PhD thesis - Institute for Space Research - University of Calgary

3.2.5: Geometric factor 43
∆h
∆α
R s
R o
θ i
x
Figure 3.9: Symbols and layout for calculation of ∆α aperture.
where f pixel can be determined by Monte Carlo analysis. For the present
calculation I assume f pixel ≃ 1. The aperture area is approximated by
the projection of the surface area of a cylinder with radius R s and height
∆h onto a plane. This gives
A aperture ≈ 2R s ∆h. (3.25)
Equation 3.14 can be used to put r in term of E i :
r =
( ) 1
Ei b
. (3.26)
a 1
Substituting these into the expression for the geometric factor gives
G pixel ≈ 2(∆r) 2 (∆h) 2
a 2 b
1 + Ro 1 bE b−2
b
i . (3.27)
R s
cos θ i
Recalling that a 1 = −q∆V a, for simplicity I let a 1 = a 2 ∆V , where
a 2 = −qa. The geometric factor takes the final form
G pixel ≈ 2b
(∆r∆h)2 (a 2 ∆V ) 2 b−2
b E
b
1 + Ro
i . (3.28)
R s
cos θ i
It has the properties that it is proportional to the square of the pixel
size, (∆r) 2 , and the square of the aperture width, (∆h) 2 , which make
sense. These properties are shared with the top-hat analyzer.