27. Camera image stabilisation

Cameras are devices for capturing photographic images. A camera is basically a box with an opening in one wall that lets light enter the box and form an image on the opposite wall. The earliest such “cameras” were what are now known as camera obscuras, which are closed rooms with a small hole in one wall. The name “camera obscura” comes from Latin: “camera” meaning “room” and “obscura” meaning “dark”. (Which is incidentally why in English “camera” refers to a photographic device, while in Italian “camera” means a room.)

A camera obscura works on the principle that light travels in straight lines. How it forms an image is easiest to see with reference to a diagram:

Camera obscura diagram

Diagram illustrating the principle of a camera obscura. (Public domain image from Wikimedia Commons.)

In the diagram, the room on the right is enclosed and light can only enter through the hole C. Light from the head region A of a person standing outside enters the hole C, travelling in a straight line, until it hits the far wall of the room near the floor. Light from the person’s feet B travels through the hole C and ends up hitting the far wall near the ceiling. Light from the person’s torso D hits the far wall somewhere in between. We can see that all of the light from the person that enters through the hole C ends up projected on the far wall in such a way that it creates an image of the person, upside down. The image is faint, so the room needs to be dark in order to see it.

If you have a modern photographic camera, you can expose it for a long time to capture a photo of the faint projected image inside the room (which is upside down).

Camera obscura photo

A room turned into a camera obscura, at the Camden Arts Centre, London. (Creative Commons Attribution 2.0 image by Flickr user Kevan, from Flickr.)

The hole in the wall needs to be small to keep the image reasonably sharp. If the hole is large, the rays of light from a single point in the scene outside project to multiple points on the far wall, making the image blurry – the larger the hole, the brighter the image, but blurrier it is. You can overcome this by placing a lens in the hole, which focuses the incoming light back down to a sharper focus on the wall.

Camera obscura photo

Camera obscura using a lens to focus the incoming light for a brighter, sharper image. (Creative Commons Attribution 2.0 image by Flickr user Willi Winzig, from Flickr.)

A photographic camera is essentially a small, portable camera obscura, using a lens to focus an image of the outside world onto the inside of the back of the camera. The critical difference is that where the image forms on the back wall, there is some sort of light-sensitive device that records the pattern of light, shadow, and colour. The first cameras used light-sensitive chemicals, coated onto a flat surface. The light causes chemical reactions that change physical properties of the chemicals, such as hardness or colour. Various processes can then be used to convert the chemically coated surface into an image, that more or less resembles the scene that was projected into the camera. Modern chemical photography uses film as the chemical support medium, but glass was popular in the past and is still used for specialty purposes today.

More recently, photographic film has been largely displaced by digital electronic light sensors. Sensor manufacturers make silicon chips that contain millions of tiny individual light sensors arranged in a rectangular grid pattern. Each one records the amount of light that hits it, and that information is recorded as one pixel in a digital image file – the file holding millions of pixels that encode the image.

Camera cross section

Cross section of a modern camera, showing the light path through the lens to the digital image sensor. In this camera, a partially silvered fixed mirror reflects a fraction of the light to a dedicated autofocus sensor, and the viewfinder is electronic (this is not a single-lens reflex (SLR) design). (Photo by me.)

One important parameter in photography is the exposure time (also known as “shutter speed”). The hole where the light enters is covered by a shutter, which opens when you press the camera button, and closes a little bit later, the amount of time controlled by the camera settings. The longer you leave open the shutter, the more light can be collected and the brighter the resulting image is. In bright sunlight you might only need to expose the camera for a thousandth of a second or less. In dimmer conditions, such as indoors, or at night, you need to leave the shutter open for longer, sometimes up to several seconds to make a satisfactory image.

A problem is that people are not good at holding a camera still for more than a fraction of a second. Our hands shake by small amounts which, while insignificant for most things, are large enough to cause a long exposure photograph to be blurry because the camera points in slightly different directions during the exposure. Photographers use a rule of thumb to determine the longest shutter speed that can safely be used: For a standard 35 mm SLR camera, take the reciprocal of the focal length of the lens in millimetres, and that is the longest usable shutter speed for hand-held photography. For example, when shooting with a 50 mm lens, your exposure should be 1/50 second or less to avoid blur caused by hand shake. Longer exposures will tend to be blurry.

