These mock data are designed to mimic the Milky Way disc - either locally or globally. Key questions include, for the local problem:

(1) What quality of data are required to determine the local dark matter density? (2) Do multiple populations (e.g. split by abundance/age) help? (3) Is 1D modelling sufficient?

And for the global problem:

(1) Can global models simultaneously recover the disc phase space distribution function and the gravitational potential? (2) What are the key degeneracies in the problem and how can these be broken?

If posting new tests, please try to approximately follow the template set out for the “spherical collisionless tests” here.

Key working group coordinator: Daisuke Kawata

These mock data give positions and velocities for stars in and around the Solar neighbourhood, relevant for recovering the local Galactic potential and e.g. dark matter content.

These first data assume a 1D population with precisely zero “tilt” term. The velocities are sampled from a 1D distribution function assuming <latex>$f \equiv f(E_z)$</latex> as in Kuijken & Gilmore 1989. This should be an easy test, but highlights:

• The importance of systematic errors due to over-restrictive assumptions about the potential;
• The role of sampling - i.e. how many stars are needed for what type of error;
• The importance of having different populations split by chemistry (and the errors introduced if we get this wrong);
• The importance of sampling high or low as compared to the disc plane; and
• The role of uncertainties in the baryonic mass model

We use a disc model:

<latex> $\nu(z) = \exp(-z/z_0)$ </latex>

and vertical force law:

<latex> $K_z = -\left[\frac{K z}{\sqrt{z^2 + D^2}} + 2Fz\right]$ </latex>

The vertical velocity dispersion (assuming no tilt) can be derived from the Jeans equations:

<latex> $\sigma_z^2(z) = \frac{1}{\nu(z)} \int_0^z \nu(z') K_z(z') dz' + \frac{C}{\nu(z)}$ </latex>

where <latex>$C$</latex> is a constant that sets the velocity dispersion at <latex>$z = 0$</latex>. However, we do not use this. Instead, we use the distribution function:

<latex> $f(E_z) = -\frac{1}{\pi}\int_{E_z}^{\infty} \frac{\mathcal{F}(\Phi) d\Phi}{\sqrt{2(\Phi - E_z)}}$ </latex>

where:

<latex>$\mathcal{F} = \frac{1}{z_0}\frac{\nu}{K_z}$</latex>

This is made more numerically tractable by the trigonometric substitution:

<latex>$\Phi = E_z \sec^2\theta$</latex>

which gives:

<latex> $f(E_z) = \frac{-\sqrt{2 E_z}}{\pi} \int_{0}^{\pi/2} \sec^2\theta \mathcal{F}(\theta,E_z) d\theta$ </latex>

We set up several models by drawing stars from the above distribution function. These are detailed in the following table. For completeness I list also the normalisation constant for the Jeans equation solution <latex>$C$</latex>, though this is not required when using the full distribution function method.

Model	Parameters	Data files	Plots
Simple	<latex>z_0 = 0.4\,{\rm kpc}</latex>; <latex>K = 1500</latex>; <latex>F = 267.65</latex>; <latex>C = 22.85	2</latex>; <latex>D = 0.18</latex>; <latex>n_* = 10	4</latex>; <latex>\Delta z_{\rm bin} = 0.05\,{\rm kpc}</latex>	simplenu_sigz_bin.dat simplenu_sigz.dat simplenu_sigz_raw.dat
Simplelow	As Simple, above, but with <latex>n_* = 10	3</latex>	simplelownu_sigz_bin.dat simplelownu_sigz.dat simplelownu_sigz_raw.dat
Simple2	<latex>z_0 = 0.9\,{\rm kpc}</latex> and <latex>C = 38.978	2</latex>; otherwise as Simple, above	simple2nu_sigz_bin.dat simple2nu_sigz.dat simple2nu_sigz_raw.dat
High	As Simple2, above, but with <z_0 = 0.65\,{\rm kpc}</latex>; <latex>C = 31.734946	2</latex>; data cut on 2 < z < 4 kpc; just ~500 stars; and <latex>\Delta z = 0.25\,{\rm kpc}</latex>	highnu_sigz_bin.dat highnu_sigz.dat highnu_sigz_raw.dat

where <latex>n_*</latex> is the number of stars, and <latex>\Delta z_{\rm bin}</latex> is the bin size (for the binned data).

