Re: Formula for converting a spectrum model to counts

Hi, John.  Thanks for the response.  Sorry I'm following up on this slowly.

So it sounds like the one part I need to work out is the bin-integration 
of the model M(E).  For a powerlaw model, I was using M(E_mid)*dE, but 
are you saying that it should technically be integral{ m(E)dE } over the 
bin, which would be similar to what I used but slightly different?


On 12/22/10 6:44 AM, John Houck wrote:
> On Wed, Dec 15, 2010 at 15:52 -0800, Rob Gibson wrote:
>> I'm trying to reproduce the formula used in ISIS (XSPEC, etc.) to
>> calculate model counts given an ARF, RMF, and flux model.
>>
>> The equation is apparently:  C(h) = B(h) + \int dE R(h, E) A(E) m(E),
>> where R, A, and m are the RMF, ARF, and model respectively.
>>
>> Can anyone tell me if the following is accurate?  (It seems to work.)
> The exposure time and background scaling aren't explicit in the
> above expression, but the expression is essentially accurate.
>
>> What are the gotchas here?
>>
>> 1) Use the CIAO tool rmfimg to convert an rmf file to a 2D R(h, E) data
>> structure.
>> 2) Load A(E) directly from the ARF file.
>> 3) m(E) returns a value of photons/keV/cm^2/s; e.g., for a powerlaw
>> model it would be:
>>     m(E) = norm * Emid^(-gamma), where Emid is the center of the RMF
>> energy bin and (norm, gamma) are the usual powerlaw model parameters.
> Because the integral is discretized, the model values must be
> bin-integrated.
>
>> 4) Evaluate the integral (R*A*m dE) using energy bins (Emid, dE) from
>> the RMF grid.
>> 5) Convert channel h to Energy using the second data table of the
>> original RMF file.
>> 6) Plot energy (converted from channel h) vs. C(h); this seems to match
>> plot_model_counts in bin_integral space.
>>
>> Is that the correct specification of the model term m(E) in the
>> equation, or is does there need to be some sort of bin-integral value
>> like in add_slang_function()?
> Yes, the bin-integration is necessary.
>
>> Apparently ISIS evaluates these models on a wavelength grid, so would
>> convert the above integral to wavelength space.  Is that right?
> Isis uses a wavelength grid internally, but since it doesn't
> move the bin edges.  For the purposes of checking the
> calculation, the units shouldn't be important.
>
> Thanks,
> -John
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