v(x,y) = VSYS + VROT * cos(theta) * sin(INC) - (x-XPOS) * sin(PA) -/+ (y-YPOS) * cos(PA) ? where: cos(theta) = ----------------------------------------- r - (x-XPOS) * cos(PA) - (y-YPOS) * sin(PA) and: sin(theta) = ----------------------------------------- r * cos(INC)where the radius

r = sqrt( (x-XPOS)**2 + (y-YPOS)**2/cos(INC)**2 )

In the above formula *v(x,y)* denotes the radial velocity at rectangular
sky coordinates *x* and *y*, *VSYS* the systemic velocity, *VROT* the rotational
velocity, *INC* the inclination angle and *theta* the azimuthal distance from
the major axis in the plane of the galaxy. In itself, *theta* is a function
of the inclination (*INC*) and the position angle (*PA*) of the major axis.
*XPOS* and *YPOS* denote the position of the rotation center in pixels w.r.t.
0,0 being the lower left corner of the map. *rotcur* can fit for each ring
the 6 parameters *VSYS*, *VROT*, *INC*, *PA*, *XPOS* and *YPOS*, though any combination
of them can be fixed (see **fixed=**). The position angle *PA* of the major axis
is defined as the angle, taken in anti-clockwise direction between the north
direction on the ‘‘sky’’ and the major axis of the receding half (positive
radial velocity) of the galaxy.

Values equal the undefined value (currently
0.0) are ignored in the fit. See *ccdmath(1NEMO)*
or *fitsccd(1NEMO)*
on how
to create a velocity field with such undefined values.

By setting the keyword
**fitmode=** appropriately, *rotcur* can also be used to fit a model of pure
expansional velocities:

v(x,y) = VSYS + VROT * sin(theta) * sin(INC)use

For a detailed discussion of this method see K. Begeman, *Astr.
& Astrophys. * **223**, 47. (1989) and references therein.

See also *velfit(1NEMO)*
on the classic WWB73 (Warner, Wright and Baldwin, 1973, MNRAS, 163,163)
method.

**in**=*image_vel*- Input velocity field map, either in
*image(5NEMO)*format, or*table(5NEMO)*format. See also**imagemode=**below. No default. **radii**=*r0,r1,r2,...rN*- Inner and
outer radii for
*N*touching rings (in arcsec). Remember that the number of radii is hence one more than the number of rings. The**units=**keyword can be used to scale your physical units in the header to ‘‘human readable’’ units (arcsec). Radii must be either all increasing or decreasing. Note that the corresponding keyword in**GIPSY**has a different meaning, where they denote the ring centers, and a different keyword*WIDTHS=*controls the width of each ring. No default. **vrot**=*v1,v2,...vN*- Array of rotation (or expansion) velocities, one for each ring. If less than the number of rings is given, the trailing ones are filled with the last supplied value.
**pa**=*p1,p2,...pN*- Array of position
angles (degrees), one for each ring. As is the convention, this is the
PA of the receding side of the galaxy. See also
**vrot=**. **inc**=*i1,i2,...iN*- Array
of inclinations (degrees), one for each ring. See also
**vrot=**. **vsys**=*v0*- Systemic velocity. Only one number applies, though VSYS can vary accross rings.
**center**=*x0,y0*- Rotation center (grid units w.r.t. lower left corner being 0,0). Two numbers are required. Default: center of map.
**frang**=- free angle around minor axis
(degrees), in the plane of the galaxy, from which data is excluded from
the fit (thus the total cone size of ignored data around the minor axis
is 2*
**frang**) [Default:**20.0**]. **side**=- Choose the side of the galaxy to fit the
velocity field to. Valid options are
**receding**,**approaching**or**both**side(s). [Default:**both**]. **weight**=- Choice of geometric weighting function with which
points are weighed into the least squares solution as a function of galactic
angle away from the major axis. Valid options are:
**uniform**,**cosine**, and**cos-squared**. [Default:**cosine**]. **fixed**=- List of parameters, separated by commas,
to be kept fixed during the fit. Choose any of the following:
**vsys, vrot, pa, inc, xpos, ypos**, although at least one parameters should be kept free. [Default: none, i.e. all parameters free. **ellips**=- The names of two parameters
for which to calculate an error ellips. (see
**fixed=**). For the two parameters it shows the major and minor axis, position angle of the one sigma deviation ellipse. [Default: not used] **beam**=- The beam size (FWHM) for beam correction. One or two numbers required. Currently these are only used to correct error bars for the number of independant points per beam. If not given, each point is assumed independant. [no correction].
**dens**=*image_den|true|false*- Image containing
the density. From this local derivatives
*(dN/dx)/N*and*(dN/dy)/N*are computed numerically, and used for an estimate of beam smearing corrections. Note that for bizarre reasons this keyword can be used to trigger reading a weight column in tabular input mode (Bimagemode=). [Default: required if a beam is supplied]. **tab**=- If specified, this output table is used in
*append*mode! This table can be piped through*tabcomment(1NEMO)*and fed to*ccdvel(1NEMO)*to create model velocity fields. [Default: not used]. **resid**=- If specified, this output will either contain a residual image (OBS-FIT) if the input file was an image, or a table with X, Y, Velocity and Residual Velocities in each ring. Each ring will be separated from the previous one with a simple commented line specifying which ring it refers to. [Default: not used].
**tol**=- Tolerance for convergence of
*nllsqfit*[Default:**0.001**]. **lab**=- Mixing
parameter for
*nllsqfit*[Default:**0.001**] **itmax**=- Maximum number of allowed
*nllsqfit*iterations [Default:**50**] **units**=- Units of input axes for radius
and velocity. Valid options are
**deg, arcmin, arcsec, rad**for radius. A numeric value can also be given, in which case your image pixel separation from the image header is multiplied by this number to get to the ‘‘arcsec’’ that will be quoted in the tables. The units for velocity can only be numeric, and will be the factor by which the velocities in the map are multiplied. [Default:**deg**] **blank=**- Value of the blank pixel that needs to be ignored.
[Default:
**0.0**]. **inherit=t|f**- Logical denoting if the initial conditions for
subsequent fitted rings should be inherited from the previous successfully
fitted ring. The fixed parameters keep of course their fixed value. [Default:
**t**] **reuse=t|f**- Reuse pixels between rings. If neighboring rings have a different
geometries, it can occur that pixels will be reused. This flag will prevent
that. Note, if you don’t reuse pixels, rotcur will more likely produce a
different rotation curve if you start at outer rings,e.g. radii=100:0:-10.
[Default:
**t**] **fitmode=****cos|sin****,1****nsigma=**- Reject outlier points will fall outside nsigma times the dispersion away from the mean velocity in a ring. By default, it will not reject any outliers.
**imagemode=t|f**- Image input file mode? By
default the input file is an image, alternatively a simple ascii table
with X and Y positions in columns 1 and 2, and radial velocities in column
3, and optional errors in the radial velocity in column 4 (activated by
setting
**dens=t**). [Default: t] **wwb73=t|f**- Use a simpler WWB73 linear method of fitting? [false]

