Code Transmission and Probability

Not long ago did mankind first send rovers to Mars to analyze the planet and find out if it ever supported life. The nagging question “Are we alone?” drives us to penetrate deeper into space. A special challenge associated with such journeys is communication. There needs to be a constant flow of digital data, strings of ones and zeros, back and forth to ensure the success of the space mission.

During the process of transmission over the endless distances, errors can occur. There’s always a chance that zeros randomly turn into ones and vice versa. What can we do to make communication more reliable? One way is to send duplicates.

Instead of simply sending a 0, we send the string 00000. If not too many errors occur during the transmission, we can still decode it on arrival. For example, if it arrives as 00010, we can deduce that the originating string was with a high probability a 0 rather than a 1. The single transmission error that occurred did not cause us to incorrectly decode the string.

Assume that the probability of a transmission error is p and that we add to each 0 (or 1) four copies, as in the above paragraph. What is the chance of us being able to decode it correctly? To be able to decode 00000 on arrival correctly, we can’t have more than two transmission errors occurring. So during the n = 5 transmissions, k = 0, k = 1 and k = 2 errors are allowed. Using the binomial distribution we can compute the probability for each of these events:

p(0 errors) = C(5,0) · p^0 · (1-p)^5

p(1 error) = C(5,1) · p^1 · (1-p)^4

p(2 errors) = C(5,2) · p^2 · (1-p)^3

We can simplify these expressions somewhat. A binomial calculator provides us with these values: C(5,0) = 1, C(5,1) = 5 and C(5,2) = 10. This leads to:

p(0 errors) = (1-p)^5

p(1 error) = 5 · p · (1-p)^4

p(2 errors) = 10 · p^2 · (1-p)^3

Adding the probabilities for all these desired events tells us how likely it is that we can correctly decode the string.

p(success) = (1-p)^3 · ((1-p)^2 + 5·p·(1-p) + 10·p^2)

In the graph below you can see the plot of this function. The x-axis represents the transmission error probability p and the y-axis the chance of successfully decoding the string. For p = 10 % (1 in 10 bits arrive incorrectly) the odds of identifying the originating string are still a little more than 99 %. For p = 20 % (1 in 5 bits arrive incorrectly) this drops to about 94 %.

Code Transmission

The downside to this gain in accuracy is that the amount of data to be transmitted, and thus the time it takes for the transmission to complete, increases fivefold.

NASA’s O-Ring Problem and the Challenger Disaster

In January 1986 the world watched in shock as the Challenger Space Shuttle, on its way to carry the first civilian to space, exploded just two minutes after lift-off. A presidential commission later determined that an O-ring failure in the Solid Rocket Booster (SRB) caused the disaster. This was not a new problem, there’s a long history of issues with the O-rings leading up to Challenger’s loss.

Before the Space Shuttle was declared operational, it performed four test flights to space and back. The first O-ring anomaly occurred on the second test flight, named STS-2 (November 1981). After each flight Thiokol, the company in charge of manufacturing the SRBs, sent a team of engineers to inspect the retrieved boosters. The engineers found that the primary O-ring had eroded by 0.053”. The secondary O-ring, which serves as a back-up for the primary O-ring, showed no signs of erosion. On further inspection the engineers also discovered that the putty protecting the O-rings from the hot gas inside the SRB had blow-holes.

Luckily, the O-rings sealed the SRB despite the erosion. Simulations done by engineers after the STS-2 O-ring anomaly showed that even with 0.095” erosion the primary O-ring would perform its duty up to a pressure of 3000 psi (the pressure inside the SRB only goes up to about 1000 psi). And if the erosion was even stronger, the second O-ring could still finish the job. So neither Thiokol nor NASA, neither engineers nor managers considered the problem to be critical. After the putty composition was slightly altered to prevent blow-holes from forming, the problem was considered solved. The fact that no erosion occurred on the following flights seemed to confirm this.

On STS-41-B (February 1984), the tenth Space Shuttle mission including the four test flights, the anomaly surfaced again. This time two primary O-rings were affected and there were again blow-holes in the putty. However, the erosion was within the experience base (the 0.053” that occurred on STS-2) and within the safety margin (the 0.095” resulting from simulations). So neither Thiokol nor NASA was alarmed over this.

Engineers realized that it was the leak check that caused the blow-holes in the putty. The leak check was an important tool to confirm that the O-rings are properly positioned. This was done by injecting pressurized air in the space between the primary and secondary O-ring. Initially a pressure of 50 psi was used, but this was increased to 200 psi prior to STS-41-B to make the test more reliable. After this change, O-ring erosion occurred more frequently and became a normal aspect of Space Shuttle flights.

