Planet. Space Sci., Vol. 44, No. 11, pp. 1251-1268, 1996
Pergamon
Cotwright 0 1996 Elsevier Science Ltd
Printed-in GreatBritain.
All rights reserved
0032-0633/96
$15.00+0.00
PII: SOO32-0633(96)OOOSf%i
Theoretical seismic models of Mars
of the mantle
:
the importance of the iron content
A. Mocquet, P. Vacher, 0. Grasset and C. Sotin
Laboratoire
03, France
Received
de Gtophysique
13 October
et Planetologie,
U.F.R.
des Sciences et Techniques,
1995; revised 4 June 1996; accepted
44072 Nantes
CCdex
7 June 1996
Abstract. Present-day averaged temperature proGles of
the mantle of Mars are computed &&x4gh~numorical
convection
experiments
performed
with
: L+
symmetricalgeometiy,
for different vaIues of core r&hi
and d&rent boundary conditions at the core-mantle
boundary. Internal heating of the mantle js considered
in each case. It is found that Zhe temljerature p~diles
of the man& are very stable whatever the imposed
conditioas at the core-tiantle
boundary.’ A 300km
thick thermal lithosphere, displaying a temperature
gradients equal to 4.4ISkrW’ is followed ‘at greater
depths by a quasi-iscvthemd mantle, the temperature
of which is faukid in a 12@O-f600K temperature range.
A mean tempbature equal to 1400 K is in a goad agreement with the low Q ofMars at ti&l frequencies. These
characteristics, together with the small increase of pressure with depth, of the order of 0.01 OPakm-*, induce
tfiepresence OFa 1ow:velocity zone similar to the Earth’s one, dawn to 300 km depth. Densities and seismic
velocit&i corresponding to these thermoi$yaamical
conditions are computed using Clirtineisen’s and fhirdorder finite strain theory for diff&erit values of the iron
content of mantle r&e-ra%. B.&W SQpkm depth, the
values of d&&ties and se&r& vehxities have ‘&hesame
order of magnitude a3 within iLEe Earth’s t&&ion
zone.’ An increase of the iron conrent of the Martian
mantie with respect to the Earth’s one results (1) in an
imrease of density, and a Gecrease of seismic velo&ies,
which em reacti more t&an, 2 B/sof ttie values expected
from an Earth like composition, (2) in a homogeniz$ion of mantle structure thrmgb the smmtbing
out of seitie discontinuities over a thickxqss of .a few
hundred kilo;nretses. This smuothing pmss
is due to
the large pressure domams of coexistence between
different phases of &vine @hen the iron comeat of this
latter mineral increases. Plausible domains of core
Correspondence to: A. Mocquet
2 rue de la Houssinibre,
1. Introduction
The present-day averaged structure of a planet is intrinsically related to its accretion history, to its core formation, and to its subsequent thermal and tectonic evolutions. In the absence of preliminary seismic data,
tentative bulk compositions of the mantle and core can
be inferred from the mean density, and from the principal
moment of inertia of the planet using geochemical arguments. A choice among different plausible models can
then be achieved by constructing theoretical density and
seismic velocity profiles, using available mineral physics
data (e.g. Anderson et al., 1977; Okal and Anderson,
1978; Lognonne and Mosser, 1993) and by comparing
these latter with seismological models of internal structure
derived from observation, in the same way as this is presently done for the Earth’s mantle (e.g. Dully and Anderson, 1989; Vacher et al., 1996). In the Earth’s case, this
type of procedure has successfully shown, for instance,
that seismic velocity gradients agree with a nearly adiabatic behaviour of the upper mantle in the depth range
100-400 km, the temperature at the foot of the adiabat
being close to 1200°C (e.g. Vacher et al., 1996), but that
the transition zone looks highly subadiabatic (e.g. Anderson and Bass, 1986; Duffy and Anderson, 1989). The
1252
comparison
between
geochemical,
seismological,
and
mineral physics data enriched the debate concerning
the
bulk composition
of the Earth’s mantle (e.g. Bass and
Anderson,
1984; Weidner,
1985), leading to a compositional
model
(Ito and Takahashi,
1987) which
adequately
fits spherically
symmetric reference Earth’s
models derived from seismology (Weidner and Ito, 1987 ;
Vacher et al., 1996). Similarly, the results of tomographic
studies indicating
that the magnitude
of relative shear
velocity variations
are at least twice as large as the magnitude of relative compressional
velocity variations
led
Duffy and Ahrens (1992) to conclude that small amounts
of water-rich partial melt could exist in the Earth’s lower
mantle. In the case of planet Mars, such tomographic
studies are far beyond the scope of future planetary
missions. Nevertheless,
it may be hoped that American,
European, Japanese, and Russian projects which are presently impulsed will provide, within a few years, important
seismological
data from which preliminary
one-dimensional seismic models will be constructed and, henceforth,
important
conclusions
regarding the composition
of the
Martian mantle and state of the core will be accessible. In
particular,
a mantle composition
close to the one derived
from the study of SNC (Shergottites,
Nakhlites,
Chassigny) meteorites
is characterized
by an enrichment
in
iron relative to the Earth’s composition
(Ringwood, 1979 ;
Dreibus and Wanke, 1985; McSween, 1985). If correct,
such enrichments
should greatly influence the density of
the mantle, its seismic structure, together with the values
of the mean principal moment of inertia I (Ohtani and
Kamaya,
1992). In fact, one of the major difficulties encountered in a study of Mars is the poor knowledge of the
mean principal moment of inertia. The actual value of the
ratio I/MR’, where M is the total mass of the planet, and
R its mean radius, is still under debate (e.g. Reasenberg,
1977 ; Bills, 1989 ; Kaula et al., 1989 ; Kaula and Asimow,
1991). Johnston
et al. (1974) performed
one of the first
calculations
of the thermal history of the planet, and of
its core formation.
They computed a rather high I/MR2
value (0.377) which implied in turn high values of mantle
densities, of the order of 3.72 g cme3. Nevertheless,
more
recent studies tend towards I/MR2 values ranging from
0.345 to 0.365. It is still not possible to choose either the
lower or the upper bound of this range. For instance, Bills
(1990) emphasized that a small increase of the ratio I/MR2
could lead to a large domain of possible compositions
for
the whole planet. Bills and Rubincam (1995) even showed
that, under the assumption
of a two-layer piecewise constant density model, it was not necessary to invoke geochemically implausible
density extrema in order to reach
any IIMR’ value in the range 0.325-0.365. In particular,
the mean densities of the mantle and core are istrongly
influenced by the respective amounts of iron effectively
present in both layers. Anderson (1972) suggested, on the
basis of chemical criteria, that the amount of iron oxide
in the Martian
mantle is higher than the Earth’s one.
Similarly, Lewis (1972) estimated that the weight ratio
FeO/(FeO + MgO) of the Martian mantle is close to 0.5.
A useful constraint
on the internal structure of Mars is
provided by the significant acceleration
in the longitude
of Phobos (Sharpless, 1945) due to solid-body tides raised
by Phobos in Mars (e.g. Burns, 1977 ; Pollack, 1977 ; Lam-
A. Mocquet et al.: Theoretical seismic models of Mars
beck, 1979). The value of this acceleration is now known
with a good accuracy (Table 1). The successive improvements in its determination
have been reviewed by Burns
(1986,1992). To the first order in the angular displacement, in radians, between Phobos’ longitude
and the
maximum
of the induced tide on Mars, the dissipation
function Q’, and the Love number k2 of the planet are
related to the orbital parameters listed in Table 1 through
(e.g. Lambeck, 1979 ; Burns, 1986)
(1)
where a is the semimajor orbital axis of Phobos around
Mars, m its mass, y1 its angular velocity, ti its secular
acceleration,
and F the universal gravitational
constant.
The values listed in Table 1 lead to
k,Q-‘-(1.834+0.061)
x 1O-3
(2)
where the uncertainty
is in 20. Any model of Mars’ inner
structure must fulfil equation (2). The Love number k, is
provided by the distributions
at depth of the density and
elastic moduli. Previous estimates of 1~ range from 0.08
(Ward et al., 1979) to 0.15 (Lognonne and Mosser, 1993).
According to equation (2), this range of values correspond
to 42~ Q<85. Smith and Born (1976) and Ward et al.
(1979) proposed
50 < Q < 150. However, a more recent
determination
of ri by Chapron-Touze
(1990) argue in
favour of the lowest end of this range, because, according
to equation (2), Q values as high as 150 would require k2
values higher than 0.26. Possible explanations
for this low
Q value have been reviewed by Lognonne
and Mosser
(1993). According to the seismic absorption
band model
of Anderson and Given (1982), the frequency dependence
of Q is governed, at tidal frequencies,
by the value of
the long period relaxation time z2. This latter parameter
depends on pressure P and temperature
T through
z2
=
z. exp
where z0 is a pre-exponential
constant,
E* and V* the
activation
energy and volume, respectively, and R* the
gas constant. In equation (3), the activation energy and
volume are set equal to 60 kcalmol-’
and 10e5 m3 mol-‘,
respectively (Anderson
and Minster, 1981). At constant
pressure, equation (3) implies that a temperature
decrease
of 200 K translates into a two decade shift of the absorption band towards long periods, and, consequently,
to a
decrease of Q. Thus, the low Martian Q values might be
the signature of a cold Martian mantle, as compared to
the Earth’s upper mantle (Anderson
and Given, 1982 ;
Lognonne
and Mosser, 1993). Alternative
explanations
invoke the presence of volatiles and/or partial melting
within the mantle of Mars (Toksiiz et al., 1978 ; Lambeck,
1979 ; Lognonne and Mosser, 1993).
Since available data on the internal structure of Mars
are sparse, it would be hopeless to pretend describing in
an overdetailed way the actual fine structure of the planet.