Camera shake

A photo I took with a long exposure (0.3 seconds) on a (stationary) train. Besides the movement of the people, the background is also blurred by the shaking of my hands; the signs above the door are blurred to illegibility.

The traditional solution has been to use a tripod to hold the camera still while taking a photo, but most people don’t want to carry around a tripod. Since the mid-1990s, another solution has become available: image stabilisation. Image stabilisation uses technology to mitigate or undo the effects of hand shake during image capture. There are two types of image stabilisation:

1. Optical image stabilisation was the first type invented. The basic principle is to move certain optical components of the camera to compensate for the shaking of the camera body, maintaining the image on the same location on the sensor. Gyroscopes are used to measure the tilting of the camera body caused by hand shake, and servo motors physically move the lens elements or the image sensor (or both) to compensate. The motions are very small, but crucial, because the size of a pixel on a modern camera sensor is only a few micrometres, so if the image moves more than a few micrometres it will become blurry.

Image stabilised photo

Optically image stabilised photo of a dim lighthouse interior. The exposure is 0.5 seconds, even longer than the previous photo, but the image stabilisation system mitigates the effects of hand shake, and details in the photo remain relatively unblurred. (Photo by me.)

2. Digital image stabilisation is a newer technology, which relies on image processing, rather than moving physical components in the camera. Digital image processing can go some way to remove the blur from an image, but this is never a perfect process because blurring loses some of the information irretrievably. Another approach is to capture multiple shorter exposure images and combine them after exposure. This produces a composite longer exposure, but each sub-image can be shifted slightly to compensate for any motion of the camera before adding them together. Although digital image stabilisation is fascinating, for this article we are actually concerned with optical image stabilisation, so I’ll say no more about digital.

Early optical image stabilisation hardware could stabilise an image by about 2 to 3 stops of exposure. A “stop” is a term referring to an increase or decrease in exposure by a factor of 2. With 3 stops of image stabilisation, you can safely increase your exposure by a factor of 23 = 8. So if using a 50 mm lens, rather than need an exposure of 1/50 second or less, you can get away with about 1/6 second or less, a significant improvement.

Image stabilisation system diagram

Optical image stabilisation system diagram from a US patent by Canon. The symbols p and y refer to pitch and yaw, which are rotations as defined by the axes shown at 61. 63p and 63y are pitch and yaw sensors (i.e. gyroscopes), which send signals to electronics (65p and 65y) to control actuator motors (67p and 67y) to move the lens element 5, in order to keep the image steady on the sensor 69. 68p and 68y are position feedback sensors. (Figure reproduced from [1].)

Newer technology has improved optical image stabilisation to about 6.5 stops. This gives a factor of 26.5 = 91 times improvement, so that 1/50 second exposure can now be stretched to almost 2 seconds without blurring. Will we soon see further improvements giving even more stops of optical stabilisation?

Interestingly, the answer is no. At least not without a fundamentally different technology. According to an interview with Setsuya Kataoka, Deputy Division Manager of the Imaging Product Development Division of Olympus Corporation, 6.5 stops is the theoretical upper limit of gyroscope-based optical image stabilisation. Why? In his words[2]:

6.5 stops is actually a theoretical limitation at the moment due to rotation of the earth interfering with gyro sensors.

Wait, what?

This is a professional camera engineer, saying that it’s not possible to further improve camera image stabilisation technology because of the rotation of the Earth. Let’s examine why that might be.

As calculated above, when we’re in the realm of 6.5 stops of image stabilisation, a typical exposure is going to be of the order of a second or so. The gyroscopes inside the camera are attempting to keep the camera’s optical system effectively stationary, compensating for the photographer’s shaky hands. However, in one second the Earth rotates by an angle of 0.0042° (equal to 360° divided by the sidereal rotation period of the Earth, 86164 seconds). And gyroscopes hold their position in an inertial frame, not in the rotating frame of the Earth. So if the camera is optically locked to the angle of the gyroscope at the start of the exposure, one second later it will be out by an angle of 0.0042°. So what?

