Our planet is one of 9 that revolve around the sun. Even in ancient times, the first ideas about the shape and size of the Earth appeared.
How have ideas about the shape of the Earth changed?
Ancient thinkers (Aristotle - 3rd century BC, Pythagoras - 5th century BC, etc.) many centuries ago expressed the idea that our planet has a spherical shape. Aristotle (pictured below), in particular, taught after Eudoxus that the Earth, which is the center of the Universe, is spherical. He saw proof of this in the nature of lunar eclipses. With them, the shadow cast by our planet on the Moon has a rounded shape at the edges, which is possible only if it is spherical.
Astronomical and geodetic research carried out in the following centuries gave us the opportunity to judge what the shape and dimensions of the Earth are in reality. Today, that it is round, they know from small to large. But there were times in history when it was believed that the planet Earth was flat. Today, thanks to the progress of science, we no longer doubt that it is round, not flat. Indisputable proof of this is space photographs. The sphericity of our planet leads to the fact that the earth's surface heats up unevenly.
But in fact, the shape of the Earth is not quite the same as we used to think. This fact is known to scientists, and it is currently used to solve problems in the field of satellite navigation, geodesy, astronautics, astrophysics and other related sciences. For the first time, the idea of what the actual shape of the Earth is was expressed by Newton at the turn of the 17th-18th centuries. He theoretically substantiated the assumption that our planet, under the influence of gravity on it, should be compressed in the direction of the axis of rotation. And this means that the shape of the Earth is either a spheroid or an ellipsoid of revolution. The degree of compression depends on the angular velocity of rotation. That is, the faster the body rotates, the more it flattens at the poles. This scientist proceeded from the principle of universal gravitation, as well as from the assumption of a homogeneous liquid mass. He assumed that the Earth is a compressed ellipsoid, and determined, depending on the speed of rotation, the size of the compression. After some time, Maclaurin proved that if our planet is an ellipsoid compressed at the poles, then the balance of the oceans covering the Earth is indeed ensured.
Can we assume that the Earth is round?
If the planet Earth is viewed from afar, it will appear almost perfectly round. An observer who does not care about high measurement accuracy may well consider it as such. The average radius of the Earth in this case is 6371.3 km. But if we, taking the shape of our planet as an ideal ball, begin to make accurate measurements of the various coordinates of points on the surface, we will not succeed. The fact is that our planet is not a perfectly round ball.
Different Ways to Describe the Shape of the Earth
The shape of the planet Earth can be described in two main ways, as well as several derivative ones. It can be taken in most cases as either a geoid or an ellipsoid. It is interesting that the second option is easily described mathematically, but the first one is not described in principle, since in order to determine the exact shape of the geoid (and, consequently, the Earth), practical measurements of gravity are carried out at various points on the surface of our planet.
Ellipsoid of revolution
Everything is clear with the ellipsoid of revolution: this figure resembles a ball, which is flattened from below and from above. The fact that the shape of the Earth is an ellipsoid is quite understandable: centrifugal forces arise due to the rotation of our planet at the equator, while they do not exist at the poles. As a result of rotation, as well as centrifugal forces, the Earth has become "fat": the diameter of the planet along the equator is approximately 50 km larger than the polar one.
Features of a figure called "geoid"
An extremely complex figure is the geoid. It exists only in theory, but in practice it cannot be felt or seen. One can imagine the geoid as a surface, the force of gravity at each point of which is directed strictly vertically. If our planet were a regular ball filled evenly with some substance, then the plumb line at any point on it would look at the center of the ball. But the situation is complicated by the fact that the density of our planet is heterogeneous. In some places there are heavy rocks, in others voids, mountains and depressions are scattered over the entire surface, plains and seas are also unevenly distributed. All this changes the gravitational potential at each specific point. The fact that the shape of the globe is a geoid is also to blame for the ethereal wind that blows our planet from the north.
Who studied geoids?
Note that the very concept of "geoid" was introduced by Johann Listing (pictured below), a physicist and mathematician, in 1873.
Under it, meaning in Greek "view of the Earth", was meant a figure formed by the surface of the World Ocean, as well as the seas communicating with it, at an average water level, without disturbances from tides, currents, as well as differences in atmospheric pressure, etc. When they say that such and such an altitude above sea level, this means the height from the surface of the geoid at this point on the globe, despite the fact that there is no sea in this place, and it is several thousand kilometers from it.
