Department of Physics & Astronomy, University of Glasgow, Glasgow, G12 8QQ, UK (yuki@astro.gla.ac.uk)

February 2002

The far side of the Moon has been considered the ultimate site for radio observations since the 1960s [1,2], whether for astronomy or for the search for extraterrestrial intelligence. Its main attraction is, of course, the shielding by the Moon against strong radio interference from Earth, both natural and man-made. This effect was demonstrated for very-low-frequency waves down to 250 kHz by the Radio-Astronomy-Explorer-2 satellite around the Moon in the mid-1970s [3]. When the satellite was behind the Moon relative to Earth, the antenna temperatures dropped by a few orders of magnitude, an encouraging result for the future of radio astronomy. For higher sensitivity observations, however, we would like to verify that the interference is attenuated by much more than a few orders of magnitude. This is particularly important for interferometric observations since any coherent noise will be easily picked up. To assess the exact amount of attenuation provided by the Moon, we developed a simulation of electromagnetic wave propagation around the Lunar environment. This report identifies the purpose and goals of this simulation study, describes the method, presents the results, and finally makes conclusions. We find that even for a very long wavelength of 6 km (corresponding to 50 kHz), radio waves are attenuated by as much as 120 dB on the Lunar surface further than about 30^O behind the limb; we expect the attenuation to be only better at shorter wavelengths (higher frequencies).

We are conducting this simulation study, first to verify that the interference is significantly attenuated on the Lunar far side, but mainly to assist in site selection of radio observatories and to make recommendations for precursor measurements.

• Performance evaluation

We would like to evaluate how radio observatory on the far side of the Moon will perform in terms of the interference from the terrestrial kilometric radiation and man-made signals. Although the Radio-Astronomy-Explorer roughly verified the radio-quietness behind the Moon, more quantitative assessment will be useful to ensure that required performance is feasible.

• Site selection

In planning a mission to set up an observatory on the Moon, the choice of site is a very influential factor that affects many elements of mission design, including communication architecture and deployment method. Therefore, identifying candidate sites is a crucial first step for further progress. It is also imperative for planning precursor missions to examine candidate sites.

• Precursor measurement recommendations

To plan a productive precursor mission, we will identify required measurements and make specifications for them. Also, we should make the most out of the upcoming SELENE mission, as it is capable of making many measurements directly relevant to the future of astronomy from the Moon.

Here we list specific goals of this simulation study. The main aim is to constrain the location of sufficient shielding. Since any position on the Moon remains roughly constant relative to Earth, we focus on shielding against interference from Earth. Terrestrial interference includes the auroral kilometric radiation and signals from communication transmitters both on the ground and in orbit. The auroral kilometric radiation is in a very low frequency range of 50 ~ 750 kHz (0.4 ~ 6 km), but very intense, as shown in this diagram:

We aim to find the level of attenuation of such radiation on the Lunar far side at various angles from the limb. We may then very roughly constrain the allowable location for the observatory by comparing the attenuated noise to the galactic background level. Since the magnitude is not everything, particularly in interferometry, we will attempt to find the directivity and coherence of the terrestrial noise. Also important is the wavelength-dependence of the attenuation. Finally, we will examine the influence of the Lunar "ionosphere".

Side note: In parallel to this particular study, we are also using the same simulation code to assess how the site selection and observation are affected by the Lunar ionosphere and the surface (its electrical properties and subsurface structures), respectively.

To investigate these and other potential matters related to radio observations from the Moon, we have developed a numerical program that can simulate wave propagation in any relevant media.

To simulate propagation of electromagnetic waves, we first derive a wave equation suitable for this numerical simulation. We begin with a Maxwell's equation in a general dielectric medium: . With , , and , this becomes , where m₀ is the magnetic permeability of free space, s is the electrical conductivity, e₀ is the permittivity of free space, and e_r is the relative permittivity (dielectric constant). Taking the time derivative, . Using another Maxwell's equation and a vector identity, . Thus, , and with we arrive at our wave equation:

For the investigation of the Moon's shielding effects, we will simulate the terrestrial interference as a plane wave incident on a spherical moon with radially symmetric electrical properties. This problem has a cylindrical symmetry with axis parallel to the direction of the plane wave propagation so that (for each vector component of the field) in the cylindrical coordinate system.

For discrete numerical simulation, we used a finite difference method to evolve the field according to the wave equation. Using the 2^nd-order finite difference in both space and time, for each vector component of the field:

Applying these to our wave equation,

Manipulating this, we finally obtain the expression used for arriving at the new time step (t+Dt):

where (making spatial step size Dr = Dz = Ds),

Using this equation, we expect the real and imaginary parts of the field (as well as each vector component) to evolve independently.

Regarding accuracy, we use a finite difference method that is 2^nd-order in space and time because no visible difference has been detected in results when we tried 4^th-order in space.

