Research Article | Open Access
Alexander M. Krot, "On the Analytical Models of Protoplanetary Formation in Extrasolar Systems", Space: Science & Technology, vol. 2022, Article ID 9862389, 19 pages, 2022. https://doi.org/10.34133/2022/9862389
On the Analytical Models of Protoplanetary Formation in Extrasolar Systems
In this work, we consider a statistical theory for a cosmogonical body formation (so-called spheroidal body) to develop the analytical models of protoplanetary formation in extrasolar systems. Within the framework of this theory, the models and evolution equations of the statistical mechanics have been proposed, while the well-known problem of gravitational condensation of infinite distributed cosmic substances has been solved. This paper derives the general equation of distribution of the specific angular momentum of forming protoplanets since the specific angular momentums (for particles or planetesimals) are averaged during a conglomeration process (under a planetary embryo formation). As a result, a new law for planetary distances (which generalizes Schmidt’s law) is derived theoretically. This paper develops also an alternative thermal emission of particles model of the formation of protoplanets in extrasolar systems. Within the framework of this model, the equation for the thermal distribution function of the specific angular momentums of particles moving in elliptical orbits in the gravitational field is derived. According to this thermal escape model, only 0.8% of the total number of particles in the solar system composing the protoplanetary cloud has angular momentum 15.6 times higher than the angular momentum of the remaining 99% of particles. This conclusion agrees completely with the known fact of a nonuniform distribution of the angular momentums in our solar system noted by ter Haar. As pointed out here, the exponential laws of planetary distances occur in some extrasolar systems.
Recently, the general problems of the formation of protoplanetary systems, the study of their dynamical behavior, and the formation and evolution of the planets have got additional attention form the scientific community in connection with the discovery of extrasolar planets which are among the greatest achievements of modern astronomy.
Our understanding of our place in the Universe changed significantly in 1995 when Mayor and Queloz from Geneva Observatory in Switzerland announced the discovery of an extrasolar planet around a star, 51 Pegasi, similar to our Sun . Geoff Marcy and Paul Butler in the United States soon confirmed their discovery, and the science of observational extrasolar planetology was born. The field has extensively grown in recent years, reporting of numerous planetary systems in 2022 (see http://exoplanet.eu/ and http://exoplanets.org/). Most of these systems usually contain one (or more) gas-giant planet close, or very close, to the parent star. Nevertheless, detection of planets with masses approximately equal to the mass of our Earth is evidence that there exist extrasolar planets with low masses. In addition to obtaining important knowledge about the formation and structure of new exoplanetary systems, these discoveries provoke genuine interest among the scientific community regarding the prospects for finding life in the Universe.
However, the questions considered in this article deal mainly with the problems of cosmogony and only partially touch upon cosmology. Cosmological bodies include large-scale space objects (for example, galaxies and their clusters) based on the fact that cosmology is a science that studies the properties and evolution of the Universe as a whole. In this context, cosmogonical bodies stay for stars, protostars, interstellar molecular clouds, planetary systems, protoplanetary gas-dust disks, planets, protoplanets, and natural satellites of planets . Generally speaking, the cosmogonical theories, according to Schmidt [3, 4], include both planetary cosmogony and stellar cosmogony.
Several cosmogonical theories are known to explain the formation of the solar system, the formation of planets, and the estimation of planetary orbits [1–36]: (i)Electromagnetic theories [8, 23](ii)Gravitational theories [3, 4, 6, 11, 13](iii)Nebular theories [5, 9, 17–22, 24](iv)Quantum mechanical theories [25–30]
Despite a large amount of research and a huge number of works aimed at studying the formation of the solar system, none of these theories can fully explain all the observed phenomena—in particular, the four groups of facts following ter Haar [5, 24]. Among those is the distribution of angular momentum: although the Sun has more than 99% of the total mass of the solar system, only 2% of the total angular momentum belongs to the Sun, while the remaining 98% belongs to the planets.
In this context, the statistical theory [16, 31–36] for a cosmogonical body formation has been developed. It is based on the so-called model of a spheroidal body, which is formed through numerous gravitational interactions of its parts and particles.
