The Cosmic Distance Ladder

Home , Distance modulus, Light echo, Surface brightness, Velocity dispersion

The Cosmic Distance Scale Physics of Galaxies 2012 part 4

1 How big is the Universe?

We want to understand how big the Universe is! Why? Scale: how big are distant things, like galaxies? Time: we’ll see that recession speed is proportional to distance, so our estimate of the age of the Universe depends on how far apart galaxies are!

2 Absolute vs. relative distance measures

Absolute methods: those that can determine a precise distance (usually through geometrical means or timing) generally only work nearby! Relative methods: those that refer to a standard candle or standard ruler — e.g., comparing stars with similar brightnesses or galaxies with similar sizes work out to very large distances!

3 Direct distance measures

Parallax: apparent motion of distant stars caused by orbital motion of earth currently limited to ~100 pc for bright stars after Gaia (~2018 or so), will reach out to ~150 kpc for the brightest stars and ~3 kpc for the faintest (V~20) stars

4 Consider the Earth, 1 AU away from the Sun at position E1. Six months ϖ later, the Earth is at position E2, but the star has remained in the same d place relative to the Sun. Then, as seen from Earth, the star appears to have 1 AU subtended an angle 2ϖ E1 E2 Sun on the sky.

5 Then if r (=1 AU) is the radius of the Earth’s orbit, r we ﬁnd = tan rad d because ϖ is clearly small; then converting to seconds of arc, = 206265 rad 206265 Deﬁning 1 AU such that d = AU and 1 parsec as the distance at which a star would have a parallax of 1″: 1 pc = 206265 AU = 3.086 1013 km = 3.26 light years The distance to a star with observed parallax ϖ″ 1 is then d = pc

6 Secular parallax:

Instead of using the Earth’s motion around the Sun, use the Sun’s motion relative to nearby stars: v⊙≈20 km/s, so the Sun moves about 4 AU/year

twice as large a distance as the Earth moves in six months!

use long time baselines to get very long distance baselines

Problem: the other stars move too!

7 By averaging over proper motions (i.e., a component of space velocity) in the right way, we can determine an average secular parallax for a group of stars µ v An extension of this is statistical parallax, where the stars are distant enough to have roughly random motions (see BM2.2.4)

8 Baade–Wesselink method

2 4 Recall that the luminosity of a star is L =4R Te

So its absolute magnitude is

M = 10 log T 5 log R + C bol e So if we know the apparent magnitude m, temperature Teff, and radius R, we have a direct distance measurement

9 Now, we can generally ﬁnd Teff from colors or the spectrum of a star

How do we ﬁnd R?

In a variable star (or a supernova), the spectral lines shift as a function of time due to the expansion (or contraction) of the atmosphere with some velocity vlos(t), so the change in size of the atmosphere is

t1 r = p v (t)dt this is just rate x time = distance! 1 los t0 where p corrects for the fact that the expansion happens all over the atmosphere, not just along the line of sight, and the negative sign comes from fact that the lines are blue-shifted

10 Therefore the apparent magnitude must change between t0 and t1: m m = 5[log(r +r ) log r ] 10[log T log T ] 0 1 0 1 0 e,1 e,0 So if we can measure the change in magnitudes and the effective temperates, we have r0 and therefore M0.

This method doesn’t work perfectly, but detailed stellar atmosphere modeling of supernovae atmospheres has made it useful for distances to some supernovae, like SN 1987A in the LMC: d=55±5 kpc

11 Distances from time delays

If you can measure the same signal at two different times — the “delay time” — then you can measure a distance: d ct

12 Supernova light echo Consider a supernova explosion at some time t that is seen at position A

Now imagine that some time t0 later a signal (say, strong spectral lines) is seen at position B, and at C an even later time t1 exactly the same signal is seen at position C A i this is light from the SN reprocessed by an inclined ring,

as seen by HST for SN1987A To observer B

13 The time delay from A to B is r t = ring (1 sin i) 0 c while the time delay from A to C is r t = ring (1 + sin i) 1 c

The ring radius rring and C inclination i are then easy to ﬁnd: for SN1987A, t0~90 days and t1~400 days, so i=(42±5)º A i and rring=(0.42±0.03) pc r

Now, the size of the ring is To observer B measured to be θ=1.66±0.03 arcsec, so d=52±3 kpc

14 Gravitational lensing

Quasars lensed into multiple images by foreground galaxies (usually ellipticals) show correlated brightness variations in the light of each component

