Lecture 18Chaotic obliquity variations

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Lecture 18Chaotic obliquity variations

• Meteo 466

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Definition of chaos

• Chaos Mathematically, this term is used to
  describe dynamical systems in which small changes
  in initial conditions lead to large changes in
  the solution after some period of time
• Examples of chaotic systems
• The weather (the Butterfly effect)
• Double pendulum
• The geodynamo (or solar dynamo)
• The inner Solar System ?

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Chaos in the Solar System

• 25 b.y. integration of planetary orbits
• Semi-analytic numerical method
• Series expansion of Laplaces equation to
  150,000 terms. Keep the first 50,000

J. Laskar, AA 287, L9 (1994)
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Mercury escapes from the Solar System
Laskar (1994)
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Causes of chaotic obliquities

• A planets obliquity can vary chaotically if the
  the planets spin axis precesses at the same rate
  as do orbital characteristics of other planets
• Two types of secular resonances
• Precession of perihelion
• Precession of the line of nodes

(1)
a
(2)
?
S
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Secular forcing frequencies in the Solar System

• Orbits in the inner Solar System are chaotic, so
  the forcing pattern is complex
• Orbits in the outer Solar System are periodic, so
  the forcing pattern is more regular

Laskar and Robutel (1993)
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Chaotic regions for Mars

• 18 m.y. simulations beginning at specified
  precession constants and obliquities
• Solution can wander horizontally within the
  fuzzy region
• Precession constant inversely related to spin
  period
• Martian daylength 24.6 hr

Chaotic region
Precession constant, /yr
Present Mars
Obliquity
Laskar and Robutel (1993)
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Mars recent obliquity
Inclination
? 8.26/yr
? 8.32/yr
Ward and Rudy, Icarus 94, 160 (1991)
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Mars recent obliquity(alternative calculation)
Touma and Wisdom, Science 259, 1294 (1993)
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Possible consequences of martian obliquity
variations

• Could obliquity variations be responsible for the
  martian gullies?
• Gullies (small-scale fluvial features) are found
  predominately at high latitudes on
  poleward-facing slopes

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Recent Martian Gully Locations(from Mars Global
Surveyor)
M.C. Malin and K.S Edgett, Science (2000)
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MOC Images (Mars Global Surveyor)
Fig. 2 Crater wall (54.8oS, 342.5oW)
M. Malin and K. Edgett, Science (2000)
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MOC Images
Fig. 4 Crater wall (37.3oS, 168.0oW)
Malin and Edgett (2000)
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Nirgal Vallis apron
MOC Images
Malin and Edgett (2000)
Fig. 9
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• Unfortunately for the obliquity hypothesis, some
  of the gullies appear to have formed essentially
  yesterday

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New Martian Gullies (from Mars Global Surveyor)
http//www.nasa.gov/vision/universe/solarsystem/mg
s-092005-images.html?msource15205tryauid11148
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• Back to Earth
• Earths obliquity is stable, but that might not
  be the case if it did not have a large Moon
• Ward and Brownlee have emphasized this in their
  book Rare Earth
• Moon forming impacts are rare hence, planets
  with stable obliquities may be rare

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Secular forcing for Earth

• Earths precession constant is 55/yr, which is
  just outside the chaotic region chaos occurs
  when the secular forcing is minus (or plus?) the
  precession rate
• If the Moon were not present, however, the
  precession constant would be lower (15/yr), and
  Earths spin axis would precess in resonance with
  the secular forcings

Earth with Moon
No Moon
J. Laskar et al., Nature (1993)
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Earths obliquity with and without the Moon
Chaotic region
Daylength (with no moon)
Laskar and Robutel, Nature (1993)
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• Does this mean that a Moon-less planet would have
  a chaotic obliquity?
• Earths spin rate has been slowed over time by
  tidal evolution of the Earth-Moon system
• Initial spin rate (after the Moon-forming impact)
  was probably 4-5 hours
• Spin rates faster than 12 hours lead to stable
  obliquity
• What would the initial spin rate have been,
  though, in the absence of a Moon-forming impact?

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Would a high-obliquity planet be habitable?

• Continents at high latitudes would be alternately
  baked and frozen
• Tropical continents would experience 2 summers
  and 2 winters

Williams Kasting, Icarus (1997)
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Would a high-obliquity planet be habitable?

• Planets near the outer edge of the HZ that
  develop dense, CO2-rich atmospheres would
  experience much smaller temperature variations
• Marine life would be virtually unaffected by high
  obliquity

2 bars CO2
Williams Kasting (1997)
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Conclusions

• Mars obliquity varies chaotically, but that
  probably isnt what causes the gullies
• Earths obliquity varies regularly, but that is
  more or less accidental
• Other (extrasolar) planets could have either
  regular or chaotic obliquities. Need to observe
  them in concert with their planetary companions
• Because most O2 comes from marine photosynthesis,
  the search for microbial life on extrasolar
  planets should be virtually unaffected by
  obliquity