Is Trump’s Lead Significant?

Polling results among Republican voters over the past three months [click to embiggen]
Snapshot of polling results among Republican voters over the past three months [click to embiggen]
At the moment, The Donald leads nationally among Republicans, with 23.5% favorability. Roughly 25% of polled Republicans currently favor Trump over the rest of the Republican Field of Clowns. People argue that 25 percent is not terribly impressive. Are they right?

You have to interpret more carefully than that. Roughly 25% of polled Republicans currently prefer Trump over the others. That last bit is important: that many other Clowns are vying for the prize matters in the interpretation of Trump’s 23.5 percent.

Since there are fifteen Clowns, an even distribution of favorability would be 6.7% per Clown, so 23.5% is in fact a pretty big outlier. How big? The mean of this favorability distribution is $\mu = 6.0$%, pretty close to the 6.7% expectation. The standard deviation of this distribution of Clown favorability ratings is $\sigma = 5.6$%. Trump’s $p = 23.5$% therefore is a $\Delta = \dfrac{\left|p\, – \mu\right|}{\sigma} = 3.1$-sigma outlier, which is statistically significant. What this means is that the chance of that being just a statistical fluke (i.e., the likelihood that a random choice from among a Gaussian distribution with $\mu = 6.0$% and $\sigma = 5.6$% would land you at 23.5% or higher) is $1 – \mathrm{erf} \left(\dfrac{\Delta}{\sqrt{2}}\right) = 0.0018 = 0.18$ percent.

In the physical sciences, a result lying three or more standard deviations away from the null hypothesis is the typical bar for publishable significance. $\mathrm{erf}$ is the error function:

$$\mathrm{erf}(z) = \dfrac{1}{\sqrt{\pi}} \int_{-z}^z e^{-t^2} dt$$

and is the probability of a random variate lying between $-z$ and $+z$ in a distribution with zero mean and standard deviation $\frac 12$. Now, the 0.2% result above would hold if the favorability distribution were a normal (i.e., Gaussian) distribution, which it almost certainly is not. But the conclusions should correspond closely enough to reality to use as a rough guide.

The next candidate down is Carson at 10.3% (and Bush at 9.8% — essentially the same). Carson is only 0.8 sigma out from the mean, which corresponds to the likelihood of his favorability rating being where it is or higher due to random chance is 45 percent (for Bush it’s 50%).

Conclusion: Carson and Bush’s $\sim10$% scores represent, for each, a coin toss in terms of preference. In other words, for Republicans, Bush and Carson are both meh for now. But Trump’s 23.5% favorability, given this particular Field of Clowns, signifies a clear and large preference.

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 ¾ 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