!"#$%&'())*!"#$%&'())* +,-./!0'01$2312'4225$36'..'+,-./!0'01$2312'4225$36'..' 7(7(

!"#$%"&'$()*&

Indirect Detection of Dark Matter with Gamma Rays

James BuckleyWashington University, St. Louis

Hunt for Dark Matter SymposiumFermilab, May 10, 2007

!"#$%&'())*!"#$%&'())* +,-./!0'01$2312'4225$36'..'+,-./!0'01$2312'4225$36'..' 7(7(

!"#$%"&'$()*&

Gamma-Ray Observatories

VERITAS MAGIC

CANGAROO IIIH.E.S.S.

The world of IACTs

VERITAS

MILAGRO

launched April 23,2007!

!"#$%&'())*!"#$%&'())* +,-./!0'01$2312'4225$36'..'+,-./!0'01$2312'4225$36'..' 7(7(

!"#$%"&'$()*&

Atm. Cherenkov Technique

Image Cherenkov light from atmospheric e/m showers - gives direction and energy

10⁵ m² effective area

Stereoscopic imaging gives the direction to several arcminute resolution

!"#$%&'())*!"#$%&'())* +,-./!0'01$2312'4225$36'..'+,-./!0'01$2312'4225$36'..' 7(7(

!"#$%"&'$()*&

GLAST

!"#$%&'()*$+,'-.$"/*0*($1'2'($

3*&4'(. )5--'65&',75(89-':();9<==>

3?#"@9)'( A*'&);9B5&90'&C94',,*&9A7D('-A9

:29,59E==93*FG9H+,97A9'-A5 -7C*-.99,590*,*),9'9

B*/ ,;5:A'(09(*/93*F96-'I'&A9JK

In December, 2007 the Gamma Ray Large Area Space Telescope (GLAST) will be launched to map the Gamma Ray sky.

!"#$%&'())*!"#$%&'())* +,-./!0'01$2312'4225$36'..'+,-./!0'01$2312'4225$36'..' 7(7(

!"#$%"&'$()*&

Newly commissioned 4-telescope VERITAS array (first light April 28, 2007)

Gamma-Ray Observatories

(Crab detected at 30 σ/√hr)

Sources already detected/discovered:

Markarian 421, Markarian 501, Crab

LSI+61 303 (pulsar in binary system)

Blazar 1ES1218+304 (highest redshift)

M87 (non BL Lac blazar)

!"#$%&'())*!"#$%&'())* +,-./!0'01$2312'4225$36'..'+,-./!0'01$2312'4225$36'..' 7(7(

!"#$%"&'$()*&

VERITAS Collaboration

Smithsonian Astrophysical ObservatorySmithsonian Astrophysical Observatory Adler Planetarium Adler Planetarium Purdue University Purdue University Barnard College, NY Barnard College, NY Iowa State University Iowa State University DePauw University, IN DePauw University, IN Washington University, St. Louis Washington University, St. Louis Grinnell College, IA Grinnell College, IA University of Chicago University of Chicago University of California, Santa Cruz University of California, Santa Cruz University of Iowa University of Iowa

University of UtahUniversity of Utah University of University of MassachussettsMassachussetts University of California, Los AngelesUniversity of California, Los Angeles Cork Institute of Technology Cork Institute of Technology McGill University, Montreal McGill University, Montreal Galway-Mayo Institute of Technology Galway-Mayo Institute of Technology University College, Dublin University College, Dublin National University of Ireland, Galway National University of Ireland, Galway University of Leeds University of Leeds Argonne National Lab Argonne National Lab Associate Members Associate Members

Project office: Whipple observatory SAOProject office: Whipple observatory SAO

VERITAS CollaborationVERITAS Collaboration~65 members in more than 20 institutions~65 members in more than 20 institutions

Funding fromFunding fromNSF/DOE/Smithsonian/PPARC/SFI/NSERCNSF/DOE/Smithsonian/PPARC/SFI/NSERC

!"#$%&'())*!"#$%&'())* +,-./!0'01$2312'4225$36'..'+,-./!0'01$2312'4225$36'..' 7(7(

!"#$%"&'$()*&

!"##"$%"&'

()*"+,"-./01/-"##"/

%"&'2/304+,/5"67/,0/,8./

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'" %#&'#($)

'" %#*++&($)

,-#."#/-/0&!

