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Origin of the Solar System Hans Rickman Dept. of Astronomy & Space Physics, Uppsala Univ. PAN Space Research Center, Warsaw This compendium deals with the initial stages of the process that led from an interstellar cloud (called the presolar cloud) to the Sun and the planetary system that we are living in. We will thus describe the current state of knowledge about the formation and evolution of protoplanetary disks including the one that surrounded the infant Sun – also called the solar nebula. The basics of the physical mechanisms thought to have been at work during those stages will be explained. Historical introduction Scientific thinking about the origin of the Solar System started soon after its structure had been revealed and Kepler’s laws of planetary motion had been formulated. The most remarkable property to be explained was the nearly coplanar orbits of all the planets and the common sense of orbital motion, meaning that the whole system is rotating around a common axis. In the mid-17th century the French philosopher René Descartes speculated that the Solar System might have formed out of a universal gas stream that developed rotating eddies like the water on a river surface. The vortex of such an eddy would act as a point of concentration of matter and would form the Sun, while the rest of the eddy would rotate around it. The planets might form in the vortices of smaller eddies developing inside the large one. Just half a century later, following the work of Isaac Newton, celestial mechanics was becoming a scientific pursuit of high esteem. An era of great progress in the physical understanding of the motions of heavenly bodies had started, and this had consequences also for the development of new ideas concerning the origin of the Solar System. 1 Using simple concepts of newtonian mechanics, in the mid-18th century the German philosopher Immanuel Kant formulated the idea that a contracting, rotating cloud would develop into a flattened disk, whose central plane is perpendicular to the spin axis. Planets might have formed from local concentrations in such a rotating disk. Later in the same century the French natural philosopher and mathematician Pierre-Simon de Laplace went further along this line by suggesting that further contraction upon cooling of such a disk would be made possible by – or indeed necessitate – the shedding of rotating rings from the disk edge, where the centrifugal force would equal the force of gravity. He thus imagined that a planetary system like ours might result from a series of such shedding episodes, eventually leaving the slowly rotating, massive Sun in the middle. The works of Kant and Laplace are often referred to in common as the nebular hypothesis. It means that the Sun and the planets arose from the same formation process, more or less at the same time. However, there was no observational evidence or strict analytic proof that the model was correct, and therefore it was met with a sound amount of scientific skepticism. In fact, there was also a completely different idea that kept attracting attention for a long time. We may call it the catastrophic scenario. The first cometary orbit to be determined – the one of the comet of 1680 – had turned out to be Sun-grazing, so the idea arose that a massive object passing close to or hitting the solar surface would be able to destroy the Sun to some degree by tearing large amounts of material off it. Thus count George Louis Leclerc de Buffon advanced the idea that the planetary system had formed out of such torn-off material. At that time (the 1770’s) there was practically no way to determine cometary masses, so the fact that they often subtend immense volumes led to the suspicion that they could be extremely massive, and count de Buffon’s massive intruder was pictured as a comet (see the picture below). Not much later it was already becoming clear that the masses of comets are negligible compared to the solar mass. But after more than a century the catastrophic scenario would reemerge in a different shape, since the Kant-Laplace mechanism seemed to have unsurmountable problems. This time the model was a close encounter between two stars and the tidal stripping of near-surface material that would occur. In the early 20th century papers were written about this scenario, e.g., by Jeffreys and by Jeans, exploring the possibility of thus forming a planetary system around the Sun. However, it was later realized that a time scale of at least ∼ 10 14 yr is required in order to make such a chance close encounter between the Sun and another star a likely event. 