 30.1 AU Orbital period Porb = 165 yr Rotation period Prot = 16h 03m (slightly faster than Uranus) Physical Characteristics Mass M = 17.2 M( Radius R = 3.9 R( Mean density <(> = 1600 kg/m3 (slightly more than Jupiter) Composition similar to Jupiter & Sun T ( 59 K (about same as Uranus, therefore must possess ) Internal heat souce, like Jupiter & Saturn Bright cirrus clouds (methane ice crystals) Great Dark Spot vortex of High pressure, observed by Voyager Magnetic Field Tilted ~47o from rotation axis Pluto Discovered 1930 by C. Tombaugh at Lowell Observatory, Flagstaff, AZ Satellite Charon discovered 1978 by J. Christy at U.S. Naval Observatory, Flagstaff, AZ Orbital period Pluto-Charon system 6.4d. Motions Orbital radius a = 39.4 AU (range 30 - 49 AU) Orbital period Porb = 249 yr Rotation period Prot = 6.5d Eccentricity e = 0.25 Physical Characteristics Mass M = 0.0018 M( (order magnitude less than Earths moon) Radius R = 0.18 R( (smaller than Earths Moon) Mean density <(> ( 2100 kg/m3 (may be less) Composition perhaps 75% rocky material (more than Jovian moons, mostly ices) T ( 40 K PHYS 228 Astronomy & Astrophysics Lecture Notes from Zeilik et al. Chap. 7 Small Bodies and the Origin of the Solar System Moons and Rings (omit, students read on own) Asteroids Irregular, rocky body smaller & less massive than planet Typical sizes < 100 km Over 2000 known, ~106 estimated total in asteroid belt Types S-type asteroids lighter colored, silicate composition, high albedo (15%) found mainly within orbit of Mars C-type asteroids darker, carbonaceous composition, low albedo (2-5%) exist mainly in asteroid belt & outer solar system M-type asteroids rare type (only 5% belong), metallic composition, medium albedo (10%) Comets Appears as fuzzy star in telescope Major Components Coma bright cloud visible Nucleus bright starlike point near center of nucleus Tail ion tail & dust tail Cometary Tails Ion tail straight, consisting of plasma of CO, CO2, N2, , CH4, NH3 small gas particles affected directly by solar wind, points away from Sun Dust tail curved, larger grains more massive dust grains following Keplerian orbits, affected by photons Cometary Nuclei Comet consists of small, 10-20 km diameter icy chunk, on very elliptical orbit about Sun Comet approaches perihelion, frozen gases begin to sublimate to form coma Dirty Snowball Cometary Model Fred Whipple (1950) Relation Between Observed Comet Brightness and Distances from Sun & Earth B ( r-n d-2 where r = distance from Sun d = distance from Earth n = index which ranges from 2 to 6, n = 2 far from Sun (no fluorescence) n ( 4 - 6 near Sun (moderate to strong fluorescence) The Oort Comet Cloud proposed by astronomer J. Oort (1950) large reservoir comets (~1011 - 1014)at ~50,000 AU from Sun extreme elliptical orbits, P ~ 107 yr An Example: Halleys Comet estimated mass ~3 x 1014 kg, loses ~1011 kg each passage P = 75-76 yr, Most recent close passage April 1986, rp = 0.59 AU, d = 0.42 AU Visited by Europes Giotto spacecraft, w/ results: most of coma gas H2O (80%), CO, formaldehyde (H2CO) size of nucleus 8 x 16 km extremely dark, albedo ~4% jets gas (H2O) & dust Meteoroids, Meteors and Meteorites Terms: meteoroid meteor meteorite Viewing Meteors (see Fig. 7-23) before midnight meteors must catch up to Earth, speeds only 12 km/s after midnight facing toward direction of orbital motion, speeds up to 42 km/s best viewing Origin of 99% meteors cometary debris, remainding 1% asteroid fragments Most are sand-grain size, burn up ~ 60 - 100 km altitude Most meteorites, rocks which reach Earths surface, originate from asteroids Types of Meteorites Irons highest densities (7500 - 8000 kg/m3) contain 90% Fe, 9% Ni most common finds, but in reality very rare Widmanstatten figures large crystalline patterns appear when etching w/ acid perhaps result from impact shock (which broke it apart) Stones lowest densities (3000 - 3500 kg/m3) composed light silicates, similar to Earths crustal rocks (therefore difficult to distinguish from ordinary rocks) most stones contain spherical chondrules, such stones called chondrites carbonaceous chrondrites contain much carbon & water Stony-Irons medium densities (5500 - 6000 kg/m3) rarest type Meteorites found in Antarctica origin Mars or Moon Collisions 1 km object each 105 yr, 10 km object every 108 yr Interplanetary Gas & Dust Zodiacal Light scattered sunlight off tiny dust particles in plane of ecliptic seen before sunrise & after sunset Source disintegrating comets & colliding asteroids Earth accretes ~106 tons of this dust /yr Radiation Pressure due to fact that electromagnetic radiation carries both radiation & momentum Suns radiation (photons) exerts force on small grains Frad = PA = AE/c where P = momentum flux E = radiant energy flux Frad = radiation force Putting in specific quantities, Frad = (((a2Rsun2Tsun4/c)r2 where a = radius of particle, r = distance from Sun. Also, grav. force is Fgrav = GMSun (4(a3(/3) r2 Ratio becomes then, substituting values, Frad/Fgrav = 3(Rsun2Tsun4/4cGMSun (a = 5.8 x 10-5 /(a Dust grains smaller than 1 (m blown out of solar system, since Frad ( Fgrav Poynting Robertson Effect Affects larger particles, where Frad << Fgrav Particle in circular orbit w/ speed v experiences aberration sunlight, giving rise to impeding force component causing to eventually spiral into Sun in time t = (7 x 105) (ar2 yr *Note: disintegrating comets & colliding asteroids increase amt dust in solar system, radiation pressure & Poynting-Robertson effect rid dust from solar system Formation of Solar System (Students should read most on own) Accretion Process (see Fig. 7-28 & discussion on p. 144) S = [R2 + 2GMR/Vo2]1/2 gravitational impact parameter where R = radius of target body M = mass of target body Vo = speed of approaching particle at infinity Special case: Vo = Vesc = [2GM/R]1/2 ( S = [R2 + R2]1/2 = (2R2)1/2 = 1.44 R PHYS 228 Astronomy & Astrophysics Lecture Notes from Zeilik et al. Chap.8 Electromagnetic Radiation & Matter BlackBody Radiation Blackbody ideal absorber & emitter of radiation Energy spectrum of blackbody per unit area per wavelength given by: E(= (((, T) (W m-2 nm-1) Power emitted per unit area of blackbody given by: ( E = ( E( d( = (T4 Stefans law for ideal radiator 0 Relation also exists between wavelength at which emission curve peaks and temperature of glowing object: T = 2.898 ( 10-3 m.K / (max Wiens displacement law PLANCK RADIATION FUNCTION E((,T) = 2(hc2/(5 [exp(hc/(kT) - 1] -1 Planck distribution function where h = 6.626 ( 10-34 J.s Planck's constant Wiens law can be derived by taking derivative dE/d(, set = 0, then solve for ( = (max Stefans law can be derived from this by integrating E((,T) over ( from 0 to ( LIGHT QUANTA (PHOTONS) assumes energy emitted in wave packets energy of each wave packet given by : E = hf = hc/( Energy of single photon The Bohr Atom KIRCHOFFS RULES OF SPECTRA Incandescent solid (or gas under high pressure) ( continuous spectrum Incandescent gas ( emission line spectrum Light from continuous source passed though low density gas ( absorption spectrum; Solar absorption lines investigated by Fraunhofer J. Balmer investigated empirical formula relating wavelength & integers Spectral wavelengths for element w/ atomic # Z: 1/( = RZ2 (1/nf2 - 1/ni2) Rydberg-Ritz formula (ni < nf) where R = 1.097 ( 107 m-1 Rydberg constant BOHRS QUANTUM MODEL OF THE ATOM N. Bohr (1913) proposed model of H atom to explain observed spectra: H atom consists of e(-) orbiting p(+) in circular orbit, bound by Coulomb attraction Only certain orbits (stationary states) are stable; e does not radiate while in stable orbit Radiation emitted by atom only when e jumps from high state to lower state Radiation absorbed by atom when e jumps from low state to higher state Terminology: n = 1 ground state n = 2, 3, 4, excited states n ( ( ionized state Energy of Emitted (or Absorbed) Photons hf = hc/( = Ei - Ef Ei > Ef SPECTRAL SERIES Hydrogen (Z=1) Lyman series nf = 1, ni = 2, 3, 4, (UV) Balmer series nf = 2, ni = 3, 4, 5, (visible) Paschen series nf = 3, ni = 4, 5, 6, (IR) Brackett series nf = 4, ni = 5, 6, 7, (IR) Orbital Angular Momentum of Electrons L = mvr = n (h/2() = n( n = 1, 2, 3, ( = h/2( Potential, Kinetic, & Total Energy of H Atoms U = -kZe2/r PE for system w/ one e- and nuclear charge +Ze K = 1/2 mv2 KE of orbiting electron Total energy of system therefore: E = K + U = 1/2 mv2 + -kZe2/r By Newton's 2nd law, Felectrostatic = Fcentripetal kZe2 / r2 = mv2 / r Then, 1/2 mv2 = K = 1/2 kZe2 / r Substituting for K above, E = 1/2 kZe2/r - kZe2/r = -1/2 kZe2/r Total energy of system Note: E < 0 Frequency of Emitted Photon (in terms of initial & final orbital radii) f = (Ei - Ef) / h = 1/2 (kZe2/h) [1/rf - 1/ri] frequency of emitted photon Eliminating v between expressions for L and K yields expression for orbital radius rn: Radii of Atomic Orbits rn = n2 (2 / mkZe2 = n2 a0/Z Radius of nth level a0 = (2 / mke2 = 0.0529 nm 1st Bohr radius Combining expressions for rn and f yields: f = Z2 (mk2e4/4((3) [1/nf2 - 1/ni2] freq of emitted photon in terms of ni & nf By direct comparison w/ Rydberg-Ritz formula, obtain R = mk2e4/4((3 Rydberg constant Substituting rn in expression for total energy E above, obtain: En = -(k2e4m/2(2) Z2/n2 = -Z2 E0/n2 Energy levels for atomic number Z where E0 = k2e4m/2(2 = 13.6 eV ground state energy of H Stellar Opacity Line blanketing stellar absorption lines remove flux primarily in B & V Thermodynamic equilibrium (TE) net flow of energy into region = net flow outward Local thermodynamic equilibrium (LTE) mean free path of particles << distance over which temperature changes significantly Ways to measure star temperature: Excitation temperature from Boltzmann law Ionization temperature from Saha equation Kinetic temperature from Maxwell-Boltzmann distribution of particle speeds Color temperature from fitting flux curve w/ Planck BB function Mean Free Path ( ( = 1/n( where n = number density particles ( = cross section for collision Absorption any process which removes photons from a beam of radiation true absorption (e.g., by atomic transitions w/in atoms) scattering (e.g., Compton scattering) Absorption coefficient (Opacity) ( Let beam parallel light rays of initial intensity I pass through tiny thickness ds of gas w/ density (. Then amount by which intensity reduced given by dI = -((I ds where ( = absorption coeff (units cm2/g) *Note: dI negative since energy removed quantities I, ( depend on (, which should appear as subscript ( = f(composition, (, T) also Mean free path can be expressed in terms of ( & ( ( = 1/n( = 1/(( Optical depth ( d( = -(( ds If take ( = 0 at top of atmosphere, then difference in optical depth final (=0 at top) minus initial (deeper inside) is 0 - ( = ( -(( ds (limits 0 to s) or ( = ( (( ds (limits 0 to s) Then functional dependence of intensity on ( is combining above result w/ assumption of pure absorption: I = Io e-( *Note: e.g., if ray starts out at ( = 1, then I decreases by factor 1/e by time it reaches surface & escapes from star optical depth ( may be considered as the number of mean free paths from the starting point to the surface, as measured along the path of the ray Definitions: ( >> 1 ( gas called optically thick ( << 1 ( gas called optically thn *Note: ( depends on (, so atmospheric gas may be optically thick at one (, but