Subsections


Introduction

The question of the origin of the Solar System, especially the origin of the Earth, is one which mankind has long strived to answer. Throughout history, many theories have been suggested. The earliest of these had religious roots, and usually placed the Earth at the centre of the universe. As scientific thought progressed, the egocentricity of man was put to one side, and theories based on observation emerged. With the advent of modern computers, it has become possible to test the plausibility of even the most computationally challenging of theories. This thesis is concerned with the development of modern sophisticated tools, to the point that the plausibility of one such theory may be tested.

The Capture Theory for the origin of planetary systems, was originally proposed by Woolfson (1964) to address the question of angular momentum distribution in the Solar System, a question which nebula theories have yet to convincingly solve. A two body encounter is not as unlikely as was once believed, as it is now thought that stars form in dense clusters, and that close encounters between them may be common (e.g. Gaidos, 1995). Woolfson proposed that such an encounter may cause a compact star, the Sun, to draw out a filament of material from a less massive and more diffuse star, a protostar, in such a way that a part of the filamentary material drawn from the protostar is captured into orbit about the compact star. Gravitational instabilities within the filament leads to its fragmentation into several protoplanetary condensations. The subsequent gravitational collapse of the central regions of these are associated with the major planets of the system. The material of the captured filament which does not condense into protoplanets, forms a resistive medium, which serves to round off the initial highly eccentric orbits of the protoplanets. In addition, the rounding-off process causes the orbits of the protoplanets to precess, such that they may collide, and it is postulated that the terrestrial planets and their satellites are the nonvolatile products of such a collision.

The first numerical simulations of the Capture Theory (Woolfson, 1964) demonstrated that material from a less massive star could be tidally drawn out, and subsequently captured into orbit around a more massive compact star. Due to the nature of these early experiments, little more than the plausibility of the theory could be demonstrated. Subsequent work (Dormand and Woolfson, 1971; Coates, 1980) involved little increase in spatial resolution, but advances in numerical techniques meant that the underlying physical processes could be modelled more accurately. In all previous work to date, the spatial resolution has been such that it has not been possible to simulate the capture of material, and the subsequent fragmentation and condensation into protoplanets, during the same simulation - auxiliary simulations, the starting parameters of which were obtained from the properties of the material in the filament at the end of capture simulations, were necessary when investigating protoplanetary evolution. This led to an increase in the scope available for making incorrect assumptions and estimations. Indeed, in the very first simulations, it was assumed that once the filament had been drawn from the protostar, the tidal field from the Sun, which was responsible for the tidal disruption in the first place, was subsequently unimportant, and could therefore be omitted in further analysis.

The main achievement of this thesis is the advancement of an existing numerical method, a spatial tree implementation of SPH, to include radiation transport. With the inclusion of this, and also the increase in computational power and general advances in numerical techniques, the spatial resolution and physical accuracy is now such that for the first time, it is possible to perform continuous simulations of the capture and fragmentation mechanisms.

Thesis Outline

The remainder of this chapter is devoted to theories of cosmogony. Two early prominent theories that serve to demonstrate the nature of the problems faced by such theories are discussed, along with the criticisms which led to their abandonment. In the course of this investigation, concepts and analysis relevant to the work presented in later chapters are developed. A brief critical review of the most widely accepted planetary formation mechanism, the Solar Nebula Theory, is given, followed by a rèsumè of the Capture Theory.

The smoothed particle hydrodynamics (SPH) technique is developed in Chapter 2. Attention is drawn to several new additions to the SPH method, developed during the course of this thesis, namely, an efficient method of enclosing an exactly constant number of particles within $2h$, a method of including a more physically realistic equation of state, which is also suitable for changes of state, and the development of an efficient second order integration scheme with automatic time-step control.

In Chapter 3 the concept of a spatial tessellation tree is introduced, the implementation of which can significantly reduce the computational run-time, without incurring an appreciable decrease in physical accuracy. The major computational achievement of this thesis, is the development of a new radiation transport algorithm, which is particularly suited to spatial tessellation tree implementations of SPH. A preliminary investigation, which reveals the nature of the problems faced in incorporating radiation transport into SPH, is first presented, followed by an in-depth presentation of the new radiation transport algorithm.

Chapter 4 is a presentation of the tests performed whilst developing the code. Standard dynamical tests, both gravitational and hydrodynamical, demonstrate that the code is clearly capable of reproducing analytical results. Whilst studying the behaviour of polytropes, a novel method for determining their experimental radius is implemented, and it is shown that the results obtained from these experiments, contrary to those presented elsewhere in the literature, agree remarkably well with theory. During the testing of the radiation transport algorithm, attention is drawn to its deficiencies as equally as much as to its strengths.

In Chapter 5 the code that is developed and tested in earlier chapters, is used to perform investigations into two areas of the Capture Theory. An investigation into the behaviour, especially the lifetime, of slowly contracting bodies is first presented. This is particularly relevant to the Capture Theory, as potentially both the protostar and the newly formed protoplanets could be in this state. It is then discussed why the filament drawn from a slowly contracting protostar is not conducive to planetary formation. Finally, preliminary results from a new type of encounter, which is conducive to planetary formation, are presented.

Chapter 6 summarises the main contributions and conclusions of this work, and suggests the possible directions in which future work may take.

Introduction to Cosmogony

Theories of planetary formation fall into two categories: Monistic theories, and dualistic theories. Those which are monistic, propose that planets are formed from the same local material which collapsed to form the central star, a star and its planets effectively forming in isolation. In this co-genetic formation of planets around stars, the process of forming planets is strongly coupled to that of forming stars, a premise which shall be presently discussed.

In the second category of planetary formation theories, dualistic formation, it is proposed that the planet forming mechanism is separate from the star forming one, and that planets result following an interaction with a secondary body (or bodies) at a later time. The planetary formation process is therefore almost completely decoupled from the stellar formation process. Most theories of this type simplify the model considerably, by assuming that the primary body (the central star) resides on the main sequence, and, in the case of modelling the formation of the Solar System, assume that at the time of interaction with the secondary body, the Sun was in its present state.

