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FRACTIONATED ACCRETION AND THE SOLAR NEUTRINO PROBLEM
By C. S. Jeffery, M. E. Bailey, & J. E. Chambers
Armagh Observatory, College Hill, Armagh BT61 9DG
During the course of its evolution, the Sun has accreted
substantial quantities of material in the form of asteroids,
comets, planetesimals and proto­planets. This debris is poor
in hydrogen and helium and has substantially enriched the
metallicity of the solar surface which must consequently be
higher than that of the unobserved core. It is well known
that compared with standard solar models a lower core metal­
licity results in a lower core temperature and a reduced
flux of high­energy neutrinos. We estimate how much high­
metallicity material could have been accreted during the so­
lar lifetime, and how such accretion will affect the evolution
of the Sun and other stars. We conclude that a few tens
of Earth masses of fractionated material may have accreted
onto the solar surface, and that this may provide a partial
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resolution of the solar neutrino problem. The solution may
also explain the Faint Young Sun paradox and the inclina­
tion of the solar equator relative to the ecliptic.
Over the past thirty years, the deficit of observed neutrinos from the deep inte­
rior of the Sun has become one of the major problems confronting particle physics
and astronomy [1, 2]. The key question is whether the deficit is a consequence of
poorly understood neutrino or solar physics. A popular conjecture is that neutrinos
may undergo transitions to states not observable by current detectors [3], but mea­
surements of the low­energy neutrino flux [4, 5] from the p­p reaction are consistent
with the observed solar luminosity. Unless this is a remarkable coincidence, the
explanation of the high­energy neutrino deficit must lie at least partially in the Sun
[6].
A long­standing proposal to explain the deficit is to consider an inhomogeneous
solar model in which the `metallicity' or heavy­element abundance Z of the deep
interior is systematically lower than that of the surface [7]. The effect of a lower
metallicity in the solar interior is to reduce the opacity and hence the central tem­
perature T c . This shifts the balance of the nuclear energy generation from the pp­II
and pp­III chains towards the pp­I chain. The high­energy neutrinos detected by the
early experiments of Davis and others are produced mainly by 7 Be electron­capture
and 8 B positron­decay reactions in the pp­II and pp­III chains, which have a much
higher temperature threshold than the pp­I chain.
Models with low central metallicity are in apparent contradiction to standard
cosmogonic arguments which predict a chemically homogeneous initial Sun following
a fully convective Hayashi­track contraction. Nevertheless, inhomogeneous models
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have been seriously considered and a variety of internal mixing mechanisms have
been investigated [8]. For example, particle diffusion --- the sedimentation of species
heavier than hydrogen towards the core --- has been shown to be a critical factor
in understanding the helioseismological results [9]. In a recent review Bahcall [10]
has emphasized that precise non­standard models, for example with reduced central
metallicity, should be developed.
A plausible explanation for an inhomogeneous, low central­metallicity Sun is
the likelihood of substantial accretion of fractionated material from the interstel­
lar medium or in the form of comets or planetesimals during early main­sequence
evolution [6, 11, 12]. Recent results on the delivery of asteroids from the main belt
to near­Earth space show that the Sun­colliding end­state is extremely common, as
indeed it is for long­period and Halley­type comets [13], and rough estimates of the
amount of material available for accretion from the solar nebula are consistent with
a fraction of the surface metallicity of the Sun (and presumably of other stars as
well) originating from such accretion. The agreement between solar and meteoritic
abundances may simply reflect this fact. Moreover, evidence for an early cometary
bombardment in the inner solar system [14, 15] indicates that over an interval of
500 million years, the Sun would have accreted high Z cometary material at a rate
AE10 times that observed in the present epoch.
Looking further afield, observations of fi Pic indicate the presence of substan­
tial cometary accretion [16, 17], and more recently the discovery of giant planets
(51 Peg, Ü Boo, AE And, ae 1 Cnc) in sub­Mercurial orbits contrasts sharply with model
predictions that they cannot form closer than about 3 AU from the parent star
[18]. This has led to the suggestion that a protoplanetary disk can remove most
or all of the orbital angular momentum from the planet [19], allowing it to spiral
3

