The Astrophysical Journal, 850:129 (7pp), 2017 December 1 https://doi.org/10.3847/1538-4357/aa94cd
© 2017. The American Astronomical Society. All rights reserved.
Turbulent Density Fluctuations and Proton Heating Rate
in the Solar Wind from 9–20Re
K. Sasikumar Raja1 , Prasad Subramanian1, R. Ramesh2, Angelos Vourlidas3,5 , and Madhusudan Ingale4
1 Indian Institute of Science Education and Research, Pashan, Pune—411 008, India; sasikumar@iiserpune.ac.in
2 Indian Institute of Astrophysics, 2nd Block, Koramangala, Bangalore—560 034, India
3 Applied Physics Laboratory, Johns Hopkins University, Laurel, Maryland, USA
4 Plot No. 2, Near RSS office, Bamb Colony, Jammer Road, Bhusaval—425 201, India
Received 2017 September 20; revised 2017 October 16; accepted 2017 October 17; published 2017 November 27
Abstract
We obtain scatter-broadened images of the Crab Nebula at 80 MHz as it transits through the inner solar wind
in 2017 and 2016 June. These images are anisotropic, with the major axis oriented perpendicular to the
radially outward coronal magnetic field. Using these data, we deduce that the density modulation index
(dNe Ne) caused by turbulent density fluctuations in the solar wind ranges from 1.9 ´ 10-3 to 7.7 ´ 10-3
between 9 and 20 Re. We also find that the heating rate of solar wind protons at these distances ranges
from 2.2 ´ 10-13 to 1.0 ´ 10-11 erg cm-3 s-1. On two occasions, the line of sight intercepted a coronal
streamer. We find that the presence of the streamer approximately doubles the thickness of the scattering
screen.
Key words: occultations – scattering – solar wind – Sun: corona – Sun: radio radiation – turbulence
1. Introduction observations of density turbulence with kinetic Alfvén waves
The solar wind exhibits turbulent fluctuations in velocity, that get resonantly damped on protons, consequently heating
magnetic field, and density. Traditionally, researchers have them (Chandran et al. 2009; Ingale 2015a).
attempted to understand this phenomenon within the framework In this paper, we investigate the characteristics of turbulent
of incompressible magnetohydrodynamic (MHD) turbulence density fluctuations and the associated solar wind heating rate
(e.g., Goldstein et al. 1995). However, density uctuations are from 9–20 R using the anisotropic angular broadening offl
not explained in this framework, and remain a relative enigma radio observations of the Crab Nebula from 2017 and 2016
despite noteworthy progress (e.g., Hnat et al. 2005; Shaikh & June 9 to 22. The Crab Nebula passes close to the Sun on these
Zank 2010; Banerjee & Galtier 2014). While most of the data days every year. Since its radiation passes through the
used for solar wind turbulence studies are from in situ foreground solar wind, these observations give us an
measurements made by near-Earth spacecraft, density fluctua- opportunity to explore the manner in which its angular extent
tions can often be inferred via remote sensing observations, is broadened due to scattering of turbulent density fluctuations
typically at radio wavelengths. Examples include angular in the solar wind. Anisotropic scatter broadening of back-
broadening of point-like radio sources observed through the ground sources observed through the solar wind has hitherto
solar wind (Machin & Smith 1952; Hewish & Wyndham 1963; been reported only for small elongations (»2–6 R; e.g.,
Erickson 1964; Blesing & Dennison 1972; Dennison & Blesing Anantharamaiah et al. 1994; Armstrong et al. 1990). Imaging
1972; Sastry & Subramanian 1974; Armstrong et al. 1990; observations of the Crab Nebula (e.g., Blesing & Dennison
Anantharamaiah et al. 1994; Ramesh et al. 1999, 2001, 2006, 1972; Dennison & Blesing 1972) offer us an opportunity to
2012; Kathiravan et al. 2011; Mugundhan et al. 2016; Sasikumar investigate this phenomenon for elongations 10R. On 2016
Raja et al. 2016), interplanetary scintillations (IPS; Hewish June 17, and 2017 June 17 and 18, a coronal streamer was
et al. 1964; Cohen & Gundermann 1969; Ekers & Little 1971; present along the line of sight to the Crab Nebula; this gives us
Rickett 1990; Manoharan et al. 2000; Bisi et al. 2009; Tokumaru an additional opportunity to study streamer characteristics. The
et al. 2012, 2016), spacecraft beacon scintillations (Woo & Parker Solar Probe (Fox et al. 2016) is expected to sample
Armstrong 1979), interferometer phase scintillations using the solar wind as close as 10 Re. In situ measurements from the
Very Long Baseline Interferometers (VLBI; Cronyn 1972), SWEAP instrument on board the PSP can validate our findings
spectral broadening using coherent spacecraft beacons (Woo & regarding the density turbulence level and the proton
Armstrong 1979), and radar echoes (Harmon & Coles 1983). heating rate.
