Fractal Theory

KOSMA-\(\tau\) Properties

It is imperative to understand the functionality and assumptions used in KOSMA-\(\tau\) when using kosmatau3d [1]. KOSMA-\(\tau\) is an isotropically-radiated, spherically symmetric PDR code. It is maintained by Markus Röllig at the Universität zu Köln, and utilises chemical models from Amiel Sternberg at Tel Aviv University.

Density

It assumes a piece-wise density profile approximating that of a Bonnor-Ebert sphere, that is, a gaseous sphere in pressure equilibrium with its environment. For the total hydrogen surface density of the PDR \(n_s\), we define the density profile as,

\[\begin{split}n_\mathrm{H, cl}(r) = n_\mathrm{s} \left\{ \begin{aligned} \left( \frac{r}{r_\mathrm{cl}} \right)^{-\gamma} & \hspace{0.5cm} & r > r_\mathrm{core} \\ \left( \frac{r_\mathrm{core}}{r_\mathrm{cl}} \right)^{-\gamma} & \hspace{0.5cm} & r \leq r_\mathrm{core} \end{aligned} \right. \hspace{1cm} ,\end{split}\]

for a given core radius \(r_\mathrm{core}\) and powerlaw index \(\gamma\). The for the KOSMA-\(\tau\) grids in kosmatau3d, we use the values \(0.2 r_\mathrm{cl}\) and \(1.5\), respectively. With this definition, we can also write the total hydrogen number as,

\[\begin{split}N_\mathrm{H, cl} &= \int_0^{r_\mathrm{cl}} \mathrm{d}r\, 4 \pi\, r^2\, n(r), \\ &= \frac{4 \pi}{3 - \gamma}\, n_\mathrm{s}\, r_\mathrm{cl}^3 \left( 1 - \frac{\gamma}{3} \left( \frac{r_\mathrm{core}}{r_\mathrm{cl}} \right)^{3-\gamma} \right) \hspace{1cm} .\end{split}\]

For \(\gamma\! >\! 0\), the number density increases towards the core, and is constant and maximal for \(r\! <\! r_\mathrm{core}\) with corresponding density \(n_\mathrm{core}\). It is also possible to derive the total hydrogen number as a function of density, but the piece-wise nature of our density profile means we will need to separate our equation for \(n\! <\! n_\mathrm{core}\) and \(n\! =\! n_\mathrm{core}\). Momentarily neglecting the core, so \(n\! \in\! \left[ n_\mathrm{s}, n_\mathrm{core} \right)\), we can write the dependence of the total number of hydrogen atoms as a function of density:

\[\begin{split}\frac{\mathrm{d}N_\mathrm{H, cl}}{\mathrm{d}n} &= \left( \frac{\mathrm{d}n_\mathrm{H, cl}}{\mathrm{d}r} \right)^{-1} \frac{\mathrm{d}N_\mathrm{H, cl}}{\mathrm{d}r} \hspace{1cm} , \\ &= \left( -\gamma\, r_\mathrm{cl}\, n_\mathrm{s} \left( \frac{n}{n_\mathrm{s}} \right)^{\frac{\gamma + 1}{\gamma}} \right)^{-1} 4\pi\, r_\mathrm{s}^2 \left( \frac{n}{n_\mathrm{s}} \right)^{- \frac{2}{\gamma}} n_\mathrm{s} \frac{n}{n_\mathrm{s}} \hspace{1cm} , \\ &= - \frac{4\pi\, r_\mathrm{cl}}{\gamma} \left( \frac{n}{n_\mathrm{s}} \right)^{-\frac{3}{\gamma}} \hspace{1cm} ,\end{split}\]

where we have expressed the radius as a function of density. For \(n_\mathrm{H, cl}\! =\! n_\mathrm{core}\), we can simply perform a spherical integration with a constant density to derive the total number of hydrogen atoms. Since the core has constant density, we can write the final form of the density dependence of the total number of hydrogen atoms:

\[\begin{split}\frac{\mathrm{d}N_\mathrm{H, cl} (n)}{\mathrm{d}n} = \left\{ \begin{aligned} \frac{4\pi\, r_\mathrm{cl}}{\gamma} \left( \frac{n_\mathrm{cl}}{n_\mathrm{s}} \right)^{-\frac{3}{\gamma}} & \hspace{0.5cm} & n_\mathrm{s} < n < n_\mathrm{core} \\ 0 & \hspace{0.5cm} & n = n_\mathrm{core} \end{aligned} \right. \hspace{1cm} .\end{split}\]