On the other hand, our goal is to derive a plausible range
of mineralogy
and composition
of the Martian mantle,
putting emphasis on the associated range of seismic struc-
A. Mocquet et al.: Theoretical seismic models of Mars
1253
Table 1. Parameters used in the study of the secular acceleration in the longitude of Phobos
(equation (1))
Parameter
Mars
mass M (x 1O23kg)
mean radius R
Phobos
semimajor axis a
angular velocity n
secular acceleration A
mass m
Uncertainties
Value
Reference
6.4179 +0.0031
3389.916* 0.038 km
Born (1974)
Bills and Ferrari (1978)
9378.5 km
2.28 x 10m4rad s-’
(4.45kO.03) x 10-20rads-2
(10.81-0.1) x 10r5kg
Chapron-TouzC (1990)
Lambeck (1979)
Chapron-TouzC (1990)
Avanesov et al. (1989)
are in lo.
tures eventually detected by future missions on Mars. In
a first part, three different temperature
profiles of the
mantle are constructed,
corresponding
to different radii
of the core-mantle
boundary,
boundary conditions at the
base of the mantle, and liquid or solid state of the core.
These input parameters are taken from Sotin et al. (1996).
In order to take into account the small radius of the
planet, the mean temperature
profiles of the mantle are
derived from two-dimensional
convection
experiments
performed
with axisymmetrical
geometry.
Griineisen’s
and third-order finite strain theories are subsequently used
to compute the mean densities and seismic velocities as a
function of depth, following the methods of Duffy and
Anderson
(1989) and Vacher et al. (1996). At this stage,
the experimental
work on the phase transition
of olivine
(Katsura
and Ito, 1989) is used to assess crude mineralogical models of the Martian mantle as a function of
the amount of iron. The reliability
of the temperature
profiles and seismic models of the upper mantle is subsequently checked back by computing
their associated
moment
of inertia ratio, Love numbers
and Q values
(equations (2) and (3)).
2. Convection experiments
Previous studies (e.g. Stevenson
et al., 1983; Schubert
and Spohn, 1990) considered
parameterized
convection
to describe the heat transfer throughout
the Martian
mantle. In a preliminary
study (Mocquet et al., 1994),
we conducted
the whole procedure described here using
Cartesian coordinates,
and a parameterized
convection
formalism,
in order to get a range of plausible starting
values for subsequent
time-dependent
numerical
calculations in a two-dimensional
spherical geometry, and
for a study of the Martian core thermal history (Sotin et
al., 1996). The present study only describes the final stage
of the computations.
It is entirely devoted to the mantle,
and references to the Martian core are only made when
they are necessary for the discussion of the mantle structure. The time-dependent
numerical calculations
are performed in a similar way to that employed by Zhou et al.
(1995), except that internal heating is taken into account
in our study, and that we do not include the influence of
the phase transitions
of olivine
on the convection
geometry. Sections 2.1 and 2.2 will describe the numerical
procedure, and its application,
respectively.
2.1. Numerical
aspects
The general form of the equations
describing
convection can be written as (e.g. Busse, 1989)
g+v.(p”)=o
thermal
(4)
P[~+(uTu]=
-vp+pg+v.z
(5)
,cF(~+“.vT)-aT(~+“.vP)
=
pFzps
I-Lx(T-TT,)-f
[
v
(kvT)+pH+p@
(6)
$(P-Ps).
(7)
’
T
1
In equations (4)-(7), p is the density, t the time, II the fluid
velocity, P the pressure, g the gravity, z the stress tensor,
C, the heat capacity, T the temperature,
CI the thermal
expansion coefficient, k the thermal conductivity,
H the
rate of heating by radioactive
disintegration,
CDthe dissipation function which gives the amount of heat released
by viscous forces, and KT the bulk modulus at constant
temperature.
Subscripts
S refer to adiabatic
values. A
present-day value of H = 6.2 x lo-l2 W kg-’ is assumed.
Equations
(4)-(6) express the conserva.tion
of mass,
momentum,
and energy in their most general form, respectively. Equation (7) is the equation of state. A number of
simplifications
are done. First, the Boussinesq approximation
is used. It means that a constant
density is
assumed, except in the gravity term of equation (5), and
that the pressure dependence
of the density is negligible
compared to the temperature
dependence.
Second, the
thermal conductivity
is constant, and following Stevenson
et al. (1983), and Schubert and Spohn (1990) the viscosity
only depends on the mean temperature
of the mantle. It
implies that a calculated dissipation
function would not
be realistic. It is therefore neglected in the present study.
The effect of pressure is also neglected in the equation
describing the conservation
of energy. It means that models are valid at surface pressure conditions.
This peculiarity will be used later on for the computation
of seismic
wave velocities. Finally, the Prandtl number is very large
(> 10”) and the left-hand side of equation (5) is negligible.
A. Mocquet et al.: Theoretical seismic models of Mars
1254
stream function
of the following
As long as a liquid outer core is present, the temperature
at the core-mantle
boundary
(CMB) is equal to the temperature of the liquidus at the inner core boundary (ICB)
since the adiabatic
gradient in the core is negligible in
Mars’ condition.
Once the core is solid, the temperature
at the CMB may vary with latitude. The temperature
of
the upper surface is equal to 220K. Thermal convection
is calculated by time-stepping
the conservation
equation
for thermal energy (equation (6)) recasted in non-dimensional variables on a unit sphere. Equation (6) becomes
. VT’
=
V2T’+
a2Y!’
I a2yl
ar”+r”ae?_---=
The scales for temperature,
T-
To
H"
(10)
U
(12)
lc
respectively,
where K is the thermal diffusivity
of the
mantle.
The energy equation (8) is approximated
with central
differences for the conduction
terms, and upwind differencing of the advection terms. The velocity at each grid
point of the mantle is calculated
at each time step by
solving the continuity
and momentum
equations describing thermal convection
of an infinite Prandtl number,
isoviscous fluid :
v*u=o
(13)
-vP+pV2U+f
= 0
(14)
where ~1is the fluid viscosity, and f the buoyancy force.
Under the assumptions
stated previously, equations (13)
and (14) correspond to equations (4) and (5), respectively.
The three unknowns
are the pressure, and the radial and
colatitudinal
components
of the velocity, u, and us, respectively. Following Zebib et al. (1980) and Machete1 (1986)
equation
(14) is more easily solved in two-dimensional
spherical
geometry
by defining
the non-dimensional
Table 2. Constant
values used in the calculations
Thermal conductivity,
k (W m-’ K-l)
4
Thermal expansivity,
CI(K-l)
2 x 10-5
(15)
(16)
(17)
where q = p/p is the kinematic viscosity. The temperature
dependence of the viscosity is similar to equation (3). It is
given by
are
AT
R - Y,
wWa - rJ3
Krl
” = (R:;c)2
u’ -
=
~==kexp
time, and velocity
T'=
Ra
(9)
’
k
,
--w
where r ’ is the non-dimensional
radial distance, and 0 the
colatitude.
The Rayleigh number Ra on the right-hand
side of equation (16) is equal to
p@;;)2f$
~ff(R--r,)~
=
c0teayf
r‘2 a8
dT’
= Ra . sin8 . x
where T’, t’, and u’ are non-dimensional
temperature,
time, and velocity, respectively,
and Y, the core radius.
When a liquid core is present, AT is the temperature
difference between the temperature
at the CMB and the
temperature
at the surface To. For a solid core, we replace
the temperature
difference by
AT
!I” and vorticity w ’ which are solutions
set of equations :
( 1
~~
(18)
where H* is the activation enthalpy (H* = E* + PV*), T
the mean temperature
of the mantle, and ,V = 1021Pas for
i’ = 1350°C. This latter value is chosen to be consistent
with the value of the Earth’s upper mantle viscosity. At
the pressure conditions of the Martian mantle, the values
of activation energy and volume used in equation (3) yield
H*-300kJmol-‘.
Equations (15) and (16) are solved using finite difference
approximations.
A multigrid
algorithm
is used. It is
derived from the one described by Stuben and Trottenberg
(1982). This solver is based on alternate zebra relaxation
(smoothing
operator), transfer from fine to coarse grids
(restriction operator) by half injection, and transfer from
coarse to fine grids (extrapolation
operator)
by linear
interpolation.
The upper boundary is a rigid surface. The
lower boundary is a free-slip surface in the case of a liquid
core, and a rigid surface in the case of a solid core. The
solution of equations (8), (15), and (16) is time-dependent
because there is no steady-state pattern of convection. The
calculations
are carried out for a large number of time
steps, and the heat flux coming out of the planet oscillates
about a value which remains constant.
The mean temperature profile of the mantle is chosen such that the
heating term aT’/dt’, and the misfit between the left- and
right-hand
sides of the conservation
equation of thermal
energy (equation (8)) are minimum. At the final stage of
the numerical
experiments,
the results are subsequently
recasted in a dimensional
reference frame using the values
listed in Tables 2 and 3.
of the temperature
profiles
Surface gravity,
9 (m sP2)
3.7
Thermal diffusivity,
IC(m’s_‘)
1o-6
A. Mocquet et al.: Theoretical seismic models of Mars
Table 3. Input parameters
of the three studied limit cases (Sotin
et al., 1996)
Model 1
Model 2
Model 3
Mean mantle density p,
kg rn-’
3416
3471
3471
Liquid core radius, km
1841
1546
Solid core radius, km
1338
1546
Total mass of the core M,,
1.7 x 1023 1.3 x 1023 1.3 x 1023
kg
CMB temperature T,, K
1600
1700
Liquid core bulk
modulus, GPa
145
200
T, values at the core-mantle
boundary (CMB) include an adiabatic contribution
equal to 0.1 Kkm-i.
The values of bulk
moduli were estimated by Sotin et al. (1996) after the experimental works of Boehler et al. (1990), Boehler (1992), and Fei
et al. (1995).
2.2. Temperature
andpressure projiles
Following the studies by Schubert and Spohn (1990) and
Sotin et al. (1996) on the crystallization of the Martian
core, three limit cases are considered. The first case considers that the liquid core is enriched in sulfur, namely
that the averaged mass fraction xFeS is equal to 0.608
(Schubert and Spohn, 1990). This hypothesis induces the
presence of a large liquid core, 1841 km in radius, surrounding a solid core 1338 km in radius (Sotin et al.,
1996). Conversely, both last limit cases consider that the
core is uniquely composed of iron, either liquid (case 2)
or solid (case 3). Since the density is unaffected by the
liquid or solid state of iron, both cases correspond to a
core radius equal to 1546 km. Additional values of the
constant parameters are listed in Table 2, and the input
parameters corresponding to the three different cases are
listed in Table 3. At the final stage of the calculations, the
values of Rayleigh number and kinematic viscosity are
equal to 105, and 101gm2s-‘, respectively, for cases 1 and
2. In the solid core case, the final values of Rayleigh
number and kinematic viscosity are equal to 8 x 105, and
5.3 x 10” m2 s-‘, respectively.
