A6523 Signal Modeling, Statistical Inference and

Data Mining in Astrophysics

Spring 2015 http://www.astro.cornell.edu/~cordes/A6523

Lecture 6 –  A puzzle: PDF of Fourier-based power spectrum –  Stochastic processes: definitions and characterization –  Fundamental relations between 2nd order statistics –  A simple Bayesian inference: Poisson process

Application: –  Fresnel-Fraunhofer diffraction simulations = convolution problem

Notes on web site: (soon) - StochasticProcesses

- Generating correlated random variables - Correlation functions as diagnostic tool - Structure functions and Allan variance.tex - Wave propagation through random media

A Puzzle about Power Spectra

•  We found that a noise-only spectrum (or of a very weak signal + noise) that the PDF of the power spectrum is a one-sided exponential.

•  A feature of this spectrum is that the errors are 100%: rms / mean = 1.

•  This statement is independent of the data set length, N = number of samples.

•  But don’t larger data sets mean better statistics? What gives?

Stochastic Processes on One Page

1. Stochastikos = proceeding by guesswork, literally skillful in aiming

2. X(y, ζ) = sequence of random variables ordered by y and associated with an ensemble {ζ}.

(y continuous or discrete)

3. Ergodic ← Strictly stationary ← Wide sense stationary (WSS) ← Stationary increments ← Nonstationary

4. WSS = stationarity up to second moments

5. Sample averages �= ensemble averages (general) Ergodic: equivalence as sample → ∞6. Second order moments (general) “t” can be time, spatial, wavelength, frequency, . . .

(a) Autocorrelation function: RX(t1, t2) ≡ �X(t1)X∗(t2)�Autocovariance function: X(t1,2) → δX(t1,2) = X(t1,2)− �X(t1,2)�

(b) Crosscorrelation function: RXY (t1, t2) ≡ �X(t1)Y ∗(t2)�Crosscovariance function: CXY (t1, t2) ≡ �δX(t1)δY ∗(t2)�i. Orthogonal: RXY (t1, t2) = 0 for all t1, t2

ii. Uncorrelated: CXY (t1, t2) = 0 for all t1, t2(c) Structure function: DX(t1, t2) = �[X(t1)−X∗(t2)]2�

7. WSS processes:

(a) ACF, ACV, CCF, CCV, SF: depend only on argument differences, τ = t2 − t1 (“time lag”)

(b) Wiener-Kinchin theorem: Power spectrum = Fourier transform of the ACF: S(ω) =�dt e−iωτRX(τ )

8. Stationary increments: SF: DX(t1, t2) → DX(τ ) even if ACF, etc. do not.

9. Second-order quantities ubiquitous in modeling, simulations, mining, and inference.

10. Third moment and bispectrum: γX(t1, t2, t3) = �X(t1)X(t2)X(t3)�3rd−order stationary

−→τ1=t2−t1,τ2=t3−t1)

γ(τ1, τ2)

Bispectrum: S(ω1,ω2) =�dτ1 e−iω1τ1

�dτ2 e−iω2τ2 γX(τ1, τ2)

1

Wiener-Kinchin theoremThe power spectrum S(f ) is simply the Fourier transform of the autocorrelation function (some-times the autocovariance function).

S(f ) =�dτe2πifτRX(τ ).

As such it (as well as the ACF) is an ensemble average quantity. With finite measurements ofrealization(s) of a process, the best we can do is to estimate the power spectrum. Properties ofS(f ) are:

1. S(f ) ≥ 0.

2. Real since R(τ ) hermitian.

3. Is the distribution of the second moment (or variance) in frequency space.

4. Partakes of the analogy S(f ) : R(τ ) :: fX(x) : ΦX(ω).In some contexts (e.g. maximum entropy spectral estimation), it is convenient to view thepower spectrum as a probability distribution of frequency components. In some Bayesiantreatments, the PDF of the frequency is explicitly calculated.

7

Correlation Functions and Power SpectraRecall from Fourier transform theorems for deterministic functions we have the relationships:

f (t) FT⇐⇒ F (f )

irreversible ↓ ↓ irreversible

�dt f (t)f ∗(t + τ ) FT

⇐⇒ |F (f )|2

For stochastic processes the situation is different. We need to distinguish the power spectrumof a realization from the ensemble-average (true) power spectrum:

x(t) FT⇐⇒ X(f )

irreversible ↓ ↓ irreversible

�x(t)x∗(t + τ )� FT⇐⇒ S(f ) =

�|X(f )|2

�

8

Utility of Correlation Functions

1. As a means for estimating power spectra (e.g. a correlator + WK theorem).

2. For establishing characteristic time scales in time series (width of the ACF or ACV).

3. For testing whether two time series are related (CCF, CCV).

4. As a basis for calculating correlation matrices used in estimation, principal componentanalysis, etc.

5. As the measurement basis in interferometry (CCF of signals for different antennas, the“visibility” function).

6. etc.

14

Correlation Function Example

Consider the autocorrelation function (ACF) of a zero-mean WSS process. We want to considerhow the ACF converges as well as understand what the ACF actually quantifies.

The figure shows a time series and its ACF along with an ACF averaged over 10 realizations ofthe time series.

