# ANNULUS: Compute a Radial Profile by Azimuthal Averaging

Other relevant VISTA commands to be used prior to running ANNULUS:

ANNULUS computes the azimuthally-averaged radial surface-brightness profile of a 2-D image, storing the profile as a spectrum (1-D image). The value at each radius is defined as the azimuthal average of the pixel values along a suitably defined annulus (ring) of that radius centered on the object. High accuracy Sinc interpolation or faster Bilinear interpolation may be used.

By default, the annuli are concentric circles, but it is possible to define elliptical annuli to be used as the averaging paths by specifying the major axis position angle and inclination angle of the ellipses with keywords.

The centers of the annuli are computed before using ANNULUS by running the AXES command, although this step may be bypassed using a keyword.

It is also possible to forego creating a radial profile, and simply use ANNULUS to compute the average along a single annulus of a given radius.

Unlike the PROFILE command, which computes the surface brightness profile of an object by fitting ellipses to the image isophotes, ANNULUS takes averages along pre-defined annuli without regard to the actual isophotes of the image. ANNULUS is more applicable than PROFILE to those cases where strong non-axisymmetric structure in the object makes fitting ellipses impractical. For example, ANNULUS might be used to compute the average radial surface brightness profile of a Spiral Galaxy, in which the isophotes, especially in the spiral arms, are manifestly not ellipses.

In cases where isophotes may be defined with PROFILE, reflecting structural properties like isophotal rotation or variable eccentricity, the results of a PROFILE calculation may be used as averaging paths in annulus via an optional keyword. This is a fairly advanced application of ANNULUS.

ANNULUS produces the radial profile of a 2-D image in the "source" buffer and stores it as a 1-D image (spectrum) in the "dest" buffer.

The following is a detailed description of each of the keywords:

N=n:
This is the number of concentric annuli to be used. In general, n > 1. If you wish to find the average along only one annulus, then use the "RAD=" keyword.

STEP=dr:
This is the spacing between successive annuli. The first point of the profile is always the image center (defined either by AXES or by using the "CEN=" keyword). By default a STEP of 1 pixel is used. In general, STEP has units of pixels, but if the "SCALE=" keyword is used, then STEP will have units of length as defined by the images scale (usually arcseconds).

PA=pa:
This is the Position Angle of the major axis of elliptical annuli in degrees. PA is measured from the top of the image counter-clockwise, so that an ellipse with PA=0.0 will have its long axis oriented along an image column, while PA=90.0 will have its long axis oriented along an image row. PA=45.0 is oriented diagonally from the upper left to lower right as seen on the TV display.

INC=i:
This is the ellipse Inclination Angle in degrees. It is defined such that:

  Face-On: i = 0 degrees (circular annuli)
Edge-On: i = 90 degrees

INC must always be LESS THAN 90 degrees. It is related to the ellipse eccentricity (e) by: e=sin(i)

CEN=r0,c0:
By default, ANNULUS expects that you've run the AXES command on the image to find the appropriate object centroid (called AXR and AXC by the AXES command). You may have ANNULUS ignore the results of the most recent AXES command by using the "CEN=" keyword to specify the center of each of the annuli. R0 is the row number of the center, and C0 is the column number. R0 and C0 may be real numbers (e.g., R0=35.4 is allowed).

SCALE=s:
This defines the linear scale of the image. Typically the scale is in units of arcseconds/pixel, but any relevant units are OK. The program printout assumes arcseconds/pixel, but it is only cosmetic. If the SCALE and STEP options are used, then the STEP value has the units of SCALE. For example:

   ANNULUS ... SCALE=0.54 STEP=1

where the units of SCALE are arcsec/pixel implies an annular spacing of 1 arcsecond (or 1.85 pixels).

FAST:
For annular radii smaller than 15 pixels, ANNULUS will use a slow, high accuracy 2-D Sinc interpolation algorithm to find the intensity at a given point along the annulus. Beyond 15 pixels, a somewhat faster Sinc interpolation scheme is used. However, using the FAST keyword tells ANNULUS to use a much faster, but somewhat lower accuracy 2-D bilinear interpolation scheme beyond 15 pixels radius. The time savings is quite noticeable.

This option allows you to use ANNULUS to find the azimuthal average along a single, given annulus with radius "r" (and PA and INC as appropriate), without creating an entire profile. When using the "RAD=" keyword, the "N=", "STEP=" are ignored. RAD has units of Pixels, unless the "SCALE=" keyword is used, at which point it has units of "SCALE".

The "PROF" keyword::
Elliptical annuli used by ANNULUS all share the same center, position angle, and inclination (eccentricity). It is possible, using the PROF keyword, to use a set of annuli in which the position angle, eccentricity, and even the centers, may vary as a function of radius. This may be done by first using the PROFILE command to fit ellipses to the isophotes of an image, and then using ANNULUS with the PROF keyword to use the ellipse parameters contained in the PROFILE common block in VISTA to define the averaging paths. The PROFILE common block may also be loaded using the GET command (see GET and SAVE) before running ANNULUS, which suggests all sorts of possibilities.

Other details:

Output may be redirected to the printer or external files using the '>' option in VISTA.

For future reference, the FITS header of the spectrum containing the radial profile generate by ANNULUS contains a number of HISTORY cards recording the relevant annular parameters (PA, INC, STEP, CEN, etc). These may be reviewed by using the "BUF FULL FITS" command, or by using "HEDIT". The FITS "STATUS" card is changed to read "Azimuthal Average", and the "CTYPE1" card reads either Pixels or Arcseconds, depending on whether or not the "SCALE=" option has been used.

Examples:

An example of a procedure using the PROF keyword:

  AXES 1 BOX=2
PROFILE 10 1 N=30 ITER=25 SCALE=0.54
ANNULUS 12 2 PROF

In this case, PROFILE is run to fit 30 elliptical isophotes to the image in buffer 1 whose centroid is found in BOX=2 using the AXES command. The azimuthally averaged radial surface brightness profile is then stored in buffer 10 (PROFILE also azimuthally averages along the best fit isophotes). The best fit elliptical isophotes from image 1 are then used to compute the radial surface brightness profile of the object in image 2 using ANNULUS, and that profile is stored in buffer 12.

A Note about selecting values of the "PA=" and "INC=" keywords:

A circular disk inclined to the line-of-sight of an observer appears in projection as an ellipse with its major axis rotated by some amount from vertical. It is the usual convention in astronomy that position angles on the sky are measured from North towards the East, with the usual orientation being North Up, and East to the Left.

The ANNULUS command defines the position angle in a similar fashion. The position angle for the "PA=" keyword is the angle between the major axis of the projected ellipse and a vertical line passing through the center of the ellipse (i.e., along the direction of a single image column). The angle is measured from the top of the image towards the left in a counterclockwise sense. If the image is oriented so that North is Up, East is to the Left, then the position angle for ANNULUS is the same as the conventional position angle.

The angle of inclination, i, is defined as the angle between the line-of-sight from the observer to the center of the disk, and a line perpendicular to the plane of the disk. For a face-on disk, the inclination is 0 degrees, and for a perfectly edge-on disk, the inclination is 90 degrees.

This angle is related to the more easily measured axial ratio of the ellipse. If "a" is the length of the major axis of the ellipse, and "b" is the length of the minor axis, then the inclination "i" is given by

   cos(i) = b/a

It may also be related to the eccentricity of the ellipse, "e", such that
   e = \sin(i).

NOTE: For ANNULUS, i must always be LESS than 90 degrees. For cases where i = 90 degrees, use the VISTA command CUT instead.