zondag 3 juli 2016

The day the Earth changed its rotation-rate

This week I saw a great article in Wired.com about the static test of one of the solid boosters for the SLS rocket. So I tweeted: "The Earth rotated a little bit faster that day! #rocketengine". I should have known that I cannot make these kind of statements without proving them first. One of my friends called my bluff.

So, I replied with "Challenge accepted!". The blog post will report on my investigation and show how much the Earth speed up/slowed down by the QM-2 test of the SLS booster rocket. My investigation consists of three parts: the model, the search for reliable information, and my conclusions.

Model of the Earth with an attached booster
By igniting their propellant, rockets can produce thrust that normally propel them upwards towards space. If you fix them on the Earth they could speed up or slow done the rotation of the Earth, depending on  the direction of its thrust. Ok, hold your horses! Of course this will have a small effect, but I still need to solve the challenge proposed by my friend. So, lets sketch the situation:

Sketch of the situation of the tied down rocket on a sphere, while letting it burn. top view: Blue circle represents the Blue Marble, or Earth, with its rotation illustrated by 𝝎. The thrust is represented by F and the arm of the torque due to the thrust is r. The moment of inertia of Earth is depicted by IEarth. side view: the latitude of the test location is given by 𝜃.
The thrust or force, F, that is pointing in the direction of the rotation can speed up the rotation, and forces pointing away from the rotation can slow down the rotation. Forces parallel to the rotation vector (the rotation vector is seen in the side view, sticking out the north pole) will not contribute to the change of rotation. So, we need to determine the direction and magnitude of the booster's thrust and its latitude on Earth to determine F and r. In this way we can calculate the torque that is responsible for changing the rotation-rate of Earth:

After we have determined the torque, it is possible to calculate the angular acceleration, 𝞪, with help of the moment of inertia of Earth, IEarth.

We're almost there! If the angular acceleration is obtained, we can calculate the new rotational velocity by using the following equation:

I have placed a minus sign, because in my sketch the force due to the rocket counteract the rotation of Earth. The angular acceleration is multiplied by the period of the rocket burn. The rotation rate of the Earth is around 7.29 ×10−5 rad/sec. With these equations we can find the change in rotation due to the QM-2 booster test.

Information investigation
From the model we can see that there are some numerical parameters needed to calculate the actual change in rotational velocity of the Earth. They are listed bellow:
  • location of test (latitude), 𝜃
  • direction of firing
  • thrust of the booster, F
  • period of firing during test, Δt
  • moment of inertia of the Earth, IEarth
So, lets find them on the internet. We have the Wired.com article with the movie of the firing: link. In this article, I found out that the test is conducted by Orbital ATK. Googling "Orbital ATK QM-2 ground test" gave me some more information. First of all, the expected thrust is reported to be 3.6 million pounds of thrust, which is 16 million Newton of thrust (lets keep it metric). In one of the documents on the website of Orbital ATK I found some more information:

Here, also the operational time of the test was listed as 126 seconds. Somewhere else they wrote about a two minute test, so I am confident this value is ok. This means, two down, three to go. 

The moment of inertia of the Earth can be found anywhere on the web, we will use 8.0 ×1034 kg m2. This is the integrated moment of inertia around the rotation axis of a differentiate sphere. My students need to calculate this for Earth, Moon, and Jupiter. Its a nice mathematical exercise, maybe for a different post.

