2. On Scaling Relations for Spiral Galaxies Riccardo Giovanelli Aug 2003

3. Menu of the Day Scaling Relations for Spirals : from theoryScaling Relations for Spirals : from theory Scaling Relations : as observedScaling Relations : as observed Cosmology applications - brieflyCosmology applications - briefly Kinematical scale length: extinction and inner disk M/L Kinematical scale length: extinction and inner disk M/L Dwarf halos: are they really cuspy?Dwarf halos: are they really cuspy?

4. Disk Formation: a primer The spin parameter  quantifies the degree of rotational support of a system : For E galaxies,  ~ 0.05 For E galaxies,  ~ 0.05 For S galaxies,  ~ 0.5 For S galaxies,  ~ 0.5 Angular momentum Mass Total Energy Protogalaxies acquire angular momentum through tidal torques with nearest neighbors during the linear regime Protogalaxies acquire angular momentum through tidal torques with nearest neighbors during the linear regime [Stromberg 1934; Hoyle 1949] As self-gravity decouples the protogalaxy from the Hubble flow,  d /d t)] becomes v.large and the growth of  ceases As self-gravity decouples the protogalaxy from the Hubble flow,  d /d t)] becomes v.large and the growth of  ceases N-body simulations show that at turnaround time values of range between 0.01 and 0.1, for halos of all masses N-body simulations show that at turnaround time values of range between 0.01 and 0.1, for halos of all masses The average for halos is  = 0.05 The average for halos is  = 0.05 Only 10% of halos have  0.10 Only 10% of halos have  0.10 halos achieve very modest rotational support halos achieve very modest rotational support Baryons collapse dissipatively within the potential well of their halo. They lose energy through radiative losses, largely conserving mass and angular momentum Baryons collapse dissipatively within the potential well of their halo. They lose energy through radiative losses, largely conserving mass and angular momentum Thus of disks increases, as they shrink to the inner part of the halo Thus of disks increases, as they shrink to the inner part of the halo. [Fall & Efstathiou 1980][Fall & Efstathiou 1980] (mass of disk) /(total mass) If the galaxy retains all baryons  m_d~1/10,If the galaxy retains all baryons  m_d~1/10, and  disk grows to ~ 0.5, R_disk ~ 1/10 R_h

5. Scaling Laws for Disks * For starters, assume that halos are isothermal spheres, i.e. * Define as the limiting radius of the halo that which encloses a mean density equal to 200 times the critical density * Since * For standard choice of units:: where: [see Mo et al 1998; ; Dalcanton et al. 1997; Firmani & Avila-Reese 2000]

6. Scaling Laws for Disks * If the disk mass fraction is * If the disk mass assumes an exponential distribution * and if the disk angular momentum fraction is then : then

7. Scaling Laws for Disks When a more realistic density profile is adopted (e.g. NFW), the scaling relations of disks become: Where are dimensionless parameters of order 1, that take into account respectively the degree of concentration of the halo, the shape of the rotation curve and the radius at which it is measured the rotation curve and the radius at which it is measured.

8. … Assuming luminous and baryonic matter are similarly distributed in disks … …and introducing a disk mass to light ratio, we can convert disk mass surface density to surface brightness and disk mass to Luminosity, for comparison with observed scaling relations: Note that theory predicts that b is proportional to , c to and a independent on . Thus variance in halo  should affect differently the scatter in each relation. 2

9. … as for the observations… We use SFI++, a sample of 4000 spiral galaxies with I band accurate surface photometry; HI and/or Ha rotational data observed and processed by our group with the main purpose of mapping the cosmic peculiar velocity field.

