4. Cosmological Distance Scale

04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale ObservationalObservational Cosmology:Cosmology: 4.4. CosmologicalCosmological DistanceDistance ScaleScale

“The distance scale path has been a long and tortuous one, but with the imminent launch of HST there seems good reason to believe that the end is finally in sight..”

— Marc Aaronson (1950-1987) 19851985 PiercePierce PrizePrize LectureLecture).. 1 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.1:4.1: DistanceDistance IndicatorsIndicators Distance Indicators • Measurement of distance is very important in cosmology • However measurement of distance is very difficult in cosmology • Use a Distance Ladder from our local neighbourhood to cosmological distances

✰Primary Distance Indicators ➠ direct distance measurement (in our own Galaxy) ✰Secondary Distance Indicators ➠ Rely on primary indicators to measure more distant object. Rely on Primary Indicators to calibrate secondary indicators!

2 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.1:4.1: DistanceDistance IndicatorsIndicators Distance Indicators

Primary Distance Indicators • Radar Echo • Parallax • Moving Cluster Method • Main-Sequence Fitting • Spectroscopic Parallax • RR-Lyrae stars • Cepheid Variables • Galactic Kinematics Secondary Distance Indicators • Tully-Fisher Relation • Fundamental Plane • Supernovae • Sunyaev-Zeldovich Effect • HII Regions • Globular Clusters • Brightest Cluster Member • Gravitationally Lensed QSOs • Surface Brightness Fluctuations 3 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.2:4.2: PrimaryPrimary DistanceDistance IndicatorsIndicators Primary Distance Indicators

PrimaryPrimary DistanceDistance IndicatorsIndicators

• Radar Echo • Parallax • Moving Cluster Method • Main-Sequence Fitting • Spectroscopic Parallax • RR-Lyrae stars • Cepheid Variables • Galactic Kinematics

4 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.2:4.2: PrimaryPrimary DistanceDistance IndicatorsIndicators Radar Echo

• Within Solar System, distances measured, with great accuracy, by using radar echo • (radio signals bounced off planets). • Only useful out to a distance of ~ 10 AU beyond which, the radio echo is too faint to detect.

1 d = c Δt 2 1 AU = 149,597,870,691 m 5 € 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.2:4.2: PrimaryPrimary DistanceDistance IndicatorsIndicators

Trigonometric Parallax • Observe a star six months apart,(opposite sides of Sun) • Nearby stars will shift against background star field • Measure that shift. Define parallax angle as half this shift

6 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.2:4.2: PrimaryPrimary DistanceDistance IndicatorsIndicators

1 AU 1 d = ≈ AU tan prads p

d p € 1 AU

1 206265 1 radian = 57.3o = 206265" d = AU = AU prads p ′′

Define a parsec (pc) which is simply 1 pc = 206265 AU =3.26ly. A parsec is the distance€ to a star which has a parallax angle of 1"

Nearest star - Proxima Centauri is at 4.3 light years =1.3 pc ➠ parallax 0.8" Smallest parallax angles currently measurable ~ 0.001" ➠ 1000 parsecs ➠ parallax is a distance measure for the local solar neighborhood. 7 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.2:4.2: PrimaryPrimary DistanceDistance IndicatorsIndicators Trigonometric Parallax The Hipparcos Space Astrometry Mission Precise measurement of the positions, parallaxes and proper motions of the stars. •Mission Goals - measure astrometric parameters 120 000 primary programme stars to precision of 0.002” - measure astrometric and two-colour photometric properties of 400 000 additional stars (Tycho Expt.) •Launched by Ariane, in August 1989, • ~3 year mission terminated August 1993. •Final Hipparcos Catalogue • 120 000 stars •Limiting Magnitude V=12.4mag •complete fro V=7.3-9mag •Astrometry Accuracy 0.001” •Parallax Accuracy 0.002”

8 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.2:4.2: PrimaryPrimary DistanceDistance IndicatorsIndicators Trigonometric Parallax • GAIA MISSION (ESA launch 2010 - lifetime ~ 5 years) • Measure positions, distances, space motions, characteristics of one billion stars in our Galaxy. • Provide detailed 3-d distributions & space motions of all stars, complete to V=20 mag to <10-6”. • Create a 3-D map of Galaxy.