I:J:K/L:M<<40/N/O:/J967<.&/PQQR

(Jungman and Kamionkowski, 1994)

!"#$%&'( #&(&'($)"*+"&,(%-.$") &/-01.&

2-3)%-"- 1-%($'.&4*+5&.$'$(6

7-'()%+!8+"079*+:4,-..6;+(5&+

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7$"-.+4(-(& #)0$"-(&4+>?+)%+(+

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F)(&*+&@,-. -0),"(4+)7+0-((&%+-"#+

-"($0-((&%+$"+-""$5$.-($)"4+G 4),%'&+)7+

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H)0$"-"(+4),%'&+)7+')"($",,0+

I-00-4+$"+5-.)+-""$5$.-($)"4

(Bergstrom 2006)

In regions of the highest dark matter density, dark matter particles and their antiparticles are expected to annihilate into gamma-rays, either directly into a gamma-ray line (with energy equal to the mass of the dark matter particle times the speed of light squared Eγ=mχc²) or a broad spectrum of gamma-rays.

Background

!"#$%&'())*!"#$%&'())* +,-./!0'01$2312'4225$36'..'+,-./!0'01$2312'4225$36'..' 7(7(

!"#$%"&'$()*&

Primordial fluctuations in dark matter density led to gravitational collapse.

Structure formed in a hierarchy, from smallest to largest dark matter halos.

When ordinary matter fell into these dark matter halos, it dissipated heat and collapsed to form stars and galaxies - cusps in density may persist in some galaxies, if not washed out by dynamics

N-body simulations of ΛCDM structure formation

Where should we look?

!"#$%&'())*!"#$%&'())* +,-./!0'01$2312'4225$36'..'+,-./!0'01$2312'4225$36'..' 7(7(

!"#$%"&'$()*&

View towards the center of our galaxy as seen in the infrared waveband - Simulations of structure formations predict cusps in the dark matter profile near the center.

Galactic Center

Largest signal typically predicted at the GC (e.g., Baltz simulated sky plot, 2006)

!"#$%&'())*!"#$%&'())* +,-./!0'01$2312'4225$36'..'+,-./!0'01$2312'4225$36'..' 7(7(

!"#$%"&'$()*&

Whipple GC Detection

Kosack et al. (astro-ph 0403422; 2004, ApJ, 608, L97)26 hr data taken over 10 years at LZA, 3.7 sigma, 2.8 TeV

!"#$%&'())*!"#$%&'())* +,-./!0'01$2312'4225$36'..'+,-./!0'01$2312'4225$36'..' 7(7(

!"#$%"&'$()*&

HESS has obtained beautiful measurements of the GC region, revealing rich astrophsyics.

While the Galactic Center provides one of the best potential sources for gamma-ray emission from dark-matter annihilation, substantial astrophysical backgrounds from other sources make it very difficult to extract the dark matter signal. HESS image of GC region (Aharonian et al., 2006,

Nature 439, 695) E-2.3 spectrum, correlation with molecular clouds - first indication of CR near source?

HESS GC Detection

!"#$%&'())*!"#$%&'())* +,-./!0'01$2312'4225$36'..'+,-./!0'01$2312'4225$36'..' 7(7(

!"#$%"&'$()*&

GC Spectrum

Whipple GC spectrum (Kosack et al. Ph.D. thesis, 2004) agrees well with newer HESS result.

1e-13

1e-12

1e-11

1e-10

1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0.001

0.01

0.01 0.1 1 10 100 1000

Flu

x (m

-2 s

-1 T

eV-1

)

Energy (TeV)

Flux

LZA CrabSZA Crab

Old SZA CrabWhipple Sgr A*

HESS SgrA*Hillas Crab result

1e-13

1e-12

1e-11

1e-10

1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

0.001

0.01

0.01 0.1 1 10 100 1000

Flu

x (m

-2 s

-1 T

eV-1

)

Energy (TeV)

Flux

LZA CrabSZA Crab

Old SZA CrabWhipple Sgr A*

HESS SgrA*Hillas Crab result

GC spectrum

Crab spectrum

!"#$%&'())*!"#$%&'())* +,-./!0'01$2312'4225$36'..'+,-./!0'01$2312'4225$36'..' 7(7(

!"#$%"&'$()*&

(Annihilation spectrum plotted over GC data points assuming 15 TeV mass, decay through t and τchannels, 25% energy resolution assuming line/continuum ratio of 5x10-4 following Fornengo, Pieri and Scopel, PRD, 70, 103529, 2004)

