gcfit.probabilities.likelihood_number_density

gcfit.probabilities.likelihood_number_density(model, ndensity, *, mass_bin=None, hyperparams=False)

Compute the loglikelihood of the cluster number density profile.

Computes the log likelihood component of a cluster’s number density profile, assuming a Gaussian likelihood. The model profile is scaled to fit the shape of the observation data, and a nuisance parameter is introduced to add a constant error component and minimize the background effects present near the outskirts of the cluster.

Optionally, a background level can be provided in the dataset metadata (ndensity.mdata[“background”]) which will be subtracted from all observations before calculation of the likelihood. By default, will be assumed to have same units as Σ.

Parameters:

modelgcfit.FittableModel

Cluster model used to compute probability distribution.

ndensitygcfit.core.data.Dataset

Number density profile dataset used to compute probability distribution and evaluate log likelihood.

mass_binint, optional

Index of model.mj mass bin to use in all calculations. If None (default), attempts to read ‘m’ from pulsars.mdata, else assumes.

hyperparamsbool, optional

Whether to include bayesian hyperparameters.

Returns:

float

Log likelihood value.

Notes

As the translation between discrete number density and surface-brightness observations is difficult to quantify, the model is actually only fit on the shape of the number density profile data. To accomplish this the modelled number density is scaled to have the same mean value as the surface brightness data (K scaling factor). The chosen K factor minimizes chi-squared:

\[K = \frac{\sum \Sigma_{obs} \Sigma_{model} / \delta\Sigma_{obs}^2} {\sum \Sigma_{model}^2 / \delta\Sigma_{obs}^2}\]

References

[1]Hénault-Brunet, V., Gieles, M., Strader, J., Peuten, M., Balbinot, E.,

and Douglas, K. E. K., “On the black hole content and initial mass function of 47 Tuc”, Monthly Notices of the Royal Astronomical Society, vol. 491, no. 1, pp. 113–128, 2020.