Cometary surface layer properties: Possible approaches to radio sounding retrieval during the CONSERT experiment—Numerical simulation and discussion

Authors


Abstract

[1] The instrumentation to be carried to the comet Wirtanen in the Rosetta mission includes a radio experiment, Comet Nucleus Sounding Experiment by Radio Wave Transmission (CONSERT), designed to reveal the average dielectric properties of the nucleus, and the structure of the interior. The analysis of the data to retrieve information about the internal structure has the character of rudimentary radio tomography. We investigate mathematically the behavior of the surface field under the transition from direct propagation to geometrical shadow to determine possible effects of the electrical properties in the close vicinity of the lander. In particular we investigate the effects of a surface layer of composition different from that of the nucleus to determine whether such a layer can influence the wave propagating into the geometrical shadow zone. It is shown that the behavior of the field in the shadow zone is determined chiefly by the topmost layer properties.

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 φc = 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 αc = arcos{[(n2 − 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.

Figure 1.

The geometry of problem with diffracted wave.

Figure 2.

The geometrical dead zone on the surface of the sphere. (a) ray trajectories geometry and (b) dead zone width plot vs. refractive index.

Figure 2.

(continued)

[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 sinα1/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 R22 = R12 + Δ2 − 2R1Δcosϕ. For definitions of all the quantities involved, see Figure 3. Solving this equation with respect to Δ, we obtain

equation image

and from the sine theorem we get sinδ/Δ = sinφ/R2. 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 = R1(cosα1 − cosα), θ = θ1 − 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.

Figure 3.

The geometrical optics consideration of the subsurface echo.

Figure 4.

Geometrical optics time delay of the subsurface echo. (a) refractive index n = 2.0 and (b) refractive index n = 1.3.

[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.

2. Simulation Approach

[13] We simulate the wave propagation from the Orbiter to the Lander. Because the Orbiter is far enough from the comet, we assume the incident wave to be plane. For simplicity we consider only a scalar wave scattering problem:

equation image

with the boundary conditions in spherical coordinates

equation image

at the discontinuities of the refractive index. The more realistic vectorial formulation of the problem requires an additional scalar function U [Van de Hulst, 1957], satisfying the same scalar equation (2) but with different boundary conditions,

equation image

provided the vectorial electromagnetic field is a linear combination of these two scalar functions and their derivatives. We believe that the scalar treatment used here gives meaningful insight into the interference effects occurring in the cometary radio occultation. The scalar treatment of the problem is expected to describe interference and diffraction effects in a semiquantitative way, but we must keep in mind that polarization effects due to the transverse nature of the electromagnetic waves are not represented by our equation. We plan to include the full polarization effects in a future paper.

[14] Since we shall concentrate on the waves traveling only a short distances through the top layer of the comet, the attenuation effects are neglected. The attenuation in the cometary material is currently not expected to be high [Kofman et al., 1998; Herique et al., 1999].

[15] Let us introduce a spherical coordinate system (r, θ, ϕ), where the polar angle θ is defined as shown in Figure 1. The origin is placed at the center of the spherical scatterer. The problem is completely symmetric relative to azimuth angle ϕ. The incident wave is a plane wave

equation image

with unity amplitude. A plane wave can be expanded into a series of spherical Bessel functions:

equation image

Inside the homogeneous sphere, the field can be represented in the form

equation image

which is a general solution of equation (2), finite at the origin (index i denotes the field inside the sphere). Outside this sphere, we seek the field in the form V0 + V, which is the sum of the incident wave and the field scattered from the sphere. The latter must satisfy the Sommerfeld radiation condition at the infinity, to represent the wave scattered by the spherical model of comet. Thus, outside the sphere the solution has the form

equation image

where the Riccati-Bessel function in the parentheses represents the incident plane wave of unity amplitude, and the Riccati-Hankel function represents the scattered field. The unknown coefficients αm, Am must be defined from the boundary condition on the surface of the sphere. Here we introduced the functions

equation image
equation image

which are called the spherical Bessel, Neumann and Hankel functions of the first and second kind, respectively. Their definition given here is often used in scattering problems treatment [Fluegge, 1971]. They are sometimes called Riccati-Bessel, Riccati-Neumann and Riccati-Hankel functions. They obey the recurrence relation

equation image

and

equation image

where zm is any function of (7–8), z is an argument. Note that the definition of spherical functions, given by Abramowitz and Stegun [1965] and used in some literature, are slightly different. We shall use the notation given above.