Figure 3: Top panel: Time series of Gaussian noise with unity variance and a correlation time of 21 steps. Bottom panel: ACF of thetime series in the top panel along with the ACF averaged over 10 realizations of the time series.

15

The time series was created by taking a realization of white, Gaussian noise and smoothing itwith a boxcar filter with width of 21 samples.

Features of the ACF include:

1. The maximum at zero lag has a value equal to the variance in the time series (set to unity).

2. The feature that maximizes at zero lag is the same in both ACFs.

3. The decay from the maximum is on a time scale ∼ 20 steps, which is order of the smoothingtime used to create the time series.

4. There are statistical variations centered on zero correlation. These variations are larger inthe ACF calculated from single-realization and are estimation errors in the ACF.

The width of the persistant feature in the ACF is the autocorrelation time of the process, whichwe define as Wy. This quantifies the time interval over which the process decorrelates.

16

The estimation error in the ACF at larger lags is determined by the number of independentfluctuations Ni in the time series used. In a single time series of length T , this number is

Ni ≈ T/Wy.

For the example, Ni ≈ 1024/21 ≈ 50. The estimation error in the ACF for a single time serieswill be

δCy ≈ Cy(0)/√Ni ≈ 0.14

so we expect variations about zero at approximately this level.

For the 10-realization average, we expect the estimation errors to decrease by another factor of1/√10 to about 0.045.

17

Generating Correlated Random Variables

Bivariate Gaussian Distribution

The joint (bivariate) PDF for X1,2 is

fX1X2(x1, x2) =1

2π

1

(1− ρ2)1/2exp

− 1

2(1− ρ2)

x21σ21+

x22σ22− 2ρ

x1x2σ1σ2

A more useful of writing this PDF is to use the column vector X = col (X1, X2) and thecovariance matrix

C =

σ21 σ1σ2ρ

σ1σ2ρ σ22

to write (using † to denote transpose)

fX(X) =1

2π(detC)1/2exp

−1

2X†C−1X

.

The bivariate Gaussian is used frequently in likelihood and Bayesian estimation to displaycontours for parameter estimates.

1

way        ^  

Figure 1: Scatter plots of two random variables X1,2 that have a joint Gaussian PDF for four different values of correlation coefficient, ρ.

2

Generating Correlated Random Variables

Consider a (pseudo) random number generator that gives numbers consistent with a 1D Gaus-

sian PDF ≡ N(0, σ2) (zero mean with variance σ2).

How do we create two Gaussian random variables (GRVs) from N(0, σ2) but that are correlated

with correlation coefficient ρ?

So we want

ρX1,X2 =�(X1 − �X1�) (X2 − �X2�)�

σ2.

Define Y1, Y2 as independent N(0, σ2) GRVs, so ρY1,Y2 = 0 and let

X1 = aY1 + bY2

X2 = cY1 + dY2.

Since the means of all variables are zero, we have

�X1X2� = �(aY1 + bY2)(cY1 + dY2)�= ac�Y 2

1 � + bd�Y 22 � + (ad + bc)�Y1Y2�

= (ac + bd)σ2

Therefore

ρX1X2 =�X1X2�

σ2= ac + bd (1)

3

We also want�X2

1� = (a2 + b2)σ2 = σ2

�X22� = (c2 + d2)σ2 = σ2

so

a2 + b2 = c2 + d2 = 1 (2)

A natural solution is to use a = cosφ b = sinφ

c = sinφ d = cosφ.

Then the constraint equations (1) and (2) are satisfied and

ac + bd = ρX1X2 = 2 cosφ sinφ = sin 2φ

soφ =

1

2sin−1 ρX1X2.

4

(2)  

Autocorrelation function of a random walk

Poisson Processes

If we throw points onto the time axis randomly but with a uniform average rate λ the probabilityof having k points in an interval [0, T ] is

P (k) =e−λt(λt)k

k!(1)

Note this is the limiting case of a binomial distribution

P (k) =�n

k

�

pk(1− p)n−k

where p is the probability of having one point in the interval [0, t] when n points occur in a largerinterval [0, T ]; i.e. p = t/T . In the limit where n → ∞ so that p → 0 while λ ≡ np = constant> 0, the Poisson expression follows from the binomial probability.

The mean and second moment of k are

�k� = λt

�k2� = �k�2 + λt.

1

Define a random process x(t) =number of points in [0, t]. Then

P{x(t)} =e−λt(λt)x

x!

The mean and second moment of x(t) are simply

�x(t)� = λt

�x2(t)� = λt + (λt)2

We can calculate the autocorrelation function of x as follows:

Consider the process at two particular times, x(t1) and x(t2):

x(t1) is the number of points in [0, t1] and x(t2) is the number of points in [0, t2].

Assume that t1 < t2 and write

x(t2) = x(t1)� �� �number of points in [0,t1]

+ [x(t2)− x(t1)]� �� �number of points in [t1,t2]

Since the interval [t1, t2] does not overlap with the interval [0, t1], the quantity x(t2) − x(t1) isstatistically independent of x(t1). This helps us calculate the autocorrelation!

2

In particular we use statistical independence to write

�x(t1)[x(t2)− x(t1)]� = �x(t1)��[x(t2)− x(t1)]�.

