Determination of Galaxy Massesfrom Stellar Dynamical Models

Roeland van der Marel (STScI)

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Why use Stellar Dynamicsfor Mass Determination?• All galaxies have stars• Gravitational physics well understood• Different uncertainties than other techniques• Provides lots of additional information beyond mass

– Black holes, dark halos, embedded disks, …– Memory of initial conditions preserved

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Why is Mass Determination not Trivial?• virial theorem: M = σRMS

2 rg / G– the devil is in the details

• rg is not measured: defined as mass-weighted integral– rg / reff depends on density profile

• (e.g., 50% higher for Sersic n=2 than n=6)– density profile must be modeled– M/L profile must be modeled

• (mass is required, but light is observed)

• σRMS is not measured: defined as an average over all3D directions and the full galaxy extent– only line-of-sight velocity (generally) observed

• velocity anisotropy must be modeled– only part of the galaxy is (generally) observed

• the full plane of the sky must be modeled

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Why is Mass Determination not Trivial?• velocity anisotropy is directly tied to the 3D-shape of

the galaxy through the tensor virial theorem– observed galaxy shape must be modeled/deprojected– viewing angle(s) must be modeled

• Only second-velocity moments are required tocalculate masses (pressure balances gravitationalattraction for hydrostatic eq.)– but to derive necessary ingredients such as velocity

anisotropy you generally need higher-order velocitymoments

• Phase-space distribution function must be modeled

• If you really want the correct answer, almost everylevel of detail must be modeled …

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Models

• Phase-space distribution function (DF) satisfiescollisionless Boltzman equation

• Jeans' Theorem: DF dependson integrals of motion1. E : Energy2. L : Angular Momentum

Lz : Angular Momentum around Symmetry Axis

3. I3 : Third Integral

• General: f(E,I2,I3)E

E,Lz

I3

Axisymmetry

[Cappellari et al. 2006]

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Models: Jeans• Hydrostatic equilibrium

– Predicts RMS velocities sqrt(V2+σ2)

• Not a closed set: assumptions necessary– mass-anisotropy degeneracy

• Spherical anisotropic f(E,L)– β = 1 - (σt/σr)2

• Axisymmetric two-integral f(E,Lz)– σR = σz, <vR vz> = 0– axisymmetric generalization of spherical isotropic

model (includes so-called oblate isotropic rotators)– simplest types of models to explore flattening and

rotation

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Models: Analytical DFs• Many analytical DFs in literature• Often not completely general

– Specific functional forms assumed– I3 not known analytically for general potential

• Superposition of simpler components allowsconstruction of general models

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Models:Orbit Superposition (Schwarzschild)• Numerically sample all

the orbits in a givengravitational potential.

• Find superposition thatproduces a model(consistent with thepotential) that best fitsthe data

• Loop over potentials tofind overall best fit

[Cretton et al. 1999]

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Models:Orbit Superposition (Schwarzschild)• orbit sampling and exploration of parameter space:

– complexity increases drastically with 3D shape assumptions

• orbit calculation, superposition, and fitting:– not fundamentally different for different geometries– different teams use different numerical concepts

(e.g., NNLS vs. maximum likelihood)

state of art2D33D2triaxial

established2D13D1axisymmetric

1980s-1990s1D12D0spherical

UsageProjectedspace

Orbitfamilies

Integralspace

Viewinganglesgeometry

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Models:N-body• Simulation of non-

equilibrium systems– evolving cosmological

structures– Mergers

• Not straightforward tomake an N-body modelthat reproduces a specificsystem– manual: fitting of mergers

(Antennae, Mice, …)– automated: NMAGIC (De

Lorenzi et al. 2006;extending ideas from Syer& Tremaine 1996)

– Idea: vary particle masses

[de Lorenzi et al. 2006]

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Data: Radial Extent• Mass

– Enclosed mass can only be measured out to the last datapoint– Total mass depends on some assumed large-radius behavior (NFW

better than isothermal ...)• M/L

– Dark halo ⇒ M/L is function of radius– Ellipticals inside ~Reff : M/L measues mostly stellar mass– dSphs inside ~Reff : M/L measures mostly dark mass

