SYMMETRY OF THE IBEX RIBBON OF ENHANCED ENERGETIC NEUTRAL ATOM (ENA) FLUX

The human brain is conditioned to identify symmetry (Enquist & Arak 1994; Rhodes 2006), and inspection of the distribution of ENA flux around the ribbon in the flux maps of Figure 1 indicates a single, broad flux peak at low energies with apparent reflection symmetry and two flux peaks of similar azimuthal width and largely opposing (on opposite sides of the ribbon) at the high energies.

Cross-correlation is routinely used to identify periodic patterns in data and thus can be used as an indicator of the presence of symmetric features in an image (Reichardt 1961; Neubecker 1996; Masuda et al. 1993), thus providing quantitative insight of our visual inspection of Figure 1. We examine rotational symmetry using the autocorrelation ${\cal A} ({\theta _{\rm 0},\;\phi _{\rm 0} })$ using the annular flux map F(θ, ϕ) at each energy in Figure 1 as a function of 6° increments of azimuthal (θ₀) and polar (ϕ₀) offset angles:

$$\begin{eqnarray} {\cal A}\left({\theta _{\rm 0},\phi _{\rm 0} } \right) &=& F\left({\theta,\phi } \right) \star F\left({\theta + \theta _{\rm 0},\phi + \phi _{\rm 0} } \right) \nonumber \\ &\equiv & \frac{1}{{4\pi }}\mathop \int _0^{2\pi }\! \mathop \int _0^\pi \! F\left({\theta,\phi } \right)F ({\theta + \theta _{\rm 0},\phi + \phi _{\rm 0} }) \nonumber\\ &&\times \sin\phi \,d\phi \, d\theta. \end{eqnarray} \tag{ 1 }$$

From Equation (1), we derive the autocorrelation score $\chi ({\theta _0,\phi _0}) = ({{\cal A}({\theta _0,\phi _0}) - {\cal A}_{\rm MIN}})/({{\cal A}_{\rm MAX} - {\cal A}_{\rm MIN}})$, where ${\cal A}_{\rm MIN}$ and ${\cal A}_{\rm MAX}$ are the minimum and maximum values of ${\cal A}({\theta _0,\phi _0})$ of each map. Values of χ = 0 and χ = 1 correspond to the lowest and highest autocorrelation, respectively, in each flux map.

The autocorrelation score is shown in Figure 2 for each of the annular flux maps of Figure 1 as a function of azimuthal and polar offset angles, with red corresponding to higher autocorrelation and blue corresponding to lower autocorrelation. As expected, the maximum value of χ lies at the center of the primary autocorrelation peak, (θ₀, ϕ₀) = (0°, 0°). As the polar and azimuth offset angles increase from 0°, χ slowly decreases at a rate that indicates the characteristic spatial sizes of the ribbon flux peak(s) in these angular directions.

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In polar angle, regions of higher χ appear to be narrow (⩽24°) and generally centered at polar offset angle ϕ₀ = 0°, confirming that the ribbon flux is narrow in polar angle, highly circular, and well-centered in the ribbon-centered coordinate frame.

In azimuthal angle, only a single autocorrelation peak, centered at θ₀ = 0°, is observed at 0.7 and 1.1 keV, suggesting the ribbon flux is unimodal. This autocorrelation peak narrows by ∼30% at 2.7 and ∼50% at 4.3 keV. Additionally, at 2.7 keV and 4.3 keV, a secondary autocorrelation peak centered near θ₀ = ±180° is clearly observed, suggesting the presence of two (bilateral) flux lobes that are generally located on opposite sides of the circular ribbon.

While the autocorrelation analysis clearly reveals the existence of rotational symmetry of the ribbon flux as well as the presence of multiple flux peaks with rotational symmetry, it does not provide a quantitative test for symmetry or the location of symmetry axes. For quantitative symmetry analysis, we test for reflection symmetry using correlation analysis of the annular flux maps of Figure 1. For this study, we do not separate the ribbon flux from the spatially slowly varying globally distributed flux (Schwadron et al. 2011, 2014); thus our results are characteristic of a combined azimuthally varying ribbon flux and a slowly varying globally distributed flux at each energy. We employ several input data sets for the correlation analysis:

• 1.  
  F(θ, ϕ): the 2D annular ENA flux maps of Figure 1, which span nine polar pixels in the angular range 48°–102° that are generally centered on the ribbon.
• 2.  
  F_P9(θ): at each azimuthal angle θ, the average ENA flux of all nine polar pixels (which span 48°–102°) and, at most energies, fully contains the ribbon flux. Because F_P9(θ) includes the flux of nine pixels at each θ, it is the most statistically significant data. However, it also includes regions inside and outside of the ribbon peak that are largely dominated by the globally distributed flux and is therefore comparatively less sensitive to variations of the ribbon flux over azimuthal angle.
• 3.  
  F_P2max(θ): at each azimuthal angle θ, the average flux of the two adjacent polar pixels of maximum flux within the nine polar pixels spanning 48°–102°. This flux is derived from only two pixels and thus provides comparatively poorer statistics; however, because F_P2max(θ) includes a comparatively larger fraction of the ribbon flux than F_P9(θ), it provides better insight into azimuthal variations of ribbon flux. Note that F_P2max(θ) > F_P9(θ) for any θ.

To better interpret the information provided by the symmetry tests, and with some insight into the summary results of this study, we construct the simplest flux symmetries of a circular emission structure that could represent the ribbon flux, shown in Figure 4. Schematically illustrated are a unimodal flux peak in Figure 4(i), opposing bilateral flux lobes of the same flux brightness in Figure 4(ii), partially opposing bilateral flux lobes of different brightness in Figure 4(iii), and non-opposing bilateral flux lobes of different brightness in Figure 4(iv).

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Analogous to physiological symmetry geometries, we refer to the primary axis of reflection symmetry at θ_S as the sagittal axis. In Figures 4(i)–(iv), for purposes of illustration in this section, we conveniently select the sagittal axis to lie along the BV-plane, such that the sagittal axis at θ_S = 0° also corresponds to a roll angle θ_R = 0°.

