During the superior solar conjunction of spacecraft, the transmitted signals from a spacecraft to the Earth ground station graze the region near the Sun's photospheric surface, passing through dense and turbulent regions of solar-charged particles. Phase changes due to solar coronal irregularities cause raypath wandering, wavefront tilting, and fluctuations in the apparent angle of arrival when observed from the Earth. This study presents a first theoretical investigation of solar wind and solar coronal effects on the angle of arrival fluctuations for RF signals. On the basis of the Chandrasekhar relationship between phase and angular fluctuations, an analytical integrating solution for angle of arrival fluctuations is derived by applying solar corona and plasma irregularity spectrum models. It is found that angular fluctuations rapidly decrease with increasing heliocentric distance at a rate of ∼r−5.5 and also decrease with increasing frequency at a rate of ∼1/f2. It is found that when using Ka band at α = 0.4° (r = 1.6 solar radii), there is a 19 millidegrees (mdeg) angular scattering, corresponding to a 9 dB gain degradation. In comparison, lower-frequency X and S band signals undergo much worse degradation effects. Beyond α = 2° (r > 8 solar radii), angular fluctuations at microwave frequency bands can almost be neglected (θRMS < 1 mdeg). A solution to minimize this degradation is to use Ka- or higher-frequency bands for the telecommunication link during periods of solar conjunction. This study not only quantifies the angular fluctuations caused by solar corona irregularities but also provides an effective method for diagnosing the plasma density fluctuations in a region very close to the solar photospheric surface.
 In order to explore the entire solar system, NASA/Jet Propulsion Laboratory has deployed many deep space missions. Spacecraft are sent to all planets, such as Mars, Jupiter, Saturn, etc., orbiting or flying by these planets for several years. Each planet revolves around the Sun at a different angular speed. When a planet moves to the other side of the Sun relative to the Earth, a superior solar conjunction occurs. During this period, radio signals transmitted from the distant spacecraft suffer severe degradation [Morabito and Hastrup, 2002; Morabito et al., 2003; Bokulic and Moore, 1999]. Communication links with Earth usually become worse or completely disrupted for several days or even weeks. Because solar conjunctions periodically occur for all deep space missions, we need to investigate all degradation phenomena on telecommunication links due to solar medium refractive effects. One such phenomenon is the angle of arrival fluctuation (or angular error) caused by gradients of solar coronal electron density near the solar surface, where the ionized gas (called plasma) consisting of charged particles is so dense that the signal paths grazing this area are strongly diffracted. As viewed from an Earth receiver, apparent angles of signals wander around the centeroid of the beam, causing the defocusing of the signal source. The severity of angular fluctuations depends on the distance between the solar photospheric surface and the raypath.
 The effects of angular fluctuations due to the Earth's ionosphere plasma environment have received little attention in the past. These effects are usually very small because the ionospheric electron density and its irregularities are not sufficiently high to cause significant degradation on RF signals. Effects are further reduced at a rate of 1/f2 as the radio signal frequency becomes higher. However, it is found that effects are intimately related to the cross-path dimension (or aperture) of the observing instrument.
 In the previous studies, the majority of the angle of arrival measurements were made with interferometer systems. Crane  studied effects caused by the ionosphere using a single-monopulse-tracking radar system at 150–400 MHz. In particular, the variance of angle of arrival was studied.
 Tropospheric scintillation effects on angle of arrival fluctuations also were studied at low elevation angles (mainly below 10° elevation angle) using a 37 m diameter antenna at 7.3 GHz [Crane, 1976]. Large seasonal variations of up to several 10s of millidegrees (mdeg) were measured below the 3° elevation angle. The angle of arrival fluctuation due to the tropospheric refractive index fluctuation is independent of signal frequency. Vilar and Smith  developed a technique using two antennas by offsetting one antenna by a small angle to convert angular fluctuations into amplitude fluctuations at elevation angle ranging from 9° to 30°. Then by comparing the difference in signal amplitudes between the two antennas, the angular fluctuations due to the troposphere were characterized.
