Radio astronomy at frequencies from 2 to 30 MHz challenges time tested methods for extracting usable information from observations. One fundamental reason for this is that propagation effects due to the magnetoionic ionosphere, interplanetary medium, and interstellar matter (ISM) increase strongly with wavelength. The problems associated with interstellar scattering off of small scale irregularities in the electron density are addressed. What is known about interstellar scattering is summarized on the basis of high frequency observations, including scintillation and temporal broadening of pulsars and angular broadening of various galactic and extragalactic radio sources. Then those high frequency phenomena are addressed that are important or detectable at low frequencies. The radio sky becomes much simpler at low frequencies, most pulsars will not be seen as time varying sources, intensity variations will be quenched or will occur on time scales much longer than a human lifetime, and many sources will be angularly broadened and/or absorbed into the noise. Angular broadening measurements will help delineate the galactic distribution and power spectrum of small scale electron density irregularities

Cordes, James M.

English

NASA Technical Reports Server

LOW FREQUENCY ASTROPHYSICS FROM SPACE
Proceedings, Crystal City, Virginia 1990
Namir E. Kassim and Kurt W. Weiler (Eds.)
LOW FREQUENCY INTERSTELLAR SCATTERING AND
PULSAR OBSERVATIONS /> >>
*James M. Cordes
Astronomy Department, Cornell University
Radio astronomy at frequencies from 2 to 3Q MHz challenges time tested methods for
extracting usable information from observations. One fundamental reason for this is
that propagation effects due to the magnetoionic ionosphere, interplanetary medium,
and interstellar medium (ISM) increase strongly with wavelength. In this paper I ad-
dress the problems associated with interstellar scattering off of small scale (~ 109 cm)
irregularities in the electron density.
I will first summarize what we know of interstellar scattering on the basis of high fre-
quency observations, including scintillation and temporal broadening of pulsars and
angular broadening of various galactic and extragalactic radio sources. Then I will ad-
dress those high frequency phenomena that are important or, at least, detectable at low
frequencies. The radio sky becomes much simpler at low frequencies: most pulsars will
not be seen as time varying sources, intensity variations will be quenched or will occur
on time scales much longer than a human lifetime, and many sources will be angularly
broadened and/or absorbed into the noise. Angular broadening measurements will help
delineate the galactic distribution and power spectrum of small scale electron density
irregularities. Images of scattered sources can take on the form of the distribution of the
scattering material in this low frequency regime. Spectral broadening may be a relevant
line broadening mechanism for low frequency recombination lines.
Summary of Interstellar Scattering and Scintillation Phenomena
Interstellar scattering phenomena are rich in variety. Diffraction from small scale irreg-
ularities is manifest as angular broadening (interstellar 'seeing'); temporal broadening
of pulsar pulses due to multipath propagation; and diffractive intensity scintillations.
The latter have been seen only from pulsars owing to their small size (the 'stars twin-
kle, planets do not* rule: the critical angular size for interstellar scintillation is typically
less than 1 micro arc sec.) Another effect, as yet unobserved from the ISM, is spectral
broadening, the smearing of a spectral line due to modulations imposed by the diffract-
ing medium. Refraction effects from larger irregularities include long term scintillations;
wandering of apparent source positions on the sky (angle of arrival [AOA] variations);
and time of arrival variations associated with the AOA variations and also with the
changing electron column density (the dispersion measure, DM) as turbulence is swept
across the line of sight. Figure 1 shows examples of several of these phenomena, includ-
ing the visibility function of a scattered pulsar; temporal broadening of a pulsar pulse;
diffractive intensity scintillations; and dispersion measure variations.
There is substantial evidence that the electron density variations responsible for the
phenomena span a large range of length scale and that they may be described by a
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Figure 1. Examples of scattering observables: (a) visibility function1; (b) pulsar pulse
with scattering broadening tail3; (c) grey scale plot of difiractive intensity scintillations3;
(d) dispersion measure variations4.
spatial power spectrum that is power law in form. Thus we define the power spectrum
in terms of the mean square electron density:
^2 ^-a (1)
where we introduce the coefficient C;j, wavenumber cutoffs 91,2, and spectral index a.