optically thin at another e.g., Earths atmosphere thin for visible (s, but thick for x-rays Contributions to Opacity 1. Bound-bound transitions (not important in stellar interiors & atmospheres hot stars since all atoms ionized) (bb = no simple equation exists 2. Bound-free transitions (photoionization) (bf = 4.34 ( 1025 (gbf / t) Z (1 + X) ( T-3.5 cm2 g-1 where gbf = gaunt factor for bf transitions (quantum mechanical correction) ( 1 t = guillotine factor (arises when H ionization means no longer contribute to opacity at high T) range 1 - 100 X = mass fraction H Z = mass fraction elements heavier than He 3. Free-free absorption (free e absorbs photon, causing speed of e to increase) reverse is free-free emission (bremsstrahlung) (ff = 3.68 ( 1022 gff (1 - Z) ( T-3.5 cm2 g-1 *Note: (ff has same functional form as (bf (Kramers law ( ( (T-3.5) gff = gaunt factor for ff transitions (quantum mechanical) ( 1 4. Electron scattering (not true absorption) very small cross section compared w/ H atom most effective as source opacity only at high T (es = 0.2(1 + X) cm2 g-1 Possible e scattering events: photon scattered by free e Thomson scattering photon scattered by e which is loosely bound to atomic nucleus Compton scattering (photon << ratom Rayleigh scattering (photon >> ratom (can be neglected in all but very cool * atmospheres) Primary source of continuum opacity in most stellar atmospheres is photoionization H- ion ionization energy only ( = 0.754 eV (need ( < hc/( = 16400 A) H- becomes increasingly ionized at high T for B, A stars, photoionization H & free-free absorption main sources continuum opacity for O stars, photoionization He & e scattering main sources continuum opacity Rosseland mean opacity total opacity averaged over all ( <(> = <(bb + (bf + (ff + (es > *Note: in general, as ( increases, ( increases (as expected) at low T, ( comes from bf, ff transitions at high T, ( comes from e scattering -many free es since gas ionized Radiative Transfer Random walk due to atmospheres opacity (absorption & scattering), photon traverses random path from interior of star to surface Individual vector displacements given by (1, (2, (3, , then net displacement resulting from N steps given by d = (1 + (2 + (3 + + (N To determine length vector d, take dot product d with itself, which yields scalar d2 = N(2 + terms involving cosine which sum ( 0 for large N Then, d ( ((N where d = net distance covered ( ( mean free path *Note: takes 100 steps for photon to travel distance 10 l 10,000 steps to travel 100 l Now optical depth given approximately by ( ( # mean free paths along net displacement d so, ( ( d/( = (N Then, avg # steps needed for photon to travel net distance d is N = (2 for ( >> 1 *Note: Condition for photon to escape from surface of star: ( ( 1 (approx) ( = 2/3 (exact derivation) Limb Darkening when looking directly at surface of Sun, see more deeply into high T, brighter region when looking at limb of Sun, see only lower T, darker layers Radiative flux related to radiation pressure on atmosphere Prad, opacity (, & density ( via dPrad / dr = -(((/c) Frad The Structure of Spectral Lines Now analyze profiles of sp. lines & relate to astrophysical condns in atmosphere Define: (o = central wavelength of abs. line = core of line sides going up to meet continuum are wings of line also, F( = radiant flux from stars atmosphere Fc = flux of continuum only outside sp. line then, D(() = (Fc - F()/Fc = 1 - F(/Fc relative depth of line at wavelength ( Equivalent Width W W = ( D(() d( = ( (Fc - F()/Fc d( = ( (1- F(/Fc )d( *Note: W for most lines on order 0.1 A. Full Width at Half Maximum (FWHM) = ((()1/2 = full width of line measured between two opposite points where D(() = *Note: can determine if spectral line is optically thick or thin from central depth of profile: F(/Fc > 0 at (o ( optically thin (D((o) < 1) F(/Fc = 0 at (o ( optically thick (D((o) = 1) Opacity ( of line varies in different parts of line: ( greatest at (o ( center of line formed in higher, cooler layers of star ( least in wings, far from (o ( formed in deeper, hotter layers of star 3 Major Line Broadening Processes govern shape of line profile: 1. Natural Broadening Energy levels not perfectly sharp, but have natural width, according to QM; thus, sp. lines resulting from transitions between energy levels not sharp. If let (t = lifetime of e in excited state then uncertainty in energy level E given by (E ( h/2((t *Note: in ground state only, lifetime of e (t ( ( ( (E ( 0 sp. lines arising from ground state called resonance lines Energy of photon given by standard formula: Ephoton = hc/( If let i = initial state, f = final state, then uncertainty in absorbed or emitted photons wavelength can be shown (problem) to be given by (( ( (2/2(c [1/((i - 1/((f] More precise calculation gives natural broadening in terms of FWHM: ((()1/2 = (2/(c [1/(to] where (to = typical total waiting time for specific transition to occur. Example H( line ( = 6563 A produced by transitions between n=2 and n=3 energy levels (1st & 2nd excited levels). e lifetime in either level given by 10-8 s, so total waiting time (to = 2 ( 10-8 s. This gives ((() ( 2.3 ( 10-12 cm = 2.3 ( 10-4 Doppler Broadening For gas in thermodynamic equilibrium, random motions of gas particles according to MB velocity distribution in different directions gives rise to thermal Doppler broadened lines. Doppler shifts occur for both line & continuum, but effects noticeable only in lines. For nonrelativistic motions, Doppler shift given by standard formula: ((/( = ( |vr|/c Solving this for ((, (( = ((/c |vr| Recall most probable speed for MB distribution of particles of mass m & temp T given by vmp = [2kT/m] Subst this in Doppler eqn and including additional factor 2 which results from taking particles moving both toward & away from observer at speeds vr gives approximate width of sp line due to thermal Doppler broadening (( ( 2(/c [2kT/m] More precise analysis gives Dopper width in terms of FWHM: ((() = 2(/c [2kT ln 2 /m] Example H( line ( = 6563 A produced in Suns photosphere at T = 5770 K. Thermal Doppler broadening yields FWHM ((() ( 0.43 A which is 1000 ( larger than value for natural broadening *Note: Even though (FWHM)Doppler >> (FWHM)natural, line depth for Doppler broadening decreases exponentially away from central core, much faster than decline for natural broadening. Thus, natural broadening may still important in wings of lines Turbulent velocities Doppler shifts also created by large-scale churning motions in atmospheres of certain large stars. Turbulence most common in red giants & supergiants. Modified formula: ((() = 2(/c [(2kT/m + vturb2) ln 2] 3. Pressure and Collisional Broadening (treated together here) collisional broadening orbitals of atom perturbed by collisions w/ other neutral atom pressure broadening orbitals of atom perturbed by E fields large numbers of passing ions often of same magnitude as natural broadening, although pressure profile can be ~order mag larger general shape of line from pressure broadening also similar to that for natural broadening Estimate for pressure broadening similar for natural broadening, except (to now taken to be avg time between collisions. We estimate (to from (to ( (mean free path between collisions) / (avg speed of atoms) = l/v But ( = 1/n( and v = vmp = [2kT/m] yields (to ( (/v = (/[n( (2kT/m) ] Finally use result for natural broadening with (to given above to determine width of line due to pressure broadening is on the order of (( = (2/(c [1/(to] ( ((2n(/(c) [2kT/m] *Note: (( ( number density of atoms n This is only an approximate formula, so no distinction made between (( and FWHM Example Consider again H( line ( = 6563 A produced in Suns photosphere at T = 5770 K. For typical number density of H atoms in Suns atmosphere, pressure broadening of line should be about (( ( 2.4 ( 10-4 A which is comparable to width for natural broadening Contribution to line profile: contribution near core out to ~ 1.8 ((() mainly from Doppler broadening ( Doppler core farther out, contribution shifts to that produced by pressure & natural profiles ( damping wing total line profile called Voigt profile PHYS 228 Astronomy & Astrophysics Lecture Notes from Zeilik et al. Chap.9 Telescopes and Detectors Optical Telescopes Telescopes made of lenses or mirrors Basic Terms for lenses & mirrors object & image focus focal length f Lens Makers formula: (omit, since covered in lab) 1/do + 1/di = 1/f Focal ratio: f ratio = f/D D = lens diameter Plate scale: s = 0.01745 f (units of f / degree) Basic Telescope terms objective eyepiece refracting & reflecting telescopes Light gathering power (LGP) LGP ( area of objective A = ((/4)D2 Expressed as ratio (LGP)2/(LGP)1 = (D2/D1)2 where D1,D2 are diameters lenses (or mirrors) 1 &2 Resolving power (RP) RP = 1/(min Minimum Resolvable Angle (or just Resolution) (min = 206,265 (/D ((() rectangular aperture where number 206265 = # arcseconds/radian. Therefore, can state in general: RP ( D Therefore, in ratio form (RP)2/(RP)1 = D2/D1 Above applies for rectangular apertures. For circular apertures of telescopes, must multiply by 1.22: (o,min = 1.22 (min = 1.22 x 206,265 (/D ((() circular aperture Magnifying Power (MP) MP = fobj / feyp Atmospheric Conditions seeing (seeing disk in (() transparency Invisible Astronomy (read on own) Detectors & Image Processing Quantum Efficiency QE = # photons detected / # photons incident Example: QE ( 1% for human eye in visible range Signal to Noise Ratio S/N = /(m where = mean # photons counted -signal (m = std dev from mean of counts (taken from several trials of measurements) - noise High S/N ( high quality observation, much info contained Photons obey Poisson distribution, which approx Gaussian for large N, (m = 1/2 std dev of Poisson distribution w/ mean count Thus, S/N ( 1/2 Example: = 237,899 ( S/N ( (m = (237,899 ( 490 This means that the signal is 490 times stronger than noise, or that noise is 1/490 = 0.0020 = 0.2% of signal, excellent! Relation of S/N to dimensions & QE of detector Let flux Fp photons/sec/m2 fall on detector area. Then avg total # photons detected in integration time t must be = QE x Fp x t So, S/N = (QE x Fp x t)1/2 In summary, S/N proportional to sqrt of QE, Fp, & t. To improve S/N, must increase one or more of QE, flux, or integration time. 1. Photography Image produced when photons incident on photographic emulsion cause photochemical reaction in silver bromide crystals embedded in gelatin - photographic emulsion QE ( 1% for human eye in visible range Resolution limited by size crystal grains -- > 20(m Image brightness proportional to log (density of grains), not # photons (eye responds similarly) 2. Phototubes Relies on photoelectric effect: photon (w/certain min energy) strikes surface of certain material dislodges es, which can flow in current which can be measured. Current produced ( flux (linear relation, unlike eye or photography) thus gives accurate measure of flux Image not produced, just measure of flux QE ( 10-20% (much better than eye or photography) Photomultiplier can transform 1 e ( 105 es for more rapid counting 3. CCDs (Charge-Coupled Devices) will discuss in lab exercise Consist of array tiny pixels on thin silicon wafer (chip), ~ 1 cm on side Wideners ST-6 CCD contains 375 x 242 pixels (90,750 total), NURO ~575 x 575 pixels Pixel size ~ 9 - 25(m, Widener ST-6 pixel size ~22(m Each pixel accumulates charge as photons fall on it & dislodge es; more photons, greater charge After exposure, charges accumulated in individual pixels measured digitally so that spatial pattern light falling on chip produces image. Computer then processes image. Advantages of CCDs over other detectors: Extremely high QE, ( 100% in red, 50% in blue ( even small telescopes can record faint objects CCD response to light very linear ( measures light intensity accurately CCD is an area detector not only gives intensity (flux) but also image simultaneously CCD images in digital form, ready to be processed & displayed by computer Spectroscopy (read on own, will have lab on stellar spectroscopy later) Terms: Dispersion of spectrograph produces spectrum Prism spectrograph dispersion by refraction in prism Grating spectrograph dispersion by diffraction grating Spectrophotometry New Generation of Telescopes Improved resolution by (1) bigger apertures ( smaller diffraction disk of stars, (2) adaptive optics ( deforms telescope optics to compensate for turbulence atmospheric cells, (3) aperture synthesis ( interferometry PHYS 228 Astronomy & Astrophysics Lecture Notes from Zeilik et al. Chap.10 The Sun: A Model Star Structure of Our Sun Basic Stats M ( 333,000 M( R ( 109 R( L ( 4 x 1026 W ( 1400 kg/m3 T ( 5800 K Major Internal Zones Core where energy produced 4H ( He Radiative zone innermost, energy transported by photons Convective zone outermost, energy transported by convection, extends down 30%R( Major atmospheric Layers Photosphere base of solar atmosphere, visible surface Chromosphere where emission & absorption lines produced Corona low density, high temp region, merges into Solar Wind high velocity outflow charged particles (ions) The Photosphere Granules Diameters ~ 700 km Convection cells Photospheric Temperatures T ( 5800 K Limb Darkening Opacity & Optical Depth Opacity k( -- has units m2/kg, and is defined in differential form by dF( = - k( ( F( dx or approximately, using differences, (F( ( - k( ( F( (x Then, k( ( -((F(/ F() /((x k( = 0 ( medium transparent k( < 1 ( medium optically thin k( > 1 ( medium optically thin Optical depth (( -- dimensionless quantity related to the opacity k( via d(( = k( ( dx The reduction in the flux of radiation is then given by (see Example #2) dF( = -F( d(( or dF( / F( = -d(( For a uniform medium, integration yields F((() / F((0) = e-( where F((0) is the original flux of the beam of radiation and F((() is the flux after passing through a medium with optical depth ((. H- Continuous Absorption negative H ion formed because second e can loosely attach itself to proton = 0.75 eV (much less than 13.6 eV for normal H) Absorption occurs by dissociation reaction H- ( H + e- (produces opacity) Emission when e- attaches to neutral H atom: H + e- ( H- (produces visible & IR continuum) Fraunhofer Absorption Spectrum Spectral lines Elemental abundances H (71%) He (27%) Other (2%) The Chromosphere Extends 10,000 km above photosphere Reddish color due to H( emission seen during solar eclipse The Chromospheric Spectrum Chromospheric Fine Structure plages filaments spicules The Transition Region The Corona The Visible Corona The Radio Corona Line Emission Forbidden lines Extreme UV lines Coronal loops & holes The Solar Wind Coronal ionized material escapes from Sun Extremely low density Speeds accelerate as travel outward, reach 400 - 700 km/s at 1 AU Solar Activity The Solar Cycle Sunspots Umbra Penumbra magnetic polarity Field strengths 0.1 - 0.4 T Sunspot Numbers 11 yr (22 yr) cycle Maunder minimum (1645 - 1705) Little Ice Age Sunspot Polarity Preceding & following spots Solar Rotation Differential rotation, 25 d at equator, 30 d at high latitudes Active Regions Bipolar Magnetic Regions (BMRs) & Plages magnetograph faculae brightenings mark active regions in photosphere plages bright regions, float in chromosphere, higher (, T than surrounding gas coronal streamers active regions in corona Prominences & Other Displays quiescent & active (eruptive) prominences loop prominences most active Solar Flares Solar Flares Outbursts which release large amts both high energy particles & photons X-ray & Radio Bursts A Model of the Solar Cycle (read on own) PHYS 228 Astronomy & Astrophysics Lecture Notes from Zeilik et al. Chap.11 Stars: Distances and Magnitudes The Distances to Stars Trigonometric (Heliocentric) Parallax Defn: Trig. parallax of star = angle ( subtended, as seen from star, by Earths orbit of radius 1 AU ((rad) = a/d where a = Sun-Earth distance (1 AU) d = Sun-star distance (in same units as a) If ((( = 1/d then d(pc) = 1/((( Note: best parallaxes have prob. errors ( 0.004(( 1 pc = 206265 AU = 3.26 ly Other Geometric Methods (omit) Luminosity Distances (omit) The Stellar Magnitude Scale Apparent mag. System first developed by Hipparchus & refined by Ptolemy Grouped over 1000 stars visible to eye on scale 1 to 6: 1st mag stars brightest 2nd mag stars next brightest . . 6th mag stars just barely visible to eye In 1854, verified that on this scale mag 1 star ( 100 x brighter than mag 6 star Modern mag scale defined so that difference (m = 5 ( factor 100 in brightness Then, (m = 1 ( l1/l2 = 1001/5 = 2.512 Note: on modern magnitude scale: stars may be brighter than mag 1 ( negative mags (e.g., Sun has m = -26.8) also may be fainter than mag 6 ( stars viewed through telescopes (e.g., m = +23.5) Examples of Apparent Magnitudes Object mv Sun .. -26.8 Moon (full) .. -12.6 Venus (brightest) . -4.4 Mars (brightest) -2.8 Sirius -1.5 Vega, ( Cen . 0.0 Antares . +1.0 Polaris .. +2.0 Faintest naked eye stars +6.5 (limit of 0.5-cm eye) Brighest quasar 3C 273 .. +12.8 (limit of 6-inch telescope) Pluto +14.9 (limit of 12-inch telescope) Faintest object observed +24.5 (limit of 150-inch telescope) In general, then, for two stars with magnitudes m1 & m2 having brightnesses (fluxes) l1 & l2 m1 - m2 = 2.5 log (l2/l1) = -2.5 log (l1/l2) Example: Find combined apparent mag of binary system consisting of two stars m1 = +3.0 and m2 = +4.0. m2 - m1 = 4.0 - 3.0 = 1.0 ( l1/l2 = 2.512 ( l1 = 2.512 l2 Then total brightness is l = l1 + l2 = 2.512 l2 + l2 = 3.512 l2 Using general magnitude eqn above m2 - m = 2.5 log (l/l2) or m = m2 - 2.5 log (l/l2) = 4.0 - 2.5 log 3.512 = 4.0 -1.36 = 2.64 Example: A variable star changes in brightness by a factor 4. What is the corresponding change in mag? m2 - m1 = 2.5 log (l1/l2) = 2.5 log 4 = 2.5 (0.6) = 1.5 Absolute Magnitude & Distance Modulus Absolute magnitude M gives intrinsic brightness of star Defn: Absolute mag M of any star = apparent mag it would have if placed at std distance 10 pc. May derive relation between m, M, & d. Since brightness of star obeys inverse-square law, have l ( 1/d2 So if l = brightness at distance d, L = brightness at std distance D = 10 pc L/l = (d/D)2 = (d/10)2 Now, treat star at distance d and again at distance D as if it were two stars & apply mag eqn above: m - M = 2.5 log L/l = 2.5 log (d/10)2 = 5 log d/10 = 5 log d - 5 log 10 = 5 log d - 5 Defn: Distance modulus = m - M Recall that distance & parallax related by d = 1/((( , so distance modulus eqn may be written (students should be able to derive) m - M = 5 log ((( -5 Magnitudes at Different Wavelengths In order to make mag system practical, must define mag for specific ( or range ( Magnitude Systems Photographic plates most sensitive to radiation ( ( 420 nm (blue-violet) ( yield photographic mags mpg Human eye most sensitive to radiation ( ( 540 nm (green-yellow) ( yield visual mags mv Johnson UBV System Most widely used mag system UBV system, in 3 bands, each ( 100 nm wide (see Fig. 11-3): U band centered on 350 nm (in near UV) ( mU B band centered on 430 nm (in blue part of visible sp) ( mB V band centered on 550 nm (in yellow part of visible spectrum) ( mV Additional 2 bands, each 150 nm wide: R band centered on 640 nm (in red part of visible sp) ( mR I band centered on 790 nm (in near IR) ( mI Also exist narrow-band filter systems (e.g., Stromgren) Color Index Measured mags in different color bands ( yields quantitative measure of stars color & also temperature Defn: Color index CI = difference of magnitudes at two different (s CI = m((1) - m((2) E.g., CIU-B = U - B = mU - mB = MU - MB CIB-V = B - V = mB - mV = MB - MV CIV-R = V - R = mV - mR = MV - MR Notes: By convention, one always subtracts longer ( from shorter one CI same whether apparent or absolute mags used CI does not depend on distance, unless IS reddening present CI has negative values for hotter stars, positive for cooler stars Since CI = mag difference, may express CI in terms of measured fluxes: CI = const - 2.5 log [F((1) / F((2)] If emitting star approximately BB, then can write CI = const - 2.5 log [B((1) / B((2)] CI yields measure of slope of radiation distribution (Fig. 11-4) ( stars color & temp can be deduced Assuming BB distribution, for B-V color index: B - V = -0.71 + 7090/T or T = 7090/[(B-V) + 0.71] Note: B-V index calibrated so that B-V = 0.00 for star with T = 10,000 K (A0 star) Stars not exactly BBs, so empirical work for stars in T range 4000 K < T < 10,000 K: B - V = -0.865 + 8540/T or T = 8540/[(B-V) + 0.865] Effects of IS Reddening Short (s scattered out more than long (s, produces color excess Defn: Color excess CE (will be discussed in Chap 19 on IS Medium) CE = CI(observed) - CI(intrinsic) Effects of Atmospheric Extinction Magnitudes quoted in literature corrected for atmospheric dimming, depends on optical depth of atmosphere ( F(() = F0(() exp (-(() Assume Earths atmosphere to be plane-parallel (no curvature), define h = angular height of star above horizon z = angle between zenith & star = 90( - h Geometry shows (Fig. 11-5) path traversed by light through atmosphere ( sec z Defn: Air mass = sec z Therefore, light suffers least extinction for star at zenith ( optical depth minimum = (0(() Then, optical depth ((() at any angle z from zenith given by ((() = (0(() sec z Thus, if can determine (0((), then can calculate F0(() for star. Practical rule: when measuring star fluxes, keep z < 60( (h > 30() Practical application used by astronomers: Astronomer observes two groups stars during observing run: program (target) stars w/ unknown mags to be determined selected std stars w/ known mags, observed over range in z. Note: On given night, both program & std stars suffer same atmospheric extinction (for same airmass) Must determine extinction coefficient k( at certain wavelength ( (e.g., V = 550 nm) on given night by: measure radiation flux of std star over large range of zenith angles then plot measured magnitudes vs sec z (airmass) graph should be straight line, slope ( amt extinction in mags per airmass finally correct program star magnitudes for that night w/ this extinction coeff (for that () Quantitative procedure for doing this as follows. Let m0(() = apparent mag of star above atmosphere (i.e, corrected for extinction) m (() = apparent mag of star below atmosphere Then, by defn mag: m(() - m0(() = -2.5 log [F(()/F0(()] Subst eqn for optical depth z, m(() - m0(() = -2.5 log [exp (-(()] = 2.5 (log e) (( = 1.086 (( and so m0(() = m(() - 1.086 (( = m(() - 1.086 [(0(() sec z ] = m(() - k0(() sec z Defn: where k0(() = 1.086 (0(() = first order extinction coefficient Notes: Range values for k0(() in V band typically 0.15 to 0.20 