 In contrast, the solar corona has a much higher plasma density, especially at the base of the corona. Near the solar photospheric surface, plasma irregularities (or density gradients) are enormous. High turbulence with numerous vortices exists here because of bountiful local surface structures. Highly ionized hot plasma erupting from the subphotosphere accelerates and further expands into interplanetary space but far from uniformly, as plasma moves much more easily along the Sun's field lines than across the field lines.
 When radio signals pass through the turbulent solar coronal regions, the phase is severely distorted because of refractive index changes. In order to understand how these degradation phenomena are generated, we will need to theoretically identify the significance of these effects using the definition of the angular fluctuation and its mathematical relationship to signal phase fluctuations. In this study, we will investigate the dependence of these effects on heliocentric distance and frequency, how the telecommunication disruption period can be minimized, and whether or not the signal degradation can be reduced or prevented. In section 2, we introduce the theoretical background, including physical mechanisms in terms of available solar medium models. In section 3, a complete mathematical expression and integration solution for angular fluctuations is derived. The discussion and summary of this study are presented in sections 4 and 5, respectively.
2.1. Physical Mechanism
 Fluctuations of refractive index along the nominal trajectory can cause the raypath to wander [Wheelon, 2001]. Angular fluctuations are proportional to the lateral gradient of the refractive index integrated along the nominal raypaths. The integrated effect of many small refractive bendings produces an angular error in the tangent vector relative to the nominal ray. This error varies with time and can be visualized as dancing about the centroid of the beam. This beam wandering is influenced primarily by large irregularities in plasma near the solar surface during superior solar conjunction. Beam motion and angle of arrival fluctuation are different descriptions of the same phenomenon of random ray bending or wavefront tilting. Raypath wandering can be described as a crinkled wavefront. As an irregular wavefront falls onto a receiving antenna, the angle of arrival changes by an angle Θ0 which is equivalent to a tilting in wavefront by Θ0. The wavefront tilt can be measured using an interferometer.
 The angular fluctuations also induce tracking errors in deep space navigation observables. If the receiving antenna beam is large enough to capture the wandering ray, one can describe the angular error δΘ with the Chandrasekhar equation, as shown in Appendix A [Wheelon, 2001]:
where k is wave number, ɛ is the dielectric constant (having a relation with the refractive index, n, as ɛ = n2), and Φ is phase fluctuation with time t;
where L is the path length from transmitter to receiver and Δɛ represents dielectric constant variations along the path s and in time t. Because ɛ = ɛ0 − reλ2N (where N is the electron number density), the dielectric gradient (Δɛ) is related to the electron density gradient (δN) by
where re is the classical electron radius (2.818 × 10−15 m) and λ is the wavelength of the radio signal (λ = 2π/k = c/f, where c is speed of light and f is frequency of the radio signal).
2.2. Models of Propagation Media
Figure 2 shows the geometry of a raypath from a distant spacecraft to an Earth receiver though the turbulent solar wind during a superior solar conjunction. The environment along the signal raypath includes solar wind, solar corona, Earth's ionosphere, and Earth's magnetic field. There is a need to develop better models for these media in order to more accurately evaluate angle of arrival fluctuation effects on RF signals.
2.2.1. Solar Magnetic Field
 Since the magnetic field at the base of the solar corona is frozen with the rotating Sun, interplanetary magnetic field lines have a spiral shape as r = −vsΦ/Ω, where r is the distance from Sun, vs is solar wind speed (between 100 and 300 km s−1 near the Sun), Φ is azimuth angle, and Ω is the angular velocity of solar rotation (2.7 × 10−6 rad s−1 at solar equator). Thus the magnetic field flux density B is modeled using two components along the directions of r and Φ as
where B0 is the magnetic field flux intensity at a certain distance r0 from the Sun. At the distance of the Earth's orbit (1 AU), B0 is about 10 nT. The angle γ of the magnetic field line relative to the tangential direction can be defined as
 We can see that at a distance very close to the Sun (r ≪ vs/Ω), this angle is near 90° and the magnetic fields are mainly aligned in the radial direction.