In this paper, we consider two alternative statistical models of the origin of protoplanets embedded in a flattened gas-dust protoplanetary disk. Our consideration is based on the distribution function of the specific angular momentum for a uniformly rotating spheroidal body (as a flattened gas-dust protoplanetary cloud) [16, 31, 32] and the distribution function of the specific angular momentum of the rotating outer shell formed by particles leaving the spheroidal body due to thermal emission .
The structure of the paper is as follows: in Section 2, we use Schmidt’s cosmogonical hypothesis  of evolution from the flattened protoplanetary gas-dust cloud (to the emerging protoplanets) modeled by a uniformly rotating spheroidal body with a distribution function of the specific angular momentum (which has been derived within the framework of the proposed statistical theory of the formation of cosmogonical bodies [16, 31, 32]). In other words, in Sections 2 and 3, the original (relic) flattened protoplanetary gas-dust cloud with a distribution function according to the model of a spheroidal body [16, 31] (inside which the central core (protostar) is formed) is considered given. The nature of the origin of this protoplanetary cloud is not considered, but it is assumed that this relic protoplanetary cloud formed together with the central body (protostar), although Schmidt also admitted the possibility of its capture by the protostar during motion in intergalactic space . On the contrary, in Section 4, we develop an alternative model of protoplanetary cloud formation due to thermal emission of particles from a protostar being in different thermodynamic states. The possibility of different thermodynamic state realization was substantiated by Boss , and theoretically, it is confirmed in the framework of the proposed statistical theory of the formation of cosmogonical bodies [16, 35] (by existence of the multiple states of virial equilibrium, see Corollaries 2.2, 4.3 in the monograph ). Thermal emission of particles from a protostar being in multiple thermodynamic states contributes to the additional formation of the gas-dust cloud (as well as the forming protoplanets) with a thermal distribution function of the specific angular momentum of particles leaving the central body (protostar) due to their thermal motions. It is noteworthy that both models (with distribution functions and respectively) lead to similar laws of planetary distances in extrasolar systems.
2. Equations of the Distribution of the Specific Angular Momentum in Protoplanetary Cloud
The cosmogonical hypothesis of Schmidt, on the origin of the solar system as a result of the evolution of gas-dust swarm, was used to show that the swarm condensation process takes place necessarily (even if there were no initial bunches or protoplanetary embryos), according to the following scenario [4, 38, 39]: (a)As a result of collisions, the relative velocities of particles decrease, and thereby, the system becomes denser, so that the flattening of cloud as a result of joint action of central gravitational and centrifugal forces takes place(b)After reaching a certain critical value of the density, the system cannot remain in its previous state, and under the action of gravity, the formation of condensations (the so-called planetesimals) begins [6, 11, 15](c)These condensations (planetesimals) have a flattened shape and masses of the order of the masses of asteroids(d)Then, the condensations collide (due to the small mean free path) and merge into a small number of large bodies, called protoplanets
This scenario is confirmed by numerous results of computational modeling. Indeed, the last four decades of research have shown that numerical modeling has become an important part of understanding the evolution of the solar system.
A contemporary understanding holds that the evolution from dust to planet can be divided into three successive phases (see for example ). The first main stage is the coagulation of micrometer-sized grains into kilometer-sized planetesimals (see [6, 41–45]). The planetesimals may form as a result of gravitational instability in the solar nebula, in which solids are sufficiently concentrated to enable planetesimals forming purely by self-gravity (e.g., [41, 46, 47]). Along with that, planetesimals may form by the direct collisional accretion of colliding particles (e.g., [43, 48, 49]), i.e., the grains are assumed to stick together during the impacts that occurred at a critical, threshold velocity . Due to the stochastic nature of such collisions, not all planetesimals grow at the same rate and some will become more massive than others. More massive bodies are more effectively able to accrete the surrounding planetesimals. This quickly leads to a runaway accretion process (e.g., [51–54]), which begins the second main phase. In a swarm of planetesimals, the relative velocity is governed by their frequent mutual encounters and, given their small gravity, is kept low . Runaway growth starts when the planetary embryos grow large enough and their gravitational perturbations determine of the planetesimals. As this is a case, planetesimals decouple from the gaseous disk and interact gravitationally with each other. A runaway growth leads to the formation of a planetary embryo with masses 1%–10% of that of Earth [55, 56]. These embryos accrete substance locally and form a dense population distributed throughout the solar system. In the third stage of terrestrial planet accretion which is the final one, the gravitational effect of the planetesimals begins to fade as their number decreases and the planetary embryos begin to perturb one another onto crossing orbits. The planets grow in the course of collisions between embryos and the accretion of the remaining planetesimals. This stage is characterized by relatively violent, stochastic collisions, as compared to the previous stages with dominating continuous accretion of small bodies [40, 57].
