Nutballs and the Mode

Atheist Republic's Kaaba: Love Wins
A stylized Kaaba (click to embiggen).

Recently, Atheist Republic (AR) posted this image (⇒) in response to the Supreme Court’s decision (pdf) that legalizes marriage in the U.S. It is a Photoshopped image of the Kaaba in Mecca. The reaction from noisome elements of the Muslim community has been, predictably, swift, violent, and largely incoherent (cf. the Facebook post or AR’s original Twitter post for a sampling). AR’s post is fine; I think it is timely, in good taste, and makes a good point. However, I think AR made a mistake.

AR responded to the growing shit storm in a subsequent post on their web site (WARNING: one image, about 3/4 of the way into the post, is deeply disturbing), electing to show a number of select examples of the insults and threats they’ve received to make a point:

Please keep in mind that these aren’t members of ISIS or Al-Qaeda making these statements, but rather are your everyday average Muslim.

Later:

…these aren’t extremists or jihadists, they’re just average Muslims. These are the ones who call themselves “moderate”.

And, if you are feeling particularly thick-headed:

To make it clear that these are supposed “moderate” Muslims, I’d like to point out that we know for a fact that one of these men is a US citizen. This particular commenter has specifically asked for information from one of our admins that he suspects lives in his area, and threatened said admin with physical violence against this admin and their family.

A skewed distribution (click to embiggen). Where do you think IPLs reside?

One thought kept nagging me as I read AR’s response: AR furnishes no valid evidence or argument to support the all-too-common claim that these select nutballs are “your everyday average Muslim” (as opposed to the crazies that carry out terrorist attacks in the name of their religion or, more accurately, their ignorant, deranged ideology). It seems likely to me that the cretinous whackjobs sprinkling AR’s posts with turds are neither average nor representative of Muslims in general. These whackjobs are — like our own noisome right wing nutballs — an abnormally incoherent, ignorant, and vocal minority. I’ve no doubt average Muslims are as willingly delusion-controlled as our average Christians here in the U.S., but I have to question that the infantile profane loudmouths of either organized delusion system lie anywhere near the peaks (i.e., the modes) of their respective population distributions.

The excerpts above — and, indeed, AR’s entire argument — illustrate several common logical fallacies. In the first two excerpts, the author is arguing by assertion. This is a counterproductive rhetorical tactic. It raises people’s hackles, to your disadvantage.

The third excerpt is somewhat more interesting. First, it cherry-picks an anecdotal example. (The example itself also seems hardly relevant — a red herring.) This is a surprising mistake, since cherry-picking is perhaps the most common logical fallacy for which rationalists such as AR criticize religionists and the right wing.

In this excerpt the author also equates being a U.S. citizen with being “moderate”, with no supporting argument or evidence. As recent events in the U.S. have shown repeatedly, there is nothing moderate about the beliefs of U.S. terrorists, Muslim or not. This is  a false equivalence, perhaps the second most common logical fallacy employed by the right (or maybe the third, behind strawman argument).

This is not an apology for “average” adherents to horrifically damaging organized delusion systems. From all that I’ve seen, Western religions are among the most senseless and destructive invented concepts in the history of humankind. But accuracy, precision, and validity in our claims and arguments, whatever the context, matter.

We rationalists are — or should be — better than this.

Seriously, you do not need to see this image — it cannot be unseen.

 Speaking of crazies, is there much, if any, difference between a Muslim terrorist who slaughters innocents in a medical treatment building and, say, a Christian terrorist who slaughters innocents in an African American church? Or between that (or any other) Muslim terrorist and a Christian terrorist who shoots dead a medical doctor during church services?

 

DDA 2015 – Constraints on Titan’s rotation from Cassini mission radar data

This is one of a series of notes taken during the 2015 meeting of the AAS Division on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.

