M4 mass modelling / method comparison results []

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.