9 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.2:4.2: PrimaryPrimary DistanceDistance IndicatorsIndicators Secular Parallax Used to measure distance to stars, assumed to be approximately the same distance from the Earth. Mean motion of the Solar system is 20 km/s relative to the average of nearby stars ➠ corresponding relative proper motion, dθ/dt away from point of sky the Solar System is moving toward. This point is known as the apex

For anangle θ to the apex, the proper motion dθ/dt will have a mean component ∝ sin(θ) (perpendicular to vsun ) Plot dθ/dt - sin(θ) ➠ slope = µ vsun 4.16 The mean distance of the stars is d = = pc 4.16 for Solar motion in au/yr. htm µ µ("/ yr) /distance. wright

/~ green stars show a small mean distance edu . red stars show a large mean distance ucla . € astro http://www.

Statistical Parallax Δv Δvr in pc/s If stars have measured radial velocities, d = r ➠ scatter in proper motions dθ/dt can be used to determine the mean distance. θ˙ θ˙ in rad/s 10

θ vt vr θ Observe cluster some years apart ➠ proper motion µ d

Radial Velocity (km/s) vR from spectral lines

Tangential Velocity (km/s) vT = 4.74µ d µ (“/yr) Stars in cluster move on parallel paths ➠ perceptively appear to move towards common convergence point (Imagine train tracks or telegraph poles disappearing into the distance) Distance to convergence point is given by θ v = v sinθ  v € T C  ⇒ d = R vR = vC cosθ 4.74µtanθ

Main method for measuring distance to Hyades Cluster ~ 200 Stars (Moving Cluster Method ➠ 45.7 pc). One of the first “rungs” on the Cosmological Distance Ladder c.1920: 40 pc (130 ly) c.1960: 46 pc (150 ly) (due to inconsistency with€ nearby star HRD) Hipparcos parallax measurement 46.3pc (151ly) for the Hyades distance. 11 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.2:4.2: PrimaryPrimary DistanceDistance IndicatorsIndicators Moving Cluster Method Ursa Major Moving Cluster: ~60 stars 23.9pc (78ly) Scorpius-Centaurus cluster: ~100 stars 172pc (560ly) Pleiades: ~ by Van Leeuwen at 126 pc, 410 ly)

• Hipparcos 3D structure of the Hyades as seen from the Sun in Galactic coordinates. • X-Y diagram = looking down the X-axis towards the centre of the Hyades. • Note; Larger spheres = closer stars • Hyades rotates around the Galactic Z-axis. • Circle is the tidal radius of 10 pc • Yellow stars are members of Eggen's moving group (not members of Hyades). • Time steps are 50.000 years. (Perryman et al. ) 12 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.2:4.2: PrimaryPrimary DistanceDistance IndicatorsIndicators Standard Rulers and Candles To measure greater distances (>10-20kpc - cosmological distances) ➠ Require some standard population of objects e.g., objects of • the same size (standard ruler) or • the same luminosity (standard candle) and • high luminosity can calculate • Flux (S) from luminosity, (L) L L S = 2 ⇒ DL = • Calculate distance (DL) 4πDL 4π S • Measuring redshift (z)

• ➠ Cosmological parameters Ho, Ωm,o, ΩΛ,o DISTANCE MODULUS m = −2.5lg(S /S )  (m−M€)   0 5 dL  dL =10 ⇒ M = m − 5lg  ⇒ m − M = 5lgdL,Mpc + 25 M = −2.5lg(L /L0 )   10pc 13

€ 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.2:4.2: PrimaryPrimary DistanceDistance IndicatorsIndicators Main sequence Fitting Einar Hertzsprung & Henry Norris Russell: Plot stars as function of luminosity & temperature ➠ H-R diagram Normal stars fall on a single track ➠ Main Sequence

Observe distant cluster of stars, Apparent magnitudes, m, of the stars form a track parallel to Main Sequence ➠ correctly choosing the distance, convert to absolute magnitudes, M, that fall on standard Main Sequence.

