Using phase scintillation spectral measurements to determine angle-of-arrival fluctuations during solar superior conjunction

Authors


Abstract

[1] In this study, we develop a complete theoretical approach to derive the angle-of-arrival fluctuations (AAF) of radio signals passing through the turbulent solar plasma medium during solar superior conjunctions. Using the power spectra of phase fluctuations measured at various solar elongation angles (or impact heliocentric distances) from the Cassini spacecraft, we have defined the dependence of the AAF variance on the heliocentric distance as ∼r−3.5 within a range very close to the Sun. This quantity decreases with increasing distance, with a slope significantly less steep than that previously expected. The AAF expression is theoretically derived by assuming a frozen turbulence and by converting a phase temporal variation into a spatial variation. To perform this calculation, the solar plasma medium is treated as an anisotropic ionized medium by applying the Booker electron irregularity spectrum model and the phase expression in term of the electron refractive index. Using the phase spectral measurements from the Cassini spacecraft during a solar superior conjunction, coefficients of the expression are calibrated, and the final AAF results are quantitatively obtained.

1. Introduction

[2] The angle-of-arrival fluctuation (AAF) is the apparent angle scattering of the received radio frequency (RF) signal beam around its centroid due to the spatial gradients of the propagation medium. The AAF is sometimes also referred to as angular broadening or angular variance. Solar plasma density irregularities and/or gradients can cause both spatial and temporal phase fluctuations of the radio signals passing through the solar coronal medium. During the superior solar conjunction period, the raypath of RF signals from a distant spacecraft to an Earth receiving station can pass very close to the base of the solar corona where the plasma density is high, and large turbulent irregularities and/or gradients exist. For RF signals passing through these regions, large plasma density fluctuations result such that the amplitude scintillation can become saturated (scintillation index ∼1), as shown in a study by Morabito et al. [2003], The deep amplitude fading that can occur in the region of strong scattering are associated with rapid phase variations that can degrade the phase measurements, such effects on signal phase including thermal noise have been discussed by Knepp [2004] in the context of assessing accuracy on ionospheric TEC measurements. When such effects are not significant, the phase fluctuation measurements may provide a tool in assessing the severity of communication signal degradation and in diagnosing the solar wind properties at an impact distance very close to the Sun.

[3] While a temporal phase variation causes a phase shift, a spatial fluctuation in phase will cause angle-of-arrival fluctuation for a point source. Apparent angle fluctuations of a fraction of a degree can cause signal loss and pointing errors for a large ground station antenna. For a large antenna with narrow beam width, the defocusing on apparent angular position can cause the antenna gain loss of several dBs. Several experimental studies on amplitude scintillation, phase scintillation, and spectrum broadening have been performed based on solar conjunction measurements [Woo et al., 1976; Woo, 1977; Morabito et al., 2003], but no investigations on angle-of-arrival fluctuations caused by the solar wind have been performed yet, even though some theoretical and experimental studies on angular fluctuations related to the Earth atmosphere were performed [Crane, 1978; Vilar and Smith, 1986; Wheelon, 1957, 2001].

[4] In the studies of Woo et al. [1976] and Woo and Armstrong [1979], power spectra of phase fluctuations during the solar conjunction for Mariner 10, Viking, Helios and Pioneer spacecraft were investigated. Berman [1977] also analyzed the temporal phase fluctuation spectra, power law slopes, and the heliocentric distance dependence from these measurements. Using the radio signals transmitted from the Earth satellites, Crane [1978] studied the variance and spectra of the AAF due to the ionospheric scintillation. Vilar and Smith [1986] found that the tropospheric effects on the AAF are about an order of millidegrees at the frequency band of 11.8 GHz. Both ionospheric and tropospheric effects on the angle-of-arrival fluctuations are relatively very small when compared with those due to solar plasma effects during the solar conjunction.