Well, a typical digital camera sensor contains pixels of the order of 5 μm across. With a focal length of 50 mm, a pixel subtends an angle of 5/50000×(180/π) = 0.006°. That’s very close to the same angle. In fact if we change to a focal length of 70 mm (roughly the border between a standard and telephoto lens, so very reasonable for consumer cameras), the angles come out virtually the same.

What this means is that if we take a 1 second exposure with a 70 mm lens (or a 2 second exposure with a 35 mm lens, and so on), with an optically stabilised camera system that perfectly locks onto a gyroscopic stabilisation system, the rotation of the Earth will cause the image to drift by a full pixel on the image sensor. In other words, the image will become blurred. This theoretical limit to the performance of optical image stabilisation, as conceded by professional camera engineers, demonstrates that the Earth is rotating once per day.

To tie this in to our theme of comparing to a flat Earth, I’ll concede that this current limitation would also occur if the flat Earth rotated once per day. However, the majority of flat Earth models deny that the Earth rotates, preferring the cycle of day and night to be generated by the motion of a relatively small, near sun. The current engineering limitations of camera optical image stabilisation rule out the non-rotating flat Earth model.

You could in theory compensate for the angular error caused by Earth rotation, but to do that you’d need to know which direction your camera was pointing relative to the Earth’s rotation axis. Photographers hold their cameras in all sorts of orientations, so you can’t assume this; you need to know both the direction of gravity relative to the camera, and your latitude. There are devices which measure these (accelerometers and GPS), so maybe some day soon camera engineers will include data from these to further improve image stabilisation. At that point, the technology will rely on the fact that the Earth is spherical – because the orientation of gravity relative to the rotation axis changes with latitude, whereas on a rotating flat Earth gravity is always at a constant angle to the rotation axis (parallel to it in the simple case of the flat Earth spinning like a CD).

And the fact that your future camera can perform 7+ stops of image stabilisation will depend on the fact that the Earth is a globe.

References:

[1] Toyoda, Y. “Image stabilizer”. US Patent 6064827, filed 1998-05-12, granted 2000-05-16. https://pdfpiw.uspto.gov/.piw?docid=06064827

[2] Westlake, Andy. “Exclusive interview: Setsuya Kataoka of Olympus”. Amateur Photographer, 2016. https://www.amateurphotographer.co.uk/latest/photo-news/exclusive-interview-setsuya-kataoka-olympus-95731 (accessed 2019-09-18).

24. Gravitational acceleration variation

When you drop an object, it falls down. Initially the speed at which it falls is zero, and this speed increases over time as the object falls faster and faster. In other words, objects falling under the influence of gravity are accelerating. It turns out that the rate of acceleration is a constant when the effects of air resistance are negligible. Eventually air resistance provides a balancing force and the speed of fall reaches a limit, known as the terminal velocity.

Ignoring the air resistance part, the constant acceleration caused by gravity on the Earth’s surface is largely the same everywhere on Earth. This is why you feel like you weigh the same amount no matter where you travel (excluding travel into space!). However, there are small but measurable differences in the Earth’s gravity at different locations.

It’s straightforward to measure the strength of the acceleration due to gravity at any point on Earth with a gravity meter. We’ve already met one type of gravity meter during Airy’s coal pit experiment: a pendulum. So the measurements can be made with Georgian era technology. Nowadays, the most accurate measurements of Earth’s gravity are made from space using satellites. NASA’s GRACE satellite, launched in 2002, gave us our best look yet at the details of Earth’s gravitational field.

Being roughly a sphere of roughly uniform density, you’d expect the gravity at the Earth’s surface to be roughly the same everywhere and—roughly speaking—it is. But going one level of detail deeper, we know the Earth is closer to ellipsoidal than spherical, with a bulge around the equator and flattening at the poles. The surface gravity of an ellipsoid requires some nifty triple integrals to calculate, and fortunately someone on Stack Exchange has done the work for us[1].

Given the radii of the Earth, and an average density of 5520 kg/m3, the responder calculates that the acceleration due to gravity at the poles should be 9.8354 m/s2, while the acceleration at the equator should be 9.8289 m/s2. The difference is about 0.07%.