The delay time of these variations with respect to each other is another geometrical distance measurement

But! Uncertainty in mass proﬁles of lens galaxies means large uncertainties in distance determinations; and ﬁnding correlated variations not straightforward

15 letters to nature

Maser distance The nearby active galaxy NGC 4258 possesses an accretion disk containing water masers orbiting in a thin disk, nearly on the plane of the sky as seen in ultra-high resolution VLBI radio imaging

Figure 1 The NGC4258 water maser. The upper panel shows the best-®tting centred about the systemic velocity of the galaxy. The positions (®lled circles in warped-disk model superposed on actual maser positions as measured by the upper panel) and LOS velocities of these masers imply that they subtend ,88 of VLBA of the NRAO, with top as North. The ®lled square marks the centre of the disk azimuth centred about the LOS to the central mass, and the observed disk, as determined from a global disk-®tting analysis8. The ®lled triangles show acceleration (8±10 km s-1 yr-1) of these features14,15 unambiguously places them the positions of the high-velocity masers, so called because they occur at along the near edge of the disk. The approximately linear relationship between frequencies corresponding to Doppler shifts of ,61,000km s-1 with respect to systemic maser impact parameter and LOS velocity demonstrates that the disk is the galaxy systemic velocity of ,470 km s-1. This is apparent in the VLBA total very thin17 (aspect ratio ( 0.2%) and that these masers are con®ned to a narrow

power spectrum (lower panel). The inset shows line-of-sight (LOS) velocity versus annulus in the disk. The magnitude of the velocity gradient (s) implies a mean impact parameter for the best-®tting keplerian disk, with the maser data super- systemic radius, r , of 3.9 mas which, together with the positions of the high- h si posed. The high-velocity masers trace a keplerian curve to better than 1%. velocity masers, constrains the disk inclination, is, to be ,82 6 18 (908 for edge-on). Monitoring of these features indicates that they drift by less than 1 km s-1 yr-1 Finally, VLBA continuum images7,13 are included as contours in the upper panel. The , 16 (refs 14±16) and requires that they lie within 5±108 of the midline, the intersection of 22-GHz radio emission traces a sub-parsec-scale jet elongated along the rotation the disk with the plane of the sky. The LOS velocities of the systemic masers are axis of the disk and well-aligned with a luminous, kiloparsec-scale jet18.

