An algorithm for coalescence of classical objects and formation of planetary systems

Abstract

Isaac Newton formulated the central difference algorithm (Eur. Phys. J. Plus (2020) 135:267) when he derived his second law. The algorithm is under various names (”Verlet, leap-frog,..”) the most used algorithm in simulations of complex system in Physics and Chemistry, and it is also applied in Astrophysics. His discrete dynamics has the same qualities as his exact analytic dynamics for continuous space and time with time reversibility, symplecticity and conservation of momentum, angular momentum and energy. Here, the algorithm is extended to include the fusion of objects at collisions. The extended algorithm is used to obtain the self-assembly of celestial objects at the emergence of planetary systems. The emergence of twelve planetary systems is obtained. The systems are stable over very long times, even when two “planets” collide or if a planet is engulfed by its sun.

Appendix: Molecular dynamics simulation of planetary systems

The simulations of the creation of planetary systems were performed for different numbers $N=100, 1000$ and 10000 of objects and for different start configurations of the objects. The spherically symmetrical objects are identical and interact with the gravitational force Eq. (4). All units (mass, time, length, energy/force) are given in units given by $G, m_i$ and $\sigma _i$, and the present simulations are started with $N=1000$ objects with equal masses $m_i=1$ and diameter $\sigma _i=1$. In order to simulate the orbit of a planet near the sun, it is necessary to choose a small time increment $\delta t=0.0025$ [22]. The MD simulations are with double-precision variables, the center of mass and the momentum and orbits of planets are conserved after more than $10^9$ time steps with a small time increment $\delta t=0.0025$.

The phase space diagram for the gravitational system [29] differs from a traditional phase space diagram. A collection of objects will, without fusion collapse at low temperature (velocities of the objects) and relative high concentration in the collection [30], but at high temperatures and low concentrations the collection of objects expands continuously in the space. The collapsed spherical symmetrical objects will, without fusion perform a crystal. It is as mentioned not possible to obtain a traditional phase diagram for a classical system with gravitational forces because the energy per object

$$\begin{aligned} u/kT= & {} 2 \pi \rho \int _0^{\infty } u(r) g(r) r^2 dr \nonumber \\\approx & {} -2 \pi \rho G m_1 m_2 \int ^\infty r_{12} dr_{12} \end{aligned}$$

(18)

(and free energy) diverges for an uniform distribution. The objects crystallize at low temperatures if the objects perform elastic collisions without fusion. In contrast to this behavior, the collections of free objects at low temperatures and concentrations and with fusion merge into solar systems with planets in regular orbits. The algorithm with fusions is used obtain planetary systems.

Here, we shall describe twelve simulations of the emergence of planetary systems for $N=1000$ and for diluted gas configurations of the objects at the start, which are spherically (blue dots in Fig. 3). The twelve systems are generated for different start configurations and kinetic energies. The momenta and angular momenta $\mathbf{L} _G(0)$ at the start $t=0$ are adjusted to zero.

Depending on the start configurations of positions and kinetic energy, the systems either collapse at low kinetic energy and high gas density into one heavy object (black hole), or expand in the open space for high kinetic energy and low density. But in between these ranges of velocities and concentrations some objects merge with creation of one heavy object (the sun), with planets and with unbounded objects. (A free object is characterized by the fact that it, with a constant direction with respect to the sun and with a positive energy $E=E_{\text {Kin}}+E_{\text {Pot}}$, increases its distance to the sun.)

The twelve systems are started at different mean “temperatures” T in the interval $T \in [0.1,0.5]$. The fusions started shortly after (Fig. 2) with an increase in the mean velocities. The total momenta and the center of masses and angular momenta $\mathbf{L}$ are conserved, but the angular momenta $\mathbf{L} _G$ are not conserved at the fusions, but varied within the range $\mathbf{L} _G \approx [\mathbf -10 ,\mathbf -10 ]$. The random round-off errors in the double precision arithmetic have no effect on the orbits of the planets. The simulations were performed by consecutive simulations with $2\times 10^8$ time steps. At the start of a simulation, the center of mass and momentum components were adjusted to zero, and by the end of a simulation the round-off errors had changed the center of mass components from $\approx 10^{-18}$ to $\approx 10^{-16}$. The components of the momentum were changed with the same factor. These tiny adjustments have, however, no effect on the stability, nor on the orbits of the planetary systems.

Without any approximations, the dynamics of a planetary systems is time demanding because one needs a rather small time increment $\delta t$ to obtain the orbit of a planet at Perihelion accurately, and the stability of the planetary system is given by its long-time behavior, which together implies that it is necessary to performs billions of time steps in order to obtain the long-time behavior of a planetary system. The present calculations are for $n= 1.8 \times 10^9$ time steps ($t=4.5\times 10^6$). Another complication is that the computational time without some approximations varies with the number N of objects as $\propto N^2$. It is, however, straight forward to implement different kind of time consuming approximations used in MD [13, 31] whereby the computational time varies proportional to $\approx N$.