The physics origin of the hierarchy of bodies in space

Recent progress on the physics basis of evolutionary organization in nature^1,2 continues to bring together phenomena that were previously considered unrelated. To the animate and inanimate examples (animal locomotion, river basins, turbulence) that were unified as a phenomenon of free-morphing flow design for greater access over time,^1–18 we are now adding examples that belonged traditionally to solid mechanics. For example, the natural occurrence of hexagonal basalt columns is attributed to a principle of maximum energy release.¹⁹ The occurrence of cracks in solids is based on the same principle.^20–22 Soil cracking under the drying wind was explained as a phenomenon of evolutionary design that enhances mass flow and accelerates drying.²³ The aggregation of dust particles into clusters and dendrites was shown to be the result of the same tendency, to relieve electrostatic forces of attraction faster, through the evolutionary design of configuration.²⁴ 

Here we add to this growing list of evolutionary phenomena the natural occurrence of multi-size hierarchy of bodies suspended in space. The hierarchy of sizes is researched intensely and described regularly in planetary science and astrophysics.^25–31 Hierarchy emerges in two ways, through accretion (coalescence) and fragmentation resulting from collisions. Viewed from thermodynamics,³² the system of bodies in space is in a state of internal tension because of gravitational attraction between neighboring bodies. This system “snaps” freely by flowing internally and changing its configuration. Bodies coalesce into larger bodies, and their collision (with fragmentation) dissipates the tension and resulting kinetic energy, en route to reduced body-body attraction throughout the system. This phenomenon has been studied in celestial mechanics under several scenarios,^25–27 and is recognized as the basis for the formation process of planets and the asteroid belt.

Sizes increase over all scales through accretion.³¹ Yet, the natural phenomenon is not only the growth of the body sizes but also the spontaneous hierarchy. The fundamental question that we address here is why “hierarchy” happens spontaneously, and why a uniform distribution of bodies of the same (growing) size does not happen. We show that the gravitational effect alone does not explain the hierarchy of sizes of bodies in space. The additional physics principle is the natural evolution (selection) of flow configuration during accretion such that the flow and evolution to equilibrium are facilitated.^1,2

Consider the coalescence of masses suspended in space. Forces of mutual attraction (gravitational) keep the system in a state of internal tension. In time, the tension is relieved through the creation of aggregates.

Two bodies attract each other with a force that is proportional to the product of the two masses and inversely proportional to their mutual separation distance squared.³³ The shape and relative motion of the bodies is not considered. Assume that a space is filled initially with masses of one size (m) that are motionless and distributed uniformly. The spacing between two neighboring masses is constant (r). This suspension is in a state of volumetric and isotropic tension. The tensile force between two neighboring masses is proportional to m²/r².

The coalescence of small masses into larger masses is driven by internal tension, which becomes smaller as a result. Equilibrium will be characterized by zero tension and total coalescence with all the individual bodies collapsed into one large body.

In thermodynamic terms, the system is isolated and exhibits internal changes (mass flows) that take it from an initial state of internal tension toward a final state of no tension and no movement. The classical thermodynamics of such an isolated system is well known.³² Consider the time evolution of the internal configuration of the system. Should the system evolve as equidistant masses of one size that increases through coalescence, or should the evolving design be hierarchical, heterogeneous, with few large masses and many small masses that feed the large masses?

In summary, which should be the evolutionary design that brings the initial suspension to coalesce into one mass? Note that the natural evolutionary tendency of flow organization is not covered by the first and second laws of thermodynamics. In the present case, the flow consists of masses (m) that come from everywhere and collapse (irreversibly, with dissipation) into progressively larger masses. Greater access means designs (organization, distribution of masses) that facilitate this flow, and greater forces of attraction that accelerate the decrease in the internal tension throughout the system. The question is: What kind of organization, uniform or hierarchical, will facilitate this evolution?

