O N T H E C O N S T I T U T I O N OF MARS (SECOND PAPER)
K. E. Bullen
(Received 1956 March
20)
Summary
A re-examination of the internal structure of Mars has been made in the
light of revised observational data on the radius R. Taking the reduced
value R=3330 km,and the mass of hlars as 6.442 x roZ6g , the hypotheses (a)
that the Earth and Mars have the same overall composition and (b) that Model
B gives a satisfactory representation of the Earth’s mantle, are examined. I t
is confirmed that the reduced value of the radius implies the presence of a
significant proportion of uncombined iron in the mantle of Mars. T h e
assumptions would also require the interior of Mars to deviate from a hydrostatic state, and if correct lend support to the view of Jeffreys that the material
of Mars has finite strength. It is still possible for Mars to have a small iron
(and/or nickel) core, the radius of which most probably does not exceed 700900 lim; the smaller the radius of the core, however, the less the deviation
from a hydrostatic state.
On hypothesis (a), the reduced radius further implies that the Earth
contains uncombined iron outside the inner core. A provisional view was
that this uncombined iron was located mainly in the outer core. Further
calculation, however, gives rise to the speculation that it is the mantle of the
Earth which contains this uncombined iron and not the outer core. A
coherent picture can be derived by assuming that Mars and the mantle of
the Earth both consist broadly of a mixture of uncombined iron with
silicates or FeO, MgO, SiOa phases. T h e Earth’s outer core on this view
would contain no uncombined iron, the latter having gravitated toward the
centre to form the inner core. In this way, the hypothesis ( a ) can be reconciled with the reduced Mars radius and with a hydrostatic state. T h e corresponding density distribution ( p g/cm3) in Mars is given approximately by
p=468
where s=r/R,
T
- 0.86x2,
being the distance from the centre.
The planet Mars has been much discussed lately on the relation of its internal
structure to that of the terrestrial planets. The purpose of the present paper is
partly to separate out well-established observations and inferences from less
certain results, and partly as a further step towards examining the question (I)
as to whether the most recent data permit a common overall primitive composition
for Mars and the Earth. A previous paper (2)bearing the same title as the present
one will be referred to as Paper I.
I. The mass of Mars.-In
Paper I, the mass M of Mars was taken as
(6.442 & 0.013) x Io26g. Urey (3) quotes Rabe’s investigation (4) as giving
6.390 x ~ o using
~ ~datag on the perturbations of Eros, but prefers the value
(6.454&0-013)x 1oZ6gderived from data of Woolard ( 5 ) and Burton (6) on
the motions of Phobos and Deimos. Throughout the present paper, no change
will be made from the Paper I value of M which lies inside the standard error
of Urey’s preferred value, and between the latter and Rabe’s value. If it should
272
K. E. Bullen
transpire that Rabe’s value is better,” the mass and mean density of Mars would
on this account need to be reduced by the order of I per cent.
2. Radius of Mars.-The
value quoted for the radius R of Mars in Ramsey’s
first paper (7) on planetary composition was (3396k 8) km. I n Paper I, the
~ assuming
value 3391 km was derived for R, taking M as 6.442 x 1 0g,~and
(a) that the Earth and Mars have the same overall composition ;
(b) that the Earth is represented by Model B (8) ;
(c) that the Earth’s outer core consists entirely of material which is a highpressure modification of the mantle material ;
( d ) that Mars has a core of separate chemical composition from the remainder,
in the same proportion by mass as the inner core of the Earth bears to the whole
Earth.
T h e value 3410km was derived for R when (d) was replaced by the assumption
(d’) that Mars has no core.
Urey (3) has preferred a reduced value, 3 1 1 0km, for the radius of the
solid surface of Mars, and D r G. P. Kuiper, in a personal communication, tells
me that he now regards the most probable value as 3330 km. Either of these
last values involves a reduction by at least eight times the quoted standard error
from the values given in the preceding paragraph. If sustained, the new values
will diminish the goodness of fit obtained in Paper I, and, as pointed out (I),
require some re-discussion of the argument in Paper I.