Pulsar scintillations and time of arrival variations, and limits on AOA variations of OH
masers5 imply that 3.5 < a < 4.0. The Kolmogorov value, a = 11/3, is consistent
with many lines of sight. Denning, respectively, the 'inner' and 'outer* length scales
t\ = 2ir/qi and lo = 2n/qo of the spectrum, limits from several kinds of observations
suggest that li < 109 cm and lo > 1014 cm. Recent work9'6*7 suggests that the inner
scale is of the order of 100 km for a few heavily scattered lines of sight, a length scale
consistent with the gyro radius of thermal ions in the warm (104 K) phase of the ISM.
There are indirect arguments that the outer scale extends to parsec scales, based on
studies of rotation measure variations8 and on the notion that cosmic ray diffusion
3.
requires magnetic irregularities on all scales from about 1 AU to 1 pc and that these are
accompanied by density irregularities9.
The easiest quantity to estimate, given the spectral index a, which we now assume to
be 11/3, is the line of sight integral of C2, which we call the scattering measure. The
scattering measure and estimates of it using angular scattering diameters of extragalactic
sources QFWHM and pulsar temporal broadening times T& are:
where 0o = 0.13 arc sec, Ar = 387 kpc m~20/3 for rd in seconds and D in kpc, and v
in GHz. Measured values of SM range from 10~4 to 103 kpc m~20/3. For scintillation
bandwidth measurements, SM may be estimated by first using the 'uncertainty' rela-
tionship 27rAt>rfT(f = 1 to estimate the effective temporal broadening time. For a pulsar
embedded in a homogenously turbulent medium, we have
f!6(fa2)cr,1I/a
In terms of the scattering diameter, the diffraction scale is id —
Spectral broadening is approximately equal to the reciprocal of the intensity scintillation
time; both are due to modulations of the signal due to motion of diffracting material
across the line of sight. In terms of SM , the broadening is10
, » « 27rVi[4;rr2A2C2£>]3/5 = 1.16 Hz VW6/5SM3/5,
*d
where re is the classical electron radius, Vj_ is the relevant transverse speed (a combi-
nation of pulsar, ISM, and Earth velocities), V100 is in units of 100 km s"1, and SM
has units of kpc m~20/3, and v is in GHz.
For galactic objects with known distances, we obtain the line of sight average C2 =
SM/D. The remarkable fact is that C2 varies by a factor of at least 104 between
different lines of sight. A homogenously turbulent medium would, of course, yield a
constant value for this quantity. Figure 2 shows SM plotted against galactic latitude
and longitude on an equal area projection. The plotted circles have diameters that
are proportional to logSM. It is obvious that scattering is a galactic phenomenon,
since the largest circles are on lines of sight in the galactic plane. Study of how SM
varies (work in progress) implies that the variation in C2, is due to both large scale
galactic structure and localized regions of intense turbulence, including HII regions and
supernova remnants. Enhanced scattering seems to be correlated over small angular
scales, implying that the depth is of order a few parsecs or less. This implies that the
internal C2 of enhanced regions is correspondingly larger than C2 by a factor of about
1000.
168
Figure 2. Scattering measures plotted versus 1,6; log SM = 3 for the largest circles,
the smallest circles have log SM = —4.
To make estimates of low frequency scattering phenomena, it is useful to fit an idealized
model to the data. The local value of C* at galactocentric radius R and distance z from
the galactic plane comprises two components:
C*(R,z) = d exp[-(*2M? + ,/#,)] + C2exp[-(K2Ml + */Ha)J.
