Column Density Maps

Using projections operators in the code for column density, one is performing the integral operation,

\sum n_i dx = \bar{n} L_x

which has the rather strange units of computational number density times computational length. We can not just assume that the resultant product will have units of 1/computational length squared. This is because while nscale has units of 1/lscale³, it does not equal 1/lscale³,

nscale \neq lscale

but in fact equals,

nscale = rhoscale/X

where X is the mean molecular weight.

Therefore,

n(cu)*l(cu) = \frac{n(cgs)}{nscale}* \frac{l(cgs)}{lscale} \neq \frac{n(cgs)*l(cgs)}{ lscale^{-2}}

This means that we can not simply multiply our projected quantity by lscale^-2 to get units of cm^-2. Instead, we should multiply it by lscale*nscale. This will give us the correct units for column density, when lscale and nscale are in cgs.

Note, for my setup, lscale is the number of cm in 1 pc, and my nscale is 1. This means that computational density and length can be thought of as being in units of cm^-3 and pc. Thus the quantity the code gives for 'projected column density in computational units' is for all intents and purposes, in units of pc/cm³. To convert this to 1/cm², multiply by the conversion factor 1pc = 3.08567758 × 10¹⁸ cm (which is of course equivalent to multiplying by nscale and lscale as stated above).

Also, as an aside, when the scales are all set to 1 in the code, all computational units = cgs. Using scales that DNE 1 is equivalent to scaling the computational quantities away from cgs. When all of the scales = 1, nscale = rhoscale/X means that nscale is essentially Avogadro's number (over the mean molecular weight). Also, other scales in the code behave nicer than nscale in that they are interrelated directly.