- Email: [email protected]
Corrigendum to “Coupled thermal–orbital evolution of the early Moon” [Icarus 208 (2010) 1–10]
Icarus 212 (2011) 448–449
Contents lists available at ScienceDirect
Icarus journal homepage: www.elsevier.com/locate/icarus
Corrigendum
Corrigendum to ‘‘Coupled thermal–orbital evolution of the early Moon’’ [Icarus 208 (2010) 1–10] Jennifer Meyer ⇑, Linda Elkins-Tanton, Jack Wisdom Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139, United States
In our recent paper (Meyer et al., 2010), we studied the evolution of the early Moon in a coupled thermal–orbital model. Our goal was to see if we could find an evolution that would pass through the shape solution of Garrick-Bethell et al. (2006). Garrick-Bethell et al. (2006) found that the shape of the Moon could be explained if the Moon’s shape froze in an eccentric orbit. We succeeded in finding an orbital evolution of the early Moon that passes close to the shape solution. However, we examined the elastic stability of the shape of the Moon during this epoch and found that the shape at that epoch could not be maintained to the present time. Thus the principal conclusion of our paper was that the shape of the Moon cannot be explained by an early high eccentricity phase of the lunar orbit. This does not rule out an early high eccentricity phase. The Mignard tidal model that we used in our work does not constrain the time lag parameter; we found that we needed a very dissipative Earth to fit the early lunar orbit to the shape solution. But for the tidal model to correspond to a lagged tidal bulge the time lag cannot exceed an eighth of the rotation period of the host planet (Efroimsky and Williams, 2009). This is because the tidal torque peaks for a 45° delay. Unfortunately, the time lag that we chose does not satisfy this constraint. We found that passage through the shape solution required a tidal time lag Dt of 123 min (for an assumed k2 of the Earth at that epoch of 0.97, the fluid Love number of the Earth). The rotation period of the early Earth was approximately 5.1 h, when the Moon was at a semimajor axis of 6.8Re, where Re is the radius of the Earth. Thus the tidal time lag is physically constrained to be less than 37 min. For this value of Dt, the peak eccentricity is only 0.31. The peak occurs at a semimajor axis of 16.5 Earth radii. The new evolution is shown in Fig. 1. Notice that the new evolution no longer passes through the shape solution of Garrick-Bethell et al. (2006). This means that
DOI of original article: 10.1016/j.icarus.2010.01.029
⇑ Corresponding author.
E-mail address: [email protected] (J. Meyer). 0019-1035/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2010.12.008
the coupled thermal–orbital model cannot produce the orbit capable of matching the shape solution, since Fig. 1 shows the evolution with maximum dissipation. However, as the second part of our paper showed, the Moon is not capable of recording the shape from that era anyway, so the point is moot. The high eccentricity peak in the original paper also could not have explained the Moon’s shape anomaly, so our previous conclusions remain intact. In our original paper, we also drew conclusions concerning the thermal evolution of the Moon and the origin of lunar zircons. In our paper, we found that the lunar magma ocean took approximately 217 Myr to freeze. With the new maximum tidal time lag this time is now increased to 272 Myr. We also placed constraints on the original depth of the zircon dated by Nemchin et al. (2009), based on its closure temperature. We found that this zircon originated at a depth of approximately 25–30 km. With the revised tidal time lag this depth is not substantially modified. Acknowledgment We thank Jim Williams for bringing this problem to our attention.
References Efroimsky, M., Williams, J.G., 2009. Tidal torques: A critical review of some techniques. CMDA 104, 257–289. Garrick-Bethell, I., Wisdom, J., Zuber, M., 2006. Evidence for a past high eccentricity lunar orbit. Science 313, 652–655. Meyer, J., Elkins-Tanton, L., Wisdom, J., 2010. Coupled thermal–orbital evolution of the early Moon. Icarus 208, 1–10. Nemchin, A., Timms, N., Pidgeon, R., Geisler, T., Reddy, S., Meyer, C., 2009. Timing and crystallization of the lunar magma ocean constrained by the oldest zircon. Nat. Geosci. 2, 133–136.
J. Meyer et al. / Icarus 212 (2011) 448–449
Fig. 1. The eccentricity of the lunar orbit plotted versus the semimajor axis of the orbit, for the dissipative lid model. For reference, the dot shows the orbit that gives the solution to the shape problem for synchronous rotation.
