Radiation Pressure Acting on the Neutral He Atoms in the Heliosphere

The first measurements of the solar He i 58.4 nm line of helium were done in the 1960s and 1970s using rocket experiments (Hall & Hinteregger 1970; Doschek et al. 1974; Cushman et al. 1975; Maloy et al. 1978). Depending on the method that was used during the observation, the FWHM and/or the total irradiance of the He i line were measured. Table 1 presents selected historical measurements.

The width of the solar emission line of He i (58.4 nm) is not constantly monitored. However, we have several measurements taken over the last few decades, which suggest that the width of the spectral line remains unchanged regardless of the phase of the solar cycle. Its height, on the other hand, depends on the level of solar activity. In this paper, we assume, following Lallement et al. (2004), that the width and the Gaussian shape of the line are constant over time, and normalization is based on measurements from the SDO. ¹

Since 2010, there have been constant measurements available of the solar EUV spectrum taken by the SDO/EVE instrument (Woods et al. 2012) that is orbiting Earth. The full resolution of the spectrum is 0.02 nm. Apart from the spectrum, level 3 of the data products also provides a list of spectral lines with the estimated total irradiance for each of them. Among the listed lines, there is the He i line (58.4 nm) that we are interested in. Currently, we are using version 7 of the data that are available via the website. ² The data are averaged over the Carrington rotation period (see the left panel of Figure 1). The ratio of the extreme values of the fluxes (maximum to minimum) for the most recent solar cycle is ∼1.4.

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The level of the solar spectrum background has been estimated based on measurements around the wavelength λ = 65 nm. The ratio of the He i line flux to the flux within 1 nm of 65 nm is around 32 and weakly dependent on the solar cycle phase. The background level may be significant for atoms whose radial velocities are Doppler-shifted away from the center of the He i line. However, in our case, this will be a secondary effect compared to the already weak effect of radiation pressure.

The definitions and units of the physical constants used in this paper are given in Table 2.

Table 2. Physical Constants in cgs Units

Constant	Symbol	Value	Unit
Elementary charge	e	4.8032 × 10−10	cm3/2 g1/2 s−1
Electron mass	me	9.1094 × 10−28	g
Speed of light	c	2.9979 × 1010	cm s−1
He i wavelength	λ0	58.433	nm
Astronomical unit	rE	1.496 × 1013	cm
Helium mass	mHe	6.6465 × 10−24	g
Gauss gravity constant a	k2 = GM☉	2.959122130672713 × 10−4	au3 day−1 M☉
Spectral flux	Iλ	Time-dependent	ph cm−2 s−1 nm−1
He oscillator strength b	fosc	0.27625	Dimensionless
Combination	$\tfrac{\pi {e}^{2}}{{m}_{{\rm{e}}}c}$	0.0265402	cm2 s−1

Notes. In our numerical simulations, we use more precise values based on the National Institute of Standards and Technology standards.

^a In the simulations, we use the k² combination because the Gauss gravity constant is known with much greater accuracy than the M_☉ and G separately; however, in the equations, we keep GM_☉ for clarity. ^b Drake (2006).

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We assume that the He i spectral line shape is given by the Gauss function normalized in such a way that the value in the center of the line is equal to I₀ = I_λ (λ₀), with FWHM = 0.0136 nm (following the Lallement et al. 2004 model):

$$\begin{eqnarray}&&{I}_{\lambda }(\lambda )={I}_{0}\exp \left[-\displaystyle \frac{1}{2}{\left(\displaystyle \frac{\lambda -{\lambda }_{0}}{\sigma }\right)}^{2}\right].\end{eqnarray} \tag{ 1 }$$

The Gauss function described by Equation (1) takes the value of $\tfrac{1}{2}{I}_{0}$ for ${\lambda }_{h}=\pm 2\sqrt{2\mathrm{log}2}$. From the definition FWHM = 2λ_h , the standard deviation σ can be expressed by FWHM as

$$\begin{eqnarray}&&\sigma =\displaystyle \frac{\mathrm{FWHM}}{2\sqrt{2\mathrm{log}2}}.\end{eqnarray} \tag{ 2 }$$

