The Direction of the Flow of Interstellar Neutral H Based on Photometric Observations from SOHO/SWAN

The distribution of the helioglow in the sky varies with the vantage point, but it clearly depends on the spatial distribution of ISN H. This distribution is shaped by the solar gravity and radiation pressure in the Lyα line on the one hand, and on the other hand by the solar ionization factors, which include charge exchange between solar wind protons and ISN H atoms, and photoionization by solar EUV photons. If the aforementioned solar emissions were invariable in time and spherically symmetric, and the flow of ISN H was perfectly homogeneous far away in front of the Sun (at infinity) down to the heliopause, then the distribution of the density and flow velocity of ISN H inside the heliosphere would be axially symmetric relative to the Sun, with the symmetry axis being the direction of the flow. Then, the density and velocity of ISN H could be represented by the so-called hot model of interstellar gas, developed by Meier (1977), Fahr (1978, 1979), Lallement et al. (1985), and Wu & Judge (1979), and the intensity of the helioglow as observed from the Sun would also be axially symmetric.

Even in this idealized case, the sky distribution of the helioglow would differ depending on the distance from the Sun and the angular distance of the vantage point from the upwind direction. The axial symmetry of the helioglow would be obtained only for vantage points located precisely at the upwind or downwind axis. In this case, the lightcurve obtained from scanning a Sun-centered circle in the sky would be flat, i.e., featureless.

When the vantage point is shifted away from the symmetry axis, then the scanning circle centered at the Sun is no longer perpendicular to the flow axis, and thus, the intensity of the helioglow sampled along this circle is no longer uniform. For a vantage point at a fixed distance from the Sun (e.g., 1 au), the magnitude of the max/min ratio for this observation geometry depends on the offset angle of the vantage point from the upwind direction: the greater the offset angle, the greater the ratio until the offset angle exceeds 90°. Then, the ratio begins to decrease again to reach 1 when the vantage point is located at the downwind axis.

Typically, observations of the helioglow are performed by spacecraft located close to the ecliptic plane. Since the direction of the flow of ISN H is inclined at a small angle to the ecliptic plane (Lallement et al. 2005), an ecliptic-bound observer is never precisely at the upwind or downwind axis. Consequently, even for the idealized conditions defined above, a Sun-centered lightcurve is not expected to ever be flat. However, when an observer within the ecliptic plane is at the ecliptic longitude equal to that of the upwind or downwind direction, the max/min ratio obtained for a Sun-centered lightcurve will be the lowest. Hence, when the longitude of the scanning circle for which the max/min ratio is the lowest, one can immediately adopt this longitude as the longitude of the direction of the flow of ISN H.

The baseline idea of the proposed method is the following:

• 1.  
  Take daily observations of the helioglow performed by a detector traveling around the Sun in the ecliptic plane, like SOHO/SWAN, and in the future IMAP/GLOWS.
• 2.  
  Extract lightcurves for scanning circles centered at or near the Sun with a selected angular radius.
• 3.  
  Calculate max/min ratios for these daily lightcurves.
• 4.  
  Identify the longitudes for which the max/min ratios reach their semiyearly minima.
• 5.  
  The longitudes of the centers of the scanning circles for which the minimum max/min intensity ratios are identified correspond to the longitudes of the downwind or upwind directions.

The proposed method allows deriving the longitude of the flow directly from the photometric maps of the helioglow. As we show further in the paper, it can be applied to nonidealized cases as well. It does not rely on any modeling. It is also independent of the absolute calibration of the detector. The only requirement is that the calibration is stable during a calendar year and uniform spatially. In the case of SOHO/SWAN, it requires that the two sensors of this instrument are reliably cross-calibrated, which has been shown to be true (Quémerais et al. 2006).