10. Scale-length—Velocity Width SFI and SFI+ samples: -I band CCD photometry -Sbc-Sc spirals -cz < 10000 km/s -HI and H  spectroscopy SFI : Giovanelli, Haynes, da Costa, Freudling, Salzer, Wegner 1600 galaxies SFI++ : Haynes & Giovanelli 4000 galaxies

11. Central Disk Surface Brightness—Velocity Width “Visibility” function What about Freeman’s Law? [Freeman (1970) found that the central disk blue surface brightness is ~ constant for giant spirals] Expected slope from Scaling law

12. TF Relation: Data SCI : cluster Sc sample I band, 24 clusters, 782 galaxies [Giovanelli et al. 1997] “Direct” slope is –7.6 “Inverse” slope is –7.8 …which is similar to the explicit theory-derived dependence  = 3 L  (rot. vel.) [Where L  (rot. vel.) ]  h= H/100

13. TF Scatter Measurements + 0.25 mag Measurements + 0.30 mag Measurements + 0.20 mag Measurement error Total error “Intrinsic” scatter …measurement errors contribute the lesser part of the scatter

14. Cosmological Applications of the TF relation: Measurement of the Hubble parameter Measurement of the Hubble parameter Assessment of the linearity of Hubble flow Assessment of the linearity of Hubble flow Measurement of peculiar velocities: Measurement of peculiar velocities: - cosmic density field - cosmic density field - mass density parameter - mass density parameter - cluster and galaxy velocity dispersions - cluster and galaxy velocity dispersions - cluster infall - cluster infall - convergence depth of the local Universe - convergence depth of the local Universe Evolution of M/L ratio with z Evolution of M/L ratio with z

15. Convergence Depth Given a field of density fluctuations  (r), an observer at r=0 will have a peculiar velocity: where  is  _mass The contribution to by fluctuations in the shell, asymptotically tends to zero as The cumulative by all fluctuations Within R thus exhibits the behavior : If the observer is the LG, the asymptotic matches the CMB dipole

16. The Dipole of the Peculiar Velocity Field The reflex motion of the LG, w.r.t. field galaxies in shells of progressively increasing radius, shows : convergence with the CMB dipole, both in amplitude and direction, near cz ~ 5000 km/s. [Giovanelli et al. 1998]

17. The Dipole of the Peculiar Velocity Field Convergence to the CMB dipole is confirmed by the LG motion w.r.t. a set of 79 clusters out to cz ~ 20,000 km/s [Giovanelli et al 1999 ; Dale et al. 1999]

18. So you want to measure a velocity width? 1. Get good image of galaxy, measure PA, position slit 2. Pick spectral line, measure peak  along slit 3. Center kinematically 4. Fold about kinematical center 5. Correct for disk inclination, using isophotal ellipticity 6. You now have a rotation curve. Pick a parametric model and fit it. E.g. Inner scale length Outer slope

19. A slit along the major axis yields a rotation curve … The rotation curve and the disk inclination combine to yield a set of isovelocity contours on the plane of the sky Disk opacity modifies V(r) … however, the l.o.s. may not penetrate all the way to the midplane of an inclined disk, effectively sampling regions in the foreground, off the disk major axis, as for an off-center slit, line of nodes pole-on producing a shallower rotation curve.

20. On the Inner Scale length of V(r) With increasing disk opacity, the inner scale length of the observed V(r) should increase:: Increasing inner scale length This effect had been investigated by comparing rotation curves observed in the same galaxy at very different  thus affected differently by opacity, e.g. 21 cm and H . For galaxies of given luminosity class, increasing disk inclination  increasing l.o.s. opacity Thus we query: for a given class, does the inner scale length increase with disk inclination?

21. On the Inner Scale length of V(r) More so for luminous than for intrinsically faint galaxies. It does. More so for luminous than for intrinsically faint galaxies. Reason: column density and metallicity. Reason: column density and metallicity. Giovanelli & Haynes (2002, ApJL)

22. The change of the inner scale length of V(r) with disk inclination can be modelled assuming that dust and stars have similar exponential distributions in both r and z. The observations (in this case for the brightest galaxies: M<-22) can be reproduced by a population with a scale height/scale length =0.145 and at  = 0.65  m Modelling Extinction…. Valotto & Giovanelli (2003)

23. On the Inner Scale length of V(r) Moreover note: at low inclinations, the mean inner scale length of V(r) appears to be at low inclinations, the mean inner scale length of V(r) appears to be the same for all galaxies the same for all galaxies

