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Subsections

3 Observations at 1 AU

In this section, we present results obtained from our model computations. The first subsection shows that the three populations of H atoms have clearly distinct spectral features. Those features are averaged out for multiple scattered components of the intensity. In the second subsection, we present intensities, line shifts and line widths at 1 AU. The last part outlines the differences between the 3p model and the results of Quémerais (2000) obtained for the hot model.

3.1 Population study: example of the self-absorbed case

Before detailing the results of the 3p model, it is interesting to try and separate the different terms due to the three hydrogen populations. For this, we will consider only the first order of scattering. For higher orders of scattering the spectral features tend to be averaged over the different populations.

We have computed the line profiles in the case of the self-absorbed approximation (SA). The three moments, intensity, apparent velocity and apparent temperature can be computed for each of the populations and for the sum of the three components. For each of these computations, the observer is at 1 AU from the sun and is looking radially away from the sun.

  \begin{figure}
\par\includegraphics[width=13cm,clip]{MS2079f04.ps}
\end{figure} Figure 4: Interplanetary line profile computed using the simple self-absorption approximation. The computation was made for an observer at one AU upwind from the sun looking in the upwind direction. The abscisse is in km s-1 in the solar rest frame and the ordinate is in units of rayleigh. The total line is shown by the thick solid line. The term due to the hot component is shown by the dotted line. The unperturbed interstellar component is shown by the dashed line. The component created at the interface is shown by dot-dashed line.

Figures 4 and 5 show actual line profiles for both upwind and downwind directions. The HSWA population (dotted line) is much hotter than the other two populations. Although it is optically thin, it is slightly absorbed by the other two populations. The PIA component (dash-dot line) is faster than the HIA component (dashed line).

Tables 3 and 4 show the intensity, apparent velocity and apparent temperature for the self-absorbed line profiles as a function of the angle from upwind. Those values are also computed for each of the three populations. Although the HSWA component intensity is fairly small in the upwind direction (less than 5% of total), it represents a larger fraction in the downwind direction mainly because it is much more isotropic than the other two components. The slower component (HIA) is the one which is the more depleted in the downwind cavity because ionization processes are more effective for slower atoms in the solar rest frame.

 
Table 3: Population study: self-absorbed intensity.
angle Total HSWA PIA HIA
($^{\circ}$) (R) (%) (%) (%)
         
0 862 4.3 43.2 52.5
20 839 4.4 43.2 52.4
40 770 4.7 43.4 51.9
60 670 5.2 44.0 50.8
80 561 5.9 44.6 49.5
100 459 6.9 44.9 48.3
120 367 8.1 45.3 46.7
140 281 10.0 46.8 43.2
160 210 12.9 50.8 36.3
180 179 14.9 54.3 30.8



 
Table 4: Population study: SA lineshift and linewidth.
  Apparent velocity in km s-1
angle   Total HSWA HIA PIA
           
0$^{\circ}$   -28.18 -24.90 -24.63 -31.36
30$^{\circ}$   -25.09 -23.27 -22.19 -27.65
60$^{\circ}$   -16.59 -18.08 -15.38 -17.47
90$^{\circ}$   -4.75 -10.64 -5.66 -3.14
120$^{\circ}$   7.24 -2.98 4.64 11.53
150$^{\circ}$   15.49 2.59 12.70 22.53
180$^{\circ}$   17.64 4.84 15.84 27.03
  Apparent temperature in K
angle   Total HSWA HIA PIA
           
0$^{\circ}$   17 590 18 1415 13 508 4928
30$^{\circ}$   17 927 18 0469 14 187 5307
60$^{\circ}$   19 817 18 3057 15 928 6154
90$^{\circ}$   24 104 19 0651 18 085 6978
120$^{\circ}$   30 475 20 0469 19 020 6981
150$^{\circ}$   40 026 20 8312 17 718 6483
180$^{\circ}$   48 744 21 1575 16 282 5902


Table 4 shows that the HSWA component is apparently much hotter than the other two components. When computing the line profile, we see that this component does not affect the center of the backscattered line but only contributes to the wings. In that sense, we will only consider the two cooler components when computing apparent temperatures in the rest of this work because they are the ones which correspond to line center photons. When computing apparent temperatures, we must give a temperature for the line center. Including the HSWA component term leads to overestimates of the line width because the actual profile is far from the Gaussian shape.

  \begin{figure}
\par\includegraphics[width=12cm,clip]{MS2079f05.ps}
\end{figure} Figure 5: Interplanetary line profile computed using the simple self-absorption approximation. The computation was made for an observer at one AU downwind from the sun looking in the downwind direction. The abscisse is in km s-1 in the solar rest frame and the ordinate is in units of rayleigh. The total line is shown by the thick solid line. The term due to the hot component is shown by the dotted line. The unperturbed interstellar component is shown by the dashed line. The component created at the interface is shown by dot-dashed line.


