Interstellar Levy FlightsCollaborators:

Stas Boldyrev (U Chicago: Theory)Ramachandran (Berkeley: Pulsar Observing)Avinash Deshpande (Arecibo, Raman Inst: More Pulsars)Ben Stappers (Westerbork: Crucial pulsar person)

Outline of Talk:Gambling with Pennies

StatisticsApplication to California Lotto

Interstellar Games of ChanceImagingPulse Shapes

Gamble and Win in Either Game!“Gauss”

You are given 1¢

Flip 1¢: win another 1¢ each time it comes up “heads”

Play 100 flips

“Cauchy”You are given 2¢

Flip 1¢: double your winnings each time it comes up “heads”

You must walk away (keep winnings) when it comes up “tails” (100 flips max)

Value = ∑ Probability($$)$$) = $0.50

Note: for Cauchy, > $0.25 of “Value” is from payoffs larger than the US Debt.

… for both games

Symmetrize the Games

“Core”

“Halo” or “Tail”

The resulting distributions are then approximately Gaussian or Cauchy (=Lorentzian)

Gauss: lose 1¢ for each “tails”

Cauchy: double your loss each “tails” -- until you flip “heads”

Play Either Game Many TimesThe distributions of net outcomes will approach Levy stable distributions.“Levy Stable” => when convolved with itself, produces a scaled copy.

In 1D, stable distributions take the form:

P ($)=∫ dk ei k ($) e-|k|

Gauss will approach a Gaussian distribution =2Cauchy will approach a Cauchy distribution =1

To reach that limit with ”Cauchy”, you must play enough times to win the top prize.…and win it many times

(>>1033) plays.

The Central Limit Theorem says: the outcome will be drawn from a Gaussian distribution, centered at N$0.50, with variance given by….

Fine Print

California Lotto Looks Like LevyPrizes are distributed

geometricallyProbability~1/($$)Top prize dominates all

moments:∑ Probability($$)$$)N≈(Top $)N

When waves travel through random media,do they choose deflection angles via Gauss

or Cauchy?

• People have worked on waves in random media since the 60’s, (nearly) always assuming Gauss.

Does it matter which we choose?

Are there observable effects?

Are there media where Cauchy is true?

Do observations tell us about the medium?

Isn’t this talk about insterstellar travel?

Interstellar Scattering of Radio Waves

• Long distances (pc-kpc: 1020-1023 cm)

• Small angular deflection (mas: 10-9 rad)

• Deflection via fluctuations in density of free electrons (interstellar turbulence?)

• Close connections to:– Atmospheric “seeing”– Ocean acoustic propagation

3 Observable Consequencesfor Gauss vs Cauchy

• Shape of a scattered image (“Desai paradox”)• Shape of a scattered pulse (“Williamson

Paradox”)• Scaling of broadening with distance (“Sutton

Paradox”)

Let’s Measure the Deflection by Imaging!

•At each point along the line of sight, the wave is deflected by a random angle.•Repeated deflections converge to a Levy-stable distribution of scattering angles.•Probability(of deflection angle) –is– the observed image*.

* for a scattered point source.

•Observations of a scattered point source should give us the distribution.

Simulated VLB Observation of Pulsar B1818-04

=1

=2

Desai & Fey (2001) found that images of heavily-scattered sources in Cygnus did not resemble Gaussian distributions: they had a “cusp” and a “halo”.

It Has Already Been Done

Actually, radio interferometry measures the Fourier transform of the image -- usually confusing -- but convenient for Levy distributions!

Intrinsic structure of these complicated sources might create a “halo” – but probably not a “core”!

Best-fit Gaussian model

Excess flux at long baseline: sharp “cusp”

Excess flux at short baseline: big “halo”

*”Rotundate” baseline is scaled to account for elongation of the source (=anisotropic scattering).

Pulses Broaden as they Travel Through Space

Mostly because paths with lots of bends take longer!

The Pulse Broadening Function (Impulse-Response Function)

is Different for Gauss and Cauchy

For Cauchy, many paths have only small deflections – and some have very large ones – relative to Gauss

Dotted line: =2Solid line: =1Dashed line: =2/3

(Scaled to the same maximum and width at half-max)

Go Cauchy!Observed pulses from pulsars tend to have sharper rise and more gradual fall than Gauss would predict.Williamson (1975) found that models with all the scattering material concentrated into a single screen worked better than models for an extended medium. Cauchy works at least as well as a thin-screen model -- for cases we’ve tested.

Solid curve: Best-fit model =1Dotted curve: Best-fit model =2

Is the Problem Solved?• For Cauchy-vs-Gauss-

vs-Thin Screen, statistical goodness of fit is about the same

• Cauchy predicts a long tail – which wraps around (and around) the pulse – could we detect that? – Or at least unwrap it?

Horizontal lines show zero-level

Pulse broadening is greatest at low observing frequency

(Because the scattering material is dispersive.)

We fit for pulse shape at high observing frequency, and for degree of broadening at low frequency.Fits at intermediate frequency (with parameters taken from the extremes) favor Cauchy – though neither fits really well!

From Westerbork: Ramachandran, Deshpande, Stappers, Gwinn

We have more nifty ideasThe travel time for the peak of a scattered pulse is pretty different for Gauss and Cauchy (sorta like the most common payoff in the games “Gauss” and “Cauchy”)Without scattering, the pulse would arrive at the same time as the pulse does at high frequency*.

*If we’ve corrected for dispersion.

We can measure that if we do careful pulse timing.

Pulse at 880 MHz

Pulse at 1230 MHz(no one’s keeping track of time)

Gauss arrivesCauchy arrives

Have we learned anything?

$ The lottery can be a good investment, depending on circumstances.

But, you are not likely to win the big prize.The lottery (and more boring games) have

much to teach us about wave propagation. We can tell what game is being played by

examining the outcomes carefully.

Fact: If the distribution for the steps has a power-law tail, then the result is not drawn from a Gaussian.

It will approach a Levy distribution: P ()=∫ dk ei k e-|k|

P () Gaussian for 2

Rare, large deflections dominate the path: a “Levy Flight”.

Klafter, Schlesinger, & Zumofen 1995, PhysTodayJP Nolan: http://academic2.american.edu/~jpnolan/

Levy Flights