Background
Josh Lande recently ran some consistency tests comparing the Monte Carlo point source parameters to the values fit by binned likelihood for point sources. For the spacecraft data, he used a one-day interval of the idealized +/-50 deg rocking provided by gtobssim. For one location on the sky, location 0: (RA, Dec) = (176.31, 44.28), the fitted flux in the 0.1-100 GeV band was too high relative to the MC value by ~8%, whereas at location 1: (RA, Dec) =
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(-176.31,
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-44.28),
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the
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disparity
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appeared
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negligible.
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For
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such
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a
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short
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time
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scale
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(1
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day),
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these
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two
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sky
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locations
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have
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very
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different
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off-axis
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profiles.
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Here
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are
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histograms
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of
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the
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livetime
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as
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a
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function
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of
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off-axis
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angle
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for
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the
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two
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locations.
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The
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black
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histogram
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corresponds
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to
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location
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0,
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and
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the
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red
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corresponds
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to
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location 1.
Diagnosis
The bug lay in the code that computes the exposure averaged PSF. This is given by
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1. !off_axis_histograms.png|thumbnail! h3. Diagnosis The bug lay in the code that computes the exposure averaged PSF. This is given by {latex} \newcommand{\e}{\varepsilon} \newcommand{\Int}{{\displaystyle \int}} \begin{eqnarray} P(\psi, \e) &=& \frac{\Int dt P(\psi, \e, \theta) A(\e, \theta)} {\Int dt A(\e, \theta)}\\ &=& \frac{\Int d\theta P(\psi, \e, \theta(t)) A(\e, \theta(t))|\partial\theta/\partial t|^{-1}} {\Int d\theta A(\e, \theta(t)) |\partial\theta/\partial t|^{-1}} \end{eqnarray} {latex} Here |
Here
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{latex}$P${latex} |
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{latex}$A${latex} |
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the
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PSF
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and
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effective
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area,
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respectively,
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and
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{latex}$\theta=\theta(t)${latex} |
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the
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angle
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of
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the
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source
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relative
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to
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the
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instrument
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z-axis.
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The
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phi-dependence
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of
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{latex}$P${latex} |
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{latex}$A${latex} |
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suppressed
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for
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clarity.
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{latex}$\psi${latex} |
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the
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angle
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between
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true
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and
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measured
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photon
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direction,
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and
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{latex}$\varepsilon${latex} |
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the
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photon
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energy.
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In
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the
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second
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line
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of
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the
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above
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equation,
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the
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integrals
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have
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been
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recast
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as
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integrations
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over
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theta.
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These
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integrations
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are
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performed
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using
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the
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livetime
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cubes
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computed
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with
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gtltcube.
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In
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ST-09-27-01
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and
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earlier,
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there
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is
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a
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cutoff
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at
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theta=70
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deg
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for
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the
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mean
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PSF
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calculation.
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The
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original
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reason
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for
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this
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cutoff
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is
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unclear,
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but
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it
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was
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probably
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motivated
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by
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some
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poor
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behavior
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of
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the
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PSF
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at
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large
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theta
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angles
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for
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an
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early
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version
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of
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the
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IRFs.
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As
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far
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as
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the
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shape
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of
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the
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PSF
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is
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concerned,
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this
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truncation
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does
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not
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have
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a
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significant
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effect.
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As
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part
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of
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some
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recent
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refactoring,
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this
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truncation
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was
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removed
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in
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Likelihood-17-22-04
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(22
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Feb
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2012).
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Unfortunately,
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I
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had
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thought
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that
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the
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truncation
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only
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affected
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the
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PSF
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shape
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calculation.
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In
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fact,
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for
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point
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sources,
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the
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binned
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analysis
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code
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has
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been
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using
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the
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exposure
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calculation
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in
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the
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denominator
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of
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the
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mean
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PSF
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expression
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for
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computing
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the
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model
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counts
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rather
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than
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using
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the
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exposure
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from
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the
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binned
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exposure
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map.
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Hence,
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all
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point
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source
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fits
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will
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be
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affected
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by
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this
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truncation.
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For
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location
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0
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(black
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histogram
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in
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above
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figure),
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a
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significant
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fraction
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of
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its
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time
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is
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spent
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just
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beyond
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the
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theta=70
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deg
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cutoff.
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This
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is
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the
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cause
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of
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the
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8%
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disparity
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that
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Josh
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found
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in
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his
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tests.
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Effects
One can estimate the impact of the theta=70
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deg
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cutoff
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by
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running
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gtexpcube2
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with
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the
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option
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thmax=180
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in
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the
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nominal
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case
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(this
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is
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the
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default)
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and
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thmax=70
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in
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the
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truncation
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case.
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Assuming
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a
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photon
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index
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of
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-2,
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which
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Josh
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used
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in
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his
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gtobssim
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runs,
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one
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can
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estimate
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the
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expected
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fractional
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offset
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in
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measured
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flux
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(Fmeas)
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relative
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to
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the
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MC
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value
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(Fmc),
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(Fmeas
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-
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Fmc)/Fmc,
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using
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the
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spectrally
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weighted
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exposures,
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exposure
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{latex}$= \left.\int \varepsilon^{-2} A(\varepsilon, \theta(t)) dt d\varepsilon\right/\int \varepsilon^{-2} d\varepsilon${latex} |
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and
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computing
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(exposure180
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-
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exposure70)/exposure70.
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For
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the
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one-day
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simulation,
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here
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is
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a
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map
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of
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the
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estimated
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fractional
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offset
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in
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Galactic
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coordinates:
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- One
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- day
The green point near the Northern Galactic pole is location 0 and has a fractional flux offset of 8%, and the green point nearer the Galactic plane has flux offset of 1.5%. The peak offsets can be has high as 16.3%.
On longer time scales, these offsets are considerably smaller. Here are maps for the 18 month and 2 year livetime cubes used by the catalog group for 2FGL:
- 18 months (/afs/slac/g/glast/groups/catalog/P6_V8_DIFFUSE/ltcube_18months_rock52.fits)
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- 2 years (/afs/slac/g/glast/groups/catalog/P6_V8_DIFFUSE/ltcube_2years_rock52.fits)
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I've
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computed
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livetime
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cubes
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for
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30
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day
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intervals
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and
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have
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estimated
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the
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fractional
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offset
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as
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a
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function
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of
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time
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for
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the
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Vela,
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Crab,
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and
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Geminga
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pulsars.
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There is a DC offset of about 1.5%
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and
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an
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additional
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dispersion
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of
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about
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the
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same
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size.
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NB:
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These
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estimates
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assume
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a
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power-law
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spectrum
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with
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photon
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index
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-2
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over
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the
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entire
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integration
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range
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0.1-100
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GeV.
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