One Dimensional Organization. First, assume that the initial masses (m) are distributed equidistantly on an infinitely long line. The spacing between two neighboring masses (r) is much greater than the linear dimension of a single m. In this first design (Fig. 1), each mass (m) is pulled left and right by a force (F₁) to which contribute all the other masses, near or distant

where S₁ = π²/6 = 1.645. First, assume that the coalescence is distributed uniformly. If two adjacent masses collapse into one mass of size 2 m, then the new spacing (between two masses of size 2 m) is 2r, and the tensile force felt throughout the system is

which has the same value as F₁. In conclusion, uniformly distributed coalescence does not relieve the internal tension.

Next, consider a non-uniform pattern of coalescence. One mass (m) fuses with its two neighbors and creates a mass of size 3 m. Left between the two masses of size 3 m is one small mass m. The spacing inside each (m, 3 m) pair is 2r. Each body of size m is pulled to the right and to the left by a force proportional to

where $S2=1+19+125+...=1.23$⁠. Comparing F₃ with F₁, we find that F₃/F₁ = 0.623, which indicates that the non-uniform pattern offers a significant reduction in the internal tension of the system. According to the constructal law, the natural organization should be non-uniform coalescence.

Two Dimensional Organization. Contemplate the same question with regard to coalescence in two dimensions. Initially, the masses are equidistant (r) and of the same size (m) (see Fig. 2). The square pattern is assumed for analytical simplicity. We calculate the net tensile force (F₁) felt in the horizontal (x) direction by the mass (m) positioned at the origin (O).

An infinite number of forces contribute to F₁. The force F₁₁ is aligned with x, due to a string of masses as shown in the upper row of Fig. 1 and in Eq. (1); therefore, F₁₁ = (m/r)²S₁. Focus on the string of m-masses aligned on the bisector of the x-y field. The spacing between masses is 2^1/2r, and the force in the bisector direction at the origin is F₁₂ = (m/2^1/2r)²S₁. What counts is the projection of F₁₂ on the x direction, namely, F_12x = F₁₂ cos 45° = (m/r)²2^−3/2S₁. There are two bisectors—one at y > 0 and the other at y < 0; therefore, the total x-contribution of the tension along the bisectors is 2F_12x= (m/r)²2^−1/2S₁.

The calculation continues accounting for all the in-between directions such as F₁₃ in Fig. 2. The next x-contribution of the pull in this direction is 2F_13x, where F_13x is the x-projection of the F₁₃ force felt at the origin. In sum, the total force in the x direction felt by the mass located at O is

This estimate is not exact because we truncated the above sum (4) by neglecting the contributions from some of the forces aligned close to the y direction.

The first alternative to the uniform distribution (Fig. 2) is the occurrence of uniform coalescence. Everywhere in space, two neighboring masses (m) form a larger mass of size 2 m. The resulting distribution is a new square pattern in which the spacing between the 2 m masses is 2^1/2r. The net force on one 2 m mass in one direction (x) follows from the same calculation as for Eq. (4), in which m is replaced by 2 m, and r is replaced by 2^1/2r

The conclusion is that F₂ is greater than F₁. Uniform coalescence is not the design that decreases the tension throughout the original suspension. Therefore, according to the constructal law, uniform coalescence should not occur.

The competing alternative is the phenomenon of non-uniform coalescence. Shown in Fig. 3, four of the original m masses fused with the m mass between them and created one mass of size 5 m. Spread equidistantly throughout the plane are the remaining m-masses. The distance between one m mass and the closest 5 m mass is 2r. The calculation of the total force (F₃) in the x direction at the origin in Fig. 3 begins with the contribution from the non-uniform string aligned with x

Next are the contributions from the strings on the bisectors, 2F_32x, where one F₃₂ is aligned with one bisector, and F_32x is the projection of F₃₂ on the x direction. The expressions for F₃₂, F₃₃, are omitted. The final result reduces to

This shows that F₃ is sensibly smaller than F₁. A smaller force means a smaller acceleration of neighboring masses that coalesce, and a slower coalescence. On the contrary, non-uniform (hierarchical) coalescence is the design that relieves the tension faster than no coalescence (F₁), and even faster than uniform coalescence (F₂).

