Pulsar Scintillation & Secondary Spectra The view from the Orthodoxy Jean-Pierre Macquart
Interstellar Scintillation: Executive Summary Radiation is scattered between a source S and observer O Inhomogeneous medium spatial variations φ(x) across wavefront For a planar wavefront the Fresnel-Kirchoff integral is u(x) = r F = i r F = (x)+ Z h x d x exp[i (x, X)], Dp s D p X r F i D e, D e = D sd p s D p pulsar D p-s phase screen z=z O -D Need phase statistics... D p D s O observer's plane z=z O
Phase statistics Screen phase assumed to be stochastic & normally distributed so that it is fully described by its second moment (equivalently, its power spectrum) D φ (r) = [φ(r + r ) φ(r )] D φ (r) = ( r r diff ) β Turbulence expected to be locally anisotropic in many theories (e.g. Goldreich & Sridhar 1995,7) Armstrong, Rickett & Spangler 1995
The stationary phase approximation In the regime of strong scattering the phase changes rapidly and the exp[i ] term in the Fresnel integral oscillates rapidly except near points of stationary phase x j whose locations satisfy (e.g Gwinn et al. 1998 et seq.): r (x j )+ x j r F X =0. Dp We can then write the total scattered wavefield as a sum of stationary phase points u(x) = X p µj exp[i j] j D p s where µ j is the magnification near the stationary phase point (derived from the second derivatives of ) and j is its phase. The total intensity is uu = X j,k p µj µ k exp[i( j k )].
The stationary phase approximation We can expand j to first order in time and frequency: j 0 j + @ j @t (t t 0)+ @ j @ ( 0). If we Fourier-transform to find the secondary spectrum (in the limit of large integration time and bandwidth): Ĩ(!, ) / X j,k p µj µ k exp[i( 0 j 0 k ) (! +! j,k ) ( + j,k ) +exp[ i( 0 j 0 k) (!! j,k ) ( j,k ), where the Doppler rates! j,k and delays j,k of each stationary phase point are! j,k = 1 @( j k ) @t j,k = 1 @( j k ) @ Points to note: Approach is valid when phase change between adjacent stationary phase points 1 If µ!1we need to treat the caustics associated with each stat. phase point more carefully, and (at least) third order derivatives of are required.
Aside: phase gradients along the optical axis
Strong Scattering Physical Picture Strong scattering r diff <r F : diffractive: interference between r diff -sized patches Interference between ray bundles gives fast, narrowband intensity fluctuations
Pulsar Dynamic Spectra strong diffractive scintillation the result of multipath propagation and interference between wave bundles arriving from different directions Size of scintillation pattern (in time and frequency) indicates the scale of the turbulence responsible for the scintillations. Frequency-scaling of decorrelation timescale and bandwidth is directly related to the underlying spectrum of turbulent fluctuations responsible for the scintillations. The dynamic spectrum of PSR 1933+16. The regions of higher intensity are indications of constructive interference from multipath propagation.
Dynamic Drifting bands Opposite slopes in a single spectrum Two angles of arrival spectra (cont.) as used to infer largescale (>1 AU) plasma gradients in the ISM Strong periodicities
Scattering Geometry Revealed by Secondary Spectra delay, τ f i j FFT f t i j ). v scint Doppler rate, ω
Position of Structure on the Secondary Spectrum delay (conjugate to frequency axis) τ = D s cβ ( θ i θ j ) φ i πν + φ j πν Doppler shift (conjugate to time axis) ω = 1 λβ (θ i θ j ) v Secondary Spectrum shape determined by speckle geometry. Main parabola: set θ j =0 and consider delay vs Doppler shift Inverted parabolae: now set θ j 0 this term usually unimportant Speckle image reconstruction generally not possible unless v is known.
Aside: when is the DM term unimportant?
3 1-3 - -1 1 3-1 - -3 10 p Isotropic scattering 8 6 4-4 - 4 q negative delay axis not shown since secondary spectrum of the intensity is symmetric about the delay=0, Doppler=0 axis
3 1-3 - -1 1 3 Anisotropic scattering -1 - -3 p 10 8 6 4-4 - 4 q negative delay axis not shown since secondary spectrum of the intensity is symmetric about the delay=0, Doppler=0 axis
Mechanics of speckle imaging In the regime of strong scattering the wavefield seen by an observer can be approximated by the sum of a number of speckles, each with a magnification µ and phase : u(r) = N j µj exp(i j), where the phase term contains a screen term (x) and a geometric phase delay term: (x r) = (x) + rf. Here = 1 (distance to screen/distance to pulsar) and r is the transverse position on the observing plane.
Mechanics of speckle imaging The visibility that we measure is just V (b) = u( b/)u (b/). If we substitute in our formula for u(r) and Fourier transform, we find Ṽ (b) = µj µ k exp {i [ j( b/) k(b/)]} (f D f D,jk ) ( jk). j,k Amplitude Phase Term Location of point on secondary spectrum I.e. a double sum over all pairs of stationary phase points. f D,jk = 1 ( j jk = D s c ( j k) + k ) V e j k These are our familiar expressions for the location
Mechanics of speckle imaging All the astrometric information is encoded in the phase term: antisymmetric 9 kds b k + j k + ( j + k ). symmetric j k = j Ds When we compare the > 0 part of the secondary spectrum to its < 0 counterpart and add the phases of each together, we get jk (b) = b ( 1. 11 1.6 10.5 1.4 10 The symmetric part (astrometric) of the phase on the Arecibo-GBT baseline 1 9.5 0.8 9 0.6 8.5 0.4 8 0. 40 0 0 0 40 Differential Doppler frequency fd (mhz) 150 60 1.8 7.5 Differential Delay τ (ms) 1.4 60 k ). 1 11.5 1.6 0 + This is just 1.8 the astrometric phase! Differential Delay τ (ms) j 100 50 1. 0 1 0.8 50 0.6 100 0.4 0. 0 150 60 40 0 0 0 40 Differential Doppler frequency f (mhz) D 60