A related problem is the issue of turbulent heating in the The rest of the paper is organized as follows. In Section 2,
inner solar wind. It is well known that the expansion of the we describe imaging observations of the Crab Nebula made at
solar wind leads to adiabatic cooling, which is offset by some
( Gauribidanur in 2017 and 2016 June. The next sectionsort of heating process Richardson et al. 1995; Matthaeus
et al. 1999). The candidates for such extended heating range (Section 3) explains the methodology for obtaining the
from resonant wave heating (Cranmer 2000; Hollweg & turbulence levels from these images. This includes a brief
Isenberg 2002) to reconnection events (e.g., Cargill & discussion of the structure function, some discussion of the
Klimchuk 2004). Some studies have attempted to link inner scale of the density fluctuations, followed by the
prescription we follow in computing the density fluctuations
5 Also at IAASARS, National Observatory of Athens, GR-15236, Penteli, and solar wind heating rate at the inner scale. Section 4
Greece. summarizes our main results and conclusions.
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The Astrophysical Journal, 850:129 (7pp), 2017 December 1 Raja et al.
Table 1
This Table Describes the Observational Quantities and the Derived Plasma Parameters in the Solar Wind
S.No Date R Peak Flux Density r Ne Heating Rate
(R) (Jy) (erg cm-3 s-1)
(1) (2) (3) (4) (5) (6) (7)
Line of Sight to the Crab Does Not Include a Streamer
1 2016 Jun 12 10.18 1349 1.48 2.9E-3 3.9E-12
2 2016 Jun 18 13.46 1473 1.76 5.3E-3 1.0E-11
3 2016 Jun 19 16.83 1546 1.69 7.7E-3 1.9E-11
4 2016 Jun 20 20.27 2003 1.98 1.9E-3 2.2E-13
5 2017 Jun 09 21.13 2015 1.48 L L
6 2017 Jun 10 17.68 1732 1.57 6.2E-3 9.2E-12
7 2017 Jun 12 10.97 1386 1.50 3.4E-3 4.7E-12
8 2017 Jun 22 26.34 2015 1.40 L L
Line of Sight to the Crab Includes a Streamer
9 2016 Jun 17 10.20 845 2.44 L L
10 2017 Jun 17 9.41 901 2.51 L L
11 2017 Jun 18 12.61 800 1.65 L L
Figure 1. (Left) SOHO/LASCO C3 image of the solar corona (inverted grayscale image) observed on 2016 June 17 at 06:30 UT is shown. The innermost black circle
indicates the solar disk (radius=1 Re). The next concentric circle is the occulting disk of the coronagraph and its radius is 3.5 R. The outermost circle marks a
heliocentric distance of 30 R. In both the images, the black features are coronal streamers. Solar north is up and east is to the left. The small circles superposed on the
image represent the location of the Crab Nebula on different days during the period of 2016 June 8 to June 21. Its closest approach to the Sun is on 2016 June 14 at a
heliocentric distance of »5 R. The coronal streamer in the southwest quadrant occults the Crab Nebula on 2016 June 17 at a projected heliocentric distance
»10.2 R. The position angle (PA, measured counterclockwise from north) of the streamer is»235. (Right) SOHO/LASCO C2 image of the solar corona (inverted
grayscale) on 2016 June 17 at 06:36 UT is shown. The red contours represent observations of the solar corona using the GRAPH at 80 MHz. The elongated radio
contours correspond to emission from the streamers in northeast and southwest quadrants.