Get Distance from the distance modulus m − M = 5lgdL,Mpc + 25 ) ve - AGB near stars

€ (more

Red Giant ⇒ Branch m-M

Turn Magnitude far stars off Main sequence

WHITE DWARF

⇐ temperature • Useful out to ~few 10s kpc (main sequence stars become too dim) • used to calibrate clusters with Hyades 14 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.2:4.2: PrimaryPrimary DistanceDistance IndicatorsIndicators Spectroscopic Parallax

Information from Stellar Spectra

• Spectral Type ➠ Surface Temperature - OBAFGKM RNS • O stars - HeI, HeII • B Stars - He • A Stars - H • F-G Stars - Metals • K-M Stars - Molecular Lines

•Surface Gravity ➠ Higher pressure in atmosphere ➠ line broadening, less ionization - Class I(low) -VI (high) • Class I - Supergiants L = 4πσT 4 R2 • Class III - Giants α • Class V - Dwarfs L ∝ M (α ~ 3− 4) • Class VI - white Dwarfs GM g = R2

Temperature from spectral type, surface gravity from luminosity class ➠ mass and luminosity. Measure flux ➠ Distance from inverse square Law € 15 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.2:4.2: PrimaryPrimary DistanceDistance IndicatorsIndicators Cepheid variable stars - very luminous yellow giant or supergiant stars. Cepheid Variables Regular pulsation - varying in brightness with periods ranging from 1 to 70 days. Star in late evolutionary stage, imbalance between gravitation and outward pressure ➠pulsation Radius and Temperature change by 10% and 20%. Spectral type from F-G • Henrietta S. Leavitt (1868 - 1921) - study of 1777 variable stars in the Magellanic Clouds. • c.1912 - determined periods 25 Cepheid variables in the SMC ➠ Period-Luminosity relation • Brighter Cepheid Stars = Longer Pulsation Periods • Found in open clusters (distances known by comparison with nearby clusters). ➠ Can independently calibrate these Cepheids

16 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.2:4.2: PrimaryPrimary DistanceDistance IndicatorsIndicators Cepheid Variables 2 types of Classical Cepheids

Mv = −(2.76lgPd −1.0) − 4.16

Distance Modulus m − M = 5lgdL,Mpc + 25

€ Prior to HST, Cepheids only visible out to ~ 5Mpc 17 € 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.2:4.2: PrimaryPrimary DistanceDistance IndicatorsIndicators Stellar pulsation ➠ transient phenomenon RR Lyrae Variables Pulsating stars occupy instability strip ~ vertical strip on H-R diagram. Evolving stars begin to pulsate ➠ enter instability strip. Leave instability strip ➠ cease oscillations upon leaving.

Type Period Pop Pulsation LPV* 100-700d I, II radial Classical Cepheids-S 1-6 I radial Classical Cepheids-L 7-50d I radial

W Virginis (PII Ceph) 2-45d II radial RR Lyrae 1-24hr II radial ß Cephei stars 3-7hr I radial/non radial

δ Scuti stars 1-3hr I radial/non radial ZZ Ceti stars 1-20min I non radial

• RR-Lyrae stars • Old population II stars that have used up their main supply of hydrogen fuel • Relationship between absolute magnitude and metallicity (Van de Bergh 1995) Mv = (0.15 ±0.01) [Fe/H] ±1.01 • Common in globular clusters major ➠ rung up in the distance ladder ➠ • Low luminosities, only measure distance to ~ M31 18 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.3:4.3: SecondarySecondary DistanceDistance IndicatorsIndicators Secondary Distance Indicators

