Derivation of Radiation Field
By Elijah Bernstein-Cooper, February 25, 2016, 0 comments.

Table of Contents

Mathis Field

The following steps were outlined to me by Dr. Min Young-Lee who used such calculations in Bialy et al. (2015).

Given dust grain absorption efficiency and assuming the dust grain radiative cooling is in equilibrium with the dust grain heating from an incident Interstellar Radiation Field (ISRF), we can follow the steps from Draine’s textbook to model the temperature of a dust grain $T_D$ by

$\begin{equation} T_{D,Si} = 16.4 (a_{Si} / 0.1 \mu m)^{-1/15} * U_{M83}^{1/6} K \end{equation}$

for silicate grains and

$\begin{equation} T_{D,C} = 22.3 (a_C / 0.1 \mu m)^{-1/40} * U_{M83}^{1/6} K \end{equation}$

for carbonasceous where $U_{M83}$ is the ISRF by Mathis+83 integrated from 0.09 micron to 8 micron and $a$ is the dust grain radius. $U(M83) = 1$ corresponds to $2.2e-5$ W m$^{-2}$. This assumes the slope of the dust modified blackbody spectrum, $\beta = 2.0$.

Since both $T_D$ for each grain type are proportional to $U_{M83}^{1/6}$ we could determine an average equilibrium temperature for a typical grain composition, and solve for $U_{M83}$ solely as a function of $T_D$. Specifically we assume that $T_{D,Si} = T_{D,C}$.

Boulanger et al. (1996) derived an average temperature for dust grains in the solar neighborhood of $T_D = 17.5 K = T_{D,Si} = T_{D,C}$ using IR spectra. We can then solve for $a_{Si}$ and $a_C$ individually, leading us to the normalized relation between the $T_D$ and the radiation field, $U$, in units of $U_{M83}$:

$\begin{equation} U(U_{M83}) = (T_D / 17.5 K)^6 U_{M83} \end{equation}$

Habing and Draine Fields

Chapter 12.5 of Draine’s textbook outlines the steps to calculate the Habing $U_{H68}$ and Draine field $U_{D78}$ measurements of the UV energy density in the solar neighborhood. The Habing field integrates the radiation energy density for photon energies from 10 to 13.6 eV. The Draine field field from 6 to 13.6 eV. The Mathis field, $U_{M83}$ includes radiation with wavelengths between 2460 angstrom and 912 angstrom.

To compare the three fields, Draine defines a scalar between a measured radiation field $U(6-13.6eV)$ (between photon energies of 6 to 13.6eV), $G_0$, relative to the Habing field of $U_{H68} = 5.29 \times 10^{-14}$ erg cm$^{-3}$ by

$\begin{equation} G_0 = \frac{U(6-13.6eV)}{U_{H68}} \end{equation}$

For the Draine field, $U_{D78}$, $G_0 = 1.69$, thus $1.69 = U_{D78} / U_{H68}$ or rather the Draine field is 1.69 times stronger than the Habing field. For the Mathis field, $U_{M83}$, $G_0 = 1.14$, thus $1.14 = U_{M83} / U_{H68}.$ This leads us to the relationship between the Draine field and the Mathis field of $U_{D78} = 1.48 U_{M83}$, or rather the Draine field is 1.48 times stronger than the Mathis field.

Hence $U$ in units of $U_{D78}$, as a function of $T_D$, used in the Sternberg+14 and Krumholz+09 models is given by

$\begin{equation} U(U_{D78}) = \frac{(T_D / 17.5 K)^6}{1.48} U_{D78} \end{equation}$

Adjusting for Modified Blackbody Slopes

The proportionality between the dust temperature and the radiation field depends on the modified blackbody slope, $\beta$ and the blackbody flux dependence, giving $U \propto T_D^{\beta + 4}$. Thus if we have information about $\beta$ we should calculate our radiation fields in the following way:

$U$ in units of $U_{D78}$ used in the Sternberg+14 and Krumholz+09 models is given by

$\begin{equation} U(U_{D78}) = \frac{(T_D / 17.5 K)^{4 + \beta}}{1.48} U_{D78} \end{equation}$

Radiation Field Values

I created a map of the radiation field using the $T_D$ and $\beta$ Planck maps shown in Figure 1. I show the radiation fields for each cloud in Table 1.

Table 1

Table of median Habing and Draine radiation fields.

Figure 1

Radiation field of Taurus-California-Perseus region shown in units of Mathis fields.