The data files are as follows:

• simplenu_sigz_bin.dat :: Binned data: <latex>z [{\rm kpc}], \nu(z), \nu_{\rm err}(z), \sigma_z(z) [{\rm km/s}], \sigma_{z,{\rm err}}(z)</latex>
• simplenu_sigz.dat :: Smooth model
• simplenu_sigz_raw.dat :: Raw particle data in <latex>[z,v_z]</latex>
• simple.png :: Plots of the above.

The system of units is kpc, Msun, km/s. Converting the vertical force to a surface density (<latex>{\rm M}_\odot\,{\rm kpc}^{-2}</latex>) via the Poisson equation (assuming no rotation curve contribution), we have:

<latex> \Sigma_z(z) = \frac{|K_z|}{2\pi G_1} </latex>

where:

<latex>G_1 = 6.67 \times 10^{-11} \times 1.989\times 10^{30} / 3.086\times 10^{19}</latex>. In the above system of units, the “baryonic” contribution <latex>\Sigma_b = K = 55 {\rm M}_\odot\,{\rm pc}^{-2}</latex>.

Simple code by Justin Read. Preliminary: adventures.pdf.

N-body disc models simulated with GCD+ (Kawata & Gibson 2003). These are the target disc models used in Hunt, Kawata & Martel (2013). The fixed Navarro, Frenk & White (1997) dark matter halo profile is applied:

<latex>\rho _{dm}=\frac{3H_{0}^{2}}{8\pi G}\frac{\Omega _{0}-\Omega_b}{\Omega_0}\frac{\delta_{c}}{cx(1+cx)^{2}}</latex>
where:
<latex> c=r_{200}/r_s </latex>,
<latex> x=r/r_{200},</latex>
<latex>r_{200}=1.63\times 10^{-2}\left(\frac{M_{200}}{h^{-1}M_{\odot}}\right)^{\frac{1}{3}} h^{-1},</latex>
and:
<latex>\delta_c = \frac{200}{3} \frac{c^3}{\left[\ln(1+c) - c/(1+c)\right]}.</latex>

We assume <latex>\Omega_0=0.266=1-\Omega_{\Lambda}</latex>, <latex>\Omega_{\rm b}=0.044</latex>, and <latex>H_0=71{\rm kms^{-1}Mpc^{-1}}</latex>. A live N-body disc is initially set up and simulated for several Gyr. The data file provides the position, velocity and mass of the disc particles. If you have any question, please email d.kawata (at) ucl (dot) ac (dot) uk.

ASCII data file format:
x,y,z,vx,vy,vz,mp
… format(7(1pE13.5))

x,y,z … position (kpc)
vx,vy,vz … velocity (km/s)
mp … particle mass (Msun)

Model	Description	<latex>M_{200}/M_\odot</latex>	<latex>c</latex>	<latex>N</latex>	Data file	Plot
GD1	Smooth disc	1.75e12	20	99,9982	gcdpdnf1.dat.gz
GD2	Barred disc	2.0e12	9	1,000,000	gcdpdbar1.dat.gz

Additionally, we provide the target GD2 in equatorial coordinates, with and without a Gaia-type error, treating the particles as tracers for a single stellar population, in this case M0 Giants.

The errors are computed following Gaia science performance estimates (credit to Mercè Romero Gómez & Francesca Figueras.)

The sun is located at (-8,0,0) kpc in cartesian coordinates. (Note that this does not take into account extinction)

ASCII data file format and units:
output.txt

For the GD1-RC stars for the challenge, the velocity of the sun w.r.t the GC is (0, 228.14, 0) km/s with the sun again at (-8,0,0) kpc.

If using these test data, please cite the Gaia Challenge wiki and Hunt, Kawata & Martel (2013).

