Welcome!

2012-11-27T13:48:00.000-08:00

Welcome to my blog about Stability Analysis. This page focuses on preforming a stability analysis on star formation within a model galaxy, but the techniques found here have many other applications as well.

An Introduction

2009-12-06T02:32:00.000-08:00

We can think of a physical system as a subset of the physical universe as a whole. We define what constitutes the system based on what we are interested in analyzing. Anything outside of the system we ignore for analysis except for how it affects the system we're looking at. So for example, your body can be considered as a physical system. The solar-system can be thought of as a physical system. A lagoon can be thought of as a physical system. Even an individual atom can be thought of as a physical system. It all depends on what you want to look at.

Stability properties of a physical system refers to how the system responds to some perturbation; whether the system can recover on its own after being perturbed or whether it goes haywire. Stability analysis helps us to understand what happens when we perturb a system.

A classic example of a perturbed system is a ball and hill. Let us assume that we have a ball sitting in a valley. On one side of the ball, there is a large hill that continues upwards very high. On the other side is a small hill but beyond that there is a drop that goes down to infinity.

When it's at rest within the valley, its height is stable:

What happens if we give up a small push up the side of the high hill (smaller than the height of the small hill)?

The ball rolls down the hill into the valley, then up the smaller hill, then back down into the valley and up the hill, etc. all the while losing energy to friction, travelling up the hills less high each time and eventually coming to rest back within the valley.

The height of the ball over time can be expressed as a sinusoidal pattern with an amplitude that decreases over time and eventually flattens. We say that for this small perturbation, the system is stable because it returns to a steady state.**

But what happens if we give the ball a large push up the side of the high hill?

The ball will roll down the side of the high hill, into the valley, up the small hill and past the maximum height of the small hill then continue up until it reaches a maximum height and then begins to fall into the infinite drop where it continues forever.

The height of the ball never reaches a steady state again so for a large perturbation we say that the system is unstable.

**NOTE: A continuously oscillating system may also be regarded as being stable.
Those are the basics. Stability analysis has a wide-range of applications in the real world.

Stability Properties of a Physical System

2009-11-27T18:12:00.000-08:00

Phase Transitions Between Massive Stars and Molecular Hydrogen in a Model Galaxy

By MPF (originally written November 2006)

Abstract

The stability properties of a physical system, phase transitions between massive stars and molecular hydrogen in a galaxy, can be determined by performing a stability analysis on a system of ordinary differential equations that govern the mass of the system.

Introduction

Stability analysis of systems allows us to determine whether or not a system is stable or will be stable if perturbed. This is important in a wide range of applications since many behaviours observed in the real world can be described using differential equations.

The stability analysis of phase transitions between massive stars and molecular hydrogen in a model galaxy allow us to determine whether or not the mass of molecular hydrogen and the mass of stars in a galaxy is in an equilibrium, if it ever was, or if it ever will be. This lets us model the future behaviour of an observed galaxy or lets us work backwards and theorize what might have occurred in a galaxy’s past, for example, if a galactic collision had once occurred and if so, when?

While stability analysis has many wide-ranging uses not necessarily related to astronomy, this report will examine the stability of a model galaxy.

Procedure and Results

The phase transitions between massive stars and molecular gas in a model galaxy are described by a system of ordinary differential equations. In this case, the equations are:

dM_s/dt = -rM_s + kaM_sM_H2 and dM_H2/dt = -aM_sM_H2 + A (1)

Where M_s and M_H2 are the total masses of massive stars and molecular hydrogen, respectively, a is the rate of induced star formation, k is the efficiency of star formation, 1/r is an average lifetime of massive stars and A is the mass accretion rate onto the galaxy.