3. The oblateness of Mars.-A third observational quantity of relevance t o
the internal structure of Mars is its ellipticity of figure e. T h e value used in
Paper I for ecl was Struve’s value (9)of 192,deduced from observations of the
motion of Phobos. Using additional data, Woolard has given the value as 191.8
and has also reduced the assessed standard error from 2 per cent to about
per cent. This “ dynamical” value of e-l is preferred to values about twicc
as great which are inferred from movements of surface markings.
4. The strength of Mars.-In order to apply the value of e-l in examining
the density distribution of Mars, it is necessary to consider how far the figure
of Mars may deviate from that given by assuming hydrostatic conditions
throughout the interior. Jeffreys (10) has given an argument based on uncompensated inequalities in the Earth’s surface configuration to show that strength
in the interior of Mars could behtsponsible for a significant departure cc from
the value of the difference between the equatorial and polar radii as implied
on the hydrostatic theory. T h e departure tc would be consistent with a
difference
between the extreme equatorial radii of the Earth, where cc= Op
and 0 is the ratio (2.6)of gravity on the Earth to that on Mars. I n his 1937
paper, Jeffreys assumed /3 to be of the order of 0.7 km as given in earlier work
of Heiskanen. Later estimates of p range from the order of 0.3 km (Niskanen,
(11)) to the order of 0.1km (Jeffreys (12)),which correspond to cc=o.8, 0.25 km,
respectively. On the argument of Jeffreys, the value 191.8 of e-l could
therefore deviate by at least 3 units and possibly by 9 units or so from the
value, e,-l say, given by the hydrostatic theory.
* I am indebted to the referee for pointing out to me that determinations of a planet’s mass from the
a
motions of its satellites depend on the distances, which are systematically affected by any errors of
the calibration of the micrometer screw. For this reason, determinations of the masses of Jupiter and
Saturn from the satellites are inferior to the best perturbation determinations. This seems not to be
true for Mars. The probable error given in Kabe’s determination is 0.013X I O ? ~ ~ .
On the constitution of Mars
273
‘Taking the smaller of these values of the deviation has roughly the same
effect as the procedure in Paper I which allowed for a 2 per cent uncertainty
i n e-l. Jeffreys points out that his estimate of 0.1 km for p is fairly uncertain,
so that the larger value of the deviation cannot be ruled out. Hence eO-‘ could
be as high as 201 or so.
Let M , C and L2 be the mass, moment of inertia and angular velocity of Mars,
and G the constant of gravitation. Let Sp/p be the constant proportional density
increase arising from an assumed uniform contraction given by SR/R= y . Then
if k, z and q are defined by
k =R3W(GM}-l,
(1)
C = zMR’,
(2)
the Kadau-Darwin hydrostatic theory gives
0.16(1+7)=(1-- 1 . 5 2 ) ~ .
By the principle of angular momentum, neglecting extraneous influences,
CQ = constant.
By ( I ) , ( 2 ) and (s), assuming no gain or loss of mass,
Sk
280
- =3y+ T ,
k
and
I
(4)
(5)
,
1 he mass is preserved constant by taking Sp/p= - 3 y , and this entails that
SC/C=zy.
From (6) and (7), it then follows that S k / k = - y and Sz/n=o.
By (4),Sq/v is then zero, and, by ( 3 ) ,
Se,-I
e,-l
= = Y e
According to (8), any uniform symmetrical contraction of Mars would affect
e,,-l- e-l by as much as one unit only in the unlikely event of R being reduced
by at least 0.5 per cent. Thus if Mars were initially molten, and if Urey is correct
in assuming that there have been no subsequent upheavals on Mars of the type
that build mountains, the Radly-Darwin theory would seem to be closely
applicable to Mars.
Another part of Urey’s theory, however, envisages a cold origin for Mars,
in which case q1
- e-l could initially have well been significantly different
from zero. On the assumption of no mountain-building forces, this difference
would then be sensibly diminished only if a rise in temperature, during the
early compression stages or subsequently, was sufficient to reduce the strength
of Mars substantially below what Jeffreys considers to be the present value.
Taking all these considerations into account, it appears probable that the
value of e,-l for Mars is within 3 units of 192, but that a value somewhat greater
than zoo is not precluded.