The last two parameters were a little bit more difficult to find and made me really feel like the Sherlock Holmes of the internet. On the internet webpage of Orbital ATK, I found that their Rocket Testing Facility was in Promontory, Utah. If you type this in Google maps, you still need to go to the northeast towards the road 83. There the facility is clearly visible at latitude 41.62 degrees, which is extremely huge:
The Orbital ATK rocket test facility. The red circle shows the two test areas I found, showing clear exhaust-plume evidence
After half an hour inspecting the facility, I found two sites where the evidence of rocket tests were clearly visible. In the movie of Wired.com a large exhaust plume is visible, which affects the terrain, this must be visible on Google maps. Unfortunately, there are two sites on the Promontory grounds, so I still did not know exactly the location and direction of the thrust vector. Lets zoom in:

Zoom in of the testing sites of Orbital ATK. The red lines indicate visual evidence. I indicated the terrain that is affected by the exhaust plumes by 1. Sites 2 and 3 give more visual information. 
How to pick between the two sites? I went back to the movie posted on Wired.com and made stills that gave me confidence about the correct site where the test took place. In the beginning of the movie the rocket is ignited and the background is visible. Here, we see visual evidence number 2. 

An indent in the hill is clearly visible in the video, which can be linked to a geological structure in the southern test site. However, there is also an indent in the northern test site, so you could interpret this evidence in favour of the northern test site. Luckily, several seconds later in the video a larger overview of the test was shot, both showing evidence 2 and 3.

Here, both the indent behind the test location is seen and a straight road and geologic structure in the right bottom of the picture. These visual objects can only be linked to the south test site in the map and are not seen on the northern test site. This make me confident that the location of the QM-2 test is the southern test site. Finally, this allows me to determine the direction of thrust and calculate the east-west component of the thrust vector that is responsible for change in rotation rate.

The thrust of the rocket is towards the north-ish-west, and therefore it will slow down the Earth instead of speed up. I was wrong in my tweet (always factcheck, boys and girls), but I have answered part of my friends challenge! Now for the last part, we have all the information to calculate the amount of slow-down of Earth's rotation.

Calculating the rotation-rate change
In the last figure, I made a decomposition of the thrust vector, because we only need the part in the direction of the rotation (east-west). By using the cosine rule we determine the magnitude of the thrust vector in the east-west direction:

I measured the angle (16.4 degree) and this results in an east-west force of 15.35 million Newton, slightly less than the full thrust of the rocket. Continuing, the arm of the torque, r, we also have to decompose:

With the averaged radius of the Earth, R = 6371 km, the arm of the torque becomes r = 4762.7 km, together the torque is calculated to be 7.31 ×1013 Nm. By dividing the calculated torque by Earth's moment of inertia will result in an angular acceleration of α = 9.14 ×10-22 rad/sec2. The angular acceleration is very small due to the huge moment of inertia of the Earth. Still, I am on a quest to answer my friend's last question.

From Orbital ATK's website, I found the duration of the test (Δt = 126 secondes), which results in a rotation-rate change of 𝝎 = 1.15 ×10-19 rad/sec, or 6.6 ×10-18 deg/sec. This tiny slowing-down of the Earth (14 orders smaller than its current rotation-rate) will only have a one degree difference in position after a billion years. So no, we will not have observed the change in rotation-rate due to the QM-2 testing of Orbital ATK. It did teach me to always fact check my statements on twitter, somebody could call your bluff.

woensdag 1 juni 2016

The gravimeter of professor Vening Meinesz

On a cold winter day, 21 November 1934, professor Vening Meinesz turned on his pendulum apparatus. Just a few minutes ago, the submarine K-XVIII dived to a depth of 30 meters [1]. At these depths, the motion of the surface waves was dampened such that it did not influence the delicate measurements done by the professor. This particular observation would mark the 500th measurement, observing the tiniest changes in the Earth's gravity field. This new gravity dataset would reveal many new mysteries of our home planet and would be the life’s work of Vening Meinesz. It is all documented in scientific publications of four volumes called Gravity Expeditions at Sea, followed by a fifth volume with gravity observations done by his students. Along these expeditions, the professor had brought his specially designed pendulum apparatus, or folklorised by the sailors on board the many submarines: Het Gouden Kalf (the Golden Calf).

