1 2 Distance Measurements Chapter 14 Cosmic Distance Ladder Cosmic Distance Ladder How to determine the distance of the observed objects? Absolute distance estimators Standard candles Standard rulers Relative distance estimators The Cosmic Distance All systematic errors associated with a given step are passed onto successive steps Ladder

How to construct the cosmic distance ladder?

Combes et al.: Chapter 11 Binney & Merrifield: Chapter 7

A A 3 4 R R The R A Cosmic Distance Ladder Cosmic Distance Ladder R R Cosmic R R R Distance A R A A Absolute Distance Ladder A A A Distance Measurements Determination

Objects & Scales 5 Recall Chapter 2 6 Trigonometric Parallax Moving Cluster Method (I)

Cosmic Distance Ladder Astronomical Measurements Require stationary background references Hipparcos (High-Precision Parallax-Collecting Satellite): accuracy of ~ 2 mas parsec (parallax-second; pc) = 3.09x1018 cm = 3.26 light years = 206264.8 AU

d 1 = pc p!!

Proper motions of Hyades members

Carroll & Ostlie Fig. 24.29 Copyright: Addison-Wesley

Recall Chapter 2 7 8 Moving Cluster Method (II) Baade-Wesselink Method (I)

Astronomical Measurements Cosmic Distance Ladder

Determine the distance of a star cluster with identified members For a star with radius R and effective temperature Teff , the intrinsic and the convergent point luminosity is given by 2 4 L = 4πR σSBTeff , 1 d tan φ υ µ − = r or in terms of the star’s bolometric absolute magnitude pc 4.74 km s 1 mas yr 1 − # − $ M = 10 log T 5 log R + constant. ! " bol − eff − But how to get the stellar radius R? υt = υr tan φ Studies of variable stars such as Cepheids and RR Lyrae stars can = µd help. Assume the amount by which the size of the star changes between times t0 and t1 is

t1 ∆r = p υ (t)dt, − los !t0 where p is to take into account the projection of the expansion veloc- ity. Carroll & Ostlie Fig. 24.30 9 10 Baade-Wesselink Method (II) Sunyaev-Zel’dovich Effect (I)

Cosmic Distance Ladder Cosmic Distance Ladder

If the star starts out with (unknown) radius r0 at time t0, then by time t1 its radius will be r0 + ∆r. The change in the apparent magnitude of the source over this time will be m m = M M bol,1 − bol,0 bol,1 − bol,0 r + ∆r T = 5 log 0 10 log eff,1 − r − T ! 0 " ! eff,0 "

One can select the time t1 to be a moment at which the star’s effective temperature is the same as it was at time t0. Hence

α∆r Distortion in the Cosmic r0 = , 1 α Microwave Background due − (m m )/5 to photons being inversely where α 10 bol,1− bol,0 ≡ Compton-scattered by For example, the distance to SN 1987A in the Large Magellanic Cloud electrons in the hot gas that pervades the rich clusters. (LMC) is measured to be 55 5 kpc. ± Carlstrom, Holder, & Reese 2002, ARA&A, 40, 643

11 12 Sunyaev-Zel’dovich Effect (II) Sunyaev-Zel’dovich Effect (III)

Cosmic Distance Ladder Cosmic Distance Ladder Distortion in the Cosmic Microwave Background due to pho- tons being inverse Compton-scattered by electrons in the hot gas that pervades the rich clusters. The corresponding optical depth is

τSZ = σTne(s)ds, ! 8π e2 2 where σT = 2 is the effective cross-section of the Thom- 3 mec son scattering. "The ch#ange in brightness temperature due to these interactions is given by ∆T 2kT = e dτ T − m c2 SZ ! e

Carlstrom, Holder, & Reese 2002, ARA&A, 40, 643 13 14 Sunyaev-Zel’dovich Effect (IV) Gravitational Lens Time Delay

Cosmic Distance Ladder Cosmic Distance Ladder Consider a spherical region of gas (isothermal and isodensity) of ra- Consider the image of a background source at redshift zs appears dius R at a distance D, and subtending an angle θ. The X-ray lumi- deviated by an angle αi due to gravitational lensing by a deflecting nosity of the cluster is given by mass at redshift zd, the the extra delay can be shown to be