2. Observations: Scatter-broadened Images flux density is »16,296 Jy at 80MHz. The flux density of the
of the Crab Nebula Crab Nebula (when it is far from the Sun and is not therefore
scatter-broadened by solar coronal turbulence) is »2015 Jy at
The radio data were obtained with the Gauribidanur 80MHz. We imaged the Crab Nebula at different projected
RAdioheliograPH (GRAPH; Ramesh et al. 1998; Ramesh heliocentric distances shown in column (3) of Table 1 in the
2011) at 80MHz during the local meridian transit of the Crab years 2016 and 2017.
Nebula. The GRAPH is a T-shaped interferometer array with We have used white light images of the solar corona
baselines ranging from»80 to»2600 m. The angular resolution obtained with the Large Angle and Spectrometric Coronagraph
is ≈5 arcmin at 80MHz, and the minimum detectable flux (LASCO) on board the SOlar and Heliospheric Observatory
5s level) is »50 Jy for 1 s integration time and 1MHz (SOHO; Brueckner et al. 1995) for general context, and to
bandwidth. Cygnus A was used to calibrate the observations. Its identify features like coronal streamers. Figure 1 shows the
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The Astrophysical Journal, 850:129 (7pp), 2017 December 1 Raja et al.
Figure 2. Image on 2016 June 12 shows the scatter-broadened Crab Nebula at a projected heliocentric distance of10.18 R during its ingress into the inner solar wind.
The images on 2016 June 17 (at 10.2 R), 2017 June 17 (9.41 R), and 2017 June 18 (12.61 R) depict the scatter-broadened Crab Nebula observed through coronal
streamers during its egress from the solar wind. The arrows depict the sunward direction on each day. The major axis of each image is perpendicular to the magnetic
field lines, which are directed radially outward from the Sun.
white light images of the solar corona obtained with the observed through the solar wind at 10.18 R during ingress.
LASCO C3 (left) and C2 (right) coronagraphs on 2016 The one on 2016 June 17 was observed at 10.20 R, while the
June 17. The black features in both inverted grayscale images one on 2017 June 17 at 9.41 R and the one on 18 June 2017 at
are coronal streamers. The location of the Crab Nebula between 12.61 R during egress. The Crab Nebula was occulted by a
2016 June 8 and 21 is marked by the red circles on the LASCO coronal streamer on 2016 June 17 and on 2017 June 17 and 18.
C3 images. On 2016 June 17, the Crab Nebula was observed These scatter-broadened images are markedly anisotropic. This
through a streamer in the southwest quadrant. The streamer was aspect has been noted earlier, for the Crab Nebula (Blesing &
associated with an active region NOAA 12555 located at Dennison 1972; Dennison & Blesing 1972) as well as other
heliographic coordinates S09W71. The contours superposed sources (Armstrong et al. 1990; Anantharamaiah et al. 1994).
over the LASCO C2 image are from the GRAPH observations Note that the major axis of these images is always perpend-
at 80MHz showing radio emission from the streamers in icular to the heliocentric radial direction (which is typically
northeast and southwest quadrants (Ramesh 2000). assumed to be the magnetic field direction at these distances)—
Some representative 80MHz GRAPH images of the Crab this is especially evident when the Crab is occulted by a
Nebula are shown in Figure 2. The image on 2016 June 12 was streamer. Using the Gauribidanur Low-frequency Solar
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The Astrophysical Journal, 850:129 (7pp), 2017 December 1 Raja et al.