Monistic theories are attractive in the sense that the process of forming planets is a natural by-product of forming a star, a feature which dualistic theories lack. However, although a particular process may be aesthetically pleasing, this does not mean that it is necessarily correct. For example, if it is determined observationally that only a small proportion of stars possess planetary systems, then theories which are monistic have to provide an explanation for this.

Two early dominant theories that exemplify the nature and problems associated with each type of theory are now examined. In the course of this examination, concepts and analysis relevant to the work presented in later chapters are developed.


The Laplace Nebula Theory: Monistic

Nebula theories had been proposed by the philosophers Emanuel Swedenborg (1688-1772), published in Philosophical and Mineralogical Works: Volume One, The Principia in 1734 and by Immanuel Kant (1724-1804), published in General History of Nature and Theory of the Heavens in 1755, but they were based on little more than speculation. These nebula theories followed the discovery of `fuzzy stars', coined nebulae, earlier in the $18^{th}$ century by observational astronomers. The first widely accepted scientific based theory, was that proposed by Pierre Laplace (1749-1827), published in Exposition du Système du Monde in 1796. Laplace proposed that it was the collapse of these hypothesised clouds of gas, which produced planets and stars simultaneously via a process shown schematically in Figure 1.1.

Figure 1.1: A schematic representation of the Laplace Nebula Theory. (a) A slowly rotating and collapsing cloud of gas and dust, (b) The collapsing nebula flattens along its rotation axis to form an oblate spheroid, (c) Formation of a lenticular shape, (d) A series of rings is left behind by the contracting core, (e) One planet is formed from the material in each ring.
\begin{figure}\centering {\epsfig{figure=/home/steve/tmp/hold/thesis/graphs/laplacea1.eps,angle=0,width=1.0\linewidth}} \end{figure}

Originally, a slowly spinning cloud of gas cools, and begins to collapse under self gravity to form a spherical shape, Figure 1.1(a). As it collapses, it spins faster to conserve angular momentum, and thus begins to flatten along the spin axis, Figure 1.1(b). Eventually, after an order of magnitude of collapse or so, it adopts a lenticular shape, Figure 1.1(c), within which the outermost material freely orbits the forming star. Further collapse gives spasmodic loss of material in the equatorial plane, leading to the formation of a series of annular rings, Figure 1.1(d). Gravitational attraction of gas within a ring leads to small clumps developing. Differential rotation of the clumps provides a mechanism by which they can combine to produce one planet per ring, Figure 1.1(e). A smaller version of this scenario, based on the collapse of protoplanets, leads to the formation of satellite systems.

This monistic theory results in the angular momentum of the system residing in the same place as the mass. The observational facts of the Solar System are in complete opposition to this: The Sun with $99.86\%$ of the mass, has only $0.5\%$ of the total angular momentum contained in its spin, the remaining $99.5\%$ residing in the orbits of the planets. Other less important problems with this theory became apparent with time. The strongest criticism being made by James Maxwell (1831-1879), who in an essay on the rings of Saturn in 1857, showed that if they were to be stable, they had to be comprised of small solid particles - rigid solid rings would have been torn apart by differential rotation, and gas rings would have dispersed quite readily. The same argument was applied to the rings of planetary material proposed by Laplace, necessitating that the rings should be hundreds of times more massive than the planets they were to form, in order that they may be durable.

Revival of the Laplace Model

In 1854, in an attempt to overcome the angular momentum problem, Edouard Roche (1820-1883) postulated that the initial mass distribution in Laplace's model, may in fact not be uniform, but highly centrally condensed (Roche, 1854). If a uniformly rotating cloud with a mass distribution of this type, were to collapse to produce a star and planet system as envisaged by Laplace, then the angular momentum of the central body would naturally be smaller. Indeed, given an arbitrarily steep initial density profile, the gross angular momentum distribution observed in the Solar System could be satisfactorily accounted for. The initial high density profile could be accomplished by assuming that the star formed in isolation, and subsequently captured the sphere of planetary material. This mechanism would result in the Nebula theory becoming a dualistic process.

James Jeans (1877-1946) returned to the centrally condensed initial configuration of the Laplace model envisaged by Roche (Jeans, 1919). Using an argument, for which Roche himself provided the analysis, he showed that in the centrally condensed hypothesis, the surrounding material would be so diffuse, that tidal forces from the forming star would prevent planets forming in the rings. Since these ideas of tidal disruption are relevant to this thesis, a brief overview and derivation is now given.

The Roche Limit

Conceptually, the Roche limit is the distance within which natural satellites are torn apart tidally by the body they orbit, this being the reason why there are no large natural satellites within the Roche limit of the major planets, only rings of material. A more common definition of the Roche limit, defines it to be the distance within which a gravitationally bound liquid satellite of uniform density, begins to lose material to the body it orbits. In this definition, it is assumed that there are no forces present other than gravity. If the body is held together mechanically, as is the case for man-made satellites and to some extent natural satellites, the disruption distance is much less than the theoretical Roche limit. Although the Roche limit is not a precisely defined quantity, the latter definition allows convenient simplification to be made to an otherwise very complicated analysis.

Given the latter definition, there are many approaches which can be used to derive the Roche limit. All approaches lead to the same dependence on the properties of the bodies involved, the only difference being the numerical scaling coefficients, which usually only differ from one another by a factor of two or less. The correct dependence, but with an approximate numerical scaling coefficient, can be obtained by considering Figure 1.2.