BEGIN{count=0;line="";} { if ($1 == "radius"){ if (count != 0){ printf("%s %s0,rad,line); rad=$4; }else{ count=1; rad=$4; } }else{ line=$0 } } END{printf("%s %s0,rad,line);}

% set r=‘nemoinp 0:100:5‘ % set v=‘nemoinp 0:100:5 | tabmath - - "100*%1/(20+%1)" all‘ % ccdvel out=map1.vel rad="$r" vrot="$v" pa=30 inc=60 % rotcur in=map1.vel radii=0:100:5 vrot=0:100:5 pa=30 inc=60 vsys=0 tab=map1.rotcur units=arcsec,1 % head map1.rotcur radius systemic error rotation error position error inclination error x-position error y-position error velocity velocity angle angle of center of center (arcsec) (km/s) (km/s) (km/s) (km/s)(degrees) (degrees) (degrees) (degrees) (grids w.r.t. (0,0)) (grids w.r.t. (0,0)) 7.50 0.00 0.44 27.08 0.50 30.45 1.09 59.72 2.47 63.50 0.20 63.50 0.30 96 12.50 -0.00 0.24 38.54 0.28 30.66 0.41 59.27 0.94 63.50 0.13 63.50 0.20 150 ... % tail map1.rotcur average inclination : 59.93 ( 0.043) degrees average position angle : 30.05 ( 0.044) degrees average systemic velocity: -0.00 ( 0.001) km/s average x-position : 63.50 ( 0.000) grids average y-position : 63.50 ( 0.001) grids

Errorbars quoted
in the table are only an estimate since the beam size is not known. Multiply
these numbers by the square root of the number of pixels per beam to get
a more realistic estimate. If the beamsize, **beam=**, is given, the formulae
of Sicking (1997) is used to correct the errors for:

factor = sqrt(4.PI.B_x.B_y/(D_x.D_y))

Sign of the pixelsize *Dx,Dy* in the CCD header is ignored, and an astronomical
image is assumed. See also the **reuse=** keyword.

For calculations of residuals in overlapping rings (e.g. warps) only the last ring velocity will be used.

TiRiFiC: http://www.astron.nl/~jozsa/tirific/index.html pPXF: http://www-astro.physics.ox.ac.uk/~mxc/idl/#ppxf Begeman(1989): 1989A+A...223...47B GBKFIT: http://supercomputing.swin.edu.au/projects/gbkfit/

19/jul/83original programKGB 9/mar/85revision of programKGB 23/may/86migrated to VAX-VMSKGB 27/nov/88UNIX versionpjt 8-feb-91flushed buffers ROTCUR.DAT each writepjt 30-apr-91moved to NEMO into Cpjt 10-sep-91documentation improvedpjt 17-oct-91added Npts to table outputpjt 21-may-92added Bob Gruendl’s rotcur awk scriptsPJT 12-jun-92added inherit=t as defaultPJT 13-aug-92implemented fitmode= without XPOS,YPOSPJT 15-oct-99compute residuals and add resid=PJT 14-mar-01V2.5: clarifications, added nsigma=PJT 9-may-01V2.6a: corrected error correction factorPJT 10-aug-01clarified some differences between NEMO and GIPSY versionsPJT 26-jan-02Added unit (scale factor) for velocity tooPJT 26-jun-02V2.8: added tabular input for irregular spaced data, fixed examplePJT 11-sep-02V2.9: implemented map residual velocity fieldPJT 30-jan-03V2.10: allow tables to use error in velocityPJT 2-jun-04V2.12: finally implemented the reuse= optionPJT