k0(() varies from night to night & location to location Bolometric Magnitudes & Stellar Luminosities Useful to define magnitude system which utilizes stars total energy emission (luminosity) at all (s Defn: Bolometric flux lbol = ( l(() d( (W/m2) Using general mag eqn, obtain expression for apparent bolometric mag of star: mbol = -2.5 log lbol + const (depends on choice zero pt or std) May now obtain expression for mbol of star as function of luminoisities, using Sun as std: Lsun = 4(Rsun (Tsun4 = total luminosity of Sun = 3.92 x 1026 J/s L* = luminosity of star Mbol(sun) = absolute bolometric mag of Sun = +4.7 Mbol(*) = absolute bolometric mag of star Then by general mag eqn, Mbol(sun) - Mbol(*) = 2.5 log L*/Lsun Substituting values Lsun & Mbol(sun) for Sun, & rearranging, log L*/Lsun = 1.89 - 0.4 Mbol(*) Note: In general, L* found from sp analysis or MS fitting clusters In practice, useful to define bolometric correction BC to facilitate determining stars bolometric mag Defn: Bolometric Correction BC = mbol - mv = Mbol - Mv or BC = 2.5 log (lv/lbol) May also write then mbol = mv + BC or Mbol = Mv + BC Notes: BC always negative BC large for stars whose intensity peaks outside V band E.g., for Sun BC = -0.08 PHYS 228 Astronomy & Astrophysics Lecture Notes from Zeilik et al. Chap.12 Stars: Binary Systems Over 50% (perhaps 80%) star systems binary Only direct means obtaining * masses Classification of Binary Systems Apparent (optical) binary two stars which appear close together because they lie along the same line of sight but are not physically associated. Visual binarygravitationally bound system that can be resolved into 2 stars at telescope. Astrometric binary only one star seen telescopically, but exhibits wavy proper motion in sky due to gravitational tug of companion too faint to be seen. Spectroscopic binary telescopically unresolved system whose duplicity is revealed by periodic oscillations of its spectral lines. Spectrum binary unresolved system whose duplicity is revealed by two distinctly different sets spectral lines (e.g., B + M classes) Eclipsing binary system in which two stars eclipse one another, leading to periodic changes in apparent brightness of system. Can also be visual, astrometric, spectroscopic Visual binaries Determination of Stellar Masses Measure a((, ((( a = a((/ ((( (M1 + M2) = a3 / P2 = (a((/ ((()3 / P2 Determination of individual masses requires knowledge relative distance each * from CM system M1a1 = M2a2 where a = a1 + a2 Mass-Luminosity Relation L/Lsun = (M/Msun)( where ( depends on type of star, usually in range 2-4 General working eqn: L/Lsun = 0.23 (M/Msun)2.3 low mass *s (M < 0.43 Msun) L/Lsun = (M/Msun)4.0 medium to high mass *s Spectroscopic Binaries Eclipsing Binaries Interferometric Stellar Diameters & Effective Temperatures Stellar Interferometer Use interference of light to measure stellar diameters Occultation of target star by Moon gives interference pattern (see Fig. 12-14) Speckle Interferometry Take many short exposures taken rapidly to observing distortion of seeing disk Averaged using Fourier analysis PHYS 228 Astronomy & Astrophysics Lecture Notes from Zeilik et al. Chap.13 Stars: The Hertzsprung-Russell Diagram Stellar Atmospheres Physical Characteristics Thermodynamic equilibrium particle collisions are very frequent, so energy distributed evenly throughout gas( Boltzmann & Saha eqns apply Perfect gas law: P = nkT Mean molecular wt: 1/( = mH n/( Special cases: Neutral H: ( = 1 (only in cooler regions) Ionized H: ( = (most of stellar interior) Empirical relation: 1/( ( 2 X + (3/4) Y + (1/2) Z (total ionization) where X,Y,Z = mass fractions of H, He, heavier elements. Hydrostatic equilibrium (EHE) -- typical volume of gas in star experiences no net force. In eqn form, dP/dr = -(GM/R2) ( = -g( Scale Height: H = kT/gm Barometric eqn: P(h) = P(ho) exp(-h/H) In terms of opacity & optical depth, EHE becomes dP/d( = g/( Integrating gives P = (g/() ( Temperatures Wiens law: (max = 2.898 x 10-3 / T Stefans law: F = (T4 Relation between L, R, T: L = 4(R2 (T4 or L/Lsun = (R/Rsun)2 (T/Tsun)4 Spectral Line formation Boltzmann excitation-equilibrium equation: NB/NA ( exp [(EA - EB)/kT] In log form log NB/NA = (-5040/T)EA - EB + constant Saha ionization-equilibrium equation: Ni+1/Ni ( [(kT)3/2/Ne] exp [-(i/kT] In log form log Ni+1/Ni = 1.5 log T - (5040/T)(i - log Ne + constant Classifying Stellar Spectra Observations spectrograph dispersion The Spectral Line Sequence Harvard spectral classification scheme: Spectral types: OBAFGKM early types OBA late type GKM The Temperature Sequence Temperature correlated with: spectral type (see Fig. 13-6) color index B-V = -0.71 + 7090/T (assumes BB distribution) HR Diagrams Devised independently by E. Hertzsprung (1911) & HN Russell (1913) Magnitude vs. Sp. Type Plot Mv vs. Sp. (see Fig. 13-7, -8, -11) Magnitude vs. Color Index Plot mv vs. B-V (see Fig. 13-8, -9, -10 Metal Abundances & Stellar Populations Two (actually three) distinct stellar populations, related to age Pop I. -- young, metal-rich stars (Z ( 0.01), age < 109 yr Pop II -- old, metal-poor stars (Z < 0.001), age 12-15 x 109 yr Disk (intermediate) population Examples: Pop I: Open (galactic) clusters Pleiades in Taurus Pop II: Globular clusters M3 in Bootis, M13 in Hercules Luminosity Classifications M-K (Morgan - Keenan) luminosity classification (1940s) I Supergiant II Bright Giant III Giant IV Subgiant V Dwarf (main sequence) VI Subdwarf See Fig. 13-11 for position of luminosity classes on HR diagram Classification criterion: pressure effect (also called luminosity effect, surface gravity effect) Application: Consider star Capella (( Aur) which has same surface temperature as Sun: Sun Class G2 V Capella Class G2 III R ( 10 Ro, M ( 3 Ro, L ( 100 Lo Surface gravity g ( M/R2, so surface gravity gCapella = 3/102 go = 0.03 go. Consequences: P and Ne also this much lower in Capellas atmosphere than Sun Capellas atmosphere favors ionized elements than Sun Sp. lines sharper in Capella due to less pressure broadening Color - Color Diagrams Already discussed Elemental Abundance Effects Read on own Distance Determinations Specroscopic parallaxes Main-sequence fitting X-Ray Emission PHYS 228 Astronomy & Astrophysics Lecture Notes from Zeilik et al. Chap.14 Our Galaxy: A Preview The Shape of the Galaxy Observational Evidence Milky Way Via lactica named by Romans Galileo (1610) resolved into individual stars W. Herschel (1785) star counts ( Sun center of grindstone Current model: galactic disk with nuclear bulge, halo, & spiral arms (see Fig. 14-8) The Galactic Coordinate System (omit) The Distribution of Stars (omit) Star Counting Interstellar Absorption Luminosity Function Luminous Stars & Stellar Clusters Stellar Populations Different populations characterized by observed metal abundances (X, Y, Z) X = abundance of H Y = abundance of He Z = abundance of all other elements Population types: Population I (Z ( 0.01) Found in mainly spiral arms of disk (see Fig. 14-8) youngest type, dominated by young, OB MS stars Disk Population (0.001 < Z < 0.01) Found generally in galactic disk old to middle age stars, also dominated by RGs Population II (Z ( 0.001) Found in halo & nuclear bulge (see Fig. 14-8) oldest type, dominated by RG Note: age ( , Z ( from Pop I to Disk to Pop II Galactic Dynamics: Spiral Features Sun moves in circular orbit about GC: Orbital radius rsun = 8.5 kpc = 2.62 x 1020 m Orbital speed vsun = 220 km/s = 2.20 x 105 m/s Use Keplers 3rd law extended by Newton to obtain total mass interior to Sun vsun = distance / time = 2(rsun/P so, P = 2(rsun/vsun = 2((2.62 x 1020 m) / (2.20 x 105 m/s) = 7.49 x 1015 s = 2.38 x 108 yr Also, a = rsun = 8.5 kpc = 8500 pc = 1.75 x 109 AU Then, total mass of galaxy is MG ( a3/P2 = (1.75 x 109 AU)3 / (2.38 x 108 yr)2 = 9.5 x 1010 Msun ( 1011 Msun Alternate method (see text): set Fcent = Fgrav Msunvsun2/rsun = GMGMsun/rsun2 Cancelling Msun terms & solving for MG yields MG = vsun2 rsun / G = 1.9 x 1041 kg = 9.5 x 1010 Msun ( 1011 Msun Note: actual mass believed to be closer to 1012 Msun, when outer edges & halo included A Model of the Galaxy Galactic disk diameter dG ( 50 kpc ( defines galactic plane (analog of ecliptic plane of solar system) Half-thickness of disk (scale height) ( 1 kpc Halo diameter dH ( 100 kpc contains high velocity stars & globular clusters on eccentric orbits (analog of comets in solar system) Central bulge contains ( 1011 Msun (about half the mass of disk interior to Sun) Nucleus (innermost part of Central Bulge) lies at very center of Galaxy, contains ( 106 Msun diameter of nucleus < 1 pc source intense nonthermal radio waves ( suggest primordial BH Spiral arms contain most gas & dust, young stars formed by density waves (theory) PHYS 228 Astronomy & Astrophysics Lecture Notes from Zeilik et al. Chap.15 The Interstellar Medium & Star Birth Interstellar Dust Dark Nebulae & the General Obscuration Dark nebulae distinct opaque clouds obscuring light of stars behind them General obscuration caused by dust distributed more uniformly & thinly than dark nebulae To take general obscuration into account in distance modulus formula, include absorption term A m - M = 5 log d - 5 + A where A = total amt absorption in magnitudes between Sun & star at distance d (pc). Note: when A ( 0, m becomes larger (fainter) ( distance overestimated Determination of A discussed in next section. Interstellar Reddening A = A((): short (s scattered more effectively than long (s (selective scattering) ( IS reddening (actually light is de-blued, not reddened) Reddening affects observed color index CI (it increases it) Define: color excess CE CE = CI (observed) - CI (intrinsic) where CI (intrinsic) can be estimated from sp. type (i.e., T) of star. Empirical work shows that amt absorption depends on ( of radiation: Av ( 1/( Also that amt absorption directly related to CE by Av ( 3 (CE) (actual constant 3.2) Hence, if one can find CE, then can determine Av & hence correct distance modulus Example (see text; also worksheet) One should generalize discussion to include ( dependence: m( - M( = 5 log d - 5 + A( where A( = k(d = absorption of photons of wavelength ( along line of sight k( = extinction coefficient at wavelength ( (in mag/kpc) Note: At visual wavelengths kv ( 1-2 mag/kpc Interstellar Polarization (omit) Reflection Nebulae appear bluish due to scattered light The Nature of Interstellar Grains must be deduced indirectly possibly graphite or silicate elongated, size of smoke particles formed in atmospheres of cool red giant stars serve as nuclei upon which molecules may form Interstellar Gas Amt IS gas ( 100x amt dust yet gas ( transparent, absorbs only discrete (s from UV radiation, dust absorbs cont. radiation Visible (s not affected by IS gas because H gas cold ( ground state ( absorption requires UV photons Interstellar Optical Absorption Lines Distinguish from stellar by: Temp effect ( IS lines cold, stellar lines hot (O,B stars) Doppler shifts different, esp. for binary stars IS lines very sharp (Vrms very low) Emission Nebulae: HII Regions H-line emission HII region of ionized H Stromgren sphere High-energy UV radiation from stars embedded in H gas ionizes gas out to radius RS Equilibrium occurs when rate recombination (H II + e- ( H I) = rate photoionization (H I + ( ( H II) Basic physics: Consider single * emitting photons into surrounding nebula, ionizing it w/ resulting recombinations: NUV = # photons emitted/s capable ionizing H RS = radius out to which ionization occurs In equilibrium, total # ionizing photons emitted/sec by * = total # recombinations/sec or NUV = (4(/3) RS3 nenH ((2) where ((2) = recombination coefficient (m3/s) to all states n ( 2 (produce photons w/ E < (i) Note: recombining to n=1 state produces another ionizing photon (E ( Lyman limit), gets absorbed; recombining to n(2 state produces photons which easily escape from nebula, transparent to them Nebula fluoresces ( high E UV photons converted to low E visible & IR photons Solve for RS, RS = [NUV / (4(/3) nenH ((2)]1/3 Example: O5 star, NUV ( 1049 photons/s, T ( 8000 K (nebula) ((2) ( 10-19 m3/s, ne ( 109/m3, nH ( 103/m3 RS ( 2.9 x 1018 m ( 93 pc Continuous radio emission Thermal bremsstrahlung Supernova Remnants (discussed in Crab Nebula film) Planetary Nebulae Nebular forbidden lines gas density higher than in HII, but low enough to allow forbidden transitions Interstellar Radio Lines 21-cm neutral H molecular Intercloud Gas (omit) The Evolution of the Interstellar Gas (omit) Star Formation Basic Physics: Size Scale for Collapse Consider large protostellar cloud mass M, temperature T, total # particles N By Virial Thm. 2 Ethermal = -U where Ethermal ( NkT U ( -GMm/R ( -GM2/l where M2 indicates mass attracts itself & l is size scale (length) of collapsing region Assuming pure molecular H cloud, then total # particles is N = M/2mH Subst these in Virial Thm gives 2(M/2 mH) kT ( GM2/l * kT/mH ( GM/l For uniform spherical cloud, must have M = (4(/3) (l3 Subst, kT/mH ( G(l2 or l ( (kT/mHG()1/2 ( 107 (T/()1/2 (m) Example: In giant molecular cloud, T ( 10 K, ( ( 10-15 kg/m3 l ( 1015 m ( 0.1 pc This yields M ( (l3 ( 1030 kg ( 0.5 Msun (about right order to form solar-mass star) Alternate derivation results in eqn w/ l as f(T, M) instead of f(T, () By eqn * above, derived from virial theorem, kT/mH ( GM/l Solve for l, l ( GMmH/kT = (8.07 x 10-15 m.K/kg) M/T (m) Expressing M in terms of Msun, l ( (1.61 x 1016 m.K) (M/Msun)/T (m) Example: A small star cluster of 60 stars, average star mass 1 Msun, initial cloud temperature T ( 10 K l ( 9.7 x 1016 m ( 3.1 pc If the cluster radius is currently ( 0.3 pc, this is a contraction of about factor 10. Molecular Outflows & Star Birth (omit) The Birth of Massive Stars (omit) The Birth of Solar-Mass Stars (omit) PHYS 228 Astronomy & Astrophysics Lecture Notes from Zeilik et al. Chap.16 The Evolution of Stars The Physical Laws of Stellar Structure Hydrostatic Equilibrium Hydrostatic equilibrium (EHE) -- balance between self-gravity (directed inward) & internal pressure (directed outward) dP/dr = -GM(r) ( (r)/ r2 Suns central pressure: Above eqn becomes, in approximation, averaged over entire star (P/(r = -GM(r) <(> (r)/ r2 In approximation, (0 - Pc)/(R - 0) ( -GM(r) <(>(r) / r2 or Pc = GM<(>/R For Sun, Pc = GMsun<(>sun/Rsun = 2.7 x 1014 N/m2 Equations of State For normal stars, assume perfect gas law P(r) = n(r) kT(r) where n(r) = ((r) / ((r) mH and ( = [2X + (3/4)Y + (1/2)Z]-1 ( 0.5 Then perfect gas eqn becomes P(r) = ((r) kT(r) / ((r) mH Modes of Energy Transport Conduction -- not important in most stars Convection transports energy in outer portion of Sun ( convective zone Radiation transports energy in inner portion of Sun ( radiative zone Major sources opacity in stellar interiors (gas ionized) electron scattering scattering of photons by free es photoionization absorption of photons resulting in ionization of elements Equation of radiative transport (Temperature gradient) L(r) = -[64((r2T3(r)] / [3((r)((r)] (dT/dr) or dT/dr = -[3 ((r) ((r) L(r)] / [64((r2T3(r)] Equation of convective transport (Temperature gradient) dT/dr = (1 - 1/() [T(r)/P(r)] dP/dr where ( = ratio cp/cv ( 5/3 for total ionized, ideal gas Solar Luminosity from Radiative Transfer Most of energy transport inside Sun by radiation. Since Ts << Tc, can make approximation of average temp gradient dT/dr ( (Ts - Tc) / (R - 0) = -Tc/R For Sun, dT/dr ( -0.02 K/m = -20 K/km Energy Sources Luminosity const outside core where energy produced Luminosity gradient inside core expressed by eqn thermal equilibrium: dL/dr = 4(r2 ((r) ( (r) eqn thermal equilibrium where ( = rate energy production per unit mass stellar material (J/s . kg) For Sun, ( ( Lsun/Msun = 2 x 10-4 J/s .kg Gravitational contraction (already discussed somewhat) Gravitational contraction of mass results in conversion of grav PE: half to thermal KE of particles & half to radiative photons: Ugrav = KE + PE = -2Eth where Eth = total thermal energy of gas By formal integration, found that Ugrav = -q (GM2/R) where q = 3/5 = 0.60 for uniform sphere = 3/2 = 1.5 for M-S stars Note: for Sun, Ugrav only ( 10-3 Enucl Thermonuclear Reactions Energy released by mass conversion to energy via nuclear processes: E = mc2 Reaction understood after 1938 4H (4.0312 amu) ( He (4.0026 amu) Mass defect (m = 0.0286 amu ( (E = 4.3 x 10-12 J. Mass fraction (m/m = 0.0071 = 0.71% of total mass of 4H converted to energy Fusion Reaction Processes PP chain dominates at core T < 20 x 106 K ( low-mass stars CNO cycle dominates at high core T, requires C present ( medium- to high-mass stars PP I Chain (91% of time) 1H + 1H ( 2H + e+ + ( (1.44 MeV) 2H + 1H ( 3He + ( (5.49 MeV) 3He + 3He ( 4He + 1H + 1H (12.9 MeV) Note particle neutrino ( produced in reaction, carries away small % energy PP II Chain see text p. 300 PP III Chain see text p. 300 Both occur much less frequently (9% of time) Solar Neutrino Problem Raymond Davis detection solar neutrinos in underground mine in SD Neutrino counts less than expected, probably due to quixotic nature of (, not true deficiency CNO cycle -- same basic conversion as PP but uses C as catalyist 12C + 1H ( 13N + ( 13N ( 13C + e+ + ( 13C + 1H ( 14N + ( 14N + 1H ( 15O + ( 15O ( 15N + e+ + ( 15N + 1H ( 12C + 4He Triple-Alpha Process -- 1st stage He burning after MS 4He + 4He ( 8Be + ( 8Be + 4He ( 12C + ( 3-( process plays major role in evolution stars Theoretical Stellar Models In order to investigate nature of stellar interiors, must make certain basic assumptions about mass, size, composition, then apply basic laws of physics to compute theoretical model star Recapitulating the Physics (omit just mention will cover in PHYS 310) dP/dr = -GM(r) ( (r)/ r2 eqn hydrostatic equilibrium dM/dr = 4(r2((r) Eqn continuity of mass dT/dr = -[3 ((r) ((r) L(r)] / [64((r2T3(r)] Energy transport (radiative) dT/dr = (1 - 1/() [T(r)/P(r)] dP/dr Energy transport (convective) dL/dr = 4(r2 ((r) ( (r) Eqn energy generation (thermal equilibrium) P(r) = ((r) kT(r) / ((r) mH Eqn state ideal gas The Physical Basis of the M-L Relationship (already covered briefly) Stellar Evolution Shown that star evolution depends on two major factors: Mass most important ( determines amt fuel & also core temp Chemical comp second imp. ( determines ( in model eqns General sequence of stellar evolution: protostar pre-main sequence main sequence post-main sequence Plot stars changing position on HR diagram called evolutionary track The Births of Stars: Protostars & PMS Stars Star is born from grav. contraction IS cloud gas & dust Virial thm. grav PE ( 50% thermal E, 50% radiative photons Protostar contracting cloud (in free fall initially) before hydrostatic equilib (HE) established Pre-main sequence star after HE established, but before ignition nuclear reactions Solar Mass Protostellar Collapse Example of estimate time cloud to collapse to star free fall (extreme low (, no collisions) Consider test particle mass m at edge spherical cloud mass M, radius R, initial density (o Assume particle falls straight into center, so e = 1 (parabolic), 2a = R. Then, M = 4/3 (R3(o = 4/3 ( (2a)3 (o = (32/3)( a3 (o Substituting for M in Keplers 3rd law extended by Newton, P2/a3 = 4(2/GM, P = [3(/8G(o]1/2 Time for particle to fall into center must be P, or tff = P = [3(/32 G(o]1/2 Plugging in consts, tff = 6.64 x 104 (o-1/2 s Assuming protostar composed pure molecular hydrogen (H2), m = 2mH = 3.3 x 10-27 kg, typical IS density 1010 m-3, get (o ( 3.3 x 10-17 kg/m3 and so tff = 6.64 x 104 (3.3 x 10-17)-1/2 ( 1.16 x 1013 s ( 4 x 105 yr Late in collapse, density & opacity become large, HE sets in ( pre-MS star T increases, & star shines by radiated photons, mostly IR When core T reaches few x 106 K, nuclear reactions begin, contraction ends ( ZAMS star ZAMS star has L = 1Lsun, but R ( 2Rsun From initial collapse to ZAMS takes ( 20 x 106 yr (both protostar & pre-MS stages) Evolution On and Off the Main Sequence Once on ZAMS, star spends 80% of total lifetime converting 4H ( He in core During MS lifetime, core builds up He ash & core T increases, R* increases slightly Calculate roughly MS lifetime of any star. From basic notions, lifetime of * must be directly proportional to mass M & inversely proportional to how rapidly it burns mass, luminosity: t ( M/L From M-L relation for MS stars, taking rough avg over all masses, L*/Lsun ( (M*/Msun)3.3 Then, relative to Sun have t*/tsun ( (M*/Msun) / (L*/Lsun) ( (M*/Msun) / (M*/Msun)3.3 ( (M*/Msun)-2.3 Thus, stars lifetime decreases by ( mass2 A Population I Star of 5 Msun (read on own) A Population I Star of 1 Msun Main Sequence -- H-burning in core Red giant branch inert He core, H-burning in shell Degenerate gas pressure (due to degenerate es) Helium flash / Horizontal branch He-burning in core, H-burning in shell Asymptotic giant branch inert C/O core, H- & He burning shells thermal pulses occur every few 103 yrs cause luminosity to increase 20-50% star pulsates on both short (days-months) & long (103 yr) time scales superwind triggered by pulsations, strips star outer envelope, forms shell to become Planetary Nebula White dwarf Nuclear reactions cease, degeneracy throughout star, mostly Extremely Massive Stars (50 - 100 Msun) Strong stellar winds due to high radiation pressure & low surface g stars lose over 50% of mass by end of MS lifetime Post MS: star alternates between red & blue supergiant blue ( particular nuclear fuel ignited, red ( fuel depleted Eventually blow up as supernova Low-Mass Stars Star needs 0.08 Msun to create high enough core T to sustain nuclear reactions Brown dwarf -- M < 0.08 Msun (never reaches ZAMS) Chemical Composition & Evolution (omit) Interpreting H-R Diagrams of Clusters Star cluster contains members of approx same age Most massive stars leave MS first In 108 yr, cluster has MS stars up to M ( 3 Msun & Mv ( 0 (More massive *s lie to right of MS) Turnoff pt reflects time elapsed since stars first arrived on MS Isochrones lines of const time overlaid on observed cluster HR diagrams to determine cluster age The Synthesis of Elements in Stars Mass of star determines how many different elements can be synthesized by nuclear reactions Sun (low-mass star) can burn H & He only ( produce C/O High mass stars (> 5 Msun) can produce elements heavier than O, Ne, Na First dredge up occurs when star becomes RG ( convective zone deepens to bring processed material from core to surface Second dredge up occurs for medium- high-mass stars PHYS 228 Astronomy & Astrophysics Lecture Notes from Zeilik et al. Chap.17 Star Deaths End result of stars life determined by mass at time of death M < 1.4 Msun ( WD (immediate precursor PN) 1.4 Msun < M < 3.0 Msun ( NS (immediate precursor SN) 3.0 Msun < M ( BH (immediate precursor SN) White Dwarfs & Brown Dwarfs Physical Properties Avg measured masses of WDs M ( 0.6 Msun Very high densities 108 109 kg/m3 internal T ( 106 107 K surface T ( 6000 31000 K R ( 0.01 Rsun L ( 10-3 Lsun Mbol ( +8 to +15 Comp. mostly C, some residual He, very little H on surface Nuclear reactions cease, E derived by remaining thermal energy of degenerate gas Pauli exclusion principle no more than 2 es w/ opposite spins can occupy given volume at one time Degenerate electron gas stellar material uniform density, behaves more like solid than gas, P not proportional to T ( increasing T does not increase P, as in normal *s (hot *s are bigger) Thus, radius of WD decreases w/ increasing mass see below for derivation Electron degeneracy creates pressure which supports *, instead of ordinary gas pressure Mass-radius relation for WDs In degenerate, nonrelativistic gas, relation beween P & ( given by P = K(5/3 eqn of state for degenerate, non relativistic gas Notes: P depends only on (, no T dependence cf. Nondegenerate matter, P = nkT ( P ( (T ( P depends on both (, T P = K((4/3 for degenerate, relativistic gas We know density of object mass M & radius R obeys simply: ( ( M/R3 Subst this in EOS above, P ( (5/3 ( M5/3/R5 Next, assuming hydrostatic equilibrium applies, must have P ( M2/R4 Equating previous two eqns, get M2/R4 ( M5/3/R5 or R ( M-1/3 If had used specifically the EOS for degenerate relativistic gas, P = K(5/3, would have gotten more specific result R = [4(K/G(4/3()5/3] M-1/3 Cooling Times for WDs All luminosity WD derived from thermal energy Ethermal = N (3/2 kT) where N = total # particles in star. Then, tcool ( Ethermal / L Example: Assume M = 0.8 Msun, T ( 107 K, L ( 10-3 Lsun, star made of pure C Ethermal ( 1040 J (student should derive) and tcool ( 109 yr (amt time for WD to cool to BD (black dwarf)) In general, can be shown that on avg, cooling time for WD obeys relation tcool ( L-5/7 Observations First WD discovered by A. Clark in 1862 Sirius B Many WDs near to Sun, numerous like RDs Main classes WDs: DA show strong H lines, similar to A stars DB show strong He lines, similar to B stars DC show only continuum, no sp lines White Dwarfs & Relativity High densities & surface gravities WD allow test General Relativity proposed Einstein Gravitational redshift occurs when photons lose energy as move from strong g to weaker g photon cannot decrease speed, so loss E shows up as decrease in freq., increase in ( For relatively weak g of WDs, may derive approximate relation for wavelength shift using only Newtonian physics (see derivation) ((/(i ( GM/Rc2 approximate gravitational redshift where (( = (f - (i Exact relation found using general relativity (f/(i = [1 - 2GM/Rc2]-1/2 Result typical WDs: ((/(i ( 10-4 Magnetic White Dwarfs (omit) Planets: masses < 0.002 Msun (2 millisuns) Stars: masses > 0.08 Msun (80 millisuns) Brown Dwarfs Brown dwarfs: masses in range 0.002 to 0.08 Msun Derive E from slow grav. contraction Burn H or 2H very weakly Should be very old, also very common L ( 4 x 10-5 Lsun Neutron Stars If at time of death star has mass > 1.4 Msun (but < 3.0 Msun, ): pressure of degenerate es cannot support star against gravity & collapses further es crushed into ps to form ns releasing neutrinos p+ + e- ( n + ( Star becomes ball neutron gas, supported against gravity by neutron degeneracy Physical Properties Extreme high densities 1017 kg/m3 R ( 10-5 Rsun ( 10-3 RWD ( 10 km g ( 1011 g( strong gravitational redshift ((/(i ( GM/Rc2 ( 0.2 (cf ((/(i ( 10-4 for WD) surface B field ( 108 T (cf. ( 102 T for WD) Pulsars Rotating Neutron Stars Discovery pulsars 1967 by J. Bell & A. Hewish Shown to be rotating neutron stars( rotating lighthouse effect Periods 10-3 to 4.0 s (fastest called millisecond pulsars) Radio observations show periods to be increasing ( spin slowing down ( E radiated away dP/dt ( 10-8 s/yr (very tiny, measured only w/ atomic clocks) Period slow down can give rough estimate of pulsars age: t ( P/(dP/dt) Ex. Crab nebula: P = 0.03 s, dP/dt = 1.2 x 10-13 s/s t ( 0.03/(1.2 x 10-13) = 1011s ( 104 yr Dispersion slowing down of longer ( photons more than shorter ( photons as radiation travels through interstellar medium. Can be used to determing distances Lighthouse model rotating magnetic neutron star. Main components of model: neutron star has extremely high ( & rapid rotation powerful dipolar B field transforms rotational E ( electromagnetic E Millisecond Pulsars (omit) Binary Pulsars (omit) The Supernova Connection Neutron stars/pulsars remnants of supernova explosions of massive stars Link between SN & ns the Crab Nebula Supernova visible 1054 AD in constellation Taurus, recorded by Chinese Telescopes in 19th & 20th centuries show large, expanding nebula (Crab) in this position Pulsar (neutron star) discovered in Crab Nebula in late 1960s Pulsar explains how Crab Nebula shines Total energy estimated radiated by pulsar due to spin down ( 5 x 1031 W Easily explains amt energy emitted by Crab Nebula itself ( 1 x 1031 W Black Holes BH defined as region in space-time where gravity so strong that not even light can escape ( black Formation of BH occurs naturally at end of life of massive star, w/ M > 3.0 Msun. Theory shows neutron degeneracy not strong enough to halt collapse, so star continues to shrink to zero volume & infinite density ( singularity (laws physics break down) Basic Physics of Black Holes Simple model BH spherical object w/ surface gravity so strong that Vesc > c. In general, consider a projectile mass m fired outward from any spherical body mass M & radius R, such as star, at exactly vesc. Total energy projectile must be, at instant launch, Etot = KE + PE = mVesc2 - GmM/R Its speed decreases as travels away & eventually speed v = 0 at r = infinity (by defn). Total energy must then be : Etot = mv2 - GmM/r = 0 + 0 = 0 Since Etot must be conserved, Etot = 0 at any time, so setting 1st eqn equal zero gives mVesc2 - GmM/R = 0 Vesc = (2GM/R)1/2 The fastest any object can travel is speed light c, so set Vesc = c, solve for R gives Schwarzschild radius Rs (after physicist K. Schwarzschild who worked out soln ca. 1920). Rs = 2GM/c2 Schwarzschild radius Expressing M in Msun & R in km gives Rs ( 3 (M/Msun) km Example. Sun Rs = 3 km Structure of Space-Time around a Black Hole (omit) Observing Black Holes (omit) PHYS 228 Astronomy & Astrophysics Lecture Notes from Zeilik et al. Chap.18 Variable & Violent Stars Naming Variable Stars Three types intrinsic variables: pulsating variables -- undergo periodic expansion & contraction cataclysmic or eruptive variables -- display sudden, dramatic changes peculiar variables -- do not fit into ether 1) or 2) Note: star may go through several different types variability during lifetime Variable Star Nomenclature: 1st star in particular constellation discovered variable designated R, (e.g., R Virginis) 2nd star designated S, ... , 9th star Z, 10th star RR, 11th star RS, , SS, ST, , ZZ, AA, AB, AZ, BB, BC, , QZ (334th variable) V335, V336, Pulsating Stars Includes: Cepheids, RR Lyrae, W Virginis, RV Tauri, long-period Mira variables Observations Cepheid variable stars period changes in sp. type, T, L light curves & radial velocity curves reveal variability Instability strip lies in HR diagram, Cepheids, RR Lyr stars found here A Pulsation Mechanism (omit) Period-Luminosity Relationship for Cepheids Shown Fig. 18-3 Note difference for Pop I (( Cep) & Pop II (W Vir) stars Avg luminosity L increases w/ increasing period P Note: RR Lyrae stars have Mv ( 0.5, irrespective of P Long-Period Red Variables Include RV Tauri, SR variables, Mira variables Cool red giants & supergiants, sp. K, M L ( 100 Lsun, Mv ( 0 contain both Pop I & II stars fall in asymptotic giant branch region HR diagram(He-shell burning NonPulsating Variables Includes: T-Tauri, flare stars, magnetic stars, spectrum variables, RS Can Ven T-Tauri Stars pre-MS stars, just before H-burning sets in sp show emission lines & abs lines found in association w/ dark IS clouds Flare Stars dwarf M stars Magnetic Variables RS Canum Venaticorum Stars Binary stars, Sun-like, chromospherically active Extended Stellar Atmospheres: Mass Loss (see Table 18-1) An Atmospheric Model Be and Shell Stars Mass Loss from Giants and Supergiants Wolf-Rayet Stars Very hot stars, T ( 30,000 K, sp. O Masses ( 10 - 40 Msun Nearly all binary systems He-rich, H deficient Classes WC (carbon) & WN (nitrogen) Show P-Cygni profiles Planetary Nebulae PN named because appearance similar to planet represents phase between asymptotic gianet & WD lasts 10,000 - 50,000 yr nebular expansion speeds ( 20 km/s central star T very hot, 50,000 - 100,000 K Masses central star: 0.5 - 0.7 Msun (same as WD masses) nebula: 0.1 - 0.5 Msun predecessor may be long-period Mira-type variables Cataclysmic & Eruptive Variables Includes: novae, dwarf novae, supernovae Novae From latin new star, Mv ( -6 to -9 Actually evolved star which suddenly brightens many magnitudes, then declines over weeks (Fig. 18-12) Sp. expansion speed v ( 2000 km/s Mass lost ( 10-5 Msun Progenitor: WD + RG companion, material accretes & ignites sudden H-burning on surface of WD May be recurrent: P ( 18 - 80 yr Dwarf novae: P ( 40 - 100 d Supernovae Mv ( -16 to -20 Expansion speeds ( 10,000 km/s Total E output ( 1044 J ( total output Sun during 1010 yr lifetime 99% SN energy emitted as neutrinos Classification (based on spectra) Type I evolved low- to medium mass stars ( 1 Msun appear in both elliptical & spiral galaxies sp: H absent Model: WD accretes matter from RG companion to push over Chandrasekhar limit 1.4 Msun, Star core collapses into neutron star, outer layers blown out into space Type II evolved high-mass stars, M in range 10 - 100 Msun occur only in spiral arms of spiral galaxies sp: H present Model: Massive star burns Fe in core (endothermic), core collapses into neutron star, outer layers blown out into space The Crab Nebula: A Special Supernova Remnant (discussed in film) Nucleosynthesis in Supernovae r process nuclei capture neutrons faster than beta decay ( builds up neutron-rich material s process nuclei capture neutrons slower than beta decay ( builds up proton-rich material Supernova 1987A Occurred in LMC on 24 Feb 87 Discovered by Ian Shelton of U. Toronto Type II, progenitor blue SG X-Ray Sources: Binary & Variable (omit) Cygnus X-1 Centaurus X-3 SS 433 X-Ray & Gamma Ray Bursters PHYS 228 Astronomy & Astrophysics Lecture Notes from Zeilik et al. Chap.20 The Evolution of Our Galaxy The Structure of Our Galaxy from Radio Studies 21-cm Data and the Spiral Structure (omit) The Galactic Distribution of Gas (omit) The Galactic Center Region Contains expanding, 3-kpc arm High-Velocity Hydrogen Clouds Discrete clouds neutral H move in eccentric orbits above & below galactic plane Magellanic Stream bridge of H gas connecting MW & LMC/SMC young O, B stars present The Distribution of Stars and Gas in Our Galaxy Metallicity measured relative to Sun: [Fe/H] = log (NFe/NH) - log (NFe/NH)solar Summary characteristics stellar populations in Table 20-1 Spiral Arms: Spiral Tracers Spiral arms delineated by young Pop I objects ( serve as tracers OB stars young, open clusters HII (starbirth) regions Stellar Populations: Galactic Disk & Halo Sun belongs to old, metal-rich Pop I ( lie