24. i.e. baryonic contribution (disk + bulge) to M(r_pe) increases with galaxy mass. Nothing new here Nothing new here (Broeils 1992; Moriondo Giovanardi & Hunt 1998; Sellwood 1999) If we define f_d as (Disk Mass)/(Halo Mass) : Constancy of r_pe implies that, within that radius: Over range of 100 in halo mass of normal spirals, f_d within r_pe changes by ~ 5Over range of 100 in halo mass of normal spirals, f_d within r_pe changes by ~ 5 Most luminous spirals would have M(r) Most luminous spirals would have M(r) fully baryon-dominated within inner few kpc fully baryon-dominated within inner few kpc - then inner part of their rotation curves should scale - then inner part of their rotation curves should scale ~ like that of a pure disk for which (1-1/e)V_max ~ like that of a pure disk for which (1-1/e)V_max is reached at 0.63 disk scale lengths. is reached at 0.63 disk scale lengths. M – 5 log(h) r_pe/r_d <-22 0.63 -21.7 0.68 -21.2 0.83 -20.7 0.95 -20.0 1.02 -19.0 1.28 Does it ?  Use I – band images to measure r_d However : On the Inner Scale length of V(r)

25. We can then use rotation curves and I-band photometry to estimate the mean baryonic mass--to-light ratio of the inner region of bright spirals : Which translates to a mean baryonic mass-to-bolometric light ratio of By measuring stellar velocity dispersions in NGC 7331, Bottema (1999) obtains Disk 1.6+/-0.7 Bulge 6.8+/-1.0

26. N- N-body simulations within the CDM scenario suggest a density profile for all halos of the form [Navarro, Frenk & White 1997]: where the radial and the density scale parameters are related as The family of NFW density profiles can be parameterized via a halo concentration index Note: for small r for large r About Cusps of DM halos

27. About Cusps of DM halos…

28. …are CDM halos too centrally cusped ? The standard CDM structure formation scenario has been shownThe standard CDM structure formation scenario has been shown to produce strongly cusped galaxy halos : with  =1 to produce strongly cusped galaxy halos : with  =1 Many observations suggest “softer” halo cores of nearly constant Many observations suggest “softer” halo cores of nearly constant density :  <0.5 [deBlok & McGaugh 1997; deBlok & Bosma 2002; density :  <0.5 [deBlok & McGaugh 1997; deBlok & Bosma 2002; Marchesini et al. 2002; Swaters et al. 2002] Marchesini et al. 2002; Swaters et al. 2002] Simulations also produce halos with an unrealistic amount ofSimulations also produce halos with an unrealistic amount of substructure (“halos within halos”) substructure (“halos within halos”) …revised CDM versions within the framework of “concordance cosmology” (   =0.7,  mass =0.3, h=0.7) appear to be able to attenuate the conflict ( attenuate the conflict ( Eke et al 20001) About Cusps of DM halos

29. - Invert Poisson’s Eqn. and fit a line to the inner part of log  (r) to obtain  About Cusps of DM halos - We get, as most groups, values of  averaging about 0.5 - However, when we statistically fold in the observational biases (slit offset, misalignment, seeing, turbulent motions), we find that the derived values of  are compatible with a population with  =1 - Select a sample of dwarfs from SFI++, [Spekkens & Giovanelli, in preparation]

30. Inner Halo Structure from Direct Inversion Kristine Spekkens, Riccardo Giovanelli (Cornell) Select 110 dwarf galaxies with H  rotation curves, I-band photometry: no bulges, bars, interactions Assume spherical matter distribution: density profile derived directly from RC. Fit for the inner slope  of the density profile  and simulate expected distribution from population of CDM halos: Inner slopes are consistent with CDM halo cusps