 
Table 5: Line profiles at 1 au from the sun.
$\theta$ ($^{\circ}$) $I_{\rm n}$ (R) $V_{\rm n}$ (km s-1) $T_{\rm n}$ (K) $\frac{I_{\rm ot}}{I_{\rm n}}$ $\frac{I_{0}}{I_{\rm n}}$ $\frac{I_{\rm sa}}{I_{\rm n}}$ $\frac{T_{\rm ot}}{T_{\rm n}}$ $\frac{T_{0}}{T_{\rm n}}$ $\frac{T_{\rm sa}}{T_{\rm n}}$
                   
0 1033 -27.3 11 608 1.08 0.75 0.83 0.87 0.89 0.88
10 1035 -27.0 11 436 1.07 0.74 0.83 0.88 0.90 0.89
20 1022 -26.0 11 478 1.06 0.74 0.82 0.88 0.90 0.89
30 986 -24.2 11 491 1.06 0.74 0.82 0.87 0.91 0.89
40 953 -21.9 11 471 1.05 0.73 0.81 0.88 0.92 0.90
50 896 -19.0 11 687 1.05 0.72 0.81 0.87 0.92 0.90
60 837 -15.6 11 929 1.04 0.72 0.80 0.87 0.93 0.91
70 785 -11.8 12 301 1.02 0.70 0.78 0.86 0.94 0.92
80 723 -7.8 12 715 1.01 0.70 0.78 0.86 0.96 0.93
90 666 -3.6 13 219 1.00 0.68 0.76 0.86 0.98 0.94
100 616 0.6 13 833 0.98 0.67 0.75 0.85 0.99 0.95
110 562 4.7 14 269 0.97 0.66 0.73 0.85 1.01 0.97
120 514 8.5 14 624 0.95 0.64 0.71 0.86 1.03 0.98
130 461 12.0 15 053 0.93 0.62 0.70 0.85 1.03 0.99
140 411 14.9 15 533 0.92 0.61 0.68 0.85 1.03 0.98
150 365 17.3 16 135 0.90 0.59 0.66 0.83 1.01 0.96
160 329 19.0 16 670 0.87 0.57 0.64 0.81 0.99 0.95
170 302 19.7 16 735 0.85 0.55 0.62 0.82 1.00 0.95
180 290 20.0 16 568 0.84 0.55 0.62 0.83 1.01 0.97


3.2 Full radiative transfer computation


  \begin{figure}
\par\includegraphics[width=12cm,clip]{MS2079f06.ps}
\end{figure} Figure 6: Interplanetary line profile computed using the full angle dependent partial frequency redistribution model. The computation was made for an observer at one AU crosswind from the sun looking radially away from the sun. The abscisse is in km s-1 in the solar rest frame and the ordinate is in units of rayleigh. The total line is shown by the thick solid line. Statistical errors are shown. The primary term (sum of 3 primary components) is shown by the dash-dot line. The term due to photons scattered 2 times is shown by the dotted line. Finally, the term computed by the MC code for photons scattered more than two times is shown by the dotted line. The corresponding statistical errors are shown.

An example of a 3p full radiative transfer line profile is shown in Fig. 6. This has been computed for an observer at 1 AU crosswind from the sun and looking radially away from the sun. The primary term (sum of the three populations) is shown by the the dashed line. The total line is shown by the thick line and the secondary term, due to photons that have been scattered more than once between the sun and the observer, is shown by the dotted line. From Table 5, we see that the dotted line (secondary term) represents a bit more than 30% of the total intensity.

Table 5 gives the numerical values of the 3p total intensity, apparent velocity and apparent temperature as a function of the angle from the upwind direction. These values where obtained for a observer at 1 AU from the sun looking radially away from the sun. Note that, as stated above, the apparent temperature is computed using only the PIA and HIA populations which represent the temperature of the core of the line. Intensities and apparent velocities are computed using all three populations. The next three columns of Table 5 give the intensity ratio of the three approximations mentioned above with the total intensity. We have also compared the apparent temperatures obtained from the approximation with the apparent temperature obtained for the complete calculation. In general, the temperature obtained from the primary term is in good agreement with the apparent temperature found for the complete calculation, except in the upwind direction where the apparent temperature is underestimated. The self-absorbed case also gives a correct approximation. As usual is such computations, the optically thin case tends to underestimate the apparent temperature by roughly 15%, i.e. here by 1500 K. On the other hand, the optically thin case gives the best approximation for the total intensity. Note however that the upwind to downwind ratio is significantly changed by multiple scattering effects. This result was also observed in the case of the hot model by (Keller et al. 1981; Hall 1992; Quémerais & Bertaux 1993; Quémerais 2000). Multiple scattering effects tend to fill the downwind cavity when compared to the optically thin case. This decreases the contrast between the upwind and downwind directions.


