dinsdag 14 augustus 2018

Determine the shape of the Earth 200 years ago with gravity

A colleague of mine pointed me to a PhD dissertation of Petrus van Galen. He wrote this dissertation in 1830 and two copies can be found in the Tresor of the TU Delft Library. The title of this dissertation is “Disputatio Mathematica Inauguralis de Pendulo Ejusque Applicatione ad Telluris Figuram Determinandam”. Loosely translated: A Mathematical Essay about the Pendulum and its Application to Determine the Figure of the Earth. Of course, as soon I heard about this, I made an appointment with the librarian responsible for the collection in the Tresor for a viewing session. I also found a link in Google books, showing almost the complete dissertation, but I wanted to smell the old pages. Furthermore, the appendices of the dissertation were not scanned properly, so the data, that Petrus van Galen used, were not yet available to me.

A week later, I cycled to the Library (appointed as one of the 10 most beautiful libraries of the world and known very well to me due to the Vening Meinesz project) and dropped by the Tresor. In one of the copies, the pages were still joint together, as they were printed on one big sheet of paper and not yet separated by the cutting machine. I was told that this was common for books in those days. The other copy was cut, which made browsing through the dissertation a bit more easy. As you could see in the Google books link, it was almost completely written in Latin, except the preface, written in Dutch by Petrus’ friends. The dissertation started by a quote from the Roman philosopher Seneca, whom I find to have written most intriguing letters (e.g. De Brevitate vitæ). I let you do the translating. 

Because of the language it was difficult to understand the thesis, but armed with 5 years of high-school Latin and Google translate, I do understand some of the text Petrus has written. And the equations are written in the universal language of mathematics, which I am able to read. Van Galen discusses the application of a single pendulum to determine the shape of the Earth. This is a similar experiment I asked you to do in one of my previous blogsIn those days it was known that the Earth was not completely round, but had a flattening, caused by its own rotation. This flattening can be determined by observing the gravity field on the surface of the Earth. Gravity has an influence of the period of a pendulum, so by observing the period of such a pendulum, scientists were able to determine the gravity field of the Earth. The value of this gravity is different depending on the latitude of the location of the experiment. In the appendix of the dissertation, Van Galen lists several now famous scientists that did this experiment on different locations. 

People like Bouguer, Condamine, and Picard were famous geodesists, people I learned about during my study and now teach about to my students. A great Italian explorer Malaspina is listed and contributes to the data with 16 entries. Even Celsius is on the list, famous for his temperature scale, but I know him for his pioneering work on post-glacial rebound in Scandinavia. The locations where the measurements were taken are also interesting to look at. It shows that the southern hemisphere was less well covered and no observations in the oceans, which were the arguments of Prof. Vening Meinesz (about whom I have wrote several blogposts: here, here, and here) for his voyage on the K-XVIII submarine in 1934-1935. In Paris, several observations have been performed by different parties, which gives us an opportunity to determine the uncertainty of gravity observations in those days. The values there differ from 979.8-981.4 Gal, so an uncertainty of 1600 mGal, which is huge in our perspective.

All very well, but how does the shape determined by Van Galen compares with Vening Meinesz’ data, other flattening values and one of our current definition of the ellipsoid (WGS84)? To do this, I have inserted all the data from Petrus van Galen in a little Matlab script, together with the definition of an ellipsoid. The definition is as follows

I used a similar expression to fit van Galen’s data, such that we can compare the ellipsoidal shape of the gravity field of Earth. From this ellipsoid the flattening of the Earth can be determined, which van Galen listed in his dissertation, being 1 : 286.32. The current value in WGS84 is 1 : 298.257223563, whereas Newton calculated it to be 1 : 230. So, van Galen was in the same order as the current value, with only 50 data points with huge uncertainties. The data that van Galen used are plotted in the following figure in comparison with data from Vening Meinesz and other definitions of the ellipsoid.

The red data points are observations collected by van Galen and the red line is my data fit using the equation listed above as the fitting function with three variables, g_e, A, and B. Blue denotes data and fit of the Vening Meinesz’ data. Black and green are a historical (1930, black) and current (green) definitions of the Earth’s ellipsoid. The data clearly shows the improvement in uncertainty of data within the 100 years between 1830 and 1930.