GC Spectrum

!"#$%&'())*!"#$%&'())* +,-./!0'01$2312'4225$36'..'+,-./!0'01$2312'4225$36'..' 7(7(

!"#$%"&'$()*&

Andromeda Galaxy

Virgo Galaxy Cluster (X-ray)

? Dwarf Galaxies

Galactic Minihalos

The first structures in the Universe 9

Figure 1. A zoom into one of the first objects to form in the universe. The colours show

the density of dark matter at redshift 26. Brighter colours correspond to regions of higher

concentrations of matter. The blue background image shows the small scale structure in the

top cube (cube size = [3 comoving kpc]3) which has a similar filamentary topology as the large

scale structure in the CDM universe. The first red image zooms by a factor of one hundred into

the average density high resolution region. This region was initially a cube of [60 comoving pc]3

resolved with 64 million particles with a gravitational softening of 10!2 comoving parsecs and

masses 1.2 ! 10!10M" " Mmoon/300. The final image shows a close up of one of the individual

dark matter halos in this region, again zooming in by a factor of one hundred so that the box

has a physical length of 0.024 parsecs. This tiny triaxial Earth mass halo has a cuspy density

profile and is smooth, devoid of the substructure that is found within galactic and cluster mass

dark matter halos. Even though the index of the power spectrum is very steep on these scales,

n # $3, we find that halos can collapse before merging into a larger system, rather than the

niave expectation that all scales are collapsing simultaneously thus erasing such structures.

Dark Matter - Where Next?

!"#$%&'())*!"#$%&'())* +,-./!0'01$2312'4225$36'..'+,-./!0'01$2312'4225$36'..' 7(7(

!"#$%"&'$()*&

Surface brightness depends on line of sight integral of the density squared

Calculate expected line or continuum angular distribution for N-body (NFW) halo distributions.

The solid angle which optimizes the SNR depends on the halo profile.☠Note that the sensitivity curves for ACTs are appropriate for point sources, for extended sources the curves move up by √Ωsource/Ωpsf

For steep halo profiles, one observes something like a point source with J α 1/d²

For shallow power laws, halos are resolved and have relatively constant surface brightness independent of distance

Flux calculation MethodJ(!) !

!line of sight

"2dl

!"#$%&'())*!"#$%&'())* +,-./!0'01$2312'4225$36'..'+,-./!0'01$2312'4225$36'..' 7(7(

!"#$%"&'$()*&

Object Mass (Msun) Distance Ang. Size

(vir rad/dist)(deg)

OptimumSNR

(arb. units)

Optimum Aperture

(deg)

Signal relative to GC (pt src)

Sensitivity requirement

(erg cm-2s-1)

Minihalo 10-4 0.5 pc 0.29 12 0.027 4x10-2 2.3x10-13

Dwarf Galaxy 108 75 kpc 0.15 5.8 0.020 1.1x10-2 6.6x10-14

GC 1.8x1011 8.5 kpc 47 620 1.2 1.0 6.0x10-12

Andromeda 1.8x1011 730 kpc 0.48 5.6 0.034 2.7x10-2 1.6x10-13

Virgo Cluster 1014 17 Mpc 0.39 1.5 0.034 0.6x10-2 3.6x10-14

!(r) !!

(r/rs)(1 + r/rs)2"

!1Maximal/GC compatible signal + universal NFW halo profile

Halo Gamma Signal

!"#$%&'())*!"#$%&'())* +,-./!0'01$2312'4225$36'..'+,-./!0'01$2312'4225$36'..' 7(7(

!"#$%"&'$()*&

Apples and Oranges...

Often when one compares predictions for different types of objects, parameters like the density profile are optimistically tweaked to achieve a large signal - this makes it difficult to compare model predictions.

Often constraint plots for instruments suppress the see of points stretching many orders of magnitude below the suppressed zero - this is especially bad for cases where the detection cross section is only loosely related to the decoupling cross section, or for gamma-rays, when the astrophysical uncertainties are large.