• First simple challenge can be recovering the surface density profile, mean rotation velocity profile and velocity dispersion profile as a function of radius from a partial data (e.g. using the particle data within 10 kpc from (x,y,z)=(-8,0,0) kpc.
• Recovering the surface density profile, mean rotation velocity profile and velocity dispersion profile as a function of radius from the error added data.

To start things off, we show the results of our Particle-by-particle M2M algorithm (PRIMAL) when applied to the barred galaxy target above with and without error. We start from a smooth disc with a different scale length, and recover the target while using the 10 kpc sphere of observation around the sun as suggested above.

Without error, it is easy to recover the radial profiles for the density, radial and vertical velocity dispersions, and the mean rotational velocity form the partial data. See Fig. 1, the black solid line is the target, taken from the complete data, the blue dotted line is our starting point, and the red dashed line is our final model. As you can see this produces a very good fit (as shown in Hunt, Kawata & Martel (2013)).

With error however things become more difficult. As a work in progress we are modifying PRIMAL to work with data containing error, and we show our best attempt so far in Fig. 2. We are happy with the density profile but we are continuing to make improvements to the algorithm to allow us to better recover the velocity profiles.

Code	Test	ASCII data	Plots
PRIMAL (Hunt et. al)	GD2	coming soon!	Fig. 1: Without Error	Fig. 2: With Error

See his presentation

K$_{z,1.1}$ is similar to the target galaxy.

See his presentation chemin-gaiachallenge.pdf

Recover rotation curve and pattern speed of the bar and spirals.

N-body representations of an analytical distribution function of the form used by McMillan and Binney (2012), evaluated in two different potentials. The distribution function represents a single “pseudo-isothermal” disc in each potential. The potentials are Milky Way-like and constructed from several spheroidal and disc like components (as in McMillan 2011).

The particles are tracers (zero mass), and the data files provide their exact positions and velocities (x,y,z,vx,vy,vz) in units of kpc and kpc/Myr (1 kpc/Myr = 977.775 km/s). Questions or comments to p.mcmillan1 (at) physics (dot) ox (dot) ac (dot) uk.

Data file 1 from first potential.

Data file 2 from second potential.

Again, N-body representations in the same two potentials - the distribution function is slightly more complicated. Data represent red clump stars (all having intrinsic G=1, V-I=1), with data including uncertainties as per those expected from Gaia, so if G>16, the radial velocity is unknown (N.B. unknown radial velocities are quoted as 0 with uncertainty given as -1)

Data format: gzipped space separated ascii, with explanation in header line (behind a # symbol):

RA(radians) dec(radians) true_parallax(mas) true_mu_a*(mas/yr) true_mu_d(mas/yr) true_vr(km/s) RA(radians) dec(radians) obs_parallax(mas) obs_mu_a*(mas/yr) obs_mu_d(mas/yr) obs_vr(km/s) err_parallax(mas) err_mu_a*(mas/yr) err_mu_d(mas/yr) err_vr(km/s) abs_G obs_G x(kpc) y(kpc) z(kpc) vx(km/s) vy(km/s) vz(km/s)

(78 Mb each, Data file 1b Data file 2b)

The Sun is placed at x=8kpc, y=0, z=0. The Sun's velocity with respect to the Galactic Centre is vR = -11.1 km/s, vz = 7.25 km/s and vphi = -258.7 km/s for model H or -249.5 km/s for model G.

If using these test data, please cite the Gaia Challenge wiki and McMillan and Binney (2012).

See his presentation.

These are full collisionless N-body simulations. The first two were published and analysed in http://adsabs.harvard.edu/abs/2011MNRAS.416.2318G and http://adsabs.harvard.edu/abs/2012MNRAS.425.1445G. The latter two include a perturbing satellite at low and high inclination and will be published shortly in Read et al. 2013 in prep.