To verify the idea outlined in Section 2.1, we performed a modeling study. We simulated the lightcurves of the helioglow for vantage points close to the ecliptic plane distributed around the Sun approximately every 2 weeks. We calculated the max/min ratios for the simulated lightcurves and we searched for the longitudes of the vantage points for which these ratios showed local minima. We verified that the longitudes of the max/min ratio minima agree with the assumed longitude of the direction of the inflow. We performed this study both for solar minimum and solar maximum conditions, while assuming that the ionization rate of ISN H and the solar illumination is either spherically symmetric (2D) or latitudinally structured (3D). Details of the simulations are presented in Section 2.2.1, and processing of the simulated lightcurves is discussed in Section 2.2.2.

With the simulations on hand, we followed the procedure of retrieval of the longitude of the flow from the virtual observations (Sections 2.2 and 2.3) and compared the results with the values adopted in the modeling setup.

With the results of this idealized scenario on hand, we verified that departures from spherical symmetry of the simulated ionization rate (Section 2.2.3) and masking of portions of the lightcurves to prevent contamination from extraheliospheric sourced does not introduce bias in the retrieved longitude of the flow (Section 2.2.4).

2.2.1. Simulations

The locations of the vantage points were selected from the set of the positions of SOHO/SWAN for which full-sky maps of the helioglow are available for the years 2009 (solar minimum) and 2014 (solar maximum). We took a subset of these points distributed around the Sun in approximately 2 week intervals. This interval corresponds to half of the Carrington rotation period. For these vantage points, we simulated the intensity of the helioglow along scanning circles centered at 0° latitude and the longitudes equal to those of the Sun for the given day minus 4°. The radius of the scanning circle was chosen 75°. This viewing geometry corresponds to that of the upcoming IMAP/GLOWS experiment.

The simulations were performed using the Warsaw Helioglow Model (WawHelioGlow; Kubiak et al. 2021a, 2021b). For this part of the modeling, we assumed that the ionization rate of ISN H is spherically symmetric relative to the Sun but varies with time according to the predictions of the Warsaw Heliospheric Ionization Model (WawHelioIon; Porowski et al. 2022) for the ecliptic plane, and that radiation pressure is also spherically symmetric, but varies with time and radial velocity of individual H atoms (WawHelioUV; Kowalska-Leszczynska et al. 2020). These assumptions correspond to the cases marked as 2D in Figures 1 and 2.

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We assumed that ISN H inside the heliosphere is a mixture of two populations, the primary and the secondary. In the selection of the flow parameters, densities, and temperatures of these populations, we followed Kowalska-Leszczynska et al. (2018) (see their Table 1). The primary population at the heliopause is cooler, faster, and much less dense than the secondary. The direction of the inflow of the primary population conforms with that of the primary population of ISN He obtained by Bzowski et al. (2015) based on IBEX observations of ISN He, and the secondary population flows from the direction of the inflow identical to that of the secondary population of ISN He, established by Kubiak et al. (2016).

We simulated two series of 27 lightcurves for vantage points distributed approximately equally around the Sun. One of these simulation series was performed for the solar Lyα flux evolving with time identically as in 2014, and the other one for the conditions during 2009. In Figures 1 and 2 (green lines, purple dots), we plot as an example the lightcurves for three vantage points chosen to be close to the upwind direction and for a point approximately 90° off the upwind direction.

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The small blue and green dots in Figures 1 and 2 represent the example lightcurves, plotted as a function of the position angle along the scanning circle. The thin gray lines, marked "2D fit" in the panels, represent functional approximations of the lightcurves, discussed in Section 2.2.2.

Inspection of the mentioned lightcurves supports the ideas presented at the beginning of Section 2.1. The greatest amplitude of the lightcurve is obtained for the crosswind vantage points, presented in the lower-right panel in Figures 1 and 2. For the three other vantage points, the smallest amplitude is visible in the upper-right panel, which corresponds to vantage points the closest to the upwind longitude adopted in the simulation.