SecondarySecondary DistanceDistance IndicatorsIndicators

• Tully-Fisher Relation • Fundamental Plane • Supernovae • Sunyaev-Zeldovich Effect • HII Regions • Globular Clusters • Brightest Cluster Member • Gravitationally Lensed QSOs • Surface Brightness Fluctuations

19 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.3:4.3: SecondarySecondary DistanceDistance IndicatorsIndicators Globular Clusters

Main Sequence Fitting H-R diagram for Globular clusters is different to open Clusters (PII objects!) ➠Cannot use M-S fitting for observed Main Sequence Stars ➠ Use Theoretical HR isochrones to predict Main Sequence ➠ distance ➠ Alternatively use horizontal branch fitting

Angular Size Make assumption that all globular clusters ~ same diameter ~ D ➠ Distance to cluster, d, is given by angualr size θ=D/d

Globular Cluster Luminosity Function (GCLF) (similarly for PN) Use Number density of globular clusters as function of magnitude M

2 −(M −M * ) φ(M) = Ce 2σ 2

Peak in luminosity function occurs at same luminosity (magnitude) Number density of globular clusters as function of magnitude M for Virgo giant ellipticals €

Distance range of GCLF method is limited by distance at which peak Mo is detectable, ~ 50 Mpc 20 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.3:4.3: SecondarySecondary DistanceDistance IndicatorsIndicators Tully Fisher Relationship v 2 GM Centrifugal R Gravitational = 2 Redshift R R

Assume same mass/light ratio for all spirals l = M /L 2 Assume same surface brightness for all spirals σ = L /R € v 4 L = € R ∝v 4 σ 2 G2 R Flux l €    4  L CvR In Magnitudes M = Mo − 2.5lg  = Mo − 2.5lg   Lo   Lo 

€ Δν M = −10lg(vR ) + C €  Wo  More practically M = −alg  − b Blueshift  sini Wo = spread in velocities i = inclination to line of sight of galaxy 21 € 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.3:4.3: SecondarySecondary DistanceDistance IndicatorsIndicators  W  Tully Fisher Relationship M = −alg o  − b Tully and Fischer (1977): Observations with I ≤ 45o  sini a = 6.25±0.3 DISTANCE b = 3.5 ± 0.3, Knowing M ➠ MODULUS

m = −2.5lg(S /S )  (m−M )   0 5 dL  dL =10 ⇒€ M = m − 5lg  ⇒ m − M = 5lgdL,Mpc + 25 M = −2.5lg(L /L0 )   10pc

Tully-Fisher Fornax & Virgo Members ✰ Bureau et al. 1996 Problems with Tully-Fisher Relation € • TF Depends on Galaxy Type

Mbol = -9.95 lgVR + 3.15 (Sa) Mbol = -10.2 lgVR + 2.71 (Sb) Mbol = -11.0 lgVR + 3.31 (Sc)

• TF depends on waveband Relation is steeper by a factor of two in the IR band than the blue band. (Correction requires more accurate measure of M/L ratio for disk galaxies) 22 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.3:4.3: SecondarySecondary DistanceDistance IndicatorsIndicators

D-σ Relationship Elliptical Galaxies ➠ Cannot use Tully Fisher Relation • Little rotation • little Hydrogen (no 21cm)

Faber-Jackson (1976): Elliptical Galaxies L∝σ4 M32 (companion to M31) L = Luminosity σ = central velocity dispersion Ellipticals MB = −19.38 ± 0.07 − (9.0 ± 0.7)(lgσ − 2.3)

Lenticulars MB = −19.65 ± 0.08 − (8.4 ± 0.8)(lgσ − 2.3) Large Scatter ➠ constrain with extra parameters➠ Define a plane in parameter space