So at this point let’s look at what the Earth’s gravitational field does look like. The following figure shows the strength of gravity at the surface according to the Earth Gravitational Model 2008 (EGM2008), using data from the GRACE satellite.

Earth Gravitational Model 2008

Earth’s surface gravity as measured by NASA’s GRACE and published in the Earth Gravitational Model 2008. (Figure produced by Curtin University’s Western Australian Geodesy Group, using data from [2].)

We can see that the overall characteristic of the surface gravity is that it is minimal at the equator, around 9.78 m/s2, and maximal at the poles, around 9.83 m/s2, with a transition in between. Overlaid on this there are smaller details caused by the continental landmasses. We can see that mountainous areas such as the Andes and Himalayas have lower gravity – because they are further away from the centre of the planet. Now, the numerical value at the poles is a pretty good match for the theoretical value on an ellipsoid, close to 9.835 m/s2. But the equatorial figure isn’t nearly as good a match; the difference between the equator and poles is around 0.6%, not the 0.07% calculated for an ellipsoid of the Earth’s shape.

The extra 0.5% difference comes about because of another effect that I haven’t mentioned yet: the Earth is rotating. The rotational speed at the equator generates a centrifugal pseudo-force that slightly counteracts gravity. This is easy to calculate; it equals the radius times the square of the angular velocity of the surface at the equator, which comes to 0.034 m/s2. Subtracting this from our theoretical equatorial value gives 9.794 m/s2. This is not quite as low as 9.78 seen in the figure, but it’s much closer. I presume that the differences are caused by the assumed average density of Earth used in the original calculation being a tiny bit too high. If we reduce the average density to 5516 kg/m3 (which is still the same as 5520 to three significant figures, so is plausible), our gravities at the poles and equator become 9.828 and 9.788, which together make a better match to the large scale trends in the figure (ignoring the small fluctuations due to landmasses).

Of course the structure and shape of the Earth are not quite as simple as that of a uniformly dense perfect ellipsoid, so there are some residual differences. But still, this is a remarkably consistent outcome. One final point to note: it took me some time to track down the figure above showing the full value of the Earth’s gravitational field at the surface. When you search for this, most of the maps you find look like the following:

Earth Gravitational Model 2008 residuals

Earth surface gravity residuals, from NASA’s GRACE satellite data. The units are milligals; 1 milligal is equal to 0.00001 m/s2. (Public domain image by NASA, from [3].)

These seem to show that gravity is extremely lumpy across the Earth’s surface, but this is just showing the smaller residual differences after subtracting off a smooth gravity model that includes the relatively large polar/equatorial difference. Given the units of milligals, the largest differences between the red and blue areas shown in this map are only different by a little over 0.001 m/s2 after subtracting the smooth model.

We’re not done yet, because besides Earth we also have detailed gravity mapping for another planet: Mars!

Mars Gravitational Model 2011

Surface gravity strength on Mars. The overall trend is for lowest gravity at the equator, increasing with latitude to highest values at the poles, just like Earth. (Figure reproduced from [4].)

This map shows that the surface gravity on Mars has the same overall shape as that of Earth: highest at the poles and lowest at the equator, as we’d expect for a rotating ellipsoidal planet. Also notice that Mars’s gravity is only around 3.7 m/s2, less than half that of Earth.

Mars’s geography is in some sense much more dramatic than that of Earth, and we can see the smaller scale anomalies caused by the Hellas Basin (large red circle at lower right, which is the lowest point on Mars, hence the higher gravity), Olympus Mons (leftmost blue dot, northern hemisphere, Mars’s highest mountain), and the chain of three volcanoes on the Tharsis Plateau (straddling the equator at left). But overall, the polar/equatorial structure matches that of Earth.

Of course this all makes sense because the Earth is approximately an ellipsoid, differing from a sphere by a small amount of equatorial bulge caused by rotation, as is the case with Mars and other planets. We can easily see that Mars and the other planets are almost spherical globes, by looking at them with a telescope. If the structure of Earth’s gravity is similar to those, it makes sense that the Earth is a globe too. If the Earth were flat, on the other hand, this would be a remarkable coincidence, with no readily apparent explanation for why gravity should be stronger at the poles (remembering that the “south pole” in most flat Earth models is the rim of a disc) and weaker at the equator (half way to the rim of the disc), other than simply saying “that’s just the way Earth’s gravity is.”