inwards until either the disk is dispersed (whereupon the planet survives) or the
planet collides with its parent star, dumping a 20--30 M \Phi ice­and­rock core onto the
stellar surface. Above­average surface metallicities have been measured for all four
stars cited above, and have been interpreted as evidence for the accretion of metal­
rich material during the early evolution of the planetary system [20], although the
evidence for massive planets in at least one case has been questioned [21].
Finally, substantial evidence is accruing that the interstellar medium (ISM) com­
prising both grains and gas is metal poor with respect to the Sun, perhaps with
Z ISM as low as 65% of Z fi [22]. Were the Sun formed from material typical of the
ISM, it would be necessary to enrich its surface metallicity in order to match the
value observed today.
It is instructive to estimate how much rocky material could have been accreted
onto the solar surface during its main­sequence lifetime. There are six main sources
of infalling material:
1. Earth­crossing asteroids: Farinella et al. [23] found ¸10 solar collisions per
Myr from simulations of 47 known Earth­crossing asteroids. These represent
¸2% of the current near­Earth­asteroid population leading to an estimated
¸500 solar impacts/Myr of – 1 km asteroids. For a mean asteroid mass of
10 14 kg, this gives 5 \Theta 10 16 kg/Myr on the Sun, or 4 \Theta 10 \Gamma4 M \Phi over the age
of the solar system. This estimate obviously depends on the assumed mean
mass, and higher values are attainable.
2. Short­period comets: of the known population, ¸5--10% will become Sun­
grazers in 10 5 yr [24]. The mass distribution is likely to be dominated by rare,
Chiron­sized objects (10 19 kg) which may enter short­period orbits once in
4

10 5 years [25]. Thus the delivery rate (if constant) would be ¸7.5% of 10 19 kg
per 0.1 Myr, or ¸5 \Theta 10 \Gamma3 M \Phi over the age of the solar system.
3. Long­period comets: assuming a constant long­period cometary flux of about
1 comet per year per AU brighter than a total absolute magnitude H 10 = 7, a
uniform distribution of perihelion distances (so that roughly 0.5% fall directly
onto the Sun), and a mean mass for comets brighter than H 10 = 7 in the range
10 15 --10 17 kg [26, 27], a delivery of heavy elements in the range 0.003--0.3 M \Phi
is indicated. This is a lower limit because (i) there are undoubtedly many
unseen `dark' long­period comets; (ii) the significant additional contribution
due to the infall of Halley­type comets [28] has not been counted; and (iii) the
cumulative effect of comet showers over the age of the solar system has been
ignored.
4. Early cometary bombardment: the `late heavy lunar bombardment' (LHB),
which ended ¸3.8 Gyr ago, was an epoch when the cometary impact rate
was AE10 times the present rate, providing an additional mass much greater
than that from present Earth­crossing asteroids and short­period comets. We
estimate that the total accreted mass lies in the approximate range 0.1--1 M \Phi ,
depending on the rate and duration of the LHB [15].
5. Accretion of protoplanetary disk debris: assuming a formation efficiency ¸30%,
approximately 4 M \Phi of debris in the form of planetesimals and planetary em­
bryos would have been removed from the inner solar system within ¸2AU
during the formation of the terrestrial planets. Approximately half would
have been scattered into the Sun, providing ¸ 2 M \Phi of Z ¸ 1 enrichment.
5

Similarly, material from 2--10 AU would, apart from that now in the asteroid
belt, have been scattered into Sun­grazing orbits via mean­motion and secular
resonances, e.g. the 3:1 resonance with Jupiter and the š 6 secular resonance.
An upper limit for this source is provided by the solar nebula density which
is required in order that giant planets could form before hydrogen and he­
lium were evacuated from the protoplanetary nebula. This is found to be
¸3--4 times that of the ``minimum mass'' nebula at the radius of Jupiter [29].
High­metallicity material remaining after the formation of Jupiter and Saturn
would have been lost by a combination of ejection or accretion onto the Sun.
The accretion of one half of this material, approximately equivalent to the
combined core masses of Jupiter and Saturn, would provide ¸50M \Phi .
6. Giant planet destruction: of 200 solar­type stars studied, at least 2% are
claimed to have massive planets in sub­Mercurial orbits. The probabilities
are not known for (a) the formation of massive planets and (b) the survival
of such planets beyond disk dissipation. The consequence of a Jupiter­mass
planet colliding with the Sun would be the accretion of several hundred M \Phi
in the form of a gaseous (H/He/Z) envelope plus a few tens of Earth masses
in the form of an ice­and­rock core (Z ¸ 0:9). Given an eccentric or inclined
orbit it could also (easily) account for the 7 ffi inclination of the solar equator
relative to the invariable plane of the solar system.
The potential contamination from all these sources ranges up to 10 2 M \Phi of metal­
rich material. Provided this was accreted after the Sun completed its fully convec­
tive evolution, which seems likely, this material would have remained predominantly
in the outer convection zone. Given a total accreted mass m rock of material with
6