routine, but estimates of absolute distances are rare1. In the vicinity of the Sun, direct geometric techniques for obtaining Ageometricdistancetothe absolute distances, such as orbital parallax, are feasible, but such techniques have hitherto been dif®cult to apply to other galaxies. galaxy NGC4258 from orbital As a result, uncertainties in the expansion rate and age of the Universe are dominated by uncertainties in the absolute calibra- motions in a nuclear gas disk tion of the extragalactic distance ladder2. Here we report a geometric distance to the galaxy NGC4258, which we infer from J. R. Herrnstein*², J. M. Moran², L. J. Greenhill², the direct measurement of orbital motions in a disk of gas P. J. Diamond*³, M. Inoue§, N. Nakai§, M. Miyoshik, surrounding the nucleus of this galaxy. The distance so deter- C. Henkel¶ & A. Riess# minedÐ7:2 6 0:3 MpcÐis the most precise absolute extragalac- * National Radio Astronomy Observatory, PO Box O, Socorro, New Mexico 87801, tic distance yet measured, and is likely to play an important role in USA future distance-scale calibrations. ² Harvard-Smithsonian Center for Astrophysics, Mail Stop 42, 60 Garden Street, NGC4258 is one of 22 nearby active galactic nuclei (AGN) known Cambridge, Massachusetts 02138, USA to possess nuclear water masers (the microwave equivalent of 12 ³ Merlin and VLBI National Facility, Jodrell Bank, Maccles®eld, lasers). The enormous surface brightnesses ( ) 10 K), relatively Cheshire SK11 9DL, UK small sizes ( ( 1014 cm) and narrow linewidths (a few km s-1) of § Nobeyama Radio Observatory, National Astronomical Observatory, these masers make them ideal probes of the structure and dynamics Minamimaki, Minamisaku, Nagano 384-13, Japan of the molecular gas in which they residue. Very-long-baseline k VERA Project Of®ce, National Astronomical Observatory, Mitaka, Tokyo, interferometry (VLBI) observations of the NGC4258 maser have 181-8588, Japan provided the ®rst direct images of an AGN accretion disk, revealing ¶ Max Planck Institut fuÈr Radioastronomie, Auf dem Hugel 69, D-53121, Bonn, a thin, subparsec-scale, differentially rotating warped disk in the Germany nucleus of this relatively weak Seyfert 2 AGN3±6. Two distinct # Department of Astronomy, University of California at Berkeley, Berkeley, populations of masers exist in NGC4258. The ®rst are the high- California 94720, USA velocity masers. These masers amplify their own spontaneous ...... emission and are offset 61,000 km s-1 and 4.7±8.0 mas (0.16± The accurate measurement of extragalactic distances is a central 0.28 pc for a distance of 7.2 Mpc) on either side of the disk centre. challenge of modern astronomy, being required for any realistic The keplerian rotation curve traced by these masers requires a description of the age, geometry and fate of the Universe. The central binding mass (M), presumably in the form of a supermassive 7 2 2 measurement of relative extragalactic distances has become fairly black hole, of 3:9 6 0:1 3 10 D=7:2 Mpc sin is=sin 82 By examining the rotation curve of the masers, it can be shown that the masers rotate in a thin disk with a hole in the center with an inner radius at θin=4.1 milliarcsec and a velocity of vin=1080 km/s θin By examining the masers again after a time span of a few years, accelerations were seen with a magnitudeFigure 1ofThe NGC4258 water maser. The upper panel shows the best-®tting centred about the systemic velocity of the galaxy. The positions (®lled circles in warped-disk model superposed on actual maser positions as measured by the upper panel) and LOS velocities of these masers imply that they subtend ,88 of VLBA1 of the NRAO,1 with top as North. The ®lled square marks the centre of the disk azimuth centred about the LOS to the central mass, and the observed v˙ =9.5 1.1kmsdisk,yr as determined from a global disk-®tting analysis8. The ®lled triangles show acceleration (8±10 km s-1 yr-1) of these features14,15 unambiguously places them ± the positions of the high-velocity masers, so called because they occur at along the near edge of the disk. The approximately linear relationship between frequencies corresponding to Doppler shifts of ,61,000km s-1 with respect to systemic maser impact parameter and LOS velocity demonstrates that the disk is the galaxy systemic velocity of ,470 km s-1. This is apparent in the VLBA total very thin17 (aspect ratio ( 0.2%) and that these masers are con®ned to a narrow

power spectrum (lower panel). The inset shows line-of-sight (LOS) velocity versus annulus in the disk. The magnitude of the velocity gradient (s) implies a mean impact parameter for the best-®tting keplerian disk, with the maser data super- systemic radius, r , of 3.9 mas which, together with the positions of the high- h si posed. The high-velocity masers trace a keplerian curve to better than 1%. velocity masers, constrains the disk inclination, is, to be ,82 6 18 (908 for edge-on). Monitoring of these features indicates that they drift by less than 1 km s-1 yr-1 Finally, VLBA continuum images7,13 are included as contours in the upper panel. The , 17 (refs 14±16) and requires that they lie within 5±108 of the midline, the intersection of 22-GHz radio emission traces a sub-parsec-scale jet elongated along the rotation the disk with the plane of the sky. The LOS velocities of the systemic masers are axis of the disk and well-aligned with a luminous, kiloparsec-scale jet18.