2.2.2. Plasma Densities of Solar Corona and Solar Wind
where r is the distance from Sun while rs is the radius of Sun (6.96 × 105 km). The solar corona nearer the Sun is dominated by the first term with the larger coefficient, while the second term describes the far solar wind region with a slower radial attenuation. The solar corona located near the solar surface with the higher electron density and increased turbulence produces the dominant contribution to the angle of arrival fluctuations. In this study, we make use of only the first term of the above model, that is
where N0 is a coefficient determined from measurements of electron density near the solar surface. (The model is only valid for a region of r less than 3.8 rs and probably greater than a few kilometers from solar photospheric surface.) Even though this model can describe the plasma density near the Sun over a large range, it may be inaccurate at a distance very close to the Sun. Plasma density may increase at a rate much faster than r−6 with decreasing solar distance near the solar surface. It is possible that the use of only the first term of equation (6) fails very near (a few kilometers) the surface of the Sun. If this is the case, the coefficient N0 will be much larger than that shown in equation (7) (2.6 × 1067 m6). However, at present, there are no available models that better describe the plasma density and irregularity of the solar corona near the photospheric surface.
2.2.3. Plasma Density Irregularity
 We assume that irregularities of plasma density are simply proportional to the plasma density, that is
 This suggests that the larger plasma density corresponds to the increased density fluctuations (or larger density gradients). The coefficient a0 may have an order of 1067 as indicated in equation (7). However, its exact value is unknown because this model may fail in a region very close to the photospheric surface. As will be seen in section 4 through comparison of theory and data, we find that electron density and turbulence near the photosphere are much higher than expected (a0 actually is of order 1074).
3. Theoretical Calculations
 From the Chandrasekhar equation (equation (1)) (setting = n = 1), the level of angle of arrival fluctuation seen by a receiver with a finite aperture in the transverse direction, u, is given by equation (7),
where k = 2π/λ and A is the surface area of the receiving antenna (πar2). From equations (2) and (3), we have Φ(s, u, v) = πreλdsδN(s, u, v). Thus
 The spatial covariance in refractive index fluctuation is the transform of the spectrum of turbulence electron density. Using the spectrum to represent the spatial covariance of electron density irregularities, we have
where ΨN(κ) is Booker's global model of the spectrum of electron density irregularities and is defined as [Booker and Ferguson, 1978]
where 〈δN2〉 is the variance of electron density fluctuations, ℜ is the axial ratio of the elongation of plasma irregularities along the magnetic field lines (30 < ℜ < 50), κ0 = 2π/L0 (where L0 is the outer scale of turbulence ∼2000 km), κm is the inner scale of turbulence, ν is the spectral index (3 < ν < 4), Qν is the normalization constant (when ν = 4, Qν ≈ 2ℜ /π), and Θ is the angle between the wave number vector and the magnetic field.
 The coordinate system used in this study is shown in Figure 2.
 Then the angular variance becomes
 We first perform two aperture integrations. This integral is a double-surface integration as defined below:
 For a circular antenna aperture, this two dimensional integral can be expressed using cylindrical coordinates as shown in Figure 3;
where ar is the antenna radius. Using the expression
 Thus we have
After integrating over both the s and s′ directions, only 2πδ(κs) is left in Z direction. We have
Using spherical wave number coordinates (κ, Ψ, and ω), κs = κ cos ψ , this integration takes the form
In equation (22), there are three integrations: aperture by wave number, plasma irregularity along the path, and angle relative to the magnetic field. We first determine the role of aperture smoothing in equation (22),
where the spectral index ν has the range 3 < ν < 4. In this range, the upper limit κmar can be taken to infinity and lower-limit κ0ar can be taken to zero. After simplification, we have the solution for I1 as
 Then we perform the integration over the raypath, which is the essence of this study. Because δN = a0r−6, δN2 = a02r−12, where r is the heliocentric distance (see Figure 3) and a0 is a coefficient to be determined in section 4, the integration is performed from Earth (0) to the spacecraft (l) along the raypath z.
where RE is distance between the Earth and Sun (1 AU = 1.5 × 108 km), as shown in Figure 1, and α is the angle between the Sun and spacecraft (solar elongation angle (SEP)). Let x = z − RE cos α and b = RE sin α. Thus dx = dz and
 Making a further substitution, let x = b tan χ and dx = b sec2χdχ, we have
 From the geometry displayed in Figure 2Figure 3, the limits of the integral are tan−1 (−RE cos α/b) = α − π/2 and tan−1 [(l − RE cos α)/b] = β + α − π/2, where β is the angle of Earth-Sun-spacecraft. Assuming that the spacecraft is at the same distance from the Sun as the Earth, β = π − 2α (this is a reasonable assumption, because the radio signals mainly interact with solar plasma irregularities in the region very close to the Sun). Thus β + α − π/2 = π/2 − α, and we have
which is a function purely depending on the angle α.
