## 1. Introduction

[2] The comets are thought to be the less evolved bodies in the solar system. They are now considered as documents of solar system formation. By studying the details of the cometary structure, we may obtain important information about the conditions in which comets were formed from stellar dust about 4.6 billion years ago. The information on comets, available at this moment, is based mainly on the data of remote sensing of cometary comas. There are almost no direct observations of cometary nuclei, except for recent missions to the Halley comet [*Keller*, 1990]. In addition, observations have been made of tidal disruption of comets by the Jovian gravitation field, allowing us to conclude that the strength of the cometary material is weak. It is supposed now that nuclei consist of easy sublimating volatiles such as water ice (mostly), carbon monoxide and dioxide, and also contain some fraction of mineral nonvolatiles, mainly silicates. The surface of cometary nucleus is directly exposed to the solar radiation, so it is being heated. This heating has strong time variations due to orbital and spin motion of the nucleus. On the surface of the nucleus and the nearest subsurface layers complicated phenomena take place, such as heat transfer, volatile sublimation, condensation etc. These processes must change the physical and chemical properties of the topmost layer of the cometary nucleus, such as temperature, porosity, chemical composition and so on. Thus, the properties of this layer should be different from those of the bulk of the cometary nucleus.

[3] These surface phenomena on the cometary layer have been extensively studied theoretically [e.g., *Skorov et al.*, 1999; *Espinasse et al.*, 1991, and references therein]. According to these papers, significant gradients of the temperature occurs in the subsurface layer about a few tens meter thick. Thus, we expect the physical properties of the surface layer to differ significantly from the ones of the bulk of the nucleus. The peculiarities of the surface layer and its electrical properties can provide useful information about physical and chemical properties of the comet as a whole. A technique, appropriate for investigating of the cometary interior, is through radio wave propagation. The CONSERT experiment, planned to be carried on board the ROSETTA mission, is an instrument of this kind. However, the primary objective of this instrument is the investigation of deep cometary interior. The thin layer, lying near the surface, probably cannot be resolved by means of the algorithms of ray tomography, planned for usage in this experiment. The subject of this paper is the question: to what extent is the CONSERT experiment able to infer information on the topmost layer of the nucleus? If so, which technique of data interpretation is required? The results of this paper give a positive answer to this question and propose a possible approach to the data interpretation.

[4] Searching for the surface layer electrical parameters, we use a technique similar to radio occultation where rays cross the layer close to tangentially. Such experiment geometry is planned in the CONSERT project [*Kofman et al.*, 1998; *Herique et al.*, 1999] in the Rosetta space mission. CONSERT consists of two spacecraft one of which (henceforth the LANDER) will land on the surface of the cometary nucleus while another (the ORBITER) will keep orbiting around it. The pulsed radio signal will be transmitted from Orbiter to Lander and then retransmitted back to the Orbiter. When the Orbiter crosses the cometary horizon as seen from the Lander, we have radio occultation of the Orbiter, very much like radio occultation experiments widely used for investigations of planetary atmospheres.

[5] The visual images of the Halley comet, obtained during the GIOTTO mission [*Keller*, 1990], show that the cometary nucleus is a small body of irregular shape. The surface of it is, generally speaking, not smooth and not convex. The surface layer formed due to solar heating, however, should have uniform thickness only when the surface is smooth and has a radius of local curvature great enough in comparison with layer thickness. In papers dealing with these phenomena spherical [*Espinasse et al.*, 1991] or planar [*Skorov et al.*, 1999] geometry is usually assumed.

[6] Following these papers, we shall also assume spherical symmetry of the nucleus. Since we shall deal with local behavior of the electromagnetic field in the vicinity of the Lander, actual surface need be locally smooth only in this vicinity. In fact, the surface may have two different principal radii of curvature, but the phenomena we are looking for should be clearly seen on the spherical model. There could also be a situation where the surface is not convex, i.e. landing site in a valley or even in a pit. In this case a different approach to the problem is required. Thus, the following treatment will be restricted to the spherical geometry of the cometary nucleus.