2.2.2. Processing of the Lightcurves

Reliable calculation of the max/min ratio for a given lightcurve is best done by maximization and minimization of an approximation formula fitted to the simulations. We approximate the lightcurves using the following function:

$$\begin{eqnarray}&&F(\psi )=\displaystyle \frac{1}{\sqrt{\pi }}\left(\displaystyle \frac{1}{2}{F}_{0}+\displaystyle \sum _{k=1}^{n}\left({s}_{k}\sin k\psi +{c}_{k}\cos k\psi \right)\right),\end{eqnarray} \tag{ 1 }$$

where F₀, c_k , and s_k are fit coefficients, obtained from fitting using a linear least-squares method, ψ is the position angle along the scanning circle, and n the order of the sin-cos series.

This formula is a representation of the Fourier transform of a function. The order n represents the cutoff and determines the scale of the smallest features that can be reproduced using this approximation. For example, for n = 1, the spatial scale is π, and for n = 3 the scale is π/3. Inspection of Figure 2 suggests that for the representation of a lightcurve featuring four local maxima, an order n ≥ 4 is needed. Generally, the higher the order, the better the approximation; caveats will be discussed later in the paper.

For all lightcurves simulated within a given series (small gray dots in Figures 1 and 2), we fit Equation (1) to the simulated lightcurve (see the thin gray lines in the example plots in Figures 1 and 2), find numerically their maximum and minimum values, and calculate the max/min ratio. Along with the max/min ratio, we record the simulation time and ecliptic longitude of the scanning circle center. These latter longitudes are listed in the headers of the panels shown in Figures 1 and 2.

With all lightcurves from a simulation series processed, we have a series of the max/min ratios as a function of the scanning circle longitude, discussed in Section 2.3, which can be used to retrieve the upwind longitude of the direction of the flow of ISN H, as detailed in Section 2.4.

2.2.3. Effect of Latitudinally Structured Ionization Rate

2.2.4. Lightcurves with Masks

In the real world, some regions in the sky feature a strong extraheliospheric background, which basically precludes their use in the studies of the helioglow.

This background is mostly due to EUV-bright stars and the Milky Way. Detailed contribution from these sources to the helioglow signal depends on the spectral sensitivity of individual instruments. Additionally, in some experiments, like SOHO/SWAN, a portion of the sky is obscured by the spacecraft body, and there is an avoidance region with a certain angular distance from the Sun that must be maintained to prevent instrument damage. This results in sky map masks that must be applied to the observed lightcurves. The SOHO/SWAN science team published a mask for extraheliospheric sources, relevant for photometric observations from this experiment in Quémerais et al. (2006), which we plot in Figure 3. A mask for the spacecraft body for an example sky map is shown in Figure 4.

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To verify that application of masks does not bias the determination of the longitude of the flow, we repeated the procedure presented in Section 2.2.2 for simulated lightcurves with the mask presented in Figure 3 applied. Example lightcurves with the masks applied are shown in Figures 1 and 2 with open circles, and the fitted approximation functions are drawn with thick green lines for the spherically symmetric ionization case, and orange lines for the case of latitudinally structured ionization.

In the case of lightcurves with masks, it sometimes happens that the region where the lightcurve has a maximum or a minimum, is masked. In these situations, we still obtain the maximum and minimum values from the fitted approximation. However, the fitted model gives sometimes unrealistically high or low intensities but we verified that the region of the minima is almost always reproduced correctly.

From analysis of unmasked lightcurves we found that for a yearly max/min ratio series, its maximum values is less than ∼4. Thus, we restrict the allowable magnitude of the max/min ratio to 4. The lightcurves that after masking and fitting yield max/min ratios larger than 4 are eliminated from the sample.

The evolution of the model max/min ratios during a calendar year is presented in Figure 5. The ratio features two local maxima and two minima. The positions of the minima correspond to the position of the center of the scanning circle the closest to the direction of the flow of ISN H adopted in the modeling. The magnitudes of the maxima of the ratios vary with the level of solar activity, as clearly demonstrated by the differences between the blue and orange lines in Figure 5. Also, details of the adopted model of the ionization rate affect the amplitude of the max/min variation, as illustrated by comparison of the set of lines drawn with more intense and more pale colors. The locations of the minima, however, are very stable and do not depend on the solar activity level and the latitudinal structure of the ionization rate.