There are four simulations:

Each .tar bundle contains:

• densityfalloff_dm/stars :: ASCII file data for the dark matter/star density fall-off at different radii and angle for small volumes around the disc.
• surfacedensity_bar/dm :: As above for the surface density for the baryons and dark matter.
• velocity dispersion :: As above for the stellar velocity dispersion in the z-direction (perpendicular to the disc).
• wedges :: Raw unbinned ASCII data for different wedges cut around the disc as in http://adsabs.harvard.edu/abs/2011MNRAS.416.2318G.
• *.txt.gz :: Raw ASCII data for the entire centred and aligned simulations (excluding the accreted satellite stars and dark matter). Warning: These files are large (>2 GB).

If using these test data, please cite the Gaia Challenge wiki, http://adsabs.harvard.edu/abs/2011MNRAS.416.2318G, and Read et al. 2013 in prep.

Snapshots provided by Elena D'Onghia from her simulations in http://adsabs.harvard.edu/abs/2013ApJ...766...34D, which have 100 million disk particles. Ask Jo Bovy to get these from a USB stick. The simulation and data consists of snapshots at 0 Myr (initial conditions), 250 Myr, and 1 Gyr (snap_000.hdf5, snap_005.hdf5, snap_020.hdf5).

The galaxy model consists of an Hernquist halo with total mass of 9.5×10^11 Msun computed at a radius of 160 kpc. The halo is simulated with a rigid potential. The concentration of the halo is adopted to be c=9 for an Hernquist profile which is approximately like c=12 for a NFW profile. The spin parameter of the halo is lambda=0.033. This galaxy contains a live stellar disk and a little live bulge that made more stable the disk:

Disk:

The disk fraction is: Mdisk/Mhalo=0.04 The scale length is 2.5 kpc. We introduced perturbers corotating with the stellar disk as softened particles (the total number of perturbers is 1000 and each has a mass of 9.5×10^5 Msun so that the total mass of the perturbers is ~ 9.5×10^8 Msun which is ~2.5% of the total disk mass). In the run we fixed the softening of the perturbes with size of giant a molecular cloud: ~100 pc. In the initial setup the vertical thickness of the disk is setup z0=0.1 of the scale length. Number of particles in the disk= 100 millions. In gadget snapshot stellar disk particles are type 2.

Bulge:

The bulge fraction is: 0.01. The size is 0.1 of the disk scale length. Number of particles in the bulge is : 10 millions. In gadget snapshot bulge particles are type 3.

The particles that mimic the perturbers are 1000 and in gadget are type 4.

Q parameter is initially set up to be larger than 1.3 at all radii.

If using these test data, please cite the Gaia Challenge wiki and http://adsabs.harvard.edu/abs/2013ApJ...766...34D.

Data provided by UB team (Romero-Gomez et al.). The simulation is described here.

Ascii file: 21635205 Red Clump particles with Grvs<16.1 (from a sample of 72M)

-In each row:

G ,Grvs ,Xreal (Kpc),Yreal (Kpc),Zreal (Kpc),VXreal (Km/s),VYreal (Km/s),VZreal (Km/s),Xobs (Kpc),Yobs (Kpc),Zobs (Kpc), VXobs (Km/s),VYobs (Km/s),VZobs (Km/s)

-Positions and velocities are galactocentric. -The Sun is placed at (-8.5,0,0)Kpc

-data format: (2(f7.3,1x),3(f7.3,1x),3(f10.2,1x),3(f7.3,1x),3(f10.2,1x))

-Real values = without errors -Obs values = with Gaia errors

Also available (not in the ascii file):

-Av ,V ,V-I ,GL ,GB ,dist ,xmuls ,xmub ,VR ,GLOBS ,GBOBS ,distobs ,xmulsobs ,xmubobs ,VROBS ,sigalpha ,sigdelta ,sigpi ,sigmua ,sigmud ,sigvr

-The full sample without the cut at Grvs<16.1

The simulation data is too large to upload it to the twiki. We can provide it in a USB stick or you can find it at google drive: https://docs.google.com/file/d/0B8sgZGL4aKpSc3dlUnRKdkszaHM/edit?usp=sharing .