References:

[1] “Distribution of Gravitational Force on a non-rotating oblate spheroid”. Stack Exchange: Physics, https://physics.stackexchange.com/questions/144914/distribution-of-gravitational-force-on-a-non-rotating-oblate-spheroid (Accessed 2019-09-06.)

[2] Pavlis, N. K., Holmes, S. A., Kenyon, S. C. , Factor, J. K. “The development and evaluation of the Earth Gravitational Model 2008 (EGM2008)”. Journal of Geophysical Research, 117, p. B04406. https://doi.org/10.1029/2011JB008916

[3] Space Images, Jet Propulsion Laboratory. https://www.jpl.nasa.gov/spaceimages/index.php?search=GRACE&category=Earth (Accessed 2019-09-06.)

[4] Hirt, C., Claessens, S. J., Kuhn, M., Featherstone, W.E. “Kilometer-resolution gravity field of Mars: MGM2011”. Planetary and Space Science, 67(1), p.147-154, 2012. https://doi.org/10.1016/j.pss.2012.02.006

20. Rocket launch sites

Suppose you are planning to build an orbital rocket launching facility. Where are you going to put it? There are several issues to consider.

  • You want the site to be on politically friendly and stable territory. This strongly biases you to building it in your own country, or a dependent territory. Placing it close to an existing military facility is also useful for logistical reasons, especially if any of the space missions are military in nature.
  • You want to build it far enough away from population centres that if something goes catastrophically wrong there will be minimal damage and casualties, but not so far away that it is logistically difficult to move equipment and personnel there.
  • You want to place the site to take advantage of the fact that the rocket begins its journey with the momentum it has from standing on the ground as the Earth rotates. This is essentially a free boost to its launch speed. Since the Earth rotates west to east, the rocket stationary on the pad relative to the Earth actually begins with a significant momentum in an easterly direction. Rocket engineers would be crazy to ignore this.

One consequence of the rocket’s initial momentum is that it’s much easier to launch a rocket towards the east than towards the west. Launching towards the east, you start with some bonus velocity in the same direction, and so your rocket can get away with being less powerful than otherwise. This represents a serious saving in cost and construction difficulty. If you were to launch a rocket towards the west, you’d have to engineer it to be much more powerful, since it first has to overcome its initial eastward velocity, and then generate the entirety of the westward velocity from scratch. So virtually no rockets are ever launched towards the west. Rockets are occasionally launched to the north or south to put their payloads into polar orbits, but most are placed into so-called near-equatorial orbits that travel substantially west-to-east.

In turn, this means that when selecting a launch site, you want to choose a place where the territory to the eastern side of the site is free of population centres, again to avoid disaster if something goes wrong during a launch. The easiest way to achieve this is to place your launch site on the eastern coast of a landmass, so the rockets launch out over the ocean, though you can also do it if you can find a large unpopulated region and place your launch site near the western side.

When we look at the major rocket launch facilities around the world, they generally follow these principles. The Kennedy Space Center at Cape Canaveral is acceptably near Orlando, Florida, but far enough away to avoid disasters, and adjacent to Cape Canaveral Air Force Station for military logistics. It launches east over the Atlantic Ocean.

Kennedy Space Center

Kennedy Space Center launch pads A (foreground) and B (background). The Atlantic Ocean is to the right. (Public domain image by NASA.)

A NASA historical report has this to say about the choice of a launch site for Saturn series rockets that would later take humans to the moon[1]:

The short-lived plan to transport the Saturn by air was prompted by ABMA’s interest in launching a rocket into equatorial orbit from a site near the Equator; Christmas Island in the Central Pacific was a likely choice. Equatorial launch sites offered certain advantages over facilities within the continental United States. A launching due east from a site on the Equator could take advantage of the earth’s maximum rotational velocity (460 meters per second) to achieve orbital speed. The more frequent overhead passage of the orbiting vehicle above an equatorial base would facilitate tracking and communications. Most important, an equatorial launch site would avoid the costly dogleg technique, a prerequisite for placing rockets into equatorial orbit from sites such as Cape Canaveral, Florida (28 degrees north latitude). The necessary correction in the space vehicle’s trajectory could be very expensive – engineers estimated that doglegging a Saturn vehicle into a low-altitude equatorial orbit from Cape Canaveral used enough extra propellant to reduce the payload by as much as 80%. In higher orbits, the penalty was less severe but still involved at least a 20% loss of payload. There were also significant disadvantages to an equatorial launch base: higher construction costs (about 100% greater), logistics problems, and the hazards of setting up an American base on foreign soil.

