Astronomical Distances. How far to a Star? Parallax. Astronomical Distances. The Parsec -new distance unit. Stellar Parallaxes are tiny

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1 How far to a Star? - Fundamental problem.. - How far away is a point of light?. Is it a 00-watt light bulb?. A star as bright as the Sun?. A galaxy of 0 stars? - Brightness alone is not enough. - How to measure astronomical distances? Astronomical Distances distance = speed x time 5 light speed c = 3 0 km/s 0.00 s Edinburgh-St.Andrews 0. s Earth circumference. s Earth-Moon distance 8 min Sun-Earth 0 min Jupiter 5 hr Pluto Stars & Elementary Astrophysics: Introduction Press F for Help Stars & Elementary Astrophysics: Introduction Press F for Help Astronomical Distances.3 yr nearest star (Proxima Centauri) 5,000 yr centre of our Milky Way Galaxy 6 nearest big galaxy. 0 yr (Andromeda, Messier 3, M3) 7 nearest cluster of galaxies 5. 0 yr (Virgo cluster) edge of visible part of Universe 0 0 yr. (The Big Bang) Stars & Elementary Astrophysics: Introduction Press F for Help 3 d p geometrical method p = parallax angle a = Astronomical Unit ( AU) = radius of Earth s orbit around the Sun. a = d tan p a d p d Stars & Elementary Astrophysics: Introduction Press F for Help a Parallax a p (small angle p) The Parsec -new distance unit Example: p p = arcsec ( = ") d a AU 3600 " 80 d = = p " π radian = 0665 AU Fast Method: 6 = 3. 0 m d = /p = 3.6 light yr For p in arcsec = parsec ( pc) and d in parsec a Stellar Parallaxes are tiny Bessel (838) First parallax: 6 Cygni p = 0.9 arcsec d = /0.9 = 3. pc also in 838: Henderson (á Centauri). Struve (Vega) The nearest star (beyond the Sun) Proxima Centauri p = 0.76, d =.3 pc. Stars & Elementary Astrophysics: Introduction Press F for Help 5 Stars & Elementary Astrophysics: Introduction Press F for Help 6

2 imits of Parallax Ground-based CCDs pc Hubble Telescope pc Hipparcos (all sky) pc Today: Nearby Stars Only The Distance adder Different methods for different supernovae distances cepheid variables stars 05: GAIA (whole galaxy) 0,000 pc Stars & Elementary Astrophysics: Introduction Press F for Help 7 PARAAX IS THE FOUNDATION FOR A OTHER METHODS Stars & Elementary Astrophysics: Introduction Press F for Help 8 Inverse Square aw uminosity energy = time Units: Joule / sec = Watt (e.g. 00 W light bulb) Solar luminosity: 6 sun= W Flux Units: uminosity, Flux Watts / square metre Sun viewed from Earth: energy F = time area F sun= 380 W/ m Stars & Elementary Astrophysics: Introduction Press F for Help 9 Stars & Elementary Astrophysics: Introduction Press F for Help 0 Inverse Square aw d area of sphere = π d Continuous sky images. fish-eye lens + CCD. 7 observatories Sky from Australia: Flux viewed from distance d : = πd F Magellenic Clouds Milky Way centre Stars & Elementary Astrophysics: Introduction Press F for Help Stars & Elementary Astrophysics: Introduction Press F for Help