Figure 1.2: Deriving the Roche limit dependence.
\begin{figure}\centering {\epsfig{figure=/home/steve/tmp/hold/thesis/graphs/rochesimpa1.eps,angle=0,width=0.9\linewidth}} \end{figure}

In this diagram, a satellite with mass $m$, and radius $r$, is at a distance $d$ from the body it orbits of mass $M$. To simplify analysis, it is assumed here that neither of the bodies suffer any deformation, in which case the primary can be modelled as a point mass. For the purposes of this derivation, the Roche limit is the distance whereby a piece of surface material of the satellite of arbitrary mass $\mu$, is tidally attracted to $M$, more than it is gravitationally attracted to $m$:
\begin{displaymath} F_{tidal}=F_{binding} \end{displaymath} (1.1)


\begin{displaymath} \frac{2GM\mu r}{d^{3}}=\frac{Gm\mu}{r^{2}} \end{displaymath} (1.2)


\begin{displaymath} d=0.78\Bigg(\frac{M}{\rho_{m}}\Bigg)^\frac{1}{3} \end{displaymath} (1.3)

where $G$ is the gravitational constant, and $\rho_{m}$ is the density of the satellite. In this derivation, the distance $d$ is the associated first approximation to the Roche limit of the whole satellite.

More rigorous analysis, in which the satellite is allowed to deform into an egg shape, gives a numerical coefficient almost twice that determined in Equation 1.3, approximately equal to $1.50$. This larger value was confirmed experimentally using the particle technique introduced in Chapter 2, for various primary-secondary scenarios:

An incompressible fluid is conveniently modelled using a polytropic equation of state, with a high ratio of specific heats, $\gamma\approx 20$.

Figure 1.3: Roche limit example for an incompressible fluid modelled by a $\gamma =20$ polytrope a tenth the mass of the primary. The curved arc centred on the primary is the Roche limit of the satellite, calculated using the maximum density in the satellite and the mass of the primary.
\begin{figure}\centering {\epsfig{figure=/home/steve/tmp/hold/thesis/results/roc... ...he/gammaequal20/disp0014.eps,angle=0,width=\rochewidth \linewidth}} \end{figure}

Various primary - secondary mass ratios were examined. Figure 1.3 shows the gradual deformation of a liquid satellite a tenth the mass of the primary. The approaching satellite is modelled by $4279$ particles, of which a thin slice of particles in the x-y plane, with radius proportional to the interaction distance (the smoothing length), is shown. The satellite is spin locked to simplify analysis, achieved by subtracting the centre of mass motion of the satellite from all particles, a procedure which results in a co-rotating frame of reference. The orbital distance of the satellite was reduced at a rate such that it responded quasi-statically. The shape of the satellite is the same viewed along the y-axis, as viewed along the z-axis shown, i.e. the deformation is axially symmetric along the line joining the bodies..

The fifth frame of Figure 1.3 corresponds to the closest the satellite could approach the primary without losing material to it, and as seen, the centre of the satellite is approximately located at the Roche limit: The Roche limit is indicated by the arc centred on the primary. It is calculated using the mass of the primary, and the maximum density present in the satellite of the modified ($1.50$ coefficient) Equation 1.3. An average density calculated using the volume and mass of the satellite would have perhaps been more appropriate in these experiments, but in general this information is not available. A dynamic flow of material onto the primary is the eventual result demonstrated in the bottom two frames.

Figure: Roche limit example for a gas body in equilibrium modelled by a $\gamma=\frac{5}{3}$ polytrope a tenth the mass of the primary. The curved arc centred on the primary is the Roche limit of the satellite, calculated using the maximum density in the satellite and the mass of the primary.
\begin{figure}\centering {\epsfig{figure=/home/steve/tmp/hold/thesis/results/roc... .../gammaequal1.66/disp0001.eps,angle=0,width=\rochewidth \linewidth}} \end{figure}

Figure 1.4 shows an example scenario more relevant to this work. Here, a compressible fluid (a gas) satellite, with the same primary-secondary mass ratio as above (10:1), is modelled using a $\gamma=\frac{5}{3}$ polytropic equation of state. For comparison purposes, the test conditions were set to be almost identical to the previous scenario. It is seen that the Roche limit is much smaller than previously. This is due to the maximum density of the satellite being considerably larger than the average density. Observationally it is seen that when the surface of the satellite crosses the Roche limit, material starts to be lost to the primary. As in the previous scenario, once this occurs, the complete disruption of the satellite is inevitable.

Returning to Jeans' criticism of Roche's modification to the Laplace model, at the time a planetary ring separates from the lenticular shaped nebula, it is now clear that its density must be sufficiently large, so that the material in the ring can withstand tidal forces from all material interior to it. If this is not the case, then planetary formation from the direct collapse and coalescing of gas within a ring is not possible. With this constraint, and using similar analysis to that above, Jeans showed that the density in a ring must be 0.361 times the mean density of all material interior to it, including the contribution from the central body. The nebula must therefore have a mass comparable to that of the central body, not, as Roche suggested, a thousand times less - approximately equal to that of the planets.

Roche's modification may have been able to explain the angular momentum distribution in the Solar System, but it was unable to provide planets. Subsequent to Jeans' criticism, nebula theories lost favour, and attention was turned to other mechanisms.

Jeans' Tidal Theory: Dualistic

Almost all dualistic theories were created in an attempt to explain the angular momentum distribution in the Solar System. Most early theories involved material being torn from the Sun during a close encounter with a secondary body. The solar material enters orbit around the Sun, and cools to form the planets. The first theory of this type was suggested by Georges Buffon (1707-1788) in De la formation des planètes in 1745, in which a passing comet grazed the surface of the Sun, tearing the required planetary material from it. It is now known that comets have nothing like the mass necessary to achieve this effect, although this does not invalidate the mechanism for larger sized bodies. Indeed, this type of interaction was resurrected earlier this century by Thomas Chamberlin (1843-1928)(Chamberlin, 1901), Forest Moulton (1872-1952)(Moulton, 1905), Jeans (Jeans, 1917), and Harold Jeffreys (1891-1989)(Jeffreys, 1918). From the scientific viewpoint, Jeans' description is the most complete, and hence is the one now considered.

In Jeans' Tidal Theory of 1917, a massive star passing close to the Sun pulls a filament of material from it. Unlike Buffon's original suggestion, the passing star is not required to graze the Sun. Instead, the tidal field acting on the Sun induces a distortion similar to what was demonstrated in Figure 1.3. This is shown schematically in Figure 1.5(a).