The pendulum apparatus of Vening Meinesz, also known as "Het Gouden Kalf" (the Golden Calf). Positioned on the left side is the protective casing with the recording instrument on top. On the right side is the pendulum apparatus with the three pendulums at the back. 
During the beginning of 1900, Earth's gravity field was only measured on land. The classical single-pendulum device needed a stable platform, which was impossible to achieve on ships. The swell and the shaking of the large engines made it impossible to keep the pendulum stable. Therefore, 73 percentage of the Earth's gravity field was yet unknown to the geodetic community. A young civil engineer from the Technische Hogeschool in Delft would change this. After his graduation in 1915, Felix Andries Vening Meinesz, son of a mayor of Rotterdam and Amsterdam, was given the task at the Rijkscommisie voor Graadmeting to set up Holland’s first gravimetric base station network. For this project he needed a device that could measure the gravity field with the highest accuracy possible, which in those days were pendulum instruments. Unfortunately, he found out that the soil of the Netherlands was very unstable. The waves of the North Sea, when smashed at the dunes of the Dutch coast, would generate solid waves in the soils that affect the motion of the pendulum when observing in Delft. 

Professor Vening Meinesz changed his location of research to a small town called de Bilt. In particular, he moved to the KNMI (Royal Dutch Meteorological Institute). Here, in the basement of the KNMI building at the Kloosterweg, underneath the office of the director of the institute, Van Everdingen, Vening Meinesz commenced his measurements and thorough calibrations with new type of pendulum instruments [2]. Evidence of his presence can still be found at the old KNMI building, where a historical plaque indicating the gravimetric base station is still present on the left rail of the concrete stairs on the west side of the building. The location at the KNMI was in particular useful for Vening Meinesz, because the geological subsurface made it a very stable environment for gravity observations. Due to the stable subsoil, external motions were dampened and the remote location would decrease the oscillation of lorries and inland shipping. The extreme stable surroundings made it possible for the professor to test and calibrate his equipment with extreme precision, resulting in very accurate measurements of the gravity field later during his expeditions at sea [3].

Geodetical plague of the gravity measurement at the old building of the KNMI, marking the location of the gravimetric reference point.
Due to the success of his work in the Netherlands removing external accelerations from the measurements, Vening Meinesz decided to try measuring on board a surface ship. Unfortunately, the motion of the waves and the shaking due to the steam engine were too severe and the observations were worthless. Vening Meinesz, slightly disappointed, presented his negative results in Maastricht at the 19th Nederlandse Natuur- en Geneeskundig Congres. After his presentation, Ir. F.K.Th. van Iterson (1877 - 1957), director of the Staatsmijnen, suggested to use submarines instead of surface ships [4]. Wave motion at 30 meters depth would be dampened and submarines use quiet electro-motors when diving. This touch of serendipity was the start of many submarine gravity expeditions at sea.

Improving during submarine expeditions - the true engineering spirit
The Golden Calf did not have its final form from the beginning. Vening Meinesz, being a true engineer, modified the apparatus many times during his numerous submarine voyages, always improving the design. During his work on gravimetric reference network of the Netherlands, the professor used the Von Sterneck-Stückrath gravimeter (1887), but it proved to be difficult to operate during the long K-II submarine expedition (1923). Vening Meinesz decided to design a new gravimeter from the experience during this expedition. He ‘cannibalised’ the pendulums of the old Von Sterneck gravimeter (the casing of the old Von Sterneck was in 2015 still in possession of the KNMI). Vening Meinesz used the principle of the Von Sterneck gravimeter to acquire high precision. However, his mathematical analyses of the pendulum motion showed that he only needed three pendulums for two independent measurements instead of four [5]. The pendulums were placed in an along-direction pair-wise configuration. One pair of pendulums would produce an independent gravity observation. This was done to eliminate any external horizontal motion. The differential equation to describe a pendulum’s motion attenuated by a horizontal acceleration is as follows:

The angle of deflection of the pendulum is represented by θ, whereas the length is l and gravity is noted by g. The horizontal acceleration is given by ay. With one pendulum it is impossible to decouple the value of g from the external accelerations acting on the instrument. Therefore, two pendulums are used, where the difference of their deflection angles is measured. The external acceleration, which is similar for both pendulums, is then mitigated by subtraction.

