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The effects of seasonal mass redistribution and interior structure on Length-of-Day variations of Mars
Advances in Space Research 38 (2006) 739–744 www.elsevier.com/locate/asr
The effects of seasonal mass redistribution and interior structure on Length-of-Day variations of Mars ¨ . Karatekin *, T. Van Hoolst, J. Tastet, O. de Viron, V. Dehant O Royal Observatory of Belgium, 3, Av. Circulaire, 1180 Brussels, Belgium Received 1 November 2004; received in revised form 29 March 2005; accepted 29 March 2005
Abstract The effect of MarsÕ interior structure, more precisely the influences of the size and the state of the core, on Length-of-Day variations (DLOD) is studied. We calculated the load Love numbers and the moments of inertia of the mantle and of the core for a set of models of MarsÕ interior. It is shown that the current level of precision of LOD observations does not allow to deduce details of the Martian interior. Future measurements of DLOD must have an accuracy better than 2% to determine whether or not the core is liquid, and even better to constrain the core size. The effect of seasonal mass redistribution on DLOD is also investigated. Martian DLOD is a good indicator of global scale seasonal CO2 cycle. Planetary rotation data from the tracking of Viking and Pathfinder landers are compared with the recent seasonal mass deposition information from Mars Global Surveyor and Mars Odyssey missions. The results reveal the current difficulties in the accurate modelling of the Martian seasonal CO2 cycle. Ó 2005 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Mars; Length-of-Day; Rotation; Geodesy; Planetary interior
1. Introduction The interior of Mars is modeled mainly from the extrapolation of the EarthÕs internal structure combined with only a few observational constraints such as the mass, the radius and the polar moment of inertia learned from the precession rate (Sohl and Spohn, 1997). The core size and whether or not the core is liquid are among the poorly constraint parameters. Both the liquid and the solid core states can be consistent with the Martian dipole magnetic field measurements. Recently, the second degree tidal love number, k2 was determined from the tracking of Mars Global Surveyor (MGS) which suggests that the core is at least partially fluid (Yoder et al., 2003). If measured precisely enough, determination of MarsÕ orientation parameters could remove most of the above uncertainties associated with the interior of Mars. *
Corresponding author. Tel.: +3223736732; fax: +32494034753. ¨ . Karatekin). E-mail address: [email protected] (O
Normal rotational modes such as the Free Core Nutation (FCN) depend on the rheology and the interior structure of the planet through the resonance effects with the normal modes of the planet (Van Hoolst et al., 2000; Dehant et al., 2003). A dedicated geodesy mission involving a network of landers could measure accurately those normal modes from the nutations (Dehant et al., 2004). However, presently the realization of such a mission is uncertain. On the other hand, some of the orientation parameters of Mars, such as DLOD, has been obtained directly from tracking data of the landers on the surface of Mars. Folkner et al. (1997) and Yoder and Standish (1997) estimated LOD variations through the analysis of tracking data of Viking and Pathfinder landers. In the near future, with several planned missions involving landers, the DLOD estimates can easily be improved, provided that the life duration of the landers are long enough to follow the seasonal variations. In the present study, we investigate the use of LOD variations to constrain the interior structure, more pre-
0273-1177/$30 Ó 2005 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2005.03.117
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¨ . Karatekin et al. / Advances in Space Research 38 (2006) 739–744 O
cisely the size and the state of the core. In addition, we study the effect of surface loads on DLOD. The LOD variations are associated with the seasonal redistribution of masses on planetary scale. MarsÕ surface mass distribution varies at seasonal time scales due to condensation, sublimation and precipitation of CO2, the primary constituent of the atmosphere. As much as a third of the atmosphere has been estimated to take place in this seasonal CO2 cycle (Smith et al., 2001). The sizes and masses of both polar caps as well as the atmospheric pressure change, resulting in seasonal variations of the Martian inertia tensor. In the absence of external torques, the rotation rate (DLOD) is principally affected by the changes in the inertia tensor. The LOD variations on Mars have been estimated by various authors. On the basis of surface pressure data from Viking landers, Cazenave and Balmino (1981) and Chao and Rubincam (1990) estimated the seasonal variations of LOD. With the help of Global Circulation Models (GCM) of Martian atmosphere, Defraigne et al. (2000), Van den Acker et al. (2002) and Sanchez et al. (2003) improved the previous DLOD estimates by including the atmospheric wind influence and a more realistic surface topography. In the present study, we compare the direct measurements of DLOD from the Viking and Pathfinder landers (Folkner et al., 1997) with the LOD variations calculated from the seasonal mass redistribution given by the MOLA laser altimeter onboard the MGS spacecraft (Smith et al., 2001), the High Energy Neutron Detector (HEND) onboard the Mars Odyssey spacecraft (Litvak et al., 2004), and the Doppler tracking of the MGS spacecraft (Smith et al., 2001; Yoder et al., 2003). Such a comparison is useful to reveal the current understanding of the Martian seasonal CO2 cycle.