Then the autocorrelation function becomes

Rx(t1, t2) = �x(t1)x(t2)�= �x(t1) [x(t1) + x(t2)− x(t1)]�= �x2(t1)� + �x(t1)� [�x(t2)� − �x(t1)�]= λt1 + (λt1)

2 + λt1 [λt2 − λt1]

ThereforeRx(t1, t2) = λt1 + λ2t1t2 t1 < t2

or

Rx(t1, t2) = λ2t1t2 + λt1U(t2 − t1) + λt2U(t1 − t2),

where U(t) is the unit step function, as always.

3

Poisson impulses: Define the process z(t) as the derivative of x(t):

z(t) =dx

dt=

d

dt

∞�

j=1U(t− tj) = z(t) =

�

jδ(t− tj)

where the tj are distributed in [0, t] according to the Poisson distribution.

The mean of z(t) can be found as the derivative of �x�,

�z(t)� =d�x�dt

= λ

and the ACF as the second derivative of the ACF of x(t)

Rz(t1, t2) =∂2

∂t1∂t2Rx(t1, t2)

= λ2 + λδ(t1 − t2)

4

Shot noise:

Suppose we pass z(t) through a linear filter, h(t)

z(t) −→ h(t) −→ N(t)

Then

N(t) = h(t) ∗ z(t) = h(t) ∗ �

jδ(t− tj) =

�

jh(t− tj).

Individual “shots” can correspond to individual charges in a semi-conductor or can be eventstaking place in an astronomical source.

Moments:

�N(t)� =� �

dt� h(t�)z(t− t�)�=

�dt� h(t�) �z(t− t�)�

� �� �λ

= λ�dt� h(t�)

RN(τ ) = �N(t)N(t + τ )� =� �

dt1�dt2 h(t1)h(t2)z(t− t1)z(t + τ − t2)

�

We can use Campbell’s theorem to obtain the autocorrelation function:

RN(τ ) =��dt1 dt2 h(t1)h(t2) �z(t− t1)z(t− t2)�� �� �

λ2+λδ(τ+t1−t2)

= �N(t)�2 + λ�dt1 h(t1)h(t1 + τ )

� �� �call this Rh(τ)

RN(τ ) = �N(t)�2 + λRh(τ )

5

Sampling of wavefields: aka imaging

Modeling, simulation, FTs, correlation functions

Imaging System, Spectral Analysis Comparison

Aperture A(x)

•  Point spread function (PSF) = |FT{A}|2

•  Circular aperture: PSF = Airy function

•  Optical Transfer Function (OTF) = FT{PSF}

•  Modulation Transfer Function (MTF) = |OTF|

•  Commonly used basis functions: Zernike polynomials (closely connected to aberrations)

Time series A(t)

•  Power spectrum |FT{A}|2

•  Finite time series: Frequency

response = sinc function [spectral window W(f)]

•  Autocorrelation window w(τ) = FT{W(f)}

•  No common analog

•  Sinusoids, exponentials

Utility of Structure Functions in Wave Propagation

Modeling and simulations of propagation through random phase screens are commonly done for studiesof propagation through the atmosphere, the ionosphere, the interplanetary medium, and the interstellarmedium. Though these media are quite differently physically, the underlying mathematics of wavepropagation is the same.

Consider the simple case of a plane wave propagating through a thin screen that changes the electro-magnetic (EM) phase randomly:

Plane Wave ei(kz−ωt)

Phase Screen Plane eiφ(x)

Distorted Wave Fronts ei(kx−ωt+φ(x))

Diffraction Pattern I(x)

1

Fresnel Diffraction

Huygens’ principle says that each point within an aperture A(x) illuminated by a plane wave radiates

spherical waves. At a position x the scalar electric field is given by the Kirchoff Diffraction Integral

(KDI)

ε(x,λ) =eikD

λD

�dx�A(x�) eiΨ(x−x�)

Ψ(x,x) =k

2D|x− x�|2

k = 2π/λ

k/D =1

λD=

1

r2F=

1

(Fresnel scale)2

Can be recast as a convolution problem:

ε(x,λ) ∝ A(x) ∗K(x)

K(x) = e12

�xrF

�2

= Fresnel function

• Locations near the aperture: Fresnel diffraction

• Far from aperture: Fraunhofer diffraction

• Transition distance = Fresnel distance = DF(size of aperture)

λ

2

Image  credit:  h4p://cronodon.com/Atomic/Photon.html  

OPTICAL INFORMATION PROCESSING/ APRIL 2011 1

Odak: an open-source library for wave propagationand Fourier optics calculations

Kaan Aksit

Abstract—This paper introduces an implementation of an

open-source library implemented under Python program-

ming language [1] that provides built-in functions for wave

propagation and Fourier optics calculation. Library provides

callable functions for Fraunhofer diffraction and Fresnel

diffraction under odak.diffractions class. Library contains

also a set of pre-defined aperture types under odak.apertureclass. In the near-future it is planned to implement a

graphical user interface for end-user. Section IV provides

a justification of the implemented methods.

Index Terms—Beam propagation, Fourier optics, Fraun-

hofer diffraction, Fresnel diffraction, Python, Open-source.

