IR Modelling of Coalsack

From Tauwiki

1 IRAS Image of Coalsack
2 Physical Model
3 Overall Procedure
4 Present Procedures
5 Things to do:
6 Important science points.
7 Formulae used
8 References
9 Useful (!?) links

[edit] IRAS Image of Coalsack

[edit] Physical Model

The Coalsack Molecular cloud and the two dust sheets in the line of sight towards it are simulated by giving a dust distribution in 1pc^3 spatial bins. The dust distribution in the molecular cloud is proportional to number density of CO, as given by Corradi et al. The two sheets in front are assumed to be at 60 and 110 pc, with thickness of 1pc each. Look at the wording in the paper carefully. I don't believe this is exactly what is said.

The dust grains are assumed to be non-porous spheres made up of carbonaceous and silicate material. The MRN size distribution of grains is assumed. The size distribution is a power law with index of -3.5 and grain sizes ranging from 0.005 micron to .25 microns. An albedo of 0.6 and a scattering phase function asymmetry factor of 0.3 is used. The Henyey Greenstein scattering function is used to calculate the intensity of scattered light.

Brightest stars in UV wavelength range are assumed to be the source of heating for the dust grains. As shown by Purcell (1976) absorption of UV Photons gives rise to temperature fluctuations in small grains(radius <= 0.005 microns). Cooling of these grains gives rise to emission in 1-25 Micron wavelength range. Draine and Anderson(1985) show temperature distribution function for various sizes. Draine and Li (2001) give exact statistical treatment of vibrational energies of grains.

The heating of large grains(radius > 0.01 microns) gives rise to steady equilibrium temperatures, given by the energy balance equation. These grains emit blackbody radiation given by Planck distribution. In the present study we limit to grains of size 0.1 microns. The IRAS response functions are convolved with blackbody emission of these grains to find out fluxes in 60 and 100 micron bands, for comparison with IRAS images.

[edit] Overall Procedure

1. Calculate energy absorbed by each individual grain in 1pc^3 space, from the scattering model. For this, store energy absorbed in each scattering, and divide by the number of grains.

2. For each grain size, find out the temperature of the grain, for given absorbed energy, using energy balance equation.

3. Each grain emits a Planck distribution with that steady state temperature. Calculate brightness, MJy/Sr in 60 and 100 micron IRAS bands from each spatial bin. For this, we need to multiply by grain number density to the band flux generated by single grain.

Previous derivations are being put up the derivations page and IR derivation page.

One page has been created to track the overall Modelling progress.It contains the current form of complete derivation.

Current status is updated in the page on Modelling results.

[edit] Present Procedures

1. For obtaining energy stored in single dust grain:

Verbally,

Energy stored in single grain = intensity of UV radiation * (1- albedo) * energy of UV radiation * ratio of number of photons emitted by star and number of photons used in the simulation * scalling factor for entire spectrum of star

$E_g = I \times (1-a) \times \frac{hc}{\lambda} \times f_{spectrum} \times \frac {FluxAt1117A}{N_{phot}}$

2. For obtaining temperatures of dust grains:

A perfect spherical black body will have an equillibrium temperature $T b$ given by $U = \frac {4\sigma}{c} T_b^4$ , where U is the energy density of photons in ISRF.

For a general spherical dust grain of radius $a \approx 0.1 \mu m$ , the power absorbed from the ISRF is

$W_{abs}=c(\pi a^2 )\int Q_{abs}(\lambda) u_{\lambda} d\lambda$

The power radiated by the grain is

$W_{rad}=4\pi a^2 \int Q_{em} (\lambda) B_{\lambda} (T_d) d\lambda$

where $Q e m (λ)$ is the efficiency factor for emission from the grain(usually termed emissivity) and $B λ (T)$ is the Planck function. By Kirchoff's law, $Q a b s = Q e m$ . Equating the rate of energy gain and loss,

$\int Q_{\lambda}u_{\lambda}d\lambda = \frac{4\pi}{c}\int Q_{\lambda} B_{\lambda}(T_d)d\lambda$

If $Q λ = 1$ at all wavelengths, then this equation reduces to simple equation of energy proportional to forth power of temperature.

Following Draine and Lee (1984), $< Q e m >$ is termed as Planck Averaged emissivity for Spherical grains. It is defined by(eqn 6.1 of DL1984)

. . . [1]

I presume this definition comes from taking the ratio:

$<Q(grainsize,T)> = \frac{\int Q_{\lambda}u_{\lambda}d\lambda}{\int u_{\lambda}d\lambda}$

Since, then the energy balance equation can be written as (eqn 6.2 of DL1984)

. . . [2]

Prototype size of dust grain is to be taken as .1 micron.

This is the same procedure that I had followed for calculating the temperature due to InterStellar Radiation Field, which I had used for comparing temperatures due to our ISRF and that of Mathis et al.

In the LHS of this eqn, the input is ISRF. In our simulation, input is UV radiation getting absorbed. The Q_em factor here is dependent on size of the grain. So, for a given size, I tabulate the RHS for all possible temperatures, and find out the grain temperature by comparing this list of RHS values with energy absorbed. So, at a time I take one grain size into consideration.

3. For obtaining Planck Distribution for individual dust grain temperatures:

Using the numerical constraint given in planck.pro , a subroutine is written in C. From Rybicki and Lightman, we note that PI * Intensity is Flux from a spherical source of isotropic brightness at the surface of the source. Since we need Intensity, removed the factor of PI.

4. For calculating IRAS band fluxes

12,25,60 and 100 micron IRAS band fluxex are calculated by multiplying the IRAS filter response with the Planck distributions. The webpage on IRAS documentation describes the response function tabulated on the specral responce webpage.