[16] The unknown coefficients in the solution, derived from the boundary conditions, are

equation image

where R is the radius of the sphere.

[17] If on the surface of the sphere the concentric spherical layer with different electrical properties is present, then the solution of the problem becomes a bit more complicated. In the innermost sphere the solution is sought in the form (5), in the outer space in the form (6) and in the layer we seek it in the form of general solution of the differential equation (2) in spherical coordinates

equation image

and the unknown coefficients Am, Bm, Cm, αm are found from the boundary conditions on the inner and outer boundaries of the layer. These boundary conditions make a system of linear algebraic equation for these four coefficients for each m, so it is not difficult to find a solution. Should several layers be present, we shall have to find more coefficients (two per each boundary) in similar way. We shall not write the solution down here because of complicated structure of the expression. Since we are interested primarily in the field in the outer space, only αm is of interest for us.

[18] For the sphere with arbitrary distribution of refractivity, the solution is also known. Finding the solution would require integration of ordinary differential equation instead of solving the linear equations system. This is much more time consuming, so we restrict our investigation by completely homogeneous spheres and homogeneous spheres with the uniform layer on their surface. This model shall allow us to study basic phenomena, which we expect from the preliminary considerations.

3. Pulse Propagation Simulation

[19] Except of the wave, diffracted over the surface of the sphere, and the wave, reflected from a possible subsurface interface, multiple reflection inside the sphere can occur, and these rays can also reach the point of observation. As we are interesting in local properties near the surface of the sphere, we must somehow filter out these multiple reflections, because they are not informative in our quest for surface information, and they may be less important in practice because of damping of the waves traveling inside the medium. To do this, we shall model the propagation of pulsed signals over the sphere, so that different reflections from internal borders will be well separated in time, so-called time-resolved procedure. This model is close to the actual signal propagation planned in the real experiment CONSERT [Kofman et al., 1998; Herique et al., 1999] on the Rosetta mission.

[20] We choose the incident pulse in the form f(r, t). Taking the spatial Fourier transform of it, we have an expansion of this pulse by the monochromatic plane waves exp(ikz). Denote this spatial spectrum of the pulse F(k, t). For every monochromatic plane wave we can obtain the scattered field V(k, r, ϑ) (6) and then find the solution of the problem of pulsed wave scattering by the sphere as an inverse Fourier transform of over the wave number. We performed the numerical realization of this approach. The discretization in wavenumber space necessary to perform the transform must satisfy the Nyquist criterion, 2Lh < 1, where h is an integration step over k, and L is the length of the spatial interval where we wish the spatial spectrum aliasing to be avoided. Let us choose a Gaussian profile of the pulse

equation image

The spatial Fourier transform over z is

equation image

For the sake of clearest separation of different internal reflections within sphere, the characteristic pulse width σ must be much smaller than the radius of the sphere so σ ≪ R, while R must be small in comparison with L, so an inequality

equation image

must hold. The number of sampling points N in the numerical integration must be so large that on the edges of the integration interval the integrand vanishes. Because the envelope of the spectrum is a simple Gaussian, we obtain the condition for N in form Nhσ/2 ≫ 1. We now can estimate the total computer memory required to store all the data. During the calculation, we have to compute and store an array of coefficients αm for each of N frequencies. Size of each array is about kR, i.e. 2π times radius of sphere measured in wavelength. Thus, the product kRN must not exceed the total number of real numbers which can be stored on the machine. Due to these considerations, the following values were chosen: σ = 20 m, k = 1, R = 500 m, h = 5 10−5, N = 8000. This choice of parameters provided adequate simulation, close to real comet size.

[21] The wave number k = 1 corresponds to the wavelength about 6 m, which belongs to the range of possible radio frequencies, initially proposed for usage in CONSERT experiment. Actually, if refractivity does not depend on the frequency, which is almost the case for water ice at frequencies more than a few tens of MHz, the results should be applicable to any frequency if all dimensions measured in wavelength units are conserved.