• Mass subcomponents (e.g., black hole, dark halo)– Kinematical detection depends on contrast with main body– Radial extent important (less radial extent: easier to explain observed

features with alternative models, e.g., anisotropy)• Tracers

– integrated light: difficult outside ~2 Reff

– PNs, GCs: possible to larger radii, but limited numbers– individual stars: possible to larger radii, but only in Local Group

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Data: Radial Extent

• Example: Major axis RGB kinematics to ~10 Reff intwo Local Group dEs (Geha et al. 2009)– Black Curves: constant M/L, f(E,Lz) Jeans models– Red Curves: Keplerian limit

• Findings:– Even at 10 Reff the influence of a dark halo is not as obvious

as one may have expected …– Inferred M/L does indicate dark matter!

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Data: Spatial Resolution• Spatial resolution is critical when kinematical

gradients are steep over a resolution element– center of galaxy with black hole– distant galaxy kinematics (z > 0.5)

• Components:– PSF FWHM (seeing)– slit width / fiber size– spatial pixel, spatial binning

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Data: Spatial Resolution

• Example: Dynamical modeling of distant galaxies– Solid curves: constant M/L, f(E,Lz) Jeans models– Dashed curves: spatial resolution not modeled– Resolution has major impact on V and V/σ

• Applications:– Mass modeling at high z (vdM & van Dokkum 2007b)– Detailed study of lensing galaxies (e.g, SLACS)– Rotation rate evolution and E vs. S0 evolution with z

(vdM & van Dokkum 2007a; van der Wel & vdM 2008)

[CDFS:VLT+HST

van der Wel& vdM 2008]

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Data: 1D vs. 2D coverage• Integral field spectroscopy with 2D

coverage now routine• Trade-off necessary with respect to

– radial extent / field of view– spatial resolution

• 2D benefits– detect minor axis or off-axis rotation– better anisotropy / DF constraints– find kinematical subcomponents /

decoupled cores– more accurate M/L

• 1D benefits– multiplexing (high z)– data reduction simpler [Emsellem

et al. 2004] [vdM et al. 1994]

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Data: Integrated Spectra vs. Discrete Velocities

• Integrated spectra– Uses:

• crowded regions ofgalaxies

• distant universe

– Note:• random errors limited

mostly by exposure time• yields average kinematics

of whole population

[van Dokkum & vdM 2007a]

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Data: Integrated Spectra vs. Discrete Velocities• Discrete velocities (stars, PNe, GCs, ..)

– Uses:• outer parts of galaxies• local Universe

– Note:• random errors limited by number of tracers

(dσ/σ = [2N]-1/2)• kinematics/modeling requires either

– spatial binning of data– special likelihood modeling approaches

(e.g., Chaname, Kleyna, vdM 2007)• possible to divide kinematics by

subpopulation (e.g., metallicity, color, ...)

• Most powerful constraints: modelingintegrated light and discrete velocitiessimultaneously [Geha et al. 2009]

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Data: Higher-order velocity moments

• LOSVD = Line-of-Sight Velocity Distribution– Crucial to break mass-anisotropy degeneracy

• Requires DF modeling to interpret– higher-order Jeans equations not used much

• Requires good quality data– integrated light: Δhi ~ 0.7 Δσ/ σ– discrete tracers: Δhi ~ 0.7 N-1/2

[vdM & Franx 1993]

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Data: Line-of-sight vs. Proper Motion• Line-of-sight (LOS) Doppler velocities

– all that is available for the majority of systems

• Proper motions (PMs)– HST now advancing our kinematical understanding of globular

clusters– 2 velocity components

• PMs break mass-anisotropy degeneracy, becauseanisotropy directly measured– unique relation PM_radial / PM_tangential ⇔ β(r)

(Leonard & Merritt 1989)

• Future application to dSphs holds significant promise(SIM, GAIA, future large optical/UV/NIR telescopes)– current datasets limited by numbers of stars– PMs can constrain directly ρdark (r) ⇒ ΛCDM comparison