€ Faber-Jackson Law I(r) I e (r /r )1/ 4 Intensity profile (surface brightness) = o − o (r1/4 De Vaucouleurs Law) 2 L = ∫ I ∝ Ioro

2 1 GM 2 M Virial Theorem mσ = m ⇒ σ ∝ /Academics/Astr222/Galaxies/Elliptical/kinematics.html ∑ 2 r ∑ r edu o o . cwru . € astr M α Mass/Light ratio ∝ M L http://burro. 1+α € 4(1−α ) −(1−α ) Fundamental Plane L ∝σ Io (Dressler et al. 1987) 23 €

€ 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.3:4.3: SecondarySecondary DistanceDistance IndicatorsIndicators D-σ Relationship Any 2 parameters ➠ scatter (induced by 3rd parameter) Ι Ι

Combine parameters ➠Constrain scatter ➠ Fundamental Plane

Instead of Io, ro: Use Diameter of aperture, Dn, 2 Dn - aperture size required to reach surface Brightness ~ B=20.75mag arcsec

Advantages Disadvantages • Elliptical Galaxies - bright ➠ measure large distances • Sensitive to residual star formation •Strongly Clustered ➠ large ensembles •Distribution of intrinsic shapes, rotation, presence of disks • Old stellar populations ➠ low dust extinction • No local bright examples for calibration

➠ Usually used for RELATIVE DISTANCES and calibrate using other methods 24 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.3:4.3: SecondarySecondary DistanceDistance IndicatorsIndicators Surface Brightness Fluctuations SBF method Measure fluctuation in brightness across the face of elliptical galaxies Fluctuations - due to counting statistics of individual stars in each resolution element (Tonry & Schneider 1988)

Consider 2 images taken by CCD to illustrate the SBF effect; Represent 2 galaxies with one twice further away as the other

measure the mean flux per pixel (surface brightness) µ = NS 1 rms variation in flux between pixels. σ = NS ∝ d N ∝ d 2   µ is independent€ of distance −2 S ∝ d  € σ 2 L S = = ⇒ d Compare nearby dwarf galaxy, nearby giant galaxy, far giant galaxy µ 4πd 2 Choose distance such that flux is identical to nearby dwarf. The distant giant galaxy has a much smoother image than€ nearby dwarf.’ Can use out to 70 Mpc with HST 25

€ 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.3:4.3: SecondarySecondary DistanceDistance IndicatorsIndicators Brightest Cluster Members •Assume: Galaxy clusters are similar Brightest cluster members ~ similar brightness ~ cD galaxies

•Calibration: Close clusters 10 close galaxy clusters: brightest cluster member MV = 22.82±0.61

•Advantage: Can be used to probe large distances

•Disadvantage: Evolution ~ galaxy cannibalism Large scatter in brightest galaxy Use 2nd, 3rd brightest Use N average brightest N galaxies.

26 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.3:4.3: SecondarySecondary DistanceDistance IndicatorsIndicators Supernova Ia Measurements (similarly applied to novae)

White dwarf pushed over Chandrasekhar limit by accretion begins to collapse against the weight of gravity, but rather than collapsing , material is ignited consuming the star in an an explosion 10-100 times brighter than a Type II supernova

Supernova !

Type II (Hydrogen Lines) Type I (no Hydrogen lines)

SN1994D in NGC4526

Type Ib,c (H poor massive Star M>8M ) Massive star M>8M o Type Ia (M~1.4M White Dwarf + companion) o Stellar wind or stolen by companion o 27 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.3:4.3: SecondarySecondary DistanceDistance IndicatorsIndicators Supernova Ia Measurements Supernova Ia:

10 •Found in Ellipticals and Spirals (SNII only spirals) Supernovae: luminosities ≡ entire galaxy~10 Lo 12 •Progenitor star identical (10 Lo in neutrinos) • Characteristic light curve fast rise, rapid fall, SN1994D in NGC4526 in Virgo Cluster (15Mpc) • Exponential decay with half-Life of 60 d. (from radioactive decay Ni56 → Co56 → Fe56) • Maximum Light is the same for all SNIa !!

