Consider the equilibrium structure of a rotating, self-gravitating, isothermal gas cloud. Such an object obeys the partial differential equations.

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

Consider the equilibrium structure of a rotating, self-gravitating, isothermal gas cloud. Such an object obeys the partial differential equations:

where Φ = 0 at the origin. where:

• Φ is the gravitational potential;
• ρ is the density;
• P is pressure;
• G is the gravitational constant;
• cs is the sounds speed;
• and Ω is the angular frequency of

Work in cylindrical coordinates and assume that the cloud is axially symmetric about the z axis: so that r=r(r,z), P=P(r,z), and Φ = Φ (r,z). Work in units where 4πG = cs = ρc = Pc = 1, where ρc and Pc are the density and pressure at the centre of the cloud respectively.

Assume that the cloud is truncated by an external pressure 0 < Ps < 1, such that ρ = 0 when P< Ps. The medium external cloud is “hot” gas, with pressure Ps everywhere, effectively zero density, and no rotation.

Solid-body rotation:

 The simplest model has Ω = const, but this will lead to a net outward force – and severe numerical instability – when r is large. Highly pressure-truncated cores will be OK, but there will be numerical instabilities when the core is not very truncated.

Your task is to devise an iterative numerical scheme to solve our system of equations self consistently and provide a range of solutions for varying values of Ω and Ps. I suggest the following algorithm:

 

1. Assume an initial pressure-truncated density
2. Solve eq. 1 using any method discussed in class. (Successive Over Relaxation would also be a good choice. Please do not use MATLAB’s PDE )
3. Use eq. 4 to obtain a new estimate of ρ.
4. Exit if converged, else go to

Your writeup should include a grid of figures (I would suggest contour plots) showing how the density structure changes as Ω and Ps are varied. I suggest at least a 5x5 grid over a sufficient range of values to show how the structure changes. Some combinations of Ω and Ps will not permit a solution, due to instability effects.

Also, calculate the XX and ZZ components of the moment of inertia tensor, where Z is the symmetry axis. The ratio Izz/Ixx is a good way to describe the deviation of the object from spherical symmetry. Make a well-labelled contour plot of Izz/Ixx as a function of Ω and Ps.

Related Questions

. The fundamental operations of create, read, update, and delete (CRUD) in either Python or Java

CS 340 Milestone One Guidelines and Rubric  Overview: For this assignment, you will implement the fundamental operations of create, read, update,

. Develop a program to emulate a purchase transaction at a retail store. This  program will have two classes, a LineItem class and a Transaction class

Retail Transaction Programming Project  Project Requirements:  Develop a program to emulate a purchase transaction at a retail store. This

. The following program contains five errors. Identify the errors and fix them

7COM1028   Secure Systems Programming   Referral Coursework: Secure

. Accepts the following from a user: Item Name Item Quantity Item Price Allows the user to create a file to store the sales receipt contents

Create a GUI program that:Accepts the following from a user:Item NameItem QuantityItem PriceAllows the user to create a file to store the sales receip

. The final project will encompass developing a web service using a software stack and implementing an industry-standard interface. Regardless of whether you choose to pursue application development goals as a pure developer or as a software engineer

CS 340 Final Project Guidelines and Rubric  Overview The final project will encompass developing a web service using a software stack and impleme