Re: multiplicative user-defined models

On Mon, Nov 06, 2006 at 14:28 -0500, rgibson_at_email.domain.hidden> Let's say I'm defining my own reddening fit model, similar to
> redden().  I use add_slang_function() to register it with ISIS.
> Call it myRedden().
> 
> Should myRedden return bin-integrated values (integrated over
> \Delta\lambda, the bin width in Angstroms)?

Probably not.

> 
> If I use a fit_fun("myRedden(1) * powerlaw(1)"), powerlaw(1) is already
> bin-integrated, so integrating myRedden would mistakenly throw in a factor of
> bin-width, and it appears that ISIS does not account for that.  I.e., ISIS
> assumes that only one component of a multiplicative model (term) is
> bin-integrated.
> 

Remember that writing a bin-integrated model in the form
"a(1)*b(1)" requires the assumption that the spectral bins are
"small enough". 

Consider the case of an absorbed powerlaw. Analytically, the
corresponding bin-integrated spectral model is

   S(E) = \int de p(e) * exp(-N*sigma(e))
        
where p(e) is the power-law model and the exp() factor 
accounts for the line of sight absorption.

To write this in the form "wabs * powerlaw" we pull the exp()
out of the integral to get:

   S(E) \approx exp(-N*sigma(E)) * \int de p(e) 
   
so that  wabs(1) = exp(-N*sigma(E))
     powerlaw(1) = \int de p(e)

In doing this, the multiplicative model is evaluated at a
specific energy, E, while the additive model is bin-integrated.
Clearly, this approximation is good only to the extent that the
bin-width is "small enough" for the functions involved and the
quality of the data being fitted.

Presumably your reddening function is analogous to the
line-of-sight absorption and would be treated the same way.

Thanks,
-John
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