The equivalent width (w) results from the comparison of the integral of the Gaussian function and the area of the rectangle with the base w and height equal to I₀:

$$\begin{eqnarray}&&{I}_{\mathrm{tot}}={\int }_{-\infty }^{\infty }{I}_{\lambda }(\lambda )d\lambda ={I}_{0}\sqrt{2\pi }\sigma ={{wI}}_{0},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&w=\sqrt{2\pi }\sigma =\sqrt{2\pi }\displaystyle \frac{\mathrm{FWHM}}{2\sqrt{2\mathrm{log}2}}=1.064\mathrm{FWHM}.\end{eqnarray} \tag{ 4 }$$

The total flux I_tot in the He i line is measured by the SDO/EVE instrument. Assuming the Gaussian shape and equivalent width w, we can calculate the flux in the center of the He i line as

$$\begin{eqnarray}&&{I}_{0}=\displaystyle \frac{{I}_{\mathrm{tot}}}{\sqrt{2\pi }\tfrac{\mathrm{FWHM}}{2\sqrt{2\mathrm{log}2}}}.\end{eqnarray} \tag{ 5 }$$

Spectral irradiance in the He i line can be expressed as a ratio between the solar gravity force and the force caused by the radiation pressure. A detailed derivation of the radiation pressure formula is given in Appendix A of Kubiak et al. (2021). Here we give a definition of μ and a scaling factor that can be used to convert the irradiance into the μ factor (Equation (A17) from the aforementioned paper adapted to helium atoms) in the atom's rest frame:

$$\begin{eqnarray}&&\begin{array}{l}\mu =\displaystyle \frac{{P}_{\mathrm{rad}}}{| {F}_{{\rm{g}}}| }\\ \quad =\quad {I}_{0}\displaystyle \frac{\pi {e}^{2}}{{m}_{e}\,c}\displaystyle \frac{h{\lambda }_{0}}{c}\,{f}_{\mathrm{osc}}\displaystyle \frac{{r}_{{\rm{E}}}^{2}}{{{GM}}_{\odot }{m}_{\mathrm{He}}}\\ \quad =\quad \displaystyle \frac{{I}_{0}}{{p}_{\mathrm{He}}},\end{array}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&\begin{array}{l}{p}_{\mathrm{He}}={\left[\displaystyle \frac{\pi {e}^{2}}{{m}_{e}\,c}\displaystyle \frac{h{\lambda }_{0}}{c}\,{f}_{\mathrm{osc}}\,\displaystyle \frac{{r}_{{\rm{E}}}^{2}}{{{GM}}_{\odot }{m}_{\mathrm{He}}}\right]}^{-1}\\ \quad =\,4.162472674170337\times {10}^{13}\ {\mathrm{phs}}^{-1}\,{\mathrm{cm}}^{-2}\,{\mathrm{nm}}^{-1}.\end{array}\end{eqnarray} \tag{ 7 }$$

In the Sun's rest frame, we must take into account the radial component of the atom's velocity, which, due to the Doppler effect, causes photons of slightly different wavelengths to be absorbed. The radial velocity can be easily related to the shift in wavelength (Δλ) using the following formula:

$$\begin{eqnarray}&&{v}_{r}(\lambda )=-\displaystyle \frac{{\rm{\Delta }}\lambda }{{\lambda }_{0}}c.\end{eqnarray} \tag{ 8 }$$

In our calculations, we use the radiation pressure dependent on the instantaneous radial velocity calculated at each point of its trajectory and normalized by the rescaled total solar irradiance that is time-dependent. Thus, the formula we use to calculate the μ factor is as follows:

$$\begin{eqnarray}\begin{array}{rcl}\mu (t,{\rm{\Delta }}\lambda ({v}_{r})) & = & \displaystyle \frac{{I}_{\lambda }(t,{\rm{\Delta }}\lambda ({v}_{r}))}{{p}_{\mathrm{He}}},\\ {I}_{\lambda }(t,{\rm{\Delta }}\lambda ({v}_{r})) & = & \displaystyle \frac{{I}_{\mathrm{tot}}(t)}{\sqrt{2\pi }\tfrac{\mathrm{FWHM}}{2\sqrt{2\mathrm{log}2}}}\exp \left[-\displaystyle \frac{1}{2}{\left(\displaystyle \frac{{\rm{\Delta }}\lambda ({v}_{r})}{\tfrac{\mathrm{FWHM}}{2\sqrt{2\mathrm{log}2}}}\right)}^{2}\right].\end{array}\end{eqnarray} \tag{ 9 }$$

We neglect two factors in our analysis: the absorption effect and the latitudinal structure of solar radiation in the He i line. We consider these factors to be of secondary importance compared to the influence of radiation pressure, which is the main focus of this paper.