Subsequently, the concept of the geoid was repeatedly refined. Thus, the Soviet scientist M. S. Molodensky created his own theory of determining the gravitational field and the figure of the Earth from measurements made on its surface. To do this, he developed a special device that measures gravity - a spring gravimeter. It was he who also proposed the use of a quasi-geoid, which is determined by the values taken by the gravity potential on the Earth's surface.
More about the geoid
If gravity is measured 100 km from the mountains, then the plumb line (that is, the weight on the thread) will deviate in their direction. Such a deviation from the vertical is imperceptible to our eye, but it is easily detected by instruments. A similar picture is observed everywhere: deviations of the plumb line are greater somewhere, somewhere they are less. And we remember that the surface of the geoid is always perpendicular to the plumb line. From this it becomes clear that the geoid is a very complex figure. In order to better imagine it, you can do the following: sculpt a ball of clay, then squeeze it on both sides to form a flattened shape, then make bumps and dents on the resulting ellipsoid with your fingers. Such a flattened rumpled ball will quite realistically show the shape of our planet.
Why do we need to know the exact shape of the Earth?
Why do you need to know its shape so precisely? What does not satisfy scientists about the spherical shape of the Earth? Should the picture be complicated by the geoid and the ellipsoid of revolution? Yes, there is an urgent need for this: figures close to the geoid help to create coordinate grids that are the most accurate. Neither astronomical research, nor geodetic surveys, nor various satellite navigation systems (GLONASS, GPS) can exist and be carried out without determining a fairly accurate shape of our planet.
Various coordinate systems
The world currently has several three-dimensional and two-dimensional coordinate systems with world significance, as well as several dozen local ones. Each of them has its own form of the Earth. This leads to the fact that the coordinates that were determined by different systems are somewhat different. Interestingly, in order to calculate them at points located on the territory of one country, it will be most convenient to take the shape of the Earth as a reference ellipsoid. This is now established even at the highest legislative level.
Ellipsoid of Krasovsky
If we talk about the CIS countries or Russia, then on the territory of these states the shape of our planet is described by the so-called Krasovsky ellipsoid. It was identified back in 1940. Domestic (PZ-90, SK-63, SK-42) and foreign (Afgooye, Hanoi 1972) coordinate systems were created on the basis of this figure. They are still used for practical and scientific purposes. Interestingly, GLONASS relies on the PZ-90 system, which is superior in its accuracy to the analogous WGS84 system adopted as the basis for GPS.
Conclusion
Summing up, let's say again that the shape of our planet is different from the ball. The earth is approaching in its shape an ellipsoid of revolution. As we have already noted, this question is not at all idle. Determining exactly what shape the Earth is gives scientists a powerful tool to calculate the coordinates of heavenly and terrestrial bodies. And this is very important for space and marine navigation, during construction, geodetic work, as well as in many other areas of human activity.
The earth is round. The figure of the Earth is a term for the shape of the earth's surface. So, the shape of the Earth differs from the ball, approaching the ellipsoid of revolution. GEOID - (from geo ... and Greek eidos view) the figure of the Earth, limited by a level surface, continued under the continents. The Earth is spherical, like all other cosmic bodies with a large mass. Such a surface is called the general figure of the Earth or the surface of the geoid.
Depending on the definition of the figure of the Earth, various coordinate systems are established. Even in the VI century. BC Pythagoras believed that the Earth has a spherical shape. The same discovery is given by the most authoritative author on this issue, Theophrastus, to Parmenides.
200 years later, Aristotle proved this, referring to the fact that during lunar eclipses the shadow of the Earth is always round. He suggested that it has the shape of an ellipsoid and proposed the following thought experiment. It is necessary to dig two shafts: from the pole to the center of the Earth and from the equator to the center of the Earth. These mines are filled with water. If the Earth is spherical, then the depth of the mines is the same.
For a better approximation of the surface, the concept of a reference ellipsoid is introduced, which coincides well with the geoid only on some part of the surface. In practice, several different mean earth ellipsoids and associated earth coordinate systems are used. The same ethereal wind that blows over it from the north is to blame for the fact that the globe has the shape of a geoid - a kind of pear, elongated to the North Pole.
Leveling heights are measured from the geoid. The concept of the geoid has been repeatedly refined. He also proposed the use of a "quasi-geoid" (almost a geoid), determined by the values of the gravity potential on the earth's surface. Deviations from the geoid are small, no more than 3 meters, but geodesy is an exact science, and such deviations are essential for it.
