Modelling progress

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[edit] Calculation of dust grain temperatures.

[edit] Calculation of IR flux to compare with IRAS.

Consider the dust bins of size (1 parsec)^3. The number density is given by ρ(r) in number of dust grains per cm^3. Weingartner & Draine (2001) and Li & Draine (2001) have developed a carbonaceous-silicate grain model which successfully reproduces observed interstellar extinction, scattering, and infrared emission. Their grain model for R_v = 3.1 is considered to be appropriate for the typical diffuse HI cloud in the Milky Way.

The Draine's webapge [[1]] tabulates the following quantities:

lambda = wavelength in vacuo (micron)

albedo = (scattering cross section)/(extinction cross section)

<cos> = <cos(theta)> for scattered light

C_ext/H = extinction cross section per H nucleon (cm^2/H)

K_abs = absorption cross section per mass of dust (cm^2/gram)

<cos^2> = <cos^(theta)> for scattered light

Also for this Model,

1.870E-26 = M_dust per H nucleon (gram/H),

1.236E+02 = M_gas/M_dust for this dust model (assuming He/H=0.096).


Let the temperature of the dust grains be T. The blackbody spectrum of the dust grains will get modified by the cross sections. From the Kirchoff's law, the absorption cross sections are same as the emission cross sections. Using K_abs*M_dust for emission cross section, total amount of radiation emitted by the volume of 1parsec^3 is


I_{em}=\int B_{\lambda}(T) C_{abs}(\lambda)d \lambda \times N

This is energy emitted in erg/s/Sr. We need to multiply it by the solid angle of the detector to get the energy absorbed by the detector.

E_{det}=\int B_{\lambda}(T) C_{abs}(\lambda)d \lambda \times N \times \Omega _{det}= \int B_{\lambda}(T) C_{abs}(\lambda)d \lambda \times N \times \frac{A_{det}}{r^2}

Also, due to detector response for different wavelengths, which is given as dimensionless relative response on IRAS webpage, the above energy gets modified to

E_{det}=\int B_{\lambda}(T) C_{abs}(\lambda)d \lambda \times N \times \frac{A_{det}}{r^2}\times f_{IRAS}(\lambda)

This is energy absorbed in erg/s at the detector. To get the intensity at the detector in erg/s/cm^2/A/Sr, we divide by the respective factors for the detector:

I_{det}=\int B_{\lambda}(T) C_{abs}(\lambda)d \lambda \times N \times \frac{A_{det}}{r^2}\times f_{IRAS}(\lambda) \times \frac {1}{A_{det} \Omega_{cloud} \Delta\lambda}

=\int B_{\lambda}(T) C_{abs}(\lambda)d \lambda \times \rho(r) A_{cloud} dz \times \frac{A_{det}}{r^2}\times f_{IRAS}(\lambda) \times \frac {1}{A_{det} \frac{A_{cloud}}{r^2} \Delta\lambda}

=\int B_{\lambda}(T) C_{abs}(\lambda)d \lambda \times dz \times f_{IRAS}(\lambda) \times \frac {1}{\Delta\lambda}

where, \Delta \lambda = \int f_{IRAS}(\lambda) d\lambda = 332650 A

Finally we need to convert the intensity from erg/s/A/cm^2/Sr to MJy/Sr:

IRFlux=\int B_{\lambda}(T)K_{abs}(\lambda)m_{dust}f_{IRAS}(\lambda)d\lambda\times \frac{\rho(r)dz}{4\pi\Delta\lambda}\times 3.336 \times 10^6 \times (\lambda(\mu m ))^2

An estimate of the Intensity:

Wavelengths in Anstrom are 

     950000.  1.00000e+06  1.05000e+06  1.10000e+06  1.15000e+06

temperature = 15

bbflux in erg/s/cm^2/A/Sr = 

  6.34805e-08  8.13796e-08  1.00682e-07  1.20862e-07  1.41399e-07

IRAS response for these wavelengths is 

0.824000     0.947000     0.939000      1.00000     0.631000

Absorption cross sections in cm^2/H are 

5.61000e-25  5.08640e-25  4.60020e-25  4.17010e-25  3.77740e-25

the wavelength interval for sum is 50000.0

the total effective wavelength range is 217050.

wavelengths in microns are

95.0000 100.000 105.000 110.000 115.000

the product bbflux*response*k_abs*m_dust*delta/delta_lambda*3.336*1e6*wav^2*dz*rho is 

0.183171     0.271116     0.331628     0.421794     0.308279

total IRAS flux in MJy/Sr is 1.51599

Objective To get correct order of magnitude estimate of IR fluxes and to match the simulation output with IRAS observations.

Components

  1. Formulae describing basic physics
  2. Implementing Basic Physics for real situation, an analytical calculation of how much IR Flux to expect
  3. Integrating step 2 in the simulation
  • Understanding exact definitions of quantities in the simulation.
  • Formulating how physical formuale can be used in the simulation.
  • Ensuring that the formulae are correctly entered in the simulation and they give exact results.
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