Bruce Bills (JPL)

Abstract

We present results of a new analysis of the rotational kinematics of Titan, as constrained by Cassini radar data, extending over the entire currently available set of flyby encounters. Our analysis provides a good constraint on the current orientation of the spin pole, but does not have sufficient accuracy and duration to clearly see the expected spin pole precession. In contrast, we do clearly see temporal variations in the spin rate, which are driven by gravitational torques which attempt to keep the prime meridian oriented toward Saturn.

Titan is a synchronous rotator. At lowest order, that means that the rotational and orbital motions are synchronized. At the level of accuracy required to fit the Cassini radar data, we can see that synchronous rotation and uniform rotation are not quite the same thing. Our best fibng model has a fixed pole, and a rotation rate which varies with time, so as to keep Titan’s prime meridian oriented towards Saturn, as the orbit varies.

A gravitational torque on the tri-axial figure of Titan attempts to keep the axis of least inertia oriented toward Saturn. The main effect is to synchronize the orbit and rotation periods, as seen in inertial space. The response of the rotation angle, to periodic changes in orbital mean longitude, is modeled as a damped, forced harmonic oscillator. This acts as a low-pass filter. The rotation angle accurately tracks orbital variations at periods longer than the free libration period, but is unable to follow higher frequency variations.

The mean longitude of Titan’s orbit varies on a wide range of time scales. The largest variations are at Saturn’s orbital period (29.46 years), and are due to solar torques. There are also variations at periods of 640 and 5800 days, due to resonant interaction with Hyperion.

For a rigid body, with moments of inertia estimated from observed gravity, the free libration period for Titan would be 850 days. The best fit to the radar data is obtained with a libration period of 645 days, and a damping time of 1000 years.

The principal deviation of Titan’s rotation from uniform angular rate, as seen in the Cassini radar data, is a periodic signal resonantly forced by Hyperion.

Notes

  • Titan:
    • hard to see surface
    • Cassini’s radar intended for mapping surface
      • didn’t get much by way of repeat observations (“tie points”), which are needed to constrain rotation
      • most data near poles — not terribly helpful
  • Rotation model from tie-point observations
    • Stiles et al. 2008: 50 tie points over 2.8 yr
    • Now: 2602 tie points over 10 yr
    • solve for 3 params (RA & DEC of spin pole, angular rate)
    • $P = 15.94547727 \pm 6.03 \times 10^{-7}$ d
    • spin pole precession
      • gravity model: ~250 yr
      • not clearly seen in data
    • spin rate variations
      • seen in data
      • dynamical model
        • assume Titan in synch. rotation
        • gravity torque
        • dissipation
        • $\rightarrow$ libration period ~850 d
        • Hyperion has nontrivial influence
    • fit: libration period = 645.4 d, damping time = 430 yr, rotation period slightly changed

DDA 2015 – Recent Formation of Saturnian Moons: Constraints from Their Cratering Records

This is one of a series of notes taken during the 2015 meeting of the AAS Division on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.

Session: Moon Formation and Dynamics I

Henry C. (Luke) Dones (SWRI)

Abstract

Charnoz et al. (2010) proposed that Saturn’s small “ring moons” out to Janus and Epimetheus consist of ring material that viscously spread beyond the Roche limit and coagulated into moonlets. The moonlets evolve outward due to the torques they exert at resonances in the rings. More massive moonlets migrate faster; orbits can cross and bodies can merge, resulting in a steep trend of mass vs. distance from the planet. Canup (2010) theorized that Saturn’s rings are primordial and originated when a differentiated, Titan-like moon migrated inward when the planet was still surrounded by a gas disk. The satellite’s icy shell could have been tidally stripped, and would have given rise to today’s rings and the mid-sized moons out to Tethys. Charnoz et al. (2011) investigated the formation of satellites out to Rhea from a spreading massive ring, and Crida and Charnoz (2012) extended this scenario to other planets. Once the mid-sized moons recede far from the rings, tidal interaction with the planet determines the rate at which the satellites migrate. Charnoz et al. (2011) found that Mimas would have formed about 1 billion years more recently than Rhea. The cratering records of these moons (Kirchoff and Schenk 2010; Robbins et al. 2015) provide a test of this scenario. If the mid-sized moons are primordial, most of their craters were created through hypervelocity impacts by ecliptic comets from the Kuiper Belt/Scattered Disk (Zahnle et al. 2003; Dones et al. 2009). In the Charnoz et al. scenario, the oldest craters on the moons would result from low-speed accretionary impacts. We thank the Cassini Data Analysis program for support.