Russia’s main launch facility, Baikonur Cosmodrome in Kazakhstan (former USSR territory), launches east over the largely uninhabited Betpak-Dala desert region. China’s Jiuquan Satellite Launch Centre launches east over the uninhabited Altyn-Tagh mountains. The Guiana Space Centre, the major launch facility of the European Space Agency, is located on the coast of French Guiana, an overseas department of France on the north-east coast of South America, where it launches east over the Atlantic Ocean.

Guiana Space Centre

Guiana Space Centre, French Guiana. The Atlantic Ocean is in the background. (Photo: ESA-Stephane Corvaja, released under ESA Standard Licence.)

Another consideration when choosing your rocket launching site is that the initial momentum boost provided by the Earth’s rotation is greatest at the equator, where the rotational speed of the Earth’s surface is greatest. At the equator, the surface is moving 40,000 km (the circumference of the Earth) per day, or 1670 km/h. Compare this to latitude 41° (roughly New York City, or Madrid), where the speed is 1260 km/h, and you see that our rockets get a free 400 km/h boost by being launched from the equator compared to these locations. So you want to place your launch facility as close to the equator as is practical, given the other considerations.

Rotation of Earth

Because the Earth is a rotating globe, the equatorial regions are moving faster than anywhere else, and provide more of a boost to rocket launch velocities.

The European Space Agency, in particular, has problems with launching rockets from Europe, because of its dense population, unavailability of an eastern coastline, and distance from the equator. This makes French Guiana much more attractive, even though it’s so far away. The USA has placed its major launch facility in just about the best location possible in the continental US. Anywhere closer to the equator on the east coast is taken up by Miami’s urban sprawl. The former USSR went for southern Kazakhstan as a compromise between getting as far south as possible, and being close enough to Moscow. China’s more southern and coastal regions are much more heavily populated, so they went with a remote inland area (possibly also to help keep it hidden for military reasons).

All of these facilities so far are in the northern hemisphere. There are no major rocket launch facilities in the southern hemisphere, and in fact only two sites from where orbital flight has been achieved: Australia’s Woomera Range Complex, which is a remote air force base chosen historically for military logistical reasons (including nuclear weapons testing as well as rocketry in the wake of World War II), and New Zealand’s Rocket Lab Launch Complex 1, a new private facility for launching small satellites, whose location was governed by the ability to privately acquire and develop land.

But if you were to build a major launch facility in the southern hemisphere, where would you put it?

A major space facility was first proposed for Australia in 1986, with plans for it to be the world’s first commercial spaceport. The proposed site? Near Weipa, on the Cape York Peninsula, essentially as close to the equator as it’s possible to get in Australia.

Site of Weipa in Australia

Site of Weipa in Australia. Apart from Darwin which is at almost exactly the same latitude, there is no larger town further north in Australia. (Adapted from a Creative Commons Attribution 4.0 International image by John Tann, from Wikimedia Commons.)

The proposal eventually floundered due to lack of money and protests from indigenous land owners, but there is now a current State Government inquiry into constructing a satellite launching facility in Queensland, again in the far north. As a news story points out, “From a very simple perspective, we’ve got potential launch capacity, being closer to the equator in a place like Queensland,” and “the best place to launch satellites from Australia is the coast of Queensland. The closer you are to the equator, the more kick you get from the Earth’s spin.”[2]

So rocket engineers in the southern hemisphere definitely want to build their launch facilities as close to the equator as practically possible too. Repeating what I said earlier, you’d be crazy not to. And this is a consequence of the fact that the Earth is a rotating globe.