For a unimodal flux distribution, a single symmetry axis (the sagittal symmetry axis θ_S) is observed and is located near the point of bisection of the distribution. For a bimodal flux distribution, we expect to observe up to three axes of symmetry. The sagittal symmetry axis is located where one flux peak is "folded" and co-registered with the other flux peak. The same symmetry operation used to identify θ_S yields two additional symmetry axes, which we refer to as transverse symmetry axes θ_T and are located at local ρ(θ_R) maxima and local C_V(θ_R) minima. Because the symmetry operation applies to the entirety of the ribbon flux, the location of a transverse axis is associated with the internal symmetry of an individual flux peak but is modulated by the global variation of flux around the ribbon. Therefore, a transverse axis can be located near (but not necessarily at) the angle of bisection of an individual flux peak of a bimodal distribution.

Figures 4(a)–(d) show simplistic, 1D azimuthally dependent fluxes F(θ) for each of the scenarios of panels (i)–(iv). These fluxes, analogous to the fluxes F_P9(θ) or F_P2max(θ) obtained from the IBEX data, are constructed as the combination of the constant, ubiquitous globally distributed flux with normalized flux magnitude 0.35 and an azimuthally varying ribbon flux of one (for unimodal) or two (for bilateral) Gaussian-shaped flux peaks. The parameters of the Gaussian flux distributions are listed in Figures 4(i)–(iv) and include the offset angle Θ_O relative to the symmetry axis θ_S, the angular full-width-at-half-maximum (FWHM) θ_FWHM, and, for the bilateral peaks, the relative flux magnitudes F_L and F_R of the left and right flux lobes, respectively.

Figures 4(e)–(h) and Figures 4(i)–(l) show ρ(θ_R) and C_V(θ_R) calculated as a function of roll angle θ_R for the flux distribution F(θ) of each scenario. For reference, also shown are the roll angle locations of the sagittal (red dashed line) and transverse symmetry axes (purple dotted line).

As indicated by the autocorrelation analysis of Figure 2, the ribbon flux at lower ENA energies (∼1 keV) likely appears as a unimodal peak that is broad in azimuth. The unimodal Gaussian flux peak of Figure 4(a) is centered at θ = 0°. The Pearson correlation coefficient in Figure 4(e) reaches a single maximum ρ(0°) = +1 (perfect positive correlation) at the sagittal axis and a single minimum ρ(±90°) = −1 (perfect negative correlation) at the transverse axis. The corresponding CV(RMSD) correlation score in Figure 4(i) reaches a single minimum value C_V(0) = 0 at the sagittal axis and a single maximum value C_V(±90°) = 0.6. A unimodal peak is therefore characterized by (1) a single ρ(θ_R) maximum and single C_V(θ_R) minimum at the same roll angle, which marks its sagittal axis, and (2) the ρ(θ_R) minimum and the C_V(θ_R) maximum lie 90° from the sagittal axis.

At 2.7 and 4.3 keV, inspection of the flux maps of Figure 1 and the autocorrelation analysis of Figure 2 clearly indicate the presence of two distinct flux peaks that generally lie on opposite sides of the circular ribbon. We therefore construct in Figures 4(ii)–(iv) three representative (but distinctly different) cases for bilateral flux lobes. The common features across these cases include (1) the flux distribution is the superposition of a constant flux of normalized magnitude 0.35 and two Gaussian flux peaks and (2) the Gaussian flux peaks are symmetrically located at offset angles ±Θ_O relative the sagittal symmetry axis, which lies at θ = 0°.

In the first bilateral lobe case, shown in Figure 4(b), the Gaussian bilateral flux lobes are of equal width and magnitude and are offset Θ_O = ±90° and are thus on opposite sides of the ribbon, separated by 2|Θ_O| = 180°. The two ρ(θ_R) maxima in Figure 4(f) and two C_V(θ_R) minima in Figure 4(j) indicate strong symmetry along two axes: θ_R = 0° (the sagittal axis) and θ_R = ±90° (transverse axis, which is the common symmetry axes that bisects each of the individual opposing flux peaks). Another signature for opposing bilateral lobes is the locations of two pairs of ρ(θ_R) minimum and C_V(θ_R) maximum at θ_R = ±45°, midway between the sagittal and transverse symmetry axes.

In the second bilateral lobe case, shown in Figure 4(c), the Gaussian flux lobes are offset Θ_O = ±70° from θ_S and thus referred to as "partially opposing" bilateral lobes. As with the case of opposing bilateral flux lobes, two pairs of ρ maximum and C_V minimum are observed. Strong symmetry is present at the sagittal symmetry axis, with ρ(0°) = +1 and C_V(0°) = 0.12. However, the second pair of local ρ maximum (at θ_R ∼ 85°) and local C_V minimum (at θ_R ∼ 74°) are slightly offset from θ_R = 90° and are not precisely paired at the same roll angle. Additionally, the correlation score C_V(74°) at the transverse axis is significantly poorer than the score C_V(0°) at the sagittal axis. These are key signatures that distinguish partially opposing bilateral lobes from opposing bilateral lobes.

To illustrate the signatures of bilateral flux lobes of different flux magnitudes, the flux magnitude F_L of the left flux peak in Figure 4(c) is 80% that of F_R. From Figures 4(g) and (k), this results indifferent magnitudes of the two C_V(θ_R) maxima. Note also that the minimum value of C_V(θ_R) is nonzero, although a nonzero C_V(θ_R) minimum is expected for a natural system in which individual flux peaks are not identical in shape and magnitude.

The third bilateral lobe case, shown in Figure 4(d), has non-opposing flux lobes that are narrow in angular width and separated by 2|Θ_O| = 90°. For this case, the two transverse symmetry axes that bisect the individual flux lobes are well-separated and distinct. Therefore, three unique symmetry axes are present and lie near the locations of the three pairs of ρ maximum and C_V minimum. As with the other bilateral lobe cases, the sagittal axis lies at the global ρ maximum and C_V minimum (θ_S = 0°), and the transverse axes lie at the roll angles of local ρ(θ_R) maximum and C_V(θ_R) minimum, which are in the vicinity of the offset angle Θ_O of each flux peak.