That the work of Petrus van Gaalen is a great piece of science can be argued by the amount of data he collected. A nice graph in the book by Torge, “Gravimetry”, illustrated the amount of data with respect to historical periods, where it shows that in 1830 around 10-100 gravity data observations were available in the scientific community. Petrus acquired an amount of 50 data points. This means that he acquired a substantial part of the data that was available, not bad in a time without internet. Think of all the visits to the library and all the letters he must have written. Above all, it shows that the Dutch Academia was interested in gravimetric research around the beginning of the Modern Age. I am especially interested in the inter connections of scientists and explorers that did these kind of experiments. The TU Delft Library is now in contact with a translator, to see if we can find some more information in this dissertation and maybe find some hidden gems of physical geodesy history. 

woensdag 23 mei 2018

Geology and Whisky on Islay

Few weeks ago, I visited the Queen of the Hebrides, or better known as the isle of Islay. My dad and I organised the trip for some friends of ours with the main objective to show them the many beauties of the island. Two years ago my dad and I made a ‘reconnaissance’ trip to explore those beauties. 

Machir Bay viewed from the old abandon RAF radio station. Photo credit: Maurits Kok.
Being true to my blog, which is a more scientific-orientated blog, I will not make this post a travellers log, but lets just say we visited several distilleries (even one that is now under construction), enjoyed the cuisine (which was quite high standards in the hotel we stayed), and made many long walks through the fantastic and overwhelming landscapes of the Scottish island. This island keeps surprising me with its fascinating history, incredible geology, and beautiful whiskies.

Whisky Wharehouse tasting at Bruichladdich. Photo credit: Maurits Kok
Before the trip I bought a book about the “A Guide to the Geology of Islay”, which described the whole geological history, going back 1.8 billion years to evidence of the Last Glacial Maximum (last ice age), ‘only’ 25,000 years ago. A short description can be found on this website. Me, becoming more and more a geoscientist, wanted to explore and see some of this evidence. So, at the second evening of our trip, when the group went to one of Port Ellen’s finest pubs, I decided to make a short geological field trip to observe a meters thick Cenozoic Dolerite Dike in the Precambrian metabasite sills and Port Ellen Phyllite that now make Port Ellen’s terrain. Sounds quite geological to me, right?

The Dolerite Dyke location sketched in red on top of a satellite image of the Port Ellen bay.
The dyke in question looks like a jetty from the satellite image, but up close they are black rocks eroded by the constant washing of the bay’s waves. They have a completely different appearances than the rocks I past by on the little beach, when I was walking towards the ‘black jetty’. Those rocks were finely layered, illustrating the deposits that slowly build up the terrain I was walking on now. No, these black rocks were not layered, but emanated volcanism; big, tough rocks, smoothed by the ocean, crumbled and covered by green algae. The outcrop started a few meters into the water of the bay and was following the NW-SE direction inland. It quickly dived beneath the surface under a row of little cottages. I wanted to see if the dyke would outcrop somewhere else behind the houses.

The Dolerite Dyke in the bay of Port Ellen. The main street of Port Ellen is visible on the other side of the bay with one of its finest pub.
So, my hike through the dunes began, observing many evidence of the sedimentary nature of the surrounding terrain. Layers, upon layers, upon layers of sand deposits, sometimes cut through by little quartz dykes. Also, the Port Ellen fishermen had constructed little sheds for them to watch the sea and hope for calm weather. In the little natural harbours between the rocks, small improvised concrete docks were constructed with little fisher boats in various states of decay were scattered around the dunes. Still, eager to explore the rocks in hoping to find outcrops of the Port Ellen dyke, I endured the growing wind, now having full effect, because I was not anymore in the shelter of the island’s cottages. The smell of rotting seaweed was execrable every time I found some shelter behind a rock formation, then blown away by the wind every time I would cross a gab in the rock formation.

After an hour of climbing the rocks, I found the expected location of the dyke, but it was used as fundament for a sewer pipe of the village. Lots of concrete was pouted to facilitated the rusted pipe, so no change of inspecting any natural rock formations. I decided to walk seaward, against the direction of the now-grown-in-size wind. At the coastline I saw a spectacular view of the ocean, a battle between the winds and the ocean water. If the weather would have been clear, I could see Ireland, but the view I saw was white-topped waves and dark skies. One very brave fisher-boat was hoping to catch some fish, but seeing the exhaust fumes coming out the exhaust pipe, suggested that the diesel motor was working in overdrive. A constant howling of the wind…

The other end of the Dolerite Dyke. In the back, the little isle Texa can be viewed. The dyke is easily found, because of the lack of layering on the rock structure.
There I found the outcrop of the dyke leaping into the ocean as the black snake it was. A clear distinction between the structure of the layered and the volcanic rock was visible. On one side, the ocean had carved out the boundary between the two rocks and made a small pebble beach for me to examine the two different rock sides. 