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Inspection of the ratios for the full lightcurves (solid lines) informs that the maximum of the max/min ratio rarely exceeds 4. The largest value of the maximum ratio is obtained for a spherically symmetric ionization rate and a high level of solar activity. During epochs of lower activity the maximum is lower, and the introduction of a realistic latitudinal structure of the ionization rate reduces the height of this maximum even more.

Application of masks results in modified max/min series in their maximum regions. However, the magnitudes and locations of the minima are little affected, as demonstrated by comparisons of the masked ratios with their unmasked counterparts in Figure 5.

Retrieving the positions of the minima is done similarly as it was for the lightcurves themselves. The yearly series of the max/min ratios are fitted using the approximation function defined in Equation (1), and the positions of the two minima are numerically searched for. From each calendar year, we obtain the downwind and upwind longitude. The downwind longitude is converted to the upwind longitude by the addition of 180° to the result.

The objective of the fitting is finding the phases of the minima, and not a precise reproduction of the shape of the max/min ratio. The max/min ratio features two yearly minima, and thus the minimum order in Equation (1) is n = 2. For the two simulated years and two cases of the ionization rate model for both of them, we investigated the results for the order n in Equation (1) varying from 3 to 7. We performed this study for the cases of lightcurves with and without gaps. The results of the minimization show a slight variation with the order of the approximation formula defined in Equation (1). We found that on average, the retrieved upwind longitude was equal to 253°, varying between 251° and 254°. This is in excellent agreement with the assumed mean direction of the inflow equal to 25321 (see Section 2.2.1).

Based on this study we concluded that the proposed method of determining the upwind longitude for ISN H is suitable for application to actual observations. This is discussed in the following section.

Solar Wind ANisotropies (SWAN; Bertaux et al. 1995) is an instrument on board the ESA space mission Solar and Heliospheric Observatory (SOHO; Domingo et al. 1995), which is used to observe the helioglow. For the purposes of this paper, we focus on photometric maps of the helioglow, collected since the beginning of the mission in 1996.

SOHO operates on an orbit around the solar-terrestrial L1 point. The vantage locations are close to those planned for the IMAP mission. The sky maps are collected using a twin periscope system, which scans two hemispheres of the sky with a narrow overlap between the fields of view of the southern and northern SWAN detectors to facilitate cross-calibration. During the first portion of the mission, maps were available every several days (with a break of several weeks in 1999 due to a mission disturbance), and after stopping spectroscopic observations with the use of a deuterium cell, they became available almost daily. We used a total of 6172 sky maps. The data are publicly available at http://swan.projet.latmos.ipsl.fr/data/L2//FSKMAPS/. An example sky map is presented in Figure 4.

For this paper, from each of the available maps, we extracted a circular belt in the sky with an angular radius of 75°, centered at points in the ecliptic plane with longitudes lower by 4° than the longitudes of the Sun. In this way, we extracted data for identical observation geometry to that planned for the IMAP/GLOWS experiment. The published maps are sampled using bilinear approximation to retrieve the measured intensities every 01 along the scanning circle.

In Figure 6, we show as an example the observed lightcurves for the dates and vantage points close to those discussed in Section 2.2.1. The tiny dots represent the lightcurves retrieved from the original SWAN maps. It can clearly be seen that they have a substantial contribution from extraheliospheric astrophysical sources. The narrow spikes correspond to individual bright stars. The wider bands of enhanced emission, visible in the spin angle ranges of ∼180° to 240° and close to ∼30° in the upper row and the lower left panel, as well as close to 90° to 120° in the lower-right panel, correspond to the Milky Way emission and bright stars seen against its background. These extraheliospheric emissions have fixed positions in the sky and, therefore, can be masked.