The right panel of Figure 1 shows a few example shapes of the He i profile line calculated using our model. There are two vertical axis labels; the left one shows the value of the spectral irradiance in physical units (ph s⁻¹ cm⁻² nm⁻¹), while the right one shows the value of the μ factor that is essential to estimate the radiation pressure strength. In our models, we use radiation profiles as a function of the radial velocity of the interacting helium atom, but for better comparison with the values given in Table 1, we present both horizontal axes (the lower shows radial velocity, while the upper shows the corresponding wavelength calculated from the center of the line). The total irradiance for those profiles was taken from the SDO/EVE measurements shown in the left panel as colored stars. The colors of the stars in the left panel correspond to the colors of the lines in the right panel. The gray dots in the left panel are the daily SDO/EVE measurements, while the gray solid line shows the average over the Carrington rotation period. We choose stars to represent different levels of solar activity during the last cycle.

Even in the high-activity period, when the He i flux is strong (red line), the radiation pressure is at a level of 2.2 × 10⁻³, which means that the force caused by radiation is only 0.22% of the gravitational force acting on the helium atom. At the same time, the radiation pressure acting on the H atoms is of the order of 1 and can compensate for or even overcome the Sun's gravity force. Therefore, we do not expect radiation pressure to significantly affect the He trajectories.

Simulations of the density distribution of helium in the heliosphere were conducted for the minimum solar activity in 2019 and the maximum solar activity in 2015. The computational grid covered three planes: the ecliptic plane, the polar plane passing through the upwind and downwind directions, and the polar plane passing through the crosswind directions. The nodes were spaced at 5° intervals in the azimuthal angle. Distances from the Sun ranging from 0.1 to 54 au were equally spaced on a logarithmic scale.

In situ measurements of ISN He are done by IBEX (McComas et al. 2009). The ISN atoms are observed using the IBEX-Lo instrument (Fuselier et al. 2009), which is a time-of-flight mass spectrometer. It was designed to detect energetic neutral atoms in Earth's orbit and measure the direction from which they came and the time of arrival. The line of sight of the IBEX-Lo detector is inclined to the spacecraft spin axis at an angle of 90° and scans a strip along a great circle with every spin. The spin axis is pointing toward the Sun, and its position changes once per week. Our simulations take into account the gas distribution, geometry of the observations, velocity of the spacecraft, and collimator effects (Sokół et al. 2015). Following Kubiak et al. (2016) and Bzowski et al. (2019), we choose spin angles for each orbit within a 204°–306° range to focus on the peak of the helium signal.

IMAP-Lo is one of the 10 instruments of the IMAP mission (McComas et al. 2018) that will be launched in early 2025. IMAP-Lo will be mounted on a pivot platform that will allow it to change the elongation angle (an angle between the rotation axis of the spacecraft and the detector's axis) during the mission within a range of 60°–160°. The detector's line of sight will be adjusted to observe the most interesting signals depending on the actual spacecraft position. IMAP will be placed in a stable orbit around the Sun–Earth L1 Lagrange point. The spin axis will be pointing 4° behind the Sun in the ecliptic and adjusted daily to maintain the designed position. The expected sensitivity of the instrument is better than that for IBEX-Lo. A longer exposure time on IMAP-Lo will ensure much larger count numbers, and as a consequence, the statistical errors will be smaller than those on IBEX-Lo in the same regions of observation. Our simulations cover a broad range of possible observation configurations. The elongation angle is sampled from 48° to 180° with a cadence of 4°, with an additional node for 90°. The longitude of the spacecraft, which can be translated into time expressed by days of the year (DOY), is sampled every 4 days. We use a full range of spin angles within each geometry of observation, sampled every 6°.

Using the WTPM code, we can track the trajectories of individual atoms depending on their measured energy, detector location, ionization rate, and solar activity phase. We have analyzed the trajectories of five atoms detected during solar minimum and five atoms detected during solar maximum. We selected a few types of atoms.

