### Table of Contents

### 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^{+0.21}_{-0.16} | 53158^{+3394}_{-3165} |

1B. GG79 | 3.48^{+0.38}_{-0.41} | 89924^{+7957}_{-7371} |

1C. Actual | 2.64^{+0.36}_{-0.30} | 68006^{+3965}_{-4217} |

##### 1A. Single mass LIMEPY model

##### 1B. 3-component model: Gunn & Griffin 1979 model A

##### 1C. 3-component model: actual mass function

## 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.

#### 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.

#### 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.

### 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.