5 . Modijication of calculatiori of Paper I to jit Kuiper’s radius.--In the
remaining calculations, the implications of taking
M = 6.442 x 10“ g,
R = 3330 km
(9)
274
K. E. Butlen
will be examined. The region outside any core which Mars may possess will
be called the mantle of Mars.
By Section 2, the values (9) require at least one of the assumptions ( a ) , (b),
(c) and ( d ) to be modified. It has already been pointed out (I) that a formally
sufficient modification is to replace ( c ) by the assumption
(cl) that the Earth‘s outer core consists of a mixture of the materials of the
mantle (in high-density form) and the inner core,
If the Earth’s inner core is supposed to consist of iron (and/or nickel), the
effect of (cl) is to admit the presence of some uncombined iron in the outer core.
On the hypothesis (a), this would raise the proportion of uncombined iron in
Mars above the value assumed in Paper I. A further effect (I) is that e,-l would
be raised above the Paper I value of 188 units.
As a step towards incorporating the modification of the last paragraph, a
formal density distribution for the mantle of Mars was first computed, assuming
( a ) , ( b ) and (9), and using the relations
dp = - Grnr2pdr ,
(10)
and
dm = 4rrr2pdr,
(11)
where r is the distance from the centre 0 of Mars, p is the pressure, and m is
the mass contained within the sphere of centre 0 and radius Y. The use of (10)
of course assumes compatibility with hydrostatic conditions. As in Paper I,
the crustal layers were assumed to consist of 30 km of material of density
2.8 g/cm3, and the mantles of Mars and the Earth were assumed to be identical
in composition.
The distribution obtained gave p = 4.24 g/cm3 and rn = 0.77 x 1oZ6g at
r = 1225 km. These values correspond to a mean density j of about 10g/cm3
for the material inside the sphere of radius 1225 km. This value of is close
to the value taken for the mean density of the core of Mars in Paper I. Hence
the assumptions here made would imply that Mars has a core of radius about
1225km and mass 0.77 x IoZ6g.
The corresponding value of e,-l was found to be 2 1 1 . In the light of the
discussion in Section 4, this value is, however, improbably great. In order to
reduce the value of e,-l to a suitable extent, it becomes necessary to assume that
the mean density of the mantle of Marawhen reduced to zero pressure is greater
than that for the Earth Model B. Th& an implication of Kuiper’s reduced
value of R is that the mantle of Mars contains relatively more uncombined iron
than the mantle of the Earth Model B.
The above argument adds some detail to the process of inference of Harrison
Brown (13) and Urey that the material of Mars may be less differentiated than
that of the Earth. It will be shown, however, that the argument does not require
Mars to be necessarily devoid of an iron core, and does not require the assumptions
( a ) and (6) to be necessarily modified. A very different interpretation will,
however, be suggested in Section 9.
6 . Modification to jit ellipticity requirements.-A revised calculation was now
carried out assuming the density at ali points of the mantle of Mars to be I + E
times that given on the calculation of Section 5 , E being taken constant. Various
values of the radius R ,of the Mars core were assumed, and the corresponding
values of E were deduced on the formal assumption that the core is of constant
On the constitution of Mars
275
density, 1og/cm3. Values of x and e-l were then derived using the formulae
of Section 4. T h e results are shown in Table I.
TABLE
I
-1
Rc
E
I225
0‘000
21 I
I200
0‘005
210
I I00
0’021
1000
0.034
0.045
0.053
206
203
900
800
700
600
500
400
0
0.059
0.064
0.068
0.070
0.073
Pg
20 I
I99
I97
196
195
I95
I94
T h e implications of Table I are as follows:
(i) I n the light of Section 4,the values shown for e,-l make it very improbable
that Mars has a core of radius greater than 900km.
(ii) T h e values of 6 for core radii between gookm and zero show that the
mantle of Mars, if composed of a mixture of uncombined iron and the material
of the Earth’s mantle, would contain between 2 and 34 per cent more uncombined
iron than the Earth’s mantle as given by Model B.
(iii) A striking result is that, for all values of R,, the value of e0-l exceeds the
observational value 192 or e-l, the least excess being twice the standard error
of the observational value. Thus, assuming ( a ) and (b), the reduction in R to
3330 km would imply that Mars almost certainly deviates from a hydrostatic
state to an extent that would accord with the latest estimate by Jeffreys of the
order of uncompensated inequalities in the figure of the Earth. T h e deviation
would incidentally be increased if Urey’s preferred value of 3310 km for R were
used. T h u s the reduced value of R, if correct, would, on this line of argument,
lend some support to the view Qf Jeffreys that the Earth and Mars have finite
strength.