The Earth together with the Sun now and already 3-4 billion years is in such a region of the spiral arm of the Galaxy, in which it is blown by the ether flow from the north. Going around the Earth, the ether flow creates various areas of pressure on it. According to the laws of the boundary layer, after 110 degrees, counting from the point at which the ether flow strikes at a right angle, that is, slightly below the equator, this flow begins to break away from the surface.
It is now that every schoolchild knows for sure that the planet is round, that the gravitational force acts on all of us, which does not allow us to fall “down” and fly out of the atmosphere ... However, the hypothesis that our planet has the shape of a ball existed for a very long time. This idea was first expressed in the 6th century BC by the ancient Greek philosopher and mathematician Pythagoras.
Back in the 17th century, the famous physicist and mathematician Newton made a bold assumption that the Earth is not a ball at all, or rather, not quite a ball. Assumed - and mathematically proved it. Be that as it may, now we know for sure that the Earth is flattened at the poles (if you like, stretched at the equator). It turns out that the Earth has not quite the correct shape, it resembles a pear, elongated to the North Pole.
physical surface of the earth
Therefore, scientists have proposed a special name for the shape of the Earth - the geoid. The geoid is an irregular stereometric figure. Strong earthquakes also affect the shape of the globe. University of Milan professors Roberto Sabadini and Giorgio Dalla Via believe that it left a "scar" on the planet's gravitational field, causing the geoid to sag significantly.
We hope that soon he will send us accurate information about the shape of the Earth today. The shape of the Earth can be described in two basic and several derivative ways. The geoid is an extremely complex figure, and it exists only theoretically, but in practice it cannot be seen or “felt”.
The concept of the shape and surface of the Earth
And we remember that the surface of the geoid is always perpendicular to the plumb line, hence it becomes clear that the geoid is not only a complex figure, but also a cunning one. In general, why is it necessary to know the shape of our planet so precisely?
Each of them has its own form of the Earth, which leads to some differences in the coordinates determined by different systems. And if you answer the question why our planet is still round, it will be necessary to consider several significant facts.
The influence of the composition of the planet Earth on its shape
All large planets of near-Earth space (the Moon, the Sun, etc.) have a grandiose mass, which implies an increased gravitational force. Without this, the force of gravity would not have such an impact on the creation of the shape of our planet - for this, the cosmic body must be optimally plastic, for example, gaseous or liquid.
And there is some significant evidence for this. The polar radius of the Earth is 6357 kilometers, its equatorial radius is 6378 kilometers, which is a difference of as much as 19 kilometers. Therefore, it would be a little wrong to call the planet an absolute ball, since rather it has the shape of a ball, slightly flattened at the poles and stretched along the Equator line.
Also, the Earth cannot be ideally round due to the fact that hot magma as a kind of liquid is present only under the crust of the earth's surface, and the crust itself is a solid. But it is worth noting that certain phenomena also affect the liquid located on the surface of the Earth - more precisely, the gravitational force of other celestial objects.
See what "Geoid" is in other dictionaries:
Geoid - a geometrically complex surface of equal values of the gravity potential, coinciding with the undisturbed surface of the World Ocean and extended over the continents. About four hundred years ago, people were sure that the Earth was flat and rested on three whales. All those who disagreed were dragged to the fires, so there were not many of them. A hundred years later, it was already possible to convince others with impunity that the Earth is a ball. A little time passed, and again they began to persecute for this belief.
In reality, the figure of the Earth is even more complicated. Yes, the Earth is not an exact ellipsoid, but a more complex body. Then they decided to call the shape of the Earth the geoid. The European satellite GOCE saw the Earth in the shape of a potato. The fact that the shape of the Earth should be different from the ball was first shown by Newton. In reality, the Earth's surface can differ significantly from the geoid in different places.
In the first approximation, the earth can be considered a sphere. In the second approximation, the Earth is taken as an ellipsoid of revolution; in some studies it is considered a biaxial ellipsoid. geoid- the body is taken as the theoretical figure of the Earth, limited by the surface of the oceans in their calm state, continued under the continents. Due to the uneven distribution of masses in the earth's crust, the geoid has an irregular geometric shape, and its surface cannot be expressed mathematically, which is necessary for solving geodetic problems. When solving geodetic problems, the geoid is replaced by geometrically regular surfaces close to it. So, for approximate calculations, the Earth is taken as a ball with a radius of 6371 km. Closer to the shape of the geoid is an ellipsoid - a figure obtained by rotating an ellipse (Fig. 2.1) around its minor axis. The dimensions of the earth's ellipsoid are characterized by the following main parameters: a- major axle shaft b semi-minor axis, polar compression and e is the first eccentricity of the meridian ellipse, where and.