In Fig. 1, the three temperature profiles of the mantle
are plotted as a function of depth z, and compared to the
previous result of Johnston and Toksiiz (1977). One of
0
P
400
1255
the main results of the computations is that the nature of
the core (either liquid or solid) and the boundary conditions at the CMB have a minor influence on the values
of the temperature over more than 65% of the mantle
radius, namely from the surface down to 1200 km depth.
Going from the surface downwards, the three profiles
are first characterized by a linear increase of temperature
with depth, with an associated slope equal to
dT/dz = 4.4K km-‘. This linear increase of temperature
defines a 300 km thick conductive thermal boundary layer.
This latter value can be interpreted as representative of
the lithospheric thickness. Both values of temperature
gradient and lithospheric thickness are identical to the
results of Johnston and Toksiiz (1977) in this depth range.
A 300 km thick thermal lithosphere is also in good agreement with the thermal history calculations of Breuer et al.
(1993), who included the effects of mantle differentiation
by crustal growth. Breuer et al. (1993) obtained values of
thermal lithosphere thickness as large as 15&250 km, and
up to 500 km, in the northern and southern hemisphere
of Mars, respectively. A maximum value of temperature
(- 1600 K) is reached at a depth equal to1 600 km. This
maximum is followed deeper by a negative gradient of
temperature, of the order of -0.4 Kkn-‘.
Down to
1200 km depth, the maximum difference between the three
profiles hardly reaches 100 K. At greater depths, down to
the CMB, more significant differences appear, due to the
different core radii and boundary conditions at the CMB.
When a liquid core is present (cases 1 and 2) the CMB is
overlain by a thermal boundary layer, about 250 km thick,
which is induced by the fixed temperature condition at the
CMB. This thermal boundary layer vanishes in the case
of a solid core (case 3), and the temperature remains
almost constant at a value close to 1300 K, from 1200 km
depth down to the CMB. These temperature profiles indicate that, below a depth of 300 km, the Martian mantle
temperature is bracketed in a 1200-1600 K range of temperature. These latter values contrast with[ the model of
Johnston and Toksdz (1977), which does, not include a
decrease of temperature below the lithosphere, and where
the temperature at the CMB reaches a high value equal
to 2100K. The temperatures that we obtain are also up
to 500 K colder than the values of Schubert et al. (1992).
Since the three cases that we studied are limit cases, the
stability of the profiles whatever the core radii or the
boundary conditions strengthen our opinion that the
mean temperature of the Martian mantle must be close to
a value equal to 1400 K. The temperature profiles displayed in Fig. 1 can be reproduced using the fifth-order
polynomial coefficients listed in Table 4.
For a constant density Martian mantle, the radial
dependence of pressure P(r) is given by l(Turcotte and
Schubert, 1982)
1600 L
i
t
/
400
800
/
,I
1200
Tempwature,K
1600
Fig. 1. Temperature
profiles in the mantle of Mars. Plotted
values include an adiabatic gradient of temperature
equal to
0.1 Kkn-’
where M, is the total mass of both liquid and solid cores.
In fact, it can be easily verified that, using equation (19)
and the values listed in Table 3, the relatively small radius
A. Mocquet
1256
Table 4. Fifth-order
Core
Liquid + solid
Liquid
Solid
The temperature
polynomial
fit to the temperature
Fig. 2. Relaxation
-..._._---.
profiles (Fig. 1)
Cl, 105
c2, lo5
cg, 105
cq, 105
cg, 105
-0.9312
0.8347
0.2587
6.5817
- 5.9847
- 1.9590
- 17.6860
17.4590
6.3040
22.8560
-25.5170
- 10.2290
- 14.1190
18.6930
8.3332
3.3007
- 5.4823
-2.7055
T is given in Kelvin by T(x) = I’= ,,c,x’, where x is the normalized
38.193rlR
(20)
where the unit of P is GPa. The approximate
relationship
(20) does not give exactly P(R) = OGPa, but the discrepancy
(0.407 GPa) has absolutely
no effect on the
results of the following sections. The pressure reached at
the CMB is equal to 18 GPa in case 1, and 2 1.4 GPa in
cases 2 and 3.
In order to check the plausibility
of a 1400 K Martian
mantle, the relaxation
time ratio r2/r,, (equation
(3)) is
computed as a function of pressure for five cases (Fig. 2) :
the three models obtained in this study, the temperature
profile proposed
by Johnston
and Toksoz (1977), and
an adiabatic profile representative
of the Earth’s upper
mantle (Vacher et al., 1996). This latter linear profile is
constrained
using values of temperatures
equal to 1523 K
(Davaille and Jaupart, 1994), and 1873 K (y-spinel+perovskite + magnesiowtistite)
at the base of the Earth’s lithosphere (80 km depth), and at 660 km depth, respectively.
The curve corresponding
to the temperature
profile of
Schubert
et al. (1992) would
follow the curve corresponding
to the Earth’s case. Figure 2 shows that our
temperature
profiles are consistent
with a low Martian
mantle Q. Between 5 and 15 GPa, the relaxation time ratio
z&, increases to values up to lo2 higher than in the Earth’s
/
seismic models of Mars
co, 10s
of planet Mars induces that, in any case, the radial dependence of pressure can be approximated
by a straight line
with sufficient accuracy. Moreover, the pressure profile of
the first studied case differs by only 0.1 MPa km-’ from
the pressure profile corresponding
to cases 2 and 3. Therefore, we use a single pressure versus radius law whatever
the studied case. This law is approximated
by
P(P) = 39.2-
et al.: Theoretical
Johnsto;:&To!isoz
(1977)
time ratios (equation (3)) corresponding
to
the three temperature
profiles displayed in Fig. 1. Similar calculations for the model of Johnston and Toksiiz (1977) and an
adiabatic model of the Earth’s upper mantle are shown for
comparison.
The temperature
profile proposed by Schubert et
al. (1992) would lead to a curve similar to the Earth’s case at
pressures higher than 5 GPa
radius (x = v/R).
case. At higher pressures, the differences in temperature
between the three models involve large variations of relaxation time ratios. For model 1 (liquid and solid cores case)
zz/zo remains constant,
whereas for model 3 (solid core
case), z2/zo reaches values up to lo5 higher than the Earth’s
values at P = 20 GPa. These are extreme cases. A mean
mantle temperature
of the order of 1400K (liquid core,
case 2) leads to a maximum of a three decade shift of z2
with respect to the Earth’s value, as would be expected
from the tidal Q values of the respective planets. On the
other hand, mantle temperatures
higher than 1800 K cannot explain the low Q value of Mars in a straightforward
way. The temperature
profile proposed by Schubert et al.
(1992) would provide z2/zo ratios similar to the Earth’s
values, and the model of Johnston
and Toksiiz (1977)
predicts a Q value higher than the Earth’s one. In both
cases, the presence of volatile and/or partial melting in
the mantle would be required to explain the low Q value
of Mars at tidal frequencies. This additional explanation
is not required by our models. Note that our three cases
display a minimum zZ/zo value at a pressure close to 5 GPa,
which corresponds to the base of the Martian lithosphere.
Such a minimum should induce a minimum of attenuation
in this depth range (Lognonne and Mosser, 1993).
3. Seismic structure of the Martian mantle
The main result of the previous section was the very stable
behaviour of both temperature and pressure profiles whatever the limit cases considered. Moreover, since the pressure gradient inside Mars is very small, it is expected
that temperature
will constrain more heavily the seismic
structure of the mantle than pressure. With temperature
and pressure profiles at hand, density and seismic velocity
profiles are computed as a function of depth using Grtineisen’s and third-order
finite strain theory (Duffy and
Anderson,
1989; Vacher et al., 1996). At this stage, a
compositional
model of the mantle is necessary. Section
3.1 will briefly describe the method used for the computation of density and seismic velocity profiles as a function of temperature
and pressure. The choice of the mineralogical models will subsequently
be argumented,
with
emphasis on the iron content of the mantle. In Section 3.3
the seismic profiles corresponding
to the different thermal
and compositional
states of the mantle considered will be
presented.
3.1. Method
The computation
of seismic velocities and density from
known thermodynamical
conditions and mineraIogies has
A. Mocquet et nl.: Theoretical seismic models of Mars
been extensively explained by previous investigators (e.g.
Davies and Dziewonski, 1975 ; Anderson, 1988 ; Duffy
and Anderson, 1989; Vacher et al., 1996). Therefore, the
method will be only briefly outlined in this paper. For
specific formulas and computational details, we refer the
reader to Duffy and Anderson (1989) and Vacher et al.
(1996). First, available data on densities and thermo-elastical properties of mantle minerals are corrected for temperature effects at surface pressure conditions using Grtineisen’s theory. The corrected values are subsequently
projected adiabatically at depth using third-order finite
strain theory. In the convection experiments, the compressibility of mantle materials and hydrostatic pressure
conditions at depths are neglected. The temperature profiles obtained previously are hence valid at zero pressure,
and the individual temperature values can be used directly
as a series of temperatures To(z) which correspond to the
feet of the adiabats valid at the respective depths z. Duffy
and Anderson (1989) showed that the uncertainties introduced by the approximations of the theory are much smaller than the uncertainties on the elastic and thermal parameters of mantle minerals provided by mineral physics
experiments performed at surface pressure conditions.
The maximum uncertainties on the computed densities
and elastic moduli do not exceed 1 and 2% of their values,
respectively. The first two steps of the procedure are performed separately for each individual mineral. The last
step is to compute the elastic properties of the mantle from
the properties of individual minerals. This is achieved by
bracketing the averaged values of the elastic moduli for
the composite material between highest and lowest
bounds. The Voigt-Reuss-Hill (VRH) averaging method
is often used for this purpose (e.g. Bass and Anderson,
1984). However, we prefer to use the more accurate procedure (HS) proposed by Hashin and Shtrikman (1963)
which is based on energetic variational principles. Indeed,
Watt et al. (1976) showed that the VRH average can be a
poor approximation which can eventually lie outside the
two HS bounds. The better accuracy of the HS procedure
with respect to VRH has also been verified by Vacher et
al. (1996).