10 MB,max = −18.33+ 5lgh100 {L ~ 10 Lo}

€

28 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.3:4.3: SecondarySecondary DistanceDistance IndicatorsIndicators

Supernova Ia Measurements 10 MB,max = −18.33+ 5lgh100 {L ~ 10 Lo}

Gibson et al. 2000 - Calibration of SNIa via Cepheids €

lgHo = 0.2(MB,max − 0.720 ± 0.459)

((ΔmB,15,t −1.1) −1.010 ± 0.934) 2 Lightcurves of 18 SN Ia z < 0:1 (Hamuy et al ) Δm −1.1 + 28.653± 0.042 (( B,15,t ) )

ΔmB,15,t = ΔmB,15 + 0.1E(B −V)

ΔmB,15 = 15 day decay rate € E(B −V) = total extinction (galactic +intrinsic)

€ Distance derived from Supernovae depends on extinction

after correction of systematic effects Supernovae distances good out to > 1000Mpc and time dilatation (Kim et al., 1997). ➠ Probe the visible Universe ! 29 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.3:4.3: SecondarySecondary DistanceDistance IndicatorsIndicators Gravitational Lens Time Delays

• Light from lensed QSO at distance D, travel different distances given by Δ=[Dcos(θ) - Dcos(φ)] • Measure path length difference by looking for time-shifted correlated variability in the multiple images

source - lens - observer is perfectly aligned ➠ Einstein Ring source is offset ➠ various multiple images Can be used to great distances

Uncertainties •Time delay (can be > 1 year!) and seperation of the images • Geometry of the lens and its mass /classes/phys240/lectures/lens_results/lens_results.html edu

. • Relative distances of lens and background sources rit

http://spiff. 30 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.3:4.3: SecondarySecondary DistanceDistance IndicatorsIndicators Gravitational Lens Time Delays • Light from the source S is deflected by the angle a when it arrives at the plane of the lens L, finally reaches an observer's telescope O. •Observer sees an image of the source at the angular distance h from the optical axis •Without the lens, she would see the source at the angular distance b from the optical axis. •The distances between the observer and the source, the observer and the source, and the lens and the source are D1, D2, and D3, respectively. http://leo.astronomy.cz/grlens/grl0.html

Small angles approximation Assume angles b, h, and deflection angle a are <<1 ⇒ tanθ∼θ Weak field approximation Assume light passes through a weak field with the absolute value of the perculiar velocities of components and G<

4GMD3 Where ε is the Einstein Radius ε = 2 c D1D2

€ 2 2 1/ 2 2 2 1/ 2 Lens equation - 2 different solutions (b + (b + 4ε ) ) (b − (b + 4ε ) ) corresponding to 2 images of the source: h1 = h2 = € 2 2

For perfectly aligned lens and source (b=0) - two images at same distance from lens h1 = h2 = e 31 € 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.4:4.4: TheThe DistanceDistance LadderLadder The Distance Ladder

TheThe DistanceDistance LadderLadder

32 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.4:4.4: TheThe DistanceDistance LadderLadder The Distance Ladder

Comparison eight main methods used to find the distance to the Virgo cluster.