4. Simulation Results and Interpretation

[22] By the formulae derived in previous sections, we made calculations for various models of cometary nuclei. First, we assumed the nucleus to be a homogeneous sphere with constant refractive index. In Figures 5a and 5b the signal amplitude is plotted in three dimensions versus angle on the spherical surface and time delay. In Figure 5a the plot is for a refractive index of n = 1.3 and with pulse length σ = 20 m. The center of the figure corresponds to the moment, when the main pulse crosses the border between light and shadow. As one can see, the amplitude of the signal rapidly decreases but smoothly going into the shadow zone. On all the figures, the amplitude is normalized by the incident wave amplitude. Thus, amplitudes are given everywhere in dimensionless units. Recall that values of angle θ larger than >π/2 correspond to illuminated area, while angles θ < π/2 lie in shadow. Figure 5b shows the same, again for a homogeneous sphere, but with a refractive index of n = 1.65. One can see that the amplitude of the signal for equal time and angle depends on the refractive index of the sphere. In principle, this type of dependence can be used for retrieving the value of the local refractive index. In practice, this would require very careful calibration. The maximal value of the amplitude of the main pulse is plotted versus angle in Figure 6 with refractive index as parameter. For small refractive indices, amplitude peaks appear in the shadow zone. It is due to rays focusing, which for small refractive indices occurs near to the surface of the sphere.

Figure 5.

Spatial-temporal diagrams of the wave pulse amplitude on the surface of the homogeneous sphere. (a) n = 1.3 and (b) n = 1.65.

Figure 5.

(continued)

Figure 6.

Peak amplitude of the pulse on the surface of the homogeneous sphere for various refractive indices.

[23] Let us next proceed to models of cometary nuclei with a homogeneous layer on its surface. These layers have a refractive index, which is slightly different from the one of the bulk of nucleus. We assume that nuclear core in our model has refractive index 1.65, which is close to one of pure ice at frequencies planned to be used in experiment. Real nucleus can be porous, but can also contain some fraction of mineral particles, so both positive and negative deviations of refractive index from this value are possible. The refractive index of the top layer can be both greater and less than the one of the core due to changes of chemical composition and physical structure taking place in this layer. Thus, both these situations should be modeled. In Figure 7a an example is shown of the field on the surface of a sphere with a top layer. The layer is 10 m thick with a refractive index of 1.3 resting on top of a denser core with refractive index 1.65 illuminated by a pulse of length σ = 20 m. Not surprisingly the top layer is not resolved because the delay time is shorter than the pulse length. Figure 7b shows the same for a layer thickness of 20 m. In this case there is clear evidence of a surface layer because of the subsurface echo occurring in the shadow zone. In the illuminated zone, interference between subsurface echo and diffracted wave causes notable changes in peak amplitude behavior as a function of angle. In Figure 8a we show a case where the top layer has a larger refractivity of n = 2.0, also for a 10 m thick layer and in Figure 8b with a of the layer of 2.0 refractive index which is 20 m thick. In these cases the top layer is well resolved by the subsurface echo because of greater delay in the layer of high refractivity, so it is not masked by the main peak. For thicker layers, the subsurface echo delay is greater so it is more clearly seen, especially in shadow zone.

Figure 7.

Spatial-temporal diagrams of the wave pulse amplitude on the surface of the sphere of the refractive index n = 1.65 with a layer of n = 1.3: (a) 10 m thick layer and (b) 20 m thick layer.

Figure 8.

Spatial-temporal diagrams of the wave pulse amplitude on the surface of the sphere of the refractive index n = 1.65 with a layer of n = 2.0: (a) 10 m thick layer and (b) 20 m thick layer.

[24] Figure 9 shows the main peak amplitude for spheres with layers of different thickness. The core refractive index was 1.65, the one of the layer was 2.0 everywhere. One can see that for layers 10, 20, 30 and 40 m thick the amplitude curves practically coincide. This means that the behavior of this wave is determined chiefly by the properties of the layer, for the range of layer thickness predicted by thermal models [Espinasse et al., 1991]. Generally it looks like the wave diffracted over the homogeneous sphere with the same refractive index.

Figure 9.