2 Thus the formation of the planetary system would not be coeval with the formation of the Sun, and it would a priori be an extremely unlikely event. Even though one might argue that planetary systems could be very rare and thus our Solar System could be exceptional, the very low a priori likelihood was considered a serious problem in addition to many questions about how our planetary system would actually arise from the tidally stripped gaseous material. Elements of modern cosmogony The currently favoured scenario of Solar System formation has a strong though superficial resemblance to the old nebular hypothesis. However, something very important has happened. On the one hand, we now have a satisfactory theoretical understanding of the physical processes that may occur in the contracting cloud and the rotating disk. On the other hand, we also have access to a rapidly increasing amount of detailed observations of ongoing star formation including the properties of the associated protoplanetary disks. In the following we will often use the word cosmogony as a synonym for the study of the formation of the Solar System, even though the word sometimes has the more general meaning of the formation of local structures in the Universe. We can describe modern cosmogony as the union of three types of study, the first of which is the development of theoretical models based on physical knowledge and understanding. Examples of such models are: • Collapse models for molecular clouds; • Accretion disk models for the solar nebula; • Grain growth and planetesimal formation models; • Models of planetary accretion and planet-disk interaction; • Migration and planetesimal scattering models. The first collapse models came nearly a century ago, as physicists such as James Jeans considered the gasdynamic problem of the stability of an infinite, homogeneous medium. Accretion disk models build on concepts that were formulated in the 1940’s by Carl Friedrich von 3 Weizsäcker. The rest of the models deal with the formation of individual planets and are left out of this compendium. The second element of modern cosmogony is observations of star-forming regions and protostars. Some examples of the most important objects of such observations are: • The structure and chemistry of molecular clouds; • Embedded Infra-Red (IR) Sources; • Pre-Main Sequence stars (T Tauri stars) with and without disks; • Circumstellar, protoplanetary dust/gas disks. The first kind of observations belongs to the realm of radio astronomy. This means of exploration has gained momentum at pace with the development of observing facilities for mm or submm waves, where the molecules have their strongest spectral features. Thus it is a relatively young field of research, and important future developments are expected from facilities currently under construction such as ALMA (Atacama Large Millimeter Array). The embedded IR sources are regions with a large concentration of warm dust, which emit important quantities of thermal radiation. The source of energy that must feed this emission by keeping the dust grains warm enough is not seen, however, being obscured by the dust. With the aid of the IR emission we are studying collapsed objects in the process of forming stars. The same process can also be followed at later stages, when the protostar is clearly seen and may or may not have a spectrum with a large IR excess emission coming from a circumstellar disk. Such contracting stars, making their way toward their stable, main-sequence configurations, are called T Tauri stars after the prototype, a well-known variable star in Taurus. Finally, the disks in question, which may be thought of as accretion disks with various masses and accretion rates and as likely sites of planetary formation, can often be observed in more detail than just an integrated IR flux. They can sometimes be imaged, and sometimes kinetic information on the gas velocities is obtained from the molecular emission lines. The figure below illustrates some of the phenomena that can be observed in protostars. The third element is best thought of as a detective work. We try to clarify the circumstances around the formation of the Solar System – an event (“crime”) that occurred (“was committed”) about 4.5 Gyr ago – by gathering witnesses and circumstantial evidence from small body 4 populations and objects in the current Solar System. To do this we need on the one hand high-precision measurements and observations and on the other hand a good understanding of the evolutionary processes that led from the beginning into what we observe today. The most important objects used in this work are: • Meteorites with the diagnostic features of their formation; • Asteroids and comets as preplanetary remnants; • The structure of the Main Belt, the Trojan clouds, and the transneptunian population; • Planetary satellites – regular and irregular. Meteorites are studied in great detail in the laboratory so that some of their properties are found with high accuracy. For instance, they provide by far the best chronometer for the origin of the Solar System. They also provide us with almost unaltered samples of the solid material in the protosolar disk (the solar nebula). However, their messages are severely obscured by our lack of detailed knowledge on where they were formed and what history they have experienced. The study of asteroids largely aims at facilitating such interpretations and putting the meteoritical evidence into a larger framework. The comets