Lunar Radio Telescope Project

From the dawn of time, man’s curiosity has fueled advances in technology, often spawning new fields to explore. One such field is radio astronomy, which involves observing and analyzing radio waves coming from space. Since visible light accounts for a tiny percentage in the electromagnetic spectrum, radio waves collected from observatories often provide useful insight into complex celestial phenomena. Radio Astronomy has, since its inception in the early 20th century, led to substantial increases in astronomical knowledge. The discovery of celestial objects like pulsars, quasars and radio galaxies are attributed to advances in this field. Some of the most extreme and energetic physical processes in the solar system can only be viewed through a radio telescope [1]. In addition, radio astronomy (combined with distributed computing programs like SETI) is key to potential discovery of intelligent extra-terrestrial life [2].

Any radio antenna on earth will capture noise power even when not pointed towards a radio source. This noise-floor that every receiver on earth is plagued with is gradually creeping up due to the increase of man-made radio devices. As the noise-floor increases, radio telescopes find it harder and harder to correctly resolve distant and faint radio emitters in space. If advancements in the wireless industry continue to push at the current rate, soon there will be no place left on earth to observe beautiful radio signatures coming from sources in the universe.

Earth is very fortunate to have its only natural satellite, the moon, in synchronous rotation. That means that there is one side of the moon that always faces away from the earth during the full revolution period of the moon (and has shielded the earth from many asteroids). This “far-side” or “dark-side” of the moon enjoys the luxury of being largely free from man-made noise that is so prevalent on earth. A radio receiver, if designed and deployed on the far side of the moon, will not be affected by the creeping noise floor that plagues terrestrial-bound radio receivers. NASA has hired our company to design a system involving one or more satellites to relay data collected by such a receiver back to earth with fidelity.

The rest of the website is organized as follows: The “System Overview” section sketches out the solution that the company chose from a high-level view. The “Orbital Mechanics” section details the orbital information about the relay satellites that are used by the system. The “Communication Links” section will delve into the technical details of the communication links employed by the system. The “Cost” section estimates the cost required to deploy the whole system, and the references section lists the references used in this project.

Shown in Figure 1 below is the overall system that is used to faithfully relay observed data from the radio telescope to the earth. As can be seen, two satellites are used: one at the Earth-Moon L2 Lagrangian point, and another in a polar orbit around the moon. The radio telescope located at the far side of the moon relays information captured in the required band and relays it to the satellite at L2. The L2 satellite acts as a bent-pipe transponder and relays the same information to the satellite that is in orbit around the moon. This lunar-orbit satellite, in turn, relays the information to earth’s DSN (Deep Space Network). The DSN consists of three antennas located on earth positioned in a way that the satellite in lunar orbit will always have line-of-sight to at least one. The antennas are each 34 meter in diameter and are located in California, Spain and Australia. The satellites used for the system are assumed to be state-of-the-art, with at antennas at least 5 meters in radius.

Figure 1. A "big-picture" view of the system.

The system is designed to be up on average over 99% of the time. When not knocked-out due to rain fading, it will seamlessly provide data in the 1 GHz to 4 GHz bandwidth as captured by the radio observatory on the far side of the moon with a bit-error-rate of less than 10-6.

As explained earlier, 2 satellites were used for relaying information from the observatory on the moon to the DSN on earth in a bent-pipe transponder fashion. Presented below are some of the design choices that were available to choose from when designing the orbital scheme of the satellites in the system:

1. Place one satellite at the Earth-Moon L2 Lagrangian Point and another at the Earth-Moon L4/L5 Lagrangian Point. This system would have been easier to design, but it is sub-optimal because the distance from the satellite to L2 to the satellite at L4 is more than the distance between the two satellites our system uses. With higher distance, the transmit power has to increase to convey the same amount of information. Therefore, this scheme was not adopted.


2. Place three orbiting satellites around the moon such that the angular velocity of each is in a direction perpendicular to the line from the center of the earth to the center of the moon. If such a satellite constellation is properly engineered, one can ensure that the station on the moon will have a line-of-sight component to at least one satellite at any given moment in time. However, such a system suffers from a few drawbacks. Firstly, initial and upkeep cost of three satellites is more than those of two. Secondly, the observatory on the moon has to keep track of the satellites revolving around the moon and synchronize with the three satellites (this can be implemented with an almanac). The satellites themselves also have to synchronize with each other and make sure that the data being sampled at the moon-based observatory is being seamlessly relayed to the DSN on earth. Thirdly, if the data rate of the link from the satellites to earth is to be reduced, Doppler-shift is unavoidable.

The orbital scheme that was chosen is free from Doppler-shift (other than the one due to rotation of the Earth) and employs only two satellites placed relatively close to each other to reduce the transmit power.