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The masks have been developed by the SWAN science team (Quémerais et al. 2006). We applied these masks to the lightcurves retrieved from all of the used maps and merged them with the masks corresponding to the body of the SOHO spacecraft (see the dark-blue regions in the example map shown in Figure 4). Even after the masking, some contributions from extraheliospheric sources remains in the lightcurves. They make a fixed-pattern noise, which inevitably persists in the data.

The masked lightcurves were rebinned into 6° bins. Experiments we have made showed that adoption of this width of the binning is the best compromise between the need to maintain detail on the one hand, and on the other hand, avoiding too much detail in the lightcurves due to the remaining unmasked extraheliospheric sources.

The binning does not address the issues related to nonuniform solar emissions in the Lyα line. These latter nonuniformities are responsible for the so-called searchlight effect (Bertaux et al. 2000). A bright region at the solar surface illuminates a portion of the sky. The illuminated region moves eastward as the Sun rotates. A lightcurve such as that we deal with in this paper has a portion with an enhanced emission. The magnitude and spatial range of this effect in the context of lightcurves retrieved from SWAN maps were discussed by Bzowski et al. (2003). The amplitude varies between ∼1% and ∼5%, with the larger value characteristic for the solar activity maximum conditions. The searchlight effect can potentially affect the derived max/min ratios for individual lightcurves and bias the longitude of the flow derived from a yearly set of observations. The mitigation of this unwanted effect is performed statistically, by averaging the results obtained from multiple years of observations.

The masked and rebinned lightcurves are shown as purple and blue circles in Figure 6. Clearly, they feature gaps. Some of these gaps occupy a substantial portion of a lightcurve and occur in the regions where a maximum or minimum of the lightcurve can be expected. With this, it is not possible to extract the magnitude of the minima and maxima of the lightcurves directly from the masked data product. To address this issue, we approximated the lightcurves retrieved from the maps using the model defined in Equation (1) with n = 3. This prevents reproducing stars and asterisms, as well as the searchlights, and permits covering most of the masked regions, i.e., gaps in the lightcurves. However, it does not allow to cover the largest ones. Adoption of a relatively large scale in the model n = 3, which corresponds to a scale of 60° in the position angle, results in smoothing over the inevitable jitter due to measurement background, individual stars, etc., visible in the example lightcurves shown in Figure 6. Experiments have shown that using a model with this relatively low order in comparison with a similar modeling performed on the simulated lightcurves discussed in Section 2.2.2 reproduces the magnitudes and positions of the minima without reproducing smaller-scale features of the lightcurves related to the searchlights and contributions from unmasked extraheliospheric sources.

Using the fitted lightcurve model, presented in the example given in Figure 6 as solid lines, we computed the magnitudes of the maximum and minimum values and calculated their ratios. The max/min ratios retrieved from all maps were split into series corresponding to individual calendar years. Since on the one hand, analysis of model lightcurves informed that ratios significantly exceeding 4 are not expected, and on the other hand, gaps in the lightcurves fitted to the lightcurves retrieved from SWAN maps sometimes resulted in unrealistically high or low values of the approximating functions, we decided to reject the max/min ratios exceeding 4. One of the rejected cases is illustrated in the upper-right panel of Figure 6 (green line).

The series of max/min ratios corresponding to individual calendar years were fitted to Equation (1). Each of these approximation functions features two local minima as a function of the ecliptic longitude of the center of the scanning circle. These minima are separated by approximately 180°. Their magnitudes correspond to ecliptic longitudes of the upwind or downwind directions. Examples are shown in Figure 7. For two selected years, 2009 and 2014, we show the max/min ratios of the lightcurves retrieved from SWAN maps for these years. The data feature a modulation as a function of the ecliptic longitude of the center of the parent scanning circle. The fitted approximation functions are shown as solid lines, with black dots representing the magnitudes and longitudes of the local minima.

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A comparison of Figures 5 and 7 indicates that the yearly variation of the max/min ratios obtained from the model is very similar to that obtained from the data.