are seen as the crown witnesses of the earliest solid substances in relatively distant regions – even more primordial than the meteorites and perhaps largely presolar. The orbital distribution of the small bodies carries important information too. It was apparently shaped by the growth and migration of the giant planets. Thus, by exploring how cometary and asteroidal orbits are distributed and by understanding the dynamical processes that worked, we may gain insights about how the planets formed and reached their final orbital configuration. In fact, the structures of the planetary satellite systems also provide important constraints on this dynamical evolution. Cloud collapse and contraction Let us consider an interstellar cloud of arbitrary type. For simplicity we assume that the cloud is spherical and homogeneous with radius Rc and mass Mc . We also assume that it is isothermal with temperature Tc . Suppose that the cloud is isolated from its surroundings so that it is not subject to any external pressure. Intuitively, the stability of the cloud will depend on (1) its self-gravity that strives to keep it together, and (2) its internal pressure, which strives to make it expand. If the cloud is in equilibrium, the two tendencies will balande each other exactly. A mathematical expression for this balance may be obtained from the dynamics of manybody systems. We may in fact consider the cloud to be such a system made up of its individual molecules. If the total gravitational potential energy of the cloud is Ω and the total kinetic energy is K, then we know that on the average the second time derivative of the moment of inertia of the cloud can be written: ¨ = 2K + Ω hIi (1) Since the system is isolated, the total energy is conserved, i.e.: E = K + Ω = const. (2) As particles move around, exchanging their kinetic and potential energies while keeping E ¨ will be different at different times, and the short-term constant, the instantaneous value of hIi ¨ stays positive for a significant time, then the system average will also change. However, if hIi 5 ¨ stays negative, the cloud will contract. (i.e., the cloud) is bound to expand. Conversely, if hIi ¨ = 0. Inserting this equilibrium The only way to keep the cloud in equilibrium is to have hIi condition into Eq. (1), we get the so-called virial theorem: 2K + Ω = 0 (3) ¨ > 0, it will Now suppose that the cloud for some reason departs from equilibrium. If hIi expand, so Ω increases, and K has to decrease by the same amount according to Eq. (2). But ¨ decreases toward zero. If on the other hand this means that 2K + Ω will also decrease, so hIi ¨ ¨ is brought back toward zero again. hIi < 0, the opposite happens as the cloud contracts, and hIi The process of tuning K and Ω into equilibrium by dynamical mixing is called virialization. For a spherical, homogeneous cloud with our assumed parameters we can express Ω as 3 GMc2 Ω=− · 5 Rc (4) and since the mean kinetic energy per molecule is 3/2 · kTc , we have: 3 K = kTc · nc Vc 2 (5) Moreover, since nc = ρc /m = Mc /mVc , where m is the mean molecular mass, nc is the number density of molecules, ρc is the density of the cloud, and Vc is the volume of the cloud: 3 Mc K = kTc · 2 m (6) Combining Eqs. (3), (4) and (6), we get: Mc = 5Rc kTc = MJ Gm (7) Since this relation holds for a cloud at the verge of instability, we realize that any larger value of Mc for the same radius Rc and temperature Tc would cause the cloud to contract. The value given by Eq. (7) is called the Jeans mass. It is often expressed as Mc (ρc , Tc ) instead of Mc (Rc , Tc ), and using Mc = 4/3 · πRc3 · ρc , we get: s 15 kTc MJ = 5 4π Gm !3/2 ρ−1/2 c (8) Let us now consider real interstellar clouds instead of the idealized model we just treated. Real clouds are not forced to obey the equilibrium condition (7) or (8), and in fact they generally have masses much smaller than the Jeans mass. The reason they do not expand is that they are pressure-contained by their surroundings – in general, a hot, rarefied medium that exerts enough pressure to balance that of the cloud. Conditions where Mc ' MJ are only found among the dark, molecular clouds. In this case the clouds are practically self-contained by their gravity and exert only a relatively small pressure on the surrounding medium. An important property is that their internal temperature is efficiently limited by the cooling that results from molecular emission. Thus, imagine that there is a sudden increase in the outside pressure – e.g., due to a supernova shock front – and that this happens to a cloud with a mass of nearly MJ . This would of course compress the cloud, so ρc would increase and Tc would tend to increase by adiabatic heating. But the extra heat is radiated away, so T c stays 6 at its initial value. We therefore see from (8) that MJ has effectively decreased, and the cloud may find itself with Mc > MJ . This means that there is a tendency to contract further, and the result will be the same. Due to the radiative cooling, contraction stimulates itself, and the situation is clearly unstable. The