in the solar wind,
Ne (R) = 7.2 R-2 + 1.95 ´ 10-3 R-4
+ 8.1 ´ 10-7 R-6 cm-3, (1)
where “R” is the heliocentric distance in units of astronomical
units (au, 1 au = 215R). The background electron density is
used to compute the inner scale of the turbulent density
spectrum. We assume that the inner scale li is given by the
proton inertial length (Verma et al. 1996; Leamon
et al. 1999, 2000; Smith et al. 2001; Bruno & Trenchi 2014;
Chen et al. 2014), which is related to the background electron
density by
li (R) = vA(R) Wp(R) = 2p ki (R) = 228 ´ Ne (R) km,
(2)
Figure 3. Peak flux density of the Crab Nebula on different days of 2016 June
(red circles) and 2017 (blue squares). The red and blue data points shown in the where Ne is the electron density in cm
-3, ki is the wavenumber,
shaded area indicate instances when the Crab Nebula was observed through a vA is the Alfvèn speed, and Wi is the proton gyrofrequency. We
streamer in 2016 and 2017 respectively. note that our definition differs slightly from that of Coles &
Harmon (1989), Harmon (1989), and Yamauchi et al. (1998)
Spectrograph (GLOSS; Kishore et al. 2014) observations we who use li = 3 ´ vA(R) Wp(R) and ki = 3 li.
identified that there were no transient radio bursts during the
time of observations. The parameters for all observations of the 3.2. The Structure Function Df
Crab Nebula in 2016 and 2017 are tabulated in Table 1.
Figure 3 shows the observed peak flux density of the Crab The structure function Df (s) is defined by (Prokhorov
Nebula with respect to its projected heliocentric distance. The et al. 1975; Ishimaru 1978; Coles & Harmon 1989; Armstrong
red circles and blue squares are for the 2016 and 2017 et al. 1990)
observations respectively. Note that, in a given year, the data Df(s) = -2 lnG(s) = -2 ln [V (s) V (0)], (3)
points obtained during ingress and egress were plotted together
with the (projected) heliocentric distance. where the quantity s represents the baseline length, G(s) is the
The observations shown in the shaded region in Figure 3 mutual coherence function, V(s) denotes the visibility obtained
represent instances where the Crab Nebula was occulted by a with a baseline of length s, and V(0) denotes the “zero-length”
coronal streamer. Evidently, the peak flux density in these baseline visibility. The quantity V(0) is the peak flux density
instances is considerably lower (as compared to the flux when the Crab Nebula is situated far away from the Sun, and is
corresponding to a similar heliocentric distance when the Crab
) unresolved; we set it to be »2015 Jy at 80MHz (Braudeis not occulted by a streamer . This could be because the line of
sight to the Crab Nebula passes through more coronal plasma et al. 1970; McLean & Labrum 1985). The images of the Crab
during instances of streamer occultation, leading to enhanced Nebula in Figure 2 are obtained by combining the visibilities
scatter broadening. In turn, this leads to a larger scatter-broadened from all the baselines available in the GRAPH. We are
image and a consequent reduction in the peak flux density. interested in the turbulent density fluctuations at the inner scale,
which is the scale at which the turbulent spectrum transitions
3. Turbulent Density Fluctuations and Solar Wind from a power law to an exponential turnover. This is typically
Proton Heating Rate the smallest measurable scale; we therefore compute the
structure function corresponding to the longest available
The angular broadening observations of the Crab Nebula baseline (s=2.6 km), since that corresponds to the smallest
described in the previous section can be used to infer the scale.
amplitude of turbulent density fluctuations and associated
heating rate of protons in the solar wind. The main quantity 3.3. The Amplitude of Density Turbulence Spectrum (C2 )
inferred from the observations is the structure function, which N
is essentially the spatial Fourier transform of the visibility The turbulent density inhomogeneities are represented by a
observed with a given baseline. The structure function is used spatial power spectrum, comprising a power law together with
to estimate C2N, the so-called “amplitude” of the turbulent an exponential turnover at the inner scale:
density spectrum. The density spectrum is modeled as a power 2 2 2 2 -a 2
law with an exponential cutoff at an “inner scale.” We assume Pdn (k, R)= CN (R)(r kx + ky )
that the inner scale is given by the proton inertial length. We ⎡ ⎛ l 2⎤i (R) ⎞
elaborate on these aspects in the subsections below. ´ exp ⎣⎢-(r
2 k 2 2x + ky )⎜⎝ ⎟⎠ ⎦⎥, (4)2p
3.1. Background Electron Density and the Inner Scale where k = r2 k 2x + k
2
y is the wavenumber, and kx and ky are
Since our aim is to estimate the level of turbulent density the wavenumber along and perpendicular to the large-scale
fluctuations in relation to the background density (Ne), we use magnetic field respectively. The quantity ρ is a measure of
Leblanc density model (Leblanc et al. 1998) to estimate the Ne the anisotropy of the turbulent eddies. In our calculations, we
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The Astrophysical Journal, 850:129 (7pp), 2017 December 1 Raja et al.