2 1/2 3 LX neTe R i 1 d s 2 2 ∝ tlens = (1 + zd) D D αi 3 dsΦ(s) , 2c ds − c and fractional change in brightness temperature ! D "Si # where the integral of the Newtonian gravitational potential Φ is eval- ∆T 4kTe = σTneR. uated along the path S traversed by the light that produces the T −m c2 i e image. An observer sees a flux

LX 2 1/2 3 F n T θ D QSO 0957+561 X ∝ D2 ∝ e e ∆T 2 θ ∝ T 3/2 ! " Te D

Hence, measurements of FX , ∆T/T , and Te give the distance D.

15 16 Water- Water-Maser Proper Motions

Cosmic Distance Ladder Cosmic Distance Ladder

Maser NGC 4258 AGN Proper Keplerian Motions orbits Emissions at a single radius by VLBI

Argon et al. 2007, ApJ, 659, 1040 Argon et al. 2007, ApJ, 659, 1040 17 18 Water-Maser Proper Motions

Cosmic Distance Ladder Cosmic Distance Ladder Measuring the drift of water masers:

1 1 υ˙ =9.5 1.1 km s− yr− ± υ2 = in rin Obtaining r =0.12 0.01 pc and comparing with the angular size in ± Relative Distance of the inner edge of the disk θin =4.1 mas, one finds a distance of D =6.2 0.7 Mpc for NGC 4258. ± Measurements

Herrnstein et al. 2005, ApJ, 629, 719

Recall Chapter 3 19 Recall Chapter 8 20 Standard Candles Tully-Fischer Law

Cosmic Distance Ladder Cosmic Distance Ladder

Tully-Fischer law H-band luminosity is pro- portional to υ4

L σ4, H ∝ υ and can serve as a distance indicator.

Sandage & Tammann 1968, ApJ, 151, 531 Aaronson et al. 1982, ApJS, 50, 241 21 22 Surface Brightness Fluctuation (I) Surface Brightness Fluctuation (II)

Cosmic Distance Ladder Cosmic Distance Ladder For galaxies containing stars from a variety of species with different Consider a galaxy at a distance D made up from identical stars of lu- intrinsic luminosities Li with an average of N¯i stars per pixel, we minosity L distributed uniformly with a surface density n. If we make have an image of this galaxy with an angular resolution of δθ, the image ¯ 2 2 2 2 2 Nifi will contain N¯ = nD (δθ) unresolved stars in each δθ resolution σF i L = = ! "2 , element. F ! N¯ifi 4πD i The flux observed from each star is f = L/4πD2, so the average total ¯ 2 ! NiLi flux in each resolution element will be where L = ¯ . ! " NiLi nL δθ2 F = N¯f = . ! 4π The average surface brightness is independent of D; however, Pois- sion fluctuations means that F has a dispersion of N¯ 1/2. Hence, the dispersion of F is

n1/2δθL σ = N¯ 1/2f = . F 4πD Fig. 7.4, Astrophyscs in a Nutshell by D. Maoz

23 24 Surface Brightness Fluctuation (III) Surface Cosmic Distance Ladder Cosmic Distance Ladder Combine the two equalities, we have Brightness σ2 L F = f = . F 4πD2 If two galaxies are made up from identical stars, comparing thre value Fluctuation of f will immediately yield their relative distances. More realistically, a galaxy can be modeled as containing stars from (IV) a variety of species with different intrinsic luminosities Li. If there are an average of N¯i stars per resolution element from species i, the above equation generalizes to ¯ 2 2 Nifi σF i L = = ! "2 , F ! N¯ifi 4πD i ! N¯ L2 where L = i i . ! " N¯iLi Tonry et al. 1997, ApJ, 475, 399 ! 25 Surface

Cosmic Distance Ladder Brightness Fluctuation (V)

Tonry et al. 1997, ApJ, 475, 399