routine, but estimates of absolute distances are rare1. In the vicinity of the Sun, direct geometric techniques for obtaining Ageometricdistancetothe absolute distances, such as orbital parallax, are feasible, but such techniques have hitherto been dif®cult to apply to other galaxies. galaxy NGC4258 from orbital As a result, uncertainties in the expansion rate and age of the Universe are dominated by uncertainties in the absolute calibra- motions in a nuclear gas disk tion of the extragalactic distance ladder2. Here we report a geometric distance to the galaxy NGC4258, which we infer from J. R. Herrnstein*², J. M. Moran², L. J. Greenhill², the direct measurement of orbital motions in a disk of gas P. J. Diamond*³, M. Inoue§, N. Nakai§, M. Miyoshik, surrounding the nucleus of this galaxy. The distance so deter- C. Henkel¶ & A. Riess# minedÐ7:2 6 0:3 MpcÐis the most precise absolute extragalac- * National Radio Astronomy Observatory, PO Box O, Socorro, New Mexico 87801, tic distance yet measured, and is likely to play an important role in USA future distance-scale calibrations. ² Harvard-Smithsonian Center for Astrophysics, Mail Stop 42, 60 Garden Street, NGC4258 is one of 22 nearby active galactic nuclei (AGN) known Cambridge, Massachusetts 02138, USA to possess nuclear water masers (the microwave equivalent of 12 ³ Merlin and VLBI National Facility, Jodrell Bank, Maccles®eld, lasers). The enormous surface brightnesses ( ) 10 K), relatively Cheshire SK11 9DL, UK small sizes ( ( 1014 cm) and narrow linewidths (a few km s-1) of § Nobeyama Radio Observatory, National Astronomical Observatory, these masers make them ideal probes of the structure and dynamics Minamimaki, Minamisaku, Nagano 384-13, Japan of the molecular gas in which they residue. Very-long-baseline k VERA Project Of®ce, National Astronomical Observatory, Mitaka, Tokyo, interferometry (VLBI) observations of the NGC4258 maser have 181-8588, Japan provided the ®rst direct images of an AGN accretion disk, revealing ¶ Max Planck Institut fuÈr Radioastronomie, Auf dem Hugel 69, D-53121, Bonn, a thin, subparsec-scale, differentially rotating warped disk in the Germany nucleus of this relatively weak Seyfert 2 AGN3±6. Two distinct # Department of Astronomy, University of California at Berkeley, Berkeley, populations of masers exist in NGC4258. The ®rst are the high- California 94720, USA velocity masers. These masers amplify their own spontaneous ...... emission and are offset 61,000 km s-1 and 4.7±8.0 mas (0.16± The accurate measurement of extragalactic distances is a central 0.28 pc for a distance of 7.2 Mpc) on either side of the disk centre. challenge of modern astronomy, being required for any realistic The keplerian rotation curve traced by these masers requires a description of the age, geometry and fate of the Universe. The central binding mass (M), presumably in the form of a supermassive 7 2 2 measurement of relative extragalactic distances has become fairly black hole, of 3:9 6 0:1 3 10 D=7:2 Mpc sin is=sin 82 Equating this acceleration with the centripetal acceleration, we see that 2 v˙ = vin/rin and so the inner radius of the disk is rin=0.12±0.01 pc; θin combining this with θin, the distance must be d=6.2±0.7 Mpc Figure 1 The NGC4258 water maser. The upper panel shows the best-®tting centred about the systemic velocity of the galaxy. The positions (®lled circles in warped-disk model superposed on actual maser positions as measured by the upper panel) and LOS velocities of these masers imply that they subtend ,88 of VLBA of the NRAO, with top as North. The ®lled square marks the centre of the disk azimuth centred about the LOS to the central mass, and the observed disk, as determined from a global disk-®tting analysis8. The ®lled triangles show acceleration (8±10 km s-1 yr-1) of these features14,15 unambiguously places them the positions of the high-velocity masers, so called because they occur at along the near edge of the disk. The approximately linear relationship between -1 7 frequencies corresponding to DopplerAnd shifts also of ,6 1,000kma black s with hole respect tomass:systemic 3.9(±0.1)x10 maser impact parameter and M LOS⊙ velocity demonstrates that the disk is the galaxy systemic velocity of ,470 km s-1. This is apparent in the VLBA total very thin17 (aspect ratio ( 0.2%) and that these masers are con®ned to a narrow

power spectrum (lower panel). The inset shows line-of-sight (LOS) velocity versus annulus in the disk. The magnitude of the velocity gradient (s) implies a mean impact parameter for the best-®tting keplerian disk, with the maser data super- systemic radius, r , of 3.9 mas which, together with the positions of the high- h si posed. The high-velocity masers trace a keplerian curve to better than 1%. velocity masers, constrains the disk inclination, is, to be ,82 6 18 (908 for edge-on). Monitoring of these features indicates that they drift by less than 1 km s-1 yr-1 Finally, VLBA continuum images7,13 are included as contours in the upper panel. The , 18 (refs 14±16) and requires that they lie within 5±108 of the midline, the intersection of 22-GHz radio emission traces a sub-parsec-scale jet elongated along the rotation the disk with the plane of the sky. The LOS velocities of the systemic masers are axis of the disk and well-aligned with a luminous, kiloparsec-scale jet18.