A grid search to minimize the mean square difference between predicted and measured
SM yields a large scale component with AI > 10 kpc and HI fas 1 kpc and a galactic
center component with A2 « 3.0 kpc and tf2 a 0.05 kpc. The strengths of the two
components are Ci w 10~4 m"20/3 and C2 ss 5 m~20/3. This model accounts for the
broad brush structure that determines SM, but there are substantial departures from
the model's predictions. Figure 3 shows the modeled SM plotted against measured
SM. Notable lines of sight that deviate from the model by more than two orders of
magnitude include the Vela pulsar11; Cygnus X-3 (ref 6); and the extragalactic source
behind the HII complex NGC6334B (ref 5). The grid search was iterated several times
so that 'deviant' sources could be recognized and excluded from the fit. The numbers
quoted above derive from the end result of this process. In a later section, the model is
used to predict observable quantities such as angular broadening, spectral broadening,
and temporal broadening.
s*.
NCC6334B
. Cyg X-3
.Vela
-5 -4 - 3 - 2 -1 0 1 2 3
log SM (octuol)
Figure 3. Modeled 5M plotted against actual SM. A few notable lines of sight are
labeled.
TABLE!
DIFFRACTION PHENOMENA
Phenomenon Quantity Typical Values
400 MHz 30 MHz 10 MHz
Intensity Scintillations At/ a A~4 4 10 kHz 10~2 10'7
o c A ~ 1 2 60*ec 3 0.7
Temporal broadening T& oc A4 4 1ms I mm 3 hr
Angular broadening 9* oc A2-2 10 m.a.3. 3" 33"
Spectral broadening At/, c< A12 1 Hz 22 83
TABLE 2
REFRACTION PHENOMENA
Phenomenon Quantity Typical Values
400 MHz 30 MHz 10 MHz
Slow scintillations <r//J a A"1-1 20% 1 0.3
Atr oc A2-2 1 yr 300 3000
Time-of-arrival A<DM oc A2 lOjxa 2 ma 16 ma
variations
Ai*ocA 1 6 1/ia 60 400
A<0» oc A3-3 1^3 5 ma 200 ma
Angular wandering A0r oc A1-6 1 m.a.3. 60 400
Interstellar Scattering at Low Frequencies
Using results from high frequency observations and our understanding of the scatter-
ing medium's spectrum and distribution, it is possible to discuss the phenomena at
low frequencies. In Table 1 the scaling laws12 for various diffraction phenomena are
presented and representative values of the observables are given. Intensity scintilla-
tions, which have typical frequency and time scales of 10 kHz and 60 sec at meter
wavelengths become sub-Hz and subsecond at 10 MHz. In order to detect intensity
scintillations, a radiometer must have sufficient signal to noise ratio for the frequency
and time resolution necessary to resolve the scintillations in both time and frequency.
At low frequencies where system temperatures are determined by the sky background,
it is clear that diffractive scintillations will never be detectable, even for the nearest pul-
sar, which has the largest scintillation bandwidth. Temporal broadening becomes much
larger than the pulse period of almost all pulsars at 10 MHz, so pulsars will be seen
as steady sources. Angular broadening becomes a few to many arc seconds. Spectral
broadening may be important for recombination line observations.
Figure 4 details the observed scattering broadening times of pulsars, showing TJ plotted
against DM . Data are taken from the literature and have been scaled to 1 GHz assum-
ing the Kolmogorov scaling r«f a A22/5. To assess the level of temporal broadening at
low frequencies, horizontal lines have been drawn (again using the Kolmogorov scaling)
to show 1 sec of broadening at 10 and 30 MHz. The meaning of these lines is that
all pulsars above each horizontal line will be essentially smeared out, since the average
pulsar period is ~ 0.7 sec. At 10 MHz, there are only 2 pulsars below the line, while at
7.
30 MHz, a few tens of objects are below the line. At 2 MHz, all pulsars are well above
the 1 sec line (not shown). It is amusing to consider the most broadened pulsar (PSR
1849+00; ref 13) which has 1 sec of broadening even at 1 GHz. The formal scaling to 1
MHz, yields a broadening of 0.5 Myr! Note, however, that the scaling law T& « A22/3 is
based on a small angle approximation14, which breaks down for heavily scattered lines
of sight at low frequencies. Relaxing the small angle approximation gives a more reason-
able result: only about 20 kyr! This means that the distribution of paths along which
radiation propagates subtends a major fraction of the galactic disk. Of course, this is
an academic exercise, since no source will be observable along such heavily scattered
lines of sight due to free free absorption (see below).