3 Magnitudes Hipparcos (nd century BC) stars visible by eye: st mag = brightest 6th mag = faintest Herschel (780s) st mag stars are 00 times brighter than 6th mag stars. 800s -- eye responds to flux ratios true flux : 0 : 00 perceived : : 3 Apparent Magnitude (measures apparent brightness) Pogson (856). apparent magnitude: m =.5log( F / F F = F 0 0 F 0 = flux of ( m /.5) 0 ) 0 mag star (Vega) Stars & Elementary Astrophysics: Introduction Press F for Help 3 Stars & Elementary Astrophysics: Introduction Press F for Help Naked Eye Stars brightest: Sirius (mag -.5) bright mag - star faint mag stars Stars & Elementary Astrophysics: Introduction Press F for Help 5 Magnitude Difference (measures brightness ratio) m m =.5 log( F/ F ) m > m F < F m m F / F (0%) (%) Stars & Elementary Astrophysics: Introduction Press F for Help 6 Apparent Magnitude Sun m = -6.8 full moon -.5 venus - Sirius -.5 Vega 0.0 faintest galaxies detected by HST mag= 0 30 mag= 0 Stars & Elementary Astrophysics: Introduction Press F for Help 7 Absolute Magnitude (measures true brightness) m = apparent mag at distance d. M = aparent mag at 0 pc. F (0 pc ) = F ( d ) M =.5 log = m 5 log d ( ) ( F (0 pc ) / 0 ) ( d / 0 0 pc ) pc d known (e.g. parallax) for nearby stars. Stars & Elementary Astrophysics: Introduction Press F for Help 8 F

4 Absolute Magnitude m = apparent mag at true distance d. M = aparent mag at 0 pc. d known (parallax) for nearby stars star m M d(pc) Vega Sirius Sun /0665 Which star is truly brighter? m M Distance Modulus (measures distance) = 5 log = 5 log ( d ( d /0 pc) / pc ) 5 Stars & Elementary Astrophysics: Introduction Press F for Help 9 Stars & Elementary Astrophysics: Introduction Press F for Help 0 Electromagnetic Radiation (EMR) EMR = ight (of any wavelength) speed of light in vacuum (slower in air, glass, ) c = 3 0 m /s wave properties (interference,diffraction). frequency = speed / wavelength. Hz = cycles/s = (m/s) / m particle properties (photons). discovery by Planck (900). photon energy :. h = Planck s constant (6.6x0^(-3) Joule/Hz) Stars & Elementary Astrophysics: Introduction Press F for Help 8 f = c / λ E = h f Thermal (Blackbody) radiation Characterised by temperature T. blackbody radiation small hole furnace temperature T light reaches equilibrium with hot walls of furnace. Stars & Elementary Astrophysics: Introduction Press F for Help Stellar Spectra Resemble blackbody spectra temperature T at star surface (hotter interior invisible) To Earth photon escapes photon trapped in gas near star surface Stars & Elementary Astrophysics: Introduction Press F for Help 3 First clue to quantum mechanics Blackbody Spectra Planck function (900) B( λ, T ) hot cool wavelength Stars & Elementary Astrophysics: Introduction Press F for Help λ. Peak shifts. Area increases

5 Wien s aw Star Colours λ max T = constant m K λ max 0 K 300 nm T cool hot Stars & Elementary Astrophysics: Introduction Press F for Help 5 Stars & Elementary Astrophysics: Introduction Press F for Help 6 Wien s aw in action Sun: T 6000 K, λ max 500 nm intensity peak at visible wavelength looks yellow or white to us hot star: T = 000 K, λ max 50 nm ultraviolet peak, looks blue cool star: T = 3000 K, λ max 000 nm infrared peak, looks very red B( λ, T ) λ energy units : time area σ = Blackbody Flux B( T ) = σ T W W = m m K Stefan Boltzmann constant Area increases with T. W m K 8 - ( ) K Stars & Elementary Astrophysics: Introduction Press F for Help 7 Stars & Elementary Astrophysics: Introduction Press F for Help 8 magnitudes : light speed : frequency: photon energy Review m =.5 log ( F / F 0 ) M = m 5 log ( d /0 pc) : 8 c = 3 0 m / s f = c / λ E = h f blackbody peak: λ max 0 K 300 nm T flux : B( T) = σ T Stars & Elementary Astrophysics: Introduction Press F for Help 9 R,T d F Review temperature : T flux at star surface: B( T) = σ T radius : R area : π R luminosity : = π R σ T distance: d area : π d luminosity : = π d F flux : F = π d = σ T R d Stars & Elementary Astrophysics: Introduction Press F for Help 30