[7] There are some obstacles, which prohibit application of traditional occultation techniques to the cometary subsurface layer. In particular, the sharp boundary of the layer gives strong reflections leading to problems in the application of ray tracing algorithms usually used in limb sounding of the atmosphere or ionosphere. However, we can consider a problem of wave scattering on a curved surface and study the wave field in the transition region between illuminated and shadowed zones and the relation with local properties of the surface. Consider a spherical body with given refractive index n. Simple ray tracing shows that the tangent ray touching the top of the curved surface of the body is refracted at the critical angle φ_{c} = arcsin(*n*^{−1}) and continues inside the body. Immediately beyond the tangent point there will be an area of geometrical shadow. Assume a simple spherical shape of the scatterer and plane incident wave, as shown on Figure 1. Angle θ denotes the angle between an arbitrary direction and the direction of the incident plane wave and denotes the position of any arbitrary point on the spherical surface. The local angle of incidence of the plane wave to the surface in any point is α = π − θ and sinα = sin(π − θ) = sinθ. As one can see from Figure 2a, the angle subtended at the center between the top, the tangent point of the rays, and the exit point of a particular ray is given by ψ = 3π/2 − θ − 2φ = 3π/2 − θ − 2 arcsin(sinθ/n). The angle of incidence corresponding to the smallest value of ψ is given by by α_{c} = arcos{[(n^{2} − 1)/3]^{1/2}}. This defines the width of the dead zone on the surface for small values of the refractive index, for larger refractive index account must be taken of rays coming from the opposite hemisphere (lower on the Figure 2a). In this case rays coming from the opposite hemisphere may arrive in the shadow zone and the width of the dead zone is determined by these rays. These rays, however, have quite different travel time and can be filtered out in time domain. The width of dead zone is plotted versus the refractive index n on the Figure 2b, where the first order rays from the opposite hemisphere taken into account.

[8] One can see that the dead zone width reaches its maximal value when the first order rays from both hemispheres meet at the edge of the dead zones. The critical value of refractive index, on which this happens, equals approximately n = 1.56….

[9] If a sharp boundary between the top layer and the bulk of the nucleus is present, the situation is different. From geometrical optics, we expect the signal to reflect from this boundary (Figure 3). The time of delay is easy to calculate. The ray, incident on the surface, by Snell's law bends so that sinα_{1}/sinφ = n, where n is a refractive index of the layer, α = π − θ is the local incidence angle. Further, from the cosine theorem we find that R_{2}^{2} = R_{1}^{2} + Δ^{2} − 2R_{1}Δcosϕ. For definitions of all the quantities involved, see Figure 3. Solving this equation with respect to Δ, we obtain

and from the sine theorem we get sinδ/Δ = sinφ/R_{2}. The time difference between the ray arriving directly at a point with position angle θ, and one reflected from the inner layer boundary, is equal to (d-2nΔ)/c, where d = R_{1}(cosα_{1} − cosα), θ = θ_{1} − 2δ. In this way we calculate the time difference as an implicit function of position angle of the incident ray θ. For certain values of thickness and refractivity of the layer, the time delay is plotted in Figure 4. This delay time can provide information on the topmost layer thickness. Whether one can observe this echo in a real experiment, must be determined by the numerical simulations described below.

[10] The amplitude of reflected wave can also be calculated with the use of the Fresnel formulae for a plane wave, assuming the wave and the surface to be locally flat. This comparison will be made in next paragraphs. If there is no abrupt change of refractive index between the top layer and the bulk there will be no reflection.

[11] A diffracted wave traveling around the spherical surface also does reach the geometrical shadow area, however. For the homogeneous sphere the behavior of this wave can be studied with the use of the asymptotic techniques [e.g., *Houdzoumis*, 2000]. In the general case of arbitrary spherically symmetrical distribution of refractivity a numerical solution is required.

[12] Thus, we have sketched the main features of the field to be expected in our model. We next have to validate our expectations by numerical calculation of the field.