The pair θ12 is observed by an ingenious design of light rays, mirrors and prisms on the top of the pendulum apparatus. This second-order differential equation is easy to solve. For small initial amplitudes of the virtual pendulum, this will result in the famous pendulum relation of Christiaan Huygens:

The period of the virtual pendulum (T1-2) can be determined from the recordings of the light rays. The recording instrument, a small ‘dark chamber’ with photographic roll of paper, was situated on top of the pendulum casing. A clockwork contraption unrolled the photo paper during the observations, such that the defections of the pendulum pair were recorded. 

Top view of the pendulum apparatus illustrating the locations of the three pendulums. The coloured dashed lines depict the path of the light rays from the recording apparatus. Red and green show the recording of the motion of two paired-pendulums, whereas blue depicts only the recording of the middle pendulum. The yellow light ray was observing the tilt and temperature changes of the instrument.
This unrolling of the photographic paper was not accurate enough, so Vening Meinesz designed another approach to accurately determine the time periods of the pendulum. The professor always took the state-of-the-art chronometers on board the submarine expeditions. One chronometer, the Nardin 212, was taken on almost all the expeditions and was accurate up to 0.04 sec/day. He asked for alternations made to the chronometers, such that they were able to open and close an electrical circuit every 0.5 seconds. The electric pulse was then used to control a shutter in the recording instrument to shortly interrupt the light ray. This resulted in small 0.5 markings in the final recording sheets and could be used to determine the time period with extreme accuracy.

During the submarine expeditions, Vening Meinesz always kept alternating the device and improving on its accuracy. The smallest details were taken into account. For example, when the submarine dived to 30 meters depth, the pressure of the air inside the enclosed vessel increased with sudden temperature changes of a few degrees. Because the Golden Calf had thermal insulation in the form of sheep’s wool, the air temperature inside the pendulum apparatus did not experience these sudden changes. However, during the 45 minute long observations, gradually the temperature would change due to leakages in the cover. In turn, this would effect the very sensitive measurements. A small electric heater in the bottom of the device would be turned on before the dive to heat up the air a few degrees to simulate the temperature of the air after dive. In this way, during the dive there would be no temperature offset between the air in the submarine and inside the pendulum device. It needed some practice of the operator, but it was effective.

A 3D computer model of the submarine Hr. Ms. K-XVIII made from the old engineering drawings of the shipyard Fijenoord
The name "Golden Calf" was given to the pendulum apparatus by the submarine crew. The story goes that during the gravity observation all the non-essential personnel had to lie down in their bed-bunk to create a very stable submarine. The Dutch Navy declared that this was a degradation of personnel life and well-being and therefore paid the submarine crew 1 guilder (currency of the Netherlands at the time) per dive extra wage for compensation. So, when the crew members saw the pendulum apparatus carried on board, they rubbed their hands and cheered on the coming of the Golden Calf, because this meant good wages. Of course, the bronze platting will have had some influence in the creation of the name. The abundant use of bronze in the casing has given it a gold colour, which could also lead to the name Golden Calf.

Results and legacy
One of the well-known theories of Vening Meinesz is his model to explain the stable situation of continents, mountains, and volcanic islands. Previous researchers assumed that these large masses were floating on a liquid mantle, like an iceberg floats in the water. From the observations with the Golden Calf, Vening Meinesz could deduce that the solid crust was partially responsible for holding up the mountains. Gravity observations of coastal regions and volcanic islands showed that the crust acted as a plate and experienced elastic bending due to the loading of the extra topography. This theory is now called Vening Meinesz isostasy and is especially successful in explaining the gravity field of oceanic islands.