2. Rotation rate and interior structure 2.1. Theory The rotation of a deformable planet about its spin axis is described by Liouville equations (Munk and MacDonald, 1960). If the external torques are neglected, the instantaneous rotation rate variations of a planet are due to the angular momentum changes between its fluid layers and solid body. To formulate the rotation rate, one customarily introduces the atmospheric angular momentum excitation function, v, consisting of the matter and the motion terms. The matter term is due to the redistribution of the atmospheric mass while the motion term depends on the variations of the atmospheric zonal winds (see Barnes, 1983). Z R4 vmatter ðtÞ ¼ ðq ðh; k; tÞ þ qatm ðh; k; tÞÞsin3 h dh dk; C S ice ð1Þ
v
motion
R3 ðtÞ ¼ gCX
Z
usin2 h dh dk dp;
ð2Þ
V
where t is the time, R the equatorial radius, g the surface gravity, X the mean rotation rate, h the colatitude, k the longitude, q the load expressed as mass per surface area, and u the zonal wind speed. The relative variation in the rotation speed is given by m¼
DX C DLOD ¼ . ½ð1 þ k 02 Þvmatter þ vmotion ¼ X Cm LOD ð3Þ
In Eq. (3), C and Cm represent polar principal moments of inertia of the whole planet and mantle, respectively. k 02 is the load Love number of degree 2 and it represents the response of a deformable planet to a unit surface load, through the combined effects of pressure and attraction (Munk and MacDonald, 1960; Defraigne et al., 2000). DLOD is linked to the planetÕs interior through these two parameters, C/Cm and k 02 . Their values for the planet Mars are discussed in the next section. 2.2. Effect of Martian core on DLOD We calculated the Love number k 02 and the moment of inertia ratio, C/Cm, for a set of models of the interior of Mars. We assumed the same density, rigidity and compressibility profiles in the mantle and crust as in the models by Sohl and Spohn (1997) which satisfies the observed mass, radius and moment of inertia. The core is assumed to be composed of a Fe–S alloy. From the content of S, we determine its radius. Neither the exact composition nor its state are known although the recent measurements of tidal Love numbers suggest an at least partially liquid core, and a core radius in the range of 1520–1840 km (Yoder et al., 2003). In the present study, we considered a slightly larger range of possible core radius (1268–1868 km) and both solid and liquid cores. The models with larger or smaller core radius have been obtained by changing slightly the core density in order to keep the total mass unchanged (Van Hoolst et al., 2003). The results are presented in Figs. 1 and 2. A larger absolute value of k 02 indicates a larger deformation of Mars. According to Fig. 1, the response of the planet to surface loads is stiffer with a smaller liquid core. The absolute value of k 02 increases with increasing core radius. It shows variations lower than 50% for the range of core radii considered. If the core is solid, the absolute value of k 02 is smaller (i.e., less easily deformable) and its variation as a function of core radius is less prominent. The ratio of polar moment of inertia, C/Cm, on the other hand, increases with the radius of the liquid core (Fig. 2). Its amplitude varies less than 10% between
¨ . Karatekin et al. / Advances in Space Research 38 (2006) 739–744 O
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0.98
0.04
liquid Core Solid Core
0.97
0.06
0.96
C/Cm (1+k2’)
k 2’
0.08
0.95 0.94
–0.1
0.93
–0.12 0.92
–0.14
–0.16 1200
Liquid core Solid core
1300
1400
0.91
1500
1600
1700
1800
1900
0.9 1200
1300
1400
Fig. 1. Variation of the second degree load love number k 02 as a function of core radius.
1.1 1.09 1.08
C/C
m
1.07 1.06 1.05 1.04 1.03 1.02 1.01 1200
1300
1400
1500
1600
1500
1600
1700
1800
1900
Core radius
Core radius
1700
1800
1900
Core radius
Fig. 2. Variation of the polar moment of inertia ratio (C/Cm) as a function of liquid core radius.
the limits of the assumed core radius. For a solid core, the ratio is equal to one and the interior structure affects DLOD only through k 02 . The combined effect, C=C m ð1 þ k 02 Þ, determines DLOD in response to a surface load excitation. It is presented in Fig. 3 for both solid and liquid cores. For a liquid core, C=C m ð1 þ k 02 Þ results from two opposing effects which essentially cancel each other since both C/Cm and 1 þ k 02 show opposite relative variations with increasing core radius. Accordingly, the resulting C=C m ð1 þ k 02 Þ does not vary significantly (<0.5%). Its variation is more important for a solid core for which C/Cm is unity and the variation is only a function of k 02 . For a small core radius the response of the planet with solid and liquid cores are similar. The difference grows with increasing core radius but remains always less than 2% within the range of core sizes considered. This implies that the uncertainties associated with the interior struc-
Fig. 3. Variation of C=C m ð1 þ k 02 Þ as a function of core radius and state.
ture have only a very small effect on DLOD. Measurements of DLOD must have an accuracy better than 2% to determine whether or not the core is liquid, and even better (<0.5%) to constrain the core size. In the present study, we assumed that the excitation of LOD due to zonal wind variations is perfectly known. Seasonal zonal winds which are the primary cause of DLOD on Earth, have been thought to be much less important on Mars (Folkner et al., 1997). Because of the lack of direct measurements, their only estimates are from Martian GCMs, which suggest, nevertheless, annual amplitudes as large as one third of the total DLOD (Van den Acker et al., 2002; Sanchez et al., 2003). This makes necessary the evaluation of zonal wind effects and more cumbersome the determination of the interior structure from DLOD measurements.