figure below illustrates what has just been said. We see a diagram of Ω vs K, where the stability line of the virial theorem has been drawn. A cloud in exact equilibrium (Mc = MJ ) has been marked at point A. Note that as long as the cloud does not lose or gain any mass, its value of K only depends on the temperature Tc . The line AB denotes the initial compression, which makes Ω more negative, and BC shows the adiabatic heating that would follow from the compression, thereby increasing K. The line CD illustrates the radiative cooling that brings the cloud away from equilibrium again, etc. Finally, let us note two consequences of the above scenario. One is that, as contraction proceeds, the gravitational energy released will be partitioned equally between radiation and heating. If this continues long enough, the radiative flux from the protostar will become important in shaping its surroundings, e.g., by heating the surrounding dust. The second is that the continued decrease of the Jeans mass may eventually lead to fragmention of the initial cloud, since smaller subunits will be able to start contracting on their own. This process may be quite important in Galactic evolution by shaping the mass distribution of stars, and in local cosmogony by favouring star formation in clusters rather than as individual supermassive objects. In fact, there are several pieces of evidence that the Sun formed as a member of a tight though short-lived cluster of stars and that extinct radioisotopes were implanted into the presolar cloud from nearby red giants or supernovae. Disk formation Let us now investigate the consequences of a finite angular momentum (L) in a contracting cloud. It is trivial to realize that any cloud – even before starting to contract – must have at least a minimum amount of angular momentum due to a very slow rotational motion. How could anything have L = 0? In real terms, one may think of such a rotation emerging simply from the rotation of the Galaxy, which has an angular velocity on the average of ω G ∼ 10−15 s−1 . Therefore any cloud that separates out from the galactic disk is expected to have at least this amount of rotation. 7 Obviously, a rotating, contracting cloud that is dynamically isolated from its surroundings must conserve its angular momentum. Since L = Iω, where I ∝ Rc2 is the moment of inertia, the progressive decrease of Rc must then be accompanied by an increase of ω ∝ Rc−2 . So the cloud spins up in the same way as the pirouetting ice skater pulling in his arms. One consequence of this is that the cloud gains energy in the shape of rotational energy, which for a uniformly spinning, homogeneous sphere can be written: L2 5 L2 1 = Krot = Iω 2 = 2 2I 4 Mc Rc2 (9) This is kinetic energy, which adds to the thermal energy of Eqs. (5) and (6). Since it grows as Rc−2 during contraction, in due course it will gain importance in the virial equilibrium condition. Note that according to Eq. (4), the absolute value |Ω| of the gravitational potential energy increases only as Rc−1 , i.e., more slowly. In fact it is worth noting that the kinetic energy term in the virial theorem does have contributions from bulk motion in addition to the thermal motions of individual molecules. Rotational energy is one such contribution, and turbulent energy is another, whose origin is in the whirling eddy motions of turbulent gas flow within the cloud. However, these phenomena become of prime importance only during the late stages of contraction, after the cloud has assumed a disk shape. This occurs, when the spin of the cloud is fast enough that the centrifugal force equals the force of gravity at the equator. The condition of centrifugal equilibrium is: Rc ω 2 = GMc Rc2 (10) Solving for Rc as a function of L and Mc , we get: 25 L Rc = · 4GMc Mc !2 (11) This is the so-called centrifugal radius, which expresses the radius of the cloud, as the contraction perpendicular to the spin axis is stopped. But along the spin axis the contraction continues, and thus a disk is formed. An order-of-magnitude estimate of Rc can be made from the observed values of ω and R in contracting cores within molecular clouds. We can thus take Ro ∼ 5000 AU and ωo ∼ 2 · 10−14 s−1 , and using (11) with Mc = M we obtain Rc ∼ 10 AU. This is clearly in rough agreement with the dimensions of our planetary system, i.e., those of the solar nebula. 