# Beam Convolution

Dealing with data from different telescopes is a common activity for astronomers. Here is a rough method for convolving a beam (i.e. PACS, Herschel) to another beam (i.e. SCUBA-2 at 850um).

A very rough and ready method is to assume that both beams are Gaussian. Let’s say PACS has an FWHM of “A” arc-seconds and SCUBA-2 has an FWHM  of “B” arc-seconds. If “A” is larger than “B”, then you need to smooth the SCUBA-2 map using a Gaussian kernel of FWHM equal to sqrt(A*A – B*B) arc-seconds. If the SCUBA-2 map has a pixel size of Pa arc-seconds, then first convert the above size into pixels by calculating:

width = sqrt( A*A – B*B )/Pa

and then smooth the SCUBA-2 map using the “gausmooth” command in the starlink kappa package:

```kappa
```
`gausmooth in=<your scuba-2 map> out=<smoothed map> fwhm=<your "width" value>`

The smoothed map is put into the file specified by the “out” parameter. Alternatively, if “B” is larger than “A” smooth the PACS map inthe same way, using a width of:

width = sqrt( B*B – A*A )/Pb

where Pb is the pixel size in the PACS map.

You will need to know what the A and B values are (at 850 um “B” is about 13.5 arc-seconds). If your maps have point sources in them, then you could determine A and B by measuring the widths of the point sources in your maps. For instance, the “psf” command (“Point Spread Function”) in the starlink kappa package allows you to determine a mean beam shape from one or more point sources in an image. It does this by fitting a generalized Gaussian function to the mean radial profile of the indicated point sources.

The above assumes that both beam shapes are Gaussian. The SCUBA-2 beam shape is not quite Gaussian and so the above method can be improved, but it involves a lot more time and effort. You need first to get accurate models for the two beams (either as analytical functions or as 2D images), then you smooth the SCUBA-2 map using the PACS beam, and then smooth the PACS map using the SCUBA-2 beam. This approach requires no deconvolution, but results in maps that have lower resolution than either the PACS or SCUBA-2 maps. The kappa package includes the “convolve” command that will smooth a map using a beam shape specified as a 2D image (the kappa “maths” command can be used to generate a 2D image if your beam shape is expressed as an analytical expression). The details of this method depend on the form in which you obtain the beam shape information.