I. INTRODUCTION

THERE are few optics simulation software available on themarket and in the open-source world. One of the most

known optics simulation software on the market is ZEMAX[2]. In open-source world there are also quality softwaresuch as Opus [3]. The full list of available optics simulationprograms can be found under [4]. The aim of this projectis to provide an optics alternative simulation software. Withsuch an alternative, it will be a lot more easier to implementwave propagation and Fourier optics inside a Python script,thus easiness is provided to Python users. Python is the mostpreferred programming language for academical purposesbeside MATLAB and Octave. Syntaxes of Python is similarin a sense to ones in MATLAB and also provides far morebetter code flow control. It is also aimed to provide a licencefree work bench for different platforms (currently supportsonly on Linux and Microsoft Windows).

The introduced system in this paper is able to providecalculations for Fourier calculations. But in near future aray tracing extension with a graphical user interface andray-tracing ability will be provided to the end-users.

II. HOW TO MAKE IT WORK

A. Under LinuxOnce sample.py and odak.py are downloaded from [5] and

saved under the same folder; Linux users can download thedependencies of matplotlib, numpy and SciPy from their de-fault package managers and they can execute the sample.pyusing python sample.py command under the terminal. Notethat running this script will show you the results of thesample questions in Section IV.

B. Under Microsoft WindowsWindows packages of Python 2.7 from [1], numpy from [6],

SciPy from [7] and matplotlib from [8] should be downloaded

The author is with the Optical Microsystems Laboratory, KocUniversity, Istanbul, 34450 TURKEY (e-mail: kaksit@ku.edu.tr).

and installed in the targeted computer. Once sample.py andodak.py are downloaded from [5] and saved under the samefolder; double clicking on sample.py will directly run or openan Idle window, if Idle window is opened simply press run(F5) from the above menus of the Idle. Note that runningthis script will show you the results of the sample questionsin Section IV.

III. MANUAL

This parts describes available functions inside the librarydescribed in this paper.

A. Class:odak.apertureThis class contains pre-defined apertures. Currently avail-

able callable aperture function are as in Table III-A.

Available callable aperture functionsodak.aperture.circle()odak.aperture.lens()odak.aperture.rectangle()odak.aperture.sinamgrating()odak.aperture.twoslits()

Figure 1 shows different available apertures fromodak.aperture class.

(a) odak.aperture.circle(). (b) odak.aperture.rectangle().

(c) odak.aperture.twoslits(). (d)odak.aperture.sinamgrating().

Fig. 1: Sample apertures.

Explanation of each function is given in subsectionsprovided in below. Note that each variable corresponds to a

0000–0000/00$00.00 c� 2011 Optical Information Processing

OPTICAL INFORMATION PROCESSING/ APRIL 2011 3

After the implementation of the fraunhoferintegral func-tion or in other saying 1D DFT algorithm with a 2D input;using the flow chart in Figure 2, Equation 3 is implementedas in below code. Implementation is made using fft approach.

def fresnel fraunhofer ( se l f , obj , wavelength ,distance , pixeltom ) :nx , ny = obj . shapeh = zeros ( ( nx , ny ) , dtype

=complex )k = 2∗pi / wavelengthd i s t a n c e c r i t i c a l = pow( f l o a t ( ( nx∗ny−

s e l f . count ( abs ( obj ) ,0 ) ) ∗pixeltom ),2 ) / distance / wavelength

print ( ’ Distance:%s ’% distance )print ( ’ C r i t i c a l distance :%s ’%

d i s t a n c e c r i t i c a l )i f distance < d i s t a n c e c r i t i c a l :

for x in xrange ( nx ) :for y in xrange ( ny ) :

h [ x , y ] =exp (1 j ∗k∗ ( xˆ2+y ˆ 2 )/ ( 2∗ distance ) )

resul t = f f t s h i f t ( ( 1 / ( 1 j ∗distance∗wavelength ) ) ∗h∗ s e l f .fraunhofer integral ( transpose (nx∗ny∗ s e l f . f raunhoferintegral (obj ∗h) ) ) )

else :for x in xrange ( nx ) :

for y in xrange ( ny ) :h [ x , y ] =exp (1 j ∗2∗pow( pi , 2 )

∗ ( xˆ2+y ˆ 2 ) ∗distance / k )resul t = f f t s h i f t ( s e l f .

fraunhofer integral ( transpose (nx∗ny∗ s e l f . f raunhoferintegral (obj ∗h) ) ) )

return resul t

Fig. 2: Way to implement Fraunhofer integral, [9].

B. Sample question ISample question I is as follows: Starting with a rectangu-

lar aperture of size 1mm, observe the diffraction pattern atdifferent critical propagation distances.

