and Galaxy Dynamics STELLAR STREAMS 

The Aquarius Project www.mpa-garching.mpg.de/aquarius 

THE MASS OF A GALAXY how to measure Bulge Stars Gas & Stellar Disk Stellar Halo Stars Globular Clusters Satellite Galaxies using a kinematic snapshot nearby ^ 

THE MASS OF A GALAXY how to measure Bulge Stars Gas & Stellar Disk Stellar Halo Stars Globular Clusters Satellite Galaxies using a kinematic snapshot STELLAR STREAMS nearby ^ 

THE MASS OF A GALAXY how to measure Bulge Stars Gas & Stellar Disk Stellar Halo Stars Globular Clusters Satellite Galaxies using a kinematic snapshot STELLAR STREAMS nearby ^ 

STELLAR STREAM? what is a 

The PAndAS Survey — this figure: Richardson+ (2011) The Astrophysical Journal, 732:76 (14pp), 2011 May 10 Richardson et al. Figure 1. Surface density map of stars with colors and magnitudes consistent with belonging to metal-poor red giant branch populations at the distance of M31. The almost uniform underlying background is mainly contributed by foreground stars in the Milky Way together with a small residual contamination from unresolved compact background galaxies. All of the previously known M31 dwarf spheroidals in this region covered by the survey are readily visible as well-defined overdensities and are marked with blue circles. The five new dwarf spheroidals are highlighted in red. (And XIV is the dwarf spheroidal just south of the present survey area, while And VI and VII lie, respectively, well to the west and north of the region shown.) NGC147 and NGC185 appear at the top of the map and M33 at the bottom left. The green circle lies at a projected radius of 150 kpc from the center of M31 within which most of the survey lies. In addition to the satellite galaxies, numerous stellar streams and substructures are visible. Although the majority of small overdensities are satellite galaxies of M31, a few to the southern end of the map (not circled) are resolved globular cluster systems picked out surrounding nearby low-redshift background elliptical galaxies. (A color version of this figure is available in the online journal.) 

The Sagittarius Stream — this figure: Belokurov+ (2004) 

The GD-1 Stream — this figure: Grillmair & Dionatos (2006) MAIR & DIONATOS Vol. 643 Fig. 1.—Smoothed, summed weight image of the SDSS field after subtrac- tion of a low-order polynomial surface fit. Darker areas indicate higher surface densities. The weight image has been smoothed with a Gaussian kernel with . The white areas are either missing data, clusters, or bright stars that j p 0Њ .2 have been masked out prior to analysis. star ight star tude been rder, ap- actic oth, On the and we tion aint- 

STELLAR STREAMS why are AWESOME? so 

STELLAR STREAMS why are AWESOME? so not 

Majewski+ (2003) FIND STELLAR STREAMS how to 

Majewski+ (2003) FIND STELLAR STREAMS how to northward, is a ‘‘ southern arc.’’ In x 6 we explore the dis- tance distribution of the arc and show that it extends physically from the main body of Sgr. The distance modulus of the southern arc is more or less constant for more than 100 from the Sgr center toward the Galactic anticenter The arc may, in fact, cross the Galactic plane at the anti center and cross into the northern Galactic hemisphere albeit at very low surface density in M giants. Fig. 2.—Solid lines indicate the color-color selection criteria adopted to find M giants for most of this paper. (a) The distribution of stars in the control field from Fig. 1e. (b) The distribution of stars from the statistically subtracted sample in Fig. 1f. Note that the control field, selected to be a Galactic longitud match to the Sgr center field, still contains about a 1% contribution from the Sgr dwarf itself. star by star subtraction of (b) from (a). Panels (d )–( f ) show the corresponding (JÀKs , JÀH ) two-color diagrams for the samples shown in (a)–(c). All source are dereddened using the Schlegel et al. (1998) maps. 