Figure 1.5: A schematic representation of Jeans' Tidal Theory. (a) A tidal bulge is induced, (b) A filament of material is drawn out in which condensations form, (c) The produced protoplanets orbit the Sun with high eccentricities.
\begin{figure}\centering {\epsfig{figure=/home/steve/tmp/hold/thesis/graphs/jeansa2.eps,angle=0,width=1.0\linewidth}} \end{figure}

If the passing star is sufficiently close, a filament of material is drawn out from the Sun. Within this filament, gravitational instability leads to several condensations forming, Figure 1.5(b). These evolve into isolated bodies, identified as protoplanets, Figure 1.5(c). The remainder of the filament forms a resisting medium also in orbit about the Sun, serving to round off the initially high eccentricities of the planets. A scaled down version of the planet formation process results in satellites forming: During the first perihelion passage of a protoplanet, the tidal field of the Sun draws a filament of material from the protoplanet, in which protosatellite condensations are formed.


The Jeans Mass

Whilst developing his tidal theory, Jeans investigated the conditions necessary for gravitational instability. In particular, given that a filament of material is drawn out from the Sun during a close encounter with a massive star, he investigated under what conditions the material is able to fragment and collapse to form planets. There are several contributing factors that need to be considered in order to make an attempt at understanding this problem. For the moment, two of these factors, tidal disruption and energy loss by radiation, are neglected, and a first approximation to the properties of the filament are obtained by considering the thermal and gravitational energies present.

In deriving the properties of the filament necessary for fragmentation, it is first useful to consider the contracted problem of an isolated spherical distribution of gas, of uniform density, and in static gravitational and thermal equilibrium. For this configuration, the virial theorem states that the kinetic energy of the system (contained in the translational thermal motion of individual molecules) is equal to minus half the gravitational potential energy of the system:

\begin{displaymath} \frac{3kTM}{2\mu}=-\frac{1}{2}\Bigg(-\frac{3GM^{2}}{5R}\Bigg) \end{displaymath} (1.4)

where $k$ is Boltzmann's constant, $T$ is the temperature, $\mu$ is the mean molecular mass, $M$ is the mass and $R$ is the radius of the gas. Rearranging Equation 1.4 to solve for the mass, and substituting for the density ($\rho$) results in:
\begin{displaymath} M_{J}=\Bigg(\frac{375k^{3}T^{3}}{4\pi G^{3}\mu^{3}\rho}\Bigg... ... 5.1\times 10^{21}\bigg(\frac{T^{3}}{\rho}\bigg)^{\frac{1}{2}} \end{displaymath} (1.5)

where the subscript $J$ on the mass indicates that the quantity is the Jeans mass, as a similar expression was first derived by Jeans. The approximate value on the righthand side of Equation 1.5 was calculated using a dissociated hydrogen ($70\%$ by mass) - helium ($30\%$ by mass) mix of gases1.1. The Jeans mass is the minimum mass necessary for a gas of uniform density to be stable against expansion. In practice, the type of body just described cannot be in equilibrium, and has a subsequent evolution which depends on the relationship between the temperature and the density (the equation of state). For example, an isolated body of uniform density, of mass equal to the Jeans mass, neither radiating or receiving energy, will initially expand at the surface and contract in the central regions, and if damped will oscillate until a stable static equilibrium with a centrally peaked density profile is formed1.2.

Equation 1.5 is a general result, with many astronomical applications ranging from the formation of galaxies, to the formation of satellite systems, and is used often in this thesis. To first approximation, Equation 1.5 can also be applied to non-spherical distributions of uniform density. For the case of a gravitationally unstable uniform distribution of gas, $M_{J}$ corresponds to the approximate mass contained in each fragmentation. In relation to the filament of planetary material, given that it is gravitationally unstable, Equation 1.5 can be used to determine the approximate mass contained in each condensation forming, and hence the distance between them and the number formed.

For a numerical example of the fragmentation mechanism relevant to the current discussion, consider Figure 1.6.

Figure 1.6: Condensations forming in an initially uniform density filament.
\begin{figure}\centering {\epsfig{figure=/home/steve/tmp/hold/thesis/results/jea... ...s/results/jeans/disp0010.eps,angle=0,width=\rochewidth \linewidth}} \end{figure}

In the top frame of this figure, a uniform density cylinder with a length $11$ times its diameter, is used to approximate the filament of material drawn out from the Sun. To simulate the tidal acceleration, and more importantly to prevent collapse along the x-axis, a uniform gradient velocity field is applied to the initially static $16310$ particles. The velocity field is set so that the velocity of the material at the ends is of the order of the escape velocity (achieved by trial and error). The ratio of thermal energy to gravitational energy is such that the material in a circular slice perpendicular to the x-axis is Jeans critical, i.e. the surface of the cylinder initially expands, and the inner regions initially contract (also achieved by trial and error). Only gravity, pressure, and viscous forces are present, other effects, such as tidal forces from external bodies, and radiative energy transfer, are omitted here.

The above information is essentially all that is required to recreate a numerical experiment which evolves with the same characteristics of Figure 1.6. The actual temperature, mass, and density present, is quite arbitrary, so long as the relationship between these is consistent with the above description. Assuming that the radius of the cylinder is equal to the solar radius, and that the mass is one hundredth that of the Sun, a temperature approximately equal to $5600\,K$ satisfies the above criteria. The length of the filament is then determined to be approximately equal to $0.1\,AU$, and the initial end velocities approximately equal to $\pm\,30\,km\,s^{-1}$. With these numerical values, the elapsed time between frames in Figure 1.6 is approximately $2.2$ days.

As can be seen, $4$ distinct condensations are formed. The number of condensations predicted using Equation 1.5 is approximately $8.7$, i.e. there are $8.7$ Jeans masses contained in the initial configuration of the filament. One reason why the number predicted is greater than the number observed is due to the amalgamation of condensations early in the fragmentation process. One possible way of preventing this amalgamation is to apply a larger initial velocity gradient to the filament.