Peak amplitude of the pulse on the surface of the sphere of n = 1.65 with layer for various values of layer thickness. In the layer n = 2.0.

[25] The time delay between the main peak and first reflection remains in good agreement with the geometric optics calculations. One can see that the subsurface echo is observed even in the dead zone, which is not accessible for it from the geometrical optics arguments. This fact gives us evidence that the diffractional phenomena play significant role in the subsurface echo propagation as well as it is for the main echo. The amplitude of the subsurface echo is in qualitative agreement with the one predicted by geometrical optics, which is shown on Figure 10. Since we are discussing behavior of the function v, satisfying the boundary conditions (3), there were assumed amplitude reflection and transition coefficients for locally plane wave corresponding to this boundary condition:

equation image
equation image

where α is the local incidence angle, φ is the local refraction angle.

Figure 10.

Subsurface echoes amplitude on the surface of the sphere of n = 1.65 with layer for various thickness and refractive indices of layer: (a) n = 2 and (b) n = 1.3.

[26] The main pulse, which diffracts over the curved surface, propagates with the wave velocity in vacuum. The amplitude of this pulse depends only on the refractive index of the topmost layer not on its thickness. This statement applies to the case where the main pulse and the internally reflected pulse can be separated.

[27] One can see that for the top layer thickness ten meters and more, the amplitude of main pulse diffracting over the surface is the same that for the homogeneous sphere with the same refractive index. In other words, the amplitude of this main pulse is determined chiefly by the top layer properties. On the other hand, the time of delay of the subsurface echo is in good agreement with geometrical optics predictions, while for its amplitude the qualitative agreement with the geometrical optics also takes place. This allows us, at least, to determine the thickness of the top layer, if it has a sharp boundary.

5. Conclusions and Remarks

[28] The approach to a problem of retrieval of the cometary surface layer parameters from the measurements of CONSERT instrument. It is shown that the properties of the wave diffracting over the cometary surface, are determined mostly by the properties of the topmost cometary layer. If the sharp boundary at the bottom of this layer is present, the reflection from this boundary is observed. The delay time depends on the angle of wave incidence very weak, and it is in good agreement with the delay time predicted by the geometrical optics. This in principle allows us to determine the thickness of this layer, provided its refractivity is known.

[29] The principal possibility of the surface layer parameters retrieval is shown here using simple spherical surface model and scalar wave equation. More detailed treatment of the problem should include analysis of the more common surface geometry and full vectorial problem treatment including the polarization effects.

Notation
n

-refractive index.

α

-local incidence angle, rad.

φ

-local refraction angle, rad.

φc

-critical refraction angle, rad.

θ

-polar angle in spherical coordinates, rad.

r

-radius, m.

R1

-external radius of the crust, m.

R2

-internal radius of the crust, m.

Δ

-auxiliary quantity defined by (1).

Δ

-auxiliary angle defined in Figure 3, rad.

R

-radius of the sphere, m.

k

-wavenumber, 1/m.

V

-wave field.

m

-index of the spherical harmonic.

Am

-amplitude of the spherical harmonic.

jm(.)

-spherical Bessel function.

nm(.)

spherical Neumann function.

hm(1)(.)

-spherical Hankel function of the first kind.

Jm+1/2(.)

-Bessel function of fractional order.

Ym+1/2(.)

-Bessel function of fractional order of second kind.

αm

-partial amplitude of scattered wave.

Cm

-partial amplitude of scattered part of spherical harmonic within the crust.

Bm

-partial amplitude of spherical harmonic within the crust.

f(r, t)

-incident wave packet.

F(k, t)

-spatial spectrum of incident wave packet.

c

-the speed of light, m/s.

t

-time, s.

σ

-characteristic length of incident wave packet, m.

k0

-central value of wave number of incident wave packet, 1/m.

h

-integration step over k, 1/m.

L

-full length of interval of numerical simulation, m.

N

-number of sampling points of the spectrum.

ρ

-wave reflection coefficient.

τ

-wave transition coefficient.

Acknowledgments

[30] This work has been partially supported by the DAAD stipendium Ref.325 PKZ A/99/45081 and the Presidential Stipendium of Russian Federation. The authors thank reviewers for useful comments and W. Kofman for helpful personal discussion.

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