use the axial ratio of the scatter-broadened images at 80MHz streamer. On 2016 June 17, the Crab Nebula was situated at a
(shown in Table 1) for ρ. The quantity C2N is the amplitude of similar projected heliocentric distance (10.2R), but the line of
density turbulence, and has dimensions of cm-a-3, where α is sight to it passed through a coronal streamer. From
the power-law index of the density turbulent spectrum. At large Equation (5), it is evident that this ratio is equal to the ratio
scales, the density spectrum follows the Kolmogorov scaling of theDLs in the two instances. In other words, the presence of
law with a = 11 3. At small scales, (close to the inner scale, a streamer approximately doubles the path length along the line
» a = of sight over which scattering takes place. This is nearly thewhen s li) the spectrum flattens to 3 (Coles & same as the density enhancement in a streamer estimated from
Harmon 1989). Since we are interested in the density low frequency radio observations (Ramesh et al. 2000).
fluctuations near the inner scale, we use a = 3. Although we show 80MHz observations in this paper, we
Many authors use analytical expressions for the structure also have simultaneous observations at 53MHz. The structure
function that are applicable in the asymptotic limits s  li or function (Equation (5)) is proportional to the square of the
s  li (Coles et al. 1987; Armstrong et al. 2000; Bastian 1994; observing frequency (i.e., Df (s) µ l2). This predicts that
Subramanian & Cairns 2011). However, these expressions are the ratio of the structure functions at 80 and 53.3 MHz should
not valid for situations (such as the one we are dealing with in be 0.44. Our observations yield a value of 0.43 for this ratio,
this paper) where the baseline is comparable to the inner scale; and are thus consistent with the expected scaling.
i.e., s » li. We therefore choose to use the General Structure
Function (GSF), which is valid in the s  li and s  li 3.4. Estimating the Density Modulation Index (N = dNk Ne)
regimes as well as when s » li (Ingale et al. 2015b). In the
e i
present case, largest baseline length »2.6 km is comparable to The density fluctuations dNki at the inner scale can be related
the inner scale lengths »4.56 km. The GSF is given by the to the spatial power spectrum (Equation (4)) using the
following expression: following prescription (Chandran et al. 2009).
8p2r2
2 3 2 3-a
2 -1
( ) e l DL ⎜
⎛ a - 2 2⎟⎞ CN (R)l a-2i (R) dNk (R) ~ 4pki PdN (R, ki) = 4pCN (R)ki e , (6)iDf s = G 1 -
r 2a-2(a - 2) ⎝ 2 ⎠ (1 - f 2p (R) f 2 ) where ki º 2p li. We estimate dNki by substituting C2N calculated
⎧ ⎡ ⎛ ⎞2⎤ ⎫ in Section 3.3 and using a = 3 in Equation (6). We then use this⎪⎨ ⎢ a - 2 s ⎪´ F - , 1, -⎜ ⎟ ⎥ - 1⎬ rad2, dNki and the background electron density (Ne, Section 3.1) to⎪⎩1 1⎣⎢ 2 ⎝ li (R) ⎠ ⎦⎥ ⎪⎭ estimate the density modulation index (Ne) defined by
(5)
 ( dNR) º ki (R)Ne . (7)
where F is the con uent hyper-geometric function, r is the Ne (R)1 1 fl e
classical electron radius, λ is the observing wavelength, R is the The density modulation index in the solar wind at different
heliocentric distance (in units of Re), DL is the thickness of heliocentric distances is computed using Equation (7). The
the scattering medium, fp and f are the plasma and observing results are listed in column (6) of Table 1. The numbers in
frequencies respectively. Substituting the model densities and Table 1 show that the density modulation index (Ne) in the
a -3 -3= 3 in Equation (5) enables us to calculate C2 . Following solar wind ranges from 1.9 ´ 10 to 7.7 ´ 10 in theN
Sasikumar Raja et al. (2016), we assume the thickness of the heliocentric range »10–20 R. We have carried out these
scattering screen to beDL = (p 2)R , where, R is the impact calculations only for the instances where the Crab Nebula is not0 0 occulted by a streamer.