within disk but not necessarily in spiral arm such objects make up thin disk Metal-rich (Z ( 25% solar) Pop II stars ( make up most of mass of Galaxy make up thick disk Globular clusters, RR Lyrae stars found in halo on eccentric orbits Dark halo component of halo containing dark, faint, low-mass objects Mass ( 2 x 1011 Msun The Galactic Bulge & Galactic Nucleus Galactic bulge radius ( 2 kpc mass ( 1010 Msun contains mix Pop I & II K, M giants, Strong IR sources ( AGB stars Galactic nucleus Infer nature from observing star-like nucleus Andromeda Galaxy Spectra ( low-mass dwarfs & metal rich giants, young O & M supergiants, molecular clouds, HII Radio observations ( compact radio source ( 140 AU diameter ( concentration mass at center (BH) Sgr A complex of radio sources at & surrounding nucleus, characteristics HII regions The Distribution of Mass in the Galaxy Mass of galaxy ( 90-95% in stars ( 5 -10% in H gas (( 1% ionized H, 4-9% neutral H) Rapid rotation near center indicated by IR observations ( few 106 Msun confined to diameter 0.04 pc Evolution of the Galaxys Structure (read on own) Density-Wave Model & Spiral Structure Developed by Lin & Shu The Galaxys Past Cosmic Rays & Galactic Magnetic Fields (read on own) Observations of Cosmic Rays The Source & Acceleration of Cosmic Rays The Galactic Magnetic Field PHYS 228 Astronomy & Astrophysics Lecture Notes from Zeilik et al. Chap.21 Galaxies Beyond the Milky Way Galaxies shown to be separate Island Universes in 1924 by E. Hubble, determined distance to Andromeda using Cepheid variables. Galaxies as Seen in Visible Light Visible Light Imaging of Galaxies Classification Scheme Devised originally by E. Hubble 3 major classes: elliptical, spiral, irregular galaxies Tuning fork diagram (see Fig. 21-1) Ellipticals E shape of oblate spheroid subclasses according to degree of ellipticity: 10(a-b)/a E0, E1, E2, , E7 No axis rotation contain only Pop II stars, little or no dust Supergiant Galaxies cD additional type elliptical, c ( supergiant, D ( diffuse diameters ( few 106 pc (Mpc) formed by galactic cannibalism at center of large galaxy cluster Spiral Galaxies S (normal) and SB (barred) possess spiral arms emanating from nucleus both subclasses according to how tightly wound spiral arms are Sa, SBa large nucleus, smooth, poorly defined spiral arms, tightly wound Sb, SBb smaller nucleus than Sa, more open spiral arms Sc, SBc tiny nucleus, extended open spiral arms Note: 1. all spirals contain both Pop I & II stars 2. Pop I/Pop II ratio ( from Sa to Sc 3. bar in SB galaxies develops if galaxy halo mass low S0 Galaxies (also called lenticulars) Intermediate between E7 and Sa Flatter than E7, w/ thin disk & more spheroidal nuclear bulge Resemble S class, but contain only Pop II stars, like E Irregular Galaxies Irr I and Irr II Show no symmetrical or regular structure Irr I have distinct OB stars & HII regions ( strong Pop I component Irr II unresolvable into stars, strong IS dust absorption Dwarf Galaxies dE and dIrr Faint, but numerous galaxies in universe dE dwarf elliptical most common type galaxy in universe dIrr dwarf irregular Peculiar Galaxies Do not fall into any of previously discussed categories, rare Some tidally disrupted, radio sources AGNs active galactic nuclei The Morphological Mix Observed galaxy distribution (biased toward bright galaxies) Spirals 77% Ellipticals 20% Irregulars 3% Representative sample (w/in radius 9.1 Mpc) Spirals 33% Ellipticals 13% Irregulars 54% Photometric Characteristics of Galaxies Integrated Colors Direct correlation between (B-V) & galaxy type: Ellipticals reddest, Irr bluest Within spirals, S0 reddest, Sc bluest Color indicates dominant Pop type Sizes Determine linear size s of galaxies by measuring angular size arad & distance d (pc): arad = s/d Results measurements linear diameters: Dwarf galaxies dE & dIrr ( 3 kpc largest giant ellipticals E ( 60 kpc largest supergiant cD ( 2000 kpc = 2 Mpc (> distance MW to Andromeda) TYPICAL (median all galaxies) ( 15 kpc (( 1/3 size MW) Luminosities (absolute magnitudes M) Must be corrected for: Extinction by dust in MW Extinction by dust within galaxy itself (not needed for E galaxies, have little dust) K-correction needed for distant, redshifted galaxies, emitted light shifted out of rest frame filter band Results from measured fluxes & known distances: dE M ( -8 ( L ( 2 x 105 Lsun cD ellipticals M ( -25 ( L ( 1012 Lsun MW M ( -21 ( L ( 2.5 x 1010 Lsun Masses (Round 1) Simple estimate assumes each star on average has 1 solar mass & emits 1 solar luminosity. With some refinements depending on morphology, get mass range ( 105 to 1010 Lsun The Visible Light Spectra of Galaxies Spectral types (integrated) Composite spectrum light from billions stars, usually F,G,K spectra ( most light comes from late-type *s Galaxies at Radio Wavelengths (omit) Continuum Imaging Line Radiation and Neutral Hydrogen Content Infrared Observations of Galaxies (omit) Few E or S0 galaxies emit IR ( little dust X-Ray Emission from Normal Galaxies (omit) Some Basic Theoretical Considerations Implications of the Classification Scheme Note three important observations: Color of galaxy depends strongly on morphological type Integrated sp type of nuclear region depends strongly on morphological type Spheroidal components of galaxies follow r-1/4 law & disk components exponential law independent of morphological type Most useful classification criteria for galaxies: Disk/bulge ratio Degree winding of spiral arms The Energetics of Galaxies the Virial Theorem Virial theorem can be applied to galaxies: 2 = - For galaxies, putting in actual variables, becomes = 0.4GM/rh where rh = radius enclosing half the mass = mean square of individual velocities of component stars Masses of Galaxies (Round 2) Determined using dynamical methods, w/Keplers 3rd law: P2 = a3 where a = r & P of orbit given by P = 2(r/V Subst V & r into Keplers law & solving for V reveals that V(r) ( r-1/2 I.e., V ( with increasing distance r from nucleus At large distance from nucleus, found that M(r) ( r PHYS 228 Astronomy & Astrophysics Lecture Notes from Zeilik et al. Chap.22 Hubbles Law & the Distance Scale The Period-Luminosity Relationship for Cepheids Discovered by H. Leavitt (1912), discussed in Chap. 18 Measure P ( L ( Mv. Combine w/ measured mv ( m-M ( d Hubbles Law One of most important results of observational spectroscopy galaxies Based on Doppler shifts obtained 1912-25 by V.M. Slipher, Lowell Obs. Most galaxies have redshifts, few blueshifts (only local) ( universe expanding Hubbles paper based on these results, appeared 1929 For galaxy at distance d, redshift defined by z = ((/(o = (( - (o) /(o where z negative for approach, positive for recession If this is a velocity Doppler effect, then speed of recession of galaxy is v = cz = c(((/(o) Hubble used apparent brightness (magnitude) of each galaxy to estimate relative distance bright ( nearby , faint ( distant Plotting radial velocity V vs distance d yielded straight line (see Fig. 22-2) so that v = Hd * or cz = Hd where H = Hubble const = in range 50 - 100 km/s/Mpc Redshift, Distance, & the Age of Universe Importance of Hubble law: gives direct correlation between galaxys redshift (from sp) & distance almost all galaxies have redshifts, only blueshifts from nearby galaxies galaxies at greater distances moving away faster than nearby ones ( uniform expansion Rearranging above, d = cz/H (Mpc) Example: for z = 0.2, H = 50 km/s/Mpc, d = (0.2)(3 x 105 km/s) / (50 km/s/Mpc) = 1200 Mpc On other hand, if H = 100 km/s/Mpc, then d = 600 Mpc Note: simple distance - redshift relation given above holds only for small z (< 0.8) Assuming space-time is Euclidean (flat), correct relationship given by d = (cz /H) [(1 + z/2) / (1 + z)2] Age of Universe Can use Hubble law to estimate expansion age of universe By simple physics, distance & time related by velocity by d = V/t, or t = d/V Eliminating V &d between this & original Hubble law eqn * above, t = 1/H Hubble time or Expansion age of universe Example: For H = 50 km/s/Mpc, get t = 1/H = 1/(50 km/s . Mpc) = 1/(50 km/s.Mpc) x (106 pc/Mpc) x (3 x 1013 km/pc) = 6 x 1017 s = 2 x 1010 yr Note: H very uncertain, so for H = 100 km/s.Mpc, obtain t = 1010 yr Also, method very crude, does not take into account deceleration of universe Parametrizing Equations with H (omit) The Physical Meaning of the Cosmic Expansion Expansion of universe ( beginning to universe ( Big Bang All observers in universe should measure same Hubble const Note: H not really constant: H ( as universe ages & expansion decelerates Ho = present expansion rate Evaluating Hubbles Constant Best observational evidence gives H in range 50 - 100 km/s/Mpc Distances to Galaxies the Distance Scale Building Up the Scale (see Fig. 22-3) Standard distant bright galaxies used std candles to determine H: supergiant S & giant E The Sandage-Tammann Process to Obtain H Relies heavily on distances to Cepheids in MW & nearby galaxies, also sizes HII regions in Sc I galaxies Yields H ( 50 km/s/Mpc The Aaronson-Huchra-Mould Way to H Relies on Tully-Fischer relation between Mb of spiral galaxies & spread in their 21-cm emissions wider 21-cm line ( greater galaxys luminosity Yields H ( 90 km/s/Mpc PHYS 228 Astronomy & Astrophysics Lecture Notes from Zeilik et al. Chap.23 Large-Scale Structure of the Universe Clusters of Galaxies Types of Clusters Regular Clusters have spherical symmetry & high degree central concentration giant systems, contain ( 104 members many member galaxies brighter than M = -15 almost all members ellipticals & spirals Irregular Clusters little symmetry or central concentration usually smaller systems, contain < 103 members contain mixture all types galaxies elliptical, spiral, irregular Local Group (MW & Andromeda members) is Irr cluster The Local Group of Galaxies (see Fig. 23-1) Contains ~20 members, dominated by spirals MW, M33, & Andromeda other members dwarf E & Irr Diameter ~1 Mpc Other Clusters of Galaxies (see Table 23-2) Example of Regular Cluster Coma Cluster 7 Mpc diameter, contains 104 galaxies Example of Irregular Cluster (other than Local Group) Virgo cluster --- 3 Mpc diameter, 205 members Abell Richness Classification based on number galaxies within 2 mag of 3rd brightest cluster member Clusters and the Galaxian Luminosity Function (omit) Galaxian Cannibalism (discuss briefly since mentioned earlier) occurs when supergiant E galaxy tidally disrupts & grav attracts smaller galaxies grows larger cD galaxies located at centers of clusters were once E galaxies, but grew by tidal disruption & grav infall other galaxies nuclei may be multiple, normal E galaxies have extensive halos up to 1 Mpc diameter Note: comparison galaxian spacing w/ those planets & stars In solar system, planets spaced out ( 105 x their diameters In Galaxy, stars spaced out ( 106 x their diameters In galaxian clusters, galaxies spaced out only ( 102 x their diameters (Thus, tidal forces very important in clusters) Cluster Redshifts & Velocity Dispersions Velocity dispersion ( of cluster galaxies obtained from Doppler shifs individual galaxies Expression for line-of-sight velocity dispersion given by ( = [(vp2/(n-1) - (c()2/(1-vp/c)2]1/2 where vp = velocity comp of galaxy parallel to line of sight ( = uncertainty of individual redshifts n = number galaxies in cluster Note: Above