The MeshSource.paint()
function produces the values
of the field on the mesh, returning either a RealField
or
ComplexField
. In this section, we focus on the
process of interpolating a set of discrete objects in a CatalogSource
on to a mesh and how users can customize this procedure.
The paint()
function paints mass-weighted
(or equivalently, number-weighted) fields to a mesh. So, when painting a
CatalogSource
to a mesh, the field \(F(\vx)\) that is painted
is:
where \(V(\vx)\) represents the field value painted to the mesh and \(\delta'(\vx)\) is the (weighted) overdensity field, given by:
where \(\bar{n}'\) is the weighted mean number density of objects. Here, quantities denoted with a prime (\('\)) indicate weighted quantities. The unweighted number density field \(n(\vx)\) is related to its weighted counterpart via \(n'(\vx) = W(\vx) n(\vx)\), where \(W(\vx)\) are the weights.
Users can control the behavior of the value \(V(\vx)\) and the weights
\(W(\vx)\) when converting a CatalogSource
object to a mesh
via the to_mesh()
function. Specifically, the
weight
and value
keywords allow users to indicate the name of
the column in the CatalogSource
to use for \(W(\vx)\)
and \(V(\vx)\). See Additional Mesh Configuration Options for more details
on these keywords.
The default behavior is \(W(\vx) = 1\) and \(V(\vx) = 1\), in which case the painted field is given by:
In the CatalogSource.to_mesh()
function, the default
values for the value
and weight
keywords are the Value
and
Weight
columns, respectively. These are
default columns that are in all
CatalogSource
objects that are set to unity by default.
The Painting Recipes section of the cookbook contains several more examples that change the default behavior to paint customized fields to the mesh.
For example, users can set \(V(\vx)\) to a column holding a component of the velocity field, in which case the painted field \(F(\vx)\) would represent the momentum (mass-weighted velocity) field. See the Painting the Line-of-sight Momentum Field recipe for further details.
Another common example is setting the weights \(W(\vx)\) to a
column representing mass and painting multiple species of particles
to the same mesh using the
MultipleSpeciesCatalog
object.
See the
Painting Multiple Species of Particles
recipe for more details.