Verbally,

band_intensity = Integral of (Planck distributio at T_g * IRAS reponse) over entire wavelength range * number of dust particles per 1pc^3 bin

5. For comparison of 60 and 100 micron simulations with observed IRAS images.

I use IDL functions ADXY and XYAD to get the pixel to pixel comparison of the images generated by simulation with observed IRAS images.

6. About the solid angle calculations involved.

Rybicki and Lightman show the constancy of intensity in the units of erg/cm^2/s/A/sr as follows:

Suppose the emission is coming from an area dA1 and receiver area is dA2. Consider the energy carried by the rays passing through both dA1 and dA2. This is given by

$d E = I 1 d A 1 d t d Ω 1 d ν 1 = I 2 d A 2 d t d Ω 2 d ν 2$

Since $d Ω 1 = d A 2 / R 2$ and $d Ω 2 = d A 1 / R 2$ , we get $I 1 = I 2$ , where I is expressed in erg/cm^2/s/sr/hz. So if we are calculating the IR flux in MJy/Sr, it will be detected in the same units, and this quantity is a useful invariant along rays in space.

While I agree that intensity is invariant, you cannot measure it. One always measures energy and converts to other things.

Yes. IRAS documentation gives Level one data description. In the header description of the level one data, they say the flux is in W/m^2. Final IRAS data is in MJy/Sr which they must have obtained after suitable divisions of factors.

The procedure above also calculates first the energy absorbed by the grains, and then the temperatures of the grains and then the energy emitted by the grains, then the same detected at the observer.

[edit] Things to do:

1. Which component of dust sheet or molecular cloud gives rise to IR excess w.r.t. UV.

2. Calculation of scaling factor for Stars spectrum.

Scaling factor = integration of Planck distribution of star temperature / product of value at the wavelength of simulation and unit wavelength.

$f_{spectrum}=\frac{\int{B(\lambda)d\lambda}}{B(\lambda_0,T)\Delta\lambda}=\frac{2\pi^4 k^4 T^4 }{15 c^2 h^3 B(\lambda_0,T)\Delta\lambda} = \frac{\sigma T^4}{ \pi B(\lambda_0,T)\Delta\lambda}$

Replacing Planck distribution by Kurucz model.

3. Including grain size distribution.

[edit] Important science points.

1. How does one get an estimate of grain size and temperatures from 60 to 100 micron ratio.

2. Extra brightness in IR is a combination of both the dust density and presence of bright stars together, not any factor alone.

[edit] Formulae used

Conversion Formulae

$1 W / c m 2 / μ m = 10 18 (λ(μ m) 2) / 299.8 J y$

$107 e r g / s / c m 2 * 10 - 4 / A = 3.336 * 10 9 * (λ(μ m) 2) M J y$

$103 e r g / c m 2 / s / A = 3.336 * 10 9 * (λ(μ m) 2) M J y$

$1 e r g / c m 2 / s / s r / A = 3.336 * 10 6 * (λ(μ m) 2) M J y / s r$

The calculation of planck function:

$c 1 = 2 * h * c * c = 3.7417749 d - 5$ in CGS units

$c 2 = h * c / k = 1.4387687 d$ in CGS units

$v a l = c 2 / w [i n d e x] / t e m p$ w is wavelength.

$b b f l u x [i n d e x] = C 1 / (w [i n d e x] 5 * (e x p (v a l [i n d e x]) - 1.))$

Formulae for energy absorbed:

$e n e r g y [d u s t i n d e x] + = i n t e n s * (1 - a l b e d o) * 3 * 6.62 * 10 - 16 / 1117 * 10 7$ ergs ; where,

$h = 6.626068 \times 10^{-34} m^2 kg / s$

$λ = 1117 A = 1117 * 10 - 10 m$

$c = 3 * 10 8 m / s$

so $h c / λ = 6.62 * 3 / 1117 * 10 - 16 J o u l e s = 6.62 * 3 / 1117 * 10 - 9 e r g s .$

Formulae for energy balance used to calculate temperature:

$E g [i n d e x] = 4 * 5.67 * p o w (10., - 5) * 1 * Q e m * p o w (T [r e a d],4)$ ;

This is right hand side of Eq. 6.2 in reference DL1984. Q_em is Planck Averaged Emissivity as defined in Eq. 6.1 of the same reference.

Stephan Boltzmann constant used in above equation is

$= 5.67 * 10 - 8 W / m 2 / T 4$

$= 5.67 * 10 - 5 e r g / s / c m 2 / T 4$

Formulae for flux calculation:

$b a n d 100 + = (B B f l u x [i n d e x] + B B f l u x [i n d e x + 1]) / 2. * I R A S r e s p o n s e * 3.336 * p o w (10,6) * p o w ((λ[i n d e x] + λ[i n d e x + 1]) / 2,2)$ ;

$f l u x = b a n d 100[i n d e x] * d u s t a r r [i e + j e * d u s t . n x + k e * d u s t . n x * d u s t . n y] * 1 p c 3 (c m 3)$

[edit] References

1. Draine and Lee, Optical Properties of Interstellar Graphite and Silicate grains Draine, Lee. The Astrophysical Journal, vol 285,89-108,1984.

2. Draine's webpage on properties of interstellar dust has files tabulating the various cross sections.

3. Corradi et al, MNRAS, 347, 1065,2004.

4. Rybicki and Lightman, Radiative processes in Astrophysics.

5. Shalima and Murthy, MNRAS, 352,1319, 2004.

6. Purcell, ApJ, 206, 685, 1976.

7. Draine and Anderson,ApJ, 292, 494, 1985.

8. Draine and Li , 551,807, 2001.

[edit] Useful (!?) links

1. Link for unit conversion.

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