Fluid Core Size of Mars from Detection of the Solar Tide
Abstract
The solar tidal deformation of Mars, measured by itsk 2 potential Love number, has been obtained from an analysis of Mars Global Surveyor radio tracking. The observedk 2 of 0.153 ± 0.017 is large enough to rule out a solid iron core and so indicates that at least the outer part of the core is liquid. The inferred core radius is between 1520 and 1840 kilometers and is independent of many interior properties, although partial melt of the mantle is one factor that could reduce core size. Ice-cap mass changes can be deduced from the seasonal variations in air pressure and the odd gravity harmonicJ 3, given knowledge of cap mass distribution with latitude. The south cap seasonal mass change is about 30 to 40% larger than that of the north cap.
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REFERENCES AND NOTES
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They (5) find maximum north and south ice-cap volumes are both ∼3 × 1015 m3 and cap density is ∼910 kg/m3. They also find δJ2(1/2 year) ≈ (2.2 ± 0.8)cos(2ℓ′ – 20°).
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J. Wahr, in A Handbook of Physical Constants: Global Earth Physics, T. J. Ahrens, Ed. [American Geophysical Union (AGU), Washington, DC, 1995], vol. 1, pp. 40–46.
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The MGS, Pathfinder, and Viking Lander data are processed with the same least-squares estimation technique and observational models as described in
Yuan D. N., Sjogren W. L., Konopliv A. S., Kucinskas A. B., J. Geophys. Res. 106, 23377 (2001);
. The MGS data is processed in segments of 4- to 6-day intervals; for each segment, the epoch spacecraft position, atmospheric drag, spacecraft maneuvers, and observational calibrations are estimated that pertain specifically to that data segment. All the MGS segments are then combined together with one Pathfinder data arc and one Viking Lander data arc. For the landers, the arc-specific parameters include lander position and range bias estimation. In addition to the segment or arc-specific parameters, the solution for parameters common to most arcs includes the Love number, rotation parame ters, Mars satellite parameters, the mean spherical harmonic gravity field to degree 85, and seasonal J2 and J3 as
9
Starting from an orientation model of Mars similar to that of Folkner et al. (2), the rotational parameters we estimate are the epoch obliquity (ε) and longitude of the Mars pole (ψ), the precession rate of the pole (dψ/dt), the obliquity rate (dε/dt), the rotation rate (dφ/dt), and the seasonal variations in rotation angle as a periodic series
where ℓ′ is the Mars mean anomaly. The rigid-body nutation model is fixed to that of (10).
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The drag model uses six flat plates to represent the spacecraft bus, antenna, and solar arrays with orientation provided by spacecraft telemetry. The atmospheric density is given by the Mars GRAM 3.4 model in C. J. Justus, D. L. Johnson, B. F. James, NASA Tech. Memo. No. 108513 (1996). We also tested solving for a drag coefficient only once per day, and this did not change the results.
13
The orbit inclination is chosen such that dΩ/dt = –3/2cosIJ2(R/a′)2n = n′. Also, the MGS orbit elements are: a = 3796 km, e = 0.0084, period = 1.96 hours, and I = 92.9°.
14
The nutations of Mars pole (δε and δψ) result in the following changes in orbit inclination relative to a space-fixed reference frame (15, 22):
The principal annual term for rigid-body response is δε = 0.0 and εδψ = –0.259"sinℓ′. This reduces the secular rate by about 10%. The fluid core can further reduce rate by up to 1%.
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A phase lag (= 1/Q) can be introduced in the angle arguments in Eq. 2 in order to account for solid friction. For example 2(Ω – L′) is replaced with 2(Ω – L′) – 1/Q. This effect reduces the observed signature by about 1%.
17
The inclination drift does cause a secular acceleration in the node (dΩδI/dt ≅ –n′tanIδI) that has about the same amplitude as the secular change in inclination after ∼20 days. Thus, both node and inclination drift contribute to the global Love number solution. However, space craft maneuvers aimed at desaturation of the momentum wheels and maintaining orbit geometry may limit the sensitivity to long-term orbit changes.
18
dΩk22/dt ≅ k22n′TcosI[cos4(ε/2)cos2(Ω – L′) + ½sin2(ε/2)cos2(Ω – ψ)], dIk21/dt ≅ k21n′TcosIsinε[cosεsin(Ω – ψ) + ½cos2(ε/2)sin(Ω + ψ – 2L′)]
19
Let Ω = L′ + θ. It happens that L′ – ψ ≈ 260° during face-on (August 1999 and June 2001). We find [cosεcos(Ω – ψ) + 1/2cos2(ε/2)cos(Ω + ψ – L′)] = 0 for sinθ ≈ 0.5, corresponding to 2:00 p.m. local Mars time.