I7
V-f
*I
1 sec at \ GHz - 0.16 yr at 30 MHx
0.3 2.31 1.3 2
loo. DM (pe em"J)
Figure 4. Pulsar temporal broadening times plotted against dispersion measure,
DM. The solid line is a least squares fit to the data; the dashed lines are ±\a devia-
tions from the fit. Downward arrows denote upper limits, which were excluded from
the fit.
Refraction phenomena are presented in Table 2. Refractive scintillations occur on time
scales measured in hundreds of years, so even the most dedicated of monitorers will
be thwarted in variability studies. Angle of arrival variations and .the associated time
of arrival variations scale more slowly with wavelength than do angular and temporal
broadening. As a consequence, these variations will be proportionally smaller than the
angular size or pulse width and thus extremely difficult to measure at low frequencies.
Predicted Observables at Low Frequencies
The net implications for low frequency radio astronomy are that: (1) Diffractive in-
tensity scintillations will be quenched for realistic instrumental bandwidths and time
constants. This means that the equivalent of speckle interferometry in the radio will
not be possible in imaging observations. (2) Pulsar periodicities will be quenched by
temporal broadening. This statement holds for all pulsars at 2 MHz, all but a few at
10 MHz, and for most at 30 MHz. (3) The time scale of refractive phenomena is much
longer than a scientific career. The radio sky indeed will appear quiesent compared to
high frequencies. (4) The dominant observable will therefore be the angular broadening
of all kinds of radio sources. For a few pulsars in the 10 to 30 MHz band, temporal
broadening can be measured, leading to determinations of the scaling law of the tem-
poral broadening, in turn leading to further constraints on the slope and cutoffs of the
wavenumber spectrum. Also, for a few pulsars, it will be possible to test whether time
of arrival variations that depend on wavelength follow the A2 dependence of the cold
plasma dispersion law, or whether TOA variations due to AOA variations are also im-
portant. These last tests will help constrain the wavenumber spectrum for 6ne on scales
much larger than 1 AU. (5) Sources viewed at low galactic latitudes will be so scattered
that their shapes will take on the form of the distribution of scattering material, viz.
the galactic plane. This is because the maximum traversal of a ray transverse to the
propagation direction is limited by the extent of the scattering medium. This source of
image asymmetry has not been addressed before and competes with asymmetries due
to anisotropies of the small scale irregularities themselves and distortion of a scattered
image by refraction15'16.
Using the idealized model for C£(/Z, z) discussed above, it is possible to predict the
values of the most important observables at low frequencies, namely the angular broad-
ening diameter BPWHM an<i the spectral broadening At/c. Figure 5 shows the angular
broadening plotted against galactic latitude for several cuts in longitude.
o •»i
«
i
0.001 GHz
••••«•••••••••
0.01 OGHi
20 40 60 80
GALACTIC LATITUDE b
Figure 5. Predicted angular broadening of extragalactic sources vs. galactic latitude
for different frequencies and for three longitudes (0, 60, and 180°.)
fSpectral broadening takes on values of 300, 20, and 5 kHz at observing frequencies
of 1, 10, and 30 MHz for the largest observed scattering measure (103 kpc m'™1* for
NGC6334B, ref 5). This broadening corresponds to velocity spreads of 105,500, and
50 km s"1 . At high latitudes, the spectral broadening is a factor of 10* smaller and is
therefore negligible.
