==== M4 mass modelling / method comparison results ====



=== 1. Comparison between single component and 3-model(s) ===
Here we compare isotropic, single-component LIMEPY models to isotropic, 3-component LIMEPY models. Gunn & Griffin 1979 posed that the complexity of real GCs with various mass components, all with their own $M/L$, could be captured by 3-component models (1) (invisible) low-mass stars, (2) (visible) turn-off stars and (3) invisible remnants. We tried their Model A and also fit a model in which we derived the mass function from the snapshot.

Mass function:
| Model            | M_j             | M_j/M_tot         | m_j                 | m_j/m_1 |
| 1B. GG79 Model A | [5.0, 1, 0.1] |[0.82, 0.16, 0.02] |                     | [0.50, 1, 1.5]|
| 1C. Actual MF    | [3.1, 1, 3.0] |[0.43, 0.14, 0.42] | [0.37, 0.78, 0.674] | [0.47, 1, 0.86]|

Results:
| Model       | Half-mass radius [pc]                | Mass [Msun]    |
| True        | 3.21                                 |  64255.9      | 
| 1A. Single  | 1.87<sup>+0.21</sup><sub>-0.16</sub> | 53158<sup>+3394</sup><sub>-3165</sub> |    
| 1B. GG79    | 3.48<sup>+0.38</sup><sub>-0.41</sub> | 89924<sup>+7957</sup><sub>-7371</sub> |
| 1C. Actual  | 2.64<sup>+0.36</sup><sub>-0.30</sub> | 68006<sup>+3965</sup><sub>-4217</sub> |


== 1A. Single mass LIMEPY model ==
{{:tests:collision:gc4:triangle_single.png?200}}
{{:tests:collision:gc4:sb_sigma_single.png?200}}
{{:tests:collision:gc4:ml_single.png?200}}

== 1B. 3-component model: Gunn & Griffin 1979 model A ==
{{:tests:collision:gc4:triangle_gg79.png?200}}
{{:tests:collision:gc4:sb_sigma_gg79.png?200}}
{{:tests:collision:gc4:ml_gg79.png?200}}

== 1C. 3-component model: actual mass function ==
{{:tests:collision:gc4:triangle_actual_mf.png?200}}
{{:tests:collision:gc4:sb_sigma_actual_mf.png?200}}
{{:tests:collision:gc4:ml_actual_mf.png?200}}



===== Laura's Results =====

**Dynamical models**: Spherical Jeans Anisotropic MGE (JAM) models.

**Data-model comparison**: discrete maximum likelihoods.

**Assumptions**:
  - Models are spherical.
  - Anisotropy is beta=1-v_theta^2/v_r^2.
  - Models assume no rotation.
  - Surface brightness profile is known.
  - No background contamination, all stars are cluster members.

**Additional comments**:
  - Surface brightness and surface mass density are input as Multi-Gaussian Expansions (MGEs). I fit an MGE to the SB profile on the wiki and then used this for all my models, so SB is fixed (see Assumption #4). Unless explicitly stated below, I assume that the surface mass profile is a scaled version of the SB profile. If I assume a constant M/L then all SB MGE components are multiplied by the same M/L value to get the surface mass profile. If I assume a variable M/L then each Gaussian component of the MGE is multiplied by a different value.
  - Anisotropy is specified for each Gaussian component of the SB MGE. If I assume constant anisotropy, then all SB components have the same anisotropy. If I assume variable anisotropy, I then each component is given a different anisotropy value.
  - I actually fit beta'=beta/(2-beta). This has the appealing property of being symmetric about 0 and finite in extent. beta'=0 is isotropy, beta'=1 is purely radial orbits and beta'=-1 is purely tangential. I only allow beta to vary between 1 and -50 to prevent extremely tangential orbits as this can cause my code to crash.


==== Line-of-sight velocities only ====

=== Model 1: constant M/L ===

Extra assumptions:
  - distance is known
  - model is isotropic
  - M/L is constant

Fit for constant M/L only: 1 free parameter.

{{:tests:collision:gc4:rv_ml_fit_mass.png?200|}} {{:tests:collision:gc4:rv_ml_fit_ml.png?200|}} {{:tests:collision:gc4:rv_ml_fit_rv_disp.png?200|}}

=== Model 2: constant M/L, constant anisotropy, distance ===

Extra assumptions:
  - anisotropy is constant
  - M/L is constant

Fit for constant M/L, constant anisotropy, distance: 3 free parameters.

{{:tests:collision:gc4:rv_mlad_fit_mass.png?200|}} {{:tests:collision:gc4:rv_mlad_fit_ml.png?200|}} {{:tests:collision:gc4:rv_mlad_fit_rv_disp.png?200|}}

=== Model 3: variable M/L ===

Extra assumptions:
  - distance is known
  - model is isotropic

Fit for M/L per Gaussian component of SB MGE: 8 MGE components --> 8 parameters.

{{:tests:collision:gc4:rv_mlvary_fit_mass.png?200|}} {{:tests:collision:gc4:rv_mlvary_fit_ml.png?200|}} {{:tests:collision:gc4:rv_mlvary_fit_rv_disp.png?200|}}



==== Line-of-sight velocities and Proper motions ====

=== Model 1: constant M/L, constant anisotropy, distance ===

Extra assumptions:
  - anisotropy is constant
  - M/L is constant

Fit for constant M/L, constant anisotropy, distance: 3 free parameters.

{{:tests:collision:gc4:all_mlad_fit_mass.png?200|}} {{:tests:collision:gc4:all_mlad_fit_ml.png?200|}}

{{:tests:collision:gc4:all_mlad_fit_pmr_disp.png?200|}} {{:tests:collision:gc4:all_mlad_fit_pmt_disp.png?200|}} {{:tests:collision:gc4:all_mlad_fit_rv_disp.png?200|}}


=== Model 2: M/L, anisotropy, distance ===

Fit for variable M/L (8 components), variable anisotropy (8 components), distance: 17 free parameters.

{{:tests:collision:gc4:all_mlavaryd_fit_mass.png?200|}} {{:tests:collision:gc4:all_mlavaryd_fit_ml.png?200|}}

{{:tests:collision:gc4:all_mlavaryd_fit_pmr_disp.png?200|}} {{:tests:collision:gc4:all_mlavaryd_fit_pmt_disp.png?200|}} {{:tests:collision:gc4:all_mlavaryd_fit_rv_disp.png?200|}}