**by Paul Hellings,
6.00" by 9.00",
Softbound,
$19.95. **

Astrophysics With a PC is the first book to appear in the English language for amateur astronomers who want to explore astrophysics with a personal computer. Among the subjects covered are the morphology of comet tails, meteor dynamics, distance calculations of wide binary stars, polytropes, homogeneous stellar models, stellar atmospheres, the structure of white dwarfs, star formation in the galaxy, individual stellar orbits in the galaxy and cosmological models for the Universe. Each of these subjects is first introduced to illustrate its importance in the global framework of astronomy. Then the relevant formulae and physical processes are discussed with qualitative physical arguments. In subsequent sections, the author provides numerical expressions, flowcharts and sample IBM-PC programs in QuickBasic that demonstrate these processes. Softbound, approximately 250 pages.

**From the Sky & Telescope Review**

…A run-of-the-mill PC has more power than the most advanced "supercomputers" of 30 years ago. With the number-crunching capacity of today’s systems, even an amateur stargazer should be able to explore the equations professional astronomers use to understand the cosmos. This is the premise behind Paul Hellings’ effort to give people a hands-on tool for doing theoretical astrophysics.

According to Hellings, astrophysics is "the discipline which studies the internal structure and evolution of celestial bodies." His book is a short tour, oriented for the layperson, through selected parts of the theoretical universe.Astronomers tell their cosmic stories in the language of mathematics—that is, equations, more specifically nonlinear differential equations. The "differential" refers to equations from the mathematical arena of calculus. The "nonlinear" part means the equations can only rarely be solved with pencil and paper. They can however be solved with a computer by taking thousands or millions of tiny time steps. If you have a fast enough computer you can explore the stories of astrophysics by exploring the solutions to the equations.Hellings provides a concise and generally well-written introduction to several astrophysical stories in a language that walks a fine line between a popular account and a mathematical description.

Among the topics he includes are the shape of comet tails, the evolution of binary-star orbits, the structure of stars, and cosmology. In each section he describes the problem and formulates the equations in such a way that the problem can be solved using a computer program (pr presents these programs too, included on a floppy disk).

He then shows how the program can be used to explore the physics behind the problem.This book is very interesting and quite well done. However, I’m a computational astrophysicist and used to the language Hellings communicates with. I fear that this book will not make much sense to someone who is not fairly well versed in, at least, first-year college calculus. A good basis for first-year college physics is probably required too.

But if you have the knowledge under your belt, or are willing to learn it from a couple of introductory textbooks, then Hellings’ work will give you a grand tour through the world of theoretical and computational astrophysics. By reading the text and playing with the programs you should be left with a deeper and broader understanding of the hows and whys of what you see in the sky.