References
Canup, R. M. (2010). Nature 468, 943
Charnoz, S.; Salmon, J., Crida, A. (2010). Nature 465, 752
Charnoz, S., et al. (2011). Icarus 216, 535
Crida, A.; Charnoz, S. (2012). Science 338, 1196
Dones, L., et al. (2009). In Saturn from Cassini-Huygens, p. 613
Kirchoff, M. R.; Schenk, P. (2010). Icarus 206, 485
Robbins, S. J.; Bierhaus, E. B.; Dones, L. (2015). Lunar and Planetary Science Conference 46, abstract 1654
(http://www.hou.usra.edu/meetings/lpsc2015/eposter/1654.pdf)
Zahnle, K.; Schenk, P.; Levison, H.; Dones, L. (2003). Icarus 163, 263

Notes

  • Can cratering records constrain moon ages?
    • see http://space.jpl.nasa.gov
    • small inner moons (and Mimas) interact strongly with rings — the so-called “ring moons”
      • migrated from outer edge of rings ~100 Myr
    • regular moons (Mimas-Iapetus) are (assumed?) primordial
    • transition is abrupt where tidal forces prevent formation
    • formation of moons from spreading rings:Charnoz et al. 2010,Canup 2010,Charnoz et al. 2011,Crida &Charnoz 2012
      • ring spreads viscously
      • outside Roche limit, formation
    • Lainey et al. 2012: dissipation stronger than thought
      • decreases timescale considerably
  • Impact rates
    • $R_{moon} = R_J \dfrac{R_S}{R_J} \dfrac{R_{moon}}{R_S}$
    • Crater scaling: diameter vs. velocity
    • impacts/$10^9$ yr: Mimas 8.5, Rhea 48
    • Mimas & Rhea counts: Robbins et al. 2015 (LPSC)
    • plot: #craters larger than D vs. D
    • Mimas: saturated up to $D \sim 20-45$ km
    • Rhea: saturated up to $D \sim 25$ km
  • Summary
    • Mimas: Craters are near saturation for diameters < 20 km
    • Rhea: saturation < 25 km
    • Ages may be underestimated

DDA 2015 – On the Spin-axis Dynamics of the Earth

This is one of a series of notes taken during the 2015 meeting of the AAS Division on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.

Gongjie Li (Harvard) (Duncombe award winner)

Abstract

The variation of a planet’s obliquity is influenced by the existence of satellites with a high mass ratio. For instance, the Earth’s obliquity is stabilized by the Moon, and would undergo chaotic variations in the Moon’s absence. In turn, such variations can lead to large-scale changes in the atmospheric circulation, rendering spin-axis dynamics a central issue for understanding climate. The relevant quantity for dynamically-forced climate change is the rate of chaotic diffusion. Accordingly, here we reexamine the spin-axis evolution of a Moonless Earth within the context of a simplified perturbative framework. We present analytical estimates of the characteristic Lyapunov coefficient as well as the chaotic diffusion rate and demonstrate that even in absence of the Moon, the stochastic change in the Earth’s obliquity is sufficiently slow to not preclude long-term habitability. Our calculations are consistent with published numerical experiments and illustrate the putative system’s underlying dynamical structure in a simple and intuitive manner. In addition, we examine if at any point in the Earth’s evolutionary history, the obliquity varied significantly. We find that even though the orbital perturbations were different in the past, the system nevertheless avoided resonant encounters throughout its evolution. This indicates that the Earth obtained its current obliquity during the formation of the Moon.