31. Conclusions Disk scaling relations are moderately well understood within the framework of hierarchical galaxy formationDisk scaling relations are moderately well understood within the framework of hierarchical galaxy formation Disk scale and surface brightness relations with rotational V are noisy and of limited utility (largely because of zero-point dependence on  silver lining Disk scale and surface brightness relations with rotational V are noisy and of limited utility (largely because of zero-point dependence on  silver lining  After extinction effects are taken into account, the inner scale length of the rotation curve of spirals has a mean value that is independent on luminosity class. This indicates that the inner parts of bright spirals are baryon dominated, yielding a baryonic mass—to-light ratio of 1.9After extinction effects are taken into account, the inner scale length of the rotation curve of spirals has a mean value that is independent on luminosity class. This indicates that the inner parts of bright spirals are baryon dominated, yielding a baryonic mass—to-light ratio of 1.9 The observed Halpha rotation curves of dwarf galaxies are consistent withThe observed Halpha rotation curves of dwarf galaxies are consistent with NFW halo density profiles (  1/r ) NFW halo density profiles (  1/r ) Most of the scatter of the TF relation is related to a galaxy’s formation history, rather than to observational errorsMost of the scatter of the TF relation is related to a galaxy’s formation history, rather than to observational errors A template TF relation at low z can be determined with an accuracy of better than 0.03 mag, and profitably used to measure the rate and deviations of the Hubble flow in the local Universe, as well as other cosmological parametersA template TF relation at low z can be determined with an accuracy of better than 0.03 mag, and profitably used to measure the rate and deviations of the Hubble flow in the local Universe, as well as other cosmological parameters At high z, the TF relation is better suited to investigate the evolutionary history of disksAt high z, the TF relation is better suited to investigate the evolutionary history of disks

32. TF Relation: Data SCI : cluster Sc sample I band, 24 clusters, 782 galaxies [Giovanelli et al. 1997] “Direct” slope is –7.6 “Inverse” slope is –7.8 …which is similar to the explicit theory-derived dependence  = 3 L  (rot. vel.) [Where L  (rot. vel.) ]  h= H/100

33. Verrazzano Bias Map by Gerolamo da Verrazzano (1529) Pacific Ocean

34. On the Inner Scale length of V(r) - 3 What does the near constancy of r_pe mean? Consider Mass within r_pe total mass at r_200 Radius within which the mean density is 200 times the critical density Mean density within r_pe Since and, by definition, we get : and, a constant r_pe yields Since

35. Scale length and Central Disk Surface Brightness… …follow observed scaling relations with rotational velocity that are in agreement with theoretical predictions. However, these two parameters are not best suited for quantitative tests, because they * are poorly determined observationally * are subject (especially SB) to severe observational bias * exhibit strong dependence of scatter on galaxy formation history … Freeman’s law is apparent only when giant disks are seen in blue light. This results mainly because : (1) of internal extinction (which flattens the photometric profile and depresses central SB); (2) of sample selection biases; (3) the intrinsic scatter in the SB scaling relation is large, thus obscuring trend.

36. On the Inner Scale length of V(r) - 4 if it is assumed that Can we explain DM dominates M (r) at all radii ? Take a NFW DM halo : At small r : And if r_pe is constant: (*) (**) Relns. (*) and (**) match only if the concentration index c increases with total mass. However, N-body simulations show the opposite : they show c to decrease with mass.

37. The Luminosity-Linewidth Relation In 1977, R.B. Tully and J.R. Fisher proposed a relation between Luminosity and line width as a means to determine redshift-independent distances to galaxies. It can be derived from the scaling relation Defining a disk mass-to-light ratio we can write, we can write where:

38. TF Relation: Observed Properties Observers present the relation as one between absolute mag. and linewidth. The slope depends mildly on the photometric band, increasing to the red. A mild type dependence is observed; even S0 gals exhibit a TF relation, albeit of > scatter [Neistein et al 1999; Hinz et al. 2001] Observationally, it is the best established among scaling relations for disks. In a PCA, the TF relation is an edge-on view of the fundamental plane of disks. No meaningful third-parameter dependences found. The average scatter of the relation is ~ 1/3 mag near I band; it increases towards the blue. Scatter depends on line width, and is not fully due to measurement errors.

39. TF Relation: best color The quality of the TF relation depends on the band used in a variety of ways: (a) Extinction increases towards the blue (b) Bulge light increases towards the red (c) Disk definition and estimate of its inclination degrade rapidly redwards of 1 micron Point (c) is most important and advises for use of R or I band photometry

40. Can Theory Reproduce the Observed TF Relation? -1 TF slope The observed TF slope is explicitly returned by the theoretical expectation which yields  = 3. However, that expectation matches observations only in the measure in which the combination of parameters that make is luminosity (rotational speed) independent.

41. Can Theory Reproduce the Observed TF Relation? -2 The disk mass-to-light ratio is observed to be [Broeils 1992] N-body, gas-dynamical simulations yield [Navarro 1998] Assuming that the redshift of disk assembly is not largely different across the observed range of TF parameters, neutrality of A requires that the disk mass fraction (e.g. the ability of halos to retain their baryons) increase significantly with luminosity.