8 Moreover, observed protoplanetary disks such as HH30 (seen edge-on in the left-hand picture above) or the “proplyds” (abbreviation for protoplanetary disks) in the Orion nebula (righthand picture above) have radii of ∼ 100 AU. This may be consistent with our rough estimate of Rc , since we shall see below that redistribution of angular momentum within protoplanetary disks leads to radial expansion of the outer parts. But before we proceed, note that as far as the formation of the Sun (or any star) is concerned, we have come practically nowhere, since our rotating disk needs to get rid of its angular momentum in order to deposit nearly all the mass at the center. Physical processes in the solar nebula So far we have described how the presolar cloud could have evolved into a rotating disk, which we call the solar nebula. Even though the cloud may initially have had a uniform rotation, or at least approximately so, there is no reason to expect the disk to have rotated like a rigid body. Since its own gravity was controlling the evolution, one may rather imagine something reminding of keplerian rotation (like in the current planetary system), where the whole disk is in centrifugal equilibrium. This means a state of differential rotation, where the angular velocity decreases with increasing distance from the center. We hence conclude that there was a radial shear in the disk. A gas parcel with a radial extent would become distorted while orbiting around the center, since the outer part would lag increasingly behind the inner one. Now, if those parts did not interact at all, one may imagine a laminar gas flow with a continuously varying angular velocity, and the gas parcel would gradually evolve into a tightly wound spiral as shown in the figure below. However, it is reasonable to expect that different gas elements actually did interact to some degree – thus opposing the longitudinal stretching imposed by the differential rotation. Such an interaction that tends to suppress the relative motion of neighbouring gas elements is referred to as viscosity. 9 The viscosity of a differentially rotating disk has three important consequences, which can be described by the words heating, turbulence and accretion. First, as the relative motion is damped, some of its energy is dissipated and thus heats the gas. Second, the coupling of gas elements in relative motion disturbs the laminar flow and creates turbulence. Hence we have to expect both that the solar nebula may have reached high temperatures and that it contained rotating eddies with a characteristic size spectrum. These eddies in turn made gas parcels move between somewhat different radial distances, and this radial coupling helped to maintain the viscosity. The third phenomenon is extremely important. Consider that an inner gas element A and an outer one B interact, as A overtakes B, so that the relative velocity is decreased. Thus the orbital motion of A is decelerated, and that of B is accelerated. This means that angular momentum is transferred from A to B, i.e., there is an outward transport of angular momentum. As a consequence A tends to move inward and B outward, so there is everywhere a local tendency for the disk to break up radially. The global result is that the innermost part of the disk is the permanent loser of angular momentum and thus collects at the center. On the other hand, the outermost part of the disk will expand by aquiring more and more angular momentum. This is how we can imagine the development of a large central concentration of mass with very little specific angular momentum. The possibility to form a slowly rotating star like the Sun in spite of the angular momentum problem is thereby guaranteed, and the solar nebula should have acted as a so-called accretion disk. Actually the accretion disk should not be expected necessarily to reach all the way to the surface of the forming star. There may be a small clear zone in between, so that the accreting gas is governed during the final stages by the magnetic field and flows onto the star along dipole-like field lines. One also should not think of the disk as an instantaneous phenomenon, which arises at one particular moment in time through the collapse of the initial cloud and then simply fades away by delivering mass to the star and getting more dispersed in its outskirts. In fact the shear size of the initial cloud means that there is a range of free-fall times between different parts. Thus the initial disk may be formed by the inner parts that collapsed with the shortest time scale, and then the disk served as an intermediate link between the continuing infall from the outer parts of the cloud and the accretion onto the star. Its overall lifetime should be measured by the duration of the infall, and the final mass of the star should be the integral of the accretion rate over the disk lifetime. The origin of the viscosity in the solar nebula has been subject to a lot of discussion. But there is now a widespread consensus that the solution is given by the so-called magnetorotational instability (MRI). Suppose that the gas disk is partially ionized and penetrated by a frozen-in magnetic field. This means that the field lines are coupled to the gas that they cross. Hence, if the orientation of the field is such that the lines join gas parcels at different radial distances (see the figure below), the differential rotation tends to stretch the field lines 10 longitudinally and thus creates magnetic tension. This tension opposes the drifting apart of the parcels and thus provides a viscosity. An instability of the laminar flow results and leads to both turbulence and angular momentum transport. Accordingly, the primary source of viscosity is magnetic viscosity, but the MRI mechanism also leads to turbulent