Objections to Jeans' Tidal Theory

The above example of fragmentation within a filament, although not corresponding exactly to Jeans' description, demonstrates that the mechanism is at least plausible. As already mentioned, the above example is completely scalable, scaling as:

\begin{displaymath} C=\frac{M}{TR} \end{displaymath} (1.6)

where $C$ is a numerical constant determined to be approximately equal to $5.1\times 10^{15}\,kg\,m^{-1}K^{-1}$ for the mix of gases, dimensions of the cylinder, and the number of Jeans masses present in this example. $T$, $R$ and $M$ are the temperature, radius and mass of the cylinder of gas respectively. Equation 1.6 was derived by taking the reciprocal of Equation 1.5 and multiplying by $M$.

The properties of the filament were chosen so as to give favourable conditions for fragmentation, and at a first glance seem fairly realistic: A temperature close to the surface temperature of the Sun, and sufficient material in the filament, so that a Jeans mass is only slightly larger than a Jupiter mass. Perhaps the dimensions of the cylinder are rather large, especially when compared to the filament formed experimentally in Figure 1.3, and to that shown schematically in Figure 1.5. A more realistic value might be around $5$ times smaller. In order to retain the same number of Jeans masses in the filament, this change requires that the temperature be $5$ times larger, or the mass $5$ times smaller. An increase in temperature is desirable, as it is likely that the filament would be hotter than suggested, having most likely been drawn from regions interior to the surface of the Sun.

The temperature issue forms one of the early major objections to this theory. Spitzer (1939) demonstrated that if sufficient material were to be drawn from the Sun to form the planets, then the material would have had to have originated from depths within the Sun, where the temperature would be of the order of millions of degrees. Clearly, with temperatures as high as this, no amount of modification to the mass or radius using Equation 1.6, can maintain the plausibility of the fragmentation process under the current interaction assumptions. Another important objection to planets forming from hot solar material, comes from the distribution of the light elements lithium, beryllium and boron. All these are rare in the Sun, as they are consumed in nuclear reactions, but they are comparatively abundant in the Earth's crust.

Objections relating to the dynamics of the interaction have been put forward by Jeffreys (1929) and Russell (1935) amongst others. In a rather tenuous argument involving circulation, Jeffreys noted that if Jupiter did indeed form from solar material, then since in its present state it has a similar mean density to the Sun, it should also currently have a similar spin period, not the observed value of $60$ times as fast. Russell's objection is more substantial:

Figure 1.7 shows the orbital characteristics of an elliptical orbit.

Figure 1.7: The characteristics of an elliptical orbit. Shown to scale for e=0.5.
\begin{figure}\centering {\epsfig{figure=/home/steve/tmp/hold/thesis/graphs/russella2.eps,angle=0,width=0.75\linewidth}} \end{figure}

For a semi-major axis equal to $a$, and an eccentricity of $e$, the distance to aphelion is:
\begin{displaymath} Q=a(1+e) \end{displaymath} (1.7)

and the distance to perihelion is:
\begin{displaymath} q=a(1-e) \end{displaymath} (1.8)

The intrinsic angular momentum of material in an elliptical orbit about the Sun is given by:
\begin{displaymath} h=\sqrt{GM_{\odot}p} \end{displaymath} (1.9)

where $M_{\odot}$ is the mass of the Sun, and $p$ is the semi-latus rectum:
\begin{displaymath} p=a(1-e^{2}) \end{displaymath} (1.10)

It follows that since the orbital angular momentum is a conserved quantity, the semi-latus rectum distance of the orbit is also, i.e. if an elliptical orbit were to be rounded off in a process where the original angular momentum of the material is retained, the final circular orbit would have a radius equal to the semi-latus rectum of the initial elliptical orbit.

Russell pointed out that protoplanets originating from solar material, would initially have perihelion distances approximately equal to the radius of the Sun. For planets to be bound to the Sun, they must have eccentricities less than $1$, and in the most favourable condition, from the point of view of the tidal theory, $e=1$, it can be shown that the semi-latus rectum distance is only twice as large as the perihelion distance:

\begin{displaymath} \frac{p}{q}=\frac{a(1-e^{2})}{a(1-e)}=1+e \end{displaymath} (1.11)

Even if the attraction of the retreating massive star is accounted for, insufficient angular momentum can be imparted to the planetary material, to account for the orbit of Mercury, let alone the other planets. In fact, if the orbits of the protoplanets are not modified considerably, at first perihelion passage they will be reabsorbed by the Sun.

Further Comments on Jeans' Tidal Theory

In the above analysis, the effects of tidal forces and energy loss by radiation have not been included. It is relatively easy to show that for the dimensions and timescales present here, the amount of cooling due to radiation is practically zero. The tidal forces acting on the filament are however not negligible. Clearly in the presence of the massive star, tidal forces acting on the filament will be sufficiently large to prevent protoplanets from forming. Even once the massive star no longer influences the evolution of the filament, it can be shown that tidal forces from the Sun prevent condensations from forming.

In order to overcome many of the issues raised against Jeans' Tidal Theory, Jeans suggested that the interaction may have occurred at an early stage in the Sun's evolution, at a time when the Sun was much cooler and larger in extent, perhaps extending out to the orbital distance of Neptune. Indeed, most of the objections against the hot condensed Sun interaction would not apply to a cool extended body. However, this type of interaction is not without its problems. One such problem is how the newly formed planets would interact with the collapsing Sun. The analysis for this was beyond the mathematical tools available at the time, and consequently little progress was made on the development of the modified theory.

A final point to be made about Jeans' original Tidal Theory, is that if the Sun were to be tidally disrupted in the manner shown schematically in Figure 1.5, the parameters of the interaction would be similar to those in Figure 1.4. This implies that the star would have around $10$ times the mass of the Sun, and that the largest possible interaction distance would be around $4$ solar radii. If it is assumed that the massive star is on the main sequence, then its radius would be slightly greater than the interaction distance, suggesting that the interaction would be more collisional in nature than tidal, a return to Buffon's original suggestion. Obviously this argument does not apply to exotic compact bodies, such as massive black holes.

Jeans' Tidal Theory gradually lost favour due to the many complications which could not be resolved.


