*Mapping star density out to 100 pc*

Kevin Jardine

Star density mapping was pioneered for the Hipparcos dataset in a paper published by Bouy and Alves
in 2015. [1]

[1] Bouy, H., and J. Alves. "Cosmography of OB stars in the solar neighbourhood." Astronomy &
Astrophysics 584 (2015): A26. (Article here: https://www.aanda.org/articles/aa/abs/2015/12/aa27058-
15/aa27058-15.html)

I have produced numerous maps for Gaia DR2 using a similar technique (for example for the solar
neighbourhood and environs here: http://gruze.org/galaxymap/map_800pc/no_stars.html)

Star point cloud data is converted into a scalar field for every element of a 3D grid using a gaussian
kernel density estimator (KDE) which essentially smooths the star count. It can then be displayed in a
voxel viewer or converted into 3D meshes using the marching cubes algorithm. The meshes can be
visualized using 3D applications like Blender or displayed in virtual reality headsets.

*Parameter selection*

Producing a mesh from point cloud data requires several parameters. The bandwidth (often measured in
parsecs) defines the amount of data smoothing. The higher the bandwidth, the larger the smoothing.
Producing meshes using the marching cubes algorithm also requires a density parameter. In addition it
is possible to filter connected mesh subcomponents to remove smaller components or components
containing a small number of stars. (Because it is typically derived from a smoothed scalar field,
smaller subcomponents in a mesh produced by the marching cubes algorithm can sometimes contain no
stars at all.) So a minimum connected component volume or star count can also be used to create the
final meshes.

The Bouy and Alves paper does not make recommendations for determining the values of these
parameters but in fact the parameter selection is quite important. A bandwidth that is too low can
introduce unnecessary noise into a map. A bandwidth that is too high can completely smooth out
smaller density regions such as star clusters.

Moreover, applying the Bouy and Alves algorithm to random point clouds will also produce density
regions, especially for lower density values.

*Drimmel principle*

In the maps I have produced for the 100 pc project, I have used the “Drimmel principle” to select the
bandwidth, based on a recommendation from Gaia astronomer Ronald Drimmel. The principle requires
selecting objects required for the map and choosing the maximum bandwidth at which these objects are
visible.

*Poisson distribution*

Although stars do not have a Poisson distribution, we can nevertheless use Poisson probability as a null
hypothesis to associate a probability value for density. This will give a way to determine the
significance of a given density value. Here is an example with a bandwidth of 9 parsecs.

There are 300,458 stars in the data set with a distance less than 100 pc. The volume of a sphere with a
radius of 100 pc = (4/3)*π*r**3 = 4,188,786.67 cubic parsecs. Given a bandwidth of 9 parsecs, the
volume of a cube of radius 9 parsecs is (2*9)**3 = 5,832 cubic parsecs. Therefore we would expect a
cube of radius 9 parsecs to contain a mean of (5,832/ 4,188,786.67)* 300,458 = 418.32 stars.

The standard deviation of a Poisson distribution with mean λ is just sqrt(λ). Thus we can easily a
compute a table of sigma values.

Sigma Star count
0 418
1 439
2 459
3 480
4 500
5 521
6 541

After applying the KDE, we can then sum up the star count within a cube of radius 9 parsecs centred on
each grid point and use this to associate a Poisson sigma probability with the star density.
This allows us to map probability rather than simply density.

*Final map*

In the final map we decided to choose a bandwidth of 3 parsecs. This was because it made the known overdensity
around the Ursa Major moving group visible. The map shows the regions associated with sigmas 3, 4 and 5.

There were a few small density clumps not associated with any known cluster or moving group. As these only
contained a small number of stars each, we decided to remove those from the map.

*Star overlay*

The poster uses the "missing_10mas" list to supplement the GCNS for an overlay of about 4400 stars with high absolute magnitude (< 3).

The density images were created using the full GCNS alone.