• 1.  
  Typical atoms from the primary population, where the spin angles correspond to the IBEX-Lo peak of the simulated flux during one orbit (pr orbit 436b and pr orbit 275b).
• 2.  
  Atoms from the primary population, where the spin angles correspond to the small IBEX-Lo flux, but the radiation pressure effect is relatively strong (pr orbit 438b and pr orbit 278a).
• 3.  
  Atoms from the secondary population, where the simulated IBEX-Lo flux is small, but the radiation pressure effect is relatively strong (sc orbit 438b and sc orbit 278a).
• 4.  
  Atom from the secondary population detected by the IMAP-Lo detector for DOY = 265 and elongation angle _FOV = 60°, where the simulated IMAP-Lo flux is high, but the radiation pressure effect is small (sc _FOV = 60°).
• 5.  
  Atom from the secondary population detected by the IMAP-Lo detector for DOY = 265 and elongation angle _FOV = 90°, where the simulated IMAP-Lo flux is small, but radiation pressure effect is relatively strong (sc _FOV = 90°).

The results are shown in Figure 2. The left panels show how the radial distance of the atom from the Sun is changing over time. Solid lines correspond to the calculations without radiation pressure, and dashed lines show how the distance would change if we included radiation pressure. The difference between the solid and dashed lines is visible only far away from the Sun (more than 50 au) for the secondary population detected by IBEX-Lo (blue lines) and IMAP-Lo with the elongation angle _FOV = 90° (dark violet lines). Those atoms travel slower through the heliosphere; therefore, their trajectories are more affected by the radiation pressure than those from the primary population. In the case of the primary population atoms (green lines) or even atoms from the secondary population detected by IMAP-Lo with elongation angle _FOV = 60° (magenta lines), the effect of radiation pressure is negligible.

The right panels of the same figure show the strength of the radiation pressure expressed as a percentage of the Sun's gravity force that is acting on the atoms in the Cartesian heliographic coordinate frame. This means that the radiation pressure is 0.22% of the Sun's gravity in the red parts of the trajectories and below 0.11% for the dark blue parts of the trajectories. The trajectories end at the detector position (IBEX or IMAP, depending on the atom). The strongest radiation pressure is seen close to the Sun, but the exposure time is relatively short, so the overall effect on the trajectory is small. The trajectory marked _FOV = 90 looks like it is getting closer to the Sun than that for _FOV = 60, so the radiation pressure should be bigger there. But in fact, it is just an effect of projection on the xy-plane; the _FOV = 90 atom is much farther from the Sun in the direction of the z-axis, hence the small magnitude of the radiation pressure. By comparing panels (b) and (d), we can see the time dependence caused by the solar cycle. Radiation pressure as a whole is higher during solar maximum (panel (d)) because the solar line He i is stronger at that time. But we do not observe any important differences between the trajectories detected during solar maximum and solar minimum. For all selected atoms, even the strongest effect is still much below 1% of the gravitational force.

Using the WTPM numerical code, we obtained the distribution of He neutral atoms in the heliosphere. We compared density distributions calculated with and without radiation pressure for low and high solar activity (2019 and 2015, respectively). The results show that the difference in density is smaller than 0.2% regardless of the solar cycle phase. The biggest changes can be seen in the case of atoms that have passed close to the Sun, as they were exposed to the strongest radiation pressure. That is why the downwind direction was slightly more affected due to the atom's longer exposure to the effect of radiation pressure. However, in both cases, direct observation of these atoms will be impossible. Therefore, we can conclude that radiation pressure affects neither studies of the pick-up ions (PUIs) nor the ionization rate estimates.

The flux of atoms measured by IBEX-Lo is sensitive to the ionization rate and—for hydrogen—radiation pressure. It was shown that the details of the radiation pressure model for hydrogen atoms can change the estimated flux significantly (Tarnopolski & Bzowski 2009; Kowalska-Leszczynska et al. 2018). One of the main goals of this work is to investigate the sensitivity of the observed signal to the radiation pressure for helium. We analyzed simulations of 12 observational seasons of IBEX from 2009 to 2020. This time span covers more than a full cycle of solar activity. Figure 3 shows two observational seasons that are representative of low (season 2019) and high (season 2015) solar activity. The upper panels in each season ((a) and (c)) show the simulated IBEX-Lo flux of the primary (green line) and secondary (blue line) populations of the ISN He atoms. The lower panels ((b) and (d)) show the relative differences between the fluxes calculated with and without radiation pressure. The red horizontal lines mark levels of 1% and 2%, and the crosses on the flux plots mark the points where the difference is above 1%.

