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Getting this plot using the BE_Field in AstroBEAR

As a measure of magnetic field strength in the pdf, I am going to use the magnetic pressure.

Magnetic pressure is an energy density, and in free space in cgs, is given by:

P_B = \frac{B^2}{8 \pi}

In the code, there is a field called 'BE_field' which is the magnetic energy (or pressure) field, given by:

BE\_field = \frac{1}{2} (B_x^2 + B_y^2 + B_z^2)

The B's in the above are in computational units (cu), implying -

BE\_field = \frac{1}{2} (||\vec{B}(cu)||^2) \equiv \frac{1}{2} B^2 (cu)

= \frac{1}{2} \frac{B^2(cgs)}{Bscale^2}

where

Bscale = \sqrt{4\pi *pscale}

Putting this into BE_field gives,

BE\_field = \frac{B^2(8\pi)^{-1}}{pscale} = \frac{P_B}{pscale} = P_B(cu)

aka - scaled magnetic pressure.

Therefore - using BE_field as is for my pdf - means I will be plotting scaled magnetic pressure vs. scaled number density.

What is pscale?

pscale = nscale(cm^{-3})*K_B(cgs)*tempscale(K)

For my problem set-up,

nscale=tempscale=1

which means -

P_B(cu) = \frac{P_B(cgs)}{nscale*K_B*tempscale}

which implies

P_B(cu)*nscale*tempscale = P_B(cu)[\frac{K}{cm^3}] = \frac{P_{B}(cgs)}{K_B}

The units of P(cgs) are Kb(K)/cm³ (from ideal gas law). This means, the output from the code is equivalent to P(cgs)/Kb and this ratio has units of K/cm³.

That is, I am free to either call the quantity,

P_B(cu)(K/cm^3) ~or~ \frac{P(cgs)}{K_B}[\frac{K}{cm^3}]

The right hand quantity seems more intuitive, so will go with that. If someone were to instead plot the left hand quantity — hopefully a reader would recognize the lack of the boltzmann constant and realize that it was in fact a scaled quantity being plotted.

This is plotted against number density, which again given nscale = 1, is already in units of cm^-3.

Code

The difference between binning by mass and by volume