viscosity. The energy budget in a typical situation for the solar nebula is sketched below, where the numbers show energy fluxes in an arbitrary unit. Energy is tapped from the Keplerian orbital motion and goes into magnetic energy and turbulent kinetic energy. From both of these it is finally converted into heat. Disk evolution Thus, viscous energy dissipation causes heating of the gas disk. We may draw a few conclusions regarding the temperatures that result from this heating. First, the heating rate of a circular annulus of the disk between radial distances r and r +∆r will be proportional to the accretional mass flux Ṁ through that annulus. If the disk is close to a steady state, the mass flux must be nearly independent of r. However, the heating rate is also proportional to the specific amount of energy loss (per unit mass), as the disk material traverses the annulus. This can be found by comparing the specific energy EK of Keplerian, circular orbital motion at the outer and inner edges of the annulus. Since EK ∝ −1/r and thus |∆EK | ∝ r−2 ∆r, we find that the heating rate of the annulus is G ∝ r−2 Ṁ ∆r (12) This has to be balanced by a cooling rate L due to thermal radiation into space, which we can express using the Stephan-Boltzmann las as 4 L = 2 · 2πr∆r · σTef f, (13) assuming the disk to be a black-body radiator. The first factor 2 comes from the fact that the disk has two sides. The second is the surface area of the annulus on either side, and the third is the emitted energy flux per unit area. From the balance condition G = L we thus find: Tef f ∝ r−3/4 11 (14) The temperature Tef f is the effective temperature of the disk analogous to the effective temperature (i.e., roughly the photospheric temperature) of a star. It corresponds to the actual temperature of the gas in the surface layer, where the emitted thermal radiation emanates. But what is the temperature inside the disk, e.g., near the equatorial plane? This does not come directly from the above energy balance condition, but it has to be consistent with the surface temperature Ts ' Tef f and the heat transport mechanisms between the interior and the surface. The disks that we are discussing – e.g.., the solar nebula – are not purely gas disks, but they contain dust as well. Analogous to the case of dark molecular clouds this dust is quite important in providing opacity to radiative transfer. The difference is that in the present case the radiation to be blocked does not come from outside the disk but from the inside. Therefore, as long as the temperatures are high enough for the Planck spectrum to peak at short wavelengths, where the opacity is high, purely radiative transfer might need an unreasonably large vertical thermal gradient. By this we mean that convection results and transports the heat more efficiently, limiting the thermal gradient to a value near the adiabatic gradient of the gas. In any case we can expect that the central temperature Tc (r) is substantially larger than Tef f (r) as long as small grains are present in the disk. But it is very difficult to ‘predict’ exactly which temperatures were prevalent in the solar nebula at given distances from the protosun. Note also that, as indicated in the above picture, the solar nebula may have had a concave surface that could have been illuminated by the protosun, thus providing yet another source of heating. In view of the uncertainties of thermal model predictions, it is very important that we have a rough indication of the actual midplane temperatures in the ‘asteroidal’ region of the solar nebula from chondritic meteorites. The fact that these are as high as around 500 − 600 K at several AU distance from the center proves that there was significant viscous heating going on at the time of their formation. So far we have considered the structure of the disk at a certain moment, when Ṁ can be assumed independent of r. But Ṁ may change with time for the following reason. In the beginning – during the collapse of the parental cloud core – there is a continuous addition of mass at a high rate. The time scale of free fall from a region the size of R ∼ 5000 AU is only ∼ 105 yr, and during this time a major fraction of the Sun’s mass should have been processed through the disk. Then, for a few Myr more, there would have been continued infall from more distant regions of the presolar cloud at a much slower but still significant rate. A crucial question is how the disk could cope with this – which amount of viscosity did it possess, and thus, which was the viscous time scale for accretion onto the protosun? From a theoretical point of view this is difficult to specify1 , so let us rather study the problem using observational 1 A very common model of accretion disks – the so-called α-disks – was introduced in 1973 by Shakura and Sunyaev. This involves a free parameter denoted α, which expresses the strength of the viscosity in dimensionless form. 12 evidence. The ages of T Tauri stars can be estimated by model fitting. One observes the location of the star in the theoretical HR diagram, i.e., its