Chapter 5
The model

The model under consideration was a central star with a circumstellar dust envelope surrounding it. While analytic models consider a star to have formed from a cloud of gas and dust of infinite radius, it was necessary, due to the nature of numerical modelling to limit the size of the dust cloud. For this reason an outer radius at which the envelope is considered to terminate was specified, as was an inner radius where the envelope begins. This inner radius was required because within a certain distance of the star no dust can exist due to the extreme temperatures vaporising all matter. The value of this inner radius is used as a normalising constant, and all radii within the program are measured in terms of it.

The equation originally used to simulate the distribution of dust around the star was:

,    EQ.32

• : The dust density.
• : A normalising constant equal to the dust density at the inner edge of the equatorial plane.
• : The distance under consideration.
• : A normalising constant, and is the distance from the central star to the inner edge of the dust cloud.
• : The angle between the position vector and the axis of symmetry.

Figure 9: The co-ordinate system.

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The disc geometry is characterised by both the radial power law index , which implies how fast the dust distribution falls off with radius, and the index , which tells the degree to which the dust distribution is flattened. ( is not related to that of EQUATION 5.)

The simulation of spherically symmetric dust shells was achieved by setting both and to 0, and the generation of full colour isophotal maps was achieved.

The architecture of the disc was to be controlled by the indices & however, it was found that regulation was hard to achieve, especially in terms of how to obtain a flattened disc-like appearance. It was decided that a function that more accurately controlled the "flatness" profile of the dust distribution was needed. The function that was introduced was the "step function", . The function was written in to the program in place of . It enabled the modelling of discs with a better-defined surface. FIGURE 10 shows how this was achieved.

Figure 9: The theroretical dust distribution.

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determines how rapidly the density falls near the surface of the disc, while is the half thickness of one hemisphere of the disc. The function takes the following form:

,    EQ.33

A & B are both constants which take the following form:

,    EQ.34

,    EQ.35

where:

,    EQ.36

,    EQ.37

and controls the amount of dust above the surface of the disc.

Most dust discs are not completely flat. They have a tendency to "flair" at the edges.

Figure 9: An image captured by the HST showing the flaring of a disc in the infra-red.

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To model this, a simple function which controls the top surface of the dense dust disc is added:

,    EQ.38

Figure 9: Schematic of how the flaring function will control the disc surface.

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The function simulates a parabolic shaped disc. , a function of , controls the collimation of the bipolar outflows exhibited by the star, a larger value of gives a more collimated jet. controls the flaring of the disc, a large value of means the disc is very flared at the edges.