20
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21
The scalar factors fn(x) are x = sin2I, f4 = 5/8(R/a)2(7x – 4), and f6 = –5/8(R/a)4[7 – 54x + (62 + 5/8)x2]. The effect of odd Jn on spacecraft orbit e and argument of pericenter ω is obtained from the equation dp/dt + iwop = 3/2nsinI(R/a)3f3δJodd
The variable p = eexp(–iω) while i =
, wo ≅ 3n(R/a)2f3, and the scalars are f3 = (1 – 5/4x), f5 = –5/16(R/a)2(8 – 28x + 21x2), and f7 = 35/64(R/a)4[8 – 54x + 99x2 – (53 + 5/8)x3]. Also, the mean orbit eccentricity e ≈ (R/a)(J3/J2) is due primarily to this forcing.
22
Yoder C. F., et al., J. Geophys. Res. 102, 4065 (1997).
23
Solutions for k21 range from –0.1 to 0.1.
24
The latest estimate for (unnormalized)J2 is 1.985683(3) × 10−3 (Mars Re = 3394.2 km), and the precession rate is [see (2, 10, 22)] dψ/dt = [–7606(0.365MRe2/C) × 10−3 arc sec/year + dψ/dtp + dψ/dtg], where the contribution from Jupiter is dψ/dtp = –0.2 × 10−3 arc sec/year and geodetic precession is dψ/dtg = +6.7 × 10−3 arc sec/year.
25
In the estimation process, the Doppler data are treated as uncorrelated measurements, i.e., white noise. Solar plasma, troposphere, and ionosphere can cause correlations in the measurements that are impractical to account for in a least-squares estimation process.
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27
The semidiurnal surface pressure field due to solar heating can be approximated by δP22|sinϑ|3cos(2α + 41°), where δP22 is the equatorial amplitude (Pathfinder pressure data indicate δP22 ≅ 0.011P0). The global mean pressure P0 ≅ 5.6 mbar, and the hour angle α = 0 at midnight (see SOM Text for more details). However, the estimate from one GCM (29) has the same phase, but ½ the amplitude. The atmospheric k2a ≅ 3.08(1 + k′2)R2/gMsun(R/a′)3δP22sin2(Ω – L′ + 41°)/sin2(Ω – L′). Surface gravity g = 0.372 m/s2 and the load k′2 ≅ –0.8k2. Thus, k2a ≈ 0.008.
28
Groten E., et al., Astron. J. 111, 1388 (1996).
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Sohl F., Spohn T., J. Geophys. Res. 102, 1613 (1997).
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V. N. Zharkov, Gudkova, Phys. Earth Planet. Inter. 117, 407 (2000).
31
C. F. Yoder, in A Handbook of Physical Constants: Global Earth Physics (AGU, Washington, DC, 1995), vol. 1, pp. 1–31.
32
Ray R. D., et al., Nature 381, 585 (1996).
33
Compressed mean core density at core temperature ranges from 5.2 g/cm3 [high pressure Fe(0.9)S (and χc = 1)] to 7.5 g/cm3 (χc = 0), and this poses limits on core size represented by the upper and lower bounds on core radius shown in model calculations in Fig. 2.
34
Dickey J. O., et al., Science 265, 482 (1994).
35
Konopliv A. S., et al., Icarus 150, 1 (2001).
36
37
The mass change in the southern cap ranges from 5.7 × 1015 (model C) to 11.3 × 1015 kg (model A). The corresponding density (5) is 1900 to 3800 kg/m3. This suggests that at least the south cap volume is larger than reported.
38
The ratio J5/J3 is 0.224 (model A), 0.436 (model B), and 0.561 (model C). The ratio f5/f3 is 0.97 for MGS (21) (table S1).
39
Smith D. E., et al., J. Geophys. Res. 104, 1885 (1999).
40
Van den Acker E., et al., J. Geophys. Res. (Planets) 105, 24563 (2000).
41
We thank J. G. Williams, C. K. Shum, and V. Dehant for thoughtful reviews and J. T. Schofield for advice and Pathfinder pressure data. The research described in this paper was carried out by the Jet Propulsion Laboratory, California Institute of Technology, under a contract with NASA.
SOM Text
Figs. S1 to S6
Tables S1 to S3
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Volume 300 | Issue 5617
11 April 2003
11 April 2003
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Received: 22 October 2002
Accepted: 19 February 2003
Published in print: 11 April 2003
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