449
Contents lists available at ScienceDirect
Icarus journal homepage: www.elsevier.com/locate/icarus
Corrigendum
Corrigendum to ‘‘Coupled thermal–orbital evolution of the early Moon’’ [Icarus 208 (2010) 1–10] Jennifer Meyer ⇑, Linda Elkins-Tanton, Jack Wisdom Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139, United States
In our recent paper (Meyer et al., 2010), we studied the evolution of the early Moon in a coupled thermal–orbital model. Our goal was to see if we could find an evolution that would pass through the shape solution of Garrick-Bethell et al. (2006). Garrick-Bethell et al. (2006) found that the shape of the Moon could be explained if the Moon’s shape froze in an eccentric orbit. We succeeded in finding an orbital evolution of the early Moon that passes close to the shape solution. However, we examined the elastic stability of the shape of the Moon during this epoch and found that the shape at that epoch could not be maintained to the present time. Thus the principal conclusion of our paper was that the shape of the Moon cannot be explained by an early high eccentricity phase of the lunar orbit. This does not rule out an early high eccentricity phase. The Mignard tidal model that we used in our work does not constrain the time lag parameter; we found that we needed a very dissipative Earth to fit the early lunar orbit to the shape solution. But for the tidal model to correspond to a lagged tidal bulge the time lag cannot exceed an eighth of the rotation period of the host planet (Efroimsky and Williams, 2009). This is because the tidal torque peaks for a 45° delay. Unfortunately, the time lag that we chose does not satisfy this constraint. We found that passage through the shape solution required a tidal time lag Dt of 123 min (for an assumed k2 of the Earth at that epoch of 0.97, the fluid Love number of the Earth). The rotation period of the early Earth was approximately 5.1 h, when the Moon was at a semimajor axis of 6.8Re, where Re is the radius of the Earth. Thus the tidal time lag is physically constrained to be less than 37 min. For this value of Dt, the peak eccentricity is only 0.31. The peak occurs at a semimajor axis of 16.5 Earth radii. The new evolution is shown in Fig. 1. Notice that the new evolution no longer passes through the shape solution of Garrick-Bethell et al. (2006). This means that
DOI of original article: 10.1016/j.icarus.2010.01.029
⇑ Corresponding author.
E-mail address: [email protected] (J. Meyer). 0019-1035/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2010.12.008
the coupled thermal–orbital model cannot produce the orbit capable of matching the shape solution, since Fig. 1 shows the evolution with maximum dissipation. However, as the second part of our paper showed, the Moon is not capable of recording the shape from that era anyway, so the point is moot. The high eccentricity peak in the original paper also could not have explained the Moon’s shape anomaly, so our previous conclusions remain intact. In our original paper, we also drew conclusions concerning the thermal evolution of the Moon and the origin of lunar zircons. In our paper, we found that the lunar magma ocean took approximately 217 Myr to freeze. With the new maximum tidal time lag this time is now increased to 272 Myr. We also placed constraints on the original depth of the zircon dated by Nemchin et al. (2009), based on its closure temperature. We found that this zircon originated at a depth of approximately 25–30 km. With the revised tidal time lag this depth is not substantially modified. Acknowledgment We thank Jim Williams for bringing this problem to our attention.
References Efroimsky, M., Williams, J.G., 2009. Tidal torques: A critical review of some techniques. CMDA 104, 257–289. Garrick-Bethell, I., Wisdom, J., Zuber, M., 2006. Evidence for a past high eccentricity lunar orbit. Science 313, 652–655. Meyer, J., Elkins-Tanton, L., Wisdom, J., 2010. Coupled thermal–orbital evolution of the early Moon. Icarus 208, 1–10. Nemchin, A., Timms, N., Pidgeon, R., Geisler, T., Reddy, S., Meyer, C., 2009. Timing and crystallization of the lunar magma ocean constrained by the oldest zircon. Nat. Geosci. 2, 133–136.
J. Meyer et al. / Icarus 212 (2011) 448–449
Fig. 1. The eccentricity of the lunar orbit plotted versus the semimajor axis of the orbit, for the dissipative lid model. For reference, the dot shows the orbit that gives the solution to the shape problem for synchronous rotation.
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