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Astronomical Applications Department, U.S. Naval Observatory - Sensitivity Integral (Page 2)

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Astronomical Applications Department, U.S. Naval Observatory - Sensitivity Integral
J.D. Phillips (SAO) is currently examining the photon sensitivity in the focal plane of the FAME
instrument (SAO TM97-xx). Among other things, the calculation involves the Planck function (as
an approximation of stellar spectra) convolved with the intensity envelope due to diffraction of
starlight through the FAME starlight aperture. Therefore, integrals of the form
=
(
)
S , , , ,
,
r s g T
1
2
d
1
2
1
(
)
sin s
2
(
)
-
(
)
sin r
(
)
sin r g
2
-
e




h
k T
1
need to be evaluated in a computationally efficient manner. (Here, r, s, and g are instrument
geometry parameters, and
-
2
1
is the frequency bandpass.) This integral form is unavailable in
Gradshteyn and Ryzhik, and Maple is unable to evaluate it.
The inability of the Risch algorithm, which Maple implements fully, to solve this integral proves
that it has no closed-form solution in terms of elementary functions. Therefore, we are forced to
make an approximation for the Planck function. A simple power series seems appropriate and has
been shown by Phillips (SAO TM97-xx) to be satisfactory for the current application:
=
1
-
e




h
k T
1
k
a
k
(
)
-
k
1
Inclusion of terms to fourth order in frequency
appears to be adequate. The question then
becomes: can we evaluate the resulting integral and present the result in a useful form?
This memo addresses this evaluation issue. In
section 1
the integral is defined, and it is shown
that a simple, brute-force approach is inadequate. In
section 2
the indefinite integral is solved and
coerced into simplified form. A fortran subroutine is created that performs the integral evaluation
in as computationally efficient a manner as is probably possible. In
section 3
, I differentiate the
integral and recover the integrand, lending some confidence in the integration result. We would
like a more rigorous and independent check, however. So in
section 4
I solve the definite integral
by successive application of integration by parts. After a lengthy calculation, I obtain a useful
result, though it is not quite as computationally efficient as the indefinite integration result. In
section 5
, I compare the definite and indefinite integrations and show that they are indeed equal.
This lends great confidence that the answer is correct. In
section 6
we consider the fact that the
found solutions are invalid when the geometrical parameters s, r, and g lie on certain surfaces in (
, ,
s r g
) space. In general, series expansions across these singular surfaces are difficult to integrate.
For the special case s = r = 0, I provide the series expansion that covers this region, an error
function for deciding when to use the series expansion, and fortran subroutines for corresponding
numerical calculations. Finally, in the
Appendix
, I present for reference the full expressions of the
indefinite and definite integrations, as well as the optimized fortran subroutine for the full integral
Page 2

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