Majewski+ (2003) FIND STELLAR STREAMS how to 

Koposov, Rix & Hogg (2010) FIND STELLAR STREAMS how to No. 1, 2010 CONSTRAINING THE MW POTENTIAL 261 Figure 1. Number density of SDSS DR7 stars with 0.15 < g − r < 0.41 and 18.1 < r < 19.85, shown in the rotated spherical coordinate system that is approximately aligned with the GD-1 stream. The map was convolved with a circular Gaussian with σ = 0. ◦2. The gray arrows point to the stream, which is barely visible in this representation, extending horizontally near φ2 = 0◦, between φ1 = −60◦ and 0◦. significant constraints on the MW potential. See also Eyre & Binney (2009) for theoretical discussion of using thin streams in order to constrain the MW potential. Figure 2. One-dimensional stellar density profile across the stream using the stars with 0.15 < g − r < 0.41 18.1 < r < 19.85 across the φ2 = 0◦ axis, integrated along the stream in the interval −60◦ < φ1 < −10◦. The Gaussian fit with ∼ 600 stars and σ = 12 is shown in red. (A color version of this figure is available in the online journal.) color–magnitude diagram (CMD) of M13 observed in the same filters. Not presuming a particular metallicity (e.g., that of M13), we start our analysis with a simple color–magnitude box selection for stars (0.15 < g−r < 0.41 and 18.1 < r < 19.85). 

Streams are not on a single simple orbit. There are always observational uncertainties. You'll never have 6D phase space measurements. REMEMBER 

USE STELLAR STREAMS how to for dynamics Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 10 June 2011 (MN L A TEX style file v2.2) Stellar Streams as Probes of Dark Halo Mass and Morphology: A Bayesian Reconstruction A. Varghese,1⇤ R. Ibata,1 G. F. Lewis,2 1Observatoire Astronomique de Strasbourg (UMR 7550), 11, rue de l’Universit´ e, 67000 Strasbourg, France 2Sydney Institute for Astronomy, School of Physics, A28, University of Sydney, NSW, 2006 Australia 10 June 2011 ABSTRACT Tidal streams provide a powerful tool by means of which the matter distribution of the dark matter halos of their host galaxies can be studied. However, the analysis is not straightforward because streams do not delineate orbits, and for most streams, especially those in external galaxies, kinematic information is absent. We present a method wherein streams are fit with simple corrections made to possible orbits of the progenitor, using a Bayesian technique known as Parallel Tempering to e ciently explore the parameter space. We show that it is possible to constrain the shape of the host halo potential or its density distribution using only the projection of tidal streams -ph.GA] 9 Jun 2011 2011 — arXiv:1106.1765 The state of the art? 

Varghese, Ibata & Lewis (2011) USE STELLAR STREAMS how to for dynamics 10 A. Varghese, R. Ibata, G. F. Lewis Figure 9. Correcting for tidal tails: The left and right panels show stream B in the XZ and VxVz planes, respectively. The black dots are particles from an N-body simulation of the stream. The grey dotted curve is the orbit of the progenitor, the remnant of which is the concentrated sphere. The blue and red dots are the corrected points for the leading and trailing arms, respectively. The very close correspondence between the computationally-expensive N-body simulation and the locus of the “corrected stream” points is evident. streams. However, itself is not found to be constrained by fitting the projections of these streams. Another significant parameter of the halo is its mass, measure with current instrumentation, which provides in- formation on the inner mass profile of the galaxy. Figure 14 shows the e↵ects of adding the rotational velocities (here 

Varghese, Ibata & Lewis (2011) USE STELLAR STREAMS how to for dynamics ases e q ting: ne of eful rbit the 7). (red and blue dots in Figure 9), which we the corrected points for a given set of para We find empirically that the o↵set ra Q at a radial distance r can be calculated a 2 . 88 times the theoretical Jacobi radius: rcutoff = 2 . 88 ⇥ ✓ msat 3 e M ( r ) ◆1 / 3 r , where msat is the mass of the satellite an lated from the circular velocity Vc of the ho e M ( r ) = rV 2 c ( r ) /G , where G is the gravita For a perfectly spherical potential, e M ( r ) the mass contained within radius r . For n 