3. Effect of seasonal CO2 cycle The effect of surface loads on LOD variations are studied from recent DLOD estimates based on independent seasonal mass redistribution observations as well as those from direct measurements. Variations in rotation about the spin axis due to surface mass redistribution are proportional to changes in the gravitational zonal harmonic of degree 2, J2 (Chao and Rubincam, 1990): DLOD 2MR2 ¼ DJ 2 . LOD 3C
ð4Þ
This only applies to the matter term in Eq. (3). The relationship provides a means to infer rotation changes from the long wavelength gravitational variations. Temporal variations of the low degree gravity field coefficients of Mars were recently determined from the MGS tracking data by Yoder et al. (2003) and Smith
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et al. (2001). The corresponding LOD variations are obtained from Eq. (4) with LOD = 88,775 s and C/MR2 = 0.365 (Yoder et al., 2003). These DLOD results are given in Fig. 4 in comparison with Lander tracking analysis from Folkner et al. (1997) who estimated seasonal rotation variations of Mars from the 3 months of Pathfinder tracking data in combination with much longer tracking data of Viking landers. Although the Pathfinder data covered a relatively short period of time, thanks to its higher signal/noise ratio, the previous DLOD estimates based only on Viking landers (Yoder and Standish, 1997) were improved. Nevertheless the associated errors remain high (about 40% for annual amplitudes and 50% for the semi-annual amplitudes). The errors given for the gravity solution of J2 are of about the same order of magnitude (Yoder et al., 2003; Smith et al., 2001). If one of the caps would increase symmetrically with respect to the decrease of the other cap (i.e., if North and South polar caps would have identical seasonal deposits and be 180° out of phase), there would not be any change in the axial moment of inertia or in rotation rate. But, due to North-South asymmetry, the contributions from the two hemispheres do not cancel and the resulting DLOD has annual and semi-annual variability (see Fig. 4). DLOD from the lander tracking data have a smaller semi-annual variation. The peak-to-peak amplitude, on the other hand, is closer to that from DJ2 of Smith et al. (2001). Part of the approximately 20% difference in peak-to-peak DLOD between the lander tracking estimates and those from DJ2 of Yoder et al. (2003) could be due to wind contributions that are not included in solutions from DJ2 observation. However, it is not possible to draw definite conclusions as the experimental uncertainties are larger than the expected wind contributions. 0.6 0.4
Δ LOD (ms)
0.2
0 –0.2
The variations of J2 and DLOD can also be obtained from the estimates of seasonal deposits given by the MOLA laser altimeter onboard the MGS and by the HEND onboard the Mars Odyssey. Smith et al. (2001) reported temporal changes in the elevation of Martian polar caps and the planetÕs global mass redistribution due to the seasonal CO2 cycle, using the data from MOLA. They showed elevation changes up to 1.5 ± 0.25 m in the North cap and 0.9 ± 0.30 m in the South cap. In the present study, we use the temporal elevation variations from MOLA, with a density of 1000 kg m3, as suggested by Smith et al. (2001), to approximately match the amplitude of the observed time-variable gravity signal. The seasonal changes in CO2 height cause variations in neutron flux measured by the HEND onboard the Mars Odyssey. These variations have been used to estimate the surface density of seasonal deposits at different latitudes by using a model-dependent technique (Litvak et al., 2004). The CO2 surface densities as a function of time were given in kg/m2 over the latitude bands of 60–70°, 70–80° and 80–90° for both hemispheres as a function of time. By multiplying these data with the corresponding surface areas, the mass variations over polar caps are obtained. From temporal variations of surface densities, it is straightforward to estimate DJ2 (Chao and Rubincam, 1990) and hence DLOD through Eq. (4). DLOD estimates from observations of seasonal deposits as well as from lander tracking data are plotted in Fig. 5. Solutions from MOLA cover more than one Martian year and the corresponding solutions overlap for 120° < Ls < 150°. DLOD depends on the sum of the loads in the two hemispheres. The minima for the three sets of data occur around Ls = 0° and Ls = 180°, which correspond to the end of the winter season in the North and South hemispheres, respectively. Large quantities of CO2 are then condensed on one of the two polar caps close to the rotation axis, leading to faster rotation of the planet. In the vicinity of Ls = 180°, the solution based on MOLA has an amplitude about four times smaller than the solution from HEND data. Its maximum around Ls = 90° is in advance by 30°, and around Ls = 210° delayed by the same amount, with respect to DLOD solutions based on HEND data. A comparison with Fig. 4, reveals a similar behavior for the DLOD from J2 variations and those from HEND data.
–0.4 Yoder et al. (2003) Smith et al. (2001) Folkner et al. (1997)
–0.6 –0.8
0
30
60
90 120 150 180 210 240 270 300 330 360
Ls (deg)
Fig. 4. DLOD deduced from J2 solutions of the MGS spacecraft tracking (Yoder et al., 2003; Smith et al., 2001) and the direct measurements of DLOD from the tracking of the Viking and the Pathfinder landers (Folkner et al., 1997).
4. Discussion and conclusions We investigated the effect of the interior structure of Mars, more precisely the size and the state of the core, on LOD variations. We calculated the load Love number k 02 and the moment of inertia ratio C/Cm for a set of models of the interior of Mars. A range of possible core radius and both solid and liquid cores are consid-
¨ . Karatekin et al. / Advances in Space Research 38 (2006) 739–744 O 0.6 0.4
LOD (ms)
0.2
0 –0.2 –0.4 HEND MOLA Folkner et al. (1997)
–0.6 –0.8 0
30
60
90 120 150 180 210 240 270 300 330 360
Ls (deg) Fig. 5. DLOD deduced from the temporal variations of seasonal mass deposits computed from HEND data (Litvak et al., 2004) and MOLA data (Smith et al., 2001) onboard of Mars Odyssey and MGS spacecrafts respectively. The direct measurements of DLOD from the tracking of the Viking and the Pathfinder landers (Folkner et al., 1997) are also given in the figure.