The ocean had formed a little pebble-beach between the two distinct rocks. Left layered rock and right the Dolerite Dyke.
The dyke had a similar width as in the calm bay of Port Ellen and the appearance of the rock was the same. The other side of the dyke showed some more detailed structures of the volcanic intrusion. A small dyke perpendicular to the main dyke direction was intruded between the layered rock, possible through an old fault, pushing away the layering of the rock terrain. 

This little finger of the main dyke was situated perpendicular to the main dyke formation (NW-SE). 
This dyke was a result of the opening of the North Atlantic Ocean just 60 million years ago, I read in the little booklet about the geology of Islay. Due to this opening of the ocean I was now seeing in battle with the wind, the incredible rock formation I was standing on was created, maybe seen and appreciate by a hand-full of people. The high energy buildup of the event had pushed hard solid layered rock several meters apart, all happening in the deep depths of the solid Earth. Many centuries had eroded the surface, such that I could now be whiteness to this magnificent rock formation. That evening the whisky tasted just that much more comprehensive.

The Lagavullin distillery. Photo credit: Maurits Kok
List of distilleries we visited:
  • Bruichladdich: Warehouse tasting (very nice Port Charlotte)
  • Kilchoman: Distillery Tour (the Loch Gorm stays my favourite)
  • Bunnahabhain: Premium tasting plus Tour (Toiteach!!! pronounced toy-chuck)
  • Coal Ila: Distillery Tour (12 years old as usual)
  • Laphroaig: Distillery Tour (Laphroig Lore was a surprise, new barrels!?!)
  • Lagavullin: Core Range Tasting Tour (16 year old was still the best) 
The summary of the trip to the island of Islay with music (credit video: Kjartan Majdandzic)!

maandag 24 juli 2017

Paving the way for our presence on the Moon

Reading the books and watching the TV series “the Expanse” awoken my old boyish-dreams of our presence on the Moon. Growing up with a dad that introduced me to the fantastic worlds of Larry Niven, Brian Aldiss, Isaac Asimov, Jack Vance, and Arthur C. Clark, gave me hope for the future and it motivated me to start my studies in Space Engineering. The science fiction became real-life science and engineering and I began to understand why the future was not yet achieved. We have made space engineering very complicated and expensive, which allowed only nations to perform this type of activities. But, I am living in the future of my old heroes of words and spaceflight has a permanent role in our daily life. 

And more and more young engineers want to participate, because space is not that complicated anymore. We evolve as humans, such that complicated problems in the past are simple issues in the future. James S. A. Corey (A.K.A. Daniel Abraham and Ty Franck) hints to this as well in the novels of "the Expanse”, where a 'simple’ mechanic makes nuclear fusion and complex astrodynamics calculations in the back of his mind. To make this reality, we need to educate our students with more ‘complicated’ techniques and tools than the guys from the Apollo program had at the time. In other words, if we want to go to the Moon again, we need to pave the way by educating our students in new and different theories and practises then we did before.

Artist's impression of project IRIS
And this is what I try to do in my role as lecture. Last month, ten of my students presented their Design Synthesis Exercise (DSE), this is their final project to complete the Bachelor of Aerospace Engineering at the TU Delft in the Netherlands. Staff members present an idea or problem for which students can prescribe and design a solution (some sort of aircraft or space vehicle). Thanks to "the Expanse” I was envisioning ways to bring humanity back to the Moon. A difficult problem, so how to do it? I thought of the French president de Gaulle, who decided to invest in the French highway infrastructure in a time of poverty (just after WWII), but ultimately this infrastructure boosted the French economy, tourisme, and mobility in ways that baffled de Gaulle’s critics.  

