Download Origin of the Solar System

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.
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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.
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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
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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
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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
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¨ 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:
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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:
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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
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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.
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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:
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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.
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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.
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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
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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
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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
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(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.
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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
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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
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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
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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.
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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 ∼
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