(iv) For o<R,<700km, the range of variation of e,-l is only three units.
Hence the present calculations do not sharply discriminate between the possibilities that Mars has no core and has a core of radius up to 700 km, although the
larger R, is, the greater is the implied deviation from hydrostatic conditions.
7. The hypothesis of a common overall composition of Mars and the Earth.We now consider the hypothesis ( a ) in a little more detail. If Mars has no core,
the calculations of the last section imply that Mars contains 33 per cent of
uncombined iron. T h e hypothesis ( a ) would then entail that the Earth contains
0.21 x 1 0 ~ of
~ guncombined iron. Taking the mass of the Earth’s inner core
as 0-15x 1 0 ~ as
~ gin Paper I (this value includes half of the region F ) it would
in turn be entailed that there is 0.06 x 1 0g ~of ~uncombined iron in the Earth
outside the inner core. T h e proportion by mass of this uncombined iron, if
it were all located in the outer core, would be of the order of I in 30.
275
K. E. Bullen
If, on the other hand, Mars has a core of radius 700 km, then a calculation
based on the results (ii) of Section 6 would give the mass of uncombined iron
in Mars as 0.27 x Io2‘jg, or 4 per cent of the total mass. On the procedure of
the last paragraph, a further 0.03 x 102’g would then need to be added to the
mass of uncombined iron outside the Earth’s inner core, and the proportion of
uncombined iron in the outer core would need to be increased from about I in 30
to about I in 20.
T h e presence of iron in the Earth’s outer core to the extent of I part in 30
or I part in 20 does not constitute a very serious departure from the hypothesis (c).
T hus a reduction in the radius R of Mars to 3330 km, while it would remove the
remarkable precision of fit obtained in Paper I, would modify the hypotheses (c)
and (d) only to the extent of requiring the mantle of Mars and the outer core of
the Earth to contain small proportions of uncombined iron. A further implication is that, whereas the calculations of Paper I had slightly favoured thc
presence of a core in Mars, the reduced radius slightly favours the absence of
a core in Mars, unless the deviation from hydrostatic cofiditions is indced
appreciable.
8. The relevance of the Earth Model B.-It needs to be emphasized that
the calculations in the preceding sections have continued to assume that Model I3
gives a fair representation of the Earth’s density distribution in the mantle.
T h e agreement originally found in Paper I arose in fact by using the Earth
Model B in preference to Model A ; Ramsey (7) had found some difficulty in
reconciling the hypothesis ( a ) with the use of Model A.
T h e revised calculations therefore focus attention on the intimate connection
between :
the hypothesis that the Earth and Mars have a common overall composition ;
the degree of reliability of Model B in representing the Earth’s density variation
(and therefore, by implication, of the writer’s pressure-compressibility
hypothesis) ;
the radius of Mars ;
the extent of deviation from hydrostatic conditions in Mars.
New information on any of these items will have repercussions on thc others.
For example, should it become well estTblished that the radius of the solid surface
of Mars does in fact not exceed 3330 krn and that the Radau-Darwin theory is
adequate for Mars, then it would follow either that the overall compositions of
Mars and the Earth are different or that the Earth Model B needs amending.
As yet, there is of course no crucial evidence on these issues, and there remains
a fair probability that the hypothesis of a common primitive overall composition
of the terrestrial planets is correct. I n fact, the relatively small quantity of iron
required in the Earth’s outer core on the revised calculations indicates that the
hypotheses ( a ) , (6) and (c) must still be regarded as being in fairly good accord
with one another.
At the same rime, it is well to appreciate that it is possible to construct a
density distribution in Mars which is fully compatible both with a hydrostatic
state and with a radius of 3330 km.
T o show this, assume a density law of the form
p.=
3a - 5bx2,
On the constitution of Mars
277
where x = r / R , so that
dP
dx
- = - iobx,
and
m = 4nR3(ax3- bxs) .