Because direct measurements generally lie well within our own MW, we need other means of reaching out to cosmic distances These methods are called indirect distance measures and they fall into two classes: Standard candles --- based on objects with known (or calibratable) luminosities Dynamical measures --- based on scaling relations like TF and FP

19 Main-sequence ﬁtting G. Piotto et al.: Globular cluster HST color-magnitude diagrams 953

By ﬁtting isochrones to the MSTO, the distance to a cluster can be measured:

0.2[(mMSTO MMSTO)+5]) dGC = 10

Fig. 4. The F555W vs. F439W F555W (flight system) color magnitude diagrams from the combination of the 4 WFPC2 cameras of 2 clusters of the database. Note that the magnitude− and color ranges covered by each figure are always of the same size (though magnitude and color intervals start at different values). Heavier dots correspond to stars with an internal total error less than 0.1 mag. Main-sequence fitting

Alternately, ﬁrst identify

local “subdwarfs” – metal- 1997AJ....114..161R poor MS dwarfs – with precise distances Then shift the GC MS to match the absolute magnitudes of these stars this gives you m–M directly

21 Main-sequence ﬁtting

Alternately, ﬁrst identify local1997AJ....114..161R “subdwarfs” – metal- poor MS dwarfs – with precise distances Then shift the GC MS to match the absolute magnitudes of these stars this gives you m–M directly

21 Standard candles

All indirect distance measurements depend on the calibration of standard candles, and one kind of standard candle in particular: pulsating stars In fact, the entire cosmic distance scale (i.e., beyond the Local Group) is based on one kind of pulsating star: the Cepheid variables

22 Cepheid variables 1938HarCi.431....1S

Cepheids are pulsating stars which “breathe” due M to their surface opacities and speciﬁc heats δ Cephei increasing as they compress, due to partial ionization of H and He

This means the pulsations v that occur in all stars are much bigger in Cepheids

pulsation phase

23 The nearest Cepheid is Polaris (the pole star), about 200 pc away, too distant for parallax measurements until the Hipparcos satellite in the 1990s

In 1913, Hertzsprung (of the HRD!) came up with the ingenious idea to use the variation of the Sun’s motion with respect to the Local Standard of Rest as a secular parallax and determined the distances to Cepheids that way

24 1968ApJ...151..531S

In the beginning of the 20th century, Henrietta Swan Leavitt, a “computer” at Harvard Observatory, discovered that the Cepheids in the Small Magellanic Cloud pulsated with periods that were directly proportional to their average luminosities Recent period-luminosity relations for Cepheids give M = 2.76 log P 1.46 V The trend is even tighter when the colour of the Cepheid is included

25 1992A&AS...96..269S

Cepheids are massive

Cepheids stars with temperatures near ~7000 K (~F stars)

26 Because Cepheids are so bright — the brightest Cepheids reach ⟨MV⟩~-8 (!) — they can be used for indirect distances out to ~20 Mpc

Note that this is well beyond this distance of NGC 4258, allowing for a direct comparison of the Cepheid distance with the maser distance for this galaxy — the difference is within the error bars of both methods, conﬁrming that the Cepheid scale is reasonable for local galaxies

27 There are also even fainter Population II (i.e., metal- poor) pulsating stars that can be used as standard candles: these are called RR Lyrae stars

The calibration for the Cepheid distance scale actually rested in large part on the calibration of the RR Lyrae distance to the LMC until recently

28 Supernovae

By using the Baade–Wesselink method for the expanding photospheres of supernovae, combined with Cepheid distances to supernova host galaxies, it is clear that type Ia supernovae (SNIae) are good standard candles — under certain circumstances

SNIae are the explosions of low-mass (≤5 M⊙) stars — but we really don’t understand the physics of the explosion!

29 For type Ia supernovae, this method allows us to estimate that their peak brightness is ⟨MB⟩≈⟨MV⟩ ≈-19.3±0.03 But it has been determined that not all SNe Ia have this peak brightness! However, the width or shape of the light curve of the SN is correlated with the peak brightness, allowing for a correction

30 Type Ia supernovae can be used as distance indicators to (staggeringly!) high distances: the record holder for the redshift of a SN Ia is at z~1.8 (!)

In fact, the distances to SNe Ia are the reason we believe the Universe is presently accelerating and therefore is one of the observations that suggest that dark energy is a major contributor to the energy density of the Universe!

31 Other sort-of-standard candles

Two other classes of objects used to be used: Planetary nebula have a sharp(ish) cutoff in their luminosity function — that is, the number of objects per decade in luminosity — with a maximum brightness that appears to be ~constant However, it’s really only possible to see PNe out to distances of ~50 Mpc, and SNe Ia are easier to observe!