Varghese, Ibata & Lewis (2011) USE STELLAR STREAMS how to for dynamics Initial conditions + model parameters Current Progenitor backward integrate progenitor orbit Points on Tidal Tails forward integrate Figure 5. Fitting shorter versions of the orbit AS1 ( q = 1). BS1 has two turning points and CS1 only one. The middle panels show the q distributions obtained by fitting only the projected positions of these orbits. The spread in the distribution increases with decreasing turning points. The bottom panels show the q distribution obtained by adding more information to the fitting: the line of sight velocities vy for BS1, the distances y and line of sight velocities vy for CS1. The fitting mechanism also turns out to be extremely useful in approximating the line of sight distances along the orbit as revealed by a grayscale plot of the distances along the various trial orbits on the coldest MCMC chain (Figure 7). 5 TESTING WITH STREAMS IN A SPHEROIDAL POTENTIAL Having demonstrated the power of the technique in con- straining the parameters of a density distribution by fitting only the projected positions of orbits, we test its ability to robustly estimate the same by fitting streams which, as men- tioned in § 1, do not delineate the orbit of the progenitor or any other exact orbit in the potential. A consistent and fast mechanism is required to derive the positions of stream stars from the progenitor’s orbit, without using N-body integra- tion. The stars which make up the stream are those which were tidally ripped from the satellite during pericenter pas- sages. Based on this, it is possible to formulate a simple fore, the trajectory of a stream star can be integrated w its initial position o↵set from a certain point in the proge tor’s past orbit by a certain distance rcutoff , o↵set outwa for trailing tail stars and inwards for leading tail stars. T is shown diagrammatically in Figure 8. Starting at the c rent position and velocity of the progenitor P , its orbi integrated backwards in time, marked by the grey dots. any point on its backward orbit, Q , at time tQ , the in and outer escape points are approximated as rQ rcut and rQ + rcutoff , indicated respectively by the blue and dots at Q . These provide the initial positions for the orb of stars that escape the satellite at Q . For their initial ve ity components, we use the velocity of the progenitor or at Q . With these initial phase space coordinates, we in grate forward for the same amount of time tQ . These orb are indicated by the blue and red dotted curves, with th final points on the leading and trailing tails at A and B spectively. Repeating this process for several points on backward integrated orbit (say, for every 50 Myr) yield set of points which lie closely on the tidal tails of the stre (red and blue dots in Figure 9), which we shall refer to the corrected points for a given set of parameters. We find empirically that the o↵set radius for a po Q at a radial distance r can be calculated approximately 2 . 88 times the theoretical Jacobi radius: rcutoff = 2 . 88 ⇥ ✓ msat 3 e M ( r ) ◆1 / 3 r , ( where msat is the mass of the satellite and e M ( r ) is cal lated from the circular velocity Vc of the host galaxy at r e M ( r ) = rV 2 c ( r ) /G , where G is the gravitational consta For a perfectly spherical potential, e M ( r ) is equivalent the mass contained within radius r . For non-spherical tentials, it is only a crude approximation to the total m inside r , but as we show below, the correction we obtain ing rcutoff calculated in this manner is su ciently accur for our purposes. We find that by fitting the stream data with these c rected points it is indeed possible to recover the paramet of the potential as well as the orbit. It is interesting to n that through this correction mechanism, one has more in mation on the progenitor’s past orbit than one would h with only the local orbit of the progenitor (i.e. the red a blue dotted curves in Figure 9 contain more informat than the grey curve does). In this sense, the tidal stre Time since infall known? 

Varghese, Ibata & Lewis (2011) USE STELLAR STREAMS how to for dynamics 10 A. Varghese, R. Ibata, G. F. Lewis Figure 9. Correcting for tidal tails: The left and right panels show stream B in the XZ and VxVz planes, respectively. The black dots are particles from an N-body simulation of the stream. The grey dotted curve is the orbit of the progenitor, the remnant of which is the concentrated sphere. The blue and red dots are the corrected points for the leading and trailing arms, respectively. The very close correspondence between the computationally-expensive N-body simulation and the locus of the “corrected stream” points is evident. streams. However, itself is not found to be constrained by fitting the projections of these streams. Another significant parameter of the halo is its mass, measure with current instrumentation, which provides in- formation on the inner mass profile of the galaxy. Figure 14 shows the e↵ects of adding the rotational velocities (here COMPARE TO DATA — SOMEHOW. 

UNCERTAINTIES MATTER. in The Real Worldtm 

Varghese, Ibata & Lewis (2011) USE STELLAR STREAMS how to for dynamics Various lengths of streams Projected position + line-of-sight velocity + circular velocity Flattened logarithmic potential or NFW-style density + baryons short is HARD good constraints on galaxy parameters 

Varghese, Ibata & Lewis (2011) USE STELLAR STREAMS how to for dynamics major shortcomings The algorithm relies on: Unrealistic knowledge of the progenitor. Axisymmetric host galaxy. A poorly defined likelihood function. Well known stream membership classifications. 

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