42. Can Theory Reproduce the Observed TF Relation? -4 N-body, gas-dynamical simulations can reproduce the slope and the scatter of the observed TF reln. They have difficulties reproducing the observed “zero point” offset, which are alleviated in  CDM simulations. [Navarro & Steinmetz 2000; Challenges for N-body+hydro : -make halos w/low concentration - Force baryons to settle slowly - thus forcing disks to assemble relatively late Saiz et al. 2001]

43. Can Theory Reproduce the Observed TF Relation? -3 Star Formation Rate (solar masses per year) Lookback time (Gyr) Evolution of the Mass-to-light ratio [ Steinmetz & Navarro 1999 ]

44. Does Evolution Affect the TF Relation? Palomar Keck Palomar: Bershady, Haynes & Giovanelli Keck: Vogt et al.

45. Collaborators At Cornell: Daniel Dale Martha Haynes Terry Herter Marco Scodeggio Nicole Vogt At Dartmouth: Gary Wegner At ESO: Luiz Da Costa W. Freudling At Jerusalem: Avishai Dekel Yehuda Hoffman Idit Zehavi At NRAO: Eduardo Hardy At U. de Chile: Luis Campusano At Rome: Enzo Branchini At Trieste: Stefano Borgani At Wisconsin: Matthew Bershady

46. Measuring the Hubble Constant A TF template relation is derived independently on the value of H. It can be derived for, or averaged over, a large number of galaxies, regions or environments. When calibrators are included, the Hubble constant can be gauged over the volume sampled by the template. From a selected sample of Cepheid Calibrators (e.g. Ferrarese et al. 2000), Giovanelli et al. (1997) obtain H_ not = 69+/-6 (km/s)/Mpc averaged over a volume of cz = 9500 km/s radius. The HST key-project team [Sakai et al 2000] gets 71+/-4+/-7

47. TF and the Peculiar Velocity Field Given a TF template relation, the peculiar velocity of a galaxy can be derived from its offset  m from that template, via For a TF scatter of 0.35 mag, the error on the peculiar velocity of a single galaxy is typically (0.15-0.20) cz For clusters, the error can be reduced by a factor, if N galaxies per cluster are observed

48. No local Hubble Bubble Zehavi et al. (1999) : Local Hubble bubble within cz = 7500 km/s ? Giovanelli et al. (1999) : No local Hubble Bubble to cz ~ 15000 km/s

49. The Peculiar Velocity Field to cz=6500 km/s SFI SFI [Haynes et al 2000a,b] Peculiar Velocities in the LG reference frame

50. The Peculiar Velocity Field to cz=6500 km/s SFI SFI [Haynes et al 2000a,b] Peculiar Velocities in the CMB reference frame

51. Cosmological Parameters from the Peculiar Velocity Field VELMOD analysis of SFI and PSCz yields a value of = 0.42+/-0.04 [Branchini et al. 2001] Analysis of the velocity correlation function yields: With the result  0.44+/- 0.05 [Borgani et al. 2000]

52. Potent Reconstruction Potent allows the reconstruction of the density field from the l.o.s. peculiar velocities [Dekel, Bertshinger] … and yields [Kolatt et al. 2000] high values of Density Velocity field

53. That’s all folks!

54. The  Parameter Angular Momentum: Circular velocity: (obtained by equatingand ) If E is the total energy of the system, with We define quantifying the degree to which the body is rotationally supported against collapse. For Ellipticals :For Spirals :

55. Torques build up Angular Momentum Torque: Angular Momentum Change …after some math (e.g.Landau & Lifchitz “Mechanics” Section 32): R (Stromberg 1934, Hoyle 1949 )

56. Distribution of among halos Protogalaxies acquire angular momentum through tidal torques with nearest neighbors during the linear regime As self-gravity decouples the protogalaxy from the Hubble flow,  d /d t)]  becomes v.large and the growth of  ceases N-body simulations show that at turnaround time values of range between 0.01 and 0.1, for halos of all masses The average of halos is  = 0.05 Only 10% of halos have  0.10 halos achieve very modest rotational support Censored