luminosity and effective temperature, and one compares with evolutionary tracks of contracting objects – one track for each stellar mass. Thus one can read off the mass of the star as well as the amount of time that has elapsed since the initial collapse according to the model used. The spectra of the stars yield information on the presence of accretion disks around them. In particular, the amount of radiation from the disk indicates the mass accretion rate onto the star. When plotting such accretion rates vs the ages of the stars, as in the above diagram, one finds a decreasing trend with rates approaching 10−6 M /yr among the youngest stars but orders of magnitude smaller for ages ∼ 10 Myr. In fact, the older T Tauri stars often do not show any evidence of disks (so-called weak-emission T Tauri stars or “naked T Tauri stars”), and the current thinking is that previously existing disks have been blown away from them by intense stellar winds called “T Tauri winds”. The general conclusion is that the lifetime of the solar nebula was only about 5 − 10 Myr. During most of this time the accretion rate was rather low, and a large majority of the Sun’s mass was accreted early on via a hot and violently turbulent disk. Finally, let us mention the phenomenon of “dead zones”, which is often discussed in connection with protostellar disks. The point is that the magneto-rotational instability will work only if the gas is ionized to some degree. In the innermost parts of the disk the temperature is high enough for thermal ionization, but in most parts one has to rely on X-rays coming from the surroundings. These can only penetrate through a limited amount of material, so the outer disk regions may be fully penetrable due to their low density, while at intermediate radial distances 13 there may be a part near the midplane that is not ionized at all. This is called the dead zone, because it would not participate in the viscous accretion. Probably it would tend to grow in mass by accumulating material from the outside, but it would also tend to be dragged inward by the accretional flow. Thus there may be episodic bursts of accretion, when such clumps fall onto the protostar. The curious phenomenon of FU Orionis stars or “fuors”, which undergo dramatic flares in brightness during periods of ∼ 10 − 1000 yr – probably related to bursts in accretion rate reaching ∼ 10−4 M /yr – could be related to the accretion of dead zones. This is observed frequently enough that it has to be a general phenomenon affecting most protostars. But we cannot be sure that it happened also in the solar nebula. The mass of the solar nebula What inferences can we make from the masses and compositions of the planets, concerning how massive the solar nebula must have been? We will only illustrate this analysis at a very basic level, because the result is in any case useful only as an order-of-magnitude estimate. Table 1 shows a cosmochemical subdivision of the elements into three groups according to the volatile vs refractory nature of the main species where they existed in the solar nebula at low temperatures, when no further chemical reactions took place. Table 1: Cosmochemical groups of elements Representative Main nebular Fraction of the elements material nebular mass H, He Gases 98.4% (H2 , He) C, N, O Ices 1.2% (CH4 , NH3 , H2 O) Si, Mg, Fe Refractories 0.3% (metals, minerals) The most volatile group can be called the gases, because they form species that would only condense at temperatures close to absolute zero. Here we find hydrogen in the shape of H 2 14 molecules and the noble gases, which are dominated by helium (He). Altogether, this group carries 98.4% of the mass of the solar nebula. Next we find the ices, i.e., mainly the hydrides of carbon, nitrogen and oxygen (CH4 , NH3 , H2 O). These molecules are relatively volatile so that they remain in the gas phase unless the temperature falls below a few hundred Kelvin. Since both C, N and O are among the most common elements after H and He, this group carries most of the remaining mass: 1.2% of the solar nebula. Note that the amount of hydrogen used in these hydrides is negligible compared to the total amount available. The third group can be called the refractories, because they go into the solid phase at much higher temperatures. They consist of metals, silicates, oxides and sulfides, and the main typical elements are silicon (Si), magnesium (Mg) and iron (Fe). Note that the amount of oxygen used in these compounds is negligible compared to the total amount available. The mass fraction of the solar nebula carried by this group is just the small remainder (0.3%) left on top of the gases and the ices. Now consider the chemical constitution of the planets. They can roughly be divided into three groups according to which of the three cosmochemical groups they are mostly made up of. The terrestrial planets are dominated by the refractories. Jupiter and Saturn are dominated by the gases, and thus they are called the gas giants. Uranus and Neptune, on the other hand, appear to be mostly made up