Chapter 1
Introduction

A "circumstellar disc" is the disc or torus, which can surround a star during both its formation and evolutionary stages. The disc is typically made up of gases and dusts. We shall primarily concern ourselves only with the dust grains within the disc. Radiation from the star must first travel through the disc before we can detect it. The path of radiation through the disc is not a simple one; its way is impeded by the grains and the interactions it suffers. Having travelled through the disc the escaping radiation has become attenuated, and therefore will have a different appearance to when it entered the disc. This project intends to computationally model circumstellar discs using the processes of radiation transport and the microphysics of dust grains. What we know about dust is based mainly on observations of its interaction with electromagnetic radiation. The interactions that are easiest to detect are scattering and absorption of starlight by dust grains and the emission of radiation by the grains themselves. To define the difference between scattering and absorption/emission, we must recall the conservation laws of quantum mechanics:

• In an absorption event, we can define a before stage where there is a dust grain and an incident photon, and we can define an after stage where only the grain remains, in an excited or heated state. The energy from the photon from has gone to heat the grain. Energy conservation, before and after the interaction, is satisfied.
• In an emission event, we define a photon-less before stage which involves a heated grain and an after stage which has a cooled-grain and an outgoing photon, the photon being produced from the energy lost by the grain whilst cooling. Energy conservation laws are satisfied. The outgoing photon need not be singular; an absorbed photon may be re-emitted as several lower energy photons as long as there is no net change in the energy of the system.
• In a scattering event, there is no stage at which we can define a photon-less system. The before stage is a grain and an incident photon, while the after stage is a grain and an outgoing photon. In a scattering event, the outgoing photon is identical to the incident photon in every way except direction. Energy conservation laws are satisfied. The direction of scattering is defined by the microphysics of the dust grain.

A common grain system configuration involves a central star surrounded by an envelope of dust. This envelope will usually take the form of a disc or torus. The degradation of electromagnetic radiation caused by the dust takes following form:

Radiation emitted by the star in the form of photons, typically in the ultra-violet or visual part of the spectrum, is scattered and/or absorbed by the grains. Dust grains tend to absorb high energy, short wavelength radiation and emit low energy, long wavelength radiation . Most other radiation is scattered. On absorption, a high-energy photon (ultra violet or visual) is converted into several low energy photons (infrared or sub-millimetre). This conversion and re-emission must take place since the grain has absorbed energy from the incident radiation. Failure to do so will result in the continuous heating of the grain, which is not observed. It is the counterbalance of these two processes, absorption and emission, which determines the temperature of the dust grain.

1.1 The Micro-Physics of Dust

If a grain is exposed to a parallel beam of monochromatic radiation with wavelength and number-flux and gives the rate at which these photons are absorbed by the grain, where is the absorption cross-section.

Equally, gives the rate at which these photons are scattered by the grain, where is the scattering cross-section.

Extinction is the combined effect of absorption and scattering and measures the capacity of a grain to remove radiation from a parallel beam. The total cross section for extinction is, .

The albedo is the ratio of scattering to extinction, it determines whether an incident photon will be absorbed or scattered from a grain on interaction.

When the grain scatters all light, if all light is absorbed.

There is no correlation between the direction of an incident photon, which is absorbed, and the direction of the associated emitted photons. However, photons that are scattered are subject to a characteristic "g" of the dust grain. The mean scattering cosine is a weighted mean of over all solid angles.

The value of ranges from -1 to 1, corresponding to spherically symmetric scattering and corresponding to uniquely forward scattering. The parameter is a measure of the forward or backward-scattering properties of the grain. So in terms of ;

• Predominantly forward scattering.
• Predominantly backward scattering.
• Symmetrical scattering (not necessarily isotropic).

The scattering phase function:

gives the probability that the photon is scattered through an angle between and .

1.2 Dust Temperatures

Of all the processes which lead to an energy exchange between a grain and the InterStellar Medium (ISM), it has been argued, by Van de Hulst (1946), that the dominant roles in determining the temperatures of the dust grains are radiative processes.

Other contributions can generally be neglected. The temperatures of a dust grain is determined by a balance between radiation absorption (heating, H) and the rate at which it emits radiation (cooling, C). A grain’s cooling rate is given by:

where is the emission efficiency. From Kirchhof’s law of radiation, the emission efficiency is equal to the absorption efficiency. This can be approximated by:

where is the geometric cross-section of a grain, and is the mean frequency of the visual pass-band. Substituting EQUATION 5 into EQUATION 4, we obtain:

where

is a constant, and the integral within has come from making the substitution .

A grain does not radiate like a blackbody, ; its emission efficiency is not 100% => at all . This is the reason why a grain’s cooling rate depends on the fifth or sixth power of the temperature:

The reason for this extra term, , is that the mean frequency at which a grain radiates is proportional to its temperature,

Therefore, from EQUATION 5 the mean emission efficiency is proportional to the ^th power of the temperature:

A dust grain’s heating rate in the general interstellar radiation field is:

where is the mean temperature of the general stellar radiation field in the ISM and is the amount by which the this radiation field is weaker than a blackbody radiation field, at temperature . Putting C=H, we obtain:

near a star can be much higher than that in the general interstellar medium. At a distance from a star of radius and surface temperature , the monochromatic flux is:

the analogous heating rate is then:

this should be compared with EQUATION 11. Putting C=H we obtain:

For an O5 star, at a distance , , for and for .

---

The temperature of a grain is dependent upon its lattice vibrations, the excitation of the internal vibrational modes of the grain. Small grains can have thermal capacities so small that if an individual grain absorbs a single ultraviolet photon its temperature will increase discontinuously; this implies the grain has no well-defined equilibrium temperature.