6 White Dwarf vs Red Giant R = 0. 0Rsun T =30,000K (Earth-sized star) = R R = 0 8 T T 0 = 0 = π R σ T log = log( πσ ) + log. R + log Stars & Elementary Astrophysics: Introduction Press F for Help 3 R = 00 R sun T = 3,000K T log = log R + log T + const bright log faint hot R/Rsun log T cool Stars & Elementary Astrophysics: Introduction Press F for Help 3 Our Sun s Future How a Star works Nuclear furnace H --> He --> C, N, O, Fe log R/Rsun Hot gas pushes out. Force: k T M F out ~ m R Gravity pulls in. Force: G M R F in ~ log T Sun s Core Temperature: T G M m 0 K k R ~ 7 Stars & Elementary Astrophysics: Introduction Press F for Help 33 Stars & Elementary Astrophysics: Introduction Press F for Help 3 Our Sun s Future He --> C,N,O,... Planetary Nebulae (PN) (PN) envelope ejects (Planetary Nebula) envelope expands (red giant) He core H --> He Ashes cool (white dwarf) 75% H, 5% He (made in the Big Bang) Stars & Elementary Astrophysics: Introduction Press F for Help 35 Stars & Elementary Astrophysics: Introduction Press F for Help 36

7 Dumbell Nebula (M7) Dumbell Nebula (M7) Egg Nebula (polarised) Egg Nebula (polarised) Stars & Elementary Astrophysics: Introduction Press F for Help 37 Stars & Elementary Astrophysics: Introduction Press F for Help 38 NGC 33 NGC 33 Hourglass Hourglass Stars & Elementary Astrophysics: Introduction Press F for Help 39 Stars & Elementary Astrophysics: Introduction Press F for Help 0 Spirograph Spirograph Ring (M57) Ring (M57) Stars & Elementary Astrophysics: Introduction Press F for Help Death of a Star, Birth of a White Dwarf Stars & Elementary Astrophysics: Introduction Press F for Help

8 Broadband Photometry Spectrum Vega Spectrum of Vega B(λ,T) λ U B V R nm measure flux in each bandpass I bandpass filters Flux ratios e.g. F(B)/F(V) change with temperature T. F( U) F( B) F( V) F( R) F( I) U B V R I Stars & Elementary Astrophysics: Introduction Press F for Help 3 Stars & Elementary Astrophysics: Introduction Press F for Help U Sun-like Spectrum Spectrum of Sun B V R I colour index : Colour Indices F( V ) apparent mag: V = mv =.5 log 0 ( V) F absolutemag: M V = mv 5log( d /0 pc) (andsimilarly for U, B, R, I,etc.) F( B) ( ).5 log F0 B B V = F( V) F0 ( V) (andsimilarly for U-B, V-R, R-I,etc.) Stars & Elementary Astrophysics: Introduction Press F for Help 5 Stars & Elementary Astrophysics: Introduction Press F for Help 6 Theory Observation flux apparent mag F U, B, V,... luminosity absolute mag M V = V 5log( d /0pc) temperature colour index T B V Stars & Elementary Astrophysics: Introduction Press F for Help 7 Bright M V Faint Blue hot Hertzsprung-Russell (or H-R) Diagram B V Red cool Stars & Elementary Astrophysics: Introduction Press F for Help 8

9 Most stars are born in Star Clusters. Same distance, age Globular Cluster (M55) H-R Diagram red giants white dwarfs Stars & Elementary Astrophysics: Introduction Press F for Help 9 main sequence Stars & Elementary Astrophysics: Introduction Press F for Help 50 Spectral Analysis Kirchhoff s laws 3 types of spectra light is dispersed into a spectrum using a diffraction grating (or prism) in a spectrograph blackbody gas cloud 3. Continuous + absorption-line spectrum Fraunhofer (85) first extensive study of the Sun - identified about 600 dark lines prism Fraunhofer lines - strongest named A,B,.... Continuous spectrum. Emission-line spectrum (+ weak continuum) Stars & Elementary Astrophysics: Introduction Press F for Help 5 Stars & Elementary Astrophysics: Introduction Press F for Help 5 Kirchhoff s laws A hot opaque body, e.g. a hot dense gas, produces a continuous spectrum, e.g. a black-body spectrum. A hot transparent gas emits an emission-line spectrum, bright spectral lines, sometimes with a faint continuous spectrum. A cool transparent gas in front of a continuous spectrum source produces an absorption-line spectrum - a series of dark spectral lines. (Kirchoff and Bunsen laboratory experiments, 850s) Stars & Elementary Astrophysics: Introduction Press F for Help 53 Vega - absorption lines SS Cygni - emission lines Stars & Elementary Astrophysics: Introduction Press F for Help 5

