Hope
College Physics Department Research Experiences for Undergraduates Summer 2011Project Summary |

Project Title: |
Using the Maximum Likelihood Method to find the Optimal
Parameters of the Models Describing the Pulsar Population Statistics |

Student Name: |
Caleb Billman |

Student's Home Institution: |
Hope College |

Research Advisor: |
Dr. Peter Gonthier |

Source of Support: |
This material is based upon work supported by the Michigan Space Grant Consortium, Hope College Department of Physics endowed funds (Frissel), NASA Astrophysics Theory and Fundamental Physics Program (NNX06AI32G), and NASA - Fermi Guest Investigator Cycle 3 (NNX10AO41G). |

We seek to maximize the likelihood of a set of parameters describing the distributions
of the initial period, magnetic field, and radio luminosities in our pulsar
population simulation code (Harding, Grenier, and Gonthier 2007) and in order
to better understand the confidence region of the model. Our pulsar population
code simulates pulsars at birth using Monte Carlo techniques and evolves them
to the present assuming different models describing the birth distributions,
spin down, and emission. One problem with this method is that it is difficult
to explore the parameter space due to high computational time. Since we are
dealing with comparisons of low counts, we need to describe the distributions
using Poisson statistics, which is best accomplished within the maximum likelihood
method (MLM). The popular least-squares method is an instance of the MLM that
assumes Gaussian statistics, a valid assumption when the number of counts is
high. In this scenario, we seek to maximize the likelihood using a Monte Carlo
Markov Chain algorithm that randomly explores the parameter space. An efficient
method to accomplish this is the Hybrid Monte Carlo algorithm (HMC), a means
of searching the parameter space by making jumps towards areas with higher
likelihood. The observed pulsar characteristics that we will be trying to fit
consist of the period, period derivative, the dispersion measure, and the radio
flux. However, before the application to the main goal associated with the
population study, a simple case study is presented as an illustration of the
method.

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