Input bar pattern speed: $\Omega_p=50$ km/s/kpc

V$_{\phi}$ vs. V$_R$ map at the various location, and looking for OLR feature. Using flat circular velocity curve model (v0), and $\Omega_p$ as free parameters I get $v0\sim210$ km/sec and $\Omega_p\sim$50km/sec/kpc, with and without errors, nearby the Sun. ps_monari.pdf

- Based on the usual Fourier decomposition in rings

- Based on moment of inertia

- (semi-local TW method)

test_methods_2.pdf

Conclusion: a not too wrong value 35-42 [km/s/kpc] is found around 4-5 kpc, with a systematic lower value for the data with noise in both the moment of inertia and the Fourier methods. However, the inner (0-2 kpc) values are very noisy, and no obvious plateau in R is found.

This is the Milky Way scaled version of Model R1 of http://adsabs.harvard.edu/abs/2005ApJ...628..678D, which is selected one of the best Milky Way model in http://adsabs.harvard.edu/abs/2013arXiv1306.4694G.

If using these data, please cite the Gaia Challenge wiki and Debattista et al. (2005) and Gardner et al. (2013).

These are ideas and initial results of the challenges discussed in the Surrey workshop week. However, please feel free to post your mock data, grab the data and do your own challenge. Please also feel free to discuss off-line and build up collaborations.

Sharing the mock data.

Comparison among different dynamical modelling methods using various mock data.

Bovy, McMillan, Chemin, Hunt, Buedenbender, Sharma, Roca-Fabrega, Monari, Inoue, Read, Steger

deriving 3D force fields from the mock data: measure force in 3D, F$_R$, F$_{\phi}$, F$_z$ as a function of R, $\phi$, z grid points

Grid in Galactic R, $\phi$, z

R: 3-20 kpc in radius

|z|: 0-4 kpc

$\phi$: +-45 deg (for Paper 1 below, no $\phi$ grid)

whoever got the first results can decide bin size.

Smooth disc: McMillan model

Smooth N-body simulation: GD1 (1M particles, see above), D'Onghia disk (100 M)

- heliocentric equatorial coordinate: R.A. DEC, parallax, proper motion (mas/yr), radial velocity (km/s)

- position and motion (w.r.t. GC) of the Sun (R$_{Sun}$=8 kpc, z=0)

- selection function

• RC tracer, solar metallicity with and without Gaia error G < 20.

• Use radial velocity data for only G_RVS < 17

• RC stars: M_G=1 mag, V-I=1.0, sig M_G=0

• no extinction

ascii data

Pfenniger, Monari, Hunt, Roca-Fabriga, Chemin

Recovering the pattern speed of the bar.

N-body Barred galaxy

-GD1 (Hunt)

-Debattista bar model

UB Test particle simulation with a bar potential

* RC tracers

* with or without Gaia error

* with or without extinction

1D vertical force field reconstruction

Cosmological simulations, multiple populations, dust extinction

Mock data available?

Cosmological simulations:

N-body+gas:

N-body: smooth, structured, barred, multi-compents, bulge

Test particles: bar potential, spiral arms

Analytic: smooth disk

Galaxia:

GUMS: kinematic model, multiple populations, dust extinction

Challenges: what we want to analysis?

starting: GD1 without error

selection function - cutting within sphere - cutting |z|<1~0.1 kpc - known dust extinction

1. - smooth disk exponential
2. - Schlegel(?) map, Galaxia type dust
3. - 3D extinction

- multiple populations

- Who is interested in what?

L. Chemin … derive rotation curve, DM vs. disk potential. Location of bar, resonance.

P. McMillan … N-body with error, the local+whole potential

S. Roca-Fabrega … Moment of distribution function, vertex deviation, bar pattern speed, spiral arm (number) position and perturbation

J. Bovy … vertical force.

J. Hunt … M2M modelling. disc structure, DM potential?

S. Sharma … analysing rotation curve

S. Inoue … vertical disc structure, local DM density

G. Monari … velocity distribution, local and radial gradient, with structures

Read/Steger … vertical force; local DM density