of the ices and hence are called the ice giants. This classification suggests a way to translate each planetary mass into the total nebular mass that corresponds to the material of the planet. Take the Earth as an example. Its amount of volatiles is negligible in relative terms, and we can consider it to be made up only of refractories. Thus the correction factor translating the Earth’s mass into the nebular mass is 1/0.3% ' 300. The other terrestrial planets have about the same correction factors. For the giant planets the analysis is a bit more complicated, since they contain mixtures of all the groups of elements, and none of them is negligible. Even the gas giants turn out to have compositions that differ from the Sun or the solar nebula, i.e., they are less “gassy”. Therefore they need corrections too, although the factors are much smaller than for the terrestrial planets. They of course depend on models for the structure of the planetary interiors. The smallest one (' 3) is found for Jupiter. Table 2: Actual planetary masses and derived nebular masses Planet Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune All Mass (ME ) Nebular mass (ME ) 0.05 15 1 300 1 300 0.1 30 320 1000 95 500 15 500 17 500 450 3000 Table 2 summarizes the results of this analysis. By adding up all the nebular masses we arrive at a value ∼ 3000 Earth masses. This is about 1% of the Sun’s mass. We can interpret it as the mass the solar nebula must at least have had in order to provide enough material for the 15 building of the planets. A model of the solar nebula with this mass is thus called the minimum mass solar nebula (MMSN). The term implies that Msn ∼ 0.01M should not necessarily be taken as the real mass of the solar nebula – this may have been substantially larger. In order to estimate the real M sn one would have to take account of the efficiency of the planetary formation mechanism. For instance, if this was only 10%, so that 90% of the nebular mass was wasted as the planets were formed, we would in reality have Msn ∼ 0.1M . One reason to think in such terms is that gravitational accretion tends to eject a large fraction of the material encountering the protoplanet, and part of this ejected material might have been ejected from the Solar System altogether. However, we only have rough estimates of this efficiency, indicating that the mass of the solar nebula was several times the mass of the MMSN. In addition to estimating the total mass of the solar nebula from corrected planetary masses, one can derive a picture of the radial density distribution. The procedure is to associate each planet with a circle around the Sun, whose radius is the semi-major axis of the planetary orbit. Then one considers a circular annulus with inner and outer radii midway to the planet’s inner and outer neighbours. The corrected planetary mass is spread over this annulus, because it is supposed to be the “feeding zone” of the solar nebula, from which the material of the planet was gathered. Dividing the corrected planetary mass by the area of the annulus, we get the mean surface density of the MMSN around the orbit of the planet. A plot of these surface densities vs distance from the center (see the figure below) shows a general fall-off with a slope that looks reasonable plus a few conspicuous drops in the main asteroid belt and the Kuiper belt – also explicable as results of dynamical or collisional depletion of solid objects. However, we must realize that the feeding zone concept is highly questionable from the point of view of current theories of planet formation, so the detailed picture thus obtained of the density structure of the solar nebula should be regarded as unreliable. Planetary formation apparently was a much more complicated process involving, e.g., radial migration such that any given planet may have ended up at a place quite different from that where the initial accretion took place. In particular, the roughly Σ ∝ r −1 relationship suggested by the figure may be underestimating the real slope due to the outward migration of Uranus and Neptune after their formation. 16 In view of these uncertainties it is wise to consider also what evidence is brought by observations of current protoplanetary disks. Interpreting the infrared fluxes observed from the < 0.01M for protostars with masses dust component generally leads to mass estimates Mdisk ∼ of roughly 1 M . However, the real masses may be larger, e.g., due to uncertainties around the dust opacities (perhaps larger than assumed) and the size distribution of the solid particles (perhaps involving large chunks that carry a large fraction of the mass). One may also expect that a disk which supports a mass accretion rate ∼ 10−8 M /yr during ∼ 2 Myr should have a mass of at least 0.02 M . On the other hand, a crude upper limit is set by spectral mapping of the CO emission from protoplanetary disks, which shows evidence of Keplerian orbital motion. The enclosed mass derived from these results is not much larger than the stellar mass, and this > 0.1 M . speaks against disk masses ∼ 17