[5] In the previous studies [Woo et al., 1976; Woo and Armstrong, 1979], the authors treated the solar wind plasma as a neutral gas by applying an isotropic Kolmogorov turbulence model with a −8/3 slope for phase spectrum of the solar wind. However, for angular broadening, we cannot use this neutral atmospheric turbulence model to describe the solar wind plasma irregularity, because the plasma density variations are anisotropic. These elongated irregularities can be described by the Booker spectrum model [Booker and Ferguson, 1978; Wheelon, 2001]. Using the phase expression for a neutral atmospheric model will generate an angular broadening value that is independent of the signal frequency. However, for a plasma medium, the AAF effect has a strong dependence on the signal frequency.

[6] Using the solar plasma density model and the Booker spectral model, recent theoretical calculations show that angle-of-arrival fluctuations on Ka band signals during solar superior conjunction can exceed 20 mdeg at impact distances <1.6 solar radii (Rs) [Ho et al., 2008]. It is relatively difficult to measure angular fluctuations caused by solar corona plasma density irregularities. However, using the phase spectral information under an assumption of a frozen turbulence (Taylor hypothesis [Wheelon, 2001]), the power spectrum of phase scintillation can be converted into the spectrum of angle-of-arrival fluctuation. Further by integrating the spectrum, the variance in angular fluctuations can be determined. In this study, we will use this new approach to calculate the AAF by defining the relationship between the phase spectrum and the angular broadening spectrum. It also provides an alternative way to validate the theoretical results with experimental data of phase spectrum measured during solar conjunction as we showed in the following sections. Measured data used for comparison with our theoretical model were obtained during the August 2007 solar conjunction of the Cassini spacecraft. The downlink signals were received by one the 34 m diameter dish receiving antenna of the NASA Deep Space Network (DSN) at X band (8.4 GHz) and Ka band (32 GHz).

2. Relationship Between Phase and Angle-of-Arrival Fluctuations

[7] The real path of a radio signal wanders around a nominal raypath from a distant spacecraft to the Earth after passing through the solar corona region due to the plasma irregularities in refractive index. As a result, angle-of-arrival fluctuations occur relative to the receiver due to the random variation of refractive index along the paths.

[8] Under Taylor's frozen field assumption, the temporal covariance can be converted into the spatial covariance [Wheelon, 2001]. This assumption implies that the entire plasma medium is transported at a constant solar wind speed with a velocity, v, which is perpendicular to the raypath direction as shown in Figure 1. This is a proper assumption for the solar wind plasma under most situations since the solar wind usually is frozen with the magnetic field embedded at a distance very close to the Sun.

Figure 1.

Geometry illustrating the frozen-turbulent-medium assumption: Solar wind has a radial convection speed V, perpendicular to the signal path from the Cassini spacecraft in orbit around Saturn to the Earth. Solar wind plasma irregularities are frozen in and carried by the wind without changes with time.

[9] The phase variation measured for a delay time (t + τ) is assumed identical to the phase measured at t at a different position separated by Δρ = vτ. Thus we have:

equation image

where r′ = r + Δρ = r + vτ, ϕ(r, t) is the signal phase as a function of position r and time t, and 〈−〉 denotes the ensemble average. Thus, all of time changes in phase are associated with a simple translation of a spatial field distribution.

[10] By making use of geometrical identities and assuming frozen flow turbulence, we can relate the phase spectrum (Wϕ) and the spectrum of the angle of arrival fluctuations (Wθ) by [Vilar and Smith, 1986]

equation image

where λ is the wavelength of the radio signal, while ω is the angular frequency of the spectrum due to the medium's fluctuation, and v is the transverse solar wind velocity. Thus from the phase spectrum, we can infer the spectrum of the angle of arrival. Furthermore, the angle-of-arrival variance can be obtained by integrating its spectrum over all frequencies.