3.2, Mineralogical
models
Since no seismological data concerning the interior of
Mars are available yet, we base our calculations on crude
mineralogical models. Doing so, we prefer to make a
few very simple assumptions concerning the mineralogical
composition of the mantle of Mars than to use rather
detailed models (Dreibus and Wanke, 1985; Laul et al.,
1986; Longhi et al., 1992), because, as has been noted by
Bills and Rubincam (1995), these latter models are more
constrained by plausible assumptions than relevant observations. The major mantle constituents of terrestrial
planets are expected to be a-olivine, /3- and y-spine1 phases
of olivine, garnet, clinopyroxenes, and orthopyroxenes
(Ringwood, 1975). While some work has been done on
the effects of the transformation of y-spine1 to perovskite
on mantle convection of Mars (e.g. Weinstein, 1995), we
do not need to consider ultra-high pressure phases such
1257
20
15
10
100
80
Mg2
Si O4
, Mel
60
%
40
Fig. 3. Phase diagram in the system Mg,SiOb-Fe$iO, at 1473K,
after Katsura and Ito (1989).The grey shaded area indicates the
domain studied in this paper : a, a-olivine ; b and y, p- and yspine1phases of olivine, respectively
as magnesiowtistite, perovskite, and stishovite, because
the pressure attained at the CMB does not exceed
21.5 GPa in our models. On the other hand, the pressure
dependence of the phase transformations of olivine is
strongly controlled by the iron content of minerals (Katsura and Ito, 1989). In the case of planet Mars, this pressure dependence is very important because the low pressure gradient of the mantle (equation (20)) implies that a
relatively small change in pressure (e.g. 1 GPa) corresponds to a large variation of depth (> 85 km). In order
to assess the influence of the molar fraction of iron X,, on
the density and seismic structure of the mantle, we span a
wide domain varying from X,, = 10 to 40%. The corresponding phase transformations are displayed in Fig. 3,
and the mineralogical models are presented in Fig. 4. The
increase of X,, has two main effects. First, it allows for
the coexistence of two phases at the same pressure, for
instance a-olivine and y-spine1 between 8 and 12GPa
(X,, = 30%). We already pointed out that, within Mars,
a small range of pressure corresponds to a large range of
depths. Therefore, the coexistence of two different phases
can occur over a large thickness. In the latter example, CIolivine and y-spine1 coexist over 360 km, from 660 down
to 1020km (Fig. 4). Second, within the coexistence
domain of two different phases, the iron content of one
phase can be very high. For example, in the case
X,, = 30%, the iron content of the y-spine1 phase at
8.5GPa reaches a value as high as 60% (Fig. 4). Such
behaviours have important effects on densities and seismic
velocities. This point is addressed in the next sub-section.
In view of the uncertainties on the experimental data,
the coexistence domain of two different phases is only
taken into account when it spans more than 1 GPa. Therefore, the (a-olivine + fi-spinel) and (P-spine1 + y-spinel)
domains which are present around 13 and 17 GPa, respec-
1800
y -spine1
.
1600
Gt
1400
y -spine1
Gt
1400
16
p-spine1
1200
4
1000
8
2
a
800
/
/
600
cpx
wolivine
I-
400
1
a-olivine
CPX
400
4
OP
200
0
I0
20
40
60
0
80
OP X
200
I
I
I
20
40
60
0
0
100
volume fraction, %
I
lO(
volume fraction, %
f
I
1800
f80
I
IXFe=40%
.
1800
20
y-spine1
id0
.
1600
y-spine1
Gt
1400
16
1200
1400
.
1200
.
Gt
17%
i
#
A
s
k
4
1000
12
13%
”
f
\
800
8
a-olivine
600
400
2
lo*
z 2
5 5
800
“$‘O
w
600
1
400
.
200
.
a-olivine
4
200
0
O
L
20
40
60
volume fraction, %
Fig. 4. Simplified
0
0
I
20
I
I80
40
60
volume fraction, %
compositional
models, based on the phase diagram displayed in Fig. 3. The
coexistence of different phases of olivine is taken into account only if the involved pressure range
extends over more than 1 GPa. The molar fraction of iron of each individual phase is indicated in per
cent at the beginning and at the end of the coexistence domains : Gt, garnet ; Cpx, clinopyroxene ;
Opx, orthopyroxene
; cc,ol-olivine ; p and y, p- and y-spine1 phases of olivine, respectively ; X,,, molar
fraction of iron
A. Mocquet et al.: Theoretical seismic models of Mars
1259
Table 5. Thermal and elastical parameters
used in the calculations of densities and seismic velocities. Unless stated otherwise, the
values are taken from DuEy and Anderson (1989) and are valid at pressure P = 0 GPa, and temperature T = 298 K
a-Olivine
y-Spine1
Pyrope
732
27.2”
3.222+1.182X,,
130b
82-31X,,
5.32
2.0
1.00188
952
21.5
3.472+1.240X,,
174
11&41x,,
6.42
2.0
1.00256
849
19.3
3.548+1.300X,,
184
11941x,,
16.5
1.5
1.00337
981”
23.8
3.562+0.758X,,
175 +x,,
90 + 8X,,
5.1
1.8
-0.016
-0.013
4.7*
1.9*
-0.018
-0.014
4.8
1.8
-0.017
-0.014
4.9
1.4
-0.021
-0.010
5.044
2.05
QO, lO”Jmol-’
k,
1.00390
a,, lo-*
@I,, K
CI,10-6K-’
P, gcmp3
K, GPa
G, GPa
P-Spine1
aK/aP
acyap
aK/aT, GPa K-’
aG/aT, GPa K-’
CPx
3.93
1.75
1.00332
941”
25.5
3.277+0.380X,,
113+7x,,
67-6X,,
4.5
1.7
-0.013
-0.010
OPx
3.68
1.75
1.00402
935
26.0
3.204+0.799x,,
104
77-24X,,
5.0
2.0
-0.012
-0.011
Cpx, clinopyroxene ; Opx, orthopyroxene ; Q,,, ratio of heat capacity over thermal expansion at T = 0 K ; k,, anharmonic parameter
proportional to the pressure derivative of the bulk modulus at T = 0 K ; a,, volume ratio at T = 298 and 0 K, respectively; Or,,
Debye’s temperature ; cc,thermal expansivity ; p, density ; K, bulk modulus ; G, shear modulus ; X,,, molar fraction of iron.
“Isaak et al. (1989).
bIsaak (1992).
“Ita and Stixrude (1992).
*Ridgen et aE. (1992).
tively, for X,, = 10% in Fig. 3, are not taken into account
in Fig. 4, where the respective phase transitions
are considered to occur sharply, A similar approximation
is used
for the (a-olivine + y-spinel) to P-spine1 transformation
(X,, = 20%) at P = 12 GPa, where the (a-olivine + /?-spinel) coexistence domain is discarded. Since the coexistence
of two phases induces a smoothing
of the density and
seismic velocity discontinuities
(see Section 3.3), these
approximations
are equivalent
to sharpen
the discontinuities
in the seismic models.
3.3. Density and seismic velocity proJiles
The thermal and elastical parameters
necessary for the
computation
of densities and seismic velocities at given
pressure and temperature
conditions are listed in Table 5.
These experimental
data rely basically on the compilation
of Duffy and Anderson (1989) updated by later works of
Isaak et al. (1989), Isaak (1992), Ita and Stixrude (1992)
and Ridgen et al. (1992). The data available at the present
time show a strong linear dependence of density and shear
moduli values on the iron content
of minerals.
For
instance,
if X,, = lo%, the density of olivine phases
increases and their shear modulus G decreases by 3.6% of
their values at X,, = 0%. In comparison,
a 3.6% decrease
of the G value by temperature
effects alone requires a
thermal increase of the order of 270 K at surface pressure
conditions.
In addition, both density and G decrease with
temperature
at constant pressure. The bulk modulus K is
not affected by the iron content of olivine phases. Therefore, since seismic velocities are proportional
to the square
root of the elastic mod&i, and inversely proportional
to
the square root of density, it is expected that seismic
velocities are more sensitive to the X,, value than to the
temperature
difference between thermal profiles. This is
clearly visible in Figs 5-7, where the computed densities,
P-wave, and S-wave velocities, respectively, are plotted as
a function of depth for different values of X,,, and for the
three temperature
profiles obtained previously.
In each
figure, the model of Anderson et al. (1977) is shown for
reference.
For a given X,, value, the density and seismic velocity
profiles are almost identical whatever the temperature
profile. Among the three limit cases considered in Section
1, the largest difference
of temperature
(400 K) was
obtained at the CMB between cases 2 and 3. This difference of temperature
corresponds
to differences equal to
0.02 gcme3 (0.5%), 0.09 km s-’ (0.9%), an.d 0.07 km s-’
(1.3%), for density, V, and Vs, respectively. These latter
values are within the range of uncertainty
(l-2%) associated with the experimental
data on thermal and elastical
parameters.
Therefore, in what follows, we consider density and seismic velocity profiles averaged over the three
limit cases considered in Section 1. These averaged profiles
are considered to be a good first-order approximation
of
the radial upper mantle structure of Mars for a given iron
content of mantle minerals, whatever the structure of the
Martian core.
The averaged’ profiles, listed in Table 6, display the
following characteristics.
As noted previously
by Anderson et al. (1977), the values of density and seismic velocity are within the range of values attained in the Earth’s
transition
zone. The seismic structure at the CMB on
Mars is close to the structure of PREM (Dziewonski and
Anderson,
1981) at a depth of 600 km. The uppermost
300 km of the Martian mantle are characterized
by a lowvelocity zone (LVZ) which corresponds
to the shallow
conductive
thermal boundary
layer observed in Fig. 1.
This LVZ is due to the increase of temperature with depth
(4.4 K km- ‘) which is relatively large when compared with
the small pressure gradient (0.01 GPa km-‘) in this depth
range. If we take the density and P- and S-wave velocities
1260
A. Mocquet
$, 3.8
et ul.: Theoretical
seismic models of Mars
-
3.4
”
3.2t:’
0
0
”
500
’ ”
3.2”
”
”
500
”
’ ”
”
1500
1000
depth, km
’ ”
”
I’
3.2
2000
:I
’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ L ’
500
0
”
’ ”
1500
3.2
’ ”
2000
km
.’