On STS-41-C (April 1984), the eleventh overall mission, there was again primary O-ring erosion within the experience base and safety margin. The same was true for STS-41-D (August 1984), the mission following STS-41-C. This time however a new problem accompanied the known erosion anomaly. Engineers found a small amount of soot behind the primary O-ring, meaning that hot gas was able to get through before the O-ring sealed. There was no impact on the secondary O-ring. This blow-by was determined to be an acceptable risk and the flights continued.

The second case of blow-by occurred on STS-51-C (January 1985), the fifteenth mission. There was erosion and blow-by on two primary O-rings and the blow-by was worse than before. It was the first time that hot gas had reached the secondary O-ring, luckily without causing any erosion. It was also the first time that temperature was discussed as a factor. STS-51-C was launched at 66 °F and the night before the temperature dropped to an unusually low 20 °F. So the Space Shuttle and its components was even colder than the 66 °F air temperature. Estimates by Thiokol engineers put the O-ring temperature at launch at around 53 °F. Since rubber gets harder at low temperatures, low temperatures might reduce the O-rings sealing capabilities. But there was no hard data to back this conclusion up.

Despite the escalation of O-ring anomalies, the risk was again determined to be acceptable, by Thiokol as well as by NASA. The rationale behind this decision was:

  • Experience Base: All primary O-ring erosions that occurred after STS-2 were within the 0.053” experience base.

  • Safety Margin: Even with 0.095” erosion the primary O-ring would seal.

  • Redundancy: If the primary O-ring failed, the secondary O-ring would seal.

The following missions saw more escalation of the problem. On STS-51-D (early April 1985), carrying the first politician to space, primary O-ring erosion reached an unprecedented 0.068”. This was outside the experience base, but still within the safety margin. And on STS-51-B (late April 1985) a primary O-ring eroded by 0.171”, significantly outside experience base and safety margin. It practically burned through. On top of that, the Space Shuttle saw its first case of secondary O-ring erosion (0.032”).

Post-flight analysis showed that the burnt-through primary O-ring on STS-51-B was not properly positioned, which led to changes in the leak check procedure. Simulations showed that O-ring erosion could go up to 0.125” before the ability to seal would be lost and that under worst case conditions the secondary O-ring would erode by no more than 0.075”. So it seemed impossible that the secondary O-ring could fail and the risk again was declared acceptable. Also, the fact that the O-ring temperature at STS-51-B’s launch was 75 °F seemed to contradict the temperature effect.

Despite these reassurances, concerns escalated and O-ring task forces were established at Thiokol and Marshall (responsible for the Solid Rocket Motor). Space Shuttle missions continued while engineers were looking for short- and long-term solutions.

On the day of STS-51-L’s launch (January 1986), the twenty-fifth Space Shuttle mission, the temperature was expected to drop to the low 20s. Prior to launch a telephone conference was organized to discuss the effects of low temperatures on O-ring sealing. Present at the conference were engineers and managers from Thiokol, Marshall and NASA. Thiokol engineers raised concerns that the seal might fail, but were not able to present any conclusive data. Despite that, Thiokol management went along with the engineer’s position and decided not to recommend launch for temperatures below 53 °F.

The fact that there was no conclusive data supporting this new launch criterion, that Thiokol did not raise these concerns before and just three weeks ago recommended launch for STS-61-C at 40 °F caused outrage at Marshall and NASA. Thiokol then went off-line to discuss the matter and management changed their position despite the warnings of their engineers. After 30 minutes the telcon resumed and Thiokol gave their go to launch the Challenger Space Shuttle. Shortly after lift-off the O-rings failed, hot gas leaked out of the SRB and the shuttle broke apart.

If you’d like to know more, check out this great book (which served as the source for this post):

The Challenger Launch Decision: Risky Technology, Culture, and Deviance at NASA

Here you can find a thorough accident investigation report by NASA:

For the broader picture you can check out this great documentary:

More on Space Shuttles in general can be found here: Space Shuttle Launch and Sound Suppression.

Space Shuttle Launch and Sound Suppression

The Space Shuttle’s first flight (STS-1) in 1981 was considered a great success as almost all the technical and scientific goals were achieved. However, post flight analysis showed one potentially fatal problem: 16 heat shield tiles had been destroyed and another 148 damaged. How did that happen? The culprit was quickly determined to be sound. During launch the shuttle’s main engine and the SRBs (Solid Rocket Boosters) produce intense sound waves which cause strong vibrations. A sound suppression system was needed to protect the shuttle from acoustically induced damage such as cracks and mechanical fatigue. But how do you suppress the sound coming from a jet engine?