ered. For a small core radius the response of the planet with solid and liquid cores are similar. The difference becomes more prominent with increasing core radius but remains always less than 2% within the range of core sizes considered. This implies that the uncertainties associated with the interior structure have only a very small effect on DLOD. Measurements of DLOD must have an accuracy better than 2% to determine whether or not the core is liquid, and even better (<0.5%) to constrain the cores size provided that the seasonal zonal wind variations are accurately known. The current LOD data has large uncertainties compared to the required ones, but it can be improved with future Martian missions involving landers. For example, the expected accuracy of a geodesy experiment with a network of landers, such as NEIGE (Dehant et al., 2004), is less than 1%. We compared the direct measurements of DLOD from the Viking and Pathfinder landers (Folkner et al., 1997) with the LOD variations calculated from the seasonal mass redistribution observations by the MOLA laser altimeter onboard the MGS spacecraft (Smith et al., 2001), the HEND onboard the Mars Odyssey spacecraft (Litvak et al., 2004), and the Doppler tracking of the MGS spacecraft (Smith et al., 2001; Yoder et al., 2003). The results show differences both in amplitude and phases due to measurement and modelling errors. DLOD calculated from the seasonal mass redistributions does not include the wind contribution, which can be as large as one third of the total signal. The differences in MGS tracking solutions (Fig. 4) result mainly from the errors in the determination of time variable J2. The quality of time variable gravity from radio-science data depends on the achieved orbit preci-
743
sion. The perturbations on the orbit due to non-gravitational forces, especially the atmospheric drag, deteriorates mainly the determination of nodal rate, hence the even zonal harmonic coefficients (Yoder et al., 2003). As shown in Fig. 6 in Smith et al. (2001) and in Fig. 3 in Yoder et al. (2003), the J2 signal is weak and it is hard to reconcile it with the expected semi-annual variation. Seasonal CO2 thickness variations measured by HEND and MOLA show significant differences (Fig. 5). HEND has a much larger footprint covering a radius of several hundred kilometers. Measurements with such a low surface resolution suffer from spatial and temporal artifacts (Prettyman et al., 2005). In addition, the interpretation of CO2 thickness variations from neutron flux measurements demands a model for the source of neutrons which depends on the water contents of the regolith and of the martian atmosphere. An atmospheric circulation model had to be used for the atmospheric corrections. Two models of layered regolith with different water content are assumed for the south and north poles. Accordingly, HEND results would change with the assumed regolith and atmospheric models. The MOLA altimeter, on the other hand, has a much smaller footprint (approximately 168 m). The vertical resolution is reported to be 37.5 cm from laser point to laser point (Aharonson et al., 2004). The uncertainties in the positioning of the spacecraft are not only restricted to the vertical, but significant discrepancies occur also in the along-track and cross-track directions. On a rough terrain with steep slopes these effects on the altimetry measurements are non-negligible (Rowlands et al., 1999). The resulting accuracy of MOLA measurements are about 10 m. When the corrections from statistical cross-over analysis are applied, the observed rootmean-square values are reduced significantly (1.8 m) but it remains still high with respect to maximum seasonal CO2 height variations of the order of 1.5 m (Aharonson et al., 2004). The density of the Martian seasonal deposits are known only approximately. The recent effort of Aharonson et al. (2004) by combining MOLA and Gamma Ray Spectroscopy observations yield unexpectedly low densities, about 0.5 g/cm3. The measurements of seasonal surface mass distribution by HEND have required the priori knowledge of not only the Martian atmospheric dynamics but also the modelling of the subsurface regolith. MOLA altimetry on the other hand, had relatively low vertical resolution and, similar to the MGS tracking data, have suffered from orbit determination errors. Accordingly the overall agreement between the different DLOD solutions is not excellent (Figs. 4 and 5). But it is reassuring that the HEND and the tracking data show relatively better agreement. More accurate observations of DLOD could constrain the interior structure and allow a better understanding of the Martian seasonal CO2 cycle.
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Acknowledgements We would like to thank to M.L. Litvak from SRI for providing the HEND data. This study was funded by a ESA/Prodex contract and supported by the European CommunityÕs Improving Human Potential Programme, MAGE, under contract RTN2-2001-00414.
References Aharonson, O., Zuber, M.T., Smith, D.E., Neumann, G.A., Feldman, W.C., Prettyman, T.H. Depth, distribution, and density of CO2 deposition on Mars. J. Geophys. Res. 109 (E18), 2004. Barnes, R.T.H., Hide, R., White, A.A., Wilson, C.A. Atmospheric angular momentum fluctuations, length-of-day changes and polar motion. Proc. R. Soc. London Ser. A 387, 31–73, 1983. Cazenave, A., Balmino, G. Meteorological effects on the seasonal variations of the rotation of Mars. Geophys. Res. Lett. 8 (3), 245–248, 1981. Chao, B.F., Rubincam, D.P. Variations of Mars gravitational field and rotation due to seasonal CO2 exchange. J. Geophys. Res. 95 (B5), 14755–14760, 1990. Defraigne, P., de Viron, O., Dehant, V., Van Hoolst, T., Hourdin, F. Mars rotation variations induced by atmosphere and ice caps. J. Geophys. Res. 105 (E10), 24563–24570, 2000. Dehant, V., Lognonne´, P., Sotin, C. Network science, NetLander: a European mission to study the planet Mars. Planet Space Sci. 52 (11), 977–985, 2004. Dehant, V., Van Hoolst, T., de Viron, O., Greff-Lefftz, M., Legros, H., Defraigne, P. Can a solid inner core of Mars be detected from observations of polar motion and nutation of Mars?. J. Geophys. Res. 108 (E12), 2003, 2003. Folkner, W.M., Yoder, C.F., Yuan, D.N., Standish, E.M., Preston, R.A. Interior structure and seasonal mass redistribution of Mars from radio tracking of Mars Pathfinder. Science 278, 1749–1751, 1997.