Old gravity results from Vening Meinesz of Indonesia (top) compared with current knowledge of the gravity field (bottom) from combined ground, seaborne, airborne, and satellite gravimetry. Please appreciate the accuracy that Vening Meinesz already obtained almost 100 years ago.
The Golden Calf revealed many secrets of the deep ocean. For example, the gravity signal at the Mid Atlantic Ridge differs from the gravity anomalies at the famous Vening Meinesz belts (now known as subduction zones). Vening Meinesz found strong negative and positive gravity anomalies situated parallel to the volcanic arc in the East Indies (Indonesia), which could not be explained by isostasy. This indicated a dynamic process along the southwest shore of the East Indies. Similar gravity anomalies were found in the West Indies, where Harry Hess, a young American scientist, was responsible for most of the gravity surveying. Harry Hess is mostly known for the founding father of the geophysical model for the spreading ridge, which occurs at the Mid Atlantic Ridge [6]. This model is believed to be the first step in accepting the tremendous powerful theory of plate tectonics. At the time of Vening Meinesz this was not yet known and both subsurface structures showed similar volcanic geology and seismic activity, but because of the different gravity anomalies Vening Meinesz and Harry Hess knew that different geological processes were at play. Other results thanks to the Golden Calf and Vening Meinesz were the first gravity measurements of a transform fault, the Romanche Trench (at the time theorised as volcanic craton). Also, the gravitational signatures of subsurface structures like the Walvis Ridge and the Rio Grandes Rise were observed during the submarine expeditions. 

Up until 1960, the Golden Calf was the only instrument that could measure the gravity field with such precision. One of the last scientific expeditions with the instrument was made in 1960 [7], to measure the gravity field in the South Atlantic and Indian Ocean. The instrument was succeeded by the Graf-Askania gravimeter, which was a spring gravimeter on a stable platform [8]. Overall, the Golden Calf was responsible for 37 years of ocean gravimetry.

The original Golden Calf is now in possession of the TUDelft Library, section Heritage. The apparatus is loaned to the museum of TUDelft, the Science Centre, where it will be placed in the geodesy section, such that the public can enjoy the beauty of this incredible contraption. In 2014-2015, a project group from TUDelft documented and studied the voyage of Vening Meinesz on board the K-XVIII, where special attention was given to the Golden Calf and its measurement principle. The project was developed under the larger Expedition Wikipedia project. The results of that project can be found on an interactive website: expeditiewikipedia.nl/#vening-meinesz


[1] Wytema, M.S. (1935), Klaar voor onderwater - Met Hr. Ms. K XVIII langs een omweg naar Soerabaja, Andries Blitz, Amsterdam.

[2] Gedenkboek F.A. Vening Meinesz (1957), Verhandelingen van het Koninklijk Nederlandsch Geologisch - Mijnbouwkundig Genootschap, Geologische serie deel XVIII, Drukkerij v/h Mouton & Co, ’S-Gravenhage. 

[3] Vening Meinesz, F.A. (1921-1945), Gravity expeditions at Sea Vol. I-IV, publication of the Netherlands Geodetic Commission, Drukkerij Walkman, Delft.

[4] van Hengel, TJC, (2014), The Diving Dutchman: het marien-gravimetrisch onderzoek van F.A. Vening Meinesz (1887-1966), PhD Thesis, University of Leiden, Leiden.

[5] Vening Meinesz, F.A. (1929), "Theory and practise of pendulum observations at sea”, publication of the Netherlands Geodetic Commission, Drukkerij Waltman, Delft.

[6] Hess, H. (1962), History of Ocean Basins, In A. E. J. Engel, Harold L. James, and B. F. Leonard. Petrologic studies: a volume in honor of A. F. Buddington. Boulder, CO: Geological Society of America, 599–620.

[7] Talwani, M. (1962), Gravity Measurements on HMS Acheron in South Atlantic and Indian Oceans, Geological Society of America Bulletin, 73, 1171-1182.