On the other hand, if the Earth were flat and non-rotating (as is the case in the most popular flat Earth models), there would be no such incentive to build your launch facility anywhere compared to anywhere else, and equatorial locations would not be so coveted. And if the Earth were flat and rotating around the north pole, then you’d get your best bang for buck not near the equator, but near the rim of the rotating disc, where the linear speed of rotation is highest. If that were the case, then everyone would be clamouring to build their launch sites as close to Antarctica as possible, which is clearly not the case in the real (globular) world.

[1] Benson, C. D., Faherty, W. B. Moonport: A History of Apollo Launch Facilities and Operations. Chapter 1.2, NASA Special Publication-4204 in the NASA History Series, 1978. https://www.hq.nasa.gov/office/pao/History/SP-4204/contents.html (accessed 2019-07-15).

[2] “Rocket launches touted for Queensland as State Government launches space industry inquiry”. ABC News, 6 September 2018. https://www.abc.net.au/news/2018-09-06/queensland-shoots-for-the-stars-to-become-space-hub/10205686 (accessed 2019-07-15).

18. Polar motion

The Earth rotates around an axis, an imaginary straight line that all points not on the line move around in circles. The axis passes through the Earth’s North Pole and the South Pole. So the positions of the two Poles are defined by the position of the rotation axis.

Earth rotation and poles

The Earth’s North and South Poles are defined as the points where the axis of rotation passes through the surface of the planet. (Earth photo is a public domain image from NASA.)

Interestingly, the Earth’s rotation axis is not fixed – it moves slightly. This means that the Earth’s poles move.

The positions of the Earth’s poles can be determined by looking at the motions of the stars. As we’ve already seen, if you observe the positions of stars throughout a night, you will see that they rotate in the sky about a central point. The point on the Earth’s surface directly underneath the centre of rotation of the stars is one of the poles of the Earth.

Star trails in the northern hemisphere

Star trails above Little Hawk Lake in Canada. The northern hemisphere stars rotate around the North Celestial Pole (the point directly above the Earth’s North Pole). The bright spot in the centre is Polaris, the pole star. The circles are somewhat distorted in the upper corners of the photo because of the wide angle lens used. (Creative Commons Attribution 2.0 image by Dave Doe.)

Through the 19th century, astronomers were improving the precision of astronomical observations to the point where the movement of the Earth’s rotational poles needed to be accounted for in the positions of celestial objects. The motion of the poles was also beginning to affect navigation, because as the poles move, so does the grid system of latitude and longitude that ships rely on to reach their destinations and avoid navigational hazards. In 1899 the International Geodetic Association established a branch known as the International Latitude Service.

The fledgling International Latitude Service established a network of six observatories, all located close to latitude 39° 08’ north, spread around the world. The initial observatories were located in Gaithersburg, Maryland, USA; Cincinatti, Ohio, USA; Ukiah, California, USA; Mizusawa, Japan; Charjui, Turkestan; and Carloforte, Italy. The station in Charjui closed due to economic problems caused by war, but a new station opened in Kitab, Uzbekistan after World War I. Each observatory engaged in a program of observing the positions of 144 selected reference stars, and the data from each station were cross referenced to provide accurate measurements of the location of the North Pole.

International Latitude Service station in Ukiah

International Latitude Service station in Ukiah, California. (Public domain image from Wikimedia Commons.)

In 1962, the International Time Bureau founded the International Polar Motion Service, which incorporated the International Latitude Service observations and additional astronomical observations to provide a reference of higher accuracy, suitable for both navigation and defining time relative to Earth’s rotation. Finally in 1987, the the International Astronomical Union and the International Union of Geodesy and Geophysics established the International Earth Rotation Service (IERS), which took over from the International Polar Motion Service. The IERS is the current authority responsible for timekeeping and Earth-based coordinate systems, including the definitions of time units, the introduction of leap seconds to keep clocks in synch with the Earth’s rotation, and definitions of latitude and longitude, as well as measurements of the motion of the Earth’s poles, which are necessary for accurate use of navigation systems such as GPS and Galileo.