(X rock = 0:1; Z rock = 0:9) and a mass for the solar convection zone m conv ¸ 0:025M fi
[9] (X fi = 0:7; Y fi = 0:28; Z fi = 0:02), where Z fi is the observed surface metallicity,
then the degree of surface metal enrichment is given in terms of the initial (or inte­
rior) metallicity Z 0 by (m conv Z 0 +m rock Z rock ) = (m conv +m rock )Z fi . It is convenient
to introduce the parameter i = Z fi \Gamma Z 0 = m rock (Z rock \Gamma Z fi )=m conv , so 0 ! i Ü 1.
Since i ? 0 the question is not whether the interior metallicity is below the surface
value, but by how much.
The implications for the solar neutrino problem can be estimated by considering
a star which is homogeneous except for a convective surface zone enriched in metals.
By varying the interior composition (Y 0 ; Z 0 ), where Y 0 denotes the central helium
abundance by mass, the behaviour of the luminosity (L) and central temperature
(T c ) can be ascertained. Since an evolved solar model should always attain the
observed luminosity at age 4.6 Gyr, the test models have been constrained to have
the same luminosity as one another by adjusting Y 0 . Reducing Z 0 alone increases
both T c and L, but the latter can be offset by a small reduction in Y 0 , which
is a (relatively) free parameter in the standard solar model (SSM). Therefore the
change in T c can be evaluated to first order as a function of Z 0 and hence of m rock ,
which we have parameterized by i. The models give dT c =dZ 0 ¸ 5T c and therefore
\DeltaT c = T c \Gamma T c;0 ¸ 5T c;0 \DeltaZ 0 = \Gamma5T c;0 i, where T c;0 is the central temperature of
the unperturbed model.
The observed neutrino fluxes OE due to the 8 B and 7 Be reactions are in deficit
relative to the predictions of the standard solar model, OE SSM , and are given as
OE( 8 B) ¸ 0:43 OE SSM ( 8 B) and OE( 7 Be) ¸ 0 [2]. However both neutrino fluxes are
sensitively dependent upon the central temperature, namely OE SSM ( 8 B) / T 24\Sigma5
c ,
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and OE SSM ( 7 Be) / T 10\Sigma2
c [30]. Thus
OE( 8 B)
OE SSM ( 8 B) =
/
1 + \DeltaT c
T c;0
! 24
= (1 \Gamma 5i) 24
Solving for m rock , and adopting the representative parameters m conv = 0:025 M fi ,
Z rock = 0:9 and Z fi = 0:02, gives m rock = 65M \Phi . This estimate is within an order of
magnitude of the possible surface enrichment from infalling planets or planetesimals.
The concomitant value of i ¸ 0:007 is also in good agreement with estimates [22]
of the Z­depletion in the local ISM. Conversely, the surface­enrichment mechanism
does not address directly either the apparent OE( 7 Be)=OE( 8 B) ratio or the discrepancy
between the Homestake and Kamiokande measurements of OE( 8 B).
In contrast to this approach, helioseismological observations now match the pre­
dictions of the standard solar model to an accuracy better than 0.2% [31]. However,
a non­standard model may also reproduce the helioseismological observations. Our
test calculations yield estimates for other properties of the Sun, in particular the
internal sound speed. We find that the required change in Z 0 , and the associated
change in Y 0 (necessary to satisfy the luminosity criterion), leads to a relative change
in the sound speed at the centre of order \Gamma0:2i , or less than 0.2%. Since helioseis­
mology only directly measures the sound speed and not the composition or the state
variables, the standard solar model is not necessarily a unique solution.
Our present models adopt simple physics and the estimated mass available for
accretion by the Sun from the proto­planetary disk relies on immature hypothe­
ses of planetary formation. A complete solution must satisfy all the observational
constraints, for example it must also solve the 7 Be neutrino deficit, and consider
the removal of heavy elements from the convective zone by diffusion. In the cur­
rent standard solar model the latter produces a central metallicity enhancement
8

Z c \Gamma Z fi ¸ 0:003 during the age of the solar system [9], smaller than the potential
deficit due to accretion.
Finally, we note that a further reason for contemplating models with lower Z 0
and T c is that such models have more massive nuclear­burning cores. This means
that the initial Sun may have been more luminous than predicted by the standard
solar model and provides a possible amelioration of the Faint Young Sun paradox
[32, 33, 34]. In conclusion, recent progress, both observational and theoretical, in
understanding the origin and early evolution of planetary systems suggests that the
accretion of metal­rich material by the Sun and other stars may be more widespread
than previously thought. Fractionated accretion by the Sun may provide a partial
solution to the solar neutrino problem, with important implications for other fields.
It is a pleasure to thank the referee for helpful comments. This work was sup­
ported by DENI and PPARC.
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