Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.atnf.csiro.au/computing/software/gipsy/ellint/ellint.ps Äàòà èçìåíåíèÿ: Tue Feb 20 14:13:22 1996 Äàòà èíäåêñèðîâàíèÿ: Fri Jan 16 20:38:56 2009 Êîäèðîâêà: IBM-866 Ïîèñêîâûå ñëîâà: ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï ï

ELLINT
Integration of image data in ellipses
jan 21, 1996
1. Purpose
ELLINT integrates image data from a GIPSY set (INSET=) in elliptical
rings, it can be used to find the radial intensity distributions in galaxies or,
for example, to find the mean intensity of instrumental rings in maps. The
ellipses are projected circles, viewed at an inclination i (INCL=). Further,
the rings are characterized by a major axis (RADII=), a width (WIDTH=),
the position angle of the major axis (PA=) and a central position (POS=).
ELLINT has three options. It calculates:
1. Statistics in a ring or segment, like the sum, mean etc.
2. The projected and faceíon surface brightness in a ring or segment
3. The (scaled) mass surface density \Sigma M in a ring
1. Options
Option 1 (sum, mean, median, rms, area): Circles in the plane of an object
(e.g. galaxy) are projected onto the sky as ellipses. The ratio between major
and minor axis of an ellipse depends on the inclination at which we view
the object. Two ellipses with different major/minor axes enclose a certain
number of image pixels in a ring. In ELLINT, a pixel belongs to such a
1

Figure 1: Example of plot generated by ELLINT
ring if its center is positioned between the two ellipse boundaries or on the
boundary of the inner ellipse. Pixels in that ring either have an image value
or are undefined (blank). If a noníblank pixel with index k has image value
I k and there are N such pixels in a ring, then the sum of the image values is:
S =
N
X
k=1
I k (1)
The units of the sum are the units of the image values (e.g. W.U.). The
second ring parameter is the mean image value per pixel:
ï
I = S
N
(2)
Option 2 (surface brightness): The calculation of the surface brightness is
necessary for option 2 and option 3. For optical data the symbol ï is used.
The next table summarizes the meaning of the symbols:
2

ï = surface brightness per pixel, projected on the sky
ï 0 = faceíon surface brightness (in plane of galaxy) per pixel
ï
ï = mean projected surface brightness in a ring averaged over non blank area
ï
ï 0 = mean faceíon surface brightness in a ring averaged over non blank area
ï
ï t = mean projected surface brightness in a ring averaged over total area
ï
ï 0t = mean faceíon surface brightness in a ring averaged over total area
If a pixel has size dxdy (dx; dy in seconds of arc) then the projected surface
brightness per pixel is
ï k = I k
dxdy
(3)
If N is the number of noníblank pixels and M is the total number of pixels,
then ELLINT distinguishes two mean surface brightnesses. The first is an
average over the noníblank area and is written as
ï
ï = 1
N
N
X
k=1
ï k = 1
N
N
X
k=1
I k
dxdy
=
ï
I
dxdy
(4)
The second is an average over the total area and can be written as
ï
ï t = 1
M
N
X
k=1
ï k = N
M
1
N
N
X
k=1
I k
dxdy
= N
M
ï
I
dxdy
= N
M
ï
ï (5)
If there are no blank pixels in your rings then ï
ï and ï
ï t are equal. Otherwise,
you have to think about what the blanks in your map actually represent.
The faceíon area of a pixel is increased by a factor cos(i). Then a pixel has
a surface brightness:
ï 0 k
= I k
dxdy
cos(i)
= cos(i)ï k (6)
and therefore the mean faceíon surface brightness in a ring is:
ï
ï 0 = cos(i)ïï (7)
ï
ï 0t = cos(i) ï
ï t (8)
3

Option 3 (mass surface density \Sigma M ): For optical thin HI data, the mass
in a ring is a linear function of the total flux S. If you use this option, the
program does not calculate masses directly, but it scales a given mass M tot (in
M fi entered by MASS=) proportional to the surface densities of the rings and
converts the results to a mass surface density \Sigma M ( M fi
pc 2
). The surface densities
oe 0 are calculated in the same way as the surface brightness in option 2, and
again, we distinguish two values oe 0 and oe 0t . Before scaling any mass, ELLINT
assumes that the entire area contributes to the total flux whether there are
blank pixels in it or not. If we know the mean faceíon surface density ï
oe 0 in
a ring with inner major axis R 1 and outer major axis R 2 then the sum in a
(faceíon) ring can be written as:
S 0 = ï
oe 0 ‹(R 2
2 \Gamma R 2
1 ) (9)
Note that if there are no blank pixels in the ring and that dxdy is small comí
pared to the area of the ring, then it is justified to make the approximation:
S 0 ‹ ï
oe 0 N dxdy
cos(i)
= ï oeNdxdy =
N
X
k=1
oe k dxdy =
N
X
k=1
I k = S
which implicates that the total faceíon and projected mass, are the same. In
ELLINT we use the geometrical expression for S 0 . To scale the mass we
calculate S 0 for each ring and then do the summation
S tot =
rings X
S 0
If we divide M tot over all rings then the mass in each ring is
M = M tot
S 0
S tot
= M tot
ï
oe 0 ‹(R 2
2 \Gamma R 2
1 )
S tot
(10)
Suppose an object has distance D(pc) (DISTANCE=), size d(pc) and is
viewed at an angle ff(radians) then the relation between these parameters
is:
d(pc) = D(pc): tan(ff) ‹ D(pc):ff(rad)
4

Figure 2: d(pc) ‹ D(pc):ff(rad)
If we want to express the distance D in Mpc and the angle in seconds of arc,
then use the conversions 1 Mpc = 10 6 pc and 1 arcsec = 2‹
360:3600
radians to
obtain the relation:
d(pc) = 4:8481D(Mpc)ff(00) (11)
The previous expression for the mass in a ring can then be written as:
M = M tot
ï
oe 0
S tot
‹(R 2
2 \Gamma R 2
1 )
(4:848D) 2 (12)
The mass surface density \Sigma M has units M fi
pc 2
, i.e. a mass divided by the
enclosed area in pc 2 and therefore:
\Sigma M = M tot
ï
oe 0
S tot
1
(4:848D) 2
(13)
ELLINT lists two columns with mass densities. The first column is a density
derived from oe 0 and the second is derived from oe 0t . In radio data a blank
usually represents a zero image value. Then the second column is a better
approximation of the true densities than the first column.
5