## Other considerations in rotating models

**This discussion of the rotating dust cloud model has** so far focussed on adjusting the p parameter to fit the results. Other parameters may also produce a change in shape of the curves. The number density was assumed to be proportional to r-a, where r is the distance from the Sun, and a given the value 2, for the purpose of constructing Figure 4.5. This is a reasonable value based on measurements of intensity and polarisation as a function of elongation, made by many investigators. Bandermann and Wolstencroft (1969) show the results of calculations for other values of a (the parameter is called P in this paper). The effect of decreasing a is to raise the curve, giving higher values of (AX) for elongations in the range 75° <e< 180°, so a model could be envisaged which did not have a predominance of sub-micron particles (p = 0 or 1, say) and a number density staying constant with increasing distance from the Sun (a = 0). This latter is almost certainly physically untrue, however. Although some authors have suggested that the density may increase again towards the asteroid belt, all modern observations, including ISO (Reach et al 2003), point to a number density decreasing with radius from the Sun, as far as, and some way beyond, the Earth.

**Figure 4.5 The Hicks and Reay data of wavelength shift vs. elongation, with Reay’s predicted curves for prograde and retrograde circumsolar dust clouds, moving in elliptical orbits. Solid lines represent models with particles of size 0.2 to 100 microns; broken curves are the result of eliminating submicron particles, and dotted curves are for models in which diffraction and radiation pressure are ignored.**

**The difference between the curves for a = 1, 2, or 3 is not very great,** and increasing a beyond this point makes almost no difference. The region close to the Sun, £ < 70°, is not sensitive to radial distribution and it is in this region that the rotating model fits most satisfactorily, so the conclusion that small particles, a micron or less in radius, dominate the cloud is still valid. e

**Figure 4.6 Predictions of Vanysek and Harwit compared with both sets of HMR data. The shaded areas indicate the allowed range of wavelength shifts assuming a geocentric dust cloud.**

**Bandermann and Wolstencroft also give** predictions for elliptical rather than circular orbits. Even for eccentricities near to 1, their curves are not changed as much as for variations in a, and such high eccentricities are highly physically unlikely. And elliptical orbits in any case degenerate to circular under the influence of the Poynting-Robertson effect.

**Figure 4.7 Illustrating the movement of the Sun and the Earth through the local interstellar medium.**

**Thus the possibility of elliptical orbits** does not materially affect the validity of the conclusions already drawn here. However, we will see that Ipatov et al in 2006 came to different conclusions, based on a model in which the dust is being continuously replaced, and there will be much more to say about rotating models in sections 4.13 and 4.14. Meanwhile let us look at other possible models, which include no dust in orbit around the Sun at all.

## Geocentric Dust Cloud (GDC) model

**A thorough theoretical investigation by Shapiro**, Lautman, and Colombo (1967) asserted the absence of a mechanism by which the Earth could acquire a dust cloud of the proportions necessary to make an appreciable contribution to the Zodiacal Light. In spite of this, and the discrediting by Nilsson (1966) of the satellite measurements which gave support to the speculation, the GDC theory still had vocal advocates in 1970.

**Vanysek and Harwit (1970)** suggested that Reay and Ring’s results for £ < 70° represent the dominance of dust near the Earth. The predictions for a model based on terrestrial dust only are shown in Figure 4.6. The HMR results certainly seem to rule out the hypothesis, as can be seen from the high shifts in the observations in the region 75° <£ < 120°, both in morning and evening.

**It is worth commenting that Divari’s model (1965)** of a cloud of terrestrial dust particles in highly elliptical prograde orbits with the major axes in line with the Sun could produce a blue shift in the morning and red in the evening; but it certainly could not account for the reversal in sense of the shift at low elongations which the consensus of radial velocity measurements clearly shows to exist. It is my opinion, based on the above, that cis-terrestrial dust is not a major contributor to the Zodiacal Light.

## A uniform dust flow model

**It was my feeling in 1974** that a component of the ZC might well be interstellar dust, drifting through the Solar System, due to the Sun’s proper motion through the local interstellar medium. I believe that this was the first time such a theory was proposed, previous discussions having been centred on asteroidal and cometal origins of the dust. It was my intention to publish this view in a further paper in MNRAS, but unfortunately my work at Imperial College ceased before this could be achieved. I was recently, however, pleased to find vindication in the Ulysses and Galileo satellite dust collection experiments conducted 20 years later, by Baguhl et al (1995), in which they concluded that there was evidence in the hyperbolic speed, and the distribution of directions in the impacts on their detectors, that interstellar dust was flowing into the Solar System. Moreover Gran et al (1997) were able to estimate a proportion of 30 per cent interstellar dust at 1.3 AU, and even more at greater solar distances. This section details my simple 1973 model to simulate of the appearance of an absorption line in sunlight reflected from dust flowing uniformly through the Solar System.

**Figure 4.8 Showing the two Doppler shifts experienced by light emitted by the Sun, scattered by the dust particle, and observed at the Earth.**

### Uniform flow theory

**In this analysis:**

**a) The direction of motion** of the dust relative to the Sun is assumed to be the negative of the Sun’s motion towards the Solar Apex. This is equivalent to the assumption that the dust cloud is stationary with respect to the local community of stars, and the Solar System is moving through it.

**b) Focusing effects** due to gravity, and scattering by radiation pressure are ignored; the dust is assumed to be of uniform density and flowing linearly.