In contrast to the numerical modeling of planet formation [41–57] with enormous computational efforts, the present paper relies on analytical principles only. Although the numerical models seem to be quite useful, it is always good to check, at least one of the mentioned theoretical approaches, whether the results are still reasonable.
The evolution of a rotating flattened and gravitating spheroidal body, analyzed within the framework of the proposed statistical theory in this paper, is useful, in particular, for understanding of the origin of the solar system [16, 31, 32, 34]. Let us briefly consider the evolution of two neighboring bunches (protoplanetary embryos) being in the growth stage. As Schmidt noted  (p. 32-33), “if their orbits are very close, they will quickly exhaust the supply of bodies and particles moving in the region between their orbits. If two planetary embryos do not unite into one, then in the future they will acquire mass and momentum already predominantly from bodies turning from the outer sides of the exhausted zone. In this case, the momentum per unit mass of one planet will decrease, the other will increase, and the radii of the orbits of two planets will begin to diverge. Thus, in the process of growth of the planets at the expense of bodies and particles, it lies the principle of adjusting the distances between them.”
Following the logic of Schmidt’s reasoning, one can find the law for the distances between bunches (protoplanetary formations) in a rotating spheroidal body, bypassing the detailed kinetics of the process. Namely, his model (explaining the law of planetary orbital distances) is based on the hypothesis that each law of distribution of a specific angular momentumof particles with a distribution functioncorresponds to its law of planetary distances .
It follows from the fact that when planets are formed (for example, in the solar system), each particle (or a conglomerate of particles, e.g., a planetesimal [6, 11, 46–49]) in a gas-dust protoplanetary cloud (or a swarm of planetesimals) has the highest chance to hit the forming protoplanet, if its specific angular momentum value is close to those of the particles which form the given planet. Although individual particles may not fall into their protoplanetary forming bunch, following Schmidt  on p. 33, “these deviations are mutually compensated, so that for the calculation it can be assumed that all particles are precisely distributed over the ‘areas’ around the axes of specific angular momenta for each planet. The boundaries of these areas are assumed equidistant from the specific momenta of two neighboring planets”.
Let be a value of specific angular momentum, which corresponds the boundary between the domains of th and th bunches (or protoplanetary embryos) in the flattened gas-dust protoplanetary cloud, whose specific angular momentums are equal to and , respectively. Then, following the hypothesis of Schmidt, we can see that
During the process of a conglomeration of particles of gas-dust media in a bunch, their specific angular momentums are averaged and the specific angular momentum of the bunch (planetesimal), as a forming protoplanet, is the ratio : where is a distribution function of the specific angular momentum . However, Schmidt could not analytically derive the form of the distribution function within the framework of his model, noting only that changes with the time in the process of cloud evolution, but this is still an unsolved problem  (pp. 35-36).
Using Schmidt’s cosmogonical hypothesis of the evolution from a flattened protoplanetary gas-dust disk to an ensemble of protoplanets [4, 38], let us consider a gravitating flattened protoplanetary gas-dust cloud, rotating with a uniform angular velocity and with a distribution function of the specific angular momentum . According to the statistical theory of gravitating spheroidal bodies (see refs. [16, 31, 32, 34]), the probability density distribution function of the specific angular momentum is where is a parameter of gravitational condensation and is a squared eccentricity. The mass density function of a uniformly rotating protoplanetary gas-dust disk as a spheroidal body in a spherical frame of reference has the following form [16, 31, 32, 34]:
In the case of , the mass density function of a slowly rotating () spheroidal body becomes the following: where is a spatial (radial) coordinate and is mass density in the center of the spheroidal body of the mass and gravitational condensation parameter .
