Interpretation of results in terms of physical models (Zodiacal Dust Cloud) Part 2

General comments on the data

All the averaged experimental scans show a major absorption feature in the neighbourhood of 5183.6 A. In three cases, there is a hint of a second small absorption line, towards the red. A curious property of the data is that the scans tend to be of less overall height at large elongations. These scans have been compensated for variation in the overall level of light intensity, as monitored by the compensation channel, so this inequality is not the result of variations in intensity around the MgI wavelength. It seems likely that the ‘comp’ channel, with its relatively wide pass-band green filter, is picking up airglow lines, light pollution and faint stars, which, when the ZL intensity is low, significantly inflate the compensation channel counts. This reduces the apparent height of the corrected spectrum. At the lower elongations these contributions are swamped by ZL continuum, and the corrected height rises to its ‘normal’ value. An additional slight rise in intensity at the position of the Gegenschein is also explainable by this reasoning. In future this problem could be avoided by using a narrow filter for the comp channel, closer to the width of the pre-monochromating filter of the F-P.

The results of plotting the wavelength shifts from these fits, against elongation, are shown in Figure 4.1. There is quite a large scatter in the shift values in both observing periods, greater than that suggested by the error bars. In the case of September-October 1971, this may be the result of an unstable Fabry-Perot. However, in April 1972, the finesse stability had been significantly improved, and we might have expected a smoother curve. In fact, the points for April do follow a curve quite well, within the error bar predictions, except for a few ‘rogue’ points (a, b, c in Figure 4.0). Reference to the widths of the Gaussian fits in Figure 4.2 shows a very clear grouping of the widths into a cluster around the median (about 2 A), and a few well outside this width. I think it is quite possible that this unrealistic width in a few of the fits indicates faulty curve fitting. Figure 4.0 shows such a scan. If a median-width Gaussian fit is made to the central points only (the fringe points may be even more noisy than realised at the time, as noted in section 4.4, owing to cosmic rays), in this case at least, a much smaller predicted shift seems likely. This suggests that it might be worth re-reducing these data with a prior bound set on the Gaussian width, and that this would reduce the number of outliers. For the purposes of this thesis, the ‘raw’ output shifts of program GAUSSN are used.


Information obtainable from the Gaussian fits

Figure 4.1, then, shows the result of plotting the mean wavelength shift (AX), against elongation, for the October and April periods. In view of the possibility of Zodiacal Cloud geometry and density being seasonably variable, the two sets of data are kept separate. On each diagram, the previous results of Reay and Ring (1968) and James and Smeethe (1970) are included for comparison. The broken curve drawn through the points in each case is a third order polynomial fitted to these new data. The polynomial fits were made using a slightly modified version of LSQFIT and were weighted inversely as the square of the standard error as calculated for each point by program GAUSSN. This graph (Figure 4.1), showing wavelength shift as a function of elongation, supplemented with various theoretical curves with which my data is compared, is the basis for most of my subsequent interpretation in this study. It is also the basis on which all other studies of radial velocities in the ZL to date may best be compared.

The Hicks and Reay data (HMR) - average wavelength shifts, determined from Gaussian fits, plotted as a function of elongation from the Sun. 'Rogue' points a, b, and c may be the result of failure in curve-fitting (see section 4.2).

Figure 4.1 The Hicks and Reay data (HMR) – average wavelength shifts, determined from Gaussian fits, plotted as a function of elongation from the Sun. ‘Rogue’ points a, b, and c may be the result of failure in curve-fitting (see section 4.2).

Before we look at this graph in detail, two other experimental measurements, the line depth and line width of the absorption feature, can be extracted from parameters of the fitted Gaussian curve, for comparison with theory. The full width of the line at its ‘half-power’ points is related to V3 as follows:

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is at a minimum when

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So …

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We require the values oftmp69218_thumb[5]which bring the function half-way down to the value

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so for these points

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Now

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is just the half-width required, so

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Again, referring to the Gaussian function, an exact measure of the percentage depth of the line (residual intensity) is evidently

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In Figure 4.2 are plotted the widths of the lines calculated from k as described above, again as a function of elongation; Figure 4.3 shows the percentage depths in the same way.

