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Home , Cosmic distance ladder, Parsec

and Magnitudes

Prof Andy Lawrence

Astronomy 1G 2011-12

Distance Measurements

Astronomy 1G 2011-12

The cosmic ladder

• Distance measurements in astronomy are a chain, with each type of measurement relative to the one before • The bottom rung is the (AU), the (mean) distance between the and the • Many distance estimates rely on the idea of a "standard candle" or "standard yardstick"

Astronomy 1G 2011-12 Distances in the

• relative distances to planets given by periods + Keplers law (see Lecture-2) • distance to measured by

• Sun-Earth = 1 A.U. (average) • Sun-Jupiter = 5 A.U. (average) • Sun-Neptune = 30 A.U. (average) • Sun- (comets) ~ 50,000 A.U.

• 1 A.U. = 1.496 x 1011 m

Astronomy 1G 2011-12

Distances to nearest • against more distant non- moving stars • 1 (pc) is defined as distance where parallax = 1 second of arc

in standard units D = a/✓ (, )

in AU and arcsec D(AU) = 1/✓rad = 206, 265/✓00 in parsec and arcsec D(pc) = 1/✓00

nearest 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 away... until launch of mission in 2014.....

Astronomy 1G 2011-12

More distant stars : standard candle technique

If a star has 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 - 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 133 pc distance to Orion 389 pc distance to centre of 8 kpc

Astronomy 1G 2011-12 Distance to nearest external

• in very nearest galaxies can just see the most luminous individual stars (Cepheid variables)

• : 48.5 kpc (satellite of )

• 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 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 ) H0 = Hubble's constant

• Recession 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 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 ~ 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 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 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 of an object as the 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 2 f = f = f ⌫ ⌫2 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