# Halo Ellipsoid Analysis¶

Warning

This functionality is currently broken and needs to be updated to make use of the Halo Finding and Analysis framework. Anyone interested in doing so should contact the yt-dev list.

## Purpose¶

The purpose of creating this feature in yt is to analyze field properties that surround dark matter haloes. Originally, this was usually done with the sphere 3D container, but since many halo particles are linked together in a more elongated shape, I thought it would be better to use an ellipsoid 3D container to wrap around the particles. This way, less of the empty-of-particle space around the halo would be included when doing the analysis of field properties where the particles are suppose to occupy.

## General Overview¶

In order to use the ellipsoid 3D container object, one must supply it with a center, the magnitude of the semi-principle axes, the direction of the first semi-principle axis, the tilt angle (rotation angle about the y axis that will align the first semi-principle axis with the x axis once it is aligned in the x-z plane.)

Once those parameters are determined, the function “ellipsoid” will return the 3D object, and users will be able to get field attributes from the data object just as they would from spheres, cylinders etc.

## Example¶

To use the ellipsoid container to get field information, you will have to first determine the ellipsoid’s parameters. This can be done with the haloes obtained from halo finding, but essentially it takes the information:

1. Center position x,y,z
2. List of particles position x,y,z

And calculates the ellipsoid information needed for the 3D container.

What I usually do is get this information from the halo finder output files in the .h5 HDF5 binary format. I load them into memory using the LoadHaloes() function instead of reading in the ASCII output.

### Halo Finding¶

import yt
from yt.analysis_modules.halo_finding.api import *

halo_list = HaloFinder(ds)
halo_list.dump('MyHaloList')


### Ellipsoid Parameters¶

import yt
from yt.analysis_modules.halo_finding.api import *



Once the halo information is saved you can load it into the data object “haloes”, you can get loop over the list of haloes and do

ell_param = haloes[0].get_ellipsoid_parameters()


This will return 6 items

1. The center of mass as an array.
2. A as a float. (Must have A>=B)
3. B as a float. (Must have B>=C)
4. C as a float. (Must have C > cell size)
5. e0 vector as an array. (now normalized automatically in the code)
6. tilt as a float.

The center of mass would be the same one as returned by the halo finder. The A, B, C are the largest to smallest magnitude of the ellipsoid’s semi-principle axes. “e0” is the largest semi-principle axis vector direction that would have magnitude A but normalized. The “tilt” is an angle measured in radians. It can be best described as after the rotation about the z-axis to align e0 to x in the x-y plane, and then rotating about the y-axis to align e0 completely to the x-axis, the angle remaining to rotate about the x-axis to align both e1 to the y-axis and e2 to the z-axis.

### Ellipsoid 3D Container¶

Once the parameters are obtained from the get_ellipsoid_parameters() function, or picked at random by the user, it can be input into the ellipsoid container as:

ell = ds.ellipsoid(ell_param[0],
ell_param[1],
ell_param[2],
ell_param[3],
ell_param[4],
ell_param[5])
dens = ell.quantities['TotalQuantity']('density')[0]


This way, “ell” will be the ellipsoid container, and “dens” will be the total density of the ellipsoid in an unigrid simulation. One can of course use this container object with parameters that they come up with, the ellipsoid parameters do not have to come from the Halo Finder. And of course, one can use the ellipsoid container with other derived fields or fields that they are interested in.

## Drawbacks¶

Since this is a first attempt, there are many drawbacks and corners cut. Many things listed here will be amended when I have time.

• The ellipsoid 3D container like the boolean object, do not contain particle position and velocity information.
• This currently assume periodic boundary condition, so if an ellipsoid center is at the edge, it will return part of the opposite edge field information. Will try to put in the option to turn off periodicity in the future.
• This method gives a minimalistic ellipsoid centered around the center of mass that contains all the particles, but sometimes people prefer an inertial tensor triaxial ellipsoid described in Dubinski, Carlberg 1991. I have that method composed but it is not fully tested yet.
• The method to obtain information from the halo still uses the center of mass as the center of the ellipsoid, so it is not making the smallest ellipsoid that contains the particles as possible. To start at the center of the particles based on position will require an O($$N^2$$) operation, right now I’m trying to limit everything to O($$N$$) operations. If particle count does not get too large, I may implement the O($$N^2$$) operation.
• Currently the list of haloes can be analyzed using object parallelism (one halo per core), but I’m not sure if haloes will get big enough soon that other forms of parallelism will be needed to analyze them due to memory constraint.
• This has only been tested on unigrid simulation data, not AMR. In unigrid simulations, I can take “dens” from the example and divide it by the total number of cells to get the average density, in AMR one would need to do an volume weighted average instead.