0
1500
1000
depth,
1000
depth,
’ * ”
2000
km
’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’
500
1500
1000
depth,
2000
km
Fig. 5. Density versus depth profiles computed
for the four compositional
models displayed in Fig.
4, and for the three temperature profiles displayed in Fig. 1 and listed in Table 4. Heavy solid curve,
case 1 (liquid and solid cores) ; thin solid curve, case 2 (liquid core) ; heavy dashed curve, solid core
(case 3); dotted curve, density model of Anderson et al. (1977). When the value of X,, is fixed, the
density profiles are almost indistinguishable
whatever the case
at a depth of 60 km as reference values, these parameters
decrease by amounts
of 0.5, 2, and 3%, respectively,
within the LVZ of PREM (Dziewonski
and Anderson,
198 1). These relative decreases reach the same amounts in
our models of Mars, regardless of the X,, value, except for
S-wave velocities, for which the relative decreases reach
values as high as 5%. We can thus conclude that a LVZ
similar to the Earth’s one should exist at the base of the
Martian lithosphere.
As stated previously, X,, is the parameter which mainly
controls the variations
of density and seismic velocities
among all models. First, an increase of X,, results in an
increase of the densities, and in a decrease of seismic
velocities.
For instance,
the values of p and Vs for
X,, = 20% differ by 2% from their respective values at
Xr, = 10%. Similarly, the values for X,, = 30% vary by
4% from the values corresponding
to X,, = 10%. The
effect on V, is smaller in our calculations
(1.4%) because,
to our knowledge, there is no experimental
data relating
the bulk modulus and the iron content of olivine phases
(Table 5). This value of 1.4% should thus be regarded as a
lower bound on the decrease of P-wave velocities between
X,, = 20 and 10%. The relative variations quoted above
might appear small when compared to the relative uncertainties (l-2% ; Duffy and Anderson,
1989) associated
with the laboratory
data listed in Table 5. However, it
must be kept in mind that these relative variations
of
seismic velocities extend over the entire mantle. Therefore,
A. Mocquet et al.: Theoretical seismic models of Mars
126
8.5
8.0
8.5
-
7.5t:’”
a
8.0
.---
I’
”
500
”
”
”
”
”
1500
1000
’
I1
7.5
2000
0
500
depth, km
1500
1000
depth,
2000
km
10.0
8.5
8.0
’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’
7.&
0
500
1500
1000
depth,
2000
0
km
500
1500
1000
depth,
2000
km
Fig. 6. Same as Fig. 5 for P-wave velocities
as seismic waves travel inside the mantle, time delays are
accumulating
along ray pass. The preliminary calculations
that we are presently carrying out show that the P-wave
travel times computed
using the X,, = 30% model are
delayed by an amount of 10 s with respect to the travel
times associated with the X,, = 10% model; for epicentral
distances larger than 3.5” (Mocquet et al., 1996).
The second effect of X,, on the density and seismic
velocities is to control
the sharpness
of seismic discontinuities
associated
with the phase transitions
of
olivine. In Section 3.2, we pointed out that, according to
Katsura
and Ito (1989), an increase of the Xr, value
induces the coexistence of different phases of olivine over
a large range of depths (Figs 3 and 4). In terms of density
and seismic velocity profiles, this gradual transition from
one phase to another one smoothes out the discontinuities
over the thickness of the coexistence domain of olivine
phases (Figs 5-7). In what follows, we will only describe
the behaviour of seismic discontinuities
for P-waves (Fig.
6) because identical descriptions
can be made regarding
the behaviour of density and S-wave velocities (Figs 5 and
7, respectively) using the values listed in Table 6. For an
Earth-like
composition
(X,, = 10%) the a-olivine to fispinel, and /?-spine1 to y-spine1 phase transitions
translate
into velocityjumps
equal to 0.51 km s-l (5.5%) at 1130 km
depth, and 0.09 km s-r (0.9%) at 1425 km depth, respectively. For X,, = 20 and 30%, the velocity jumps appear
at a shallower depth, around 1000 km. The amplitude of
the X,, = 20% discontinuity
is similar to the amplitude
of the X,, = 10% discontinuity.
For X,, = 30%, this velocity jump decreases to 0.26 kms-’ (2.9%), and vanishes
for X,, = 40%. For X,, values higher than lo%, the dis-
A. Mocquet et al.: Theoretical seismic models of Mars
1262
4.2~~“““““““““,I
0
500
’ ”
4.2”’
1000
depth, km
2000
1500
0
’ ’ ’ ’ ’ ”
500
’ ’ ’ ’ ’ ’ ”
1000
depth,
’
1500
2000
1500
2000
km
5.2
4.8
4.8
4.6
4.6
4.4
4.4
4.2”’
”
0
’ ”
500
”
”
’ ”
1000
depth,
”
1500
”
I
.’
b
’ ”
2000
0
km
500
1000
depth,
km
Fig. 7. Same as Fig. 5 for S-wave velocities
continuity
at 1425 km depth disappears. The smoothing
process of the 1130 km discontinuity
is accompanied
by
an increase of the velocity gradient dV,/dz over a 300 km
thick layer, in the 83&l 130 km depth range, where dI/,/dz
increasesfr0m0.11s-‘(X,,=10%)t00.18s-’(X,,=30%).
Therefore, an iron-rich Martian mantle should rather be
characterized by large seismic velocity gradients extending
over a thickness of several hundred kilometres,
than by
sharp seismic discontinuities.
We would like to emphasize that the effects of the iron
content described in the previous paragraphs
are valid
independently
of the chosen temperature
profile. Indeed,
densities and seismic velocities computed using the temperature profile of Johnston
and Toksiiz (1977) display
identical characteristics
to those described previously : an
increase of X,, induces an increase of the densities, a
decrease of seismic velocities, and a smoothing out of the
seismic discontinuities
associated
sitions of olivine (Fig. 8).
with
the phase
tran-
4. Discussion
In the previous sections, we described the influence of the
iron content of the Martian mantle on the behaviour of
phase transitions,
but we neglected the consequences
of
these phase transitions
on the temperature
profiles,
because these latter effects had not been taken into
account in the convection
experiments.
In fact, both CIolivine to /?-spinel, and p-spine1 to y-spine1 phase transformations
are exothennic,
and hence increase the temperature of the mantle. This “Verhoogen effect” (Jeanloz
and Thompson,
1983) results in an increase of temperature
A. Mocquet
et al.: Theoretical
seismic models of Mars
1263
Table 6. Averaged density p and seismic profiles of the Martian
values are not included in this table
x,, =
Depth
&ml
0
59
119
178
238
297
356
416
475
534
594
653
712
772
831
891
950
1009
1069
1128
1188
1247
1306
1366
1425
1484
1544
1603
1662
1722
1781
1841
1852
(g &-')
3.39
3.38
3.37
3.37
3.37
3.37
3.37
3.39
3.40
3.42
3.44
3.46
3.47
3.49
3.52
3.55
3.58
3.62
3.65
3.68
3.82
3.84
3.85
3.86
3.87
3.92
3.94
3.95
3.96
3.97
3.98
3.99
3.99
10%
x,,
VP
(km s-‘)
(km s-‘)
8.32
8.26
8.18
8.13
8.10
8.09
8.11
8.15
8.20
8.26
8.32
8.39
8.46
8.53
8.61
8.73
8.84
8.95
9.06
9.19
9.70
9.75
9.80
9.84
9.88
9.97
10.02
10.08
10.11
10.13
10.16
10.18
10.19
4.82
4.75
4.68
4.62
4.58
4.56
4.55
4.55
4.56
4.58
4.60
4.62
4.64
4.67
4.70
4.76
4.81
4.86
4.90
4.97
5.32
5.35
5.36
5.38
5.39
5.42
5.44
5.46
5.47
5.48
5.48
5.49
5.49
VS
mantle,
computed
= 20%
for different
molar fractions
x,, = 30%
VP
VS
(gcz-‘)
(kms-‘)
(kms-‘)
(g c&‘)
3.46
3.45
3.44
3.43
3.43
3.43
3.44
3.45
3.47
3.48
3.50
3.52
3.54
3.56
3.59
3.62
3.67
3.72
3.84
3.87
3.89
3.92
3.97
3.99
3.99
4.00
4.01
4.03
4.04
4.05
4.06
4.07
4.07
8.20
8.14
8.07
8.01
7.98
7.98
7.99
8.03
8.09
8.14
8.21
8.28
8.34
8.41
8.49
8.61
8.76
8.92
9.38
9.49
9.58
9.63
9.70
9.79
9.83
9.86
9.89
9.95
9.98
10.01
10.03
10.05
10.06
4.72
4.66
4.58
4.53
4.48
4.46
4.45
4.46
4.46
4.48
4.50
4.53
4.55
4.57
4.61
4.66
4.74
4.83
5.14
5.19
5.23
5.24
5.26
5.30
5.31
5.32
5.34
5.37
5.37
5.38
5.39
5.39
5.39
3.52
3.51
3.50
3.50
3.49
3.50
3.51
3.52
3.53
3.55
3.57
3.59
3.61
3.64
3.69
3.76
3.81
3.87
3.98
4.00
4.02
4.03
4.05
4.06
4.07
4.08
4.09
4.11
4.12
4.12
4.13
4.15
4.15
lower than lOOK. In the case of the a-olivine to /?-spine1
phase transformation
within the Earth, at 410 km depth,
Vacher et al. (1996) computed
a value of P-wave discontinuity
(0.40 kms-‘)
very close to the value of
the
spherically
symmetric
Earth’s
model
IASP
(0.33 km s-’ ; Kennett and Engdahl, 1991), when the Verhoogen effect was taken into account. If this latter effect
was neglected, the velocity jump discrepancy was doubled.