Litvak, M.L., Mitrofanov, I.G., Kozyrev, A.S., et al. Seasonal carbon dioxide depositions on the Martian surface as revealed from neutron measurements by the HEND instrument onboard the 2001 Mars Odyssey spacecraft. Solar Sys. Res. 38 (3), 167– 177, 2004. Munk, W.H., MacDonald, G.J.F. The Rotation of the Earth; a Geophysical Discussion. University Press, Cambridge, 1960. Prettyman, T.H., Elphic, R.C., Feldman, W.C., Murphy, J.R., Nelli, S.M., Titus, T.N., Tokar, R.L. Spatial deconvolution of Mars Odyssey Neutron Spectroscopy data: analysis of Mars southern seasonal cap. In: 36th LPSC, p. 1384, 2005. Rowlands, D.D., Pavlis, D.E., Lemoine, F.G., Neumann, G.A., Luthcke, S.B. The use of laser altimetry in the orbit and attitude determination of Mars Global Surveyor. Geophys. Res. Lett. 9 (26), 1191–1194, 1999. Sanchez, B.V., Rowlands, D.D., Haberle, R.M., Schaeffer, J. Atmospheric rotational effects on Mars based on the NASA Ames general circulation model. J. Geophys. Res. 108 (E5), 2003. Smith, D.E., Zuber, M.T., Neumann, M. Seasonal variations of snow depth on Mars. Science 294, 2141–2146, 2001. Sohl, F., Spohn, T. The interior structure of Mars: implications from SNC meteorites. J. Geophys. Res. 102 (E1), 1613–1635, 1997. Van den Acker, E., Van Hoolst, T., de Viron, O., Defraigne, P., Forget, F., Hourdin, F., Dehant, V. Influence of the seasonal winds and the CO2 mass exchange between atmosphere and polar caps on MarsÕ rotation. J. Geophys. Res. 107 (E7), 2002. Van Hoolst, T., Dehant, V., Defraigne, P. Sensitivity of the Free Core Nutation and the Chandler Wobble to changes in the interior structure of Mars. Phys. Earth Planet. Int. 117, 397–405, 2000. Van Hoolst, T., Dehant, V., Roosbeek, F., Lognonne´, P. Tidally induced surface displacements, external potential variations, and gravity variations on Mars. Icarus 161, 281–296, 2003. Yoder, C.F., Konopliv, A.S., Yuan, D.N., Standish, E.M., Folkner, W.M. Mars fluid core size from detection of the solar tide. Science 296, 2141–2146, 2003. Yoder, C.F., Standish, E.M. Martian precession and rotation from Viking lander range data. J. Geophys. Res. 102 (E2), 1997.
The effects of seasonal mass redistribution and interior structure on Length-of-Day variations of Mars ¨ . Karatekin *, T. Van Hoolst, J. Tastet, O. de Viron, V. Dehant O Royal Observatory of Belgium, 3, Av. Circulaire, 1180 Brussels, Belgium Received 1 November 2004; received in revised form 29 March 2005; accepted 29 March 2005
Abstract The effect of MarsÕ interior structure, more precisely the influences of the size and the state of the core, on Length-of-Day variations (DLOD) is studied. We calculated the load Love numbers and the moments of inertia of the mantle and of the core for a set of models of MarsÕ interior. It is shown that the current level of precision of LOD observations does not allow to deduce details of the Martian interior. Future measurements of DLOD must have an accuracy better than 2% to determine whether or not the core is liquid, and even better to constrain the core size. The effect of seasonal mass redistribution on DLOD is also investigated. Martian DLOD is a good indicator of global scale seasonal CO2 cycle. Planetary rotation data from the tracking of Viking and Pathfinder landers are compared with the recent seasonal mass deposition information from Mars Global Surveyor and Mars Odyssey missions. The results reveal the current difficulties in the accurate modelling of the Martian seasonal CO2 cycle. Ó 2005 COSPAR. Published by Elsevier Ltd. All rights reserved. Keywords: Mars; Length-of-Day; Rotation; Geodesy; Planetary interior
1. Introduction The interior of Mars is modeled mainly from the extrapolation of the EarthÕs internal structure combined with only a few observational constraints such as the mass, the radius and the polar moment of inertia learned from the precession rate (Sohl and Spohn, 1997). The core size and whether or not the core is liquid are among the poorly constraint parameters. Both the liquid and the solid core states can be consistent with the Martian dipole magnetic field measurements. Recently, the second degree tidal love number, k2 was determined from the tracking of Mars Global Surveyor (MGS) which suggests that the core is at least partially fluid (Yoder et al., 2003). If measured precisely enough, determination of MarsÕ orientation parameters could remove most of the above uncertainties associated with the interior of Mars. *
Corresponding author. Tel.: +3223736732; fax: +32494034753. ¨ . Karatekin). E-mail address: [email protected] (O
Normal rotational modes such as the Free Core Nutation (FCN) depend on the rheology and the interior structure of the planet through the resonance effects with the normal modes of the planet (Van Hoolst et al., 2000; Dehant et al., 2003). A dedicated geodesy mission involving a network of landers could measure accurately those normal modes from the nutations (Dehant et al., 2004). However, presently the realization of such a mission is uncertain. On the other hand, some of the orientation parameters of Mars, such as DLOD, has been obtained directly from tracking data of the landers on the surface of Mars. Folkner et al. (1997) and Yoder and Standish (1997) estimated LOD variations through the analysis of tracking data of Viking and Pathfinder landers. In the near future, with several planned missions involving landers, the DLOD estimates can easily be improved, provided that the life duration of the landers are long enough to follow the seasonal variations. In the present study, we investigate the use of LOD variations to constrain the interior structure, more pre-
0273-1177/$30 Ó 2005 COSPAR. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.asr.2005.03.117
740
¨ . Karatekin et al. / Advances in Space Research 38 (2006) 739–744 O
cisely the size and the state of the core. In addition, we study the effect of surface loads on DLOD. The LOD variations are associated with the seasonal redistribution of masses on planetary scale. MarsÕ surface mass distribution varies at seasonal time scales due to condensation, sublimation and precipitation of CO2, the primary constituent of the atmosphere. As much as a third of the atmosphere has been estimated to take place in this seasonal CO2 cycle (Smith et al., 2001). The sizes and masses of both polar caps as well as the atmospheric pressure change, resulting in seasonal variations of the Martian inertia tensor. In the absence of external torques, the rotation rate (DLOD) is principally affected by the changes in the inertia tensor. The LOD variations on Mars have been estimated by various authors. On the basis of surface pressure data from Viking landers, Cazenave and Balmino (1981) and Chao and Rubincam (1990) estimated the seasonal variations of LOD. With the help of Global Circulation Models (GCM) of Martian atmosphere, Defraigne et al. (2000), Van den Acker et al. (2002) and Sanchez et al. (2003) improved the previous DLOD estimates by including the atmospheric wind influence and a more realistic surface topography. In the present study, we compare the direct measurements of DLOD from the Viking and Pathfinder landers (Folkner et al., 1997) with the LOD variations calculated from the seasonal mass redistribution given by the MOLA laser altimeter onboard the MGS spacecraft (Smith et al., 2001), the High Energy Neutron Detector (HEND) onboard the Mars Odyssey spacecraft (Litvak et al., 2004), and the Doppler tracking of the MGS spacecraft (Smith et al., 2001; Yoder et al., 2003). Such a comparison is useful to reveal the current understanding of the Martian seasonal CO2 cycle.