In comparing GLAST and ACTs, it is possible to produce results that show the superiority of one instrument over the other by choosing the appropriate fiducial neutralino mass, or by comparing point source sensitivities for known sources (favorable for ACTs) or all-sky survey sensitivity (favorable for GLAST)

!"#$%&'())*!"#$%&'())* +,-./!0'01$2312'4225$36'..'+,-./!0'01$2312'4225$36'..' 7(7(

!"#$%"&'$()*&

GLAST and ACTs

Bergstrom, Ullio, Buckley (Astropart. Phys1998) found that for the GC, an ACT could potentially probe more parameter space with better sensitivity for detecting a gamma-ray line

For the continuum emission, integrating down to the instrument threshold, GLAST can do better than VERITAS for masses up to a couple hundred GeV (e.g., Koushiappas 2006, Baltz 2006)

1

10

102

50 100 150 200 250 300

E ! ! GeV "

Nu

mb

er o

f p

ho

ton

s in

2 y

ears

2! line

10-16

10-15

10-14

10-13

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104

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ph

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!"#$%&'())*!"#$%&'())* +,-./!0'01$2312'4225$36'..'+,-./!0'01$2312'4225$36'..' 7(7(

!"#$%"&'$()*&

The most promising new targets for DM searches may be the local-group Dwarf galaxies. Such sources are dominated by dark matter with mass-to-light ratios as high as 1000, and are relatively free of astrophysical backgrounds.

Galactic substructure could be detected at about the same level as Dwarf galaxies. An exciting possibility is to detect the first CDM structures in the form of very nearby microhalos. These objects could exhibit proper motion and parallax. Guidance from GLAST is needed to locate the sources, or ground-based instruments with very wide field of view are needed to survey the whole sky.

!"#$%&'()&)'*+ ,'--.*

!"#$%&'!(#")'*+,-"%./#0 1#0#23'.4'5,-'6%$#0'

7"%8+9':*5.;#5-/'5%5#0';#**&'<=> ! <=? *%0#"'

;#**-*@'08;.4%*.53'AB'2'<=C 6*84 ! ;#**D5%D

0.1,5'"#5.%'<==D<===9'E4-'%)'5,-';%*5'/#"F'

;#55-"D/%;.4#5-/'1#0#2.-*'F4%(4G'H5#"+%%"

! ;8$,'$0-#4-"'%I*-"J#5.%4'$%4/.5.%4*'5,#4'

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012".-334%5$+4%6-"72'

872$*+-2$)%9-*7"2:)$%;%<<%!

*//) .-7)")-2$%6/+%=>?%

/@,$+5-2"/7,'

/!//The first structures in the Universe 9

Figure 1. A zoom into one of the first objects to form in the universe. The colours show

the density of dark matter at redshift 26. Brighter colours correspond to regions of higher

concentrations of matter. The blue background image shows the small scale structure in the

top cube (cube size = [3 comoving kpc]3) which has a similar filamentary topology as the large

scale structure in the CDM universe. The first red image zooms by a factor of one hundred into

the average density high resolution region. This region was initially a cube of [60 comoving pc]3

resolved with 64 million particles with a gravitational softening of 10!2 comoving parsecs and

masses 1.2 ! 10!10M" " Mmoon/300. The final image shows a close up of one of the individual

dark matter halos in this region, again zooming in by a factor of one hundred so that the box

has a physical length of 0.024 parsecs. This tiny triaxial Earth mass halo has a cuspy density

profile and is smooth, devoid of the substructure that is found within galactic and cluster mass

dark matter halos. Even though the index of the power spectrum is very steep on these scales,

n # $3, we find that halos can collapse before merging into a larger system, rather than the

niave expectation that all scales are collapsing simultaneously thus erasing such structures.

High resolution simulation of structure formation showing earth-sized microhalos which might pervade local space, and give an observable signal (Diemand et al., Koushiappas)

Promising Targets

!"#$%&'())*!"#$%&'())* +,-./!0'01$2312'4225$36'..'+,-./!0'01$2312'4225$36'..' 7(7(

!"#$%"&'$()*&

(Koushiappas, 2007)

Comparison of signals from Dwarf Galaxies, microhalos and substructure of the galactic halo.

Sensitivity Estimates

Uncertainties in halo structure (due to influence of baryons, e.g.) means a large sample variance ⇒ need to survey a

number of different objects

!"#$%&'())*!"#$%&'())* +,-./!0'01$2312'4225$36'..'+,-./!0'01$2312'4225$36'..' 7(7(

!"#$%"&'$()*&

For energies < 200 GeV, GLAST provides the most sensitive means of searching for continuum emission from gamma-ray sources; >200 GeV instruments such as VERITAS and HESS provide the best sensitivity. However, for detection of an annihilation line or cutoff feature, ground-based instruments are probably the only means of detecting enough photons over most of the allowed parameter space.