Let’s take a step back. What is the source of this sound? When the hot exhaust gas meets the ambient air, mixing occurs. This leads to the formation of a large number of eddies. The small-scale eddies close to the engine are responsible for high frequency noise, while the large-scale eddies that appear downstream cause intense low-frequency noise. Lighthill showed that the power P (in W) of the sound increases with the jet velocity v (in m/s) and the size s (in m) of the eddies:

P = K * D * c-5 * s2 * v8

with K being a constant, D the exhaust gas density and c the speed of sound. Note the extremely strong dependence of acoustic power on jet velocity: if you double the velocity, the power increases by a factor of 256. Such a strong relationship is very unusual in physics. The dependence on eddy size is also significant, doubling the size leads to a quadrupling in power. The formula tells us what we must do to effectively suppress sound: reduce jet velocity and the size of the eddies. Water injection into the exhaust gas achieves both. The water droplets absorb kinetic energy from the gas molecules, thus slowing them down. At the same time, the water breaks down the eddies.

During the second Space Shuttle launch (STS-2) a water injection system was used to suppress potentially catastrophic acoustic vibrations. This proved to be successful, it reduced the sound level by 10 – 20 dB (depending on location), and accordingly was used during every launch since then. But large amounts of water are needed to accomplish this reduction. The tank at the launch pad holds about 300,000 gallons. The flow starts at T minus 6.6 seconds and last for about 20 seconds. The peak flow rate is roughly 15,000 gallons per seconds. That’s a lot of water!

The video below shows a test run of the sound suppression system:

Sources and further reading:

Click to access art09.pdf

Click to access CAE_XUYue_Investigation-of-Flow-Control-with-Fluidic-injection-for-Jet-Noise-Reduction.pdf

Comets: Visitors From Beyond

The one thing we love the most in the world of astronomy is a good mystery. And if there was ever a mysterious and yet very powerful force of nature that we witness in the night skies, it is the coming of the mighty comet.

The arrival of a comet within view of Earth is an event of international importance. Witness the huge media attention that the Haley or Hale-Bopp have had when they have come within view The sight of these amazing space objects is simultaneously frightening and awe inspiring.


Above all, it is during these comet viewings that the astronomer comes out in all of us. But what is a comet? Where did it come from? And how does it get that magnificent tail?

We should never confuse comets with asteroids. Asteroids are small space rocks that come from an asteroid belt between Mars and Jupiter. While still quite stunning to see, they pale in comparison to the arrival of a comet. Asteroids also have received considerable study by the scientific community.

Not as much is known about comets. As a rule, comets are considerably larger than asteroids. The composition of a comet is a mixture of nebulous, gasses, ice, dust and space debris. One scientist called the composition of a comet as similar to a “dirty snowball” because the composition is so diverse and changeable. The center or nucleus of a comet is usually quiet solid but the “snowball” materials often create a “cloud” around that nucleus that can become quite large and that extends at great lengths behind the comet as it moves through space. That trailing plume is what makes up the comet’s magnificent tail that makes it so exciting to watch when a comet comes within view of Earth.

The origins of comets is similarly mysterious. There are a number of theories about where they come from but it is clear that they originate from outside our solar system, somewhere in deep space. Some have speculated they are fragments left over from the organization of planets that get loose from whatever gravitational pull and are sent flying across space to eventually get caught up in the gravity of our sun bringing them into our solar system.

Another theory is that they come from a gaseous cloud called the Oort cloud which is cooling out there after the organization of the sun. As this space debris cools, it gets organized into one body which then gathers sufficient mass to be attracted into the gravity of our solar system turning into a fast moving comet plummeting toward our sun. However, because of the strong gravitational orbits of the many planets in our solar system, the comet does not always immediately collide with the sun and often takes on an orbit of its own.

The life expectancy of comets varies widely. Scientists refer to a comet that is expected to burn out or impact the sun within two hundred years as a short period comet whereas a long period comet has a life expectancy of over two hundred years. That may seem long to us as earth dwellers but in terms of stars and planets, this is a very short life as a space object indeed.

Scientists across the globe have put together some pretty impressive probes to learn more about comets to aid our understanding of these visitors from beyond. In 1985, for example, the United States put a probe into the path of the comet Giacobini-Zinner which passed through the comets tail gathering tremendous scientific knowledge about comets. Then in 1986, an international collation of scientists were able to launch a probe that was able to fly close to Haley’s comet as it passed near Earth and continue the research.

While science fiction writers and tabloid newspapers like to alarm us with the possibility of a comet impacting the earth, scientists who understand the orbits of comets and what changes their paths tell us this is unlikely. That is good because some comets reach sizes that are as big as a planet so that impact would be devastating. For now, we can enjoy the fun of seeing comets make their rare visits to our night sky and marvel at the spectacular shows that these visitors from beyond put on when they are visible in the cosmos.