 Next we will perform the integration over the magnetic field angle. From the two diagrams in Figure 2, we can see that the Sun's magnetic field lines are mainly in the ecliptic plane (in the UZ plane). Near the solar surface, the magnetic fields are basically in the radial direction, that is, in the U direction.
 The magnetic field in UVZ coordinates can be expressed as
where u and z are unit vectors in the U and Z directions, respectively, and the angle relative to the Z direction is γ = tan−1 (−vs/Ωr). In the vicinity of the solar surface (r is very small), γ is near 90°.
 Only the turbulence wave number vectors in the plane of UV can play roles in generating angle of arrival fluctuations relative to the Z direction. Thus we have
 Thus the angle between the wave number and magnetic field vectors is
 The azimuth integration associated with the magnetic field angle becomes
When v = 4, using the results from Appendix B, we have
 Thus the final solution is
 Assuming that δθv has the same magnitude as δθu, RMS angle of arrival fluctuations perpendicular to the direction of propagation are given by
 In the following, we discuss these results to understand their dependence on key parameters. Figure 4 shows the dependence of the aperture-averaging term (I1) on antenna radius for various spectral indices between 3.0 and 4.0. We can see that a larger antenna generates smaller fluctuations because of the aperture average effect. Those small wavefront wanderings will be averaged out by the antenna smoothing process. For example, for ν = 3.5, I1 decreases by a factor of 2 from ar = 10 m to ar = 35 m (70 m antenna diameter). It should be noted that when the spectral index ν = 4, because arν−4 = 1, this term (I1) becomes independent of ar. Both aperture-averaging and spectral dependences will cancel each other. There is no difference between a large antenna and a small antenna under this condition.
I2 is the most important term in the entire integration, and the heliocentric distance dependence is shown in Figure 5. We can see that there is a significant decrease in I2 with increasing heliocentric distance. From 1 rs to 10 rs, I2 abruptly decreases from 107 to 10−4, a change of 11 orders of magnitude. It should be noted that because I2 ∝ 〈2〉, angle of arrival fluctuations should have a dependence of ∼r−5.5.
Figure 6 shows the dependence of the azimuthal integration term (I3) on heliocentric distance due to solar wind speed and magnetic field effects for a fixed plasma elongation ratio (ℜ = 35). At a smaller distance from the Sun, because all magnetic field lines are nearly in the radial direction, this term has a weak dependence on the distance. The magnetic field angle with turbulence wave number depends on the solar wind speed. When the solar wind speed is large (300 km s−1), this angle is near 90°, corresponding to a weak dependence on (r/rs). Also, the larger elongation ratio generates larger I3 values. When we calculate the term (I3), we do not take into account acceleration of solar wind through an adiabatic expansion process. That is, we assume that the solar wind has a constant speed through all radial ranges. Observations show that the solar wind speed abruptly increases with increasing heliocentric distances near the Sun. After including this factor, the curves in Figure 6 will become less dependent on (r/rs), similar to the curve with vs = 300 km s−1.
 To obtain quantitative results from equations (34) and (35), we need to normalize the coefficients in equation (34). However, it is relatively difficult to theoretically determine the coefficient in equation (34) because we have used uncalibrated solar coronal models here, especially for the coefficient (a0) from the irregularity model. To perform this study we have calibrated coefficients in equation (34) using previous measurements. (A correction has been made by including the second terms in equation (6) for this calibration at r = 10 rs, as the second term begins to dominate at a distance of r = 3.8 rs.) 11During the 1960s and 1970s [Woo, 1977; Ward, 1975], there were numerous interferometer observations of angular scattering (broadening) caused by solar corona using radio signals from natural sources at various frequencies. These measurements were extensively analyzed, and results were very well defined. Ward and Woo had summarized these measurements by scaling data to the frequency 2.3 GHz [Woo, 1977; Ward, 1975]. They found that the variations of tangential scattering are nearly proportional to the mean electron density of the solar corona. Angular broadening has strong heliocentric dependence. At 10 solar radii at 2.3 GHz, the average angular broadening is about 0.008 (∼0.13 mdeg). Parameters used for this calculation are listed in Table 1.