Here we have removed the negative sign from the derivative since to be valid for increasing density. The density probability distribution function (PDF) for the spherical clump is defined as,

\[\mathcal{P}_\mathrm{cl}(n) \equiv N_\mathrm{H, cl}^{-1} \frac{\mathrm{d}N_\mathrm{H, cl} (n)}{\mathrm{d}n} \hspace{1cm} .\]

Using the density profile of the KOSMA-\(\tau\) clumps, as well as manually integrating the core to derive its probability, we obtain,

\[\begin{split}\mathcal{P}_\mathrm{cl}(n) = N_\mathrm{H, cl}^{-1} 4\pi r_\mathrm{cl}^3 \left\{ \begin{aligned} \frac{1}{\gamma} \left( \frac{n}{n_\mathrm{s}} \right)^{-\frac{3}{\gamma}} & \hspace{0.5cm} & n_\mathrm{s} < n < n_\mathrm{core} \\ \frac{1}{3} \left( \frac{n_\mathrm{core}}{n_\mathrm{s}} \right)^{\frac{\gamma - 3}{\gamma}} & \hspace{0.5cm} & n = n_\mathrm{core} \end{aligned} \right. \hspace{1cm} .\end{split}\]

How this is utilised in kosmatau3d for the fractal ISM will soon be explained in the Ensembles section.

Far-UV radiation

Currently [2] KOSMA-\(\tau\) uses a modified Draine spectrum for the spectral energy distribution (SED) incident on the clump surface, which can be scaled by a given factor. We therefore denote the far-UV radiation as \(\chi\) in units of \(\chi_\mathrm{D}\).

More information about the various far-UV spectra will be provided here upon the initial publication for kosmatau3d.

Ensembles

Here I will soon explain the mathematics and statistics associated with groups of clumps in an ensemble. Most of this information will be placed here after the publication of Yanitski et al. (2024).

The basis of the fractal approximation of the ISM is the clump mass distribution:

\[\frac{\mathrm{d} N_\mathrm{cl}}{\mathrm{d} m_\mathrm{cl}} \propto m_\mathrm{cl}^{-\alpha} \hspace{1cm} ,\]

where \(N_\mathrm{cl}\) is the number of clumps of a given mass \(m_\mathrm{cl}\) and \(\alpha\) is the index of the clump mass distribution. This distribution is bourne out of turbulence, and may vary between star forming regions. The value we used in the Milky Way is \(1.84\), which was determined from the Polaris Flare (Heithausen et al. 1998).

We combine the clump mass distribution with a mass-size relation,

\[m_\mathrm{cl} \propto r_\mathrm{cl}^\varpi \hspace{1cm} ,\]

where \(r_\mathrm{cl}\) is the radius of the clump and \(\varpi\) is mass-size index (determined to be \(2.31\) in Heithausen et al. 1998), in order to approximately derive the clump surface density distribution.

Todo

This derivation is being completed at the moment. Please be patient :-)

References

Heithausen, A., Bensch, F., Stutzki, J., Falgarone, E., & Panis, J. F. 1998 Astronomy & Astrophysics, 331, L65

Stutzki, J., Bensch, F., Heithausen, A., Ossenkopf, V. & Zielinsky, M. 1998, Astronomy & Astrophysics, 336, 697

Yanitski, C. N., Ossenkopf-Okada, V., & Röllig, M. 2024, in preparation

Footnotes

[1]

The spherical KOSMA-\(\tau\) PDR models are referred to as clumps in the context of kosmatau3d. The distinction must be made that while the clumps referenced in this documentation are the astronomical clumps that will eventually collapse into stars, they are approximately in hydrostatic equilibrium. Thus we are able to compute an instantaneous synthetic observation. kosmatau3d is a fractal model to simulate the inhomogeneous structure of the ISM using a multitude of these smaller clumps (see eg. Stutzki et al. 1998).

[2]

At some point in the future this will be extended to utilise a user-defined SED, but currently there is nobody developing this. It is particularily important in order to use KOSMA-\(\tau\) to model the X-ray dominated regions (XDRs) around active galactic nuclei (AGNs).