A distinction is made between a general earth ellipsoid and a reference ellipsoid.
Centre earth ellipsoid are placed at the center of mass of the Earth, the axis of rotation is aligned with the average axis of rotation of the Earth, and the dimensions are taken such as to ensure the closest proximity of the ellipsoid surface to the geoid surface. The general earth ellipsoid is used in solving global geodetic problems, and in particular, in processing satellite measurements. At present, two general-earth ellipsoids are widely used: PZ-90 (Parameters of the Earth 1990, Russia) and WGS-84 (World Geodetic System 1984, USA).
Reference ellipsoid- an ellipsoid adopted for geodetic work in a particular country. The coordinate system adopted in the country is associated with the reference ellipsoid. The parameters of the reference ellipsoid are selected under the condition of the best approximation of a given part of the Earth's surface. In this case, the alignment of the centers of the ellipsoid and the Earth is not achieved.
In Russia, since 1946, it has been used as a reference ellipsoid Krasovsky ellipsoid with parameters: a= 6 378 245 m, a = 1/298.3.
2. Coordinate systems in geodesy. Absolute and relative heights.
Coordinate systems used in geodesy
To determine the position of points in geodesy, spatial rectangular, geodesic and flat rectangular coordinates are used.
Spatial rectangular coordinates. The origin of the coordinate system is located in the center O earth ellipsoid (Fig. 2.2).
Axis Z is directed along the axis of rotation of the ellipsoid to the north. Axis X lies at the intersection of the equatorial plane with the prime meridian of Greenwich. Axis Y directed perpendicular to the axes Z and X to the East.
Geodetic coordinates. The geodetic coordinates of a point are its latitude, longitude and height (Fig. 2.2).
Geodetic latitude pointsM called the angle V, formed by the normal to the surface of the ellipsoid passing through the given point, and the plane of the equator.
Latitude is measured from the equator north and south from 0 to 90 and is called north or south. North latitude is considered positive, and south latitude is negative.
Sectional planes of an ellipsoid passing through an axis oz, are called geodetic meridians.
Geodetic longitude points M called a dihedral angle L, formed by the planes of the initial (Greenwich) geodesic meridian and the geodesic meridian of the given point.
Longitude is measured from the prime meridian in the range from 0 to 360 east, or from 0 to 180 east (positive) and from 0 to 180 west (negative).
Geodetic elevation point M is her height H above the surface of the earth's ellipsoid.
Geodetic coordinates with spatial rectangular coordinates are related by the formulas
X=(N+H) cos B cos L, Y=(N+H) cos B sin L, Z=[(1 e 2 )N+H] sin B,
where e- the first eccentricity of the meridian ellipse and N radius of curvature of the first vertical. Wherein N= a/ (1e 2 sin 2 B) 1/2 . Geodetic and spatial rectangular coordinates of points are determined using satellite measurements, as well as by linking them with geodetic measurements to points with known coordinates. Note that along with geodesics, there are also astronomical latitude and longitude. Astronomical latitude is the angle made by a plumb line at a given point with the plane of the equator. Astronomical longitude is the angle between the planes of the Greenwich meridian and the astronomical meridian passing through the plumb line at a given point. Astronomical coordinates are determined on the ground from astronomical observations. Astronomical coordinates differ from geodetic ones because the directions of the plumb lines do not coincide with the directions of the normals to the surface of the ellipsoid. The angle between the direction of the normal to the surface of the ellipsoid and the plumb line at a given point on the earth's surface is called plumb line.
A generalization of geodetic and astronomical coordinates is the term - geographical coordinates.
Flat rectangular coordinates. To solve the problems of engineering geodesy, from spatial and geodetic coordinates, they move on to simpler - flat coordinates, which make it possible to depict the terrain on a plane and determine the position of points with two coordinates X and at.
Since the convex surface of the Earth cannot be depicted on a plane without distortion, the introduction of flat coordinates is possible only in limited areas where the distortions are so small that they can be neglected. In Russia, a system of rectangular coordinates is adopted, the basis of which is the Gaussian conformal transverse cylindrical projection. The surface of an ellipsoid is depicted on a plane in parts called zones. The zones are spherical bicagons bounded by meridians and extending from the north pole to the south (Fig. 2.3). The size of the zone in longitude is 6. The central meridian of each zone is called the axial meridian. The zones are numbered from Greenwich to the east.