In another limiting case of , equation (4) describes the mass density of a flattened gaseous disk [31, 34] in agreement with the known barometric formula (for a flat rotating disk) obtained with the usage of the hydrostatic mechanical equilibrium condition [6, 11, 39]: where is a value of mass density in a central plane (layer) of this gaseous disk in the cylindrical frame of reference , . Here, is the total mass of an initial prestellar molecular cloud, i.e., the mass of the star plus the mass of the gaseous disk [16, 31, 34].
Obviously, the probability density function (3) satisfies the normalization condition because
Let us calculate an average value of specific angular momentum based on the integration by the parts with the usage of formula (7a):
Now, let us consider the general derivation of the law of planetary distances within the framework of the statistical theory of spheroidal bodies (without any restriction like ).
3. A Uniformly Rotating Spheroidal Body Model for the Protoplanetary Cloud Formation: Derivation of the General Equation of Distribution of the Specific Angular Momentum of Forming Protoplanets
Taking equation (7b) into account, we obtain the following:
Multiplying both sides of equation (11) by , we obtain the following:
In the limiting case of , using L’ Hospital’s rule, equation (12) yields true identity:
Let us consider some particular cases of equation (15): (a)If the condition is true, then, using representation of the exponential function in equation (15) by Maclaurin’s series restricted up to the linear term only we obtain from equation (15) the following: whence follows a uniform law of distribution of the specific angular momentums of the bunches (planetesimals): (b)Still taking the condition in equation (15) and considering up to the quadratic term in the Maclaurin’s series of the exponentequation (15) yields the following: that is,
The characteristic equation for the second-order difference equation (21) has the following form (): whose solutions are two identical roots , i.e., the solution with a multiplicity . This means that the linear difference equation of the form (21) has a solution in the form of an arithmetic progression: described by Schmidt’s law . Based on the logical deduction of (21) and (23), we conclude that the following constants and should be chosen: where is a difference of the arithmetic progression.
Using Binet’s formula, the equation of an orbit of a body in a remote zone of a rotating spheroidal body has been derived in . As a result, for a protoplanet/planetesimal (or a particle) of a stellar–planetary system (including our solar system), moving in an elliptic orbit with a major semiaxis and geometric eccentricity , the value of the orbital specific angular momentum can be determined by the following well-known formula: where is a double-areal velocity of the orbital motion of a protoplanet (or a particle). So, the orbital specific angular momentum for th forming protoplanet , where is an orbital angular momentum and is a mass of the th forming protoplanet, is as follows: where is the major semiaxis and is the geometric eccentricity of orbit of the th protoplanet. Substituting (24a), (24b), and (25) into equation (23) gives the known Schmidt’s law of planetary distances :
If , then, it follows directly from equation (26) that
So, taking into account (26) and supposing (the condition of almost circular orbits), Schmidt’s law of planetary distances occurs: where , are constants. The following theorem can be formulated:
Theorem 1. If the mass density of a spheroidal body satisfies the condition:, which is equivalent to the requirement of a uniform law of angular momentum distribution function: , then, Schmidt’s law (28) takes place.