*Sky & Telescope*

**Table of Contents**

**1 Some Numerical Methods**

1.1 Introduction

1.2 Numerical Solutions of Differential Equations

1.2.1 Euler's Method

1.2.2 Midpoint Method (Cauchy Method)

1.2.3 Predictor-corrector Method (Heun Method)

1.2.4 Runge-Kutta Method of the Fourth-Order

1.2.5 Some Examples

1.2.6 System of Simultaneous First-Order Equations

1.2.7 Reducing a Second-Order Equation into Two First-Order

Equations

1.3 Computing Zeros of Functions

1.3.1 Newton-Raphson Method

1.3.2 Fixpoint Method

1.4 Numerical Computation of Integrals

1.4.1 Fitting a Parabola to Obtain a Crude Estimate

1.4.2 Simpson's Method

1.4.3 How to Program Simpson's Method

**2 The Morphology of Comet Tails**

2.1 Introduction

2.2 The Orbital Elements

2.3 Cometocentric Coordinates

2.4 The Syndynames and the Bessel-Bredechin Theory

2.5 Computational Method

2.6 Applications

2.7 The Program Listing

**3 Meteor Dynamics**

3.1 Introduction

3.2 Physical Background

3.3 Summary

3.4 Numerical Method

3.5 Applications

3.6 The Program Listing

**4 The Restricted Three-Body Problem**

4.1 Introduction

4.2 Some Astrophysical Applications

4.3 Mathematical Description

4.4 Numerical Method

4.5 Flowchart of the Program

4.6 Application: The Trojan Asteroids

4.7 Application: Orbit-Orbit Resonances

4.7.1 Nature of the Orbits

4.7.2 Practical Examples

4.7.3 The Case of Pluto/Neptune

4.8 The Program Listing

**5 Equipotential Surfaces of the Two-Body Problem**

5.1 Introduction

5.2 The Potential Energy Field

5.3 Computation of the Positions of the Collinear Points

5.3.1 Examples

5.4 Computation of the Equipotential Curves

5.4.1 Qualitative Solution

5.4.2 Mathematical Solution

5.5 Numerical Method

5.6 Some Remarks

5.7 Practical Examples

5.8 Application: Close Binary Evolution

5.9 The Program Listing

**6 The Dynamical Parallax**

6.1 Introduction

6.2 Basic Formulae and Iterative Procedure

6.3 Concerning the Program

6.4 Practical Examples

6.5 The Program Listing

**7 Polytropes**

7.1 Introduction

7.2 Physical Background

7.2.1 The Continuity of Mass

7.2.2 The Equation of Hydrostatic Equilibrium

7.2.3 The Polytrope Relation

7.3 Mathematical Background

7.4 Numerical Procedure

7.5 Some Remarks About the Program

7.6 Applications

7.7 The Program Listing

**8 Homogeneous Stellar Models**

8.1 Introduction

8.2 Physical Background

8.2.1 The Pressure, Density, and Mass

8.2.2 The Equation of State

8.2.3 The Temperature

8.2.4 The Luminosity

8.3 Boundaries and Polytrope Fitting Procedure

8.3.1 The Polytrope Index in the Core

8.3.2 The Position of the Boundary of the Convective Core

8.3.3 Fitting the Two Polytropes

8.3.4 Stopping the Iterations at the Stellar Surface

8.4 The Choice of the Free Parameters

8.5 Numerical Method

8.5.1 Initial Conditions and Central Values

8.5.2 The First Step

8.5.3 The Convective Core Iteration

8.5.4 Fitting the Two Polytropes

8.5.5 The Radiative Zone

8.6 The Final Results

8.7 Applications

8.7.1 Model of a 10M Star

8.7.2 Results for Other Masses

8.8 The Program Listing

**9 Stellar Atmospheres**

9.1 Introduction

9.2 Physical Background

9.2.1 The Absorption Coefficient kappa

9.2.2 The Optical Depth tau

9.2.3 The Temperature T

9.2.4 The Gas Pressure P_g and the Mean Molecular Weight mu

9.2.5 The Radiation Pressure P_r

9.2.6 The Equation of Hydrostatic Equilibrium

9.2.7 Summary of the Equations

9.3 Numerical Method

9.3.1 The Initial Values at tau = 0

9.3.2 Proceeding from a Level tau _i to tau_i+1

9.3.3 The Step-size d tau_i

9.4 Some Remarks Concerning the Program

9.5 Applications

9.6 The Program Listing

**10 The Structure of White Dwarfs**

10.1 Introduction

10.2 Electron Degeneracy

10.3 Equations of Structure

10.3.1 Equation of State

10.3.2 Hydrostatic Equilibrium and Continuity of Mass

10.4 Numerical Method

10.4.1 Initial Conditions

10.4.2 Computing the Density from the Pressure

10.4.3 The Iteration Procedure

10.4.4 The First Step

10.5 Some Remarks

10.6 Examples and Applications

10.6.1 Detailed Model of a White Dwarf

10.6.2 The Chandrasekhar Limit

10.7 The Program Listing

**11 Star Formation in the Galaxy**

11.1 Introduction

11.2 A Simple Model for Star Formation

11.3 Equations of Interaction Between the Components

11.4 Numerical Methods

11.5 Applications

11.5.1 Evolution Towards a Stationary State

11.5.2 Evolution Towards a Limit Cycle

11.6 The Program Listing

**12 Individual Stellar Orbits in the Galaxy**

12.1 Introduction

12.2 Schmidt's Model

12.3 Schmidt's Model Modified with an Infinite Density Law

12.3.1 The Total Mass

12.3.2 The Rotational Curve

12.4 The Motion of an Individual Star

12.4.1 The Equations of Motion

12.4.2 Operational Framework

12.4.3 Conservation of Energy

12.5 Numerical Method

12.6 Initial Conditions and Practical Hints

12.7 Practical Examples and Applications

12.7.1 Detailed Results for an Orbit

12.7.2 Types of Orbits

12.7.3 Motion around the Rotational Axis of the Galaxy

12.8 The Program Listing

**13 Cosmological Models for the Universe**

13.1 Introduction

13.2 Physical and Mathematical Backgrounds

13.3 Types and Classes of Zero-Pressure Models

13.4 Numerical Method

13.5 Some Remarks

13.5.1 On the Choice of DX

13.5.2 An Interpretation of the Curvature k

13.6 Practical Examples

13.6.1 Type A1

13.6.2 Type A2

13.6.3 Type A3

13.7 The Program Listing

**Appendix
Bibliography
Index**