parameter related to the projected heliocentric distance of the
Crab Nebula in units of centimeters. When the Crab Nebula is 3.5. Solar Wind Heating Rate
occulted by a streamer, however, this estimate of DL is not
valid. It is well known that the streamer owes its appearance to We next use our estimates of the turbulent density
the fact that the line of sight to the streamer intercepts excess fluctuations (dNki) to calculate the rate at which energy is
coronal plasma that is contained around the current sheet deposited in solar wind protons, following the treatment of
Ingale (2015a). The basic assumption used is that the density
“fold.” It therefore stands to reason that theDL along a line of
fluctuations at small scales are manifestations of low frequency,
sight that intercepts a streamer will be larger than that along a oblique (k^  k), Alfvèn wave turbulence. The quantities k⊥
line of sight that does not include a streamer. In view of this, and kP refer to components of the wave vector perpendicular
we use the formula DL = (p 2)R0 and compute the density and parallel to the background large-scale magnetic field
fluctuation amplitude and turbulent heating rate only for the respectively. The turbulent Alfvèn wave cascade transitions to
instances where the Crab Nebula is not occulted by a streamer. such oblique Alfvèn waves (often referred to as kinetic Alfvèn
In the instances where it is occulted by a streamer, we can waves) near the inner/dissipation scale. We envisage a
estimate the extra line-of-sight path length implied by the situation where the turbulent Alfvèn wave cascade resonantly
presence of the streamer. In order to do this, we first compute damps on (and thereby heats) the protons at the inner scale.
the structure function (Equation (5)) in the instances when the Since this implicitly assumes that the Alfvèn waves do not
line of sight to the Crab Nebula contains a streamer. We then couple to other modes at the inner scale, our estimate of the
estimate the ratio of this quantity to the structure function (at a proton heating rate is an upper limit. As explained in
similar heliocentric distance) when the line of sight does not Section 3.1, we assume that the inner scale is the proton
intercept, a streamer turns out to be »2. For instance, inertial length, which is expressible as li = vA Wp, where vA is
Df (s = 2.6 km, 2016 June 17)/Df(s=2.6 km, 2016 June the Alfvèn speed and Wp is the proton gyrofrequency. This way
12)=2.16. On 2016 June 12, the Crab Nebula was situated at of writing the proton inertial length emphasizes its relation to
10.18R and the line of sight to it did not pass through a the resonant damping of Alfvèn waves on protons.
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The Astrophysical Journal, 850:129 (7pp), 2017 December 1 Raja et al.
distances. These values are tabulated in column (7) of Table 1.
Figure 4 depicts the density modulation index and the solar
wind heating rate graphically as a function of heliocentric
distance.
4. Summary and Conclusions
4.1. Summary
We have imaged (Figure 2) the Crab Nebula at 80MHz
using the GRAPH in 2017 and 2016 June, when it passed close
to the Sun and was obscured by the turbulent solar wind. Since
the Crab Nebula is a point source at 80MHz when it is far from
the Sun, these images are evidence of anisotropic scatter
broadening of radiation emanating from it as it passes through
the turbulent solar wind. We calculate the structure function
with the visibilities from the longest baselines (2.6 km) used in
Figure 4. Variation of the density modulation index (red circles) and the solar making these images. The structure function is used to infer the
wind proton heating rate (blue squares) with projected heliocentric distance. amplitude of the density turbulence spectrum (C2N), which is
We note that the proton heating rate is correlated with the density modulation then used to compute the magnitude of the turbulent density
index.
fluctuations at the inner scale (Equation (6)). This is then used
to compute the density modulation index (Equation (7)).