10 Kirchoff s aws hot opaque interior, cool transparent atmosphere absorption lines Spectral Fingerprints Each element / molecule absorbs and emits only certain specific wavelengths of light. Spectral lines are diagnostic of the chemical composition and physical conditions (temperature, pressure) hot transparent gas (accretion disc) emission lines Stars & Elementary Astrophysics: Introduction Press F for Help 55 Stars & Elementary Astrophysics: Introduction Press F for Help 56 Atoms and Ions atoms: nucleus (protons + neutrons) + electrons mass charge proton + neutron 0 electron /836 - atom - neutral : equal numbers of protons (+ve) and electrons (-ve) ion - ionised : electrons removed, positive net charge Atoms and Ions Examples: Hydrogen ( H ) ( proton ) + electron H II ( p ) charge + Helium (He) ( p + n ) + e- 0 He II ( p + n ) + e- + He III ( p + n ) + Oxygen (O) ( 8p + 8n ) + 8 e- 0 OIII ( 8p + 8n ) + 6 e- + OVI ( 8p + 8n ) + 3 e- +5 Stars & Elementary Astrophysics: Introduction Press F for Help 57 Stars & Elementary Astrophysics: Introduction Press F for Help 58 e.g. Hydrogen: E = e r Atomic Energy evels I = n n n E = 3.6 ev = 0 E e = proton charge r = size of electron orbit n= Stars & Elementary Astrophysics: Introduction Press F for Help 59 3 I = 3.6eV = Ionisation Potential photon n= bound-bound 3 Atomic Transitions absorption ionisation emission bound-free recombination free-free Stars & Elementary Astrophysics: Introduction Press F for Help 60

11 Energy change associated with each transition is h c E = h f = λ ( h is Planck's constant) higher E fi higher frequency (shorter wavelength) photon is absorbed/emitted energy changes are very small measured in EECTRON VOTS (ev) ev = J Stars & Elementary Astrophysics: Introduction Press F for Help 6 bound-bound h c + E λ bound-free Energy Conservation = E h c + E = E + mv λ h c free-free m v h c m v Stars & Elementary Astrophysics: Introduction Press F for Help 6 λ + = λ + Example: H atom (see handout: energy level diagram) Energy difference between the ground state (n=) and the first excited state (n=) : E = E E = I = (3.6 ev)(0.75) = 0.eV h c ( J s)(3 0 m/s)(0 nm/m) λ= = -9 E (0.eV)(.60 0 J/eV) =.6 nm in UV part of spectrum yman α line (α) 3 n= yman Series yman α β γ 9. nm yman continuum photons ionise H from the ground state. Stars & Elementary Astrophysics: Introduction Press F for Help 63 Stars & Elementary Astrophysics: Introduction Press F for Help 6 3 n= Balmer Series Balmer Hα Hβ nm H 36 nm Balmer continuum photons ionise from n= H Hβ Balmer absorption and emission lines Hα Stars & Elementary Astrophysics: Introduction Press F for Help 65 Stars & Elementary Astrophysics: Introduction Press F for Help 66