If the mass lightest SUSY particle has a mass <500 GeV, the LHC could directly observe a neutralino. Above 500 GeV, direct detection experiments and indirect astrophysical experiments provide the only avenue for detection.

While dark matter may be detected at the LHC or in direct detection experiments, and neutrino experiments may provide a detection of the dark matter in the local halo, gamma-ray measurements provide the only possible means of observing the halo distribution and of verifying the role of such particles in structure formation of the universe.

Edward A. Baltz (KIPAC) Dark Matter 2006 February 23rd, 2006

Measuring Dark Matter Properties at High-Energy Colliders

E.A. Baltz, M. Battaglia, M. Peskin & T. Wizansky, hep-ph/0602187

!"#$%&'()*$+,'-.$"/*0*($1'2'($

3*&4'(. )5--'65&',75(89-':();9<==>

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:29,59E==93*FG9H+,97A9'-A5 -7C*-.99,590*,*),9'9

B*/ ,;5:A'(09(*/93*F96-'I'&A9JK

Complementarity

!"#$%&'())*!"#$%&'())* +,-./!0'01$2312'4225$36'..'+,-./!0'01$2312'4225$36'..' 7(7(

!"#$%"&'$()*&

!"#$%&'())*!"#$%&'())* +,-./!0'01$2312'4225$36'..'+,-./!0'01$2312'4225$36'..' 7*7*

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,%%$-$#*&(#'.$*#%/$0,%%$-$#*&(#'.$*#%/$0!"##$%&'())*'+',,,

,89:;'())<

=$#215'=25215$;3'>0$? .38$#215'=25215$;3'

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BCD BCD

EFG6$3;

4$H28

C$669$3;

Complementarity

!"#$%&'())*!"#$%&'())* +,-./!0'01$2312'4225$36'..'+,-./!0'01$2312'4225$36'..' 7(7(

!"#$%"&'$()*&

The FutureU.S. (AGIS) and European groups (CTA) are currently in the planning stages for a next generation gamma-ray experiment with km² effective area, 10% energy resolution, arcmin angular resolution, and energy thresholds as low as 40 GeV. Such instruments could achieve sensitivities of 10-¹³ down to 10-¹⁴ erg cm-² sec-¹ at 200 GeV.

Artist’s concept for a km² array of wide-field ACTs (J.Buckley, D. Braun)

Please attend ourmeeting:

“The Future of Very High Energy Gamma-Ray Astronomy”

http://future-tev.uchicago.edu/

May 13&14, Chicago, IL Wyndham Hotel

!"#$%&'())*!"#$%&'())* +,-./!0'01$2312'4225$36'..'+,-./!0'01$2312'4225$36'..' 7(7(

!"#$%"&'$()*&

(Differential sensitivity plot for points sources/50hr)

Sensitivity of km2 Array

For a wide-field (8-10deg) km2 array, this sensitivity could be obtained for every point in a 90 degree swath of the galactic plane every 2 years

!"#$%&'())*!"#$%&'())* +,-./!0'01$2312'4225$36'..'+,-./!0'01$2312'4225$36'..' 7(7(

!"#$%"&'$()*&

APS Whitepaper

Initial editorial board was formed Sept. 27, 2006 and expanded in Dec, 2006 to include external advisors

B. Dingus (MILAGRO)

H. Krawczynski (VERITAS, EXIST)

M. Pohl (Theory, GLAST)

V. Vassiliev (VERITAS)

W. Hofmann (HESS)

S. Ritz (GLAST)

F. Halzen (Ice Cube)

T. Weekes (VERITAS)

02/27/2007 07:03 PMmkwp

Page 1 of 2http://cherenkov.physics.iastate.edu/wp/

PAGE NAVIGATION

The charge from APS

Editorial board

Organizational meetings

Meeting at the GLAST

Symposium

WORKING GROUPS

Extragalactic Astrophysics

Galactic compact objects

SNR and cosmic rays

Dark matter

Gamma-ray bursts

Technology

The future of ground-based gamma-ray

astronomy

APS White Paper

The Status and Future of Ground Based Gamma-

Ray Astronomy

In the last two years ground-based gamma-ray

observatories have made a number of stunning

astrophysical discoveries which have attracted the

attention of the wider scientific community. The

high discovery rate is expected to increase during

the forthcoming years, as the VERITAS observatory

and the upgraded MAGIC and HESS observatories

commence scientific observations and the space-

based gamma-ray telescope, GLAST, is launched.

