[Part I.] Support adding the external potential for CDM halo construction

Why

The CDM halo is constructed using the Par_EquilibriumIC, which can construct the particle cloud with its velocity dispersion taking the external potential into account. Previously, in the HaloMerger test problem, this functionality was forced to be turned off internally because we didn't need it. However, in future applications (e.g., putting a particle cloud inside an FDM halo), we may need to construct the particle cloud in equilibrium with an external potential.

Changes

• Add the option HaloMerger_ParCloud_?_BuiltWithExtPot in Input__TestProb_ParCloud to decide whether to consider the external potential for constructing the particle cloud.
• Use the parameter HaloMerger_ParCloud_?_ExtPot_Filename to provide the external potential table if HaloMerger_ParCloud_?_BuiltWithExtPot is on.
• Add a Python script, Make_ParCloud_ExtPotTable.py, to generate the example external potential table.
  • This is the potential of a uniform-density sphere, which is consistent with and can be checked by the default external potential implementation in this test problem (set in Input__TestProb).

Verifications

Setups

• Run python3 Make_ParCloud_ExtPotTable.py
• Set OPT__EXT_POT = 1 in Input__Parameter.
• In Input__TestProb

HaloMerger_Halo_Num                     0
HaloMerger_ParCloud_Num                 2
HaloMerger_ExtPot_UniDenSph_M           1.08565410e-01
HaloMerger_ExtPot_UniDenSph_R           9.70381642e-03
HaloMerger_ExtPot_UniDenSph_CenCoordX   0.19407633
HaloMerger_ExtPot_UniDenSph_CenCoordY   0.12938422
HaloMerger_ExtPot_UniDenSph_CenCoordZ   0.12938422

• In Input__TestProb_ParCloud

HaloMerger_ParCloud_1_VelocityX         0.0
HaloMerger_ParCloud_2_VelocityX         0.0
HaloMerger_ParCloud_2_BuiltWithExtPot   1
HaloMerger_ParCloud_2_ExtPot_Filename   ParCloud_ExtPotTable

Results

• The left halo is constructed without the external potential. It is stable by itself.
• The right halo is constructed with the external potential. It is stable if OPT__EXT_POT = 1 (left), and it will spread out if OPT__EXT_POT = 0 (right).

Run with the external potential	Run without the external potential

Issues

The current version of Par_EquilibriumIC has some limitations (e.g., the bins of the external potential table have to be the same as the density profile table) and potential bugs with the effect of external potential.

• These issues will be addressed at Improve Par_EquilibriumIC gamer-project/gamer#315.
• Fortunately, the stability issue is not significant in simple cases based on previous experiences.

[Part II.] Change the CDM halo density profile

Why

The CDM halo in the HaloMerger test problem is constructed from the given density profile.
Previously, we used the formula rho_0 * ( x**(-gamma) ) * ( ( 1.0+(x**alpha) )**( (gamma-beta)/alpha ) ) to fit the FDM halo without a soliton and construct the CDM halo.
However, the density distribution of the CDM halo looks more extended than the FDM halo.
Although the differences are in the large-radius and small-density regions, it still makes the comparison of simulation results between CDM and FDM less straightforward.

Changes

Therefore, in this PR, we change the default CDM density profile to the density profile extracted from the FDM simulations.

• The new density profile is called "FDMHaloDensityProfile", and the old fitted one is renamed "FitHaloDensityProfile".

In this way, the density distributions (especially at the large radius) look almost the same.

Density Projections

Old CDM	New CDM	FDM

Density Profiles

Old CDM	New CDM	FDM