Rectangular aperture with size of 1mm can be examinedunder Figure 3. For the rest of the calculations wavelengthis chosen as 532nm. Below code is used to plot the diffractionpatterns at different critical propagation distances.

def question1 ( ) :

onepxtom = pow(10 ,−4)aperture = odak . aperture ( )rectangle = aperture . rectangle

(256 ,256 ,10)aperture . show ( rectangle , ’ Rectangle

aperture ’ )d i f f r a c = odak . d i f f r a c t i o n s ( )distance = 100wavelength = 532∗pow(10 ,−9)output = d i f f r a c . fresnel fraunhofer (1∗

rectangle , wavelength , distance , onepxtom)

aperture . show ( abs ( output ) , ’ Fraunhoferd i f f r a c t i o n at distance of %sm with %snm wavelength ’% ( distance , wavelength ))

nx , ny = output . shapeintens i ty = [ ]for i in range (0 , nx ) :

intens i ty . append (sum( abs ( output [ i , : ] ) )/ ny )

p l t . f igure ( )p l t . p lot ( intens i ty )p l t . show ( )return True

Fig. 3: Rectangular aperture with size of 1mm. Here 10 pixelcorresponds to 1mm.

1) Fresnel/Fraunhofer diffraction patterns at different dis-tances for sample question I: Distances of each Fraunhoferdiffraction pattern is written on top of the figures. After eachfigure an intensity profile is provided as the next figure.

Fig. 4: Fraunhofer diffraction pattern

Edge Diffraction

Diffrac?on  pa4ern  from  point  source:  

An  extended  source  a4enuates  the  fringes  à  poor  man’s  interferometry    Radio  lunar  occulta?ons  used  in  1950s-­‐60s  prior  to  inven?on  of  VLBI  

1989AJ.....98.2156P

t('ARUS 56, 122-146 (1983)

Strong Turbulence and Atmospheric Waves in Stellar Occultations

RICHARD G. FRENCH*'+ AND RICHARD V. E. LOVELACE': *Department qfEarth and Planetal3' S( iences. Massacim.~etts Institute ¢~f" TecJmology, Cambridge,

Massachusetts 02139: ,'Department qf'Astronomy. Welle.~h'y College, Wellesh,y. Massachusetts 02181: and ~-Department of Applied Physics, Department qf Astronomy, ('ornell University. Ithaca, New York 14853

Received October 7, 1982: revised April 7, 1983

Many of the problems of stellar occultation observat ions s tem from the difficulty of determining the effecls of realistic a tmospheric structure on the lightcurvcs. General techniques for producing model l ightcurves for a variety of realistic a tmospheric irregularities, including turbulence and inertia-gravity waves, are presented and applied. Using numerical s imulations which model the propagation of a wave through a phase-changing screen, the limit of strong scintillations for one- dimensional . Kolmogorov-l ike turbulence, both for a point source and for extended sources, is investigated in some detail, and significant departures from the behavior in the v, eak scintillation regime are found. The results are compared with published analytical results and recent occullation data. The effects of large-scale a tmospheric waves with realistic horizontal s tructure are examined. and the reliability of the numerical inversion method of retrieving the truc a tmospher ic vertical structure under c i rcumstances of strong ray crossing and horizontal inhomogeneit ies is assessed . The simulations confirm that large-scale layered features of the a tmosphere are accurately recov- ered: horizontally inhomogencous structures (including turbulence) with coherence scale L -z (2nRH) t-" (where R = planetary radius and H scale height) have little effect on the derived temperature profiles. It is concluded lhal analysis o f occultations may eventually allow us to determine both the quasiglobal a tmospheric structure and the statistical characterist ics of small- scale refractivity variations.

1. I N T R O D U C T I O N

Stellar occultations are sensitive probes of distant planetary atmospheres, and. in some cases, provide the only information we have about the structure of these remote upper atmospheres. Unfortunately. it has not proven easy to extract definitive results from the observations. The spiky structure of the lightcurves has been w~riously inter- preted as scintillation due to isotropic tur- bulence (Jokipii and Hubbard, 1977; Texas-Arizona Occultation Group, 1977: Hubbard, 1979a) and as evidence for lay- ered atmospheric structure (Kovalevsky and I.ink, 1969: French and Elliot, 1979; French et al. , 1982). A large (and occasion- ally acrimonious) body of literature has been generated by champions of each view. but there has been little common ground for discussion. Proponents of turbulent scintil- lation argue that the assumption that the

122 0019-1035/83 $3.00 Copyright (~, 1983 by Academic Press, lnc All rights of reprtu.luction in any form re~,erved,

atmosphere is perfectly layered is naive and unrealistic, while their critics contend that the assumptions of weak scintillation the- ory are violated by the very observations the theory is used to explain.

A new approach is called for: the investi- gation of the properties of lightcurves tbr atmospheres with more realistic properties than those considered so far. In Section 1I we relax the restrictions of weak scintilla- tion theory and construct model lightcurves for atmospheres with strong, anisotropic turbulence using wave optics. This is ac- complished by numerical simulations which model the propagation of a wave through a phase-changing screen while maintaining complete amplitude and phase information of the wave (Lovelace and Scannell, 1977; Scannell, 1980). We compare the results with available weak scintillation theory and with recent occultation data. Then, in Sec- tion lII, we examine the effects of large-

124 FRENCH AND LOVELACE

INCIDENT PLANE WAVE FROM STAR

=~

TUqBUL ENT PL ANE TARY

SPH

EMERGENT, WAVE FRONT

/

E ARTH'S ~ATF' @

OCCULTATION G E O M E T R Y

EOUIVA_EN ~ THIN CREEN

-Z-TL-7 . . . .

