Determining distance. L 4π f. d = d = R θ. Standard candle. Standard ruler

Determining distance Standard candle d = L 4π f 1 2 d L Standard ruler d = R θ θ R

Determining distance: Parallax RULER tanπ = R d π R d π R = 1AU = 1.5 10 13 cm Define new distance unit: parsec (parallax-second) 1pc = 1AU tan 1!! 1pc ( ) = 206, 265AU = 3.26ly d = 1 π

Determining distance: Parallax

Determining distance: Parallax Point spread function (PSF)

Determining distance: Parallax Need high angular precision to probe far away stars. d = 1 π Error propagation: σ d = d π 2 σ π 2 = 1 π 2 2 2 σ π = σ π π 2 = σ π π d σ d d = σ π π At what distance do we get a given fractional distance error? d = σ d d 1 σ π

Determining distance: Parallax e.g., to get 10% distance errors d max = 0.1 σ π Mission Dates σ π d max Earth telescope ~ 0.1 as 1 pc HST Hipparcos 1989-1993 ~ 0.01 as 10 pc ~ 1 mas 100 pc Gaia 2013-2018 SIM cancelled ~ 20 µas ~ 4 µas 5 kpc 25 kpc

Determining distance: Parallax 8 kpc Hipparcos 1 kpc Gaia

Determining distance: variable stars

Determining distance: variable stars Cepheid variables: Pop I giants, M ~ 5-20 M sun Pulsation due to feedback loop: An increase in T è HeIII (doubly ionized He) è high opacity è radiation can t escape è even higher T and P è atmosphere expands è low T è HeII (singly ionized He) è low opacity è atmosphere contracts è rinse and repeat

Determining distance: variable stars RR-Lyrae variables: Pop II dwarfs, M ~ 0.5 M sun

Determining distance: variable stars CANDLE Variable stars have a tight period-luminosity relation Measure lightcurves: flux(t) Get period P From P-L relation, get L Use L to get distance Very powerful method. Cepheids can be seen very far away. Used to measure H 0 P-L relation is calibrated on local variables with parallax measurements

Determining distance: Tully-Fisher

Determining distance: Tully-Fisher Δλ λ = v rot c

Determining distance: Tully-Fisher NGC 3198 rotation curve

Determining distance: Tully-Fisher HI emission line width HI line is at 1420MHz

Determining distance: Tully-Fisher HI absorption line width

Determining distance: Tully-Fisher

Determining distance: Tully-Fisher CANDLE α L = Cv rot

Determining distance: Tully-Fisher 2 v rot = Gm ( < r ) r m(< r) = r 0 ρ( r)4πr 2 dr Flat rotation curve è density profile is a singular isothermal sphere (SIS) ρ( r) = C r 2 m(< r) = r 0 C r 2 4πr 2 dr = 4πCr M = 4πCR 2 v rot = G4πCr r = 4πGC 2 v rot = GM R

M = 4 3 π R3 ρ ( < R) Determining distance: Tully-Fisher Dark matter halo definition: = 4 3 π R3 Δρ 0 Δ 200 R = 3M 4πΔρ 0 1 3 2 v rot = GM R = G 4πΔρ 0 3 1 3 M 2 3 M = 3 4πG 3 Δρ 0 1 2 3 v rot L M α 3α L v rot

Determining distance: Tully-Fisher Pizagno et al. (2007)

Determining distance: Tully-Fisher Pizagno et al. (2007) L = Cv α rot α M i = 2.5log( Cv rot ) + c = 2.5α logv rot + ( c 2.5logC)

Determining distance: Faber-Jackson

Determining distance: Faber-Jackson

Determining distance: Faber-Jackson

Determining distance: Faber-Jackson CANDLE L = Cσ v α

Determining distance: Fundamental Plane

Determining distance: Fundamental Plane

Determining distance: Fundamental Plane RULER L = Cσ v α I β R = Sσ v γ I δ

Determining distance: Fundamental Plane Bernardi et al. (2007)

Determining distance: Surface Brightness Fluctuations

Determining distance: Surface Brightness Fluctuations

Determining distance: Surface Brightness Fluctuations

Determining distance: Surface Brightness Fluctuations d=0.76 Mpc x2 M32 x4 x8

Determining distance: Supernovae type Ia SN 1572

Determining distance: Supernovae type Ia White dwarfs are made of a C/O core that is supported by electron degeneracy pressure. WD masses cannot exceed 1.4 M sun (Chandrasekhar limit) because then gravity wins. WD that accretes enough mass to surpass this limit, collapses, heats up, and fuses all its C/O in a fast runaway reaction. The energy released unbinds the star. SN Ia

Determining distance: Supernovae type Ia ΔM 15 Riess et al. (2006)

Determining distance: Supernovae type Ia CANDLE Phillips (1993)

The Distance Ladder Method Scatter Reach Systematics Parallax ~d <1 kpc Cepheids 5-10% 30 Mpc Metallicity SBF 5-10% 50 Mpc Stellar LF Tully-Fisher 10-20% >100 Mpc Mass-to-light FP/D n -sigma 10-20% >100 Mpc Kinematics SN Ia 5-10% >1000 Mpc Dust

The Distance Ladder SN Ia Tully-Fisher/D n -sigma Cepheids/RR Lyrae parallax

Redshift We can measure galaxies radial velocity using the Doppler effect. Doppler effect: when an object is moving away from (or toward) us, the frequency of light that we see from it is shifted. galaxy spectrum è Doppler shift (redshift) è radial velocity

Redshift z = λ obs λ emit λ emit Relativistic Doppler Effect 1+ z = 1+ v r c 1 v r c z v r c

Hubble Law

Hubble Law v r = H 0 d H 0 500 km s Mpc 1

The expansion of the universe is such that galaxies recessional speeds are proportional to their distance. Hubble Law

Cosmological Redshift

Cosmological Redshift z = λ obs λ emit λ emit Cosmic scale factor: ( ) = R( t) a t ( ) 1 + z = a t 0 a t 1+ z = λ obs λ emit ( ) R t 0 Wavelengths stretch with scale factor: ( ) = 1 a( t)

Hubble Law v r = H 0 d H 0 70 km s Mpc 1 H 0 100h km s Mpc 1 h 0.7

Hubble Law

Hubble Law

Hubble Law Riess et al. (2006)