3. Theoretical Expressions

[11] The generalized anisotropic Booker spectrum model [Booker and Ferguson, 1978] defines the spectrum of electron density irregularities, ΨN(κ, r), as follows [Wheelon, 2001]:

equation image

where 〈δN2〉 is the variance of electron number density fluctuations, ℜ is the axial ratio of the elongation of plasma irregularities along the magnetic field lines (30 < equation image < 50), equation image is the turbulence wave number, κ0 = 2π/L0, where L0 is the outer scale of turbulence ∼2000 km [Woo, 1977], κm = 5.91/l0 (where l0 is the inner scale of turbulence), v is the spectral index of 3-D refractive spectrum (3 < ν < 4), Qν is a normalization constant, and Θ is the angle between the wave number vector and the magnetic field.

[12] The experimental measurements [Woo and Armstrong, 1979] show that when the SEP (Sun-Earth-Probe) angle is very small (that is, the signal path grazes very close to the solar surface) the phase spectral index (ν − 1) has a value of 2.07. In this study we have assumed that ν = 37/12 = 3.0833. That is

equation image

where A(r) = QνδN2(r)〉/2π. We can see that this model is a special case of the generalized von Karman turbulence spectrum model [Vilar and Smith, 1986] of Φn(κ) = αn2 + κ02)n/2eequation image, when κ0 = (1/L0) → 0 and κm = equation image.

[13] The signal phase fluctuation as a function of time t is given by:

equation image

where k is wave number of radio signal, ɛ is the dielectric constant (related to the refractive index, n by ɛ = n2), L is the path length from transmitter to receiver, Δɛ(s, t) represents dielectric constant variations along the path s and at time t. For the case of a plasma medium, ɛ = ɛ0reλ2N (where N is the electron number density), and the dielectric gradient (Δɛ) is related to the electron density fluctuations (δN) by

equation image

where re is the classical electron radius (2.818 × 10−15 m), and λ = 2π/k. Thus, its phase expression is also a function of λ (that is, ϕ(t) = −πreλequation imageΔN(s, t)ds), which is significantly different than that for a neutral atmosphere.

[14] When the signal phase has a temporal variation over a spatial range, the phase spectrum can be calculated using the phase covariance as follows:

equation image

where the phase covariance is a function of the spatial correlation of the electron-density fluctuations.

[15] Using the Booker spectrum to represent the spatial covariance of electron-density irregularities which have a convection speed v, we have [Wheelon, 2003]:

equation image

where J is the Bessel function, ar is the antenna radius, and z is the direction along the raypath.

[16] After applying ΨN = A(r−37/12 for the case of Θ = 90°, we have the phase spectrum

equation image

A Gaussian function can be used to approximate the Airy function:

equation image

where b = 0.4832.

[17] By defining an aperture-smoothing angular frequency, ωs = v/bar, we have

equation image

where Γ(a) is Gamma function while U(a, b, z) is the second type of Kummer function. The above expression approximately has a frequency dependence of ω−25/12exp(−ω2/ωs2). We can use this model to fit the phase spectrum from the experimental measurements.

[18] For the integral along the path, z, we have a plasma irregularity model which is an average of both near and far fields of the solar plasma [Muhleman et al., 1977; Ho et al., 2008]. We assume δNa0/r4, then

equation image

where α is the angle between the Sun and spacecraft as viewed from the Earth (that is, solar elongation angle, or SEP angle).

[19] Now let us make an estimate to the ratio of ω/ωs in equation (11). When the solar wind speed, v = 100 km/s and ar = 17 m, the aperture-smoothing frequency ωs = v/bar = 100 × 103/(0.4832 × 17) = 2.0 × 103rad/s. The phase spectra measurements show that the phase fluctuations are mainly in a frequency range of 10−4–100 Hz. Thus, the aperture-smoothing frequency is much larger than the frequency of phase fluctuations, because a finite-size receiving antenna can be viewed as a point antenna, relative to the large drift velocity of solar wind plasma. The aperture smoothing effects on phase fluctuations and angle-of-arrival fluctuations are almost negligible. When ωωs, using a small argument expansion for the Kummer function, U(a, b, z), we have Γ(1/2)U(1/2, −1/24, ω2/ωs2) ≈ 1.95 and exp(−ω2/ωs2) ≈ 1 − ω2/ωs2. Thus the phase spectrum becomes:

equation image

and the spectrum for angle-of-arrival fluctuation using the relationship (2) becomes:

equation image

To obtain the value of angular variance, we can integrate equation (14) over all spectral frequencies. The integrating range for ω is from 0 to its upper limit, ωm = κmv(fm = κmv/2π), where the upper cutoff wave number κm = 0.942κs and κs is inner-scale wave number (κs = 2π/l0). When the inner-scale length l0 = 10 km [Woo, 1977], we have fm = 9.4Hz. Thus, the angular variance becomes

equation image

[20] When we use an isotropic generalized von Karman model: ΦN(κ) = αn2 + κ02)p/2 and p = 3 to replace ΨN in equation (8), the phase spectrum takes the following form

equation image

where IK(α) = equation imageαnN(r)〉dz. Using equation (2), the spectrum for the angle-of-arrival fluctuation becomes:

equation image

The angular variance is given by

equation image

We can find that there are similar dependence on spectral frequency for two types of plasma irregularity models in equations (14) and (17). This equation clearly shows the dependence on the signal's wavelength (frequency), and SEP angle (the heliocentric distance). The RMS AAF in all directions can be calculated as: Δequation imagerms = 1.414equation image.

4. Experimental Measurements

[21] In order to detect or measure the angular fluctuations due to solar plasma irregularities during the solar conjunction, techniques such as the monopulse tracking or interferometry may be employed. However, in the case of lack of these measurements, we suggest an alternative approach by examining the measured power spectrum of the phase scintillations during the solar conjunction. A basic assumption is that these phase fluctuations with time can map into spatial fluctuations when they are convected as a whole with a constant solar wind speed as provided in section 2.

[22] Doppler-frequency tracking data for the Cassini spacecraft are used for this study. The data are selected from a solar conjunction that took place during a 7 day period from 19 to 25 August 2007. Cassini is in orbit around the planet Saturn which in turn has a much larger orbit around the Sun than that of the Earth. Thus one superior solar conjunction event with Cassini occurs roughly every year relative to the Earth and thus the DSN (Deep Space Network) receiving station. During this solar conjunction, the minimum SEP angle of the Cassini raypath to the Sun was 1.247°. During this period, there are five intervals with complete measurements of dual frequency (X/Ka). The downlinks were coherent with an X band uplink (two-way) and were recorded using the DSN's closed-loop receivers. Cassini radio experiment parameters for this solar conjunction downlinks are listed in Table 1. Each track has about 6 h of data. The tracking measurements were made from worldwide DSN stations: DSS 25, 26, 34 and 55 (34 m diameter BWG antennas). The data are sampled at 1 s resolution and include Doppler frequency, phase, and receiving power to noise density (Pc/N0), etc.

Table 1. Cassini Solar Conjunction Experiment Parameters in 2007
DateDOYSEP Angle, αDistance From Sun, r/rsReceiving Station IDData Length(s)
Two-Way XTwo-Way Ka
19 Aug2312.6810.32520,70020,700
20 Aug2321.987.62521,40021,400
21 Aug2331.435.52520,90020,900
22 Aug2341.254.82621,50021,500
23 Aug2351.566.02521,60021,600
24 Aug2362.178.325/3421,40021,400
25 Aug2372.8811.15521,60021,600

[23] Instead of directly using the measured Doppler frequency, we have used the more accurate phase data to reconstruct the frequency residuals. We difference the received X band frequency and 1/3.8 of the Ka band frequency to remove all contributions which are linearly proportional to the signal frequency (e.g., Doppler shift due to spacecraft motion, instrumental effects, tropospheric effect, etc.). The differenced frequency data ideally only contains the contribution of the plasma density irregularities. The spectrum of the phase difference scintillations is dominated by the plasma density fluctuation spectrum. The following steps are used to generate the power spectra of phase fluctuations from these dual frequency measurements:

[24] At first, the phase data of two frequencies (X and Ka) are used separately to calculate the Doppler frequency by the phase difference technique. That is, the frequency at ti, is the difference between the phases at ti−1 and at ti+1 divided by the time interval Δt: fi(Hz) = (ϕi+1 − ϕi−1)/(ti+1ti−1). Then the frequency residuals at X band due to plasma can be approximated by taking the difference between X band and Ka band: ΔfX(i) = fX(i) − fKa(i)/3.8, because there is a precise ratio of 3.8 between the Ka and X band downlink frequency. By doing this, all frequency fluctuations with linear frequency dependence cancel out due to both being coherent with the common uplink. Mainly those frequency variations due to the solar plasma (with nonlinear) effects remain. Finally the phase fluctuations due to the solar wind plasma turbulence are reconstructed through: ϕX(i) = equation imageΔfXdt = ΔfX(it + ϕi−1.

[25] Figure 2 shows about 6 h of differenced frequency residual data from a Cassini track on 2007/231 (DOY), referred at X band. During this track, the SEP angle is about 2.68°. (The signal path has an impact distance of 10.3 solar radii from the center of the Sun.) We can see that most frequency residuals span a range between −0.2 to 0.2 Hz with a mean value around zero, but some large outliers runup to ±1.0 Hz. We do not know if these outliers are related to any solar activity.

Figure 2.

A time series of Cassini dual-frequency differenced residuals referred to X band acquired using DSS 25. Data were obtained on DOY 231, 2007, at an SEP angle of 2.68° during solar conjunction.

[26] Figure 3 displays the signal phases reconstructed using the frequency residuals in Figure 2 for day 2007/231. Because the dual frequency differencing removes nondispersion effects such as Doppler effect, the phase fluctuations should be mainly caused by solar plasma turbulence.

Figure 3.

Phases reconstructed from the differenced frequency residuals shown in Figure 2. These phase fluctuations are basically caused by the solar plasma turbulence.

[27] Figures 4 and 5 show the time series of the differenced frequency residuals and reconstructed phases, respectively, for the track of 2007/234 at the minimum SEP (the closest distance to the Sun) angle of 1.25° during the conjunction. During this period, we find that frequency residuals have constantly larger fluctuations with a range of ±0.2 Hz. Because all nondispersion effects have been removed using the X/Ka dual-frequency difference, the phase fluctuations shown in Figure 5 are likely dominated by phase scintillations due to solar charged particles (both temporal and spatial phase variations).

Figure 4.

Frequency residuals at X band measured by DSS 26 from a Cassini track on DOY 234, 2007. The SEP is at the minimum value of 1.247° during this solar conjunction.

Figure 5.

Time history of phases reconstructed using the frequency residuals in Figure 4. The Doppler and other nondispersive effects have been excluded in the dual-frequency difference from which the phase fluctuations were reconstructed.

[28] Figure 6 shows the phase spectra for three ingress tracks on days 2007/231, 2007/232, and 2007/234, while Figure 7 shows the phase spectra for three egress tracks on days 2007/234, 2007/236, and 2007/237, respectively. All of these spectra were fit using equation (13) where the spectral index (ν − 1) is 25/12. Data are sampled at a rate of 1 sample per second, that is, its Nyquist frequency is 0.5 Hz. To construct these spectra, we have used a window of 10000 data points (2.78 h). Thus, the spectra have a frequency range from 10−4 to 100 Hz. We can see that, in general, for signal paths with smaller SEP angles, the spectra have larger power fluctuations. However, there is an exception for day 232 (α = 1.98°) and day 234 (α = 1.25°). The day 232 spectrum has slightly higher power than that of day 234 which has the minimum distance from Sun. This may suggest that on day 232 the signal path encountered larger solar wind plasma fluctuations (or solar flare/coronal mass ejection activities) than on other days. Thus this may have been the case of the largest signal fluctuations in phase powers during this solar conjunction period. During egress, the spectral power decreases as the SEP angle increases. We can see that the Booker model with a −2.08 power law has a good fit for the spectra for the three ingress tracks. However, for the egress tracks, the slopes of the model appear slightly steeper than those of the measured spectra.

