The Estimation of Stellar Image Centers from
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Raster Scan Data
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1. Model of stellar photographic image
In case that the density of the emulsion does not reaches the
limit of maximum density the Gaussian two dimensional Normal
Distribution Function known from error theory is a good
representation of the photographic density distribution of an
well exposed stellar image.
Let's assume an area which contains the image of a star. The
scanned area might look like as outlined:
-5 -4 -3 -2 -1 0 1 2 3 4 5
┌───┬───┬───┬───┬───┬───┬───┬───┬───┬───┬───┐
5 │ 1│ 2│ . │ . │ │ │ │ │ . │ 10│ 11│ 5
├───┼───┼───┼───┼───┼───┼───┼───┼───┼───┼───┤
4 │ 12│ 13│ . │ │ │ │ │ │ │ . │ 22│ 4
├───┼───┼───┼───┼───┼───┼───┼───┼───┼───┼───┤
3 │ │ │ │ │ │ │ │ │ │ │ │ 3
├───┼───┼───┼───┼───┼───┼───┼───┼░░─┼───┼───┤
2 │ │ │ │ │ │ │ ░░░░░░░░░░░ │ │ 2 /│\
├───┼───┼───┼───┼───┼──░░░░░░░░░░░░░░░░─┼───┤ │
1 │ │ │ │ │ │░░░░▒▒▒▒▒▒▒▒▒░░░░░ │ │ 1 │
├───┼───┼───┼───┼──░░░░▒▒▒▓▓▓▓▓▓▒▒▒░░░░─┼───┤ │
0 │ │ │ │ │ ░░░▒▒▒▓▓▓███▓▓▓▒▒░░░ │ │ 0 │ y-axis
├───┼───┼───┼───┼░░░▒▒▓▓▓█████▓▓▓▒▒░░░──┼───┤ │
-1 │ │ │ │ │░░░▒▒▓▓▓███▓▓▓▒▒▒░░░ │ │-1 │
├───┼───┼───┼───░░░░▒▒▒▓▓▓▓▓▓▒▒▒░░░░┼───┼───┤ │
-2 │ │ │ │ ░░░░░▒▒▒▒▒▒▒▒▒░░░░ │ │ │-2 │
├───┼───┼───┼───░░░░░░░░░░░░░░░░┼───┼───┼───┤
-3 │ │ │ │ │ ░░░░░░░░░░░ │ │ │ │-3
├───┼───┼───┼───┼───░░──┼───┼───┼───┼───┼───┤
-4 │ │ │ │ │ │ │ │ │ │ │ │-4
├───┼───┼───┼───┼───┼───┼───┼───┼───┼───┼───┤
-5 │111│ . │ │ │ │ │ │ │ │ . │121│-5
└───┴───┴───┴───┴───┴───┴───┴───┴───┴───┴───┘
-5 -4 -3 -2 -1 0 1 2 3 4 5
────────────────────────>
x-axis
With regard to the fog of the platen we may write by
introducing a Cartesian coordinate system centered at the
center of the stellar image (a5,a7):
x-a5 y-a7
X = ──── , Y = ──── :
a6 a8
┌ ┐
│ ┌ ┐a10│
x-a5 y-a7 │ 1 │ 1 ┌ ┐│ │
d(x,y) = a1 + a2∙──── + a3∙──── + a4∙exp│-─∙│────────∙│ X² - 2∙a9∙X∙Y + Y² ││ │
a6 a8 │ 2 │ 1 - a9² └ ┘│ │
│ └ ┘ │.
└ ┘
Here are a1 : fog of platen
a2 : x-slope of fog in units of a6
a3 : y-slope of fog in units of a8
a4 : center density above fog
a5 : center x-coordinate of stellar image
a6 : measure of width in x-coordinate
a7 : center y-coordinate of stellar image
a8 : measure of width in y-coordinate
a9 : 'correlation' between x and y, │a9│ < 1
a10: accounts for flattened center, a10 > 0.
a6, a8, and a9 alone will describe any kind of elliptical
shape. However, under normal conditions we may expect that:
1. a6 = a8 and a9 = 0, i.e. a circular stellar image,
2. a10 ≈ 1, i.e. not strongly saturated stellar image,
3. a1 ≈ 0.25, i.e. no unusually high fog on plate.
In case of guiding errors the images may be somewhat elliptic.
Due to the non linearity of photographic emulsions also a
strong magnitude dependence may be introduced additionally which
has to be accounted for.
If there are bright stars also to be measured the coefficient
a10 will be significantly greater then 1 (up to 3). However,
this has no influence on the accuracy of the stellar position.
