Rotation curves of spiral galaxies

Rotation curves of spiral galaxies Rotation curves Mass discrepancy Circular velocity of spherical systems and disks Dark matter halos Inner and outer regions Tully-Fisher relation From datacubes to rotation curves Gas and stars Beam smearing

Rotation in galaxies ESO/VLT

Neutral gas Optical image Same scale Same scale! Neutral (atomic) medium Total HI map Boomsma et al. 2008, A&A

Optical Optical vs HI Same scale Atomic neutral hydrogen visibile 21 - cm

Rotation of a galactic disc Line profiles Doppler effect Velocity field

Tilted ring model Sset of concentric rings For each ring, parameters: 1) rotation centre (x 0, y 0 ) 2) systemic velocity V 0 3) circular velocity V c (R) 4) position angle P.A. of the major axis 5) inclination angle i (i = 90 o for edge-on) V(x,y) = V 0 + V c (R)sin(i)cos(ϕ) ϕ = azimuthal angle in the plane of the galaxy cos(ϕ)= [-(x - x 0 )sin(p.a.) + (y - y 0 )]/R sin(ϕ)= [-(x - x 0 )sin(p.a.) + (y - y 0 )]/Rcos(i) ϕ = 0 for major axis V LOS = v sys + v R sin(ϕ)sin(i) + v ϕ cos(ϕ)sin(i) + v z cos(i)

Velocity fields versus rotation curves V LOS = v sys + v R sin(ϕ)sin(i) + v ϕ cos(ϕ)sin(i) + v z cos(i) Flat Solid body Rising + flat Rise+decline+flat

NGC 3198: the classical case Begeman 1987

Different types of rotation curves Rotation curves can be divided in 4 classes corresponding to different types of galaxies. 1) just rising, 2) slowly rising+flat, 3) fast rising+flat, 4) fast rising +decline+flat. Casertano & van Gorkom 1991

Mass components

HI rotation curve Optical disc Begeman 1987

Discrepancies Begeman 1987

Circular speed and spherical systems Rotation curves of disc galaxies Circular speed of a spherical system According to the second Newton s theorem, the gravitational force on a test particle that lies outside a closed spherical shell of matter is the same as it would be if the matter were concentrated at its centre: 2.1) where M(r) is the mass contained inside the radius r : 2.2). The corresponding circular speed is: - 2.3). As a consequence if the entire mass of the system is concentrated within a radius r=r K, beyond that radius M(r)=M=constant. Thus, for r > r K, the circular velocity becomes: 2.4) frequently referred to as the Keplerian fall.

Circular speed of a power-law density profile Spherical power-law density profile It is instructive to derive the circular speed for a spherical body with a density profile that follows the power law: 2.5) if the mass interior to r (calculated integrating eq. 2.5 between 0 and r) is: 2.6) and the corresponding circular speed is: 2.7). Eq. 2.7 suggests that, in order to have a flat rotation velocity (as observed in the outer parts of disc galaxies), we would need, i.e. the mass density should be proportional to r -2. In this case eq. 2.5 becomes that of a so-called singular isothermal sphere.

NGC 6946 Gas (HI) σ turbulence Optical Boomsma et al. 2008, A&A

Stellar disks Bulge-disk decomposition Kent 1985

Contributions to the rotation speed Circular velocity for: an exponential disc (solid) a spherical body with the same mass profile (dashed) and a point mass (dotted) ( ) ) R M s (R) = M d (R) = 2π dr R Σ 0 e R /R d 0 [ ( = 2πΣ 0 Rd 2 1 e R/R d 1 + R ) ]. R d Binney & Tremaine 1988 v c (R) =4πGΣ 0 h R y 2 [I 0 (y)k 0 (y) I 1 (y)k 1 (y)] d I n are the m y = R/(2h R )..2 hfilippo and Fraternali decl

Effect of thickness v c (R) =4πGΣ 0 h R y 2 [I 0 (y)k 0 (y) I 1 (y)k 1 (y)] Exponential disc fit to the rotation curve of NGC 2403. Effect of different thicknesses.

Discrepancy Optical disc Begeman 1987

Dark matter halos

Dark matter profiles Dark matter haloes are often modelled using the non-singular isothermal profile: Dark matter profiles 2.14) which leads to the circular velocity: 2.15) with asymptotic velocity. Recently, cosmological simulations of Cold Dark Matter collapse have shown that there should be a universal law for DM profiles going from small structures (dwarf galaxies) to large ones (galaxy clusters). One of these universal profiles, called, after Navarro Frenk & White (1997), NFW profile takes the analytic form: 2.16) where r s is the scale radius, is a characteristic density and is the critical density of the Universe. The NFW profile leads to the circular velocity: 2.17) where v 200 is the circular velocity normalized at the virial radius r 200 e c=r 200 /r s is the so-called concentration of the halo. Other CDM cosmological simulations give profiles with a similar behaviour as the NFW. The isothermal and universal profiles differ significantly at small and large radii. The latter region is very difficult to probe in galaxies, whilst the former is in principle observable. The isothermal profile has a core towards the centre of the galaxy ( universal profiles have cusps. ) whilst the

Maximum disk hypothesis Gas disc + stellar disc + DM halo V tot =sqrt(v G 2 + (M/L)*v S 2 + v DM2 ) Begeman 1991 v G = gaseous disk v S = stellar disk v DM = dark matter halo)

Rotation curve fitting: the Mass to Light ratio Maximum disc: highest M/L possible Maximum halo: low M/L M/L=1.7 M/L=0.8 Several M/L accepted (degeneracy)

Rotation curve fitting: maximum disc Begeman 1989

Cumulative mass Van Albada et al. 1985

NFW profile From CDM simulations of structure formation (e.g. Navarro, Frenk & White 1997) Universal profiles: For most galaxies good fit as the Isothermal profiles Navarro 1998

HSB vs LSB

Photometry HSB and LSB galaxies Dynamics HSB µ B (0) = 21.65 mag arcsec -2 Freeman's LAW (1970) LSB µ B (0) > 23 mag arcsec -2 Verheijen & Tully 2003

Types

LSB galaxies: DM dominated HSB M/L 2-3 LSBs M/L 10 Begeman 1989

Inner and outer regions

Inner slopes

Cusp problem De Blok & Bosma 2002

Cusp problem De Blok & Bosma 2002 Gentile et al. 2007

Inner shape of the potential The inner potential of LSB galaxies seem to have slopes closer to γ = 0 than γ = 1. De Blok & Bosma 2002 Spekkens & Giovanelli 2005

Very extended curves NGC 5533 D ~ 50 Mpc Noordermeer 2006 Broeils 1992 Sanders 1996

Malin 1 z=0.08 D=380 Mpc Lelli, Fraternali & Sancisi 2010

Malin 1 Lelli, Fraternali & Sancisi 2010

Tully-Fisher Relation

Tully-Fisher relation between absolute magnitude and HI profile width Example global HI profile Tully & Fisher 1977

Tully-Fisher with rotation curve V flat not reached? V flat = V max V flat < V max Verheijen 2001

Tighter relation between M and V flat = Relation between the stellar mass (luminosity) and the DM halo mass Verheijen 2001

Baryonic Tully-Fisher The slope change in the TF is reconciled when the gas mass is taken into account: M d = A * V c b b=3.98 ± 0.12 McGaugh et al. 2005