Since the Verhoogen effect tends to diminish the values
of velocity jumps at seismic discontinuities,
the values
determined
in the preceding
section should thus be
regarded as maximum values. It is therefore expected that
the mantle of Mars does not display sharp seismic discontinuities,
in contrast with the model initially proposed
by Anderson et al. (1977) (dotted curves in Figs 5-Q but
that it is rather characterized
by smooth velocity gradients. Zhou et al. (1995) considered
an Earth-like
composition for the Martian mantle (X,, = 12%), and showed
that, in this case, the phase transitions
of olivine could
favour the amplification
and superheating
of megaplumes
in Mars. However, these effects remain locally distributed
in space and should not affect significantly
the average
temperature
profile of the mantle. On the other hand, if
of iron X,,. Crustal
x,, = 40%
VP
VS
(km s-‘)
(km s-‘)
(g c&-j)
(km s-‘)
(km s-l)
4.63
4.56
4.49
4.43
4.38
4.36
4.35
4.36
4.37
4.39
4.41
4.43
4.46
4.50
4.58
4.67
4.77
4.87
5.03
5.09
5.15
5.17
5.19
5.20
5.22
5.23
5.24
5.27
5.28
5.28
5.29
5.29
5.29
3.59
3.58
3.57
3.56
3.56
3.56
3.57
3.58
3.60
3.64
3.68
3.73
3.77
3.82
3.85
3.91
3.95
4.00
4.04
4.07
4.07
4.11
4.12
4.14
4.15
4.16
4.17
4.18
4.19
4.20
4.21
4.22
4.23
7.97
7.91
7.84
7.78
7.75
7.75
7.77
7.81
7.86
7.95
8.06
8.17
8.29
8.41
8.54
8.72
8.90
9.07
9.21
9.32
9.41
9.46
9.50
9.54
9.58
9.61
9.64
9.70
9.73
9.75
9.78
9.80
9.81
4.53
4.46
4.39
4.33
4.29
4.26
4.26
4.26
4.27
4.31
4.37
4.42
4.48
4.55
4.62
4.71
4.81
4.90
4.97
5.02
5.05
5.08
5.09
5.11
5.12
5.13
5.14
5.17
5.18
5.19
5.19
5.20
5.20
8.09
8.02
7.95
7.89
7.86
7.86
7.90
7.92
7.97
8.03
8.09
8.16
8.23
8.32
8.46
8.64
8.82
9.00
9.26
9.40
9.53
9.58
9.62
9.66
9.70
9.73
9.76
9.82
9.85
9.88
9.90
9.93
9.94
VP
VS
the iron content of the Martian mantle is greater than that
of the Earth, the induced smoothness and homogenization
of mantle structure should diminish the influence of phase
transitions
on the geometry of mantle convection.
4.1. Limitations
of the mineralogical
models
The mineralogical
models that we use in this study present
two major crude simplifications.
First, the volume fraction
of olivine vs. garnet and pyroxenes
remains constant
whatever X,,. Doing so, we follow the usual procedure
which separates the bulk composition
of the mineral
assemblage in two subsystems, olivine on one hand, and
the remaining
minerals on the other hand (Jeanloz and
Thompson,
1983 ; Irifune, 1987). It is hence assumed that
both subsystems do not affect each other’s phase equilibria
(Ita and Stixrude, 1992). Ita and Stixrude (1992) noticed
that this assumption
is consistent with the results of Akaogi and Akimoto (1979) which show no change in the
relative proportions
of both subsystems
up to at least
20 GPa. Second, the lack of experimental
data does not
1264
A. Mocquet et al.: Theoretical seismic models of Mars
L5 ti
4.2
c
8
tb
i:
.%
2
4
_g 0.5
4
-1
3.8
0.0
1
[i
P
;
-0.5
L
-1.0
3.6
Anderson et al. (1977)
i
3.2
0
1000
Depth, km
500
1500
2000
0
500
1000
Depth, km
1500
2000
Fig. 9. Relative variations
between the values of densities and
seismic velocities obtained
for a pure olivine mantle
(X,, = 30%), and the mineralogical model valid for X,, = 30%
(Fig. 4), using an identical amount of iron for all minerals. The
values obtained for the case of the pure olivine mantle are used
as references. All calculations are performed using a temperature
distribution averaged over the three profiles displayed in Fig. 1
..’
4.2
0
_:I
0
Anderson et al. (1977)
’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’
1000
500
1500
Depth, km
’ ’
2000
los?
Anderson et al. (1977)
0
500
1000
Depth, km
Fig. 8. Same as Figs 5-7 using the temperature
1500
2000
profile of John-
previously. The first case considers a pure olivine mantle
with X,, = 30%. In the second case, the mineralogical
model valid for X,, = 30% is used (Fig. 4), but an identical
amount of iron is equally assigned to all minerals. The
values obtained
for a pure olivine mantle are used as
references. The relative variations between both cases are
plotted as a function of depth in Fig. 9. As can be expected
from Fig. 4, the main differences (< 1.5%) occur down to
500 km depth. At greater depths, the decreasing amount
of pyroxenes induces a sharp decrease of the relative variations. Below 1000 km, these latter never exceed - 0.4%,
and l%, for seismic velocities and density, respectively.
Even though this test is performed for two extreme cases,
these latter values are smaller than the uncertainties
of
laboratory
measurements,
and more than one order of
magnitude
smaller than the variations
of densities and
seismic velocities reported in the previous section. Therefore, present-day
available data on the behaviour of pyroxenes and garnets as a function of their iron content
(Table 5 ; Duffy and Anderson, 1989) support our hypothesis that the influence of the iron content of the mantle on
its seismic structure is mainly governed by olivine.
4.2. Relationships
between mantle and core structures
ston and Toksiiz (1977)
let us take into account the effect of the increasing amount
of iron in pyroxenes and garnets. Our calculations are thus
based on the basic assumption that the phase transitions in
the olivine system are the major causes of seismic discontinuities
(e.g. Ringwood,
1975), whereas other minerals form a complex solid solution with transitions
spanning a wide depth interval (Akaogi et aZ., 1987). In order
to test the influence of this assumption
on our results, the
distributions
at depth of densities and seismic velocities
are computed for two additional limit cases, using a temperature profile averaged over the three ones obtained
In the previous section, we pointed out that the density
and seismic velocity profiles were almost independent
of
the thermal history and present-day structure of the core,
and a large domain of iron content was considered for
the mantle. Even though astronomical
constraints
on the
inner structure of Mars are sparse, we can however test
our models by computing their associated ranges of Love
numbers,
Q factors (equations
(1) and (2)), and I/MR’
ratios.
The Love numbers h, and k, are computed for y1= 2,
3, and 4 using the values of Table 6. The bulk modulus of
the liquid and solid cores (Table 3) were estimated by
Sotin et al. (1996) using the experimental
data of Boehler
A. Mocquet
et al.: Theoretical
seismic models of Mars
1265
0.35
0.30
0.25
c
0.20
0.15
0.10
0.05
LO
15
20
25
30
35
40
xFe%
0.20
= 1750 km (LM)
\\\\S\\\\\
0.15
ti
0.10
0.05
0.00
r
I
I
I
I
I
I
10
15
20
25
30
35
40
xFe%
Fig. 10. Love numbers h, and k, (n = 2, 3, and 4) corresponding
to the models of Table 6. Due to the lack of data on the shear
modulus of iron at the pressure and temperature
conditions of
a solid Martian core, a liquid core is assumed (Table 3). The
values computed by Lambeck (1979) and Lognonne and Mosser
(1993) (LM) are also shown
et al. (1990), Boehler (1992), and Fei et al. (1995). To our
knowledge, there is no experimental determination of the
shear modulus of iron at the pressures and temperatures
relevant to a Martian solid core. An extrapolation of
Poisson’s ratio of the Earth’s inner core to Martian pressure conditions yield a very low value, of the order of
12GPa. The very large difference between Martian and
Earth’s core pressure conditions makes this estimate very
suspect. Therefore, we prefer not to consider the solid
core, and focus instead on the radius of the liquid core.
The results (Fig. 10) are in a very good agreement with
previous authors’ estimates. The values increase with
increasing core radius, as it is predicted by the theory. In
addition, an increase of the iron content of the mantle
induces a small increase of the Love numbers (N 10%
between X,, = 10 and 40%). The Q values corresponding
to the k2 values shown in Fig. 10 are, following equation
(2), in the range 48 < Q < 95. Our models thus favour a
very low Q Martian mantle, with 50~ Q< 100 at tidal
frequencies.
Following Bills (1990), the mean density (p), and the
mean principal moment of inertia M,, with
M2 = ; <p)I/MR’
(21)
enable to bracket the density versus depth distribution
of the planet. Apart from petrological or geochemical
arguments on the composition of the Martian crust,
mantle, and core, an additional condition must be fulfilled: if the core is composed of pure iron and/or iron
sulfur, its mean density must be greater than 5.77 g cme3,
and lower than 8.09gcmM3 (Schubert and Spohn, 1990).
Under such constraints, possible values of outer core
radius and core density, and the corresponding moment
of inertia ratios I/MR2, can be computed for each of the
models of mantle density listed in Table 6. Three limit
cases are considered (Fig. 11). In the first case (top panel)
a two-layered model without crust is considered. A threelayered model (middle panel), where the mean crustal
density is assumed at a value equal to 2.64gcmp3 (Anderson et al., 1977) is considered in the second case. The
crustal thickness is set equal to 100 km. The two latter
cases have been chosen in such a way that they bracket
the presently unknown averaged crustal thickness of
Mars. The last case (bottom panel) considers a four-layered model. Crustal parameters are identical to the previous case, and, following Sotin et al. (1996), a high density (8.09 g cm-“) solid inner core is assumed., with a radius
equal to 1338 km. Unknown parameters are the radius
and density of the overlying liquid outer core. In Fig. 10,
the straight dashed lines correspond to .the computed
outer core radii. Their respective values are indicated in
km at the end of each line. Following Bills (1990), we
consider that I/MR2 cannot exceed 0.365. As expected
from the definition of the mean principal moment of inertia, the presence of a thick low-density crust shifts the
computed values of outer core radii towards smaller
values of IjMR” when X,, is set at a constant value. The
domains of possible values of outer core radius, outer core
density, and X,, content of the mantle are in very good
agreement with the results of previous investigators. For
example, if we assume an I/MR2 value close to 0.365, an
outer core radius equal to 1700 km is obtained either with
a very thin crust and a Martian mantle composition close
to that of the Earth (X,, = 10% ; top panel), or with a
thick crust and an iron-rich Martian mantle (X,, = 30% ;
middle panel). This value of outer core radius agrees with
the previous estimate of Longhi et al. (1992).