The motion of Earth’s poles can be broken down into three components:

1. An annual elliptical wobble. Over the period of a year, the Earth’s poles move around in an ellipse, with the long axis of the ellipse about 6 metres in length. In March, the North Pole is about 6 metres from where it is in September (though see below). This motion is generally agreed by scientists to be caused by the annual shift in air pressure between winter and summer over the northern and southern hemispheres. In particular there is an imbalance between the Northern Atlantic ocean and Asia, with higher air pressure over the ocean in the northern winter, but higher air pressure over the Asian continent in summer. This change in the mass distribution of the atmosphere is enough to cause the observed wobble.

Annual wobble of North Pole

Annual elliptical wobble of the Earth’s North Pole. Deviation is given in milliarcseconds of axial tilt; 100 milliarcseconds corresponds to a bit over 3 metres at ground level. (Figure adapted from [1].)

2. Superimposed on the annual elliptical wobble is another, circular, wobble, with a period of around 433 days. This is called the Chandler wobble, named after its discoverer, American astronomer Seth Carlo Chandler, who found it in 1891. The Chandler wobble occurs because the Earth is not a perfect sphere. The Earth is slightly elliptical, with the radius at the equator about 20 kilometres larger than the polar radius. When elliptical objects spin, they experience a slight wobble in the rotation known as free nutation. This is the sort of wobble seen in a spinning rugby ball or American football in flight (where the effect is exaggerated by the ball’s exaggerated elliptical shape). This wobble would die away over time, but is driven by changes in the mass distribution of cold and warm water in the oceans and high and low pressure systems in the atmosphere. The Chandler wobble has a diameter of about 9 metres at the poles.

The combined effect of the annual wobble and the Chandler wobble is that the North and South Poles move in a spiralling pattern, sometimes circling with a diameter up to 15 metres, then reducing down to about 3 metres, before increasing again. This beat pattern occurs over a period of about 7 years.

Annual _ Chandler wobble of North Pole

Graph showing the movement of the North Pole over a period of 4500 days (12.3 years), with time on the vertical axis and the spiralling motion mapped in the x and y axes. The motion tickmarks are 0.1 arcsecond in rotation angle of the axis apart, corresponding to about 3 metres of motion along the ground at the Pole. (Public domain image from Wikimedia Commons.)

3. The third and final motion of the Earth’s poles is a systematic drift, of about 200 millimetres per year. Since 1900, the central point of the spiral wobbles of the North Pole has drifted by about 20 metres. This drift is caused by changes in the mass distribution of Earth due to shifts in its structure: movement of molten rock in the mantle, isostatic rebound of crust following the last glacial period, and more recently the melting of the Greenland ice sheet. The melting of the Greenland ice sheet in the last few decades has shifted the direction of polar drift dramatically; one of the serious indications of secondary changes to the Earth caused by human-induced climate change. Changes in Earth’s mass distribution alter its rotational moment of inertia, and the rotational axis adjusts to conserve angular momentum.

Motion of North Pole since 1900

Plot of motion of the North Pole since 1900. The actual position of the Pole from 2008 to 2014 is shown with blue crosses, showing the annual and Chandler wobbles. The mean position (i.e. the centre of the wobbles) is shown for 1900 to 2014 as the green line. The pole has mostly drifted towards the 80° west meridian, but has changed direction dramatically since 2000. (Figure reproduced from [2].)

Each of the three components of Earth’s polar motion are: (a) observable with 19th century technology, (b) accurately measurable using current technology, and (c) understandable and quantitatively explainable using the fact that the Earth is a rotating spheroid and our knowledge of its structure.

If the Earth were flat, it would not be possible to reconcile the changes in position of the North and South Poles with the known shifts in mass distribution of the Earth. The Chandler wobble would not even have any reason to exist at close to its observed period unless the Earth was an almost spherical ellipsoid.

References:

[1] Höpfner, J. “Polar motion at seasonal frequencies”. Journal of Geodynamics, 22, p. 51-61, 1996. https://doi.org/10.1016/0264-3707(96)00012-9

[2] Dick, W., Thaller, D. IERS Annual Report 2013. International Earth Rotation Service, 2014. https://www.iers.org/IERS/EN/Publications/AnnualReports/AnnualReport2013.html