Accuracy of the fits

The program GAUSSN computes confidence limits for all four parameters, which depend on the scatter of the experimental points and on how closely the spectra approximate to a Gaussian shape. However the errors on the points are not all purely due to random shot noise. A serious source of error arises in the outside points. These are already subject to the largest shot noise, due to their small counts, but if the filter shape is only slightly misplaced in wavelength, in the dividing out process, these points are seriously affected, giving an overall asymmetry to the spectrum. To estimate the effect of this, the filter was deliberately misplaced on some test scans by a wavelength interval corresponding to the maximum drift of the etalon during a scan. To these false results the Gaussian curves were fitted as before, and the variation in parameter Xm, and therefore of the ‘average shift’, was found to be about 0.2 A maximum. This, then, is the estimate of the error bar, shown in Figure 4.1. Similar considerations apply to the error bars in Figure 4.2 and Figure 4.3. As noted in section 4.2, there is more scatter in the points than the error bars would lead one to expect. In the 1971 observations, this may be due to an F-P which was not temperature stabilised, and whose finesse was not reliable. In the April observations, with an improved F-P, I now believe that cosmic rays, not easily distinguishable from ZL signal, may have made the peripheral points in the scans even more noisy, sporadically, than previously believed, leading to some false Gaussian fits, sometimes signalled in the graphs by unrealistic widths.

Experimental line widths computed from Gaussian fits - plotted as a function of elongation.

Figure 4.2 Experimental line widths computed from Gaussian fits – plotted as a function of elongation.

 Experimental line depths of the absorption feature as a percentage of continuum level,plotted against elongation

Figure 4.3 Experimental line depths of the absorption feature as a percentage of continuum level,plotted against elongation

It is worth noting that asymmetry in itself cannot be assumed to indicate a misplacement of the filter. The effect can also be the result of not sampling over a wide enough range to reach the continuum level, coupled perhaps with a chance low placing of the end points. It could also be the result of a blend in the absorption feature at the ‘low’ end, only partially resolved. Subsequent to my researches, Clarke et al (1996) predicted the possibility of real pronounced asymmetries resulting from pure prograde models with elliptical orbits and a more complete scattering theory; effectively two shifts, due to two ‘cells’ at opposite ends of the line of sight, are superposed. Also James (1969) was able to predict asymmetries by including retrograde as well as prograde particles. Clarke’s predictions would be noticeable only at lower elongations than were observed in this study, and we will return to the question of whether the asymmetries in the HMR are significant or not, in discussing the findings of Madsen et al (2006), in section 4.13.3.

Evidence from line widths and depths

The predictions of Reay (1969) were used for a preliminary comparison with my data. Reay’s programs were used to simulate spectra from a typical prograde circular model with a = 1, p = 1, and a model similar but consisting of 50 per cent prograde material and 50 per cent retrograde. At the resolution used in the experiment, the two lines one might expect from the second model were not resolved; the result was a curve similar in shape to that due to a pure prograde model (i.e. roughly Gaussian) but slightly wider and less deep.

The predicted figures, for elongation 90 degrees, are

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Depth (%)

