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Distances and Magnitudes Prof Andy Lawrence Astronomy 1G 2011-12 Distance Measurements Astronomy 1G 2011-12 The cosmic distance ladder • Distance measurements in astronomy are a chain, with each type of measurement relative to the one before • The bottom rung is the Astronomical Unit (AU), the (mean) distance between the Earth and the Sun • Many distance estimates rely on the idea of a "standard candle" or "standard yardstick" Astronomy 1G 2011-12 Distances in the solar system • relative distances to planets given by periods + Keplers law (see Lecture-2) • distance to Venus measured by radar • Sun-Earth = 1 A.U. (average) • Sun-Jupiter = 5 A.U. (average) • Sun-Neptune = 30 A.U. (average) • Sun- Oort cloud (comets) ~ 50,000 A.U. • 1 A.U. = 1.496 x 1011 m Astronomy 1G 2011-12 Distances to nearest stars • parallax against more distant non- moving stars • 1 parsec (pc) is defined as distance where parallax = 1 second of arc in standard units D = a/✓ (radians, metres) in AU and arcsec D(AU) = 1/✓rad = 206, 265/✓00 in parsec and arcsec D(pc) = 1/✓00 nearest star Proxima Centauri 1.30pc Very hard to measure less than 0.1" 1pc = 206,265 AU = 3.086 x 1016m so only good for stars a few parsecs away... until launch of GAIA mission in 2014..... Astronomy 1G 2011-12 More distant stars : standard candle technique If a star has luminosity L (total energy emitted per sec) then at L distance D we will observe flux density F (i.e. energy per second F = 2 per sq.m. landing on our detector); so if know L and measure F 4⇡D we can calculate D but different stars have hugely different luminosities - to know L, we must know which type of star it is (red dwarf, blue supergiant, etc) - it is then a "standard candle" - use spectral features, type of variability etc but any particular type has to be calibrated by having at least one example with a parallax distance distance to Pleiades 133 pc distance to Orion Nebula 389 pc distance to centre of Galaxy 8 kpc Astronomy 1G 2011-12 Distance to nearest external galaxies • in very nearest galaxies can just see the most luminous individual stars (Cepheid variables) • Large Magellanic Cloud : 48.5 kpc (satellite of Milky Way) • Nearest large galaxy M31 : 778kpc Astronomy 1G 2011-12 More distant galaxies • could assume that they are standard candles/yardsticks – D=(4πL/F)1/2 or D=θ R – more distant galaxies are indeed fainter and smaller – but accuracy poor : large range of true L and R • occasionally can see a supernova go off in galaxy; these are good standard candles and can be seen very far away • can measure mass from rotation speed; if we know typical mass-to-light ratio can then get L • see later lectures for details Coma cluster of galaxies : ~100 Mpc Astronomy 1G 2011-12 Very distant galaxies • From Doppler shifts of local galaxies we vr = cz = H0D find all galaxies are receding (expanding Universe) H0 = Hubble's constant • Recession Velocity proportional to if v in km/s and D in Mpc -1 -1 distance (Hubble law) then H0~70 km s Mpc • eg galaxies at 100 Mpc have redshift z=v/c = 0.023 • if this applies universally, then can measure redshift z to estimate distance • eg galaxy with z=0.13 has D~560 Mpc • simple formula only applies locally • see later cosmology lectures Astronomy 1G 2011-12 Luminosities, Fluxes, and Magnitudes Astronomy 1G 2011-12 Bolometric Quantities • Bolometric Luminosity is the energy emitted by an object per second over all wavelengths 26 – Sun : Lbol = 3.826x10 W = Lsun 32 – Massive hot star ~ 300,000 Lsun ~ 10 W 11 11 37 – Milky Way with ~10 stars ~ 10 Lsun ~ 10 W 39 13 – Luminous quasar ~ 10 W ~ 10 Lsun • Flux density is energy arriving at detector/sec/sq.m 2 – Flux density of sunlight at Earth F=1360 W/m L -7 2 F = – Flux density of brightest stars F~10 W/m 4⇡D2 – faintest naked eye star F~10-10 W/m2 – sun like star at 1 kpc F~10-14 W/m2 – faint galaxies in Hubble deep field F~10-19 W/m2 • huge range ! Astronomy 1G 2011-12 Traditional magnitudes • Historically naked eye stars divided into six "magnitudes" • Brightest first mag, faintest sixth mag • Herschel showed this corresponds to roughly factor of 100 in brightness • If we divide that factor 100 into 5 steps where each step is the same multiplying factor M then M5=100 ; so M=log(100)/5=2.5 is the factor corresponding to one magnitude step Astronomy 1G 2011-12 Logarithmic fluxes : magnitude scale • Huge range of observed fluxes suggests a logarithmic scale is in fact sensible. We match tradition by defining ∆m = 2.5 log (F /F ) − 2 1 F1 , F2 are fluxes of stars 1 and 2, and Δm is the magnitude difference between them. If we pick a flux F0 as the zeropoint, then we can define a magnitude system, so that if an object has flux F, then its magnitude is : m = 2.5 log (F/F ) − 0 • an object with F=F0 has m=0 • fainter fluxes give larger (positive) magnitudes Astronomy 1G 2011-12 Logarithmic Luminosities : Absolute Magnitudes • Can define the absolute magnitude of an object as the apparent magnitude it would have when placed at distance D=10pc • Then an object with abs.mag M at distance D pc will have m=M+5logD-5 • The Sun has M=4.83; at 1kpc it will have m=14.83 • A large galaxy has M~-20; at 100 Mpc it will have m=15 Astronomy 1G 2011-12 Spectral Flux • Spectrograph spreads light into component wavelengths • Radio receiver measures brightness in a narrow frequency range • X-ray detector measures the energy of each photon arriving • Then we can measure the intensity at each wavelength : spectral flux density is fλ = energy/sq.m./sec/unit wavelength OR fν = energy/sq.m./sec/unit frequency • How are these related ? Flux in small λ range Δλ is fλΔλ Δλ corresponds to frequency range Δν= Δλ . c/λ2 = Δλ . ν2/c (Note c=ν/λ and differentiate...) So fλΔλ and fνΔλ are same amount of flux and so we find c λ2 f = f = f ⌫ λ ⌫2 λ c Astronomy 1G 2011-12 Flux in a waveband • Usually we can't actually measure bolometric flux, and a proper spectrum is too hard, especially for faint objects • Instead measure flux over a "waveband" = broad range of wavelengths • Compare flux in different wavebands = "colour" • eg brightness through different standard glass filters • most extensively done in optical-IR astronomy • can then define a magnitude system for that specific waveband Astronomy 1G 2011-12 Magnitudes at different wavelengths • The magnitude of a star depends on the wavelength interval at which we observe. - Originally photographic plates were sensitive at 420 nm. - The eye – 540 nm. • A commonly used wide-band magnitude system is the UBV system - Apparent magnitudes denoted by U, B, V. - Absolute magnitude denoted by MU, MB, Mv. • The UBV system has now been extended into the near- and far-infrared. Astronomy 1G 2011-12 Colour index • The colour index (CI) is the difference between magnitudes at two different wavelengths. - B – V and U – B are examples of colour indices • If stars radiated as perfect blackbodies then • Stars, however, deviate from a blackbody so that for 4000 K < T < 10000 K Hotter stars have B – V negative. Cooler stars have B – V positive. Astronomy 1G 2011-12.
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