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The strongest effect can be seen when the flux itself is very small. The impact of radiation pressure on the primary population flux is visible in places where it is obscured by a much stronger secondary population flux. The only area where radiation pressure plays a significant role is in the wings of the "warm breeze" in the second half of the observing season. Bzowski et al. (2019) showed that in this region, the differences between the two "warm breeze" treatments (the primary and secondary populations as separate distributions or the primary population and the interactions that create the secondary component) appear, but the errors are too big to distinguish between the two phenomena (see Figure 4 in Bzowski et al. 2019 or Figures 5 and 6 in Kubiak et al. 2016).

Figure 4 shows the relationship between the difference in fluxes calculated with and without radiation pressure as a function of the flux and the spacecraft's ecliptic longitude for the primary (panel (a)) and secondary (panel (b)) populations. The color scale allows for identification of observation directions where this effect may be significant. In both cases, the strongest effect is observed when the spacecraft is between 140° and 160° ecliptic longitude. For the primary population, a flux difference above 2% occurs for very small flux values (below 10⁻³), so it can be neglected. The secondary population is more susceptible to the effects of radiation pressure. A flux difference above 2% mostly occurs when the particle flux exceeds 100, which could potentially be significant. But all of those potentially significant points are located at the wings of the flux versus spin angle plots for each orbit (see blue crosses in Figure 3), where the IBEX-Lo measurement errors are largest (see Figure 4 in Bzowski et al. 2019 or Figures 5 and 6 in Kubiak et al. 2016 for reference). Therefore, the effect of radiation pressure is below the IBEX-Lo sensitivity, and there seems to be no chance to detect it.

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The design of IMAP-Lo will be similar to that of IBEX-Lo but with the addition of a pivot platform that will provide new observational opportunities. Details can be found in Sokół et al. (2019) and M. A. Kubiak et al. (2023, in preparation). Figure 5 shows where we expect radiation pressure to be important in the time and pivot platform elongation angle parameter space. The calculations that we show here assume high solar activity as it was during solar maximum in 2015. IMAP will be launched during similar conditions in 2025. During lower solar activity, the radiation pressure is weaker; therefore, the considered effect is smaller. Colored points show where radiation pressure is relevant; that is, the following conditions for at least one spin angle bin are fulfilled: the expected flux is above 100 cm⁻² s⁻¹ sr⁻¹, the energy of the atoms is greater than 20 eV, and the difference between the calculations with and without radiation pressure is above 1% (dark red), 2% (red), 5% (orange), or 9% (yellow). Gray dots mark all grid points. The primary population, shown in the top panel, is almost unaffected. The only regions where radiation pressure could be relevant are in the downwind direction (IMAP longitude ∼75°) for a low elongation angle of the pivot platform. For the secondary population ("warm breeze"), a large part of the available observation configurations will be affected.

For both populations, a region where we expect an indirect beam of He atoms (DOY > 250) will be especially affected. This effect is understood given that an indirect beam is formed by atoms that passed their perihelia and hence were exposed to a stronger radiation pressure close to the Sun for a longer time than atoms from the direct beam. This region can be compared to the IBEX-Lo "warm breeze" orbits. The flux is comparable, so the statistical errors of the IMAP-Lo measurements should not be larger than those on IBEX-Lo (which are shown in Figure 4 in Bzowski et al. 2019 or Figures 5 and 6 in Kubiak et al. 2016). It means that the radiation pressure effect causing a flux difference of up to 9% will likely be statistically significant.

For a given elongation angle (_FOV) and DOY, the simulated flux is changing with the spin angle of the spacecraft. Selected examples are shown in Figure 6 for the primary and secondary He populations. The elongation angles were taken as is planned for IMAP-Lo observations. Colored points mark the spin angles where all conditions are met (flux above 100 cm⁻² s⁻¹ sr⁻¹, energy above 20 eV, and flux difference greater than 1%). The part of the orbit where the spin angles are smaller than 180° is where the detected atoms are hitting the detector almost head-on. Since the velocity of the spacecraft adds to the velocity of the atom, we will observe higher energies, and the atom will be easier to detect. The other half of the orbit is less preferable due to much smaller energies (the atoms are chasing the spacecraft).