In summary, the key signatures for interpreting the flux symmetry of the IBEX ribbon include:

• 1.  
  Reflection symmetry is present only when a ρ(θ_R) maximum and C_V(θ_R) minimum pair lie at the same roll angle θ_R (Livadiotis & McComas 2013).
• 2.  
  The sagittal symmetry axis is identified at the roll angle of the global maximum ρ(θ_R) and global minimum of C_V(θ_R).
• 3.  
  The reflection symmetry is strongest when ρ(θ_R) → +1 and C_V(θ_R) → 0.
• 4.  
  A unimodal peak is characterized by a single, paired ρ maximum and C_V minimum, which defines the location of the sagittal symmetry axis. Additionally, the ρ minimum and C_V maximum are likewise paired and lie ∼90° from the sagittal symmetry axis.
• 5.  
  Bilateral lobes are characterized by two or three pairs of ρ maximum and C_V minimum.
• 6.  
  Opposing bilateral lobes are characterized by two pairs of ρ maximum and C_V minimum with similar ρ maxima values and similar C_V minima values. The transverse symmetry axis is located ∼90° from the sagittal symmetry axis.
• 7.  
  Partially opposing bilateral lobes are distinguished from opposing bilateral lobes by a single global C_V minimum at θ_S and a local C_V minimum at θ_T, such that C_V(θ_S) < C_V(θ_T). The transverse axis is slightly offset from 90° relative to the sagittal symmetry axis.
• 8.  
  Partially opposing bilateral lobes of different flux magnitudes are identified when the magnitudes of the two ρ minima are dissimilar and the magnitudes of the two C_V maxima are dissimilar.
• 9.  
  Non-opposing bilateral lobes are identified by three pairs of ρ maximum and C_V minimum.

To calculate the sagittal symmetry axis locations, we follow Livadiotis & McComas (2013) to identify the Pearson correlation maximum at each energy. We first convert the correlation coefficient to a positive definite quadratic form using R(θ_R) = 1−ρ(θ_R)², such that the maximum positive value of ρ(θ_R) is located at the same roll angle as the minimum of R(θ_R). The downward-opening parabola of ρ(θ_R) near its maximum is therefore transformed into an upward-opening parabola of R(θ_R) near its minimum. Next, the six data points of R(θ_R) closest to the estimated symmetry axis (derived from the interpolated roll angle of the ρ(θ_R) maximum) are fit to

$$\begin{equation} R ({\theta _{\rm R}}) = R_{\rm min} - R_{\rm C} ({\theta _{\rm S} - \theta _{\rm R}})^2. \end{equation} \tag{ 5 }$$

The three fit parameters include: the sagittal symmetry axis θ_S, which identifies the parabolic vertex; the minimum value R_min that lies at the parabolic vertex; and the curvature coefficient R_C that is used to derive the error of θ_S. The derived sagittal axes are shown as vertical dashed lines in the left panels of Figure 5 for each input data set F_P9(θ), F_P2max(θ), and F(θ, ϕ); at each energy, these values are generally clustered together and indicate consistent results over the input data sets.

The errors associated with the derivation of θ_S from the parabolic fits are crucial for comparing the results across data sets as well as comparing and combining the Pearson and CV(RMSD) results. This error is calculated using the curvature coefficient R_C of the fitted parabola according to (Livadiotis & McComas 2013)

$$\begin{equation} \left({\delta \theta _{\rm S}} \right)^2 = \frac{{R_{\rm min}}}{{\left({N - 1} \right)R_{\rm C}}}\end{equation} \tag{ 6 }$$

where N is the number of data points used for the parabolic fit (here N = 6).

A similar method of parabolic fit and derivation of both θ_S and δθ_S are obtained for the C_V(θ_R) results (Livadiotis 2007), whose minimum can intrinsically be modeled in positive definite quadratic form similar to Equations (5) and (6):

$$\begin{equation} \begin{array}{l} C_V ({\theta _{\rm R}}) = C_{\rm min} + C_{\rm C} ({\theta _{\rm S} - \theta _{\rm R}})^2 \\ \ms \displaystyle ({\delta \theta _{\rm S}})^2 = \frac{{C_{\rm min}}}{{({N - 1})C_{\rm C}}} \\ \end{array}\end{equation} \tag{ 7 }$$

where C_min is the minimum of C_V(θ_R) and C_C is the parabolic curvature coefficient. The derived values of θ_S for the CV(RMSD) results for N = 6 are shown as dashed lines in the right panels of Figure 5 for each of the data sets. At each energy, the sagittal symmetry axes derived using the CV(RMSD) analysis are closely clustered, indicating consistency over the data sets.

In Figure 5, the Pearson correlation results at 2.7 keV for F_P9(θ) are unique because three ρ(θ_R) maxima are observed, with the sagittal axis identified at the location θ_R ≈ 0° of the ρ(θ_R) maximum (although a second local maximum of nearly the same magnitude lies at −42°). Additionally, the ρ(θ_R) minimum lies nearly 90° from the sagittal axis. These are signatures of non-opposing bilateral lobes as indicated in Figure 4(h). However, all other correlation results at 2.7 keV consistently show a sagittal axis near θ_R ≈ 30°, and none exhibit signatures of non-opposing lobes. We infer from these results that the bilateral lobes at 2.7 keV have only weak signatures of non-opposing flux lobes and, as will be discussed later, are classified as partially opposing lobes. Because of the discrepancy in sagittal axis identification introduced by the signatures of non-opposing lobes, we do not use in any subsequent analysis the value of θ_S derived using Pearson analysis at 2.7 keV with the F_P9(θ) data.

After excluding the Pearson analysis results for the F_P9(θ) data at 2.7 keV, we derive the symmetry axis locations θ_S and their associated errors ±δθ_S for each energy, each input data set, and both Pearson and CV(RMSD) analyses. These are shown in Figure 6, for which the center black bar at each energy is the composite symmetry axis obtained by combining the Pearson and CV(RMSD) analyses; the Pearson and CV(RMSD) results lie to the left and right of the black bar, respectively. The symmetry axis locations derived using both the Pearson and CV(RMSD) analyses are consistent and clearly show a systematic rotation of θ_S through ∼65° as the energy increases from 0.7 keV to 4.3 keV.

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As previously stated, the necessary criteria for identification of strong symmetry and the sagittal axis is the presence of both a maximum of ρ(θ_R) and a minimum of C_V(θ_R) at a similar roll angle (Livadiotis & McComas 2013). Figure 6 clearly shows that both the Pearson and CV(RMSD) analyses identify consistent locations of θ_S at each energy, and Figure 5 shows that these values are associated with large, positive values of the Pearson correlation coefficient (∼0.8 to ∼0.95) and small C_V(θ_R) correlation scores (∼0.1 to ∼0.3). This is clear evidence for strong reflection symmetry associated with the ribbon flux.