The fundamental conclusion is that the hierarchical distribution of masses facilitates coalescence and brings the entire suspension to equilibrium faster. We demonstrated this in both scenarios, one dimensional (Fig. 1) and two dimensional (Fig. 2). This prediction is in accord with the constructal law^1–4 and also in accord with observations. In this way, the constructal law accounts for the origin of the spontaneous hierarchy among celestial bodies. This phenomenon of self-organization is of the same nature as the hierarchical cracking of elastic bodies in volumetric tension en route to less tension throughout the volume.

In this article, the origin of spontaneous hierarchy of bodies in space was explained on the basis of the simplest possible model that retained the most important feature of all evolutionary configurations in nature: the freedom to flow (to move) and to morph while flowing. However simple and limited, the arguments are strong in favor of hierarchy as a natural tendency rooted in physics, and the findings deserve to be further debated by the scientific community for more realistic (general) models, for example, three-dimensional with relative (orbital) motion between bodies.

The hierarchy prediction put forth in this article provides a concrete answer to Ellis and Silk's question of what is a scientific theory.³⁴ To see how, we must first review the concepts that lead to and underpin any scientific theory:

• The human observation that certain things (images, events) happen the same way innumerable times represents a distinct natural tendency, i.e., one universal phenomenon of nature (physics).

• A law of physics is a compact statement (text, or formula) that summarizes (a), i.e., the innumerable observations of the same kind everywhere. One law, for one distinct universal phenomenon, is the structure of the physics doctrine.

• To rely on the law (i.e., to invoke it) in order to experience a purely mental viewing of how things should be in a particular set of circumstances (i.e., to predict observations for that setting) is the particular theory that became available for that setting because the law is known.

In summary, there is one phenomenon, there is one law, and the theories that spring from the law are as many as the circumstances in which the thinker contemplates the phenomenon, i.e., the manifestations of the law. For example, there is one constructal law and there are many constructal theories, covering the board from biological to non-biological phenomena: lung structure, rhythm (respiration, heartbeat), animal locomotion, river basin structure, river channel cross sections, aircraft evolution, turbulent structure evolution, snowflake evolution, and many more. Ellis and Silk³⁴ argued that “…the issue boils down to clarifying one question: What potential observational or experimental evidence is there that would persuade you that the theory is wrong and lead you to abandoning it? If there is none, it is not a scientific theory.” Sure, there are many observations of evolutionary design in nature that validate the constructal law, for example:

1. A flat plume or jet should evolve into a round plume or jet, never the other way around.¹⁶ 

2. Solid bodies that grow during rapid solidification (e.g., snowflakes) should be tree-like, not spherical.³² 

3. The bigger moving bodies (animals, rivers, vehicles) should live longer and travel farther, not live less and travel less.¹⁷ 

4. The human lung should be tree-shaped with 23 levels of branching.¹⁵ 

5. All animal speeds (swimming, running, flying) should be proportional to the body size raised to the power 1/6, and for a given body size should increase from swimmers to runners and then flyers.^1,18

The examples go on, and every single one is evidence in answer to the question formulated by Ellis and Silk.³⁴ There is more to this than meets the eye. Some readers may be tempted to argue that the evidence has long been available in the past, and that the theory cannot predict “future” observations. This argument is incorrect, in two ways:

First, predicting an old phenomenon that was not recognized and questioned previously is a theory. Think of Galilei's law of gravitational fall, and Clausius' law of irreversibility (the second law). The fact that everything on earth has weight (from which the word “gravity”) and that everything flows from high to low (i.e., one way, or “irreversibility”) are phenomena that were not brought into physics before Galilei and Clausius questioned them and summarized them in the form of two succinct laws.

Second, throughout the literature there are numerous predictions that refer to future observations, such as examples no. 1 and no. 2 above, and many more evolutionary designs that occur at short time scales that are comparable with our life time, for example, the evolution of technology (e.g., airplanes³⁵) and sports.¹ This should come as no surprise, because all science is an artifact (an add-on) that empowers humans to predict the future.

Professor Bejan's research was supported by the National Science Foundation. The authors thank Allison Wagstaff for editing the manuscript.