  \begin{figure}
\par\includegraphics[width=12cm,clip]{MS2079f07.ps}
\end{figure} Figure 7: Intensities from the 3p model as a function of the angle from the upwind direction. The full radiative transfer result is shown by the solid line. It is composed of the primary (dash-dot line) and secondary (3 dots and dash line) terms. For comparison we have added the optically thin result (dotted line) and the self-absorbed computation (dashed line).


  \begin{figure}
\par\includegraphics[width=12cm,clip]{MS2079f08.ps}
\end{figure} Figure 8: Apparent velocity (line shift) as a function of the angle from the upwind direction from the 3p ADPFR model. The reference is the solar rest frame. As in the previous figure, we show the full radiative transfer result (solid line) as well as the result for primary and secondary terms. The optically thin result (dotted line) is very similar to the full computation line shift. Self-absorbed and primary results are very similar.In general line shifts are not very sensitive to the type of computation (ADPFR, OT or SA).

The results of Table 5 are shown in Figs. 7-9. Figure 8 shows that the apparent velocity is not very sensitive to the way the background profile is computed. However, the optically thin case gives an approximation of the apparent velocity which is always within 1 km s-1 of the actual value. The other two approximations are within 3 km s-1. In Fig. 9, we have displayed the apparent temperature of the line profiles as a function of the upwind direction. This apparent temperature corresponds to the sum of the two cooler populations, the PIA and HIA populations. We see in this figure that the full 3p calculation, the primary and self-absorbed calculations give very similar values in the downwind hemisphere. Yet the full 3p calculation is larger in the upwind direction, because it adds a component which is slower (less Doppler shifted) than the two primary components. In the upwind direction, the difference is roughly 1500 K. When comparing those data with optically thin or self-absorbed model computations, as done by Costa et al. (1999), one must roughly remove 1500 K to the observed value in the upwind direction to account for multiple scattering effects.

  \begin{figure}
\par\includegraphics[width=12cm,clip]{MS2079f09.ps}
\end{figure} Figure 9: Apparent temperature (line width) of antiradial line intensities (from 1 AU) as a function of the angle from the upwind direction. The full radiative transfer (3p ADPFR model) result is shown by the diamonds and dotted lines. The solid line corresponds to the primary term computation and the dashed line to the self-absorbed approximation. The statistical errors computed above are shown by the dotted line. The ADPFR model yields results similar to the Primary and Self-Absorbed cases in the downwind hemisphere but slightly larger (15%) in the upwind part of the sky.

3.3 Differences with hot model results

Our interest here is to find ways to discriminate between interface models and hot models as seen from one AU. The reader is referred to Quémarais (2000) for the computation of the Hot model UV background results.

First, it must be pointed out that the main difference between the two types of models can be seen on the spectral profile of the backscattered line (Figs. 4-6). Indeed, the three-population type line profiles are much more dissymmetric than hot model line profiles. This is due to the existence of the faster unperturbed lism component and slower HIA component which are averaged in a unique population in the case of the hot model.

Furthermore, the existence of the hot component (HSWA) in the 3p model is a very obvious tool in this study. It has no counterpart in the Hot model. This result will used to reanalyze the hydrogen cell measurements of the SWAN instrument which are very sensitive to doppler shift.

If we consider the intensity measurements, we note that the two types of models are very similar in the upwind hemisphere. The upwind to crosswind intensity ratio will not give any useful information. The upwind to downwind intensity ratio could be used here, unfortunately this requires to have a very good knowledge of the effective radiation pressure at the time of observation. Indeed, we know that changes of the radiation pressure have effects on the upwind to downwind intensity ratio which are of the same order of magnitude as the one we are seeing between the two types of hydrogen distributions. This becomes even more difficult if we consider solar cycle variation of the radiation pressure as shown by Bzowski et al. (2001). At the present time, the uncertainty on the actual radiation pressure from the sun does not allow us to discriminate between the two types of hydrogen models by simply studying the upwind over downwind intensity ratio. Let us note also that the apparent velocity is sensitive to the value of the radiation pressure.

Finally, we can compare the apparent temperature deduced from these models (Fig. 9). We see that the line width is larger in the downwind direction in the case of the existence of the heliospheric interface. However, the change is rather small (10%) and once again the uncertainty on the various interstellar and solar parameters will prevent an easy diagnostic from observations obtained at one AU from the sun. We can also note that the apparent temperature of the 3p model as a function of the angle with the upwind direction is almost constant within 50$^{\circ}$ of the upwind direction (see Fig. 9). This is not true for the hot model. In that second case, we find an increase of 600 K at 50$^{\circ}$ from the upwind value. A similar effect was described by Costa et al. (1999) from the study of hydrogen cell data. Further data analysis will be required to clarify this point.


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