1.86

30.4

Prograde

2.12

26.5

Prograde and Retrograde

The difference is about 10 per cent in each case. Our experimental values for these parameters are shown in Figures 4.2 and 4.3. The scatter on the points is quite large, as would be expected, and a difference of 10 per cent would not show up well in these conditions. The HMR results, as can be seen in Figure 4.2, may favour the prograde situation, but it is evident that more and better measurements are needed to make a reliable choice between models. Overall, the widths are similar to the theoretical estimates, but the relative depths tend to be slightly greater than the estimates. This is probably a reflection of the fact, already suspected from asymmetries, that the range of scanning did not satisfactorily reach the true continuum level, producing an under-estimate of its intensity, and hence a higher percentage depth. The possibility of either diffuse ‘wings’ to the line, or of blends, cannot be excluded, though at this point the narrowness of the pre-monochromating filter makes firm conclusions difficult. The only other observers to publish measurements of line-widths are Madsen et al (2006). Their results are shown for comparison in Figure 4.4. The resolution in these observations is much better (12km/s, compared with about 90km/sec), and they also conclude that no dependence of line-width on elongation can be established, but it is very interesting to note that the general pattern of points plotted is quite similar to ours (Figure 4.3). The full-width at half maximum (FWHM) expected for dust in typical asteroidal orbits (about plus and minus 20 km/sec, giving a maximum dispersion of 40 km/sec) is significantly less than the 77 km/s of their, and our, observations.

Madsen et al (2006) remark that to fit the observations, we need to consider a large fraction of non-asteroidal, possibly cometary particles. Future work at even higher resolutions may reveal exactly what this pattern is. Madsen et al (2006) comment that a flat bottom to their profiles, particularly in the Gegenschein area, where one might expect to see a pure daylight spectrum, indicates large numbers of particles in elliptic orbits. However the theoretical curve for elliptical orbits in Figure 4.4 does not fit their observations, or ours. The main conclusion of Ipatov et al (2006) is that asteroids alone cannot account for the WHAM observations of radial velocities, and that particles produced by comets, including high-eccentricity comets such as Comet 2P/Encke and long-period comets, are needed. The mean eccentricity of Zodiacal dust particles that best fit the WHAM observations is estimated to be about 0.5.

Evidently any inferences from these data on the extent of ‘filling in’ of the Fraunhofer line by the superposition of a very blurred spectrum (Ingham 1963) would also be unreliable, and in general, though these figures are included for the sake of completeness, I have not drawn any firm conclusions from them.

Analysis of the wavelength shift versus elongation data

Since the large probable errors in the estimates of line-depth and line-width make it difficult to analyse these graphs with confidence, we will turn our attentions now to the plots of wavelength shift as a function of elongation (Figure 4.1), for which the analysis is much more promising.

From Madsen et al (2006). Line widths as a function of elongation. The coloured lines are models which trace the motion of populations of dust particles of various origins, subject to gravity, radiation pressure, and drag forces,with a ratio of radiation pressure to gravitational force p = 0.02.

Figure 4.4 From Madsen et al (2006). Line widths as a function of elongation. The coloured lines are models which trace the motion of populations of dust particles of various origins, subject to gravity, radiation pressure, and drag forces,with a ratio of radiation pressure to gravitational force p = 0.02.

The dotted curves in this figure show a preliminary least-squares fitted polynomial, computed for each set of data, in a program adapted from the LSQFIT. Whilst the April 1972 data curve is fairly symmetric about the anti-solar point, a distinct evening-morning asymmetry appears in the September-October 1971 data. The evening radial velocity maximum at e = 110° is significantly higher than the morning maximum at e = 40°.

Here follow various attempts to synthesize the experimental data using dynamical models of the supposed Solar System dust cloud and its environs. From each model, the behaviour of the average shift (AX) with elongation is predicted and compared with the shifts extracted from my data. For a more thorough comparison, a full spectrum over the range of interest could be predicted for each elongation, sampled at a rate comparable with that of the data. This kind of analysis is deferred to a future experiment.

Forces experienced by particles in the Zodiacal Cloud

In the models discussed here, the forces acting on a particle are considered to be gravity, radiation pressure, and the Poynting-Robertson effect (Robertson 1937). Forces due to electromagnetic drag, the action of the Sun’s magnetic field on a charged particle, are neglected in this first analysis, which is of a quasi-steady-state velocity pattern.