32 Globular clusters have a ~Gaussian-shaped luminosity function that peaks at a magnitude of M~-6.6

Two problems:

Fainter than the brightest Cepheids!

Not clear if the peak is always at the same magnitude

33 Surface brightness ﬂuctuations

Even if a galaxy is too far away to resolve individual stars, the fact that galaxies are made up of individual stars has interesting consequences

Consider a galaxy at a distance d consisting only of stars of luminosity L, with n stars per unit area projected on the sky

Let’s take an image of the galaxy with a detector with a resolution of δθ; then the image will contain an average of N ¯ = n ( d ) 2 stars in each δθx δθ resolution element

34 L The ﬂux from each star is f = 4d2 So the total ﬂux in each resolution element is

this is (effectively) the ¯ 2 surface brightness of F = fN = nL /(4) the galaxy ...which is independent of distance d. How does this help us?

Poissonian statistics to the rescue! Not all resolution elements will have the same number of stars: the dispersion in the number will be N¯ 1/2

35 So the root-mean-squared fluctuation in the observed flux per resolution element will be n1/2L 1 = N¯ 1/2f = F 4 d ...so the fluctuations in the surface brightness scale inversely with distance: a galaxy at twice the distance will be twice as smooth. This is why this method was originally called the method of incipient resolution by Baum (1958)!

2 2 So we can measure F /F = f = L/ (4 d ) and thus determine the ﬂux of each star from the SBFs

36 So if you can measure the ﬂuxes of the stars via SBFs in two galaxies made up of intrinsically identical stars, then you can measure the relative distance between the two galaxies!

In reality, galaxies aren’t made up of stars with identical luminosities; so we correct our equation to read 2 ¯ 2 F iNifi L = = 2 F iN¯ifi 4d where N¯ L2 L = i i i iN¯iLi

37 Clearly is a weighted stellar luminosity – weighted towards the most luminous stars

So if two populations have the same , then SBFs can be used to measure the relative distance

otherwise, we need to calibrate : for old populations, empirically M¯ = M =3.0(V I) 4.8 I I

38 “Dynamical” distance we’ll come back later to the physics behind these scaling relations... indicators

Both the Tully-Fisher (TF) relation and the Fundamental Plane (FP) and its projection, the Dn-σ relation, can be used to measure the distances to galaxies, because they all (roughly) depend on luminosity as

L v4

39 TF as a distance indicator

706 VERHEIJEN Vol. 563

If you can calibrate a large -24 -22 enough number of spiral -20 -18 galaxies with good rotation -16 curves, you can use the TF 2 2.5 2 2.5 2 2.5 2 2.5 relation as a distance indicator -24 -22 Problems: need to have -20 -18 excellent radio and optical -16 data; need to “weed out” 2 2.5 2 2.5 2 2.5 2 2.5 galaxies that don’t ﬁt; only -24 Sc galaxies really ﬁt; need -22 -20 to choose bandpass and -18 velocity measurement -16 2 2.5 2 2.5 2 2.5 2 2.5 correctly; not that many FIG. 5.ÈTF relations for all 31 galaxies in the RC sample with measured rotation curves (circles and triangles). The crosses indicate the additional seven galaxies in the SI sample for which a reliable rotation curve could not be measured. Those seven galaxies were ignored when making the inverse least-squares i Sc’s with good Cepheid Ðts. The TF relations are constructed for each of the four available passbands using three di†erent kinematic measures: the corrected widthW R I of the global H I proÐle (upper panels), the maximum rotational velocityV measured from the H I rotation curve (middle panels), and the amplitude of the, Ñat part V of the H I rotation curve (lower panels). The open triangles indicatemax galaxies with R curves, the open circles indicate galaxies with D curves, and the Ðlled symbolsflat indicate galaxies with F curves. Solid lines show the Ðts to the 15 Ðlled circles only. The dashed lines show Ðts using all galaxies from the RC sample distances in each panel.