Table 1 lists the derived weighted mean values of θ_S (using a weighted mean with weights δθ_S⁻²) at each energy for both the Pearson and CV(RMSD) analyses. The Pearson correlation and CV(RMSD) results agree within 3°, which quantitatively demonstrates strong reflection symmetry of the ribbon flux. All Pearson and CV(RMSD) results are then combined at each energy, again using a weighted (δθ_S⁻²) average, to obtain a composite value of θ_S at each energy, which are also listed in Table 1. These values are used as the primary symmetry axes for the remainder of this study.

Table 1. Location of the Sagittal Symmetry Axis θ_S of the Ribbon Flux Relative to the Ribbon Center–Heliospheric Nose Vector in the Ribbon-centered Frame of Figure 1

ENA Energy	θS, Pearson Correlation	θS, CV(RMSD)	θS, Composite	Flux Centroid, θF
(keV)
0.7	350 ± 38	352 ± 58	350 ± 32	27°
1.1	214 ± 60	272 ± 72	238 ± 46	24°
1.7	−18 ± 25	−11 ± 46	−16 ± 22	−3°
2.7	−283 ± 66	−297 ± 81	−288 ± 51	−18°
4.3	−409 ± 37	−388 ± 57	−403 ± 31	−9°

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As indicated in Table 1, the sagittal symmetry axis location is a strong function of energy, and we observe systematic rotation from θ_S = +35° at 0.7 keV to θ_S = −40° at 4.3 keV. Notably, the sagittal symmetry axes at 0.7 keV and 4.3 keV lie generally on opposite sides of the BV-plane, which is located at θ_S = 0°.

To identify a transverse symmetry axis θ_T from the data, we use the same parabolic fit method that was used to derive the sagittal symmetry axis: Equation (5) applied to the six closest points around secondary maxima of ρ(θ_R) or secondary minima of C_V(θ_R). We calculate transverse symmetry axis locations for input data sets that satisfy two criteria: (1) a secondary maximum of ρ(θ_R) and a secondary minimum of C_V(θ_R) both exist and (2) the peak of the secondary ρ(θ_R) maximum and trough of the secondary C_V(θ_R) minimum each extend over at least 36° in θ_R (six 6° pixels) for a meaningful parabolic fit to the data.

The transverse axis locations derived from the parabolic fits are summarized in Table 2 and are shown as the solid vertical lines in Figure 5. As with the derivation of the sagittal axis, the errors δθ_T of the transverse axis are derived from the curvature coefficient of the parabolic fit, and the composite values of θ_T are calculated using a weighted mean with weights δθ_T⁻².

Table 2. Location of the Transverse Symmetry Axis θ_T of the Ribbon Flux Relative to the Ribbon Center–Heliospheric Nose Vector in the Ribbon-centered Frame of Figure 1

ENA Energy	Input Data Set	θT, Pearson Correlation	θT, CV(RMSD)	θT, Composite	|θT − θS|
(keV)
2.7	FP2max(θ)	629 ± 68	594 ± 220	626 ± 65	914 ± 83
	FP2max(θ)	556 ± 64	519 ± 152
4.3	FP9(θ)	480 ± 99	415 ± 214	518 ± 46	921 ± 55
	F(θ,ϕ)	490 ± 137	429 ± 240

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Also listed in Table 2 is the angle between the sagittal (θ_S) and transverse (θ_T) symmetry axes. At both energies, the angle between the sagittal and transverse symmetry axes is ∼90°, which is a distinguishing feature of opposing and partially opposing flux lobes.

Referring to Figure 5, at 0.71 and 1.1 keV each of the ρ(θ_R) and C_V(θ_R) correlation distributions exhibits a single maximum and single minimum that are spaced ∼90° apart in roll angle, which is characteristic of a unimodal distribution as in Figures 4(e) and (i). No signatures of bilateral flux peaks are observed at these energies.

At 4.3 keV in Figure 5, two distinct pairs of ρ(θ_R) maximum and C_V(θ_R) minimum are clearly observed in all data sets, and Table 2 shows that the resulting sagittal and transverse symmetry axes are separated by ∼90°. These are signatures of opposing or partially opposing bilateral flux lobes. Furthermore, the global minimum C_V(θ_S) at the sagittal symmetry axis is approximately half the value C_V(θ_T) at the local minimum of the transverse axis; this notable difference C_V(θ_S) ≪ C_V(θ_T) is a key signature of partially opposing bilateral flux lobes, as illustrated in Figure 4(g). Finally, the magnitudes of the two ρ minima are generally similar, and the magnitudes of the two C_V maxima are generally similar, indicating bilateral lobes of similar flux magnitudes. We therefore conclude that the ribbon at 4.3 keV is characterized by partially opposing bilateral flux lobes of similar flux magnitudes.

The results at 2.7 keV are consistent with those at 4.3 keV. In Figure 5, the ρ(θ_R) results clearly show two maxima that are ∼90° apart in roll angle and indicative of bilateral lobes. The C_V(θ_R) results also show the emergence of a second minimum that is paired with a ρ maximum; this transverse symmetry axis is particularly distinct for the F_P2max(θ) data, which are listed in Table 2 and lies ∼91° from θ_S. Finally, partially opposing flux lobes are indicated by C_V(θ_S) ≪ C_V(θ_T), and flux peaks of similar magnitude are indicated by similar values of ρ minima and of C_V maxima.

The results at 1.7 keV exhibit a mixture of unimodal and bimodal flux peaks. In Figure 5, the ρ(θ_R) results show two maxima that are ∼90° apart in roll angle and thus indicative of bilateral lobes. The C_V(θ_R) results show the emergence toward a second minimum, but not a distinct minimum which is required for classification as bilateral flux lobes.

In summary, the ribbon flux is predominantly unimodal at 0.7 and 1.1 keV, predominantly partially opposing bilateral lobes at 2.7 and 4.3 keV, and in a transition state from a unimodal to bimodal distribution at 1.7. Because the transition from unimodal to bilateral flux distributions occurs between 1.1 and 2.7 keV, we fit the natural log function specified in Figure 7 to the data over this span of energies. The resulting fit, shown as the solid red line, suggests a strong ln(E) dependence of θ_S for these energies, and the sagittal symmetry axis traverses the ribbon center–heliospheric nose vector (and thus the BV-plane) at 1.65 keV.