2. Rotation rate and interior structure 2.1. Theory The rotation of a deformable planet about its spin axis is described by Liouville equations (Munk and MacDonald, 1960). If the external torques are neglected, the instantaneous rotation rate variations of a planet are due to the angular momentum changes between its fluid layers and solid body. To formulate the rotation rate, one customarily introduces the atmospheric angular momentum excitation function, v, consisting of the matter and the motion terms. The matter term is due to the redistribution of the atmospheric mass while the motion term depends on the variations of the atmospheric zonal winds (see Barnes, 1983). Z R4 vmatter ðtÞ ¼ ðq ðh; k; tÞ þ qatm ðh; k; tÞÞsin3 h dh dk; C S ice ð1Þ
v
motion
R3 ðtÞ ¼ gCX
Z
usin2 h dh dk dp;
ð2Þ
V
where t is the time, R the equatorial radius, g the surface gravity, X the mean rotation rate, h the colatitude, k the longitude, q the load expressed as mass per surface area, and u the zonal wind speed. The relative variation in the rotation speed is given by m¼
DX C DLOD ¼ . ½ð1 þ k 02 Þvmatter þ vmotion ¼ X Cm LOD ð3Þ
In Eq. (3), C and Cm represent polar principal moments of inertia of the whole planet and mantle, respectively. k 02 is the load Love number of degree 2 and it represents the response of a deformable planet to a unit surface load, through the combined effects of pressure and attraction (Munk and MacDonald, 1960; Defraigne et al., 2000). DLOD is linked to the planetÕs interior through these two parameters, C/Cm and k 02 . Their values for the planet Mars are discussed in the next section. 2.2. Effect of Martian core on DLOD We calculated the Love number k 02 and the moment of inertia ratio, C/Cm, for a set of models of the interior of Mars. We assumed the same density, rigidity and compressibility profiles in the mantle and crust as in the models by Sohl and Spohn (1997) which satisfies the observed mass, radius and moment of inertia. The core is assumed to be composed of a Fe–S alloy. From the content of S, we determine its radius. Neither the exact composition nor its state are known although the recent measurements of tidal Love numbers suggest an at least partially liquid core, and a core radius in the range of 1520–1840 km (Yoder et al., 2003). In the present study, we considered a slightly larger range of possible core radius (1268–1868 km) and both solid and liquid cores. The models with larger or smaller core radius have been obtained by changing slightly the core density in order to keep the total mass unchanged (Van Hoolst et al., 2003). The results are presented in Figs. 1 and 2. A larger absolute value of k 02 indicates a larger deformation of Mars. According to Fig. 1, the response of the planet to surface loads is stiffer with a smaller liquid core. The absolute value of k 02 increases with increasing core radius. It shows variations lower than 50% for the range of core radii considered. If the core is solid, the absolute value of k 02 is smaller (i.e., less easily deformable) and its variation as a function of core radius is less prominent. The ratio of polar moment of inertia, C/Cm, on the other hand, increases with the radius of the liquid core (Fig. 2). Its amplitude varies less than 10% between
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0.98
0.04
liquid Core Solid Core
0.97
0.06
0.96
C/Cm (1+k2’)
k 2’
0.08
0.95 0.94
–0.1
0.93
–0.12 0.92
–0.14
–0.16 1200
Liquid core Solid core
1300
1400
0.91
1500
1600
1700
1800
1900
0.9 1200
1300
1400
Fig. 1. Variation of the second degree load love number k 02 as a function of core radius.