expression only for one dimension, along line of sight. For large n & randomly oriented orbits, total (physical) velocity dispersion is ( = (3 (p Superclusters Question: Do clusters galaxies form superclusters of clusters? Answer: Yes Discovery known since 1924 after Hubble demonstrated existence galaxies revealed more clearly on Palomar Sky Survey wide-angle views of sky taken in 1950s Local Supercluster dominated by Virgo cluster, contains Local Group (see Fig. 23-10) Other superclusters Perseus supercluster (Fig.23-9), Coma supercluster (Fig.23-11) Voids regions of few or no galaxies (e.g., Bootes void) Summary features large-scale structures: Superclusters not spherical, most have flattened, pancake structure, slight curvature All rich galaxy clusters lie in superclusters At least 95-99% (probably 100%) of all galaxies lie in superclusters Voids are predominantly spherical Voids empty of at least bright galaxies Peculiar Motions & the Great Attractor Question of whether grav attraction of superclusters could cause peculiar motions Local Supercluster moving toward direction of Hydra-Centaurus region, to still undetermined source called Great Attractor (too much obscuration to tell what it is) What is a Void? (already discussed briefly) Intergalactic Matter Intergalactic dust must be very sparse between galaxies little or no dimming of galaxies Intergalactic gas (mostly H) extremely low density ( must be totally ionized Masses Round 3: The Missing? Mass Most useful parameter in determining missing mass problem is M/L ratio M/L high for ellipticals, low for irregulars M/L studies ( presence dark matter, distribution unknown PHYS 228 Astronomy & Astrophysics Lecture Notes from Zeilik et al. Chap.24 Active Galaxies & Quasars Radiation Mechanisms Radiation from nornal galaxies dominated by thermal processes: visible starlight thermal radio emission IR radiation from heated IS dust Radiation from active galaxies produced by both thermal & nonthermal processes: synchrotron radiation Emission Lines Strong emission lines in galaxy sp imp indicator of activity Emission lines arise from downward bound-bound or bound-free transitions Two classes excitation/ionization mechanisms: Collisional from cloud-cloud collisions or IS shock waves Radiative several different processes, all require high-E (hard) photons for excitation/ionization Thermal radiation from very hot BB ( produces large #s UV photons Synchrotron radiation from relativistic es in powerful B field Forbidden lines arise from transitions between metastable states under low densities Forbidden transitions from electric quadrupole, magnetic dipole, magnetic quadrupole nature Partially forbidden lines from electric dipole Notation: forbidden lines enclosed in square brackets, e.g., [O III] 500.7 Synchrotron Radiation Requires supply of relativistic es and B field Flux nonthermal radiation has spectral form F (() = Fo(-( or log F(() = -( log ( + constant Both radio galaxies & quasars have similar synchrotron spectra Two main types sp: Extended sources: ( ( 0.7 - 1.2 Compact sources: ( ( 0.4 Moderately Active Galaxies Preludes to Activity Disk regions most normal galaxies ( abs lines dominate, some emission HII regions Nuclear regions normal galaxies ( H( & [O II] 372.7 emission lines as well as abs lines Starburst Galaxies (omit) AGNs Show all or most of characteristics: high luminosities, L >1037 W nonthermal emission, plus excessive UV, IR, radio, & radio compared w/ normal galaxies compact region of rapid variability (few light-months across at most) high contrast of brightness between nucleus & large-scale structures explosive appearance or jet-like protuberances broad emission lines (sometimes) Note: nucleus of MW possesses some of these properties, but luminosity only L ( 1035 W Seyfert Galaxies aftter K. Seyfert (1943), noted spiral galaxies w/ properties: show unusually bright nuclei, broad emission lines most Seyfert galaxies (~90%) are spirals ~1% all spirals are Seyfert galaxies, may be brief active phase tend not to be strong radio sources most tend to be in close binary galactic systems ( tidal forces may induce Seyfert activity BL Lacertae Objects Galaxies similar to prototype BL Lac, w/ properties rapid variability at radio, IR, visual (s no emission lines, just continuum nonthermal continuous radiation, most in IR strong, rapidly varying polarization Radio Galaxies Refers to those galaxies w/ radio luminosity L > 1033 W Two main types: Extended radio galaxies radio emitting region larger than optical image of galaxy Compact radio galaxies radio emitting region same size or smaller than optical image Extended radio galaxies commonly show double structure of two gigantic lobes separated by Mpcs & symmetrically Classification by structure: Classical doubles (e.g., Cyg A) have high L, lobes aligned thru center, bright hot spots at ends Wide-angle tails, or bent tails (e.g., Cen A) have intermediate L, bend thru nucleus, tail-like pro Narrow- tail sources have lowest luminosities, U shapes, rapidly moving galaxies in cluster Note: at least 50% classical doubles show jets, over 80% other two categories show jets Quasars: Discovery & Description Discovered 1960 by A. Sandage & T. Matthews quasi-stellar radio sources 3C 48, 3C 273 faint starlike object which was strong radio source Emission lines Identified 1963 by M. Schmidt as enormously redshifted H Balmer lines Characteristics similar to active galaxies strong, broad emission lines, radio sources Emission-Line Characteristics All emission lines very broad, highly redshifted Requires use relativistic Doppler formula: z = ((/(o = [(1 + v/c) / (1 - v/c)]1/2 - 1 Note: for nonrelativistic v reduces to z = ((/(o ( v/c Example: for quasar w/ z = 2, set z = 2 = [(1 + v/c) / (1 - v/c)]1/2 - 1 Get v/c = 0.8 Absorption-Line Spectra Most quasars w/ z > 2.2 also have strong abs lines in their sp Most quasars w/ z < 2 do not have any absorption lines z from abs line always ( z from em line Classes absorption line quasars: Type A: Broad Absorption Line (BAL) quasars inferred high-velocity ejection Type B: Low Velocity Sharp Line Systems C IV lines, difference ~3000 km/s between abs & em Type C: Sharp Metallic Lines difference up to 30,000 km/s between abs & em Type D: The Lyman-( Forest show sharp Ly-( lines, v differences same as Type C Continuous Emission Continuous emission from QSOs nonthermal (same as for radio galaxies) Two categories classed according to sp index (also same as for radio galaxies) range ( 0.0 to 1.6, w/ division at ( ( 0.5 Quasars also split into two categories according to polarization: Low polarization quasars (most) High polarization quasars (only 3% of bright QSOs) shown to be compact radio sources, similar to BL Lac objects Optical Appearance stellar appearance angular diameter < 1(( (some have faint nebulosity associated) optical variations on time scale hours to yrs ( set limits on physical size of emitting region strong correlation between variation & polarization Rule: If object varies w/ period t, then radius of emitting region cannot be larger than ct R ( ct Observations suggest for quasars R ( 1010 km ( 1 light day ( size of solar system Problems with Quasars Enormous z for quasars implies two imp things: they must be very far away, & redshifts due to cosmological expansion they must release vast amts energy Example: Quasar 3C 273 has z = 0.16, m = +13. For H = 50 km/s.Mpc, get d = 770 Mpc L ( 1040 W = 100x luminosity of most bright galaxies Problem: How does object generate 100x energy of galaxy in region less than 1 pc across? Energy Sources Most continuous energy of quasar from synchrotron emission: relativistic es in B field Central E source quasar must each year provide es w/ total E ( 1043 J Quasar models based on supermassive BH (107 - 109 Msun) in nucleus of young galaxy BH fueled by tidal disruption of passing stars Material then forms accretion disk around BH, Energy radiated as material spirals inward, releases gravitational PE to power quasar Efficiency energy generation (( 50% of PE is radiated from virial theorem) Model calculations show infall 1 Msun/yr provides ( 1012 Lsun Quantitative Analysis: PE of mass m brought in from ( to distance R from mass M : Ugrav ( -GMm/R Assume BH powering quasar has mass M & luminosity L given by M ( 108 Msun = 2 x 1038 kg L ( 1012 Lsun /yr = 4 x 1038 J/s R ( RS = 2GM/c2 ( 3 x 1011 m ( ( 2 AU) Schwarzschild radius Then PE available per unit mass is, from eqn above, U/m = -GM/R = 4.5 x 1016 J/kg and rate at which mass must infall is (m/(t = ((U/(t) / ((U/(m) ( L/(U/m) ( 1022 kg/s ( 3 x 1029 kg/yr ( 10-1 Msun/yr Superluminal Motions (omit) Double Quasars & Gravitational Lenses (mention briefly) Quasars Compared with Active Galaxies (omit) Non-Cosmological Redshifts (omit) Astrophysical Jets (omit) PAGE  PAGE 83 "BCDMu;J(7z:J]lo'()*.23a/ 0 = > K _ d e v H* jp jqjUmHnHuH*j5UmHnHu6>*CJ>* *6CJCJ5O"CDtu +9:; & Fh^h` & F$a$ִ(xyz"'(JKmno&'012ab & F `  % & _ ` V W M N `    Y Z      & ' 1 ; > L O P V W [ \ t     4 z { ~          ' ( 2 9 jH*jUmHnHu6H* jp jq5H*H*6VN   4 5 D E L M { | } ~    & F & F & F`  ( ) : ; O P Q R t u ` & F $a$`9 M O R s v w { | }   st4JKY4567>?@AGSU[ jm jl j jr j6 jH* j je>*5CJCJ *6CJCJH* jp5H*J%`234K!3EGTU & F  & Fh^h`df<=>NOn` 2ikpsuvwz  #%&1467;<>LN~ *+-.789:;`c j j5CJH* js5>* jmH* jl j jU)<=^|}MN`ab & F cfghijmx{ NTUbxz,.4MOXY]^_`aefhij~ *,<@BCE jp 5CJH* *565CJCJ 5>*CJ *5CJ j6 j jH* jH* jM34Vkl$%=?Z[ & F & F & F  & F EFabghklop"89IJKce-.0?@AG^z|789DEFG_`abchipr j jq56 jw *5CJH* j>*CJ 5>*CJ *6CJCJ55CJ jH*H*L89JKex%9di./ & F8^8 & F & F$a$/0@AGP2\]^ & F & F8^8 & F & F & F & F & FFG`qrY  & F & F` & F & F^ & FuvHh8 9 < = 8!9!:!;!!A![!`!j!k!m!w!x!y!z!{!|!}!!!!!!! 56CJ jrH* jCJ 5>*CJ *6H* j0H* jl jD5CJ j jp56L HIjk= ,!-!k!l!!!!!! & F & F & F & F & F & F & F!!!!!!"")"*"`"a"b"j"k"m"u"##########$$)$*$9$D$R$S$l$m${$|$$$$$$$$$$$$$$$$$$$$$$$$$$%%%&&&H&&&''''E'656H* jp jH* j jr j5CJ * 5>*CJCJ *5CJ6CJCJ5M!"")"*"`"a"b"l"m"u"|"""&#b#########$*$ & F+ & F 8^8 & F  & F*$a$*$:$;$D$S$m$|$$$$$$%%k%%%%%& &I&K&&&&` & F- & F 8^8 & F, & F & F &&-'<'P'Q'''''0(1(V(v(((((((!)")#)0)1)`)a)))`E'F'J'K'L'M'd'e''''''''''''''''''2(3(T(U(W(X(c(d(e(f(g(m(n(o(p(v(w(x((((((((((((((((((((( ) )))#)/)1)2)B)C)D)E)J)K)M)N)O)P)Q)^)_)565CJ jH*H* j j jH* j jr j jpS))))) *D*e*f*s***** +1+B+C+N++++++++++ & F/ & F% & F._))))))B*C*e*f*s********* + +B+N+]+^+i+j++++++++++++++++++++,8,A,R,z,,,,,,,,,,,- -2-:-K-V-W-X-b--------------z.{. jr>*6>* jt6 jlH*5H*5CJ 5>*CJ5>* *T+++, ,8,R,a,q,,,,,,,,,,,-2-I-J-W-X-b- & F0 & F & F & F ^ & Fb------ ...:.U.`.a.m.......'/(/:/@/\/z//////{.........////////0/0|00000000000000111111k1p1s1t1y1z111111222 2?2I2a2b2222222222333 33!3&3'3/333I35>*6CJCJ5CJ 5>*CJ *5CJ6 jH*5CJ 5>*CJ *5 jH*H* jrM//00000I0y0z0{0000000111)1A1B1h1i1u1v11 ^`` & F111111111112 2.2<2=2>2b222222222 3H3I3`$a$ & F2I3a3s3t3333333X4i4!505T5Y5[5b5c555555566+6,6m6n6p6q66 777777778!8#8$8'8(8;8?8F8G8b8c8e8~888888888 9 99/999999999::(:):3:K:L:l:m:n:5CJ566 jH* 5>*CJ *H* jr jH*5UI3b3u33334-4W4X4i44 5!5Q5R5S5T5Z5[5c5~555556-6\6`\66666 7B7S7e7777777777 8)8I8d8e8~888889`990919=9Q9n9o99999:::&:':(:L:m:n:::::::$a$ & F & F`n:w:::::::::::::b;};;;;;;;;; < <<<'<(<<<===>>> > > >>>>>!>#>$>%>&>(>)>6>8>G>H>L>M>N>T>U>V>W>`>>>>e?f???????????d@e@m@o@q@r@@ j j j jp jr jH*H*H*5 *6CJCJU:;a;b;};;;;<7<<<<<=[======M>b>c>>> ? 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