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Data: Line-of-sight vs. Proper Motion

• Example: Omega Cen(Anderson & vdM 2009; vdM & Anderson 2009)– ~105 stars– median per star uncertainty 70 mirco arcsec / yr ~ 1.7 km/s @ 5 kpc– no evidence for IMBH (MBH < 1.2 x 104 M)

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Parameter Uncertainties: Inclination

• Density deprojectionis not unique unlessedge-on (i=90)

• Inclination poorlyconstrained bystellar kinematicsobservations– When no a priori DF

constraints are given[Krajnovic et al. 2005]

V σ h3 h4 h5 h6

i=55

i=90

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Parameter Uncertainties: Inclination• Importance of the inclination

– understanding stellar disks, rotation, V/σ– Not very important for M/L estimates (little sensitivity)

• Estimating the inclination– Use dust or gas disks (assumed circular in equatorial plane)– Use stellar kinematics with “observationally motivated”

assumptions for the DF• Cappelari (2008): Jeans models, free σz/σR < 1, <vR vz> = 0

– Use proper motions (van de Ven et al. 2005)• <vLOS> = 4.74 D tan i <PMmin>• Assumes axisymmetry

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Parameter Uncertainties:Velocity Anisotropy• Mass-anisotropy degeneracy

– Radially anisotropy models:steep LOS dispersion profiles

– Tangentially anisotropic models:shallow LOS dispersion profiles

(in the same potential)

• Resolved with:LOSVDs, 2D data, PMs, ...

[vdM 1994b]

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Parameter Uncertainties:3D Shape• Spherical and axisymmetric models well studied and

characterized• Triaxial models for fitting state-of-the-art IFU data

only now starting to be explored (e.g., van den Boschet al. 2008)– essential for understanding minor axis rotation and

decoupled cores– ability to uniquely constrain viewing angles and model

properties dependent on features in the data

• Impact on mass and M/L determinations unclear– Will require detailed modeling of large sample

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How much do the details matter?

• vdM & van Dokkum (2008b)– Compare of different studies with different assumptions and

data quality, all transformed to B-band– Compare of relation between M/L and σeff (Cappellari et al.

2006)

nofixf(E,Lz,I3)axisymGr , 2DHST, ICap06

nofitf(E,Lz,I3)axisymHST, slitHST, VGeb03

yesnof(E,L)spherGr , slitGr , BKro00

nofitf(E,Lz)axisymGr , slitHST, VMag98

nonof(E,Lz)axisymGr , slitGr , RvdM91

DHBHDFGeomSpectPhotom

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How much do the details matter?

• Variation in zeropoint of relation between studies: +/- 0.02 dex– M/L determinations over visible regions of galaxies extremely robust

[vdM &van Dokkum2007b]

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Virial Theorem Revisited• Cappellari et al. (2006):

– M/L ~ 5.0 σeff2 reff / L G

• Why does something so simplework so well?– real galaxies fill only a small subset

of the parameter space of allmodels allowed by physics

– real galaxies resemble r-3 density inr-2 potential: σLOS independent of β

– the radial range for which data isavailable minimizes the mass-anisotropy degeneracy

[vdM et al. 2002; CNOC1]

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Masses at High Redshift

• So can we all just use the virial theorem at high zand forget about the details?– if you want rough mass estimates: yes– if you want to study evolution: no

• Example (van der Wel & vdM 2008):– virial theorem underestimates mass of S0 galaxies at z ~ 1,

as compared to dynamical modeling

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Conclusions

• A vast arsenal of stellar dynamical methods exists tointerpret a wide range of data

• If all you care about is the mass inside the last datapoint– even simple approaches give the right answer

• If you care about dark halos, black holes, radial profiles,internal dynamics, 3D shapes, presence ofsubcomponents, etc.– there is rich information content on galaxy formation+evolution– detailed stellar dynamical analysis is usually called for

• My advice in practice– use a method that is commensurate with the information

content of the data– think long and hard about the caveats introduced by the things

you have not modeled