Proof. According to the condition of Theorem 1 and taking into account the relation [16, 31, 34], we find the following:
where is a volume probability density function in the cylindrical frame of reference [16, 31, 34]. Let us note that a one-dimensional probability density function of particle detection along the radial coordinate can be found through the volume probability density function based on the following relation [16, 31, 34]:
where is the particle distribution function with respect to the radial coordinate in a rotating (with a uniform angular velocity ) spheroidal body, that is, one-dimensional probability density of finding a particle in a uniformly rotating spheroidal body at the distance from the axis of rotation. On the other hand, we obtain the share of particles, located at distances close to from the axis of rotation, -axis, which is equal [16, 31, 34]:
As shown in [16, 31, 34], the share of particles located near the distance close to from the -axis of rotation, i.e., within a volume of annular cylindrical layer , is equal to the share of particles rotating with a constant angular velocity around the -axis and having the values of specific angular momentum in the interval :
Evidently, relation (32) is valid for a rotating spheroidal body in the state of relative mechanical equilibrium [16, 31, 34]. Under this condition, the moving particles being at distance from the -axis have circular orbits within a uniformly rotating spheroidal body, and therefore, the -projection of the specific angular momentum
is directly proportional to the square of distance from the rotation -axis within a uniformly rotating spheroidal body. Consequently, the share of particles having a specific angular momentum value in the interval , in accordance with equations (32) and (35), is equal to
where is a specific angular momentum. So, the probability that the value of specific angular momentum belongs to the interval is equal:
Therefore, the probability density function , which expresses the mass distribution via the values of specific angular momentum, is indeed described by formula (3). According to (32)–(37), we obtain the following:
Taking into account condition (29), equation (38) becomes the following:
whence we obtain that the function must be approximated by a uniform distribution law of the following kind:
Let us also note that relation (40) can be obtained from equation (3) directly in the assumption of the smallness of an inverse average value of the specific angular momentum since (see [16, 35]):
This also justifies the assumption of used in the above-considered cases (a) and (b). Taking into account equation (40), formula (2) becomes the following:
Substituting equation (1) into (42), we obtain the following difference equation:
It is clear that equation (43) describes the well-known property of an arithmetic progression whose th term is calculated by the following formula:
where is the difference and is the first (the zeroth) term of an arithmetic progression.
Taking into account equation (44) and formula (25) for the relation of the specific angular momentum with the square root of radius for orbit for theth protoplanet (under the condition of the circular character of the planetary orbit when ) we conclude that zeroth approximation of function leads to Schmidt’s well-known law: where are some constants. The theorem is proved.
Corollary 2 (conclusion of Laskar ). For a constant surface mass density (constant distribution) , the orbital major semiaxes of formed planets satisfy the following relation: , that is, Schmidt’s law of planetary distances (28).
Proof. Indeed, taking into account that where is a surface mass density; we establish from equation (29) that whence, bearing in mind that , we derive the condition of Laskar (constancy of the surface mass density) as follows: because is a radial coordinate along which the major semiaxes of planet orbits are measured. That proves the corollary.
Now, let us consider another case investigated by Laskar , namely, when
Therefore, according to the definition in equation (1), we can determine the value of specific angular momentum along the border between th and th protoplanets: i.e., corresponds to the value whereas corresponds to the value in accordance with equation (55). According to (38), (47), and (51), we obtain the following:
According to equation (7b), and also taking into account that that is, we obtain the following:
Taking into account that equation (72) with sufficient accuracy is approximated as follows: or
Exponentiating equation (75), we find that that is, whence
Summarizing equation (76c), we find the following:
Taking the logarithm of (79), we obtain the following relation: that in the case of constants selecting and passes into the law of planetary distances of Gurevich and Lebedinsky  and Laskar : which confirms Laskar’s second proposition (50). It is clear that equation (80b) corresponds to the exponential type of law: where . Obviously, equation (80c) generalizes several empirical laws proposed by Nieto , Murray and Dermott [59, 60], Poveda and Lara [61, 62], Flores-Gutierrez and Garcia-Guerra , and others. (c)Concerning condition (41), let us now take a cubic approximation of the following exponent:
And taking into account the approximation we have whence
The characteristic equation for the difference equation (85) has the following form: or whence we find that
Characteristic equation (88) is reduced to two equations:
The solution of equation (89a) is the root of a multiplicity of , and the solutions of equation (89b) are th power roots with sufficient accuracy (under condition (41): , that is, ). As a result, nonlinear difference equation (85) at the linear approximation () has a general solution [16, 31, 34]:
Taking (25) into account, equation (90) yields the law of planetary distances of the following kind [16, 31]: if and , , are coefficients to be sought for a planetary system (for example, the solar system [16, 31, 34]) in the form of dependence on . Therefore, the proposed law for planetary distances (91) generalizes Schmidt’s well-known law (28). As shown in , this new law gives a very good estimation of real planetary distances in the solar system (0% for the relative error of estimation and 1.4% for the absolute error). In addition, its maximal value is equal to 11% for the Earth, but for Pluto, the proposed law gives too high an error according to the derived rule for the maximal number of necessary coefficients in the law (91). This fact could be considered as an additional argument for a specific nature of Pluto, which appeared in the solar system in a different way, as other known planets .