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The fit is extrapolated (dashed line) to 0.7 keV, which is a strongly unimodal flux distribution, and to 4.3 keV, which is a strongly bimodal flux distribution. At these energies the extrapolation projects symmetry axes at 50° and −56°, respectively, from the ribbon center–heliospheric nose vector, both of which are ∼15° greater than observed. This indicates that the symmetry axes of the unimodal and bilateral flux distributions are substantially less dependent on energy than at the transition energies; furthermore, the symmetry axis angle may converge toward a single value characteristic of unimodal flux distribution at low energies and, on the opposite side of the ribbon center–heliospheric nose vector (the BV-plane), toward a single symmetry value characteristic of a bilateral flux distribution at high energies.

Presuming the existence of symmetry, a rotationally symmetric flux map, such as opposing bilateral lobes as in Figure 4(ii), results in a flux centroid at the ribbon center. In contrast, a flux distribution that has reflection symmetry but is not rotationally symmetric results in a flux centroid that is displaced from the ribbon center along the sagittal symmetry axis. Examples of such a system include the unimodal flux peak in Figure 4(i), the partially opposing bilateral lobes in Figures 4(iii), and the non-opposing bilateral lobes in Figure 4(iv).

The flux centroid R_F relative to the ribbon center for each annular map of Figure 1 is

$$\begin{equation} {\bf R}_{\bf F} = \mathop \sum _i F_i {\bf r}_i {\rm d\Omega}_i \end{equation} \tag{ 8 }$$

where r_i, F_i, and dΩ_i are the vector direction, flux value, and solid angle subtended by pixel i in the sky map. If R_F lies at the ribbon center, then opposing pixels generally have similar flux, i.e., F(θ) ≈ F(θ + 180°), which is a signature for strong n-fold (n ⩾ 2) rotational symmetry such as opposing bilateral flux lobes (n = 3). Alternately, evidence of strong symmetry in the correlation analysis combined with R_F located away from the ribbon center indicates a strong reflection symmetry and weak rotational symmetry, such as a unimodal (n = 1) flux peak or non-opposing bilateral flux lobes. Partially opposing bilateral lobes will also exhibit a centroid, though at a moderate distance from the ribbon center compared to the cases of unimodal and non-opposing flux distributions.

The distance of R_F from the ribbon center is a qualitative (rather than a quantitative) indicator of nonrotational ribbon symmetry. Deviation of R_F from the symmetry axis can arise from asymmetry of a unimodal flux peak or bilateral flux lobes with different flux magnitudes or different shapes.

Nonribbon flux features that lie within the annular flux maps that do not follow the ribbon symmetry can also influence R_F. First, the globally distributed flux is spatially slowly varying, and its total flux in the annular flux maps is a significant fraction (∼0.3–0.5) of the total ribbon flux; a nearly constant flux distributed around the ribbon drives the calculated R_F toward the ribbon center. Second, the presence of asymmetric flux variations within the annular maps can drive R_F away from the ribbon symmetry axis. This may include asymmetric features of the ribbon flux as well as nonribbon flux features such as the flux from the heliotail (McComas et al. 2013) and the variation of the globally distributed flux (Schwadron et al. 2011, 2014). Therefore, the presence of the globally distributed flux acts to skew R_F toward the ribbon center, and asymmetric flux features can skew R_F away from the sagittal symmetry axis.

The flux centroids R_F calculated using the annular flux maps F(θ, ϕ) of Figure 1 are shown as the points in Figure 8 at each energy. The distance of the centroid points from the center of the plot (the ribbon center) is represented as the fractional distance (in percent) in the polar direction to the polar angle of 745 that was identified in Funsten et al. (2013) as the circular location of maximum ribbon flux at low energies. Also shown as the dashed lines are the composite sagittal symmetry axes θ_S listed in Table 1. The azimuthal angle θ_F of R_F at each energy is also listed in Table 1 for comparison with the correlation analysis results.

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Several systematic features of the calculated flux centroids are clearly observed. First, the azimuthal angles of the centroid locations at all energies except 4.3 keV lie within 12° of θ_S and therefore are consistent with the sagittal symmetry results from the correlation analysis. Therefore, with the exception of 4.3 keV, the abundance of asymmetric flux in the annular maps, which includes the flux variation of the globally distributed flux, is minimal or mutually offsetting (flux of a feature on one side of the ribbon offsets the flux of an independent feature on the other side of the ribbon). The flux centroid at 4.3 keV lies ∼30° from θ_S, and the annular flux map at this energy likely includes one or more underlying asymmetric flux features that are either not present or exist at lower relative flux magnitude at lower energies.

Second, relative to the sagittal symmetry axis, the azimuthal angle of the flux centroid at each energy is systematically biased toward the ribbon-center heliospheric nose vector (with the exception of 1.7 keV, which lies along this vector and for which θ_F ≈ θ_S). This suggests that the predominant asymmetric flux feature(s) in the annular flux maps lie in the vicinity of this vector.

Third, the polar offset of the calculated flux centroid location at 0.7 and 1.1 keV is consistently ∼17% of the characteristic circular radius (745) of the ribbon, while at higher energies is consistently ∼11% of this angular distance. The higher value at lower energies reflects strong unimodal shape; at higher energies, the offset is likely due to partially opposing bimodal flux peak, noting that R_F would lie near the ribbon center for opposing bilateral lobes and at a larger angular distance for non-opposing lobes. This result is therefore consistent with the transition from unimodal flux distribution at low energies to partially opposing bilateral flux lobes at higher energies.

With knowledge of the sagittal symmetry axis location θ_S as a function of energy, we now examine the 1D ribbon fluxes as a function of azimuthal angle Θ from this symmetry axis_. Figure 9 shows F_P9(Θ) and F_P2max(Θ) for −180° ⩽ Θ ⩽ 180°. The F_P9(Θ) and F_P2max(Θ) flux distributions are generally similar; however, because F_P2max(Θ) represents the two highest ribbon flux pixels that are adjacent, F_P2max(Θ) is larger and more variable than F_P9(Θ).

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To identify systematic trends of the symmetric flux around the ribbon, and for quantitative comparison of the observations of ribbon flux symmetry with models and simulations, we formulate a simplistic empirical representation of the ribbon flux ${\cal F}({\rm \Theta})$ using

$$\begin{equation} {\cal F}\left({\rm \Theta} \right) = {\cal F}_0 + {\cal F}_{\rm R} \left({\rm \Theta} \right).\end{equation} \tag{ 9 }$$

We assume that ${\cal F}$₀ is a constant flux that is independent of Θ and is loosely associated with the globally distributed flux, but does not account for its variation (Schwadron et al. 2011, 2014) in the annular flux maps. At each energy, ${\cal F}$₀ is defined as the average flux of the five lowest-flux pixels in the annular flux maps of Figure 1. Values for ${\cal F}$₀ are listed in Table 3 and are shown as the red-shaded regions of Figures 9(a)–(e).