Notes

  • Obliquity $\cos \epsilon$ affects climate
    • Mars obliquity variations caused collapse of Martian atmosphere
  • Obliquity variations of a Moonless Earth
    • without Moon, $\epsilon$ is chaotic (Laskar et al. 1993)
      • geostrophic winds
    • but N-body sims: $\epsilon$ constrained to $\epsilon \lesssim 45$ deg — why?
    • Sun and planetary torques: spin precession rate, inclination variation
      • model as superposition of linear modes
      • resonance overlap: two connected chaotic zones — Laskar 1993, Morby 2000, Laskar 1996
    • average over primary resonances $\rightarrow$ secondary resonances
      • overlap of secondary resonances creates the chaotic bridge (Chirikov 1979)
    • Results
      • regular at $\ge 85$ deg
      • less chaotic in bridge
      • analytic and numerical are consistent
      • Li & Batygin 2014a
      • diffusion timescale 10 Myr in primary chaotic zones, 2 Gyr in the bridge
  • Pre-late heavy bombardment evolution of Earth’s obliquity
    • Li & Batygin 2014b
    • solar system starts more compact (Nice model)
    • study evolution of mode freqs and effects on Earth’s inclination
    • also, Moon was closer
    • two freqs match prior to LHB only if $\epsilon \ge 85$ deg
    • $\therefore$ Earth’s obliquity arose during the formation of the Moon

DDA 2015 – Recent dynamical evolution of Mimas and Enceladus

This is one of a series of notes taken during the 2015 meeting of the AAS Division on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.

Session: Moon Formation and Dynamics III

Maja Cuk (SETI Institute)

Abstract

Mimas and Enceladus are the smallest and innermost mid-sized icy moons of Saturn. They are each caught in a 2:1 orbital resonance with an outer, larger moon: Mimas with Tethys, Enceladus with Dione. This is where the similarities end. Mimas is heavily cratered and appears geologically inactive, while Enceladus has a young surface and high tidal heat flow. Large free eccentricity of Mimas implies low tidal dissipation, while Enceladus appears very dissipative, likely due to an internal ocean. Their resonances are very different too. Mimas is caught in a 4:2 inclination type resonance with Tethys which involves inclinations of both moons. Enceladus is in a 2:1 resonance with Dione which affects only Enceladus’s eccentricity. The well-known controversy over the heat flow of Enceladus can be solved by invoking a faster tidal evolution rate than previously expected (Lainey et al. 2012), but other mysteries remain. It has been long known that Mimas has very low probability of being captured into the present resonance, assuming that the large resonant libration amplitude reflects sizable pre-capture inclination of Mimas. Furthermore, Enceladus seems to have avoided capture into a number of sub-resonances that should have preceded the present one. An order of magnitude increase in the rate of tidal evolution does not solve these problems. It may be the time to reconsider the dominance of tides in the establishment of these resonances, especially if the moons themselves may be relatively young. An even faster orbital evolution due to ring/disk torques can help avoid capture into smaller resonances. Additionally, past interaction of Mimas with Janus and Epimetheus produce some of the peculiarities of Mimas’ current orbit. At the meeting I will present numerical integrations that confirm the the existence of these problems, and demonstrate the proposed solutions.

Notes

  • tidal rates $\dfrac{1}{a}\dfrac{d a}{d t}$: Mimas = 59, Enceladus = 23
  • numerical integrations — brute force
    • artificial migration
    • slow
  • the trouble with Mimas
    • Mimas and Tethys in inclination-type 4:2 MMR
    • inclination of both moons affected by the resonance
    • libration amp. of resonance is large, ~100 deg $\rightarrow$ primordial Mimas inclination — doesn’t work
    • eccentricity of Tethys has complex effects
    • Mimas-Tethys evolution rate: $\dfrac{da_{moon}}{dt} \propto \dfrac{R^5_{planet}}{a^{3/2}}$
  • introduce ad hoc ring torques — artificial torque on Prometheus
    • gives Tethys resonance a kick
    • $\therefore$ don’t take Mimas-Tethys resonance too seriously
  • …more ad hoc games…
  • rings-Janus-Mimas-Enceladus-Dione system evolution is very complex

DDA 2015 – On the in situ formation of Pluto’s small satellites

This is one of a series of notes taken during the 2015 meeting of the AAS Division on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.