- - O

_L A _ I _

T _ 8 f-

Fie;. 1. Occultation geometry. The distortions in the wave front of a plane wave. due Io passage through a turbulent a tmosphere , can be represented by a thin phase-changing screen at distance 1) from the Ear th ' s path. The lightcurve as viewed from the Earth ahmg its path is compuled in overlap- ping sections Ifor example, A and B) by summing the phases of the appropriate section of the ups t ream phase screen. Edge effects are avoided by apodizing.

where u is the a tmospheric refractivity. The lateral deflection of the ray is negligible within the a tmosphcrc .

The refractivity can be decomposed as

v = vA(r) + vdx , y . z ) , (2)

where v.,x is the refractivity of the average a tmosphere (at a distance r from the center of the planet), and where v, is the variation of the refractivity from its average value. The variations may represent turbulence, density waves, or other phenomena in the a tmosphere. For the consideration of tur- bulence, we adopt an isothermal average atmosphere,

vA(r) = Vo e x p l - ( r - ro)/H], (3)

where r0 is a reference altitude, v0 is the refractivity at this level, and H is the atmo- spheric scale height, l.etting r0 be the alti- tude corresponding to the half-intensity point of the average lightcurve, the half- light refractivity is

O ?, 2 < I, (4)

. ~ - (27rR)V2 D

where R is the planetary radius, and D is the distance from the Earth to the planet.

Furthermore. we assume that the root- mean-square turbulent refractivity is pro- portional to the local mean refractivity in the region of the a tmosphere sampled by the wave (Ar ~ H , Az " - ( R t f ) v 2 L

(pt2) 1'2 :x FA. ( 5 )

It follows from Eqs. (1) and (2) that

= qbA + ¢P,, (6)

where ~A is the phase shill due to the aver- age a tmosphere , and (br is that due to the variations. For the "hal f - in tens i ty" ray, ~).,~ = 2~'(H/rf) 2. where the Fresnel radius is de- fined as r f = (XD) v2. We typically have ~,~ ~> I. Generally. the point source wave am- plitude at the reference screen is simply

E,(x,y) = exp(i(1)A + i(b0. (7)

ICARUS 129, 178–201 (1997)ARTICLE NO. IS975757

The Thermal Structure of Tritonʼs Atmosphere: Results from the 1993and 1995 Occultations

C. B. Olkin,1,2 J. L. Elliot,3,4 H. B. Hammel,1 A. R. Cooray,1,4 S. W. McDonald, and J. A. Foust

Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139–4307E-mail: colkin@lowell.edu

A. S. Bosh, M. W. Buie, R. L. Millis, and L. H. Wasserman

Lowell Observatory, Flagstaff, Arizona 86001–4499

E. W. Dunham2 and L. A. Young5

NASA Ames Research Center, Space Science, Moffett Field, California 94035–1000

R. R. Howell

Physics and Astronomy Department, University of Wyoming, Laramie, Wyoming 82071

W. B. Hubbard and R. Hill

Lunar and Planetary Laboratory, University of Arizona, Tucson, Arizona 85721

R. L. Marcialis6

Pima Community College, 2202 West Anklam Road, Tucson, Arizona 85709

J. S. McDonald

SETI Institute, 2035 Landings Drive, Mountain View, California 94043

D. M. Rank and J. C. Holbrook

University of California, Lick Observatory, Santa Cruz, California 95064

and

H. J. Reitsema

Ball Aerospace, P.O. Box 1062, Boulder, Colorado 80306–1062

Received October 11, 1996; revised April 18, 1997

This paper presents new results about Triton’s atmosphericr- structure from the analysis of all ground-based stellar occulta-u- tion data recorded to date, including one single-chord occulta-

tion recorded on 1993 July 10 and nine occultation lightcurvesfrom the double-star event on 1995 August 14. These stellaroccultation observations made both in the visible and in the

h-infrared have good spatial coverage of Triton, including thefirst Triton central-flash observations, and are the first data tos-probe the altitude level 20–100 km on Triton. The small-planetlightcurve model of J. L. Elliot and L. A. Young (1992, Astron.a,J. 103, 991–1015) was generalized to include stellar flux re-fracted by the far limb, and then fitted to the data. Values ofthe pressure, derived from separate immersion and emersionchords, show no significant trends with latitude, indicatingthat Triton’s atmosphere is spherically symmetric at �50-kmaltitude to within the error of the measurements; however,asymmetry observed in the central flash indicates the atmo-sphere is not homogeneous at the lowest levels probed (�20-km altitude). From the average of the 1995 occultation data,the equivalent-isothermal temperature of the atmosphere is47 � 1 K and the atmospheric pressure at 1400-km radius(�50-km altitude) is 1.4 � 0.1 �bar. Both of these are notconsistent with a model based on Voyager UVS and RSS obser-vations in 1989 (D. F. Strobel, X. Zhu, M. E. Summers, andM. H. Stevens, 1996, Icarus 120, 266–289). The atmospherictemperature from the occultation is 5 K colder than that pre-dicted by the model and the observed pressure is a factor of1.8 greater than the model. In our opinion, the disagreementin temperature and pressure is probably due to modeling prob-lems at the microbar level, since measurements at this levelhave not previously been made. Alternatively, the differencecould be due to seasonal change in Triton’s atmosphericstructure. ! 1997 Academic Press

r

-t)er

)

