Mass modelling of dwarf spheroidal

galaxies

Ewa L. ŁokasCopernicus Center, Warsaw

Collaborators

• Jarosław Klimentowski (Copernicus Center, Warsaw)

• Stelios Kazantzidis (Ohio State)• Gary Mamon (IAP, Paris)• Lucio Mayer (Zurich)• Francisco Prada (IAA, Granada)

The method of velocity moments

• Measure positions and velocities of stars in the galaxy

• Determine the profiles of velocity moments

• Make assumptions about the model for the mass distribution and anisotropy of stellar orbits (e.g. mass follows light, β = const)

• Fit the solutions of the Jeans equations to the observed velocity moments

• Adjust the free parameters (total mass, anisotropy)

Velocity moments

• The fourth-order moment is governed by the higher order Jeans equation, which for β = const reads

02)(

22

grdr

dr

r

• The second-order velocity moment σ is obtained from the lowest-order Jeans equation

032)( 24

4

grdr

drr

r

Line-of-sight momentsFor comparison with observations we need to project the solutions along the line-of-sight

drRr

r

r

R

RIR r

Rlos 22

2

2

22 1

)(

2)(

drRr

rRrg

RIR r

Rlos 22

44 ),,(

)(

2)(

44 /)()( losloslos RR

The simulation• The simulation traced the evolution of a two-

component dwarf galaxy on an eccentric orbit in a static Milky Way potential for 10 Gyrs

• The dwarf initially had a stellar disk and an NFW-like dark matter halo

• The dwarf was modelled with 106 stellar and 4 x 106 dark matter particles

• The progenitor had an initial mass of 4 x 109 M

• 99% of the mass is lostKlimentowski et al.

2007

Observing the dwarf

Depending on the angle of view the measurements will be different

Simulated data

Very strong contamination for observation along the tidal tails

unbound accepted rejected

Contaminated velocity dispersion

• For observation along the tails the dispersion shows a secondary increase

• After cleaning the data of interlopers the dispersion decreases with radius and traces well the profile for bound particles

Two sources of contamination

• The kinematic samples can be contaminated by tidal tails

• Additional source of contamination are the stars of the Milky Way

unbound + MW stars

Application to Fornax

Kinematic dataset of 202 stars for the Fornax dwarf from Walker et al. (2006)

Sample selection

members

interlopers

Velocity dispersion profilesfor Fornax

Models withM/L=const

Constraints on the parameters

Models withM/L=const

Summary of M/L=const models

method M/LV

[M/L]

β χ2/N

den Hartog

11.3 –0.17 3.4/4

Perea et al.

11.7 –0.24 4.2/4

3 sigma 9.7 –1.4 9.4/4

cut–off 10.2 –1.7 5.5/4

Additional constraints from kurtosis

σlos σlos + κlos

M/LV =11.3 M/L

β = –0.17 χ2/N=3.4/4

M/LV =11.4 M/L

β = –0.03 χ2/N=10.9/10

Tidal tails in Fornax?

If tidal tails are visible then they are probably not along the line of sight (Coleman et al. 2005)

Contamination from the MW

The rejected stars in Fornax have velocities consistent with the population expected from the MW according to the Besancon model (Robin et al. 2003)

Application to Draco

Kinematic sample for 207 stars from Wilkinson et al. (2004)

● accepted° rejected

Velocity moments for Draco

• Velocity moments were calculated for the sample of 194 stars cleaned of interlopers

• The interlopers most likely some from the tidal tails since no contamination from MW stars is expected at the velocity range of Draco stars

Distribution of dark matter

Small dark matter haloes orbiting bigger ones get tidally stripped, but their inner slope remains cuspy (Kazantzidis et al. 2004), Klimentowski et al. 2007)and is well fitted by

br

r

r

Cr exp)(

Dark matter

distribution

in Draco

DM profile

Mass [108 M]

rb/RS β χ2/N

cusp 5.5 7.0 –0.1 8.8/9

core 1.2 1.4 0.06 9.5/9

br

rCrr exp)(

Application to Leo I

• Kinematic sample of 328 stars from Mateo et al. (2007)

• The secondary increase of dispersion disappears after removal of interlopers

Constraints on parameters

If kurtosis is included or analysis

restricted to inner points the data are consistent with isotropy or weakly

tangential orbits

Constraints on parameters

If the data are cleaned of interlopers the agreement with isotropy or weakly

tangential orbits is even better

Weakly tangential orbits

Weakly tangential orbits are expected for dwarfs not strongly dominated by dark matter

Conclusions

• Removal of unbound stars from the tidal tails and/or Milky Way is essential in modelling dwarfs

• The kinematic data for Fornax and Leo I are consistent with mass following light and weakly tangential orbits

• Fornax and Leo I are not strongly dark matter dominated with M/L=11 and 7 solar units

• Draco is poorly fitted by mass-follow-light models and has M/L>100

References• Łokas, E. L., Mamon, G. A., Prada, F., 2005, Dark

matter distribution in the Draco dwarf from velocity moments, MNRAS, 363, 918

• Klimentowski, J., Łokas, E. L., Kazantzidis, S., Prada, F., Mayer, L., Mamon, G. A., 2007, Mass modelling of dwarf spheroidal galaxies: the effect of unbound stars from tidal tails and the Milky Way, MNRAS, 378, 353

• Sánchez-Conde, M. A., Prada, F., Łokas, E. L., Gómez, M. E., Wojtak, R., Moles, M., 2007, Dark matter annihilation in Draco: New considerations of the expected gamma flux, Physical Review D, 76, 123509

• Łokas, E. L., Klimentowski, J., Wojtak, R., 2007, The effect of unbound stars on the mass modelling of the Fornax dwarf, arXiv:0712.2372