Figure 6.

Phase power spectra for three ingress tracks on DOY 2007/231, 2007/232, and 2007/234, respectively. The spectra were fit using a model of the form A(α)ω−2.08.

Figure 7.

Phase power spectra for three egress tracks on DOY 2007/234, 2007/236, and 2007/237. The spectra were also fit using a model of the form A(α)ω−2.08.

[29] Figures 8 and 9 display the spectra of phase scintillations for ingress tracks and egress tracks, respectively, using the generalized von Karman spectral model for the fits (equation (16), that is (ω2 + κ02v2)(1−p)/2, where p = 3). We have adjusted κ0 and v in the model to fit the spectra for the various tracks. We can see that the model fits all measured spectra very well. This suggests that the slopes for these spectra basically follow a power law of −2.0 that decreases with increasing frequency. Woo and Armstrong [1979] noted that when the signal paths pass very close to the Sun (small SEP angle), the spectral index ranges from −2.0 to −2.67 expected for Kolmogorov turbulence.

Figure 8.

Power spectra for three ingress tracks on DOY 2007/231, 2007/232, and 2007/234, respectively. The spectra are fit using a model of the form I(α)(ω2 + κ02v2)−1.

Figure 9.

Power spectra for three egress tracks on DOY 2007/234, 2007/236, and 2007/237. The spectra are fit using a model of the form I(α)(ω2 + κ02v2)−1.

[30] When using equations (13) and (16) to fit the power spectra of phase scintillations as shown in Figures 69, the coefficients can be calibrated (IB(α) and IK(α)) for each SEP (α) angle. After these coefficients are determined, as a final step, we can calculate the angular variance using equations (15) and (18). Figure 10 shows angle-of-arrival fluctuations (AAF) calculated using a 34 m diameter DSN dish antenna receiver at three frequency bands (S band: 2.3 GHz; X band: 8.6 GHz, and Ka band: 32 GHz) as a function of heliocentric distance (SEP angle). The AAF has dependence on the signal wavelength as λ−2. At high radio frequencies, there is much less of AAF effect than at lower frequencies. As shown in Figure 10, the AAF at Ka band is only (2.3/32)2 = 1/193.6 of that for S band. The AAF decreases rapidly with increasing heliocentric distance following a power law with a −3.5 index. Only within a 2.0° SEP angle (or about 8 solar radii), the Δθrms effect on received power needs to be seriously considered at these frequencies. This angular fluctuation will result in the scattering of the beam center relative to the boresight of receiving antenna, thus, causing the degradation of received power. As a result, there will be attenuation due to AAF on received radio signals during the solar conjunction. The attenuation due to the angular broadening can be very large when the signal paths are very close to the solar surface (SEP = 0.26°), contributing to significant degradation and loss of the signals (combined with thermal noise due to pickup of the solar disc, and strong amplitude scintillation). The angle-of-arrival fluctuations calculated from the phase spectra for three frequency bands at various SEPs are listed in Table 2. For reference, the half-power beam widths (HPBWs) for a 34 m antenna are 230 mdeg at S band, 65 mdeg at X band, and 18 mdeg at Ka band, respectively.

Figure 10.

Angle-of-arrival fluctuation dependence on heliocentric distance (r/rs) or SEP angle (α) at three frequency bands derived from phase spectra measured during solar conjunction.

Table 2. Angular Fluctuations for Various SEP Angles at Three Frequencies
SEP Angle, α (deg)Distance From Sun, r/rsAngle-of-Arrival Fluctuations (mdeg)
at S Bandat X Bandat Ka Band
0.251.0334025017
0.31.217701329
0.41.6650484
0.52.0300222
0.62.4160121
0.72.810070.5
0.753.07050.4

5. Discussion and Summary

[31] We have presented a theoretical study and measurements using a calibration of the phase spectrum during the Cassini solar conjunction of August 2007 and have characterized the AAF magnitude and their dependence on SEP angle and frequency. We can compare our results with those of previous studies in the following aspects.