The longitude of the axial meridian of the zone with the number N is equal to:
0 = 6 N 3 .
The axial meridian of the zone and the equator are depicted on the plane by straight lines (Fig. 2.4). The axial meridian is taken as the abscissa axis x, and the equator - for the y-axis y. Their intersection (point O) serves as the origin of the given zone.
To avoid negative ordinate values, the intersection coordinates are taken equal to x 0 = 0, y 0 = 500 km, which is equivalent to an axis shift X west for 500 km.
So that by the rectangular coordinates of a point it is possible to judge in which zone it is located, to the ordinate y on the left, the number of the coordinate zone is assigned.
Let, for example, the coordinates of the point A look like:
x A = 6 276 427 m, y A= 12 428 566 m
These coordinates indicate that the point A located at a distance of 6276427 m from the equator, in the western part ( y 500 km) of the 12th coordinate zone, at a distance of 500000 428566 = 71434 m from the axial meridian. For spatial rectangular, geodetic and flat rectangular coordinates in Russia, a unified coordinate system SK-95 is adopted, fixed on the ground by points of the state geodetic network and built on satellite and ground-based measurements as of the epoch of 1995
Height systems
The heights in engineering geodesy are counted from one of the level surfaces. point height call the distance along the plumb line from the point to the level surface, taken as the beginning of the calculation of heights.
Heights are absolute if they are counted from the main level surface, that is, from the surface of the geoid. On fig. 2.5 segments of plumb lines Ah and Vv absolute heights of points A and V.
Heights are called conditional, if any other level surface is selected as the beginning of the height calculation. On fig. 2.5 segments of plumb lines Ah and Vv conditional heights of points A and V.
adopted in Russia Baltic height system. The absolute heights are counted from the level surface. The numerical value of the height is usually called mark. For example, if the point height A is equal to H A\u003d 15.378 m, then they say that the elevation of the point is 15.378 m.
The height difference between two points is called excess. So, exceeding the point V over the dot A equals
h AB = H V H A .
Knowing the height of the point A, to determine the height of a point V on the ground measure the excess h AB. point height V calculated according to the formula
H V = H A + h AB .
The measurement of elevations and the subsequent calculation of the heights of points is called leveling.
The absolute height of a point must be distinguished from its geodetic height, that is, the height measured from the surface of the earth's ellipsoid (see section 2.2). The geodetic height differs from the absolute height by the deviation of the geoid surface from the ellipsoid surface.
εἶδος - view, literally - “something like the Earth”) - a convex closed surface coinciding with the surface of the water in the seas and oceans in a calm state and perpendicular to the direction of gravity at any point. Geometric body, an ellipsoid of revolution deviating from a figure of revolution and reflecting the properties of the gravity potential on the Earth (near the earth's surface), an important concept in geodesy.Definition of "geoid"
Story
The term "geoid" was proposed in 1873 by the German mathematician Johann Benedikt Listing to refer to a geometric figure, more accurately than an ellipsoid of revolution, that reflects the unique shape of the planet Earth.
Application
The geoid is the surface relative to which the height above sea level is measured. Exact knowledge of the geoid is necessary, in particular, in navigation - to determine the height above sea level based on geodetic (ellipsoidal) height, directly measured by GPS receivers, as well as in physical oceanology - to determine the heights of the sea surface.
Quasi-geoid
The figure of the geoid depends on the distribution of masses and densities in the body of the Earth. It does not have an exact mathematical expression and is practically indeterminate, and therefore in geodetic measurements in Russia and some other countries, instead of the geoid, its approximation, the quasi-geoid, is used. The quasi-geoid, unlike the geoid, is unambiguously determined by the results of measurements, coincides with the geoid on the territory of the World Ocean and is very close to the geoid on land, deviating only a few centimeters on flat terrain and no more than 2 meters in high mountains.
see also
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Notes
Literature
- Pariyskiy N. N. On some consequences of the non-sphericity of the Earth // Slow deformations of the Earth and its rotation. M., 1985. S. 35-39.
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An excerpt characterizing the Geoid
“And you know, my dear, it seems to me that Buonaparte has definitely lost his Latin. You know that today a letter has been received from him to the emperor. Dolgorukov smiled significantly.– That's how! What does he write? Bolkonsky asked.
