The differential Tully-Fisher relation for spiral

galaxies

Irina Yegorova

SISSA, Trieste, Italy

Outline:

•Observations of spiral galaxies

•The standard Tully-Fisher relation

•Differential Tully-Fisher relation

Evidence of dark matter comes from:

WMAP

X-Ray

gravitational lensing

rotation curves

blue shift

red shift

The Tully-Fisher (TF) relation is an empirically established correlation between the luminosity L of a spiral galaxy and its rotational velocity V (Tully-Fisher, 1977)

New method of determiningDistances to galaxies

R.B.Tully and J.R.Fisher,A&A, 54. 661-673, 1977

TF-relation has two important applications:

1. It is used to obtain cosmological distances

M=m - 5logD - 25

2. It can be used for studying the dynamical properties and the evolution of galaxies

M M M mass total lumdark

lum

dark

M

M α

prametermatter dark

2c

lum0 R

Mμ

parameterdensity surface

Physical basis of the TF-relation

From the equation of centrifugal equilibrium we get:

M

V20

cR

G V0 – representative velocity, M - total mass, Rc – characteristic radius of luminous matter - structural parameter depending on the shape of the mass distribution

10

22240 1M ]μGα)([γVlum

10

22240 )]/M()1([ LGVL lum

The first equation can be written in this form:

This equation can be written in the form of theTully-Fisher relation:

)/log(5.2 sunsun LLMM

Differential Tully-Fisher relation

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0

50

100

150

200

250

V

R/Rd

UGC2405

Vmax

1st sample: 967 spiral galaxies Mathewson (1992)

2nd sample: 304 spiral galaxies Courteau (1997) 86 galaxies selected for analysis

3 sample: 329 spiral galaxies Vogt (2004) 81 galaxies selected for analysis

Samples:

1. Mathewson sample: R R/Ropt; bin=0.2

2. Courteau sample: R R/Rd; bin=0.2

3. Vogt sample: R R/Rd; bin=0.2

Ropt=3.2Rd, where R_d is the disk exponential length-scale, for Freeman (exponential) disk this corresponds to the 25 B-mag/arcsec^2 photometric radius.

1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6

-16

-17

-18

-19

-20

-21

-22

-23

-24

Mi

log V

0.2R/Ropt

0.4R/Ropt

0.6R/Ropt

0.8R/Ropt

1.0R/Ropt

1.2R/Ropt

1st sample: TF-relation for 967 galaxies

1st sample: TF-relation for 967 galaxies

2nd sample: TF-relation for 86 galaxies

1.4 1.6 1.8 2.0 2.2 2.4

-19.0

-19.5

-20.0

-20.5

-21.0

-21.5

-22.0

-22.5

-23.0

1.2 1.4 1.6 1.8 2.0 2.2 2.4

-19.0

-19.5

-20.0

-20.5

-21.0

-21.5

-22.0

-22.5

MR

log V

R/Rd=0.7

MR

log V

R/Rd=0.3

1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4

-19.0

-19.5

-20.0

-20.5

-21.0

-21.5

-22.0

-22.5

-23.0

MR

log V

R/Rd=1.3

1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5

-19.0

-19.5

-20.0

-20.5

-21.0

-21.5

-22.0

-22.5

MR

log V

R/Rd=1.7

1.9 2.0 2.1 2.2 2.3 2.4-18

-19

-20

-21

-22

-23

MR

log V

R/Rd=2.3

1.9 2.0 2.1 2.2 2.3 2.4

-19.0

-19.5

-20.0

-20.5

-21.0

-21.5

-22.0

-22.5

-23.0

MR

log V

R/Rd=2.9

Slope of the TF-relationMB = ai + bi log V(Ri)

Gap in the slopes of 2 samples:

10

22240 )]/M()1([ LGVL lum

The slope of TF-relation is related to k

k=0.1 for I bandk=0.3 for R band

No dark matter

function of the band

V;)k

(.M x log1

452

41 VL kx

klum L~L

M

Physical meaning of the slope

240 )),(1(~ LRVL

The slope of the TF-relation steadily rises with distance due to the fact that the fractional amount of the dark matter in galaxies changes with the radius.

M

M

lum

dark

increases with R

decreases with L

parmetermatter dark

this has influence on the slope

Scatter of the TF-relation

TF-relation using Vmax

slope=-5.54; scatter=0.49; number of galaxies=83

slope=-7,579; scatter=0,328; number of galaxies=843

1.4 1.6 1.8 2.0 2.2 2.4 2.6-16

-17

-18

-19

-20

-21

-22

-23

-24

Mi

logVmax

1.9 2.0 2.1 2.2 2.3 2.4 2.5

-19.0

-19.5

-20.0

-20.5

-21.0

-21.5

-22.0

-22.5

MR

log Vmax

Main results:1. We found the new method Differential TF relation.

2. The slope decreases monotonically with the distance, while the scatter increases with distance. This implies the presence of a non luminous mass component (DM) whose dynamical importance, with respect to the stellar disk (baryonic matter) increases with radius.3. We found small scatter in the TF-relation. This implies that galaxies have similar physical characteristics.4. Minimum of the scatter 0.15 (1st sample), 0.24 (2nd sample), 0.29 (3 sample) at 2.2R/Rd at the position of the maximum

disk contribution

We are planning to do mass modeling for these galaxies to reproduce the observable slope and scatter in the TF-relation

Work in progressWork in progress