Offset Functions¶
This doc describes the offset functions that are coded in astropath. A future refactor will allow for user-supplied functions.
In our formalism, the offset function is written as \(p(\omega|O)\) and it describes the true distribution of the transient within its host galaxy.
Definitions¶
Here are the key items related to \(p(\omega|O)\):
\(\omega\): 3-D true position of the transient
\(O\): galaxy properties (e.g. position, angular size)
\(\theta\): offset between the transient and the center of a galaxy.
\(\phi\): galaxy angular size
\(\theta_{max}\): the maximum separation between the transient and the galaxy allowed in units of \(\phi\). \(p(\omega|O) = 0\) for \(\theta > \theta_{max}\)
theta_prior¶
The code base uses a dict named theta_prior to handle the offset function. It currently holds three items:
method – A string defining the method used. core, uniform, exp
max – \({\theta_{max}}\) in units of \(\phi\). \(p(\omega|O) = 0\) for \(\theta > \theta_{max}\)
ang_size – An array of angular sizes \(\phi\) for all of the candidate galaxies.
Below are the 3 offset functions currently coded. Note that each of these are normalized to unit total probabilty when integrated over the full sphere.
uniform¶
Here, \(p(\omega|O)\) is given equal weighting for all locations with \(\theta < \theta_{max}\).
core¶
Here, \(p(\omega|O)\) is proportional to \(\phi / (\theta + \phi)\)
exp¶
Here, \(p(\omega|O)\) is proportional to \((\theta/\phi) \, \exp [-(\theta/\phi)]\)