Table 3. Fit Parameters Derived from Empirical Fits of Equation (10) to the Data of Figure 9

ENA Energy	${\cal F}$0	${\cal F}_{R1}$	${\cal F}_{R2}$	ΘFWHM1	ΘFWHM2	ΘO	fR
0.7 keV	103	360 ± 13	405 ± 16	154° ± 8°	128° ± 5°	61° ± 1°	3.1
1.1 keV	47	144 ± 7	141 ± 9	178° ± 12°	142° ± 8°	55° ± 2°	2.8
1.7 keV	27	63.0 ± 2.9	64.5 ± 1.7	85° ± 4°	187° ± 9°	68° ± 1°	1.9
2.7 keV	10.6	28.6 ± 1.0	30.6 ± 0.9	112° ± 6°	125° ± 6°	74° ± 1°	2.0
4.3 keV	3.8	10.1 ± 0.4	8.9 ± 0.4	118° ± 9°	129° ± 11°	79° ± 1°	1.8

Note. ENA fluxes ${\cal F}$₀, ${\cal F}$_R1, and ${\cal F}$_R2 are in units of (cm² s sr keV)⁻¹.

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The ribbon flux ${\cal F}$_R(Θ) is constructed as the superposition of two flux peaks of different width and magnitude, but offset by the same distance |Θ_O| in opposite directions relative to the sagittal symmetry axis. This construct allows representation of bilateral lobes with reflection symmetry in which the two flux peaks are separated by ∼2|Θ_O|. It also allows for a unimodal distribution, when flux peaks with small offset |Θ_O| relative to their angular widths merge into a single peak. However, representing the unimodal representation with two unimodal distributions is overdetermined (with complete degeneracy at Θ_O = 0°), and small asymmetries in a unimodal flux peak can drive large differences in the fit parameters of the two peaks that comprise ${\cal F}$_R(Θ).

For each of the two ribbon flux peaks of ${\cal F}$_R(Θ), we use a Gaussian distribution, which has the key parameters of flux magnitude ${\cal F}$_R, offset angle Θ_O, and distribution width Θ_W:

$$\begin{eqnarray} {\cal F}_{\rm R} \left({\rm \Theta} \right) &=& {\cal F}_{R1} {\rm exp}\left({\frac{{ - \left({{\rm \Theta} - {\rm \Theta}_{\rm O}} \right)^2}}{{2{\rm \Theta}_{W1}^2}}} \right)\nonumber \\ \ms && + {\cal F}_{R2} {\rm exp}\left({\frac{{ - \left({{\rm \Theta} + {\rm \Theta}_{\rm O}} \right)^2}}{{2{\rm \Theta}_{W2}^2}}} \right), - 180^\circ \le \Theta \le 180^\circ. \nonumber \\ \end{eqnarray} \tag{ 10 }$$

Here, ${\cal F}_{R1}$ and ${\cal F}_{R2}$ are flux constants of each flux peak, Θ_W1 and Θ_W2 are the Gaussian widths of each flux peak, and Θ_O is the azimuthal offset angle of maximum flux relative to the sagittal symmetry axis Θ_S. The FWHM for a Gaussian distribution is Θ_FWHM = 2.35Θ_W.

At each energy, Equation (9) was fit to the F_P2max(Θ) data, which are more sensitive to variations in ribbon flux than F_P9(Θ) data. The fit parameters are listed in Table 3, the black dashed lines of Figures 9(a)–(e) show the individual flux peaks of ${\cal F}$_R(Θ), and the green-shaded regions of Figures 9(a)–(e) show the combination of the individual ribbon peaks ${\cal F}$_R(Θ).

The last column of Table 3 lists the ratio f_R of the total ribbon flux $\int {\cal F}_{\rm R} \,({\rm \Theta})d{\rm \Theta}$ (the area shaded green) to the total flux $\int {\cal F}_{\rm O}\, d{\rm \Theta}$ (the area shaded red). The ribbon flux clearly dominates at all energies, ranging from a factor of ∼3 larger than the underlying flux at low energies and a factor of ∼2 at high energies.

Several trends are apparent in Figure 9 and Table 3. First, the flux magnitudes ${\cal F}_{R1}$ and ${\cal F}_{R2}$ of the bilateral lobes are similar at each energy, which is consistent with the Pearson and CV(RMSD) analysis results of bilateral lobes with similar flux magnitudes.

Second, the offset angle Θ_O of the two lobes increases from ∼60° at low energy to ∼80° at high energy. This increase in offset angle is expected as the flux distribution transitions from a unimodal to a bimodal distribution.

Third, the angular widths Θ_FWHM1 and Θ_FWHM2 vary substantially over the lowest energies at which a unimodal flux distribution dominates. This results from the overdetermined model ${\cal F}_{\rm R} ({\rm \Theta})$, which is attempting to fit two flux peaks to the unimodal data. Nevertheless, at 0.7 keV and 1.1 keV the resulting fit distribution ${\cal F}_{\rm R} ({\rm \Theta})$ suggests a broad unimodal peak with a slight asymmetry, yielding a maximum flux at Θ ∼ −45° rather than at the sagittal symmetry axis. This is consistent with the slight offset of the flux centroid from the sagittal symmetry axis toward the BV-plane.

The results at 1.7 keV mark the transition from a unimodal distribution to bilateral lobes. However, because the bilateral lobes are emerging and not dominant, this system remains largely overdetermined with the emergence of a bilateral lobe at Θ ∼ 60° driving the fit that results in Θ_FWHM1 ≫ Θ_FWHM2. Importantly, the results presented here using a bimodal flux model cannot distinguish whether the observed ENA unimodal and bilateral flux distributions result from a single energy-dependent process or a combination of independent processes that operate at different energies.

