Mask-Weighted Summed Count Rate

BAT Cal Memo 2004-09-30
21 July 2005
D. Hullinger

Summary

The mask-weight count rate for a particular photon energy is derived in terms of the source flux, the transparency of the lead tiles and other passive materials, and the effective area of a single detector.

Calculated Counts/s in Each Detector:

Photons/s/cm² (from source) arriving at the detector if no absorbing materials were in the way:

Photons/s/cm² arriving at the detector if absorbing material is in the way:

f_trans: fraction of photons transmitted through the carbon fiber, honeycomb, HV grid, and epoxy (everything except the lead tiles)

Photons/s/cm² arriving at the detector if a lead tile is also partly in the way:

, where

f_Pb is the fraction of photons transmitted through lead (=0 for low energy photons and approaching 1 for high energy photons)
f_illum is the fraction of the detector that is not masked by a lead tile (a number between 0 and 1)

We will now introduce a new quantity called the "mask-weighting factor", w, defined as:

where C is chosen so that w satisfies the equation:

It turns out that C≅1.
K may be chosen in a number of different ways, depending on the "corrections" options given when running batmaskwtimg or batmaskwtevt (hereafter I refer only to batmaskwtimg, but it is implied that batmaskwtevt behaves in the same way). More on this later.

Substituting w into the expression for photons/s/cm² above gives:

Counts/s measured by the detector that result from photons which propagate directly from the source:

where A_eff is the effective area of the detector.

Total Counts/s measured by the detector (including background counts):

where B is the background count rate (resulting from photons not propagating directly from the source, particle interactions, etc.)

Mask-Weighted Counts/s:

To generate a "mask-weighted" count rate that eliminates background counts, the individual count rates are added together, weighted by the mask-weighting factor:

where the subscript d indicates that the variable applies to a particular detector.

If we assume that:

A_eff is the same for every detector, and
B is the same for every detector,

Then this equation simplifies quite nicely:

since, by definition,

(because of how the constant C was chosen).

Parenthetically, is approximately equal to a constant multiplied by the number of detectors in the sum. This constant, which depends only upon the distance between the source and the array, is approximately 0.54 for a source at infinity.

By default, K is chosen so that

where cos_edge is a correction factor that approximately removes the angular dependence of A_eff (that is, Aeff / cos_edge≅A_eff,on-axis).

With K chosen in this way, the mask-weighted count rate becomes:

Note that at low photon energies, f_Pb ≅ 0, so the count rate is just the expected count rate for a detector that is not shadowed by the mask. At higher photon energies, f_Pb approaches 1, so that as the mask becomes more and more transparent, the mask-weighted count rate decreases to 0.

B does not actually have to be the same for every detector for it to cancel out of the expression. It just needs to satisfy ∑B⋅w = 0 (that is, B must not be correlated with w), which can more reasonably be assumed to be true.

To find the number of counts/s in a particular energy bin, the same analysis applies. In that case, A_eff would be the effective area of a detector corresponding to that energy bin (in other words, the number of counts/s depositing their energy in that energy bin per incident photon/s/cm²).

The Value of the "K" Constant and the MSKWTSQF Keyword

By default (when the "corrections" parameter of batmaskwtimg is set to "default", which is equivalent to "flatfield,ndets,pcode,maskwt"), the contant "K" is equal to cos_edge*NGOODPIX*PCODEFR*0.27, where

cos_edge is the cosine correction that accounts for edge illumination
(included in "K" if the "flatfield" correction is specified)
NGOODPIX is the number of active, non-noisy pixels
(included in "K" if the "ndets" correction is specified)
PCODEFR is the coding fraction (1 if the array is fully coded)
(included in "K" if the "pcode" correction is specified)
the "0.27" number is a value that results from the fact that some detectors are somewhere between fully coded and fully masked. Its value depends on the tangential distance between the source and the array (bat_z or bat_zobj), and batmaskwtimg calculates the appropriate value automatically. "0.27" corresponds to a source at infinity. It turns out that for a source with bat_z of 300 cm (the typical distance used for ground calibration), the value is about 0.33
(included in "K" if the "maskwt" correction is specified)

With these corrections, as described above, and the pha file is normalized to "the number of counts/s per fully illuminated on-axis detector"

batmaskwtimg creates a keyword called MSKWTSQF (historically short for "mask-weight squared factor") that carries over into the pha file. The keyword contains information about which corrections were applied, and batdrmgen uses it to normalize the response matrix.

In the "corrections=default" case described above, the MSKWTSQF keyword is 1/cos_edge.

The rule is:

If NO corrections are applied ("corrections=none"), MSKWTSQF is NGOODPIX*PCODEFR*0.27.,
If a given correction is applied, MSKWTSQWF is DIVIDED by the value corresponding to that correction.,

Here are some examples:

Corrections	K	MSKWTSQF
flatfield,ndets,pcode,maskwt	cos_edgeNGOODPIXPCODEFR*0.27	1/cos_edge
ndets,pcode,maskwt	NGOODPIXPCODEFR0.27	1
flatfield,maskwt	cos_edge*0.27	NGOODPIX*PCODEFR/cos_edge
flatfield	cos_edge	NGOODPIXPCODEFR0.27/cos_edge
ndets,pcode	NGOODPIX*PCODEFR	0.27
maskwt	0.27	NGOODPIX*PCODEFR
(none)	1	NGOODPIXPCODEFR0.27

How batdrmgen uses MSKWTSQF

batdrmgen really only calculates a response matrix appropriate for a pha file that is normalized to "counts/s per fully-illuminated detector". In order to make the response matrix compatible with whatever corrections might be applied to the pha file, batdrmgen multiplies the response matrix by the value of the MSKWTSQF keyword. Thus, in order to generate a response matrix appropriate for a pha file normalized to "counts/s per fully-illuminated on-axis detector", batdrmgen multiplies the calculated response matrix by 1/cos_edge.

Keywords batdrmgen does NOT use

batmaskwtimg writes a few other keywords that carry over into the pha file (notably, PCODEFR and NGOODPIX). It is important to realize that batdrmgen does not read them or use them in any way, so making changes to them will have no effect on the spectral fit. The only keywords batdrmgen reads from the pha file are:

BAT_XOBJ
BAT_YOBJ
BAT_ZOBJ
MASKWTSQF