Table 1. Parameters Used for Angle of Arrival Fluctuation Calculation
2.8 × 10−15
Distance between Sun and Earth (1 AU)
1.5 × 1011
6.96 × 108
Solar rotation angular speed at the equator
2.7 × 10−6
Speed of light
3.0 × 108
Outer scale wave number
κ0 = 2π/L0
6.28 × 10−7
Outer scale length
2 × 107
Solar wind speed
2.0 × 105
Adjusted irregularity coefficient
8.75 × 1074
 After the coefficient in equation (34) is calibrated, the angular fluctuations as a function of heliocentric distance for three frequency bands are plotted in Figure 7. From Figure 7, we can see that the angular errors rapidly decrease with increasing heliocentric distance at a rate of ∼r−5.5 and also decrease with increasing frequency at a rate of ∼1/f2. We can use this solution to extrapolate the angular fluctuation values at a region very close to the Sun where measurements have never been made. We find that at α = 0.25° (r/rs = 1.0), θRMS = ∼41.8° for S band, θRMS = ∼3.1° for X band, and θRMS = ∼245 mdeg for Ka band, while at α = 0.5° (r/rs = 2.0), θRMS = 0.9° for S band, θRMS = 69 mdeg for X band, and θRMS = 5.4 mdeg for Ka band. At S and X bands, the angular fluctuations are much larger than those at Ka band as expected. Beyond α = 2° (r/rs = 8), angle of arrival fluctuations at these frequency bands can almost be neglected (θRMS < 1 mdeg). The angular errors at near solar regions (α < 0.7°) are listed in Table 2.
Table 2. Angular Fluctuations for Various Solar Elongation Angles (SEP) at Three Frequencies
SEP Angle, α°
Distance from Sun, r/rs
Angle of arrival Fluctuations
At S Band
At X Band
At Ka Band
 At a larger heliocentric distance, the effect of angle of arrival fluctuations probably cannot be modeled by a single-term model as we used here. Another slow variation term may be required, as shown in equation (6). This may explain why previous measurements beyond 10 rs show a weak dependence on heliocentric distance. However, we believe that at regions near the photospheric surface, angle of arrival fluctuations may increase with decreasing distances of r/rs at a much faster rate.
 Because angle of arrival fluctuation is sensitive to the electron density gradient (irregularity), we can use it to adjust the previous model we used, to deduce the irregularity intensity, and to diagnose the plasma environment near the solar surface. After calibrating angular fluctuations using measurements at a distance of 10 rs, we can further define the coefficient in the irregularity model of the solar corona. As a result, we find that a0 = 8.75 × 1074, far higher than that originally defined in equation (7). That is, if we use an irregularity model with this newly adjusted coefficient, the calculation results will fit all previous measurements. This suggests that electron density and irregularity near the solar surface either have higher values or higher-order dependence on r/rs than previously modeled. The model we used may be good for describing the solar corona near the Sun but not good for the photospheric region.
Figure 8 shows the reduction in antenna gain caused by angle of arrival fluctuations relative to a receiving antenna. At three frequency bands and assuming a 70 m diameter Deep Space Network dish antenna, each maximum gain is calculated first using International Telecommunication Union F.699 antenna gain model [International Telecommunication Union Radiocommunication Sector, 2006]. Then using random angular fluctuations as off-boresight angles, antenna gains at such off-boresight angles are also found. The gain reductions (the difference between boresight gain and off-boresight gain) are calculated at three frequency bands and plotted in Figure 8 as a function of SEP angle of raypaths. Angle of arrival fluctuation effects can cause losses of 36 dB at Ka band, 53 dB at X band, and 70 dB at S band at 1.0 rs. S and X bands have larger gain reduction than at Ka band. The Mars Reconnaissance Orbiter had a superior solar conjunction at a minimum of 0.4° SEP angle (1.6 rs) last time in January 2007. From this study, we predicted that at Ka band, there would have been a 19 mdeg RMS angular fluctuation and a 9 dB antenna gain reduction.