In order to derive an equilibrium for both masses, both equations of system (1) must be set equal to 0. The equilibrium mass of massive stars, M_s⁰, can be expressed as:

M_s⁰ = Ak/r

While the equilibrium mass of molecular hydrogen, M_H2⁰, can be expressed as:

M_H2⁰ = r/ak

System (1) can be expressed in the following form:

M_s = M_s⁰ + m_s exp(λt) and M_H2 = M_H2⁰ + m_H2 exp(λt) (2)

Where m_s and m_H2 are small perturbations in the masses of massive stars and molecular hydrogen, respectively. In order to determine if the system is stable or not, we must look at λ. When we solve for λ (see appendix), the resulting expressions are:

λ = 0.5(kaM_H2⁰ – r – aM_s⁰ ± √(( -kaM_H2⁰ + r + aM_s⁰)² – 4arM_s⁰ ))

If any of the following conditions are met, the system will be stable:

[(-kaM_H2⁰ + r + aM_s⁰)² > 4arM_s⁰ AND (kaM_H2⁰ – r – aM_s⁰) < 0]

OR [0 < 0.5(kaM_H2⁰ – r – aM_s⁰ ± √(( -kaM_H2⁰ + r + aM_s⁰)² – 4arM_s⁰))]

Since the mass functions are proportional to exp(λt), stability can only occur when either λ < 0 or λ is complex with a real part < 0 and an imaginary part ≠ 0.

If we make the following assumptions:

1/r = 1, k = 0.02, M_H2⁰, A = 3, a = 0.5 and M_s⁰ = 0.06 (3)

We get the following roots:

λ = -0.0150 + 0.172554i

λ = -0.0150 - 0.172554i

Since λ is a complex number and the real part of λ < 0 and the imaginary part of λ ≠ 0, we know that the system is oscillatory with a maximum amplitude that dampens over time and is therefore stable.

When the substitutions in (3) are made, and a non-equilibrium occurs with a small perturbation in the mass of molecular hydrogen, m_H2, over time we should expect amplitudes of the masses for both the massive stars and the molecular hydrogen to oscillate and eventually dampen out into an equilibrium state. An increase in the amount of molecular hydrogen in the system from, say, a collision with a small galaxy, would increase the total molecular hydrogen of the system and induce star formation. The induced star formation would then decrease the amount of molecular hydrogen in the system and thus decrease the rate of induced star formation, etc. This cycle would continue until an equilibrium in the mass of massive stars and the mass of molecular hydrogen is reached.

If we assume the system is initially in a slightly non-equilibrium state where

M_{H2 =} 1.1M_H2⁰ and M_{s =}M_s⁰ and use the same values as (3) and adopt the units of mass 10⁷M and time 10⁷ yr for a numerical simulation model we observe the behaviour of the masses of massive stars and molecular hydrogen in the system over a period of 5 billion years as follows:

The masses exhibit an oscillatory behaviour. The model shows that the initial non-equilibrium caused induced massive star formation. This led to a decrease in the amount of molecular hydrogen, which in turn, led to a decrease in the rate of induced star formation which continues to oscillate and dampen until equilibrium is reached.

The predictions of the stability analysis agree with the numerical simulation.

Conclusion
By considering the ordinary differential equations that govern the phase transitions between massive stars and molecular hydrogen in a model galaxy, the stability of the system can be determined by knowing and/or assuming certain characteristics of the system.

In a case where 1/r = 1, k = 0.02, M_H2⁰, A = 3, a = 0.5 and M_s⁰ = 0.06, and an initial non-equilibrium of M_{H2 =} 1.1M_H2⁰ and M_{s =}M_s⁰, the system is oscillatory and becomes stable over time.

Appendix
Matlab code:

l = -0.0150 - 0.1726i;
r = 1;
k = 0.02;
A = 3;
Ms0 = 0.06;
Mh0 = 100;

dMsdt = 0;
dMhdt = 0;

Ms = Ms0;
Mh = Mh0*1.1;
a = 0.5;
ms = 0;
mh = 0;

for t = 1:500;
dMsdt = -r*Ms+k*a*Ms*Mh;
dMhdt = -a*Ms*Mh+A;
ms = Ms - Ms0 + dMsdt;
mh = Mh - Mh0 + dMhdt;
a = real(-(dMhdt-A)/(Ms*Mh));
Ms = real(Ms0 + ms*exp(l));
Mh = real(Mh0 + mh*exp(l));

Msf(t) = Ms;
Mhf(t) = Mh;
end;