[32] 1. The first aspect is heliocentric distance dependence. Berman [1977] analyzed the phase fluctuation spectrum from the Mariner 10 solar conjunction. He found that the heliocentric distance dependence for the average phase fluctuation spectra includes two terms: one is R−2.6 (far field), while another is R−10 (near field). Woo and Armstrong [1979] examined Viking I and II solar conjunction data over an extensive solar elongation range (from 2.17 Rs to 212.9 Rs) using S and X band phase difference scintillation measurements. They found that the variations of spectral power at a constant heliocentric distance could vary as large as 2 orders of magnitude. Within a range of 20 to 100 Rs, the phase difference spectra have the dependence:

equation image

The corresponding one-dimensional density spectrum has a dependence of ∼R−3.45. Within 20 Rs, it appears that the slopes of dependence with heliocentric distance become steeper in their spectra.

[33] A previous study [Ho et al., 2008] used a near Sun plasma density model (NeR−6) to derive the angle-of-arrival fluctuations by assuming that electron density fluctuations (irregularities) are proportional to the mean electron density. A dependence of the AAF on the radial distance as ∼R−5.5 was obtained. This corresponds to a phase spectral dependence of Wϕ(ω) ∼ R−11. This slope on heliocentric distance dependence for a near solar range is much steeper than the results obtained by Woo and Armstrong [1979] from a far solar range as shown above. If we use a far field solar wind density model (NeR−2.3), a dependence of Wϕ(ω) ∼ ω−2R−3.6 is obtained. In this study, based on measured phase spectra at various solar distances, using an average density fluctuation, ΔNeR−4 (averaging both fields), we find that the power spectra of phase scintillations have a dependence with heliocentric distance of the form Wϕ(ω) ∼ ω−2R−7. This corresponds to an AAF dependence of ΔθrmsR−3.5 as shown in Figure 10. This power index (−3.5) lies between −5.5 (obtained in our previous study) and −2.45 (obtained from the study by Woo and Armstrong [1979]). An accurate plasma density irregularity model is not yet available that adequately describes the region very close to the Sun.

[34] 2. The second aspect is power index on spectral frequency. Using the Viking and Helios measurements, Berman [1977] found that at an SEP angle of 11.5°, the average phase spectrum dependence with frequency is Wϕ(f) = 3.7 × 10−3f−2.42 (rad2/Hz). Woo and Armstrong [1979] found that between 20 and 215 Rs, the mean spectral index for phase difference power spectrum is 2.65, which is very close to the expected Kolmogorov spectrum (ν − 1 = 8/3). However, they found that in a range near the Sun the phase spectrum has a noticeable change. The spectral index decreases from 2.65 to 2.0. In the range between 2 and 7 Rs, the spectral index (ν − 1) has an average value of 2.07, which indicates the plasma irregularity features are possibly associated with energy dissipation and acceleration of solar wind in the near-Sun region. In this study, we find that in all five cases of power spectra of phase scintillations the spectral index lies very close to 2.0.

[35] 3. The third aspect is frequency range of phase spectrum. Woo and Armstrong [1979] found that the frequency range of phase difference spectra is typically 10−4–10−2 Hz for observations outside 20 Rs and 10−3–101 Hz for observations within about 20 Rs. For some of the measurements very close to the Sun, the frequency of the spectra can extend up to 100 Hz. The upper limits of the frequency in the spectrum are related to the intrinsic features of plasma in these areas. In the near-Sun region, it is expected to have significant amounts of small-size (high-frequency) irregularities or turbulence associated with energy deposition. Furthermore, the frequency range is also limited by the data sample resolution, data duration (length). The upper frequencies of the spectrum may be dominated by system thermal noise. During solar conjunction, even though phase scintillation is not saturated as amplitude scintillation, its spectrum is limited by the instrumental noise at higher frequencies. Since the data used in this study have a 1 s time resolution, we could not see the portions of the spectrum at frequencies higher than its Nyquist frequency (0.5 Hz), even though its upper frequency may extend to above the 1000 Hz at this range of solar radii.