Satellite₁ is placed at the Earth-Moon Lagrangian Point L2. Initial analysis of the 3-body problem will yield the orbit to be unstable. However, in practice a satellite at L2 can be quasi-stable by placing it in a Lissajous orbit. A little station-keeping has to be done to keep it on course if it is about to become unstable.

Satellite₂ is placed in a polar orbit around the moon. The direction of angular velocity points in the vector from the Earth’s center to the moon’s center. The height of the orbit was decided at 5400 km above the surface of the moon. It was found that low-moon orbits have a tendency to go unstable, while high-moon orbits would require more transmit power by the relay satellite. A compromise of 5400 km gives a stable orbit while keeping the transmit power required by the satellite at L2 reasonably low.

Figure 1. The orbital scheme used.

Following is a table that summarizes the orbital mechanics of the two satellites employed by the system.

There are six links used in this system. Link1 is the link from the moon to the satellite at the Earth-Moon Lagrangian point L2. Link2 is the link from the aforementioned satellite to the satellite in polar lunar orbit. Link3 is the link from the polar lunar orbit to DSN on earth. Note that Link4, Link5 and Link6 are used for controlling the satellites and are not considered here.

All the links shown are digital links. A design decision to choose digital over analog was made because of the superior performance of digital links (partly because digital links can be coded). The moon-based observatory samples and quantizes the input at its antenna. It then transmits the quanitzed data digitally to the satellite at L2. The L2 satellite relays the information to the lunar orbit satellite, which transmits it to the DSN on earth.

The system is designed to capture data in the 1 GHz to 4 GHz spectrum. In order to obtain a digital bit-stream of this information, the observatory at the moon must sample the radio spectrum at 8 GHz in accordance with Nyquist’s theorem. A design decision to use 6-bit quantization was made based on the following logic. The system temperature on the moon is about 20 K. This means that the radio receiver on the moon will be getting kT_sysB noise power. This comes to -120 dBW. 6-bit quantization means that the digitizing SNR will be about 36 dB. Since typically the power values of radio sources in the outer galaxy are not more than 36 dB more than the noise floor of the observatory at the moon, the decision to use 6-bit quantization is justified.

Once the quantization rate was determined, the data rate could be found out by simply multiplying the sampling rate with the quantization number. The data rate of the system came out to be 36 Gbps. A design decision was taken to use rate ½ convolutional “turbo” code with 256 QAM to provide redundancy in the information bit-stream so that the Bit Error Rate (BER) could be reduced. Using higher QAM modulation scheme meant having to transmit more power while utilizing lesser bandwidth. Since bandwidth was not a major issue as long as it was under 35 GHz (after which the earth receivers become electromagnetically rough), the decision to use 256 QAM was justified. Figure 1 shows a graph of the Bit Error Rate vs. E_b/N₀. As can be seen from the graph, a channel using turbo code comes very close to the ideal Shannon-Capacity channel.

Figure 1. Graph of  E_b/N₀ vs. BER for coded 64 QAM.

Once the data rate and the type of modulation were determined, the link-budget equation was employed to compute the required transmit power for the both satellites as well as the station on the moon. The equation is shown in Figure 4 below. G_T and G_R, the transmit and receive antenna gains respectively, were determined using the parabolic antenna approximation given in Figure 3.

Figure 2. The link-budget equation. P_R and P_T are the received and transmitted power in decibels respectively. G_T and G_R are the transmitter and receiver antenna gains in decibels respectively. r is the distance between the transmitter and receiver and λ is the carrier frequency.

Figure 3. The antenna-gain equation. η_A is the aperture efficiency, λ is the frequency of interest, and A_e is the electromagnetic area (often can be approximated by the physical area).

 Notice that the link budget only holds for a particular frequency. However, due to the high data rate in this system, the bandwidth is large relative to the carrier frequency. Therefore calculations were done using a lambda value near the higher end of the band as well as the lower end. The average of those two results was found to be about the same as a calculation using a frequency that is at the middle of the band. The results obtained by using the link-budget equation using the center frequency are tabulated in Table 1 below.

Table 1. Results of applying the link-budget equation on the three links.

A raised-cosine pulse with a roll-off factor of 0.1 was used for digital communication between the satellites and stations. A general representation of a raised-cosine pulse is shown in Figure 4 below.

Figure 4. A general representation of a raised-cosine pulse. A lower roll-off factor would mean a smoother pulse.

The cost of the project was determined using the project fact sheet provided. Both satellite orbits are considered High-Earth Orbits (HEO). Table 1 below summarizes the total cost of the project.

The websites referenced are provided below:

1. http://en.wikipedia.org/wiki/Radio_astronomy
2. http://setiathome.berkeley.edu/
3. http://science.nasa.gov/headlines/y2006/30nov_highorbit.htm
4. http://www.coseti.org/paper_01.htm

The MATLAB script used to generate the data in this website can be downloaded [here].