# Calculate the uncertainty in measured flux density from an integrated intensity map

I have some ALMA data, in the form of a spectral cube, which I have integrated along the velocity axis to create an integrated intensity ('moment 0') map.

The integrated intensity map shows emission from a protoplanetary disk. I measure the total flux density to be 3.0 Jy km/s. I would now like to attach some uncertainty to that result. I can measure the RMS, $$\sigma$$, in an emission-free region of the map, in units of Jy/beam km/s. I would like to know how to go from this, to an uncertainty in units of Jy km/s.

I begin by calculating the beam area:

$$\displaystyle \Omega_\text{beam} = \frac{\pi \theta_\text{maj}\theta_\text{min}}{4 \text{ln}2} \approx 0.07 \text{ arcsec}^2$$

I then calculate number of pixels per beam:

$$\text{pixel width} = 0.03 \text{ arcsec}$$
$$\text{pixel area} = (0.03 \text{ arcsec})^2 = 0.0009 \text{ arcsec}^2$$
$$\text{pixels per beam} = 0.07 \text{ arcsec}^2/0.0009 \text{ arcsec}^2 \approx 78$$

I then convert the measured RMS from Jy/beam to Jy/pixel:

$$\sigma_\text{Jy/beam} = 0.004 \text{ Jy/beam}$$
$$\sigma_\text{Jy/pixel} = \sigma_\text{Jy/beam} \; / \; 78 \approx 0.00005 \text{ Jy/pixel}$$

Finally, I use the following equation to calculate the uncertainty (from Appendix in Beltran et al. 2001)

$$\text{uncertainty} = \sigma_\text{Jy/pixel} \sqrt{A}$$

where A is the total area where emission is measured, in pixels. In my case, A=38531 pixels, giving:

$$\text{uncertainty} = 0.0005 \times \sqrt{38531} \approx 0.01$$

I don't trust this result, since there is a lot of noise present in the image, and the uncertainty calculated here is only around $$\pm 0.3\%$$

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