12 Rydberg formula λ = R nl nu - Rydberg constant R = m = 9. nm R n l - principal quantum number of lower level n u - " " " of upper level YMAN series n l = n u =, 3, UV BAMER 3,, 5 visible PASCHEN 3, 5 near IR BRACKETT 5, 6 IR PFUND 5 6, 7 IR Stars & Elementary Astrophysics: Introduction Press F for Help 67 Hydrogen is simplest. Multi-electron atoms have more complicated energy-levels. ions with electron are like Hydrogen but with larger Ionisation Potential due to higher charge of nucleus Stars & Elementary Astrophysics: Introduction Press F for Help 68 ine Strengths and Widths Equivalent Width Ca II Fe I, II Measures line strength, NOT line width. E.W. = width of rectangle with same area as line. Units: nm H H each line has different strength (quantum mechanics) 0 Same width, different equivalent width. more ions --> stronger line Stars & Elementary Astrophysics: Introduction Press F for Help 70 Stars & Elementary Astrophysics: Introduction Press F for Help 69 λ= λ 0 Doppler Shift. λ< λ 0 rest wavelength: λ0 velocity: v redshift: λ λ 0 v z = = λ λ0 λ 0 c redshifted: z > 0 receding blueshifted : z < 0 approaching Doppler (83) for sound. λ > λ 0 Stars & Elementary Astrophysics: Introduction Press F for Help 7 Example: v = 00 km s -, l 0 = 500 nm v 00 km s λ= λ 0 = 500 nm = 0.3 nm c km s small shift, so no colour changes. unless v ~ c (near a black hole, or relativistic jet) Cosmological redshifts can be large: λ= λ z Big Bang Doppler Shift ( + ) = ( nm) (+ 6) 88 nm 0 = T 0 T = + z 3000 K 00.7 K Stars & Elementary Astrophysics: Introduction Press F for Help 7

13 Hubble Deep Field Thermal Broadening mv random motions kt of atoms Typical redshift: z ~ High redshift: z > 5, UV light shifts to red wavelengths Stars & Elementary Astrophysics: Introduction Press F for Help 73 λ0 Stars & Elementary Astrophysics: Introduction Press F for Help 7 Rotational Broadening Pressure Broadening Energy levels shift when particles are nearby λ0 Stars & Elementary Astrophysics: Introduction Press F for Help 75 high pressure gas Stars & Elementary Astrophysics: Introduction Press F for Help 76 Zeeman Shift Energy levels shift and split in Magnetic field Quantum Uncertainty fuzzy energy levels short visit -> uncertain E magnetic field E t h h =. 0 5 ev sec Stars & Elementary Astrophysics: Introduction Press F for Help 77 Stars & Elementary Astrophysics: Introduction Press F for Help 78

14 G M O Spectral Types Spectral Types A B V Stars & Elementary Astrophysics: Introduction Press F for Help 79 M G A O Why spectra differ ine strengths (EW ratios) change mainly due to SURFACE TEMPERATURE (hot-> high ionisation and excitation cool-> neutral atoms and molecules) Some line widths and ratios change with UMINOSITY Very little range of abundances (7% H + % He % everything else) Stars & Elementary Astrophysics: Introduction Press F for Help 80 Spectral Classification 890s first photographic spectra 98- Henry Draper Catalogue "spectral classes" of ~ 5,300 stars!!! (star names HD 353, HD 0958, etc) original classification: A,B, R,S from simple to complex lines many letters later dropped or merged. 90s photometry (colour indices) revealed correct temperature sequence confirmed by atomic physics 90s Morgan & Keenan (MK spectral types) Stars & Elementary Astrophysics: Introduction Press F for Help 8 handout: spectral classes provide a "short-hand" description of the appearance of a stellar spectrum. Stars & Elementary Astrophysics: Introduction Press F for Help 8 Spectral Types hot cool O B A F G K M ( Oh! Be A Fine Girl Guy, Kiss Me! ) ( "early-type" "late-type" ) sub-class 0-9 e.g. B0, B9, G Spectral Types hot cool O B A F G K M ( Oh! Be A Fine Girl Guy, Kiss Me! ) ( "early-type" "late-type" ) sub-class 0-9 e.g. B0, B9, G (N R S ( No Romeo, Scram )) Stars & Elementary Astrophysics: Introduction Press F for Help 83 Stars & Elementary Astrophysics: Introduction Press F for Help 8

