Download Interstellar Levy Flights

Interstellar Levy Flights
Collaborators:
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
Statistics
Application to California Lotto
Interstellar Games of Chance
Imaging
Pulse Shapes
Gamble and Win in Either Game!
“Gauss”
“Cauchy”
You are given 1¢
You are given 2¢
Flip 1¢: win another 1¢ Flip 1¢: double your
each time it comes up
winnings each time it
“heads”
comes up “heads”
Play 100 flips
You must walk away (keep
winnings) when it comes
up “tails” (100 flips max)
Value = ∑ Probability($$)($$) = $0.50
… for both games
Note: for Cauchy, > $0.25 of “Value” is from payoffs larger than
the US Debt.
Symmetrize the Games
Gauss: lose 1¢ for each “tails”
Cauchy: double your loss each “tails” -- until
you flip “heads”
“Core”
The resulting
distributions are then
approximately
Gaussian or
Cauchy (=Lorentzian)
“Halo” or
“Tail”
Play Either Game Many Times
The distributions of net outcomes will approach
Levy stable distributions.
“Levy Stable” => when convolved with itself,
produces a scaled copy.
In 1D, stable distributions take the form:
P b($)=∫ dk ei k ($) e-|k|
Gauss will approach a Gaussian distribution b=2
Cauchy will approach a Cauchy distribution b=1
Fine Print
The Central Limit Theorem says: the outcome
will be drawn from a Gaussian distribution,
centered at N$0.50, with variance given by….
To reach that limit with
”Cauchy”, you must play
enough times to win the top
prize.…and win it many times
(>>1033) plays.
California Lotto Looks Like Levy
Prizes are distributed
geometrically
Probability~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 Consequences
for 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*.
b=1
b=2
* for a scattered point source.
•Observations of a scattered point
source should give us the
distribution.
Simulated VLB Observation of
Pulsar B1818-04
It Has Already Been Done
Desai & Fey (2001) found that images of heavily-scattered sources in Cygnus
did not resemble Gaussian distributions: they had a “cusp” and a “halo”.
Actually, radio interferometry measures the Fourier transform of the
image -- usually confusing -- but convenient for Levy distributions!
Excess flux at long
baseline: sharp “cusp”
Excess flux at short
baseline: big “halo”
Best-fit Gaussian model
*”Rotundate” baseline is scaled to account for elongation of the source (=anisotropic scattering).
Intrinsic structure of these complicated sources might create a “halo”
– but probably not a “core”!
Pulses Broaden as they Travel
Through Space
Mostly because paths with lots of bends take longer!
The Pulse Broadening Function (ImpulseResponse Function)
is Different for Gauss and Cauchy
Dotted line: b=2
Solid line: b =1
Dashed line: b =2/3
(Scaled to the same maximum and width at half-max)
For Cauchy,
many paths
have only
small
deflections –
and some
have very
large ones –
relative to
Gauss
Go Cauchy!
Solid curve: Best-fit model b=1
Dotted curve: Best-fit model b=2
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 thinscreen model -- for
cases we’ve tested.
Is the Problem Solved?
• For Cauchy-vs-Gaussvs-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 ideas
The travel time for the
Cauchy arrives
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.

Gauss arrives
Pulse at 1230 MHz
(no one’s keeping track of time)
Pulse at 880 MHz
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.
Levy Flights
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:
b
i
k
Dq
-|k|
P b(Dq)=∫ dk e
e
P b(Dq) Gaussian for b2
Rare, large deflections
dominate the path:
a “Levy Flight”.
Klafter, Schlesinger, & Zumofen 1995, PhysToday
JP Nolan: http://academic2.american.edu/~jpnolan/