Though we did not use geochemical arguments in our
modelling approach, the density versus depth profiles
computed for X,, = 20 and 30% (Table 6) support previous propositions that the mantle of Mars contains a
large proportion of iron with respect to the Earth’s mantle
(Dreibus and Wanke, 1985 ; McSween, 1985). According
to Fig. 10, iron contents higher than Xr, = 30% should
be avoided, because, whatever the considered case (two-,
three-, or four-layered model), these contents lead to
I/MR2 values higher than 0.365. This is particularly visible
for X,, = 40%, for which I/MR2 values as high as 0.380
(top panel) and 0.370 (middle panel) are obtained for the
two- and three-layered models, respectively. Iron contents
higher than 20% are also impossible if a four-layered
planet is considered (bottom panel), because these contents involve unrealistic liquid core densities lower than
5 gem-3.
Two additional conclusions can be drawn from Fig. 10.
First, ifwe consider that an Earth-like mantle composition
A. Mocquet
1266
0.350
0.360
0.370
0.380
outer
core
should
et al.: Theoretical
be small,
seismic models of Mars
8.0
of 30&400 km.
the weakness (or
magnetic field on
of the order
This small thickness
might explain
absence?) of an internally
generated
Mars.
5. Conclusion
8.0
5.0
j
t
I/Ma2
Fig. 11. Domains of compatibility
between profiles of mantle
density (Table 6), outer core radius (labelled dashed curves, km)
and density, and principal moment of inertia ratio I/MR’. The
rectangles delimit the domain of realistic possibilities. They are
defined such that the I/MR2 ratio is lower or equal to 0.365
(Bills, 1990), and that the averaged outer core density is comprised between 5.77 and 8.09 gcme3 (Schubert and Spohn, 1990).
Open circles, X,, = 10% ; black dots, X,, = 20% ; open squares,
X,, = 30% ; black squares, X,, = 40%. Top panel, two-layered
planet (mantle and core); middle panel, three-layered
planet
(crust, mantle, and core) ; bottom panel, four-layered
planet
(crust, mantle, outer and inner core). A 100 km thick crust with
an average density equal to 2.64 gcmm3 (Anderson et al., 1977)
is considered in both latter cases. An inner core radius equal to
1338 km and an inner core density equal to 8.09 g cme3 are used
for the four-layered planet case
(X,, = 10%) represents
of the Martian
mantle,
the lowest limit of the iron content
the ratio I/MR2 cannot
be lower
than 0.355. Second, if a solid inner core is present
centre of the planet, the thickness of the overlying
at the
liquid
The present-day averaged temperature profile of the Martian mantle is found to be almost independent
of the
thermal history of the core. A 300 km thick thermal lithosphere,
displaying
a temperature
gradient
equal to
4.4 K km-’ is followed at greater depths by a quasi-isothermal mantle, the temperature
of which is bracketed in
a 1200-l 600 K temperature range. These temperature and
associated seismic models favour a very low Q Martian
mantle, with 50 < Q < 100 at tidal frequencies. The lithospheric gradient of temperature,
together with the small
increase
of pressure
with depth,
of the order of
0.01 GPa km-‘, induce the presence of a low-velocity zone
similar to that of the Earth, down to 300 km depth. At
greater depths, computed densities and seismic velocities
have the same order of magnitude
as within the Earth’s
transition
zone. An increase of the iron content of the
Martian mantle with respect to that of the Earth results
(1) in an increase of density, and a decrease of seismic
velocities, which can reach more than 2% of the values
expected from an Earth-like composition,
(2) in a homogenization of mantle structure through the smoothing out
of seismic discontinuities
over a thickness of a few hundred kilometres.
It is therefore expected that a composition of the mantle of Mars similar to the one derived
from the studies of SNC meteorites (Ringwood,
1979;
Dreibus and Wanke, 1985 ; McSween, 1985) should be
detectable by seismic methods.
Aclznowledgements.
This work has been conducted
as a contribution to the preparation
of future exploratory
missions to
Mars, as a collaboration
between European,
American,
and
Russian space agencies. Financial support by French Ministry
of Research,
Centre National
de la Recherche
Scientifique
(C.N.R.S.)
and Programme
National
de Planetologie
are
acknowledged.
Constructive reviews by P. Lognonne and H. C.
Nataf were helpful in improving this paper. The program necessary for the computation
of Love numbers was kindly provided
by P. Lognonne.
References
Akaogi, M. and Akimoto, S., High-pressure
phase equilibria
in a garnet lherzolite, with special reference to Mg’+-Fe*+
partitioning among constituent minerals. Phys. Earth Planet.
Inter. 19, 31-51, 1979.
Akaogi, M., Navrotsky, A., Yagi, T. and Akimoto, S., Pyroxenegarnet transformation
: thermochemistry
and elasticity of
garnet solid solutions and application to a pyrolitic mantle,
in High-pressure Research in Mineral Ph)&s (edited by M.
H. Manghnami
and Y. Syono), pp. 251-260. Terra Scient.
Publ. Co. (TERRAPUB),
Tokyo/Am.
Geophys.
Union,
Washington,
DC, 1987.
Anderson, D. L., Internal constitution of Mars. J. Geophys. Res.
77,789%795,1972.
Anderson, D. L., Temperature and pressure derivatives of elastic
A. Mocquet
et al.: Theoretical
seismic models of Mars
constants with application
to the mantle. J. Geophys. Res.
93,4688-4700,
1988.
Anderson, D. L. and Bass, J. D., Transition region of the Earth’s
upper mantle. Nature 320, 321-328, 1986.
Anderson, D. L. and Given, J. W., Absorption
band Q model for
the Earth. J. Geophys. Res. 87, 3893-3904, 1982.
Anderson, D. L. and Minster, J. B., The physics of creep and
attenuation in the mantle, in Anelasticity in the Earth (edited
by F. D. Stacey, M. S. Paterson and A. Nicholas), Geodynamics Series 4, pp. 5-12. Am. Geophys. Union, Washington, DC, 1981.
Anderson, D. L., Miller, W. F., Nakamura, Y., Toksiiz, M. N.,
Dainty, A. M., Duennebier, F. K., Lazarewicz, A. R., Kovach,
R. L. and Knight, T. C. D., Seismology on Mars. J. Geophys.
Res. 82,45244546,
1977.
Avanesov, G. A., Bonev, B. I., Kempe, F., Bazilevsky, A. T.,
Boycheva, V., Cbikov, K. N., Danz, M., Dimitrov, D.,
Duxbury, T., Gromatikov, P., Halmau, D., Head, J., Heifetz,
V. N., Kolev, V., Kostenko, V. I., Kottsov, V. A., Kratsevtev,
V. M., Krasikov, V. A., Krrnnov, A., Knzmin, A. A., Losev, K.
D., Lumme, K., Mishev, D. N., Miihlmann, D., Muinonen, K.,
Murav’ev, V. M., Murchie, S., Murray, B., Neumann, W.,
Paul, L., Petkov, D., Petucbova, I., Piissel, W., Rebel, B.,
Sbkuratov, Yu. A., Simeonov, S., Smith, B., Totev, A., Uzunov,
Yu., Fedotov, V. P., Weide, G.-G., Zapfe, H., Zhukov, B. S.
and Ziman, Ya. L., Television observation of Phobos. Nature
341, 585-587,1989.
Bass, J. D. and Anderson, D. L., Composition
of the upper
mantle: geophysical tests of two petrological
models. Geophys. Res. Lett. 11,237-240, 1984.
Bills, B. G., The moment of inertia of Mars. Geophys. Res. Lett.
16,385-389, 1989.
Bills, B. G., Geodetic constraints
on the composition
of Mars.
J. Geophys. Res. 95, 14,131-14,136, 1990.
Bills, B. G. and Ferrari, A. J., Mars topography
harmonics and
geophysical
implications.
J. Geophys. Res. 83, 3497-3508,
1978.
Bills, B. G. and Rubincam, D. P., Constraints
on density models
from radial moments:
applications
to Earth, Moon, and
Mars. J. Geophys. Res. 100, 26,305-26,315, 1995.
Boehler, R., Melting of the Fe-Fe0
and the Fe-FeS systems at
high pressure : constraints on core temperature.
Earth Planet.
Sci. Lett. 111,217-227,
1992.
Boehler, R., Von Bargen, N. and Chopelas, A., Melting, thermal
expansion, and phase transitions of iron at high pressures. J.
Geophys. Res. 95,21731-21736,
1990.
Born, G. H., Mars physical parameters as determined from Mariner 9 observations of the natural satellites and Doppler tracking. J. Geophys. Res. 79,48374844,
1974.
Breuer, D., Spohn, T. and Wiilluer, U., Mantle differentiation
and the crustal dichotomy
of Mars. Planet. Space Sci. 41,
269-283, 1993.
Burns, J. A., Orbital evolution, in Planetary Satellites (edited
by J. A. Burns), pp. 113-156. University of Arizona Press,
Tucson, Arizona, 1977.
Burns, J. A., The evolution of satellite orbits, in Satellites (edited
by J. A. Burns and M. S. Matthews), pp 117-I 58. University
of Arizona Press, Tucson, Arizona, 1986.
Burns, J. A., Contradictory
clues as to the origin of the Martian
moons, in Mars (edited by H. H. Kieffer, B. M. Jakosky, C.
W. Synder and M. S. Matthews), pp. 1283-1301. University
of Arizona Press, Tucson, Arizona, 1992.
Busse, F. H., Fundamentals
of thermal convection,
in Mantle
Convection, Plate Tectonics and Global Dynamics, Vol. 4, The
Fluid Mechanics of Astrophysics and Geophysics (edited by
W. R. Peltier), pp. 23-96. Gordon & Breach, New York,
1989.
Chapron-Touze,
M., Orbits of the Martian satellites from ESAPHO and ESADE theories. Astron. Astrophys. 240, 159-172,
1990.
1267
Davaille, A. and Jaupart, C., Onset of thermal convection in
fluids with temperature-dependent
viscosity : application to
the oceanic mantle. J. Geophys. Res. 99, 19853-19866, 1994.