Fourth, the bilateral flux peaks at 2.7 and 4.3 keV become distinct, and the fit parameters of the two model flux peaks of ${\cal F}$_R(Θ) agree within 15%. This reinforces the strong reflection symmetry and partially opposing locations that were obtained by the correlation analysis. At these highest energies, the largest deviation of the model results ${\cal F}$_R(Θ) from the F_P2max(Θ) data lie near the one side of the transverse axis Θ ≈ −90°. At this location, which corresponds to the most northern extent of the ribbon in heliographic latitude, one flux lobe appears to split into two sublobes or, equivalently, a notch of flux depletion appears in the lobe.

Most ribbon models (e.g., Schwadron & McComas 2013) (1) attribute the source population of the ribbon ENA flux to the solar wind and its processing in the heliosheath and (2) assume or derive by modeling that the journey of ENAs from their source to IBEX uniquely retains information about properties of the source plasma. Therefore, we should expect some imprint of the latitudinal structure of the solar wind on the ribbon symmetry or at locations of ribbon asymmetry, and particularly in the vicinity of heliospheric latitude transition between slow and fast solar wind.

Figure 9(f) shows the latitude Ψ_HCI of the ribbon in the Heliocentric Inertial (HCI) frame (Fränz & Harper 2002), color-coded for energy and assuming, as before, the ribbon is located 745 from the ribbon center. The yellow shaded region corresponds to the HCI latitudes generally associated with the slow solar wind, for which we have used Ψ_HCI ∼ 36° as the latitude of transition between slow and fast solar wind, although this interface location is substantially blurred by solar variability and by the offset of the Sun's global magnetic field orientation and its spin axis (McComas et al. 2000).

At low energies, we find no obvious association between the symmetry axis of the unimodal distribution and heliocentric latitude. However, at 2.7 and 4.3 keV, we observe that one lobe lies at the highest northern extent of the ribbon, well within the latitude of the fast solar wind, while the other lobe lies at the most southern extent of the ribbon, largely embedded in the region of slow solar wind but reaching the transition latitude between slow and fast solar wind. Therefore, the transverse symmetry axes at 2.7 and 4.3 keV, which lie nearly perpendicular to the sagittal axis, scribe a circle of fixed HCI longitude through the ribbon center. We further investigate this symmetry in the next section.

As viewed from the inner heliosphere, the sagittal symmetry axis is an arc segment of a great circle in the sky that traverses the ribbon center. We therefore represent the symmetry axis as a symmetry plane that contains the Sun and the ribbon center. Figure 10 shows the projection of the symmetry plane (black dashed lines) onto the ENA flux maps in HCI coordinates, whose north pole is tilted 725 relative to ecliptic north and thus is slightly offset from the ecliptic reference frame. Also shown as white diamonds are the location (and its antipode) of the vector normal that defines the symmetry plane. For reference, Table 4 shows one vector normal coordinate (but not its antipode).

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Table 4. Key Parameters of the Sagittal Symmetry Plane of the Ribbon in the Heliocentric Inertial (HCI) Frame

ENA Energy (keV)	Vector Normal to Symmetry Plane		Tilt of Symmetry Plane Relative to HCI Equator
	HCI Longitude	HCI Latitude
0.71	−113°	22°	678
1.1	−105°	31°	591
1.7	−80°	48°	423
2.7	−37°	55°	347
4.3	−18°	53°	366

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Table 4 also lists the tilt angle of the symmetry plane relative to the HGI equator. Because the symmetry plane contains the ribbon center, the tilt angle must always lie between a maximum value of 90°, which corresponds to a symmetry plane that contains the HGI poles, and a minimum value of 346, which corresponds to the HCI latitude of the ribbon center. The derived tilt angles of the symmetry plane systematically shift from a maximum of 68° at low energies to ∼35° at 2.7 keV and 4.3 keV.

Since the ribbon center direction indicates, in principle, the interstellar field direction, this result suggests that the symmetry planes define stages in the deformation of the interstellar magnetic field around the heliosphere. The mean free path of ENAs sampled by IBEX is larger at higher ENA energies, thus higher energy ENAs contain information of ENA emission from larger distances from the Sun compared to lower energy ENAs. Therefore, the energy-dependent deflection of the ribbon center (Funsten et al. 2013) and energy-dependent symmetry axes locations are both likely signatures of the magnetic field deformation at different distances from the Sun, with the higher energy ENAs sampling deeper into the ISM where the ISM magnetic field is less perturbed by the heliosphere.

We find an interesting heliospheric latitudinal ordering at 2.7 and 4.3 keV that was qualitatively discovered in Figure 9, in which the tilt angle of the symmetry plane lies within 2° of the heliographic latitude of the ribbon center. Thus, the symmetry plane corresponds closely to the heliographic equator tilted up to the location of the ribbon center. The bilateral flux lobes at these energies therefore lie at the most southern and northern latitudes scribed by the ribbon and may provide insight into the deformed interstellar magnetic field at comparatively large distances from the heliosphere.

Figure 11 shows simulations of the ENA flux that would be measured in the inner heliosphere based on the secondary emission hypothesis (McComas et al. 2009b; Chalov et al. 2010; Heerikhuisen et al. 2010; Schwadron & McComas 2013; Möbius et al. 2013). The ENA flux results are computed by post-processing 3D, time-dependent, MHD-plasma/kinetic-neutral simulation results of the heliosphere based on the Multi-Scale Fluid-Kinetic Simulation Suite (MS-FLUKSS) framework (Pogorelov et al. 2008b, 2009; Heerikhuisen et al. 2013; Zirnstein et al. 2014). The interstellar magnetic field magnitude and direction at the simulation boundary (∼1000 AU) are 3 μG and ecliptic (45°, −44°), respectively. The results are simulated at a measurement time of 2009.5, with solar wind boundary conditions derived from latitudinal and time-dependent equations from Sokół et al. (2013), while accounting for the delay times between primary ENA creation and secondary ENA detection at 1 AU (see Zirnstein et al. 2014 for more explanation).

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The simulations show circular arcs of ENA flux that generally coincide with the location of the ribbon in the IBEX ENA flux maps. At 1.1 keV shown in Figure 11(a), the ribbon flux is distinctly unimodal and peaked close to the BV-plane, which lies along the ribbon center–heliospheric nose direction. The unimodal ribbon flux distribution that is a dominant feature in the simulation output is directly attributed to the slow solar wind as the primary source population. Furthermore, alignment with the BV-plane indicates the strong, coupled influences of hydrodynamic interactions that are ordered by the direction of the Sun's motion through the ISM and processes that are driven by the ISM magnetic field.