15 one example of a luminosity criterion: uminosity Classes H lines of Balmer series are affected strongly by pressure broadening pressure gradient surface gravity of star g GM g = R (M = mass, R = radius, G = gravitational constant) dwarf star, small R, large g broadened H lines giant star, large R (~00 ), small g narrow H lines (handout: spectra showing luminosity effects) Stars & Elementary Astrophysics: Introduction Press F for Help 85 main-sequence V most common subgiants IV red giants III common bright giants II rare supergiants Ia,b very rare (blue to red) white dwarfs DA quite common DB,DO Stars & Elementary Astrophysics: Introduction Press F for Help 86 MK Spectral Types e.g. Sun G V Vega A0 V. Betelgeuse M Iab Rigel B8 Ia Aldebaran K5 III Review Multicolour Photometry Use filters (e.g. U B V R I ) measure flux densities : (f B, f V,...) apparent magnitudes : (B, V, ) colour indices : B V =.5 log f f + ( ) [ ] constant absolute magnitudes (d from parallax): = V 5 log( d /0 pc ) M V B V Stars & Elementary Astrophysics: Introduction Press F for Help 87 Stars & Elementary Astrophysics: Introduction Press F for Help 88 Bolometric Magnitude Spectral Types M bol = absolute bolometric magnitude total flux over the entire spectrum. gives M V gives M bol O B.C. vs Spectral Type M A F Difficult to measure M bol. Easy to measure M V. λ Stars & Elementary Astrophysics: Introduction Press F for Help 89 G M B V Stars & Elementary Astrophysics: Introduction Press F for Help 90 G A O

16 Bolometric Corrections B.C. = M bol - M V < 0 to make star brighter. Sun: M V =.83, B.C. = 0. M bol =.69 F0V B.C. = 0 (most optical). O5V = 3.8 (mostly UV M8V =.0 mostly IR) = 0 (sun) M bol M bol 0. M bol M (sun) (sun) =.5 log (sun) Stars & Elementary Astrophysics: Introduction Press F for Help 9 bol Calibrations Calibrations of colour indices, temperatures, absolute visual magnitude (M V ), and (less precise) spectral types very well defined for most main-sequence stars (5,000 T 30,000 K) less so for hotter O stars (> 0,000 K) and cooler M stars (<,500 K) (handout: example of relevant calibrations) Stars & Elementary Astrophysics: Introduction Press F for Help 9 The Hertzsprung-Russell (HR) Diagram first presented independently by H (9) and R (93) to show links between spectral types (or colours) of stars and absolute magnitudes now recognised as one of the most important diagrams for all astronomy, because of its importance for understanding the evolution (ageing) of stars (handout: HR diagram) Alternative versions of the H-R diagram: uminosity or M bol Theory vs Observations theory Temperature T M v observations colour index (e.g. B-V) or Spectral Type Stars & Elementary Astrophysics: Introduction Press F for Help 93 Stars & Elementary Astrophysics: Introduction Press F for Help 9 To calculate R: = π R σt Observe:. parallax p --> distance d=/p.. spectral type or colour index --> T. apparent magnitude, e.g. V. V M = V 5log ( d /0 pc) M = M B.C. M bol M bol Stellar Radii Not highly accurate (0-50%) bol V + ( / (sun) ) ( sun) =.5 log Typical Radii Solar radius: Rsun=7x0^5 km main-sequence stars: R ~ Rsun giants: R ~ up to 00 Rsun supergiants: red: R ~up to 000 Rsun blue: R ~0-50 white dwarfs: R ~ 0.0 Rsun Stars & Elementary Astrophysics: Introduction Press F for Help 95 Stars & Elementary Astrophysics: Introduction Press F for Help 96

