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> $\sigma_z^2(z) = \frac{1}{\nu(z)} \int_0^z \nu(z') K_z(z') dz' + \frac{\sigma_z^2(0)\nu(0)}{\nu(z)}$ </latex>

where <latex>$\sigma_z^2(0) = -\frac{1}{\nu(0)} \int_0^\infty \nu(z') K_z(z') dz'$</latex> 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.

Model	Parameters	Data files	Plots
Simple	<latex>z_0 = 0.4\,{\rm kpc}</latex>; <latex>K = 1500</latex>; <latex>F = 267.65</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>; otherwise as Simple, above	simple2nu_sigz_bin.dat simple2nu_sigz.dat simple2nu_sigz_raw.dat
High	As Simple2, above, but with <latex>z_0 = 0.65\,{\rm kpc}</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 pc}^{-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 = 55.53 {\rm M}_\odot\,{\rm pc}^{-2}</latex>.