The Capture Theory

Before continuing, it should be stated that at the time of writing, the Capture Theory is far from being a well recognised theory of planetary formation, despite it being more than just plausible - many, if not all of the major features of the Solar System are explained by a logical sequence of events, which follow naturally from the Capture Theory scenario. It seems that the main reason that the Capture Theory, a dualistic theory, does not command popular support, is due to the overwhelming popularity of one of its competitors, the monistic Solar Nebula Theory. The Capture Theory as first suggested by Woolfson (1964), even predates the origin of the Solar Nebula Theory by several years, yet it has gained little support over the years, whilst support for the Solar Nebula Theory has increased spectacularly in recent years, despite that many problems faced by early nebula theories have not been satisfactorily resolved. The reason for the continued growth in popularity of the Solar Nebula Theory, comes from the belief held by many proponents of that theory, that planets are formed in that way, and that any problems encountered in developing the theory will therefore naturally be resolved. This belief has propelled the theory into modern literature as being the theory of planetary formation, and it is widely believed, both in the scientific community and in the general public, that the age old problem of the origin of the Solar System has been solved.

The Solar Nebula Theory

By far the most popular theory in recent times, the Solar Nebula Theory, like most nebula theories, proposes that planets are formed simultaneous to the formation of the Sun. Unlike the original Laplacian Theory, where planets are formed from the direct collapse of nebula material, the planets form from the ground up, in a process called accretion (e.g., Cameron, 1978). Observations of dust disks around new stars show that they last only a few million years, so solar-nebula theorists accept the constraint that the planet forming process must take less than $10^{7}$ years. The first stage in forming planets is that grains settle towards the mean plane of the disk. For very fine grains this might take too long, so Weidenschilling et al. (1989) suggested that the grains are sticky, and so form aggregations which settle more quickly. Once a dust disk has formed, it will break up due to Goldreich-Ward instability (Goldreich and Ward, 1973) to form planetesimals with dimensions from $100\,m$ to $1\,km$ or so. These planetesimals coalesce to form either terrestrial planets or the cores of major planets. The mechanism for this last stage was first described by Safronov (1972), who found formation timescales of order $10^{6}$ years for Earth, $10^{8}$ years for Jupiter, and $10^{10}$ years for Neptune. The problem is that there are two conflicting requirements - the planetesimals should come together quickly, to reduce the timescale, and gently, to enable coalescence to take place. Stewart and Wetherill (1988) suggested that local density and viscosity enhancements of the material in the disk, could lead to a runaway growth of planetesimals. Under these conditions, the formation timescales were reduced by two orders of magnitude. This still gave times too long for Uranus and Neptune, and there was also the problem that planetesimals in the Jupiter region did not efficiently coalesce, but were scattered throughout the Solar System.

Methods so far suggested have not resolved the angular momentum problem. Mostly they involve material spiralling in to join the central condensation, and this would give hundreds, if not thousands, of times more angular momentum than is possessed by the Sun, in fact the Sun could not form at all. The linkage of outflowing ionized material to the solar magnetic field, could remove a considerable amount of angular momentum, but the rate of outflow, and the required dipole moment for the Sun, are extremely implausible. In any case, for the mechanism to operate, it is first necessary for the Sun to form.

The Capture Process

In similarity to Jeans' Tidal Theory, Woolfson's Capture Theory is a dualistic theory, created in an attempt to explain the angular momentum distribution in the Solar System. The difference between Woolfson's theory and Jeans', is that the roles of the Sun and the interacting star are reversed. In Jeans' theory, a massive star was required so that the Sun could be sufficiently tidally disrupted to give rise to the planets. In Woolfson's theory, material torn from the passing star forms the planets, and thus the star is required to be less massive than the Sun for favourable tidal conditions. Woolfson demonstrated that if the interaction was to produce a planetary system with dimensions consistent with that of the Solar System, the less massive star must be in an early stage of its formation, so that it has a low density, and a large radius - a protostar.

In Woolfson's paper of 1964, the protostar was taken as having a mass $0.15$ times that of the Sun, and a radius equal to $14.7\,AU$. Clearly under these conditions, the Sun can be realistically modelled as a point mass object, even though in fact it may be a protosun, with a radius tens of times its current value. In these early simulations, the Sun was assumed to only interact gravitationally with the protostar. In later chapters, the effects of solar radiation heating the protostar, the filament, and the planets, is investigated.

Figure 1.8 shows a schematic representation of the capture mechanism.

Figure 1.8: A schematic representation of the capture mechanism (Dormand and Woolfson, 1989).
\begin{figure}\centering {\epsfig{figure=/home/steve/tmp/hold/thesis/scans/ct.eps,angle=0,width=1.0\linewidth}} \end{figure}

The protostar approaches the Sun on a hyperbolic orbit, and a filament of material is tidally drawn from it close to perihelion. The captured material enters orbit about the Sun, and the depleted protostar departs on a slightly more hyperbolic orbit than that on which it approached. In the early two dimensional particle simulations designed to produce the Solar System, several particles were captured into elliptical orbits, with angular momentum similar to those of the major planets, see Figure 1.9.

Figure 1.9: The loss and capture of material from a model star. Orbits marked $H$ are hyperbolic, and represent material lost from the star and not captured by the Sun. The figures marked on the orbits are perihelion distances ($10^{12}\,m$) and, in brackets, the orbital eccentricity (Woolfson, 1964).
\begin{figure}\centering {\epsfig{figure=/home/steve/tmp/hold/thesis/scans/capture642.eps,angle=0,width=0.85\linewidth}} \end{figure}

In this simulation, the protostar approached the Sun on a parabolic orbit ($e=1$).

In these early simulations, not much more than the plausibility of the capture mechanism could be reliably demonstrated using particle techniques. This was because computer resources had only just progressed to the point where using particle methods was a viable option, and consequently, the methods were new, and the resolution in them low. Calculations of whether or not the captured filament could fragment into protoplanets, were therefore performed separate to the capture simulations, using analysis similar to that presented in Section 1.3.1. However, in moving from particle simulations to Jeans type analysis, the tidal field from the Sun had to be omitted in order to numerically contract the problem. Under this simplification, it was concluded that if the captured material moved sufficiently far from the Sun before returning to perihelion, it would be able to fragment and condense into compact protoplanets, which would subsequently be able to withstand disruption at first perihelion with the Sun.