17 Most accurate radii (<%) from. ECIPSING BINARY STARS and (for nearby stars). INTERFEROMETRY and (for a few stars). UNAR OCCUTATIONS. Accurate R improves T via T Accurate Radii = πr σ if distance (hence ) also known. Stars & Elementary Astrophysics: Introduction Press F for Help 97 Binary Stars -- two stars in mutual gravitational attraction, orbiting their common centre of mass -- only source of empirical masses for stars -- accurate sizes, shapes, temperatures, luminosities (hence distances) Stars & Elementary Astrophysics: Introduction Press F for Help 98 primary star m Centre of Mass a secondary star m a a Orbit Velocities a a a V V a m a m = a m = a m + m a m = a m + m V V = a a = m m V + V = P = orbit period π a P Stars & Elementary Astrophysics: Introduction Press F for Help 99 Stars & Elementary Astrophysics: Introduction Press F for Help 00 Elliptical Orbits Visual binary Model binary binary Stars & Elementary Astrophysics: Introduction Press F for Help 0 Stars & Elementary Astrophysics: Introduction Press F for Help 0

18 visual binary Types of Binaries Spectroscopic binary spectroscopic binary. SB SB lines from or stars eclipsing binary Stars & Elementary Astrophysics: Introduction Press F for Help 03 Stars & Elementary Astrophysics: Introduction Press F for Help 0 Eclipsing binary Star sizes from timing Proximity Effects ellipsoidal variations heating effects Stars & Elementary Astrophysics: Introduction Press F for Help 05 Stars & Elementary Astrophysics: Introduction Press F for Help 06 Types of Binaries Binary Star with Accretion Disc detached semi-detached contact Stars & Elementary Astrophysics: Introduction Press F for Help 07 Stars & Elementary Astrophysics: Introduction Press F for Help 08

19 ight curves for close binary stars Computer Models of Eclipsing Binary Stars ight ight curve curve of of Contact Contact Binary Binary Orbital Phase Stars & Elementary Astrophysics: Introduction Press F for Help 09 Stars & Elementary Astrophysics: Introduction Press F for Help 0 Velocity curves for binary stars Velocity curve redder Radial Velocity K K bluer Stars & Elementary Astrophysics: Introduction Press F for Help Orbital Phase Stars & Elementary Astrophysics: Introduction Press F for Help Orbit inclination i = 0 for face-on orbit i = 90 degrees for edge-on orbit Doppler shifts measure. K = V sin i To measure masses: C of M - visual radial velocity and light variations - spectroscopic - eclipsing 3 3 velocity orbital phase (time) radial velocity curves Stars & Elementary Astrophysics: Introduction Press F for Help 3 Stars & Elementary Astrophysics: Introduction Press F for Help

20 Observe: K = V K = V Calculate masses: m K = m K Kepler s aw: m + M Masses a m P = yr AU sun sin i sin i P π a sin i = ( K + K ) P 3 Analysis of RV curves gives "minimum masses" 3 3 ( Msin i ), ( M sin i ) and projected sizes of orbits ( a sin i ), ( a sini) Stars & Elementary Astrophysics: Introduction Press F for Help 5 Stars & Elementary Astrophysics: Introduction Press F for Help 6 Analysis of light curves of eclipsing binaries gives orbital inclination i radii of both stars, relative to the size of the orbit r r, a a Hence, for eclipsing, spectroscopic binaries, we obtain: masses M and M radii R and R luminosities and (if T or T known) Stars & Elementary Astrophysics: Introduction Press F for Help 7 Stars & Elementary Astrophysics: Introduction Press F for Help 8 used as tests of theoretical models of stars HR Diagram high mass main sequence low mass T Empirical MASS-UMINOSITY relationship for main-sequence stars:. M i.e. M = for 0. Msun < M < 0 M sun M sun see log M vs.log plots (handout) sun Stars & Elementary Astrophysics: Introduction Press F for Help 9 Stars & Elementary Astrophysics: Introduction Press F for Help 0

21 Mass-luminosity and HR diagram (T-) for detached eclipsing binaries Mass-radius plot for detached eclipsing binaries Stars & Elementary Astrophysics: Introduction Press F for Help Stars & Elementary Astrophysics: Introduction Press F for Help Mass-uminosity for Main Sequence Slope = + M M sun Msun Stars & Elementary Astrophysics: Introduction Press F for Help 3 Energy supply: Rate of burning: ifetime: E t ~ = ~ 0 0 Star ifetimes E = M c (Joules) M c Stars burn for a long time. Big stars burn out faster. M (W = Joule/s) M = M M c M / M M yr M 3 Stars & Elementary Astrophysics: Introduction Press F for Help sun The Celestial Sphere Celestial Pole N 5. THE MOTIONS OF STARS IN SPACE Celestial Equator Stars & Elementary Astrophysics: Introduction Press F for Help 5 Stars & Elementary Astrophysics: Introduction Press F for Help 6 S