FDNGC 3992 sample and subsequently calculating the scat- farther away than the UMa Cluster as a whole, which is not ters including NGC 3992 yields prms \ 0.35 mag and inconceivable given the velocity crowding in the UMa40 pbi \ 0.31 mag, a small but signiÐcant di†erence. Removing region, would put NGC 3992 and UGC 6969 back on the the most deviating galaxy in the RC/FDNGC 3992 sample TF relation while UGC 6923 moves slightly farther away (NGC 3953, *M \[0.51 mag) does not signiÐcantly from the relation, making it stand out more as a galaxy with reduce the scatter any further. an R curve (open triangle). It is conceivable that NGC 3992 is a background galaxy Finally, removing NGC 3992 from the RC/FD sample given its high recession velocity ofV \ 1139 km s~1. Its and considering the remaining 21 galaxies in the RC/ companions UGC 6923 and UGC 6969sys also have high sys- FDNGC 3992 sample, the correlation becomes robust, pro- temic velocities of 1151 and 1210 km s~1, respectively, gressively tighter, and steeper and displays less scatter going straddling the high-velocity edge of the clusterÏs 700È1210 from the blue to the near-infrared (see Table 4). Further- km s~1 velocity window. Furthermore, note that UGC more, for the RC/FDNGC 3992 sample, the correlation also 6969, the lower open triangle, also lies signiÐcantly below tightens when usingV instead ofV from the rotation the relation. Assuming that the NGC 3992 group is 50% curve. flat max 1996MNRAS.280..167J The FP as a distance indicator

The Fundamental Plane is a relation between velocity dispersion, surface brightness, and effective radius The ﬁrst two of those are distance- indepedent, so the FP actually gives you a standard ruler, not a standard candle

41 1987ApJ...313...42D 1996MNRAS.280..167J

In fact, a combination of surface brightness and radius, called Dn, the radius at a surface brightness of 20.75 in the B band, is very well correlated with the velocity dispersion:

log Dn =1.33 log + C

This makes Dn a standard length The problem is that Cepheids don’t occur in elliptical galaxies (which have old populations) and other variables (like RR Lyraes) are too faint Need either clusters of galaxies with spirals with known Cepheid distances or SN Ia to calibrate

42 In fact, distances to individual galaxies using Dn-σ (or the full FP) or TF are pretty uncertain, so entire clusters of galaxies (or at least groups of galaxies) are used to get accurate distances

43 An important digression: Extinction and reddening

Many galaxies, like the Milky Way, are full of dust Dust absorbs and scatters light, with blue light scattered and absorbed more than red light

We quantify the dust extinction by AX, the difference between the observed X-band magnitude m(X) and the magnitude m0(X) that would have been observed in the absence of dust: A (m m ) X 0 X

44 The reddening or color excess E(X-Y) is the difference between the observed color (X-Y) and the intrinsic color (X-Y)0: E(X Y ) [m(X) m(Y )] [m0(X) m0(Y )] = AX AY Then the distance modulus (in some band) needs to be corrected by the extinction: m M = 5 log d + A 5 X X X If mX and MX are known for some star with a known distance, then AX can be determined directly

It is often useful to know the ratio of total to selective absorption, which measures the slope of the extinction A curve: R = X X E(X Y )

45 The Hubble Law and the Hubble Constant

In 1912, V.M. Slipher announced that he had computed radial velocities for 12 spiral nebulae and found that they were all (except for M31) moving away from the Milky Way, because their spectra were redshifted by 1925, he had found that nearly every one of 40 nearby galaxies with redshifts were redshifted and moving away from us

46 In 1929, Edwin Hubble presented a paper in which he showed that the distances to 18 galaxies inferred from Cepheid variables were directly proportional to their velocities (as measured by Slipher): v = H0d

47 68 FREEDMAN ET AL. Vol. 553

sample, prolate clusters could be selected on the basis of brightness.) Cooling Ñows may also be problematic. Fur- thermore, this method assumes hydrostatic equilibrium and a model for the gas and electron densities. In addition, it is vital to eliminate potential contamination from other sources. The systematic errors incurred from all of these e†ects are difficult to quantify. Published values ofH based on the SZ method have ranged from D40 to 800 km s~1 Mpc~1 (e.g., Birkinshaw 1999). The most recent two-dimensional interferometry SZ data for well-observed clusters yieldH \ 60 ^ 10 km s~1 Mpc~1. The systematic uncertainties are0 still large, but the near-term prospects for this method are improving rapidly (Carlstrom 2000) as additional clusters are being observed, and higher resolution X-ray and SZ data are becoming available (e.g., Reese et al. 2000; Grego et al. 2000).