 The Earth's troposphere may also play a role at low elevation angles (<10°). At very low elevation angles (<2°), tropospheric effects cannot be neglected. Angle of arrival fluctuations of about 10 mdeg RMS may be generated. However, a difference between angle fluctuations caused by solar corona and by the troposphere exists. The latter has an obvious elevation angle dependence as δθRMS ∝ , where is the apparent elevation angle of signals observed from Earth. By observing at high elevation angles, we can exclude the Earth's tropospheric effects from the observed angle fluctuations. We do not expect angular fluctuations from ionospheric or magnetospheric effects in the microwave frequency range, because electron density and its fluctuations within the Earth's ionosphere are so low.
5. Summary and Conclusions
 We have investigated the significance of angular scattering for a deep space telecommunication link at varying microwave frequency bands. A complete theoretical study of angle of arrival fluctuations using turbulence theory and a simplified model of the solar corona is performed for the first time. At first, we describe the physics and propagation mechanism for microwave signals though the solar corona base regions. The angle of arrival fluctuations for a deep space path are theoretically derived using the Chandrasekhar equation and available solar corona model. After applying Booker's global model for the plasma irregularity spectrum, we obtain the relationship between the angle of arrival fluctuations and the electron density irregularities in the solar corona. Through a complicated integration process, analytical solutions are finally developed. The magnitudes of angular errors, especially when the path is very close to the solar surface, are calculated as a function of the Sun-Earth-spacecraft angle after normalizing a coefficient using previous measurements made at larger solar distances.
 The solution includes three parts. The aperture-averaging part shows that large antennas reduce angular fluctuations because of the aperture-smoothing effect. The angular integration shows the dependence on solar wind speed and elongation ratio near the Sun. Integration over the irregularity spectrum shows the strong heliocentric distance dependence of angular fluctuations. We find that the fluctuations rapidly decrease with increasing heliocentric distance at a rate of ∼r−5.5. The fluctuations also decrease with increasing frequency at a rate of ∼1/f2. At Ka band, at α = 0.25° (r/rs = 1.0), θRMS = 245 mdeg. The gain reduction is 36 dB. At α = 0.4° (r/rs = 1.6), there is a 19 mdeg angular error and a 9 dB gain reduction. Compared with Ka band, S and X bands have much worse angular fluctuations and gain reductions. These gain reductions discussed here are caused by angle of arrival fluctuations, not including other degradation effects due to inhomogeneous plasma, such as amplitude scintillation, phase scintillation, spectral broadening, etc. Using the results, in turn, we can adjust the coefficients of the irregularity model we used. Thus our result provides a valuable tool in diagnosing characteristics of the electron density in the solar corona near the photospheric surface where raypaths graze. It should be noted that this result is strongly dependent on the simplified solar coronal model we used for near–solar surface regions, especially on the electron density irregularity model for the photospheric surface. Actual irregularities near the solar photospheric surface are so complicated that they may not be modeled so simply.
 Through this study we can conclude that a way to minimize the angle of arrival fluctuation effects due to solar corona irregularities during superior solar conjunction is to employ Ka band or higher frequencies (such as W band) for deep space telecommunication links. Use of the higher frequencies can greatly reduce the degradation effects on the signals and make telecommunication links less disrupted.
Appendix A:: Chandrasekhar Equation
 Starting from the ray trajectory equation,
Letting the dielectric constant have a perturbation, we have
Integrating both sides of equation (A2) along the nominal raypath and ignoring Δɛ, we have the instantaneous cross-plane angular error
because Φ(s, t) = Δɛ(s, t)ds.
 In general, we have the Chandrasekhar equation as
Now let b = 0 and expend all values of cos2x into four quadrants:
Now differentiating both sides of the equation with respect to c, we have
 Letting c = 1, we have
Applying the above result in the angular integration, we want to solve
 We are also 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 are grateful to the referees for their review. 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.