The continuation of these achievements into the

next decade will require a new generation of

ground-based observatories. In view of the long

lead time for developing and installing new

instruments, the Division of Astrophysics of the

American Physical Society has requested the

preparation of a White Paper on the status and

future of ground-based gamma-ray astronomy.

Scientists from the entire spectrum of astrophysics

are invited to contribute to the concepts and ideas

presented in the White Paper. We wish to stress

that international participation is encouraged.

The charge from APS Editorial board Organizational meetings

Meeting at the GLAST Symposium

http://cherenkov.physics.iastate.edu/wp/

!"#$%&'())*!"#$%&'())* +,-./!0'01$2312'4225$36'..'+,-./!0'01$2312'4225$36'..' 7(7(

!"#$%"&'$()*&

Working Groups

Extragalactic Science - Henric Krawczynski (Washington U.)

Dark Matter Science - Jim Buckley (WU)

Gamma-ray Bursts - David Williams/Abe Falcone (Santa Cruz)

Galactic Diffuse Emission, SNR and Origin of Cosmic Rays - Martin Pohl (Iowa State U.)

Galactic Compact Objects - Phil Kaaret (U. Iowa)

Technology - Karen Byrum (ANL)

!"#$%&'())*!"#$%&'())* +,-./!0'01$2312'4225$36'..'+,-./!0'01$2312'4225$36'..' 7(7(

!"#$%"&'$()*&

APS Whitepaper

Ted Baltz (SLAC) Savvas Koushiappas (LANL)

Jim Buckley (Wash U) Henric Krawczynski (Wash U)

Karen Byrum (ANL) Stephan LeBohec (U. Utah)

Brenda Dingus (LANL) Martin Pohl (ISU)

Stephen Fegan (UCLA) Stefano Profumo (Caltech)

Paolo Gondolo (U. Utah) Joe Silk (Oxford)

Geonfranco Bertone (INFN) Vladimir Vassiliev (UCLA)

Dan Hooper (FNAL) Scott Wakely (U. Chicago)

Deirdre Horan (ANL) Matthew Wood (UCLA)

Theorist ExperimentalistExperimentalist/Theorist

Dark Matter Working Group

!"#$%&'())*!"#$%&'())* +,-./!0'01$2312'4225$36'..'+,-./!0'01$2312'4225$36'..' 7(7(

!"#$%"&'$()*&

Planning of next generation gamma-ray observatory will be driven, in part, by dark matter science.

Even if other experiments detect dark matter in the lab or local halo, gamma-ray measurements provide a unique means to measure the halo structure

Gamma-ray measurements are complementary to GLAST, Direct, Neutrino, LHC, ILC measurements

Gamma-ray continuum signal is closely tied to total annihilation cross-section and relic abundance - primary uncertainty comes from halo model

Much work remains to be done, more collaborators are welcome in planning the next generation experiment!

Conclusions

!"#$%&'())*!"#$%&'())* +,-./!0'01$2312'4225$36'..'+,-./!0'01$2312'4225$36'..' 7(7(

!"#$%"&'$()*&

Dwarf spheroidal galaxies in the local group have ~107-109 solar masses, but very low

surface brightness with high mass-light rations of 100-1000

From rotation curves, it appears that many have a flat core (M/L~1), but a few LSB galaxies

may show evidence for cusps

Formation of a steep stellar cusp or slow growth of a BH may create a DM spike

Fig. 6.— LSB rotation curves from the datasets of de Blok et al. (2001b) (B01), de Blok and Bosma (2002)(B02) and Swaters et al. (2003) (S03). Solid curves show best fits using the Courteau (1997) fitting formulagiven by eq. 7. Best fits with ! ! 1 correspond to rotation curves that rise and turn over gently as a functionof radius. Curves with ! > 2 feature a much sharper transition from the rising to the flat part of the curve.See text for full discussion.

25

!"#$%&'())*!"#$%&'())* +,-./!0'01$2312'4225$36'..'+,-./!0'01$2312'4225$36'..' 7878

!"##$%&'())*'+',,,

(Woods and Vassiliev, 2006)

Dwarf Galaxies