[36] 4. The fourth aspect is radio signal wavelength dependence and anisotropy effects. Previous studies [Berman, 1977; Woo, 1977; Woo et al., 1976; Woo and Armstrong, 1979] treated the solar wind plasma as a neutral atmospheric medium, that is also an isotropic medium. The phase calculation used a neutral gas refractive index model. Their results show that the AAF does not have a wavelength dependence of the wavelength. Since the plasma can more easily move along the magnetic field lines rather than crossing field lines, we expect that there is large anisotropy in the direction perpendicular to the field line by a factor of the elongation index. In the study of Woo and Armstrong [1979], they found that the effects of anisotropy on the shape of the phase scintillation spectrum are not significant. Using the near-Sun estimates of (small-scale) anisotropy deduced from angular broadening measurements, they found that the spectral level increases by no more that a factor of two. In our study, we view the plasma as an ionized medium and use a plasma refractive index in the phase expression. The signal phase fluctuation is a function of λ2 and ΔN. Thus, the resultant AAF is a function of λ2 as expected. The AAF effects quickly decrease with increasing frequency of the received radio signals. Our calculations show that in the direction along the magnetic field lines (Θ = 0°), due to less density gradient, its phase spectral power level will be smaller than that in other directions at most by a factor of equation image37/12.

[37] In addition to the above aspects discussed, we realize that the frozen flow assumption may not be perfectly valid through all heliospheric space. Over such a vast distance range, the solar wind speed is not uniform and also may not always lie in the same direction. This will affect the accuracy of the results. In this study we have only examined one solar conjunction case with spectra for five tracks. Our results need to be confirmed using more solar conjunction measurements in the future.

[38] During solar conjunctions, turbulences in the solar wind plasma can cause many degradation phenomena on the radio signals that pass through it. The angular fluctuations can cause received signal power degradation because of the scattering of the signal in angle and defocusing of the receiving antenna. For angle-of-arrival fluctuations (angular broadening), we do not have direct measurement at this stage. However, we have acquired a lot of high-resolution phase scintillation data from deep space missions during solar conjunctions. Through this study, we have used the power spectra of phase scintillation during the Cassini August 2007 solar conjunction to successfully derive the angle-of-arrival fluctuations. We have first performed a theoretical investigation to derive the relationship between the phase spectrum and angular variance. We then used the theoretical model to fit the spectrum measured from Cassini's dual frequency radio link and calibrated the coefficients of the model. Finally the AAF is calculated using both theoretical model and experimental measurements.

[39] In summary, we have used power spectra of phase scintillations to determine the AAF variance during the Cassini August 2007 solar conjunction, based on the theoretical relationships we developed between the phase spectra, the AAF spectra, and AAF variance. The solar plasma is treated as an anisotropic ionized medium by applying the Booker electron irregularity spectrum model and the phase expression of the electron refractive index. This technique differs from that used in previous studies. On the basis of a frozen-in assumption, all phase variations with time can be converted into the spatial variations using the transverse solar wind speed. We then use the models to fit the measured phase spectra to calibrate the model's coefficients. We use both theoretical models and measurements to define the AAF magnitude as a function of SEP angle and frequency. The results show that the AAF magnitude decreases with increasing heliocentric distance as r−3.5, with a much slower slope than that obtained from the previous study which was derived purely based on the solar plasma density for the near field. In the future, we expect to examine additional solar conjunction cases using the method presented here, and to compare these results with those acquired from monopulse tracking or interferometry to directly measure the AAF during solar conjunctions.

Acknowledgments

[40] We are indebted to Albert D. Wheelon, a CalTech trustee and pro bono consultant to JPL, for his valuable consulting in the theoretical aspects of the work. The authors thank Miles Sue of JPL for his comments and suggestions. We are grateful to the referees for their valuable review comments. The research described in this paper was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.

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