1.1 1.09 1.08
C/C
m
1.07 1.06 1.05 1.04 1.03 1.02 1.01 1200
1300
1400
1500
1600
1500
1600
1700
1800
1900
Core radius
Core radius
1700
1800
1900
Core radius
Fig. 2. Variation of the polar moment of inertia ratio (C/Cm) as a function of liquid core radius.
the limits of the assumed core radius. For a solid core, the ratio is equal to one and the interior structure affects DLOD only through k 02 . The combined effect, C=C m ð1 þ k 02 Þ, determines DLOD in response to a surface load excitation. It is presented in Fig. 3 for both solid and liquid cores. For a liquid core, C=C m ð1 þ k 02 Þ results from two opposing effects which essentially cancel each other since both C/Cm and 1 þ k 02 show opposite relative variations with increasing core radius. Accordingly, the resulting C=C m ð1 þ k 02 Þ does not vary significantly (<0.5%). Its variation is more important for a solid core for which C/Cm is unity and the variation is only a function of k 02 . For a small core radius the response of the planet with solid and liquid cores are similar. The difference grows with increasing core radius but remains always less than 2% within the range of core sizes considered. This implies that the uncertainties associated with the interior struc-
Fig. 3. Variation of C=C m ð1 þ k 02 Þ as a function of core radius and state.
ture have only a very small effect on DLOD. Measurements of DLOD must have an accuracy better than 2% to determine whether or not the core is liquid, and even better (<0.5%) to constrain the core size. In the present study, we assumed that the excitation of LOD due to zonal wind variations is perfectly known. Seasonal zonal winds which are the primary cause of DLOD on Earth, have been thought to be much less important on Mars (Folkner et al., 1997). Because of the lack of direct measurements, their only estimates are from Martian GCMs, which suggest, nevertheless, annual amplitudes as large as one third of the total DLOD (Van den Acker et al., 2002; Sanchez et al., 2003). This makes necessary the evaluation of zonal wind effects and more cumbersome the determination of the interior structure from DLOD measurements.
3. Effect of seasonal CO2 cycle The effect of surface loads on LOD variations are studied from recent DLOD estimates based on independent seasonal mass redistribution observations as well as those from direct measurements. Variations in rotation about the spin axis due to surface mass redistribution are proportional to changes in the gravitational zonal harmonic of degree 2, J2 (Chao and Rubincam, 1990): DLOD 2MR2 ¼ DJ 2 . LOD 3C
ð4Þ
This only applies to the matter term in Eq. (3). The relationship provides a means to infer rotation changes from the long wavelength gravitational variations. Temporal variations of the low degree gravity field coefficients of Mars were recently determined from the MGS tracking data by Yoder et al. (2003) and Smith
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et al. (2001). The corresponding LOD variations are obtained from Eq. (4) with LOD = 88,775 s and C/MR2 = 0.365 (Yoder et al., 2003). These DLOD results are given in Fig. 4 in comparison with Lander tracking analysis from Folkner et al. (1997) who estimated seasonal rotation variations of Mars from the 3 months of Pathfinder tracking data in combination with much longer tracking data of Viking landers. Although the Pathfinder data covered a relatively short period of time, thanks to its higher signal/noise ratio, the previous DLOD estimates based only on Viking landers (Yoder and Standish, 1997) were improved. Nevertheless the associated errors remain high (about 40% for annual amplitudes and 50% for the semi-annual amplitudes). The errors given for the gravity solution of J2 are of about the same order of magnitude (Yoder et al., 2003; Smith et al., 2001). If one of the caps would increase symmetrically with respect to the decrease of the other cap (i.e., if North and South polar caps would have identical seasonal deposits and be 180° out of phase), there would not be any change in the axial moment of inertia or in rotation rate. But, due to North-South asymmetry, the contributions from the two hemispheres do not cancel and the resulting DLOD has annual and semi-annual variability (see Fig. 4). DLOD from the lander tracking data have a smaller semi-annual variation. The peak-to-peak amplitude, on the other hand, is closer to that from DJ2 of Smith et al. (2001). Part of the approximately 20% difference in peak-to-peak DLOD between the lander tracking estimates and those from DJ2 of Yoder et al. (2003) could be due to wind contributions that are not included in solutions from DJ2 observation. However, it is not possible to draw definite conclusions as the experimental uncertainties are larger than the expected wind contributions. 0.6 0.4
Δ LOD (ms)
0.2
0 –0.2
The variations of J2 and DLOD can also be obtained from the estimates of seasonal deposits given by the MOLA laser altimeter onboard the MGS and by the HEND onboard the Mars Odyssey. Smith et al. (2001) reported temporal changes in the elevation of Martian polar caps and the planetÕs global mass redistribution due to the seasonal CO2 cycle, using the data from MOLA. They showed elevation changes up to 1.5 ± 0.25 m in the North cap and 0.9 ± 0.30 m in the South cap. In the present study, we use the temporal elevation variations from MOLA, with a density of 1000 kg m3, as suggested by Smith et al. (2001), to approximately match the amplitude of the observed time-variable gravity signal. The seasonal changes in CO2 height cause variations in neutron flux measured by the HEND onboard the Mars Odyssey. These variations have been used to estimate the surface density of seasonal deposits at different latitudes by using a model-dependent technique (Litvak et al., 2004). The CO2 surface densities as a function of time were given in kg/m2 over the latitude bands of 60–70°, 70–80° and 80–90° for both hemispheres as a function of time. By multiplying these data with the corresponding surface areas, the mass variations over polar caps are obtained. From temporal variations of surface densities, it is straightforward to estimate DJ2 (Chao and Rubincam, 1990) and hence DLOD through Eq. (4). DLOD estimates from observations of seasonal deposits as well as from lander tracking data are plotted in Fig. 5. Solutions from MOLA cover more than one Martian year and the corresponding solutions overlap for 120° < Ls < 150°. DLOD depends on the sum of the loads in the two hemispheres. The minima for the three sets of data occur around Ls = 0° and Ls = 180°, which correspond to the end of the winter season in the North and South hemispheres, respectively. Large quantities of CO2 are then condensed on one of the two polar caps close to the rotation axis, leading to faster rotation of the planet. In the vicinity of Ls = 180°, the solution based on MOLA has an amplitude about four times smaller than the solution from HEND data. Its maximum around Ls = 90° is in advance by 30°, and around Ls = 210° delayed by the same amount, with respect to DLOD solutions based on HEND data. A comparison with Fig. 4, reveals a similar behavior for the DLOD from J2 variations and those from HEND data.