Spatial step size was chosen so that the intensity of the wave remains reasonably un-dispersed while it propagates in free space. With our algorithm, at least 6 spatial intervals per wavelength were desired: . Temporal step size was chosen to be half of the spatial step size (in units where c=1) so that the algorithm is stable: (stability condition for finite difference method).

As with any simulation in a finite grid space, the boundary to simulate infinity was a challenge. This program employs a relatively thin damping zone in which the conductivity gradually increases so that the waves are absorbed very gradually to minimize reflections. Since our problem is cylindrically symmetric, only half of the cross-section of the moon is required in the simulation. This half was put against the left side of the grid space and a symmetric boundary was employed for that side.

To produce the plane wave, the top side of the grid space was excited by superposing a sinusoidally-varying field on this line of grid points throughout the period of simulation.

Finally, the accuracy and reliability of our algorithm were tested by verifying that the simulation reproduces expected patterns of reflection, refraction, edge diffraction, and attenuation.

We would expect the electromagnetic properties of the Moon to influence the propagation of radio waves. Examining our wave equation, we must specify the relative electric permittivity e_r (dielectric constant) and the electrical conductivity s of the medium.

To model the body of the Moon itself, we refer to the Lunar Sourcebook [5] for these properties as functions of depth. Both the permittivity and the conductivity of the lunar material depend strongly on the density. The density in turn varies with depth (roughly as r = 1.39 z^0.056), so we can estimate the depth-profile of these electrical properties. For the relative permittivity, data compiled in the Lunar Sourcebook give the approximation:

The electrical conductivity is much more difficult to specify because it varies quite significantly depending on the temperature and also on the frequency. Since we expect that the shielding effect would be inferior with less conductive moon (smaller loss tangent), it is safest to assume the lowest reasonable values for conductivity. For approximation, we work with the DC conductivity (as it is generally lower than the AC conductivity we desire), which is related to the loss tangent L as:

For the loss tangent, one of the approximations given for all data in the Lunar Sourcebook is being used for now:

These expressions were used to construct a moon with radially varying electrical permittivity and conductivity in our simulation.

Above the Lunar surface, the dual-frequency radio occultation experiments by the Luna 19 and the Luna 22 orbiters detected electron concentrations on the sunlit side in the 1970s [6,7]. Depending on the number density, such electron layer could significantly affect the propagation of radio waves, especially at low frequencies. The electron concentration will alter the refractive index n, resulting in effective changes to the permittivity in our wave equation: ,

where n_p is the plasma frequency, which can be approximated as: to give:

In this simulation, a very simple linear profile was used for the electron number density, with the maximum at the surface and linearly decreasing to zero at altitude of 50 km.

A continuous plane wave was incident from the top to simulate terrestrial interference, and the simulation was run long enough for the energy density distribution to reach equilibrium. Here we present the results of simulations using 6-km (50-kHz) very-low-frequency waves. Below is an example showing how the energy density of the wave is attenuated on the far side. (The absorbing zones are apparent in the edges.)

The space surrounding the Moon was filled with an electron density of 5/cm³ to simulate the local plasma due to the Solar wind. Even such tenuous electron concentration will affect the propagation of very-low-frequency waves. Due to this interplanetary scattering, around 50 kHz is the lower limit for astronomical observations from nearby Earth.

Below are plots showing how much the waves are attenuated at various angles from the farthest point on the Moon. The different series are results at various altitudes from the Lunar surface. The altitudes of 100 km and 200 km were included to show the attenuation levels for orbiters (orbiters for precursor measurements or orbiting telescope).

The first plot shows the results without the Lunar "ionosphere":

The next plot shows the results with an "ionosphere" with a maximum electron concentration of 10/cm³ at the surface and linearly decreasing to zero at 50-km altitude:

We see that the ionosphere actually helps to reduce the received energy density within about 30 degrees from the limb.

We find that, even for a very long wavelength of 6 km (50 kHz), radio waves are attenuated by as much as 120 dB on the far-side locations over ~30 degrees from the limb. If we are interested in observing at such low frequencies, the results seem to suggest that we should choose an observatory site at least 30 degrees from the limb. At higher frequencies, the attenuation should only get better because the waves will not be expected to diffract as much.

The result at higher altitudes indicate that, at these low frequencies, an orbiting observatory will be able to take advantage of the shielding by the Moon only during a very small fraction of its orbit. Also, a precursor orbiter mission to verify the shielding effect of the Moon should keep in mind that attenuation of the terrestrial interference on the far side is likely 1~2 orders of magnitude better on the surface than in orbit.

Even if the terrestrial interference is attenuated by 120 dB, an interferometer may pick it up if the noise is coherent. We are currently trying to determine the directivity and coherence of this attenuated interference on the far side of the Moon. We plan to simulate observations by an array on the Lunar far side surface, using the same algorithm for wave propagation.

I would like to thank Dr Graham Woan for his valuable input and advice. I would also like to thank Dr Claudio Maccone for his encouragement. Finally, I would like to thank my friend Luis Armendariz for tips with programming.