How do I get flux density when I have mJy/beam , beam size and pixel size in an astronomical image I have a fits file of the galaxy m87 produced by SCUBA 2 and I am wondering how I obtain the fluxes at 450 and 850 microns. the output image gives pixel magnitudes in mJy/beam where I want to find eh flux for a certain part of the galaxy. I am as far as summit the values of all pixels within an aperture but I am not sure where to go from there to obtain the flux as there are more than one pixel within the FWHM. say I have a an aperture of radius 1.5 pixels that gives a sum of 6985.04885203 mJy/beam with a pixel size of 3.99999999 arc seconds and a beam size of 13 arc seconds how do I obtain the flux for that aperture at that wavelength?
In essence read of the peak flux value and if a point source this will be the same as your integrated flux.

to elaborate on this:

In this question your source appears to be comparable to your beam. so I would treat as a point source. - note you might also want to consider that many people apply a "match filter" to their data if dealing with faint point source emission - this smooths the map - increasing the effective beam size within the map and also there may be some flux loss (see for example: http://adsabs.harvard.edu/cgi-bin/bib_query?arXiv:1707.00990)

You will see this (mJy/beam to mJy) is also discussed here: http://www.eaobservatory.org/jcmt/help/ 

For a real point source, a peak brightness value reported in units of mJy is the same as a peak brightness value reported in mJy/beam.

But what happens if we have a map in mJy/beam and we want to obtain an integrated intensity value, a total flux value? We first sum up a number of pixels and now we want to get our units correct from mJy/beam to mJy…

Total Flux = flux summed over a number of pixels/(number of pixels in a beam)

Then your units are:

[mJy*pixels/beam] / [pixels/beam] = [mJy].

Now we know that For a Gaussian:

Beam Area =  2 × π × σ² [arcsec]

where the σ of the Gaussian beam can be calculated from the JCMT FWHM values at 850 and 450 microns (reminder the beam components are provided in Dempsey’s 2013 paper).

FWHM = 2 σ √(2 ln 2)   [arcsec]

So we can use the FWHM to obtain σ to calculate the Beam Area and report the beam area in terms of pixels:

number of pixels in a beam = Beam Area [arcsec] / (pixel length)²

Finally I should add you will also want to do an aperture correction - see Dempsey et al 2013's calibration paper if you are using an aperture of a different size than 60".