4. Thermal Emission of Particles Model of Protoplanetary Cloud Formation: The Derivation of Distribution Functions of Moving Particles in the Gravitational Field due to Thermal Escape of Particles in the Outer Protoplanetary Shell under Formation
As already noted, the main challenge of modern cosmogonical theories is the problem of angular momentum distribution in the solar system. While the Sun constitutes more than 99% of the total mass of the solar system (the total mass of all planets is equal to only 1/745 or 0.13% of the Sun’s mass), it has less than 2% of the total angular momentum, i.e., the remaining 98% belong to the planets exclusively [5, 24]. In our opinion , a possible explanation for the angular momentum shift to the periphery of the solar system is caused by thermal and gravitational instabilities of a forming spheroidal body. Indeed, the rapid increase of gravitational field in an evolving gravitating spheroidal body leads to disturbances of the particle distribution function of the rotating spheroidal body and in passing from one virial equilibrium state to another (at the given temperature) [16, 35]. As already stated here, the transition between different thermodynamic states is explained by existence of the multiple states of virial equilibrium of the gravitating spheroidal body under its formation [16, 35].
As noted regarding equations (29) and (30), the equilibrium distribution functions, e.g., volume density of probability function and one-dimensional probability density of finding a particle in a uniformly rotating spheroidal body in the cylindrical frame of reference , are described by the following:
Jeans  found (on p.68) that under transition to a new state of virial equilibrium, it is possible that the temperature of gravitating gas (nebulae) increases due to the energy of the gravitational field, whose potential in the case of a rotating spheroidal body is described by the following formula (see in , p. 299):
This means that an unstable state of virial equilibrium can be violated with the increasing temperature of a spheroidal body. Moreover, the temperature increase leads to an increase in the kinetic energy of the thermal motion of particles, so that many of them acquire the mean square velocity of thermal movement becoming greater than the escape velocity from the spheroidal body, that is, . The separation process of the spheroidal body leads to the formation of its inner zone I (a stellar core) and remote zone II (an exterior shell) . While leaving the core of the spheroidal body, these particles begin moving on Keplerian elliptical orbits (see ): where is a major semiaxis of protoplanets and is an eccentricity of a particle orbit. Obviously, we can find the critical velocity of the escape of particles from this core of spheroidal body in a remote zone II when ( is a point of mass density overfall [16, 34] considering as a radius of the inner core) excluding the potential of centrifugal force [16, 31, 33]:
If we take into account the potential of the centrifugal force [34, 64], the escape velocity of particles from the core of a rotating spheroidal body can be calculated through the potential of gravity as follows:
In connection with this, we need to estimate a number of particles which leave the rotating spheroidal body (due to their thermal chaotic motions) and move along Keplerian elliptical trajectories in its gravitational field. According to  (p. 364), the particle distribution function of a self-gravitating gas-dust protoplanetary cloud obeys Jeans equation. As shown by Jeans  (p. 371), the joint distribution function of the spatial coordinates and the corresponding velocity components for such particles is described by the following expression (see also [16, 34]): where . Substituting the integration constants and into Eq. (97) in accordance with  (p.130-132) gives us the joint Jeans distribution function of spatial coordinates as well as velocity components for particles in a gravitational field: where is the Boltzmann constant. In particular, as follows from equation (98), the Jeans distribution function of the spatial coordinates of particles in the gravitational field of a uniformly rotating spheroidal body with the gravitational potential at an interior point  (p. 455)