This decision needs some justification. Parker (1964) calculated that U/V and soft X-rays might produce a charge of +10 Volts on a particle in orbit in the Solar System. The Sun’s magnetic field, that of a dipole, but with the lines of force massively stretched out along the ecliptic plane by the Solar Wind, reverses in sign every few days for a particle in orbit, and the subsequent Lorentz force on sub-micron particles tends merely to disperse them slowly from the invariable plane.

The constant drag force has been estimated as a small force comparable with the Poynting-Robertson force by Singer (1967), and in a co-rotating plasma may be a small accelerating force.

Belton (1966), in a fuller analysis of all the charging and discharging processes, concludes that the force due to the equilibrium charge per particle at 1 AU from the Sun is of the same order of magnitude as both radiation pressure and gravity for a particle of 0.1 microns radius, although he admits to a lack of information especially with regard to discharge, which depends strongly on the shape of the particle. However, Belton has underestimated the radiation pressure at 0.1 microns by using the geometrical cross-section, so the effect of the Lorentz force is over-estimated; in any case the latter decreases sharply with particle radius, so for a particle of 0.3 microns it is proportionally nearly an order of magnitude less, even on this reckoning.

Thus the typical scattering time of these forces is long, and the forces do not significantly modify the momentary trajectory of a particle, even though they play a vital part in the evolution of the interplanetary medium.

Rotating dust cloud models

Many programs have been written to predict the Doppler effects of radial velocities on the absorption line spectrum generated by various rotating dust cloud models. Some recent simulations will be discussed in this topic, but for an initial assessment, the predictions of Reay (1969) were used. Predictions were made from prograde and retrograde models of dust in orbit around the Sun. The orbits are assumed to be circular, and the Poynting-Robertson effect is for this first analysis ignored.

The model assumes a spatial density which falls off as r-a at a distance r from the Sun, and a distribution of particle radii a given by:

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The radiation pressure exerted on a particle by sunlight is proportional to the flux at the location of ethe particle. The force it experiences is therefore, like the gravitational force, inversely proportional to the square of its distance from the Sun, but in the opposite direction. The effect of radiation pressure is thus to reduce the effective gravitational mass of the Sun for the dust particle, reducing its orbital velocity for a given distance from the Sun. But the force is also proportional to the surface area offered, i.e. to the square of its radius, whereas the gravitational force is proportional to its cube, so radiation pressure becomes more important the smaller the particle is. Hence we can expect the magnitudes of the wavelength shift in the predicted curves to vary strongly with p, and on whether the orbits are prograde or retrograde. In Reay’s treatment, both reflected and diffracted components of the scattered light are evaluated, and an integral along the line of sight gives an effective wavelength shift

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For the geometry of this method, which follows that of Ingham, see section 1.2.5 and Figure 1.5; and a fuller description of the line-of-sight integral process, as applied to a continuous flow model, is given in section 4.11.

Figure 4.5 shows the results of Reay’s calculations for a = 2. In both periods of observations the agreement at low elongations (less than 70 degrees) between the new experimental points and a prograde model with a high value of p look promising. For the April results, the agreement continues to be of the right general shape at all high elongations, but: for the September-October results, the model clearly predicts a curve at odds with the Hicks and Reay (HMR) observations, which indicate a crossover at about 80 degrees elongation on the morning side, as opposed to the symmetrical cross-over at 180 degrees predicted in the curves.

It is noticeable that the value p = 5 in the prograde predictions seems to be rather too small for both sets of results. This indicates that we are dealing primarily with the very smallest particles, around 0.3 microns, assuming a density of 3 grams per cubic centimetre, which can exist in orbit around the Sun. As this size is approached from above, the repulsive force due to radiation pressure increases to the point where the effective solar attracting mass approaches zero. A particle which is thus ‘just’ in orbit has a low orbital speed. In other words for particles of such a size, prograde and retrograde models become more and more similar to models in which the dust is moving slowly towards or away from the Sun, or drifting en masse through the Solar System. A linear drift model is described separately in 4.11.

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