–0.4 Yoder et al. (2003) Smith et al. (2001) Folkner et al. (1997)
–0.6 –0.8
0
30
60
90 120 150 180 210 240 270 300 330 360
Ls (deg)
Fig. 4. DLOD deduced from J2 solutions of the MGS spacecraft tracking (Yoder et al., 2003; Smith et al., 2001) and the direct measurements of DLOD from the tracking of the Viking and the Pathfinder landers (Folkner et al., 1997).
4. Discussion and conclusions We investigated the effect of the interior structure of Mars, more precisely the size and the state of the core, on LOD variations. We calculated the load Love number k 02 and the moment of inertia ratio C/Cm for a set of models of the interior of Mars. A range of possible core radius and both solid and liquid cores are consid-
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LOD (ms)
0.2
0 –0.2 –0.4 HEND MOLA Folkner et al. (1997)
–0.6 –0.8 0
30
60
90 120 150 180 210 240 270 300 330 360
Ls (deg) Fig. 5. DLOD deduced from the temporal variations of seasonal mass deposits computed from HEND data (Litvak et al., 2004) and MOLA data (Smith et al., 2001) onboard of Mars Odyssey and MGS spacecrafts respectively. The direct measurements of DLOD from the tracking of the Viking and the Pathfinder landers (Folkner et al., 1997) are also given in the figure.
ered. For a small core radius the response of the planet with solid and liquid cores are similar. The difference becomes more prominent with increasing core radius but remains always less than 2% within the range of core sizes considered. This implies that the uncertainties associated with the interior structure have only a very small effect on DLOD. Measurements of DLOD must have an accuracy better than 2% to determine whether or not the core is liquid, and even better (<0.5%) to constrain the cores size provided that the seasonal zonal wind variations are accurately known. The current LOD data has large uncertainties compared to the required ones, but it can be improved with future Martian missions involving landers. For example, the expected accuracy of a geodesy experiment with a network of landers, such as NEIGE (Dehant et al., 2004), is less than 1%. We compared the direct measurements of DLOD from the Viking and Pathfinder landers (Folkner et al., 1997) with the LOD variations calculated from the seasonal mass redistribution observations by the MOLA laser altimeter onboard the MGS spacecraft (Smith et al., 2001), the HEND onboard the Mars Odyssey spacecraft (Litvak et al., 2004), and the Doppler tracking of the MGS spacecraft (Smith et al., 2001; Yoder et al., 2003). The results show differences both in amplitude and phases due to measurement and modelling errors. DLOD calculated from the seasonal mass redistributions does not include the wind contribution, which can be as large as one third of the total signal. The differences in MGS tracking solutions (Fig. 4) result mainly from the errors in the determination of time variable J2. The quality of time variable gravity from radio-science data depends on the achieved orbit preci-
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sion. The perturbations on the orbit due to non-gravitational forces, especially the atmospheric drag, deteriorates mainly the determination of nodal rate, hence the even zonal harmonic coefficients (Yoder et al., 2003). As shown in Fig. 6 in Smith et al. (2001) and in Fig. 3 in Yoder et al. (2003), the J2 signal is weak and it is hard to reconcile it with the expected semi-annual variation. Seasonal CO2 thickness variations measured by HEND and MOLA show significant differences (Fig. 5). HEND has a much larger footprint covering a radius of several hundred kilometers. Measurements with such a low surface resolution suffer from spatial and temporal artifacts (Prettyman et al., 2005). In addition, the interpretation of CO2 thickness variations from neutron flux measurements demands a model for the source of neutrons which depends on the water contents of the regolith and of the martian atmosphere. An atmospheric circulation model had to be used for the atmospheric corrections. Two models of layered regolith with different water content are assumed for the south and north poles. Accordingly, HEND results would change with the assumed regolith and atmospheric models. The MOLA altimeter, on the other hand, has a much smaller footprint (approximately 168 m). The vertical resolution is reported to be 37.5 cm from laser point to laser point (Aharonson et al., 2004). The uncertainties in the positioning of the spacecraft are not only restricted to the vertical, but significant discrepancies occur also in the along-track and cross-track directions. On a rough terrain with steep slopes these effects on the altimetry measurements are non-negligible (Rowlands et al., 1999). The resulting accuracy of MOLA measurements are about 10 m. When the corrections from statistical cross-over analysis are applied, the observed rootmean-square values are reduced significantly (1.8 m) but it remains still high with respect to maximum seasonal CO2 height variations of the order of 1.5 m (Aharonson et al., 2004). The density of the Martian seasonal deposits are known only approximately. The recent effort of Aharonson et al. (2004) by combining MOLA and Gamma Ray Spectroscopy observations yield unexpectedly low densities, about 0.5 g/cm3. The measurements of seasonal surface mass distribution by HEND have required the priori knowledge of not only the Martian atmospheric dynamics but also the modelling of the subsurface regolith. MOLA altimetry on the other hand, had relatively low vertical resolution and, similar to the MGS tracking data, have suffered from orbit determination errors. Accordingly the overall agreement between the different DLOD solutions is not excellent (Figs. 4 and 5). But it is reassuring that the HEND and the tracking data show relatively better agreement. More accurate observations of DLOD could constrain the interior structure and allow a better understanding of the Martian seasonal CO2 cycle.
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Acknowledgements We would like to thank to M.L. Litvak from SRI for providing the HEND data. This study was funded by a ESA/Prodex contract and supported by the European CommunityÕs Improving Human Potential Programme, MAGE, under contract RTN2-2001-00414.
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