Davies, G. F. and Dziewonski, A. M., Homogeneity
and constitution of the Earth’s lower mantle and outer core. Phys.
Earth Planet. Inter. 10, 336-343, 1975.
Dreibus, G. and Wanke, H., Mars, a volatile-rich planet. Meteoritics 20, 367-381, 1985.
Duffy, T. S. and Ahrens, T. J., Lateral variations in lower mantle
seismic velocity, in High-pressure Research : Application to
Earth and Planetary Sciences (edited by Y. Syono and M. H.
Manghnami),
pp. 197-205. Terra Scient. Publ. Co. (TERRAPUB), Tokyo/Am.
Geophys. Union, Washington,
DC,
1992.
Duffy, T. S. and Anderson, D. L., Seismic velocities in mantle
minerals and the mineralogy of the upper mantle. J. Geophys.
Res. 94,1895-1912,
1989.
Dziewonski, A. M. and Anderson, D. L., Preliminary reference
Earth model. Phys. Earth Planet. Inter. 25,297-356,
1981.
Fei, Y., Prewitt, C. T., Mao, H. and Bertka, C. M., Structure
and density of FeS at high pressure and high temperature
and the internal structure of Mars. Science 268, 1892-1894,
1995.
Hashin, Z. and Shtrikman, S., A variational
approach to the
theory of the elastic behaviour of multiphase materials. J.
Mech. Phys. Solids 11, 127-140, 1963.
Irifune, T., An experimental
investigation of the pyroxene-garnet transformation
in a pyrolite composition
and its bearing
on the constitution
of the mantle. Phys. Earth Planet. Inter.
45,324-336,
1987.
Isaak, D. G., High temperature
elasticity
of iron-bearing
olivines. J. Geophys. Res. 97, 1871-1885, 1992.
Isaak, D. G., Anderson, 0. L. and Goto, T., Elasticity of singlecrystal forsterite to 1700K. J. Geophys. Res. 94, 5895-5906,
1989.
Ita, J. and Stixrude, L., Petrology, elasticity, and composition
of the mantle transition zone. J. Geophys. Res. 97, 68496866, 1992.
Ito, E. and Takahashi,
E., Ultrahigh-pressure
phase transformations and the constitution
of the deep mantle, in Highpressure Research in Mineral Physics (edited by M. H.
Manghnami and Y. Syono), pp. 221-229. Terra Scient. Publ.
Co. (TERRAPUB),
Tokyo/Am.
Geophys.
Union, Washington, DC, 1987.
Jeanloz, R. and Thompson, A. B., Phase transition and mantle
discontinuities.
Rev. Geophys. Space Phys. 21, 51-74, 1983.
Johnston, D. H. and Toksiiz, M. N., Internal structure and
properties of Mars. Icarus 32, 73-84, 1977.
Johnston, D. H., McGetchin, R. T. and Toksiiz, M. N., Thermal
state and internal structure of Mars. J. Geophys. Res. 79,
3959-3971, 1974.
Katsura, T. and Ito, E., The system Mg,SiO,-Fe,SiO,
at high
pressures and temperatures
: precise determination
of stabilities of olivine, modified spinel, and spinel. J. Geophys. Res.
94, 15,663-15,670, 1989.
Kanla, W. M. and Asimow, P. D., Tests of random density
models of terrestrial planets. Geophys. Res. Lett. 18, 909912,199l.
Kaula, W. M., Sleep, N. H. and Phillips, R. J., More about the
moment-of-inertia
of Mars. J. Geophys. Res. 16, 1333-1336,
1989.
Kennett, B. L. N. and Engdahl, E. R., Traveltimes for global
earthquake location and phase identification.
Geophys. J. Int.
105,429%465, 1991.
Lambeck, K., On the orbital evolution of the Martian satellites.
J. Geophys. Res. 84, 5651-5658, 1979.
Lard, J. C., Smith, M. R., War&e, H., Jagoutz, E., Dreibus, G.,
Palme, H., Spettel, B., Burgbele, A., Lipschutz, M. E. and
Verkouterue, R. M., Chemical systematics of the Shergotty
1268
meteorite and the composition
of its parent body (Mars).
Geochim. Cosmochim. Acta 50,909-926,
1986.
Lewis, J. S., Metal/silicate
fractionation
in the solar system.
Earth Planet. Sci. Lett. 15, 286, 1972.
Lognonne, P. and Mosser, B., Planetary
seismology.
Surv.
Geophys. 14,239-302, 1993.
Longhi, J., Knittle, E. and Holloway, J. R., The bulk composition, mineralogy and internal structure of Mars, in Mars
(edited by H. H. Kieffer, B. M. Jakosky, C. W. Snyder and
M. S. Matthews), pp. 184208. University of Arizona Press,
Tucson, Arizona, 1992.
Machetel, P., Convection
dans le manteau terrestre, etude en
geometric spherique, These Univ. Toulouse, France, p. 156,
1986.
McSween Jr, H. Y., SNC meteorites : clues to Martian petrologic
evolution? Rev. Geophys. Space Phys. 23,391416,
1985.
Mocquet, A., Grasset, O., Sotin, C. and Vacher, P., Modele
sismique du manteau de Mars, Coil. Nat. Plan. I.N.S.U.,
Toulouse, France, Vol. 2, S17-7, pp. 1-14, 1994.
Mocquet, A., Vacher, P. and Sotin, C., Iron content of the Martian mantle : the potential
of seismological
investigations,
Lunar Planet. Sci. Conf. XXVII, Houston, Texas, pp. 895896, 1996.
Ohtani, E. and Kamaya, N., The geochemical model of Mars :
an estimation from the high pressure experiments.
Geophys.
Res. Lett. 19, 2239-2242, 1992.
Okal, E. and Anderson, D. J., Theoretical models for Mars and
their seismic properties. Icarus 33,514528,
1978.
PoUack, J. B., Phobos and Deimos, in Planetary Satellites
(edited by J. A. Burns), pp. 319-345. University of Arizona
Press, Tucson, Arizona, 1977.
Reasenberg, R. D., The moment of inertia and isostasy of Mars.
J. Geophys. Res. 82,369-375,
1977.
Ridgen, S. M., Gwanmesia, G. D., Kackson, I. and Liebermann,
R. C., Progress in high-pressure
ultrasonic interferometry,
the pressure dependence of elasticity of Mg,Si04 polymorphs
and constraints on the composition
of the transition zone of
the Earth’s mantle, in High-pressure Research : Application
to Earth and Planetary Sciences (edited by Y. Syono and
M. H. Manghnami),
pp. 167-182. Terra Scient. Publ. Co.
(TERRAPUB),
Tokyo/Am.
Geophys. Union, Washington,
DC, 1992.
Ringwood, A. E., Composition and Petrology of the Earth’s
Mantle, p. 618. McGraw-Hill,
New York, 1975.
Ringwood, A. E., Origin of the Earth andMoon, p. 295. Springer,
New York, 1979.
Schubert, G. and Spohn, T., Thermal history of Mars and the
sulfur content of its core. J. Geophys. lies. 95, 14,095-14,104,
1990.
A. Mocquet
et ul.: Theoretical
seismic models of Mars
Schubert, G., Solomon, S. C., Turcotte, D. L., Drake, M. J. and
Sleep, N. H., Origin and thermal evolution of Mars, in Mars
(edited by H. H. Kieffer, B. M. Jakosky, C. W. Snyder and
M. S. Matthews), pp. 147183. University of Arizona Press,
Tucson, Arizona, 1992.
Sharpless, B. P., Secular accelerations
in the longitudes of the
satellites of Mars. Astvon. J. 51, 185-195, 1945.
Smith, C. J. and Born, G. H., Secular acceleration
of Phobos
and Q of Mars. Icarus 21, 51-53, 1976.
Sotin, C., Labrosse, S. and Mocquet, A., Nusselt-Rayleigh
relationship for a fluid heated from below and from within :
application
to the cooling rate of the Martian core, Lunar
Planet. Sci. Cant XXVII, Houston, Texas, pp. 1247-1248,
1996.
Stevenson, D. J., Spohn, T. and Schubert, G., Magnetism and
thermal evolution of the terrestrial planets. Icarus 54, 466
489, 1983.
Stnben, K. and Trottenberg, U., Multigrid
methods:
fundamental algorithms,
model problem analysis and applications. Lecture Notes Appl. Mech. 960, 1-176, 1982.
Toksoz, M. N., Tsni, A. T. and Johnston, D. J., Thermal evolution of terrestrial planets. Moon and Planets 18, 281-320,
1978.
Turcotte, D. L. and Schubert, G., Geodynamics. Wiley, New
York, 1982.
Vacher, P., Mocquet, A. and Sotin, C., Comparisons
between
tomographic
structures
and models of convection
in the
upper mantle. Geophys. J. Int. 124,45-56, 1996.
Ward, W. R., Burns, J. A. and Toon, 0. B., Past obliquity
oscillations of Mars : the role of the Tharsis uplift. J. Geophys.
Res. 84, 243-259, 1979.
Watt, J. P., Davies, G. F. and O’Connell, R. J., The elastic
properties of composite materials. Rev. Geophys. Space Phys.
14,541-563, 1976.
Weidner, D. J., A mineral physics test of a pyrolite mantle.
Geophys. Res. Lett. 12,417420,
1985.
Weidner, D. J. and Ito, E., Mineral physics constraints
on a
uniform mantle composition,
in High-pressure Research in
Mineral Physics (edited by M. H. Manghnami and Y. Syono),
pp. 439446.
Terra Scient. Publ. Co. (TERRAPUB),
Tokyo/Am. Geophys. Union, Washington,
DC, 1987.
Weinstein, S. A., The effects of a deep mantle endothermic phase
change on the structure of thermal convection
in silicate
planets. J. Geophys. Res. 100, 11,719-l 1,728, 1995.
Zebib, A., Schubert, G. and Strauss, J. M., Infinite Prandtl number convection in a spherical shell. J. Fluid Mech. 97, 257277, 1980.
Zhou, H., Breuer, D., Yuen, D. A. and Spohn, T., Phase transitions on the Martian mantle and the generation of megaplumes. Geophys. Res. Lett. 22,1945-1948,
1995.