**c) Scattering is assumed** to be isotropic – i.e. diffraction effects are ignored.

The method is to find the shift in wavelength of sunlight reflected from a general element in space, and integrate along a line of sight for various elongations. Only the ecliptic plane is considered.

**Figure 4.8 illustrates the double** Doppler shift which occurs in this configuration. Referring to the figure, light from the Sun is scattered by the dust particle, and observed at the Earth. On the way, the relative radial velocities of Sun and particle,and Particle and Earth,both produce Doppler shifts,Considering only components of velocity in the ecliptic plane, radiation emitted by the Sun at wavelength X is ‘seen’ by the particle in its own frame of reference at a wavelength is given by e e

where c is the speed of light.

Thus

An observer on the Earth sees a further shiftgiven by

Hence the total shift in wavelength is

**Figure 4.9 Showing the geometry of the scattering, in the plane of the ecliptic only. The diagram shows roughly an October configuration.**

We require this as a function of elongation e from the Sun in terms of the known ephemeris positions and velocities.

**In Figure 4.9**, which is based on the positions for the October 1st configuration, Vd is the (apparent) component of the dust particle’s velocity in the Ecliptic plane. Its direction is 91° ecliptic longitude. Since all speeds are small relative to the speed of light, the effect of Special Relativity on the angles can be ignored.

**If the solar apex has ecliptic latitude B,** and the speed with which the Sun is moving through the interstellar medium is S, the apparent velocity of the dust Vd is —S cos B, representing the negative of the Sun’s velocity component in this plane. It has a numerical value of 11.88 km/s.

It remains to find cos0d, and cos0s in terms of £, the elongation of the dust particle as viewed from the Earth, and x, the distance of the particle from the Earth.

**If L is the ecliptic longitude of the Sun as seen from the Earth at the time in question, and since the direction of the dust flow has ecliptic longitude 91 degrees:**

**To obtain 0 s, referring again to Figure 4.9,**

and, by dropping a perpendicular from the Sun to the line joining the Earth and the particle,

This gives enough information to define 0 for all configurations, and 0 s is then given by 0 —0d (the program STREAM used the ARCTAN function with the signs determined by the two functions above). By this method th€e Doppler shift for the particle was determined.

**To find the contribution to the line-of-sight** integration from an element containing such particles, referring to Figure 4.10 (overleaf), the illumination of dust in the element is inversely proportional to d2, and if we call the scattering area per unit volume p, the light reflected from the element is proportional to

The illumination at the Earth due to this element is inversely proportional to x , so the eventual light intensity per unit solid angle of view is proportional to

So the intensity dE due to this element, arbitrarily scaled, is given by

**Figure 4.10 Showing the geometry of the scattering, in the plane of the ecliptic only. The diagram shows roughly an October configuration.**

We can now determine the ‘average shift" in the usual way as

Thus

**Program STREAM (topic 4)** uses this method to make predictions for elongations at 10 degree intervals. The integral is approximated by dividing x into 1/10 AU steps and adding up the trapezia, the series being truncated when contributions become less than one per cent of the existing total. The output of STREAM was then processed by program CONVL (topic 5), which convolved the predicted spectrum with the instrumental profile of the F-P.

**Figure 4.11 Showing radial velocity predictions as a function of elongation from the Sun for a linear dust flow model, in Spring and Autumn configurations.**

The predicted curves are shown in Figure 4.11, for 1st October and 1st April, along with the HMR experimental points for each case.

**The region around e = 0 exhibits** one of the relatively few examples in Physics of a finite discontinuity. This is to be expected because at low elongations only dust near the Sun needs to be considered, this dust being so highly illuminated by the Sun, as dictated by the inverse square law of illumination. The effect is, however, not a point discontinuity, because of course the Sun appears not as a point, but with a finite angular diameter.

**For a particle orbiting between us and the Sun**, the Doppler shift due to the particle’s motion with respect to the Sun is in opposite senses on opposite sides of the Sun, as viewed from the Earth, and as we look at smaller and smaller orbits, the Sun begins to mark a sharp, discontinuous change in Doppler shift.

**In fact,** the low elongation region is omitted from the diagram for a different reason. If the particles in this region are small with respect to the wavelength of light, as we have reason to believe they are, diffraction becomes important. Diffraction becomes an important consideration in general for any particles with small scattering angles – which occur closest to the Earth along a given line of sight. The half-width of the diffraction peak is given in radians by:

where d is the diameter of the particle.

For aparticle and forthis gives

(radians), or about 100 degrees.

**The light diffracted in this case is thus still scattered over a wide range of angles,** though somewhat less than the 360 degrees assumed for isotropic scattering. This, coupled with the fact that the statistical weighting for these particles is small, means that this roughly zero-shifted component does not seriously compete with the reflected components in the integral along a line of sight in the regions we have been able to observe. Very close to zero elongations this component would eventually become important enough to mask the discontinuity around the Sun, by bringing the curves back towards zero, the effect being more marked if the particles were extremely small. Perhaps, when spectroscopy of the ZL is performed from a space vehicle, it will be appropriate to return to this discussion. In the meantime there is no need to pay any further attention to diffraction in any of these models. As a final note on the situation close to the Sun, there is also reason to believe (Over 1958) that at less than a few solar radii the particles will be vaporised – another reason to ignore this region for these calculations.