By 2.7 keV, the ribbon flux is distinctly bimodal, with partially opposing bilateral lobes as shown in Figure 11(b). In the simulation, these bilateral flux lobes result from fast solar wind that is associated with higher heliographic latitudes feeding the high latitude extents of the ribbon. The substantial difference between the bilateral lobe flux magnitudes of the simulation results are based on the solar wind properties for the selected epoch up to 2009.5. For example, simulation results using solar wind boundary conditions at time 2013.5 indicate an opposite flux magnitude asymmetry in which the northern lobe is brighter than the southern lobe (see Figure 4 in Zirnstein et al. 2014). These results, which are the subject of a future study, predict that the intensities of bilateral lobes change with the evolution of the solar wind structure and properties over the solar cycle. Nevertheless, this flux magnitude asymmetry of the lobes is notably different than the observations, for which the lobe fluxes are similar in both shape and magnitude. We note that the Figure 1 flux maps used for this study have been acquired over five years, during which the asymmetric brightness of the bilateral lobe fluxes may dynamically change.

Also shown in Figure 11(b) is the equatorial plane (solid yellow line) as projected in the ribbon-centered frame. The bilateral flux lobes appear to be symmetric at the dashed yellow line, which connects the ribbon center and the point of intersection of the equatorial plane and the ribbon. This dashed line marks the inferred symmetry axis and lies below the BV-plane. This is consistent with the 2.7 and 4.3 keV results of Figure 10 in which the symmetry axis corresponds to a tilt of the HCI equator to the ribbon center and suggests a heliographic ordering of the ribbon at high energies.

The ribbon flux is extraordinarily circular and large, spanning ∼150° as projected in the sky (Funsten et al. 2013). This study reveals strong, spectral-dependent reflection symmetry in which the ribbon flux distribution is predominantly unimodal at low energies and predominantly bilateral lobes at high energies. While we find symmetry signatures associated with the HCI reference frame that might be associated with a deformed interstellar magnetic field outside the heliosphere, we do not find symmetry signatures uniquely associated with slow and fast solar wind. This suggests that the ribbon flux is governed by at least one strong spectral filtration process that acts on ENAs between their origin in a source plasma population and their eventual detection by IBEX.

Outside of the ribbon, the spectral information of IBEX measurements may contain more direct signatures of the source plasma properties; for example, ENA spectral measurements at the locations of Voyager 1 and 2, whose locations projected in the sky are largely outside of the ribbon, suggest contributions from multiple plasma populations (Desai et al. 2014). However, the spectral content within the ribbon, which may also contain the combined signatures of multiple plasma populations, are substantially altered by spectral filtration. Thus, understanding the source plasma populations inferred by ENA spectral measurements within the ribbon requires a global understanding of the kinematic processes and energy-dependent transmittance that underlie this spectral filtration.

Key Conclusions:

• 1.  
  The ribbon of ENA flux observed by IBEX has strong reflection symmetry from 0.7 to 4.3 keV, and the primary (sagittal) symmetry axis is a strong function of ENA energy. This study only examines the intrinsic symmetry of the circular ribbon in which the symmetry axis traverses the ribbon center.
• 2.  
  The ribbon ENA flux distribution is predominantly unimodal at 0.7 and 1.1 keV and strongly bimodal, with partially opposing bilateral flux lobes of similar magnitude, at 2.7 and 4.3 keV. The bimodal flux lobes emerge at 1.7 keV. Simulations attribute the slow solar wind as the source plasma for the unimodal flux distribution and the fast solar wind as the source plasma for the bilateral flux lobes.
• 3.  
  The simulations predict asymmetric variability of the flux magnitudes of the bilateral flux lobes over a solar cycle. While the observed flux magnitudes of the bilateral flux lobes are similar, the IBEX flux maps are an integral measurement over a five-year acquisition, which is sufficiently long to mask temporal variability on the time scale of the solar cycle. Flux variability of the bilateral lobes will be an important test for the "secondary" hypothesis of ribbon formation.
• 4.  
  The ribbon center–heliospheric nose vector (and thus the BV plane if the ribbon center defines $\hat {\boldsymbol B}_{\rm ISM}$) appears to be an organizing direction for some symmetry observations. This may indicate the superposition of symmetries associated with the hydrodynamic interaction of the Sun's motion through the ISM, which is ordered by $\hat{\bf v}$_Sun, and the interstellar magnetic field, which is ordered by the direction to the ribbon center. The sagittal symmetry axis appears to be ordered according to the log of the ENA energy, and this ordering appears to be centered at the ribbon center–heliospheric nose direction. First, the sagittal symmetry axes of the unimodal flux distribution at 0.7 and bilateral lobes at 4.3 keV are offset ∼30° in azimuth but in opposite directions from the BV-plane. Second, the sagittal symmetry axis at 1.7 keV, the transition energy between unimodal and bilateral lobe flux distributions, lies in close proximity to the BV-plane. Additionally, the flux centroid analysis indicates that asymmetric flux contributions to the ribbon are biased at ribbon locations in the vicinity of the ribbon center–heliospheric nose vector.
• 5.  
  The most striking association of ribbon symmetry with the heliocentric inertial (HCI) reference frame is that the sagittal symmetry planes at 2.7 and 4.3 keV correspond to a tilt of the HCI equator to the ribbon center, such that the bilateral flux lobes are located at the northernmost latitudes (Ψ_HCI ∼ +71°) and extreme southernmost latitudes (Ψ_HCI ∼ −40°) scribed by the ribbon. The shapes, magnitudes and locations of the bilateral lobes are similar at each energy (with the exception of a flux notch at the transverse axis in the northern lobe).
• 6.  
  The energy dependence of the unimodal and bimodal flux distributions, their locations around the ribbon, and the lack of direct correspondence of symmetry features with fast and slow solar wind strongly suggest the presence of one or more spectral filtration processes that occur between the initial formation of an ENA in the source plasma and its eventual detection at IBEX. The strong dependence of ribbon symmetry on ENA energy may reflect the evolving structure of the perturbed interstellar magnetic field with distance from the Sun. Understanding the underlying processes of spectral filtration is critical for interpreting the observed ENA spectral distributions within the ribbon.
• 7.  
  The different symmetries observed at low (0.7 and 1.1 keV) versus high (2.7 and 4.3 keV) energies may change over time as the structure and properties of the solar wind evolve through the solar cycle.