The spectrum defined in equation (1) implies a lower bound on the emission measure
for a given line of sight. The total emission measure is:
,0
d3 (nl) =1 da ((ne)2 + ((6ne)2}} = EM0 + EMS.Jo
The line of sight integral of equation (1) gives the second term on the right hand side
of the above equation. The integral of eqn (1) is
for a £ 3, where the approximate equality holds for the inner scale being much smaller
than the outer scale, t\ <C t§, which seems to be a good approximation. For a = 11/3
we have
EMg as 544 pc cm'6 SM(kpc m'20/3) [4(pc)]2/3 .
The measured values of SM therefore imply minimum emission measures of 0.05 to
103 pc cm"6 if we assume an outer scale of 1 pc. In Figure 6 we show EMs plotted
against 1, 6 using an outer scale of 100 pc as would be measured along the lines of sight
to extragalactic sources. Lines are also drawn that designate optical depth unity for
several radio frequencies. It is clear that much of the Galaxy is opaque at these low
frequencies. It is of interest to compare the predicted minimum EM with measured
values of Reynolds17. For |6| = 90°, Reynolds obtains EM w 4 pc cm"6 while the
ey /•»
scattering measurements suggest EMg(\b\ = 90°) as O.ll£0' . If we assume that EMo %
EMs, then an outer scale of IQ « 100 pc is implied. Such an outer scale is consistent
with other conjectures about the overall extent in wavenumber of the electron density
spectrum18'19.
References
1. Gwinn, C.R., Cordes, J.M., Bartel, N., and Wolszczan, A. 1988, Ap. J. (Letters),
334, L13.
2. Cordes, J.M., Weisberg, and Boriakoff, V. 1985. Ap. 7., 288, 221.
3. Cordes, J.M., Wolszczan, A., Dewey, R. J., Blaskiewicz, M., and Stinebring, D. R.
1990; Ap. J., 349, 245. See also Rawley, L.A., Taylor, J. H., and Davis, M.M. 1988,
Ap. J., 326, 947.
4. Gwinn, C.R., Moran, J. M., Reid, M. J., Schneps, M. H: 1988, Ap. J., 330, 817.
5. Moran, J.M., Rodriquez, L. F., Greene, B. and Backer, D.C. 1989, Ap. /., in press.
UJ
S
i-o
r • I at 30 MHz
1-180
r - 1 at 1 MHz
20 3040 60
GAIACTIC LATITUDE b (d«qrtt«)
Figure 6. Predicted minimum emission measure vs. galactic coordinates.
6. Molnar, L. A., Mutel, R. L., Reid, M. J., and Johnston, K. J. 1990, Ap. J., submitted.
7. Spangler, S.R. and Gwirin, C.R. 1990, Ap. J., in press.
8. Simonetti, J. H., Cordes, J.M., and Spangler, S.R. 1984, Ap. /., 284, 126; Lazio, J..
Spangler, S.R., and Cordes, J.M., Ap.J., in press.
9. Jokipii, J. R. 1988, in AIP Proceedings 174, Radio Wave Scattering in the Interstellar
Medium, 48.
10. Lazio, J. and Cordes, J.M., in preparation.
11. Backer, D. C. 1974; Ap. 7., 190, 667.
12. Cordes, J. M., Pidwerbetsky, A., and Lovelace, R.V. 1986, Ap. J., 310, 737. See
also Rornani, R., Narayan, R., and Blandford, R. D. 1986, M.N.R.A.S., 220,19; Foster,
R. S. and Cordes, J. M. 1990, Ap. J., submitted.
13. Clifton, T. 1985, Ph.D. Thesis, Univ. of Manchester.
14. Rickett, B. J. 1977, Ann. Rev. Ait. Ap., 15, 479.
15. Spangler, S. R. and Cordes, J. M. 1988, Ap. 7., 332, 346.
16. Narayan, R. and Goodman, J. 1989, M.N.R.A.S., 238, 963.
17. Reynolds, R. J. 1990, these proceedings.
18. Armstrong, J. W., Cordes, J. M., and Rickett, B. J. 1981, Nature, 291, 561.
19. Lee, L.C. and Jokipii, J. R. 1976, Ap. J., 206, 735.


Low Frequency Interstellar Scattering and Pulsar Observations

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