In these first simulations, the approaching protostar was assumed to be in a state of quasi-static collapse, and it was also assumed that the timescale for collapse was much larger than the interaction timescale. These assumptions were the reason that the radial surface velocity of the protostar could be set to zero. Subsequent three dimensional simulations by Dormand and Woolfson (1971), assumed that the protostar had a finite rate of collapse, governed by the rate at which it radiated energy from its surface. In order for this assumption to be valid, the protostar must be in a state of quasi-static collapse, a premise which is investigated in Section 5.1. In the work by Dormand and Woolfson, a more detailed analysis of the evolution of the captured material of the 1964 simulation (Figure 1.9), including the disruptive tidal field of the Sun, demonstrated that the material did not in fact have time to condense sufficiently before its first perihelion passage, essentially always being within the Roche limit of the Sun.

If the perihelion distance of the Sun-protostar interaction is considerably larger than the radius of the protostar (by a factor of three or so), the interaction is of the `slow' type. If the perihelion distance of the Sun-protostar interaction, is of the order of the radius of the protostar, the interaction is of the `fast' type. This terminology is an attempt at broadly classing the type of simulation, according to the relative velocities of the interacting stars at closest approach (perihelion). During a slow encounter, the protostar responds almost quasi-statically to the increasing gravitational field. A fast encounter is more impulsive in nature, such that the approaching protostar is close to perihelion before it has had chance to deform substantially. In typical slow encounters, material leaves the protostar near to aphelion, so that fragmentation within the filament has to compete against an increasing tidal field from the Sun. In typical fast encounters, material leaves the protostar close to perihelion, so that the tidal field is decreasing, resulting in more favourable fragmentation conditions. Also, in fast encounters, the distance to aphelion tends to be larger than in slow encounters, thus the filament has a longer period of time in which it can produce protoplanets, and subsequently, any protoplanets formed, have longer to contract before first perihelion with the Sun. Increasing the eccentricity of the encounter also increases the relative approach velocities - the speed of the interaction.

The simulations of Woolfson (1964), Figure 1.9, are of the slow type. Aside from investigating these simulations in greater detail, Dormand and Woolfson (1971) also investigated several simulations of the fast type. The results from one such encounter are shown in Figure 1.10.

Figure 1.10: A sample capture simulation (Dormand and Woolfson, 1971). Top frame: The protostar encounters the Sun, and a filament of material is drawn from it. Bottom frame: The properties of the filament are used to estimate starting parameters for two separate simulations, in which planets of mass $M=1.25M_{Jupiter}$ and $M=0.79M_{Jupiter}$ were formed.
\begin{figure}\centering {\framebox{\epsfig{figure=/home/steve/tmp/hold/thesis/s... .../tmp/hold/thesis/scans/planet71.eps,angle=0,width=0.89\linewidth}}} \end{figure}

In this simulation, a protostar with an initial radius of $16.7\,AU$, a mass of $0.25M_{\odot}$ and a surface collapse rate of $0.042\,AU\,yr^{-1}$, encountered the Sun in an almost parabolic ($e=1.01$) orbit, with perihelion equal to $20.7\,AU$. The top frame of this figure shows the first part of the simulation, in which material is demonstrated to be captured by the Sun. The bottom frame of this figure corresponds to the second part of the simulation, in which the end parameters of the first part of the simulation, are used as the starting parameters of two potential protoplanets in two separate simulations.

Although the initial starting configuration of the protostar is questionable, all of the simulations performed demonstrated that the capture mechanism is completely plausible, and that for fast encounters, subsequent condensation into planets, even in the presence of the solar field, can occur. Moreover, the distribution of mass and angular momentum at the end of the simulation, was demonstrated to be of the order of that observed in the Solar System.

Towards a Complete Theory of Planetary Formation

Since the first paper of 1964, and the follow up paper of 1971, much time and effort has been spent in developing a complete theory of planetary formation, with the capture hypothesis forming a central premise (Woolfson, 1984,1978c,1979,1978a; Holden and Woolfson, 1995; Woolfson, 1978b,1999a; Melita and Woolfson, 1996; Eggers and Woolfson, 1996; Dormand and Woolfson, 1980; Williams and Woolfson, 1983; Melita and Woolfson, 1998; Stock and Woolfson, 1983; Dormand and Woolfson, 1974,1977,1988; Connell and Woolfson, 1983; Schofield and Woolfson, 1982a,b). The sequence of events that lead to the production of a planetary system, which were designed to account for the formation of the Solar System, are currently believed to be as follows. For a complete review see Dormand and Woolfson (1989) and Woolfson (2000).

The Interstellar Medium and Star Formation

Starting with the observational facts, most theories of star formation aim to form a Jeans critical region of material, with the total angular momentum contained in the material corresponding to that of the Sun plus the planets. In the Capture Theory, only the Sun is produced from the collapse of such a region, and therefore the angular momentum contained in the region is required to be approximately two orders of magnitude less than that produced by nebula inducing theories of star formation. Woolfson (1979) has produced a model of star formation, whereby the majority of the original angular momentum present is retained in the relative motion of the stars. In this model, a supernova event triggers the collapse of the interstellar medium to form a dark cool cloud1.3. Turbulent motion within the cloud results in many Jeans critical regions developing, from which single slowly spinning stars are formed.