Method Distance Mpc

1 Cepheids 14.9±1.2

2 Novae 21.1 ±3.9

3 Planetary Nebula 15.4 ±1.1

4 Globular Cluster 18.8 ±3.8

5 Surface Brightness 15.9 ±0.9

6 Tully Fisher 15.8 ±1.5

7 Faber Jackson 16.8 ±2.4

8 Type Ia Supernova 19.4 ±5.0

Jacoby etal 1992, PASP, 104, 599 HST Measures distance to Virgo (Nature 2002) D=17.1 ± 1.8Mpc 33 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.4:4.4: TheThe DistanceDistance LadderLadder The Distance Ladder

Supernova (1-1000Mpc)

Hubble Sphere (~3000Mpc) 1000Mpc Tully Fisher (0.5-00Mpc)

100Mpc Coma (~100Mpc)

10Mpc Virgo (~10Mpc) Cepheid Variables (1kpc-30Mpc)

1Mpc M31 (~0.5Mpc) RR Lyrae (5-10kpc)

100kpc LMC (~100kpc) Spectroscopic Parallax (0.05-10kpc)

10kpc Galactic Centre (~10kpc) Parallax (0.002-0.5kpc)

1kpc RADAR Reflection (0-10AU) Pleides Cluster (~100pc)

Proxima Centauri (~1pc) 34 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.5:4.5: TheThe HubbleHubble KeyKey ProjectProject The Hubble Key Project

TheThe HubbleHubble KeyKey ProjectProject

35 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.5:4.5: TheThe HubbleHubble KeyKey ProjectProject To the Hubble Flow cz = Hod The Hubble Constant • Probably the most important parameter in astronomy • The Holy Grail of cosmology • Sets the fundamental scale for all cosmological distances €

36 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.5:4.5: TheThe HubbleHubble KeyKey ProjectProject To the Hubble Flow cz H d To measure Ho require = o • Distance • Redshift Cosmological Redshift - The Hubble Flow - due to expansion of the Universe

Must correct for local motions / contaminations vo = radial velocity of observer 1+ z = (1+ z)(1− v /c + v /c) v = radial velocity of galaxy o € G G

-1 vo - Measured from CMB Dipole ~ 220kms (Observational Cosmology 2.3)

€

vG - Contributions include Virgocentric infall, Great attractor etc…

Decompostion of velocity field (Mould et al. 2000, Tonry et al. 2000)

37 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.5:4.5: TheThe HubbleHubble KeyKey ProjectProject Hubble Key Project cz = Hod Observations with HST to determine the value of the Hubble Constant to high accuracy • Use Cepheids as primary distance calibrator • Calibrate secondary indicators • Tully Fisher •Type Ia Supernovae • Surface Brightness Fluctuations € • Faber - Jackson Dn-σ relation • Comparison of Systematic errors • Hubble Constant to an accuracy of ±10%

. Cepheids in nearby galaxies within 12 million light-years. . Not yet reached the Hubble flow . Need Cepheids in galaxies at least 30 million light-years away . Hubble Space Telescope observations of Cepheids in M100. . Calibrate the distance scale

38 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.5:4.5: TheThe HubbleHubble KeyKey ProjectProject Hubble Key Project

H0 = 75 ± 10 km=s=Mpc 39 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.5:4.5: TheThe HubbleHubble KeyKey ProjectProject Combination of Secondary Methods

Mould et al. 2000; Freedman et al. 2000

-1 -1 10 H0 = 71±6 km s Mpc → τ0 = 1.3 × 10 yr

Biggest Uncertainty • zero point of Cepheid Scale (distance to LMC) 40 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.6:4.6: SummarySummary Summary

• There are many many different distance indicators

• Primary Distance Indicators ➠ direct distance measurement (in our own Galaxy)

• Secondary Distance Indicators ➠ Rely on primary indicators to measure more distant object.

• Rely on Primary Indicators to calibrate secondary indicators

• Create a Distance Ladder where each step is calibrated by the steps before them

• Systematic Errors Propagate!

• Hubble Key Project - Many different methods (calibrated by Cepheids)

• Accurate determination of Hubble Constant to 10%

-1 -1 10 H0 = 71±6 km s Mpc → τ0 = 1.3 × 10 yr

Is the Ho controversy over ?

41 04.2.28 Chris Pearson : Observational Cosmology 4: Cosmological Distance Scale - ISAS -2004 Cosmological Distance Scale 4.6:4.6: SummarySummary Summary

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