22 Declination ( Dec, ) (Degrees North) Star Coordinates δ N δ W α Right Ascension (RA, ) (Hours East ) ( hours = 360 degrees) Stars & Elementary Astrophysics: Introduction Press F for Help 7 S α E Stars & Elementary Astrophysics: Introduction Press F for Help 8 Star Velocities star v δ v α v r star v δ v α v r observer 3 velocity components (mutually perpendicular) observer - in line-of-sight (radial direction) RADIA VEOCITY v r (km s - ) Stars & Elementary Astrophysics: Introduction Press F for Help 9 Stars & Elementary Astrophysics: Introduction Press F for Help 30 v δ v δ star v α v r star v α v r observer - components perpendicular to the line of sight TRANSVERSE VEOCITY v t (km s - ) V t = + V α V δ observer v α and v δ are the components of v t in the directions of increasing RIGHT ASCENSION (α) and DECINATION (δ) on the sky. Stars & Elementary Astrophysics: Introduction Press F for Help 3 Stars & Elementary Astrophysics: Introduction Press F for Help 3

23 v r - from Doppler shift of spectral lines v α, v δ - projected onto the sky we can measure only angular changes over time, called PROPER MOTION. µ ( arcsec yr - ) Two components of µ are µ α cos δ, µ δ (see handout ) Proper Motions tiny Speeds of stars orbiting the Galaxy v ~ 50 km s - but distances are in parsecs - pc = km thus proper motions µ are small (< 0. arcsec yr - ) Stars & Elementary Astrophysics: Introduction Press F for Help 33 Stars & Elementary Astrophysics: Introduction Press F for Help 3 V t small angle = µ d d µ Vt - 5 µ (arcsecyr ) d (pc) 3 = 0 km pc 7-06,65 (arcsec/radian) 3 0 s yr - Fast caclulation: Vt =.7µ ONY for V t in km s -, µ in arcsec yr -, d in parsecs d µ = µ α cos δ + µ δ Stars & Elementary Astrophysics: Introduction Press F for Help 35 Stars & Elementary Astrophysics: Introduction Press F for Help 36 star v r v α observer SPACE MOTION = velocity vector speed V = + V r V t + direction specified by v α, v δ, v r Stars & Elementary Astrophysics: Introduction Press F for Help 37 v δ Astrometry Satellites HIPPARCOS (997) Accurate parallax and proper motion for bright stars (V < 9) 0 3 arcsec yr stars to 00 pc GAIA planned ESA (0...) arcsec parallax 0 5 proper motion 6-0 arcsec yr distances and motions. of stars throughout the Galaxy Stars & Elementary Astrophysics: Introduction Press F for Help 38 -

24 Star Cluster at the Galactic Centre VT (Very arge Telescope) D=8m, one of, in Chile. Infrared light (to see through the intervening dust) Stars & Elementary Astrophysics: Introduction Press F for Help 39 Stars orbiting the Black Hole Adaptive Optics (corrects for seeing) 0.05 pc 8500 pc 0665 arcsec radian =. arcsec Stars & Elementary Astrophysics: Introduction Press F for Help 0 Star Motions at the Galactic Centre Stars & Elementary Astrophysics: Introduction Press F for Help Stars & Elementary Astrophysics: Introduction Press F for Help Black Hole at the Galactic Centre THE END From the proper motions, measure sizes and periods of the star orbits. Kepler s law : M a = M sun AU the black hole mass 3 P yr M M sun Stars & Elementary Astrophysics: Introduction Press F for Help 3 Stars & Elementary Astrophysics: Introduction Press F for Help