-re-

FIG. 6. Stellar occultation by a planetary atmosphere. Starlight inci-tdent from the left encounters a planetary atmosphere where refractionbends the light rays. The refracted light, dimmed by the spreading of therays, is observed in the shadow plane. Light that has crossed the x axis(from where the observer is) constitutes the far-limb flux. The resultingoccultation lightcurve is seen on the left. The observed signal is the sum.of all flux at the observer’s location. At some locations this comes fromxone ray; closer to the center of the shadow path there will be two rays(from the near and far limbs); and even closer the observer is within theevolute of the central flash and there will be more rays contributingto the observed flux. (Reproduced, with permission, from Elliot andOlkin, 1996).)

FIG. 8. Normalized Triton lightcurves and best-fitting model. The partial lightcurve fits did not include data within 20% of the occultationmidtime, as indicated by the bar.

256 CLEGG, FEY, & LAZIO Vol. 496

In the above equations the physical parameters j, a, D,and occur consistently in the combination a,N0

a 4j2r

eN0 D

na2 \AJjDaB2 1

n jreN0 . (14)

We have written a in this second form to emphasize theessential physics. The phase advance through the lens is

The Fresnel scale is (jD)1@2. Thus, the properties ofjreN0/n.

the lens are determined by the square of the ratio of theFresnel scale to the lens size and the phase advance throughthe lens. The larger the parameter a, the greater are theobservable e†ects due to the lens. Hence a weak lens,

can produce large observable e†ects if thejreN0/n > 1,

Fresnel scale is sufficiently larger than the lens size. Simi-larly, a strong lens, need not produce largejr

eN0/n > 1,

observable e†ects if the Fresnel scale is small relative to thelens size. Numerically,

a \ 3.6A j

1 cmB2A N0

1 cm~3 pcBA D

1 kpcBA a

1 AUB~2

. (15)

Upon substitution of the above dimensionless variables,the following expressions are derived for the refractiveproperties of a Gaussian lens :

hr(u)/h

l\ [au exp ([u2) ; (refraction angle) (16)

u[1 ] a exp ([u2)] [ c \ 0 ; (ray path) (17)

Gk\ [1 ] (1 [ 2u

k2)a exp ([u

k2)]~1 ; (gain factor) (18)

I(u@, a) \ ;k/1

n P~=

`=B(b

s)G

k(u@, a, b

s)db

s. (total intensity)

(19)

4. CHARACTERISTIC LIGHT CURVE PRODUCED BY A

GAUSSIAN LENS

When transverse motion between source, lens, and obser-ver is assumed, e.g., motion of the observer along the u@ axis,I(u@, a) translates into a light curve I(u@, a ; t, v), sinceu@(t) \ u@(t \ 0) ] vt, where t is time and v is the relativetransverse velocity. We will show that interpretation ofradio light curves in terms of plasma lenses that can beapproximated as Gaussian in proÐle can lead to inferencesregarding the physical properties of the lens.

4.1. General DescriptionWe will present numerical results in the following sec-

tions. Here we will Ðrst develop a physical understandingfor the basic characteristics of refraction by a Gaussianplasma lens Plane waves from an inÐnitely distant(Fig. 2).source are incident on the lens screen. The plane waves areindicated by straight dotted lines in the Ðgure, and the lensis represented by a plot showing the electron columndensity as a function of coordinate u along the lens plane.This representation is purely schematic, as the lens isassumed to have a negligible but uniform width along theline of sight. Upon emergence, the constant-phase surfacesare distorted into contours that mimic the function asN

e(u),

represented by the inverse GaussianÈshaped dotted lines.The lines are inverse GaussianÈshaped because the phasevelocity is greatest through the center of the lens, where N

ereaches its maximum value of N0.

FIG. 2.ÈSchematic diagram of refraction by a Gaussian plasma lens.See ° for a complete description of this Ðgure.4.1

In the limit of geometrical optics, the rays of energy Ñuxtravel perpendicular to the constant-phase surfaces. Thedirection of travel of the rays is indicated schematically bythe small arrows in the Ðgure. Extending this concept, wehave drawn the path of approximately 100 rays as theytravel perpendicular to the constant-phase surface afteremergence from the lens. The ray path is shown for a totaldistance D along the lens axis. An observer located close tothe lens plane but far from the lens axis (e.g., the upper leftand upper right regions of the ray trace) sees an unchangedsource : There is no change in the observed Ñux density (nochange in the number density of the rays) or in the sourceÏsposition (the direction of arrival of the rays).

Somewhat farther from the lens and slightly o†-axis, inthe ““ Focusing Regions ÏÏ marked in the rays beginFigure 2,to converge and their number density increases. An obser-ver located in this area of the ray trace would see a source ofenhanced brightness somewhat displaced from its ““ true ÏÏposition, as evident from the increased number density andskewness of the rays, respectively. At the same distance fromthe lens, but closer to the lens axis, the rays are spread apartby the lens. An observer in this region would see a source ofdecreased brightness due to the lower number density ofrays. The source would be o†set from its true position by an

262 CLEGG, FEY, & LAZIO Vol. 496

FIG. 9.ÈLight curve of the quasar 0954]658 during an extreme scat-tering event. (a) Data obtained at 2.25 GHz, compared to our lens model(dashed line) with a \ 160 and A two-component model forb

s\ 0.4.