Star-Star Interactions

In the above model of star formation, more massive stars condense out of the dark cool cloud before less massive stars - a sequence supported by observation of young stellar clusters (Williams and Cremin, 1969). Observationally, it is determined that the stellar density in young clusters is of the order of $10^{4}$ to $10^{5}\,M_{\odot}\,pc^{-3}$ (e.g. Gaidos, 1995). This leads to a distance between stars of order $5000\,AU$. For every massive star that has reached the final stages of formation ( $R\approx 0.1\,AU$), there will be a certain fraction of less massive stars, which have yet to reach this stage. It is proposed that a less massive star, still in a relatively early stage of its formation, interacts with a more massive and compact star. The original assumption was that the less massive star was in a state of quasi-static collapse ( $R\approx 10\,AU$) at the time of the interaction. Recent analysis, as presented in later chapters, has led to the proposal that the less massive star was in an earlier stage of its life ( $R\approx 100\,AU$), at a time when it was collapsing almost freely. The former hypothesis leads to small interaction cross-sections, but long interaction timescales, whereas the latter hypothesis leads to large interaction cross-sections, but short interaction timescales. Given that the stars in a cluster are continually formed over a period of order $10^{6}$ to $10^{7}$ years (Williams and Cremin, 1969), encounters between compact stars and forming less massive stars could potentially be common. A review of the above parameters, which lead to interaction probabilities, and ultimately to the predicted frequency of planetary systems, is currently in progress (Woolfson, 1999b).

Planetary Formation

Subsequent to the tidal interaction, a filament of material is captured from the protostar into a highly eccentric orbit about the Sun. This filament fragments, as suggested by Jeans and as demonstrated in Figure 1.6, into several massive protoplanets. The original simulations gave protoplanetary masses of the order of half a Jupiter mass. Recent simulations presented in Section 5.2, give masses an order of magnitude larger than this. In order to confront this anomaly, it is proposed that as the protoplanet contracts, heavy elements segregate to the centre, and a large fraction of the atmosphere, containing mainly hydrogen and helium, is lost: Perhaps at first perihelion with the Sun, the cores of the protoplanets were sufficiently centrally condensed to survive the tidal force from the Sun, but the majority of outer layers were not. Another possible mechanism for removing excess mass is that much of the material of the collapsing protoplanet forms an accretion disk about it, which at a later time disperses - even though the original angular momentum of the initial protostar may be negligible, the angular momentum imparted to the filament during the course of the interaction, can result in the protoplanets containing a non-negligible amount of angular momentum in their spin. The segregation, and subsequent loss of light material, is a tentative explanation of the compositions of the gas giants, estimated from observation and theoretical calculations.

Satellite Formation

The process of forming satellites is far from certain. Although the Capture Theory is at odds with the Solar Nebula Theory where forming planets is concerned, there is no reason why an accretion type process, in the context of the Capture Theory, could not be responsible for the formation of satellites around gas giants. Indeed, angular momentum type considerations, similar to those applied to the nebula hypothesis for forming the Sun and planets, do not apply to a nebula type hypothesis of satellite formation, as the angular momentum contained in the spin of the planets is greater by two orders of magnitude than that contained in the orbital motion of their satellites. However, a possible tidal theory for their origin has been suggested: During the first perihelion with the Sun, a tidal bulge is induced in the collapsing protoplanet. As the protoplanet collapses, this tidal bulge is enhanced, forming a filament type structure in orbit about the protoplanet, in which protosatellites are formed. Although this scenario has not been investigated here, the former explanation seems more likely in the light of current knowledge.

Orbital Evolution

A natural by-product of the capture event is the capture of a resisting medium in orbit about the compact star, formed from the material of the captured filament which does not condense into planets. This resisting medium serves to round off the initially highly eccentric orbits of the planets. The gaseous component of the medium is eventually expelled from the system by radiation pressure, whilst the solid components fall into the Sun by the Poynting-Robertson effect. In the case of the Solar System, it is proposed that angular momentum added to the Sun from the dust, would over time, gradually shift the spin axis of the Sun, originally in a completely arbitrary direction, to that perpendicular to the mean plane of the planets. This would explain the small departure of only $7^{\circ}$ for the Sun's spin axis to the normal of the mean plane of the Solar System.

The Terrestrial Planets

Since the terrestrial planets are two orders of magnitude less massive than the gas giants, it is difficult, computationally, for the formation of both class of planets to be investigated simultaneously. The simulations of Dormand and Woolfson (1971) suggested that the formation of gas giants from filamentary material closer to the Sun than that which produced Jupiter, could not occur due to large tidal forces. The formation of smaller sized condensations at this same close distance, which could perhaps survive the tidal forces from the Sun, was not investigated, and therefore was not ruled out.

It was demonstrated by Dormand and Woolfson (1977) that the resisting medium which served to round off the orbits of the planets, also resulted in orbital precession, such that a collision between planets was possible. It was proposed that the terrestrial planets are satellites and remnants of the cores of two colliding gas giants, and that the asteroids arose from such an event. A planetary collision can also convincingly explain many of the finer features of the Solar System.

Explained Features of the Solar System

(,) has compiled a list of facts that any theory of planetary formation should address. The facts are separated into groups according to whether they are gross features, or relate to details of the system.

Gross features

The distribution of angular momentum between the Sun and the planets. A planet-forming mechanism. Planets to form from `cold' material. Direct and almost coplaner orbits. The division into terrestrial and giant planets. The existence of regular satellites.

Secondary features

The existence of irregular satellites. The $7^{\circ}$ tilt of the solar spin axis to the normal to the mean plane of the system. The existence of other planetary systems.

Finer details of the Solar System

Departures of planarity of the system. The Earth-Moon system. Variable directions of planetary spin axes. Bode's law or commensurabilities linking planetary and satellite orbits. Asteroids - their origin, compositions and structures. Comets - their origin, compositions and structures. The formation of the Oort cloud. The physical and chemical characteristics of meteorites. Isotopic anomalies in meteorites. Pluto and its satellite, Charon. Kuiper-belt objects. As noted by Woolfson, all but one of the above facts are explicable in the context of the Capture Theory. The question of whether the Capture Theory is capable of producing planetary systems, similar to those recently inferred from observation (e.g. Marcy et al., 2000, and references therein), remains unanswered. Although currently there seems no reason why this should not be possible, only a fuller investigation than that presented in Chapter 5 will answer this question.

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Stephen Oxley 2002-01-19