'I'he hydrostatic assumption then gives
- 5bx2).
= - 4nGR2(ax- 6x3)(3a
Assuming a uniform chemical composition, the incompressibility R is givcn by
dp 47rGR2( a - bx2)(3a- 5 b ~ ~ ) ~
k=p=
Iob
('3)
dP
'raking ecl= 191.8 and using the Radau-Darwin theory gives x=o.3906.
Applying this value of x, the assumptions (9) and (12) then yield the density
distribution
p = 4.68 - 0 . 8 6 ~ ~ .
(14)
I n the distribution given by (14),the surface and central densities are 3-82 and
4.68g/cm3. T h e value of h derived from (14) ranges from 1 . 1 1 x 1o12 at the
surface to 1.87 x 1o12dyn/cm3 at the centre. T h e central value is not greatly
different from the value 1.7 x 1o12dyn/cm2 indicated for h inside the Earth at
m ~ is about the pressure at the centre of
a pressure of 0.35 x ~ o l ~ d y n / cwhich
Mars, assuming a hydrostatic state. Th u s the model given by (14) is fully
compatible with the observed mass and ellipticity of Mars, with a radius of
3330 km, and with hydrostatic conditions.
9. A new speculation on the distribution of uncombined iron in the terrestrial
planets.-The law (14) may be compared with the density distribution for Mars
given on the calculations of Sections 5 and 6 for the case of no core. With the
model of Section 6 for the case of no core, the density is found to range from
3.55 g/cm3 at the surface (the crustal layers being here neglected) to 4.66 g/cm3
at the centre of Mars. P'bere is, however, a 15 per cent jump in density near
a depth of zookm, corresponding to the density discontinuity formally set at
80 km depth in the Earth Model B. This distribution for Mars and the law (14)
are thus in very close accord except in the outermost zoo km.
This comparison leads to the interesting speculation that such uncombined
iron as may exist outside the Earth's inner core may occur in the mantle rather
than in the outer core. T h e speculation has the intriguing possibility of reconciling Urey's idea that the Earth's mantle contains uncombined iron with the
notion that the outer core consists of a high density modification of silicates or of
phases such as FeO, MgO and SiO, which Birch (14)has suggested for the
region D.
On this view, the terrestrial planets would be composed of a mixture of
uncombined iron and phases such as FeO, MgO and SiO, (the proportions
might change with depth) down to the depth at which a pressure of the order
1.35 x Io12dyn/cm2is reached. At this pressure there would be a polymorphic
278
On the constitution of Mars
transition accompanied by a change from the solid to the fluid state of all
ingredients except uncombined iron (and nickel), the uncombined metals sinking
towards the centre to form an inner core. T h e picture would, moreover, meet
such objections as have been raised on the question of the miscibility of iron
with other mantle ingredients in the fluid outer core. Mars would, on the new
picture, have no core since the pressure at all depths is less than 1.35 x 1oI2dyn/cm2.
Venus would have small outer and inner cores. Following a previous argument
(IS), Mercury would have a metal core, but probably no fluid outer core, a fair
proportion of the mantle having been volatilized away. T h e Moon would differ
in composition from the Earth, Mars and Venus ; but this appears to be the
case on most other theories.
It may be remarked that the miscibility argument has been applied by some
authors (though there are differences of opinion on questions of miscibility at
pressures of the order of a million atmospheres) to support an iron composition
for the outer core. T h e argument is that uncombined iron would not mix with
the other proposed ingredients in a fluid outer core, which must therefore consist
very largely of iron. T h e new picture here suggested would be compatible with
accepting the premise in this argument, but with drawing the opposite conclusion
that the outer core of the Earth is the one place where there is practically no
uncombined iron.
University of Sydney:
1956March 13.
References
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( 2 ) K. E. Bullen, M.N., 109,6S8, 1949.
(3) H.C.Urey, The Plmzets, Their Ori$rz a d Development, Yale University Press, 1952.
(4) E.Rabc, Astroiz, J., 55, 112, 1950.
(5) E. W.[Yoolard, Astr~n.J.,51, 33, 1944.
(6)H.E. Rurton, Astroii. T., 39, 154,1929.
(7)W.H.Ramsey, M.N. 108,406,1948.
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