A variational principle has been applied to one-dimensional stellar systems to show that stationary distribution functions which are always decreasing in going outward from the center of the system are stable. Other stationary distributions may be unstable as is illustrated by means of computer experiments.
Numerical experiments with a simple two-dimensional rod model show that the spiral structure and other filamentary structure of galaxies may result from purely gravitational effects.
It will now be shown that the minimum energy property which has been obtained by Hohl and Feix (ref. 1) for a special distribution can be extended to arbitrary distribution functions. The water-bag model illustrated in figure 1 is used in the analysis. The contours ν+(k)(x,t) and ν−(k)(x,t) surfaces of constant distribution function f=fk. According to the Liouville theorem the phase space bounded by the contours is incompressible so that the area bounded by the contours is conserved. In the limit of a very large number of contours the water-bag model can be used to construct arbitrary distribution functions.
To simplify the equations we assume symmetric contours ν+(k)=ν−(k)=ν(k).
It is easily shown (ref. 2) that the equations which stationary contours ν(k) must satisfy are
E is given by
where G is the gravitational constant, N is the number of mass sheets, each of mass m per unit area, in the system.
Ak is defined by the distribution function
where xs(k) is the end point of the contour k. The total energy of the system is
Extremizing the integral for W requires that g satisfy the Euler-Lagrange equation
or
which are the equations for the stationary contours.
If equations (6) are to represent a minimum energy configuration then Legendre's criterion of the second variation of g must be satisfied. That is, the quadratic form whose coefficient matrix has the elements
must not be negative. Since only the diagonal elements
are nonzero, the Legendre condition requires that
for all k. Equation (9) is equivalent to stating that the distribution function must always decrease in going outward from the center of the system where f=f1 must be the largest. If equation (9) is satisfied the system is a minimum energy configuration and is always stable. However, if Ak>0 is not satisfied for all k the system is not a minimum energy configuration. The system may then be unstable since the contours ν(k) can now be deformed while keeping the total system energy constant. Numerical experiments with a one-dimensional model have been performed for two contour systems to illustrate the interchange instability which destroys the stationary state. For the two systems investigated A1=-A2. Figure 2 shows the normalized stationary contours for four values of u, the ratio of the minimum to maximum star energy. The results of the computer experiments are shown in figure 3 and figure 4. In the first case shown in figure 3 the ratio of the minimum to maximum star energy is 0.4. The figure shows that the stationary contours of the 2000 star system are quickly distorted while the heavier outer water bag tries to displace the inner bag. Figure 4 shows the results for a 2000 star system with a ratio of minimum to maximum star energy equal to 0.25. The growth of the instability is now much slower because the central water bag or hole is much smaller.
Results of additional computer experiments have shown that certain distribution functions, F(U), that are not always decreasing with increasing energy, U, remain stable over many periods, 2π / √(4πGρ), where ρ is the mass density of the sheets per unit area. For such distribution functions the bulk of the stars are in a region where F(U) is monotonically decreasing but F(U) has a high energy bump corresponding to particles in the spiral arm that develop in the two-dimensional phase space (ref. 2).
The evolution of two-dimensional stellar systems made up of mass rods that are of infinite extent in the z-direction has also been investigated for 400 and 500 rod systems. The force acting on a particular mass rod is obtained by summing directly over the 1/r force from each mass rod. This is a time-consuming process and the application of fast methods of solving the Poisson equation would speed up the calculations. The relatively large grid size which would be required by the method of solving the Poisson equation will smooth the force due to the near neighbors of the particles and will affect the evolution of the system. The effect of the near neighbors is included if the force acting on a mass rod is obtained by summing directly the 1/r force from each particle in the system. It would be more desirable to study the evolution of a system of mass points moving in a plane. This can also be done by simply summing the 1/r2 force from each mass point. However, the more rapid divergence of the 1/r2 force for near neighbors now requires that a much smaller time step be used in the computations resulting in a considerable increase in computer time.
The system is advanced in time in the following manner. First, the force acting on all particles is computed by summing the 1/r force for all particles. Second, the system is advanced for a small time step Δt and the process is repeated. The results of the calculations are displayed in x-y coordinate space. During the calculations the total energy and angular momentum is computed to check on the accuracy of the computations. The normalizations 4πG=1 and m=1 have been used for all the calculations.
Figure 5 shows the time development of a system of 400 mass rods which has an initially rectangular distribution of uniform density in x-y space. The system has an initial thermal energy equal to 1/5 of the initial potential energy plus an initial solid body rotation equal to nearly twice that required to oppose the gravitational force towards the center of the system. It can be seen from figure 5 that the system quickly develops into a barred spiral. However, at a later time the spiral structure has almost completely disappeared and the system approaches a configuration similar to an elliptical galaxy. The time has been normalized to ωr-1, the inverse of the frequency of rotation.
The remaining two-dimensional calculations were performed for 500 particle systems which have an initially uniform circular distribution in x-y space and zero thermal velocity. The evolution of such systems is then studied for various values of initial solid body rotation. The initial positions are obtained by using a random number generator which gives a nearly uniform distribution over a circular region of the x-y plane.
We now present the results for the case where the frequency of rotation, ωr. equals ωg, the frequency required such that the centrifugal force balances the gravitational force. Thus,
where ρ is the mass density of the rods per unit length. The resulting evolution of the system is shown in figure 6. The time has been normalized to ωg-1. Figure 6 shows that the system is relatively stable. At t=6.32ωg-1 there appear four irregular spiral arms. However, at a later time the spiral arms almost completely disappear and the system takes an appearance reminiscent of an elliptical galaxy.
The results for the case of zero initial rotation are presented next. Figure 7 shows that after an initial implosion the system expands again and presents some highly irregular filamentary structure. After a second implosion the temperature of the system increases due to the randomness of the initial positions. The pressure due to the temperature then tends to reduce the oscillations and the system again takes a form similar to an elliptical galaxy.
For ωr=½ωg the system again contracts initially and then expands again. The results are shown in figure 8. An irregular structure appears initially which tends to disappear at a later time. Also at time t=4.20ωg-1 the system is clustered into two aggregates which combined again at a later time.
In figure 9 the results for the case ωr=1.3ωg are shown. The system pulsates and shows some irregular structure. The general behavior is very similar to that of the previous case for ωr=½ωg.
The simple two-dimensional model for a stellar system showed that spiral and other filamentary structure can result from purely gravitational effects. Since the spiral structure and other filamentary structure tend to disappear in time one would conclude that the spiral galaxies are young galaxies which later develop into elliptical galaxies. However, these results should be confirmed with a computer model using point masses that are confined to move in a plane.
1. Hohl, F, Feix, M R, 1967, Astrophys J, vol 147, no 3, March 1967, pp. 1164-1180.
2. Hohl, F, PhD. Thesis, 1967, College of William and Mary, Williamsburg, Virginia.