Man Yin Woo (University of Hong Kong)

Abstract

The formation of Pluto’s small satellites – Styx, Nix, Keberos and Hydra remains a mystery. Their orbits are nearly circular (eccentricity $e = 0.0055$ or less) and near resonances and coplanar with respect to Charon. One scenario suggests that they all formed close to their current locations from a disk of debris, which was ejected from the Charon-forming impact. We test the validity of this scenario by performing N-body simulations with Pluto-Charon evolving tidally from an initial orbit at a few Pluto radii. The small satellites are modeled as test particles with initial orbital distances within the range of the current small satellites and damped to their coldest orbits by collisional damping. It is found that if Charon is formed from a debris disk and has low initial eccentricity, all test particles survive to the end of the tidal evolution, but there is no preference for resonances and the test particles’ final $e$ is typically > 0.01. If Charon is formed in the preferred intact capture scenario and has initial orbital eccentricity ~ 0.2, the outcome depends on the relative rate of tidal dissipation in Charon and Pluto, $A$. If $A$ is large and Charon’s orbit circularizes quickly, a significant fraction of the test particles survives outside resonances with $e \gtrsim 0.01$. Others are ejected by resonance or survive in resonance with very large $e$ (> 0.1). If $A$ is small and Charon’s orbit remains eccentric throughout most of the tidal evolution, most of the test particles are ejected. The test particles that survive have $e \gtrsim 0.01$, including some with $e \gt 0.1$. None of the above cases results in test particles with sufficiently low final $e$.

This work is supported in part by Hong Kong RGC grant HKU 7030/11P.

Notes

  • Pluto satellite system
    • 5 known
    • Charon dominant
    • all nearly coplanar
    • all nearly circular
    • all near MMR with Charon
    • Brozovic et al. 2015
  • Formation scenarios
    • forced resonant migration
      • Ward & Canup 2006
      • Nix & Hydra formed in same giant impact that formed Charon
      • ruled out by Lithwick & Wu
    • multi-resonance capture
      • unlikely (Cheng et al. 2014)
    • collisional capture of planetesimals
      • Lithwick & Wu 2008, Dos Santos et al. 2012
      • ruled out: capture time << collisional timescale, also Walsh & Levison 2015
    • in situ formation
      • Kenyon & Bromley 2014
      • giant impact produced debris ring
      • problem: outward tidal evolution of Charon
  • Solving the migration problem
    • forced eccentricity — Leung and Lee 2013
    • for $e_C = 0.24$, $e_f \sim 0.01$ to $0.02$ for test particles (small moons)
    • integrate two tidal models, constant $\Delta t$ and constant $Q$
    • For constant $\Delta t$, no preference for resonances and $e \gt 0.01$
  • Conclusion: it is unlikely that all the small satellites formed close to their current position

DDA 2015 – Rotational and interior models for Enceladus II

This is one of a series of notes taken during the 2015 meeting of the AAS Division on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.

Radwan Tajeddine (Cornell)

Abstract

We will discuss the underlying dynamical models and the consequent interior models that pertain to our discovery of a forced rotational libration for Saturn’s moon Enceladus (Thomas et al. 2015).

Despite orbital variations that change the mean motion on timescales of several years owing to mutual satellite interactions, the rotation state of Enceladus should remain synchronous with the varying mean motion, as long as damping is as expected (Tiscareno et al. 2009, Icarus). Taking that dynamically synchronous rotation as the ground state, we construct a model that naturally focuses on the physically interesting librations about the synchronous state that occur on orbital timescales. We will discuss the differences between the method used here and other dynamical methods (e.g., Rambaux et al. 2010, GRL; cf. Tajeddine et al. 2014, Science), and we will review the rotation states (whether known or predicted) of other moons of Saturn.