The specific energy per unit time ( , erg cm-3 s-1) in the Assuming that the turbulent Alfvèn wave cascade in the solar
turbulent Alfvèn wave cascade is transferred from large scales wind dissipates on protons at the inner scale, we calculate the
to smaller ones, until it dissipates at the inner/dissipation scale. heating rate of protons in the solar wind (Equation 8). The
The proton heating rate equals the turbulent energy cascade rate density modulation index and solar wind proton heating rate
at the inner scale (k ), which is given by (Hollweg 1999;i are plotted in Figure 4 as a function of heliocentric distance.
Chandran et al. 2009; Ingale 2015a)
 ki (R) = c0r 3pki (R)dvk (R) erg cm-3 s-1, (8) 4.2. Conclusionsi
where c0 is a constant usually taken to be 0.25 (Howes
The main conclusions of this paper pertain to the anisotropy
of the scatter-broadened image of the Crab Nebula, the density
et al. 2008; Chandran et al. 2009) and r = m N -3p p e(R) g cm , modulation index of the turbulent fluctuations in the solar wind
with mp representing the proton mass in grams. The quantity and the solar wind proton heating rate from 9–20 R. Some of
ki = 2p li is the wavenumber corresponding to the inner scale the conclusions are:
(Equation (2)) and dvki represents the magnitude of turbulent
velocity fluctuations at the inner scale. The density modulation 1. The 80MHz scatter-broadened images of the Crab
index Ne and the turbulent velocity fluctuations are related via
Nebula at heliocentric distances ranging from 9 to
the kinetic Alfvèn wave dispersion relation (Howes et al. 2008; 20 R in the solar wind are anisotropic, with axial ratios
typically 2 (Table 1). The major axis of the Crab
Chandran et al. 2009; Ingale 2015a) Nebula is typically oriented perpendicular to the magnetic
⎛1 + g k 2 (R)r2 (R) ⎞ field direction, as in Anantharamaiah et al. (1994) and
d ivki (R) = ⎜⎜ i i ⎟⎝ ⎟⎠Ne(R, ki)vA(R). (9) Armstrong et al. (1990; although their observations wereki (R)li (R) at much smaller distances from the Sun).
2. On 2016 June 17 and 2017 June 17, a coronal streamer
The adiabatic index gi is taken to be 1 (Chandran et al. 2009) was present along the line of sight to the Crab Nebula.
and the proton gyroradius (ri) is given by The line of sight to the Crab encountered more coronal
r (R) = 102 ´ m1 2T1 2B-1(R) cm, (10) plasma on these days, as compared to the days when ai i streamer was not present. The axial ratio of the scatter-
where μ is the ion mass expressed in terms of proton mass (»1) broadened images on these days was somewhat larger
and Ti is the proton temperature in eV. We use Ti = 86.22 eV, (»2, see Table 1) and the peak flux density is
which corresponds to a temperature of 1 ´ 106 K. considerably lower (Figure 3), reflecting this fact. In the
The Alfvèn speed (vA) in the solar wind is given by presence of a streamer, the path length over which
scattering takes place was found to be approximately
vA(R) = 2.18 ´ 1011m-1 2N-1 2e (R)B(R) cm s-1, (11) twice that of when the streamer was not present.
and the magnetic eld stength (B) is taken to be the Parker 3. The density modulation index (Ne º dNe Ne) at the innerfi
scale of the turbulent spectrum in the solar wind from
spiral mangetic field in the ecliptic plane (Williams 1995) 9–20 R ranges from 1.9 ´ 10-3 to 7.7 ´ 10-3 (see
B(R) = 3.4 ´ 10-5R-2 (1 + R2)1 2 Gauss, (12) Table 1). Earlier estimates of Ne include Mugundhan et al.
(2017) who reported 0.006 ± 0.002 from 1.6–2.2 Re,
where, “R” is the heliocentric distance in astronomical units. Sasikumar Raja et al. (2016) who reported 0.001 
Equations (12), (11), (10), and (9) and the density modulation Ne  0.1 from 10–45 Re, 0.001  Ne  0.02 reported
index computed in Section 3.4 are used in Equation (8) to by Bisoi et al. (2014) in the distance range of 56–185 Re,
compute the solar wind heating rate at different heliocentric and 0.03  N  0.08 reported by Spangler & Spitlere
6
The Astrophysical Journal, 850:129 (7pp), 2017 December 1 Raja et al.
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