===== Local models/mocks =====

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.

==== Simple 1D data sampled from a distribution function of vertical energy ====

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 [[http://adsabs.harvard.edu/abs/1989MNRAS.239..571K|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>| {{:data:simplenu_sigz_bin.dat}} {{:data:simplenu_sigz.dat}} {{:data:simplenu_sigz_raw.dat}} | {{:data:simple.png?200}} |
| Simplelow | As Simple, above, but with <latex>n_* = 10^3</latex> | {{:data:simplelownu_sigz_bin.dat}} {{:data:simplelownu_sigz.dat}} {{:data:simplelownu_sigz_raw.dat}} | |
| Simple2 | <latex>z_0 = 0.9\,{\rm kpc}</latex>; otherwise as Simple, above| {{:data:simple2nu_sigz_bin.dat}} {{:data:simple2nu_sigz.dat}} {{:data:simple2nu_sigz_raw.dat}} | {{:data:simple2.png?200}} |
| 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> | {{:data:highnu_sigz_bin.dat}} {{:data:highnu_sigz.dat}} {{:data:highnu_sigz_raw.dat}} | {{:data:high.png?200}} |

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>.