We will also describe our measurements of the control point network on the surface of Enceladus using Cassini images, which was then used to obtain its physical forced libration amplitude at the orbital frequency. The fit of Cassini data results in a libration amplitude too large to be consistent with a rigid connection between the surface and the core, ruling out the simplest interior models (e.g., homogeneous, two-layer, two-layer with south polar anomaly). Alternatively, we suggest an interior model of Enceladus involving a global ocean that decouples the shell from the core, with a thinner icy layer in the south polar region as an explanation for both the libration (Thomas et al. 2015) and the gravity (Iess et al. 2014, Science) measurements.

Notes

  • Libration measurement
    • 3D reconstruction of coords of a network of control point (fiducial satellite surface points — e.g. craters)
    • most of Enceladus’s orbit was covered
    • Thomas et al. 2015
    • minimize RMS residual $\rightarrow 0.120 \pm 0.014$ deg
  • Solid models
    • core plus two-layer in hydro.equilib. plus south polar sea
      • measured libration amplitude rules this out
    • decoupled shell from the core (indep.librations)
      • consistent with observed libration amplitude if shell thickness 21-26 km and ocean thickness 26-31 km
  • Gravity data
    • suggests a local mass anomaly — interpreted as ocean thicker under south pole

DDA 2015 – Rotational and interior models for Enceladus I

This is one of a series of notes taken during the 2015 meeting of the AAS Division on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.

Session: Moon Formation and Dynamics II

Matthew S. Tiscareno (Cornell)

Abstract

We will discuss the underlying dynamical models and the consequent interior models that pertain to our discovery of a forced rotational libration for Saturn’s moon Enceladus (Thomas et al. 2015).

Despite orbital variations that change the mean motion on timescales of several years owing to mutual satellite interactions, the rotation state of Enceladus should remain synchronous with the varying mean motion, as long as damping is as expected (Tiscareno et al. 2009, Icarus). Taking that dynamically synchronous rotation as the ground state, we construct a model that naturally focuses on the physically interesting librations about the synchronous state that occur on orbital timescales. We will discuss the differences between the method used here and other dynamical methods (e.g., Rambaux et al. 2010, GRL; cf. Tajeddine et al. 2014, Science), and we will review the rotation states (whether known or predicted) of other moons of Saturn.

We will also describe our measurements of the control point network on the surface of Enceladus using Cassini images, which was then used to obtain its physical forced libration amplitude at the orbital frequency. The fit of Cassini data results in a libration amplitude too large to be consistent with a rigid connection between the surface and the core, ruling out the simplest interior models (e.g., homogeneous, two-layer, two-layer with south polar anomaly). Alternatively, we suggest an interior model of Enceladus involving a global ocean that decouples the shell from the core, with a thinner icy layer in the south polar region as an explanation for both the libration (Thomas et al. 2015) and the gravity (Iess et al. 2014, Science) measurements.

Notes

  • Enceladus
    • 2nd largest Saturnian moon
    • Plumes — salty jets — observed by Cassini
    • What is under the surface?
    • Rotational parameters $\rightarrow$ interior structure
  • Forcedlibrations
    • same period as orbital
    • nat. freq. $\omega_0 \approx n \sqrt{3 (B-A)/C}$
    • near-spherical: moon always points at empty focus (synchronous)
    • elongated: moon would always point at Saturn
    • Enceladus axis oscillates around empty focus (synchronous rotation)
    • as $\dfrac{B-A}{C} \rightarrow \dfrac{1}{3}$, resonance (Tiscareno et al. 2009)
    • but Enceladus $\dfrac{B-A}{C} \ll \dfrac{1}{3}$
      • Enceladus libration $0.120\pm0.014$ deg
      • rules out rigid connection between surface and core
      • hence, some kind of global subsurface ocean
  • Mean motion variations
    • Enceladus resonant arguments from interaction with Dione:
      $ILR_D = \lambda_E\, – 2 \lambda_D + \varpi_E$ (librating)
      $CIR_D = \lambda_E\, – 2 \lambda_D + \Omega_D$ (circulating)
      $CER_D = \lambda_E\, – 2 \lambda_D + \varpi_D$ (circulating)
    • As long as damping is sufficiently strong, synchronous rotation maintained
      • damping must be $\gamma_{\pi/2} = \dfrac{2 e}{1\, – \left(\dfrac{n}{\omega_0}\right)^2} \Rightarrow \tau \approx 1.0\,Q\ \mathrm{days}$
      • but $10 \lt Q \lt 100$ days
    • rot. rate varies with the CER and ILR freqs
      • not really “librations”
      • maintaining synch. rot., while the mean motion varies quasiperiodically
  • Rotational models
    • Global Fourier components have limited usefulness
    • MM variation more complex than a few periodic terms
    • Define rot.statewrt Saturn
      • base state: synch rot (expected for low triaxiality)
      • accounts for MM variation
      • easy to generate a range of kernels for many vals of $\gamma$
    • Tiscareno 2015
    • deflect $\psi(t) = (2 e+\gamma)\sin M$
    • generate kernels of $\psi(t)$ for a wide range of $\gamma$ values, check for best control-point resids
    • dissipation?