0954]658 was used : a lensed component of Ñux density 0.35 Jy, and anunlensed component of Ñux density 0.3 Jy. (b) Comparison of the 8.1 GHzdata and a two-component model with a \ 12, lensed Ñuxb

s\ 0.11,

density 0.15 Jy, and unlensed Ñux density 0.45 Jy.

sight was una†ected by the lens, or emission on largerscales, which is completely resolved by VLBI arrays but isunresolved to the much smaller Green Bank Interferometerused to obtain the light curve. Both scenarios are consistentwith the observed structure of 0954]658.

Gabuzda et al. conducted 5 GHz observ-(1992, 1994)ations between 1987 and 1989, and found the source toconsist of a compact core (\0.5 mas) with a jet extendingapproximately 5 mas to the northwest. Within the jet theyidentiÐed multiple, compact (\1 mas) knots. With the pos-sible exception of the knot K0 et al.(Gabuzda 1994),however, none of the knots they identiÐed would have beenpresent at the time of the ESE. The proper motion of theknots is approximately 0.4 mas yr~1, large enough thatmost of the knots were not ejected until after the ESE(epoch 1981.1), unless the knots have undergone substantialacceleration since their ejection. & ReadheadPearson

included this source in a Ðnding survey conducted in(1988)1978. Although they did not image it, they found it to becompact and poorly modeled as a single Gaussian com-ponent. These VLBI observations bracket the time of theESE and found indications of compact, yet complex struc-ture. We therefore conclude that it is plausible that, at thetime of the ESE, the source consisted of multiple compact

components, whose typical angular scales are milli-arcseconds.

On intermediate to larger angular scales, tens of milli-arcseconds to arcseconds, there is also emission from thissource. Gabuzda et al.Ïs VLBI observations(1992, 1994)also included simultaneous VLA observations. They foundthat approximately 15% (B0.15 Jy) of the Ñux was resolvedout by the VLBI observations. Also, using lower resolutionVLA observations et al. Ðnd a jet extend-Kollgaard (1992)ing approximately 4A to the south of the VLA core ; the jetÏsÑux density is approximately 0.22 Jy. This extended emis-sion would lie entirely within the synthesized beam of theGreen Bank Interferometer, but would be una†ected by thepassage of a Gaussian lens in front of the source. Insummary, this source displays emission on a range ofangular scales, from submilliarcseconds to arcseconds. Acombination of small-scale components, whose line of sightthe lens did not intersect, and emission on larger scalescould provide the 0.35 Jy of unlensed Ñux.

To obtain quantitative estimates of the lens parameters,we proceed as follows. We obtain approximate values for aand by appeal to the light curves. We reÐne our estimatesb

sfor a and by producing simulated light curves with slight-bsly di†erent values of a and and comparing visually theb

ssimulated and actual light curves. Due to the simple modelwe are considering here, we judged a more exhaustivesearch of the (a, parameter space not to be worthwhile.b

s)

Comparing the shape of the inner caustics and minimumof the 2.25 GHz ESE to we estimate to beFigure 5, b

sFrom the separation of the inner and outer0.1 \ bs\ 1.

caustics in the 8.1 GHz ESE and equation we estimate(23),a B 10 at this frequency. Since a P j2, at 2.25 GHz we havea B 130.

Varying the parameters and comparing simulated lightcurves to the actual 2.25 GHz light curve, we Ðnd reason-able agreement for a \ 160 and In web

s\ 0.4. Figure 9

have superposed the simulated light curve for this set ofparameters on the 2.25 GHz light curve of 0954]658. Indoing so, we have scaled the u@ axis arbitrarily since itdepends on the unknown relative transverse velocities of theobserver and lens. The agreement with the ESE minimum isgood. The simulation underestimates the amplitude of theinner pair of caustics. The outer caustics for a \ 160 are faroutside the time range of the observations plotted in Figure

We have compared our simulation to the full time range9.of the observations of 0954]658 We do not see any(Fig. 8).evidence for the existence of the outer caustics, but the pre-dicted amplitude of the outer caustics is signiÐcantlysmaller than the predicted amplitude for the inner caustics.The outer caustics are probably lost in the noise in the lightcurve. The ESE occurs during a general decrease in thesourceÏs Ñux density, probably caused by intrinsic varia-tions. This general decrease in the sourceÏs Ñux densityexplains the lack of agreement between the amplitude of theactual light curve and the model light curve after the ESE.

To compare our model with the 8.1 GHz light curve, wescale the light-curve Ðt to the 2.25 GHz ESE to 8.1 GHz.We believe such a simple scaling is justiÐed within thecontext of our one-dimensional simulations of a Gaussianlens passing in front of a single compact component of0954]658. Below we discuss brieÑy the consequences ofrelaxing the assumptions in our simple model.

We assume that the angular size of the lensed componentof 0954]658 scales as j, as is generally appropriate for

Interstellar plasma lenses