FIG. 7.ÈValues of the Hubble constant determined using the Sunyaev- 9.2. T ime Delays for Gravitational L enses Zeldovich e†ect (open squares) and gravitational lens time delays (asterisks) A second method for measuringH at very large dis- from 1990 to the present. From the compilation of Huchra (http://cfa- tances, independent of the need for any0 local calibration, www.harvard.edu/Dhuchra) for the Key Project. comes from gravitational lenses. Refsdal (1964, 1966) showed that a measurement of the time delay, and the the distance to the cluster (e.g., Birkinshaw 1999; Carlstrom angular separation for gravitationally lensed images of a et al. 2000). The observed microwave decrement (or more variable object, such as a quasar, can be used to provide a precisely, the shift of photons to higher frequencies) results measurement ofH (see also, e.g., the review by Blandford as low-energy cosmic microwave background photons are & Narayan 1992).0 Difficulties with this method stem from scattered o† the hot X-ray gas in clusters. The SZ e†ect is the fact that the underlying (luminous or dark) mass dis- independent of distance, whereas the X-ray Ñux of the tributions of the lensing galaxies are not independently cluster is distance dependent; the combination thus can known. Furthermore, the lensing galaxies may be sitting in yield a measure of the distance. more complicated group or cluster potentials. A degeneracy There are also, however, a number of astrophysical com- exists between the mass distribution of the lens and the plications in the practical application of this method (e.g., value ofH (Schechter et al. 1997; Romanowsky & Kocha- Birkinshaw 1999; Carlstrom 2000). For example, the gas nek 1999;0 Bernstein & Fischer 1999). In the case of the distribution in clusters is not entirely uniform: clumping of well-studied lens 0957 561, the degeneracy due to the sur- the gas, if signiÐcant, would result in a decrease in the value ] rounding cluster can be broken with the addition of weak- of. There may also be projection e†ects: if the clusters H lensing constraints. However, a careful analysis by observed0 are prolate and seen end on, the true could be H Bernstein & Fischer emphasizes the remaining uncertainties larger than inferred from spherical models. (In a Ñux-limited0 Today this relation as in the mass models for both the galaxy and the cluster Hubble’s Law and H0 is which dominate the overall errors in this kind of analysis. Values ofH based on this technique appear to be converg- known as the Hubble ing to about0 65 km s~1 Mpc~1 (Impey et al. 1998; Tonry & Franx 1999; Bernstein & Fischer 1999; Koopmans & Fass- Constant nacht 1999; Williams & Saha 2000). Typically, v is in km/s and d is 9.3. Comparison with Other Methods It is encouraging that to within the uncertainties, there is -1 broad agreement inH values for completely independent in Mpc, so H0 is in km s 0 -1 techniques. A Hubble diagram (log d versus log v) is plotted Mpc in Figure 8. This Hubble diagram covers over 3 orders of magnitude, and includes distances obtained locally from Cepheids, from Ðve secondary methods, and for four clus- The currently favored value ters with recent SZ measurements out to z D 0.1. At z Z 0.1, of the Hubble constant is other cosmological parameters (the matter density,)m, and the cosmological constant,) ) become important. -1 -1 " H0=100 h km s Mpc = 10. IMPLICATIONS FOR COSMOLOGY -1 -1 log(v/c) = log v log c 72±8 km s Mpc , where FIG. 8.ÈPlot of log distance in Mpc vs. log redshift for Cepheids, the Tully-FisherNote relation, how Type each Ia supernovae, method surface takes brightness you Ñuctuations, One of the classical tests of cosmology is the comparison h=0.72 is used to fundamental plane, and Type II supernovae, calibrated as part of the Key of timescales. With a knowledge ofH , the average density Project. Filledout circlesa little are further: from Birkinshaw That’s (1999), why for this nearby is Sunyaev- of matter, o, and the value of the cosmological0 constant, ", Zeldovich clusters with cz \ 30,000 (z \ 0.1) km s~1, where the choice of integration of the Friedmann equation parameterize the Hubble cosmologicaloften model called does not the have “Cosmic a signiÐcant e†ectDistance on the results. The constant SZ clusters are Abell 478, 2142,Ladder” and 2256, and are listed in BirkinshawÏs 8nGo k " Table 7. The solid line is forH \ 72 km s~1 Mpc~1, with the dashed lines H2\ [ ] (7) representing ^10%. 0 3 r2 3

48 Note that the Hubble constant has units of inverse time (s-1)!

The inverse of the Hubble constant thus gives a timescale, called the Hubble time:

1 17 tH =4.35 10 s = 13.8 Gyr H0 If the Universe has been expanding uniformly since the Big Bang (which it hasn’t), then the Hubble time should equal the age of the Universe (which it nearly does!)