The Moon needs infrastructure! Infrastructure that facilitates surface research platforms, mining missions and other human presence on the Moon. Not for the few missions and proposals that we have now, but for Moon missions and applications we have not yet thought of. So, during the DSE my students had to design an infrastructure system around the Moon that was capable of enabling 24/7 communication on the Moon surface (the complete Moon surface, so also the far-side and the poles). They had to design the constellation, the satellites themselves, a deployment strategy and the transfer trajectory. And they had to do it cheap! Some found my idea too complicated for 3rd year undergrads, but the students really showed several innovative ideas and were capable to construct a feasible design.
Logo of project IRIS
Their first innovative design is the transfer trajectory. Instead of using a simple Hohmann transfer, they opted for a low-energy ballistic transfer trajectory, in which the rocket flies to the Sun-Earth L2 equilibrium point and falls back to a Moon orbit, without the need for a large insertion thrust at Moon orbit (as you do in a Hohmann transfer). I provided the students with some basic on three-body problem mechanics, some literature, and a simple Matlab script that was able to propagate objects in space. The figure below shows the designed trajectory they came up with:

The complete low energy transfer trajectory in Sun-centred rotating frame designed by the students
This choice of transfer trajectory reduces the amount of propellant needed, extra mass you need to launch into space. Of course, this comes with a price. Instead of the 2-day travel time you have with a direct transfer, these type of trajectories could take several months. However, if you have the time  (no humans onboard), why not? Furthermore, a ballistic transfer trajectory eliminates the need for a large complicated propulsion system onboard your spacecraft, because it does not need to slow down the spacecraft at the Moon orbit. This opens up a whole new type of satellites that could orbit the Moon.

Engineering drawings of one of the designed cubesat communication satellite, showing some of the internal and external subsystems. The satellite used a parabolic and a phased-array antenna for communication, hence the large solar cell array.
This relates to the second innovative idea of my students, which is to use multiple small but identical satellites that operated in a Swarm-type of constellation. Small satellites (like Cubesats) are making their way to professional applications here around Earth. Think about a company like Planet (Labs, used to) that uses cubesats to observe the entire Earth with their goal being to generate a real-time Google Earth. The don’t need a few big complicated satellites, they need a lot of cheap, small, and simple sats with a camera and a communication system to send the data back to Earth. Other applications are Earthquake warning system (QuakeSat), global Wifi (project Outernet), or understanding our atmosphere (QB50). Cheap to manufacture and launch (small is light) and the system’s reliability could be reduced, because if one sat fails multiple others are still present to fulfil the mission. And let’s just be honest, multiple satellite deployment is just so cool to watch:

This DSE project showed that students are capable of creating innovative ideas and are able to think about realistic solutions in space engineering. A new age of space engineering is at our doorsteps and it might open up interplanetary space for humanity.

Note: all figures were obtained from my students DSE report: Information Relay InfraStructure for translunar communication. 

maandag 8 mei 2017

Delfi-C3 is feeling the power of Spring

All the trees have blossom, baby ducklings are rooming the ponds, and we get more Sun: we survived the winter! Spring is here to give us our much needed vitamine D. 
Spring on the campus of TU Delft
Not only on the surface of the Earth this is felt, but also in outer space. The Delfi-C3, now more than 9 years orbiting our Earth, notices that spring has arrived on the Northern Hemisphere. During winter, our ground station (DopTrack) only receives data during the morning and mid day (between 8:00 and 13:00 CET). Starting from April, we start receiving signal around 20:00 CET.
Fraction of a 24 hour day (UTC) at Time of Closest Approach (TCA) of the Delfi-C3 with DopTrack.
We saw this already last year, when in April the telemetry of Delfi-C3 started appearing in the evening. The Sun was already set on the ground, but at 560 km height light was still shinning on the solar cells of the satellite. The satellite is designed such that when sunlight hits the solar cells and enough power is received the onboard computer turns on and goes through its boot program. Equipment is turned on and the radio is transmitting telemetry back to Earth. The data is transmitted on a carrier frequency, that is recorded by the DopTrack at TCA, which we call Frequency at Closest Approach (FCA).
The complete dataset of DopTrack consisting of recordings of Delfi-C3's FCA for almost 2 years now. A clear distinction between mid-day and evening passes can be noticed.
We have recorded the radio transmissions of Delfi-C3 with our calibrated equipment, enabling us to accurately determining the received frequency. The mid-day passes seem more stable around an average 145888000 Hz, whereas the evening passes have a larger spread and higher averaged frequency. More analysis is needed, but we suspect that in the evening passes the thermal equilibrium is not yet obtained during recordings. Lets hope after this spring we can say more.

zaterdag 14 januari 2017

Interacting of the SDR with Matlab during my classroom tutorial

This year was the second time I gave my tutorial with an SDR dongle. The idea behind this tutorial is to simulate ground-station-to-satellite interaction in the radiowave spectrum. The ground station is replaced by the SDR dongle and the student's computer. For the satellite, I use a radio beacon designed and built by a colleague of mine. The device transmits an FSK modulated signal at a certain frequency. During the tutorial, the students learn to develope software that interacts with the SDR, such that they can process the digital stream observed by the SDR. I made a small video of the signal with GQRX software, such that you can see the signal in the frequency spectrum and listen to the FSK modulation.