However, in case of linear photosensitive photon recording, i.e.
CCD's, no flattened stellar image is expected and hence a10 = 1.
2. Data Reduction by Least Squares
Let's assume a consequent numbering of the pixels. With
pixel size DX,DY and pixel numbers NX,NY in x- and y-direction
respectively we assume our pixels (xi,yi, i=1 to NX∙NY) to be
defined with respect to the center coordinates xc,yc of the
rectangular data area as follows:
xi = xc - DX∙( ½∙(NX-1) + (i-1) mod NX )
yi = yc + DY∙( ½∙(NY-1) - (i-1) / NX ) ,
i = 1 to NX∙NY .
To solve the equations
D(xi,yi) ≡ Di = d(xi,yi; a1, ..., a10) + vi ,
_
D(xi,yi) = d(xi,yi; a) + vi , i = 1 to NX∙NY,
we first have to linearize d(xi,yi; a1, ..., a10) with respect
to initial coefficients a01, a02, ..., a010. Therefore we need
the partial derivations with respect to a1, ..., a10 at the
point a01, ..., a010. With the abbreviations:
X(x,â) = (x-a5)/a6 , Y(y,â) = (y-a7)/a8 ,
┌ ┐
_ │ a1 │
a = │ .. │ ,
│ a10│
└ ┘
1 ┌ ┐
A = ────────∙│ X² - 2∙a9∙X∙Y + Y² │ ,
1 - a9² └ ┘
_ ┌ a10 ┐
u(x,y;a) = a4∙exp│ -½∙A │
└ ┘
we derive:
dd
──── = 1 ,
da1
dd
──── = X ,
da2
dd
──── = Y ,
da3
dd u
──── = ──── ,
da4 a2
dd -1 a10 1 a10-1
──── = ── + ────∙─────∙u∙A ∙(X - a9∙Y) ,
da5 a6 a6 1-a9²
dd dd
──── = X∙──── ,
da6 da5
dd -1 a10 1 a10-1
──── = ── + ────∙─────∙u∙A ∙(Y - a9∙X) ,
da7 a8 a8 1-a9²
dd dd
──── = Y∙──── ,
da8 da7
dd 1 a10-1
──── = a10∙─────∙u∙A ∙(X∙Y - a9∙A) ,
da9 1-a9²
dd 1 a10
──── = -───∙u∙A ∙ln(A) .
da10 2
The integrated intensity of this 2-dimensional modified
Gaußfunction is given by:
Γ(1/a10) 1/a10 ½
I = ────────∙π∙a4∙a6∙a8∙2 ∙(1-a9²)
a10
1/a10
2 ½
= π∙──────∙Γ(1/a10)∙a4∙a6∙a8∙(1-a9²) .
a10
For estimation of the mean error σI it's necessary to linearize
I = I(a4,a6,a8,a9,a10) with respect to the parameters a4,.. ,a10.
1/a10
dI 2 ½ I
─── = π∙──────∙Γ(1/a10)∙a4∙a6∙a8∙(1-a9²) = ────
da4 a10 a4
dI I
─── = ────
da6 a6
dI I
─── = ────
da8 a8
-½
dI ½∙(1-a9²) ∙(-2a9) -a9
─── = ──────────────────∙I = ─────∙I
da9 ½ 1-a9²
(1-a9²)
┌ ┐
│ 1/a10 │
dI ½ d │2 │
──── = π∙a4∙a6∙a8∙(1-a9²) ∙────│──────∙Γ(1/a10)│
da10 da10│ a10 │
└ ┘
The term in brackets (u) is:
1/a10 ┌ 1/a10┐
2 ln│2 │-ln(a10)
u = ──────∙Γ(1/a10) = e └ ┘ ∙Γ(1/a10)
a10
1/a10∙ln(2)-ln(a10)
= e ∙Γ(1/a10)
It follows for the derivation of u with respect to a10:
du ┌ -1 1 ┐ u dΓ(1/a10)
──── = u∙│─── ∙ln(2) - ───│ + ─────── ∙─────────
da10 └a10² a10┘ Γ(1/a10) da10
┌ ┐
u │ -ln(2) a10 dΓ(1/a10) │
= ───∙│ ────── - 1 + ────────∙───────── │
a10 │ a10 Γ(1/a10) da10 │ ,
└ ┘
┌ , ┐
du -u │ Γ (1/a10) │
──── = ────∙│ ln(2) + a10 + ───────── │
da10 a10² │ Γ(1/a10) │ .
└ ┘