[8] Graf, A. (1958), Das Seegravimeter, Z. Instrumentenkd., 60, 151-162. 

zaterdag 20 februari 2016

Hearing the sound of a satellite

So, the last few weeks were quite exciting at the DopTrack satellite tracking station. We have updated the software, such that the station is now able to automatically record any satellite we tell it to record. Also, the radio antennas are installed higher in the sky for better visibility. Moreover, TUDelft students have designed software to automatically extract the carrier signal from raw recordings. And last but not least, the development of the website for the Virtual Laboratory DopTrack is started, but I will report on that later on, when we have a working website. All major steps to have our project become a real educational tool for satellite tracking. 

Software update: Automated satellite tracking
The DopTrack station is now able to automatically compute, when a satellite is flying over and start recording it accordingly. We just have to set a line of code in the rec.list file and DopTrack is doing the rest. An example recording list is:

Delfi-C3                          32789                145870000                  250000
Delfi-n3Xt                      39428                 145870000                  250000
UKUBE-1                      40074                 145870000                  250000
CANX-2                         32790                 437478000                  250000

The list just needs the name, NORADID number, tuning frequency for the radio, and bandwidth for the recording. Everyday, this file is checked and new recordings are added to the waiting list. We then record the selected satellite when it is flying over, which is determined by propagating its updated TLE file. An example result: the radio downlink spectrogram of the UKUBE-1 (FunCube-2 module) is seen in the following figure.
Spectrogram of the recording UKUBE-1 at tuning frequency 145.870 MHz during the satellite pass on 18th of February 2016. Start of recording was 12:12 CET.
The shift in the frequency is due to the relative motion of the satellite as seen from the ground station. The goal of the DopTrack project is to determine the actual orbit of satellites by extracting this Doppler shift and convert it to range-rate observables. These can be used in orbit determination studies.

Positioning of the radio antennas
Last week, the antennas have received new locations. Three,three meter high, RVS poles were placed on the roof of the EWI building. We have placed the GPS and radio antennas (VHF and UHF) on the top of those poles. This gives the antennas 360 degree visibility of the sky.
Final inspection of the radio antennas by the DopTrack team.
The installation of the antennas was a good opportunity to have our intern do some work on the station and have some team bounding going on. Martin helped me installing the antennas on the poles, while Joao was busy downstairs. He installed a new power switch, which makes it possible to remotely turn off and on hardware components in the station. The new location of the antennas are awesome. Its almost as if they reach for the moon!
The VHF antenna with amplifier on the left and the UHF antenna on the right, both reaching for the moon.
After some small software hiccups, the station is back up and records the radio downlink of Delfi-C3.

DopTrack's Range-Rate Extraction software: DRRE
This semester DopTrack played a role in the new Space Minor of our faculty. For ten weeks long, two groups of students were asked to develope software that is able to extract the carrier signal of Delfi-C3 from the raw recorded data of DopTrack. At the end of the project, the students were able to delivered this software. Yet again it proves that our students are not to be under-estimated.
The red dots are carrier signal data records automatically extracted from the raw recording. The gray-scale picture is the spectrogram of the raw recorded electromagnetic spectrum.
Here, the red dots are the extracted data from the raw recording. Despite, the horizontal and vertical unwanted signal, the students were able to let the computer do all the work without them telling it too much specific information. Great job, because now we are able to post-process all the recorded data we have. So, I did this and made a plot of all the frequency at Time-of-Closest_Approach, or FCA. This is exactly at the bending point in the Doppler S-curve. Here, the relative velocity of the satellite is zero compared to the ground station and we are recording the real frequency of the satellite. In other words, we are hearing the true sound of the satellite.

Six months of data showing the variation of the transmitted frequency of Delfi-C3 is around 800 Hz, with some outliers.
For now, we have six months of data and all we can deduce from this data is that the onboard oscillator has an influence on the transmitted frequency of around 800 Hz. But after more recordings, we eventually hope to see a long trend in the data that we can attribute to degradation and/or temperature change onboard the satellite.