Last year, I was using an open-source software package "gnuradio" for the signal processing part, but I found out that this introduced a lot of trouble-shouting in the first few hours, because of students having:
  • different operating systems and versions (every single student had different installation problems)
  • different level of experience with python, shell scripting, or other software language (the tutorial is part of our Space Minor, so we get a variety of students from different faculties and the learning curve of gnu radio is just to steep)
  • different level of understanding of signal processing theory
This made last year's tutorial interesting and a lot of fun, but also very chaotic. This year, I decided to streamline it a bit more. Luckily at out university, most staff and students use Matlab for programming in their education (which runs on any operating system), therefore by converting the tutorial to Matlab would solve the first two evaluation points. Furthermore, gnuradio has a lot of possibilities, but this overwelms the more beginner-experienced students. Matlab has all the basics (needed for this tutorials), but does not have large libraries where you could drown in coding. Thanks to Mathworks new toolbox "RTL-SDR Support from Communications System Toolbox", communicating to the SDR is now very simple. 

Students would get an SDR dongle to connect to their laptop via an USB port (I discussed this SDR in a previous post). They need to develop software that can process the data from the ground station. So, to replicate this I let them record the QI signal that is observed by their mini-ground station and they need to plot the corresponding waterfall plot (as is seen in the video). 

The radio transmitter built by my colleague that is used in the ground station tutorial. It transmits a call sign with an FSK modulation. Left is the transmitter and right is an GPS antenna to calibrate the oscillator of the device
The code to record the QI signal coming from the SDR dongle is listed below. I modified an m-file from the SDR Toolbox tutorial (sdrrSpectrumAnalyzer.m). 

close all;
% Set initial parameters (these settings can be changed according to the user needs)
T                  = 60;            % Length of recorded signal (sec)
fc                 = 28.126125e6;   % Center frequency (Hz)
FrontEndSampleRate = 250e3;         % Samples per second (Hz)
% Do not change the following settings
n = 1;                              % multiple of the Framelength
FrameLength        = n*256;         % samples per counted frame
dt = FrameLength/FrontEndSampleRate;% length of sample
count = ceil(T/dt);                 % number of counts
% Create receiver and spectrum analyzer System objects
hSDRrRx = comm.SDRRTLReceiver(...
    'CenterFrequency', fc, ...
    'EnableTunerAGC',  true, ...
    'SampleRate',      FrontEndSampleRate, ...
    'SamplesPerFrame', FrameLength, ...
    'OutputDataType',  'double');
% Open file to write data
fid = fopen('Test_data_GPS_250kHz_minus.32fc','wb');

%% Stream processing
if ~isempty(sdrinfo(hSDRrRx.RadioAddress))
    for count = 1 : count
        % get data from SDR
        [data, ~] = step(hSDRrRx);  % no 'len' output needed for blocking operation
        data = data - mean(data);   % remove DC component
        % Construct the QI signal in a vector
        D = zeros(size(data,1),2);
        D(:,1) = real(data);
        D(:,2) = imag(data);
        % write the QI signal to a file
% Release all System objects

This code will generate a file that is similar to the recording-data files our ground station is generating. So, if the students are able to post process their own recorded data, the ground station data files should easily be processed as well. How to plot a waterfall plot, was already discussed here. Since then, I have improved my coding, but the basics are kept the same. When following these instructions, I obtained the following waterfall plot of the FSK-modulated radio signal transmitted during class.
Waterfall plot of the recording with the SDR dongle during class. 
The radio signal is visible in the spectrogram (or waterfall plot). No characteristic S-curve is seen, because in class the transmitter is not moving with respect to the antenna's of the students. The carrier frequency is kept at the same value, because the relative velocity of the receiver and transmitter is zero. The FSK modulations is seen when we zoom in on the signal.
Zoomed in version of the waterfall plot, such that the signal can be better inspected.
Here, (not so clear as in the video) can be seen that the frequency jumps between four different values, such that it is able to communicate data. In the video this can be heard as four different tones of the signal. Note the reflections of the signal to the left and right of the zoomed in spectrogram. A good filter should be able to remove these, but I have kept it simple.

I found the interface of Matlab with the SDR very simple to use, which made it ideal to be used for my course on satellite communication and tracking.

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.