DDA 2015 – Forced libration of tidally synchronized planets and moons

This is one of a series of notes taken during the 2015 meeting of the AAS Division on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.

Session: Moon Formation and Dynamics I

Valeri Makarov (for Bryan Dorland) (USNO)

Abstract

Tidal dissipation of kinetic energy, when it is strong enough, tends to synchronize the rotation of planets and moons with the mean orbital motion, or drive it into long-term stable spin-orbit resonances. As the actual orbital motion undergoes periodic acceleration due to a finite orbital eccentricity, the spin rate oscillates around the equilibrium mean value too, giving rise to the forced, or eccentricity-driven, librations. We contend that both the shape and amplitude of forced librations of synchronous viscoelastic planets and moons are defined by a combination of two different types of perturbative torque, the tidal torque and the triaxial torque, the latter related to a permanent deformation of the distribution of mass from a perfect rotational symmetry. Consequently, forced librations can be tidally dominated (e.g., Io and possibly Titan) or deformation-dominated (e.g., the Moon) depending on a set of orbital, rheological, and other physical parameters. With small eccentricities, for the former kind, the largest term in the libration angle can be minus cosine of the mean anomaly, whereas for the latter kind, it is minus sine of the mean anomaly. The shape and the amplitude of tidal forced librations determine the rate of orbital evolution of synchronous planets and moons, i.e., the rate of dissipative damping of the semimajor axis and eccentricity. The known super-Earth exoplanets can exhibit both kinds of libration, or a mixture of thereof, depending on, for example, the effective Maxwell time of their rigid mantles. Our approach can be extended to estimate the amplitudes of other libration harmonics, as well as the forced libration in non-synchronous spin-orbit resonances.

Notes

  • Numerical sim of Lunarspinlibrations in longitude with polar torques exerted by Earth
    • spectrum of harmonics
    • dominant term: $\dfrac{\dot{\theta}}{n}\, – 1 \propto -\cos M$
  • Geometry offorcedlibrations
    • longest axis of planet tries to align with line of centers (but can’t)
    • triaxial torque (Eckhardt 1981) $\ddot{\theta}\approx 6 n^2 \dfrac{B-A}{C^3}\left[e\, – \dfrac{31}{16}e^2 + O\left(e^5\right)\right] \sin M$
  • Problem: Io
    • striking difference with moon
    • dominant term now $\dfrac{\dot{\theta}}{n}\, – 1 \propto +\sin M$
  • Tidal torques
    • torque = triaxial + tidal
    • tidal = secular + periodic
      • secular can dominate in the “linear” regime
        • plot: $\ddot{\theta}$ vs. $\dfrac{\dot{\theta}}{n}\, – 1$
      • It’s not the amplitude that’s important, but the gradient, which provides a subtle but powerful feedback.
      • tiny interval of pert freqs where secular torque can be important
  • Problem: doesn’t work for Moon
    • because Moon is not sitting on that linear portion of $\dfrac{\dot{\theta}}{n}\, – 1$
    • $\therefore$ need further harmonic terms