What can he write? Tradiridira, etc., all just to gain time. I tell you that he is in our hands; It's right! But the funniest thing of all,” he said, suddenly laughing good-naturedly, “is that they couldn’t figure out how to address the answer to him? If not the consul, it goes without saying not the emperor, then General Buonaparte, as it seemed to me.
“But there is a difference between not recognizing the emperor, and calling Buonaparte general,” said Bolkonsky.
“That's just the point,” Dolgorukov said quickly, laughing and interrupting. - You know Bilibin, he is a very smart person, he offered to address: "usurper and enemy of the human race."
Dolgorukov laughed merrily.
- No more? Bolkonsky noted.
- But still, Bilibin found a serious address title. And a witty and intelligent person.
- How?
“To the head of the French government, au chef du gouverienement francais,” Prince Dolgorukov said seriously and with pleasure. - Isn't that good?
“Good, but he won’t like it very much,” Bolkonsky remarked.
- Oh, and very much! My brother knows him: he dined with him more than once, with the present emperor, in Paris and told me that he had never seen a more refined and cunning diplomat: you know, a combination of French dexterity and Italian acting? Do you know his jokes with Count Markov? Only one Count Markov knew how to handle him. Do you know the history of the scarf? This is a charm!
And the garrulous Dolgorukov, turning now to Boris, now to Prince Andrei, told how Bonaparte, wanting to test Markov, our envoy, deliberately dropped his handkerchief in front of him and stopped, looking at him, probably expecting services from Markov and how, Markov immediately he dropped his handkerchief beside him and picked up his own without picking up Bonaparte's handkerchief.
- Charmant, [Charming,] - said Bolkonsky, - but here's what, prince, I came to you as a petitioner for this young man. Do you see what?…
But Prince Andrei did not have time to finish, when an adjutant entered the room, who called Prince Dolgorukov to the emperor.
- Oh, what a shame! - said Dolgorukov, hastily getting up and shaking hands with Prince Andrei and Boris. - You know, I am very glad to do everything that depends on me, both for you and for this nice young man. - He once again shook Boris's hand with an expression of good-natured, sincere and lively frivolity. “But you see…until another time!”
Boris was excited by the thought of the closeness to the highest power in which he felt himself at that moment. He was aware of himself here in contact with those springs that guided all those enormous movements of the masses, of which he in his regiment felt himself to be a small, obedient and insignificant part. They went out into the corridor after Prince Dolgorukov and met a short man in civilian clothes, with an intelligent face and a sharp line of protruding jaw, which, without spoiling him, gave him special liveliness and resourcefulness of expression. This short man nodded, as to his own, Dolgoruky, and began to stare at Prince Andrei with an intently cold look, walking straight at him and apparently waiting for Prince Andrei to bow to him or give way. Prince Andrei did neither one nor the other; Anger was expressed in his face, and the young man, turning away, walked along the side of the corridor.
The geoid is a model of the figure of the Earth (i.e., its analogue in size and shape), which coincides with the mean sea level, and in continental regions is determined by the spirit level. Serves as a reference surface from which topographic heights and ocean depths are measured. The scientific discipline about the exact shape of the Earth (geoid), its definition and significance is called geodesy. More information about this is provided in the article.
Potential Constancy
The geoid is everywhere perpendicular to the direction of gravity and approaches a regular oblate spheroid in shape. However, this is not the case everywhere due to local concentrations of accumulated mass (deviations from uniformity at depth) and due to height differences between continents and the seafloor. Mathematically speaking, the geoid is an equipotential surface, i.e., characterized by the constancy of the potential function. It describes the combined effects of the gravitational pull of the Earth's mass and the centrifugal repulsion caused by the planet's rotation on its axis.
Simplified Models
The geoid, due to the uneven distribution of mass and the resulting mass, is not a simple mathematical surface. It is not quite suitable for the standard of the geometric figure of the Earth. For this (but not for topography), approximations are simply used. In most cases, a sphere is a sufficient geometric representation of the Earth, for which only the radius should be specified. When a more accurate approximation is required, an ellipsoid of revolution is used. This is the surface created by rotating an ellipse 360° about its minor axis. The ellipsoid used in geodetic calculations to represent the Earth is called the reference ellipsoid. This shape is often used as a simple base surface.
The ellipsoid of revolution is given by two parameters: semi-major axis (Equatorial radius of the Earth) semi-minor axis (polar radius). The flattening f is defined as the difference between the major and minor semiaxes divided by the major f = (a - b) / a. The semi-axes of the Earth differ by about 21 km, and the ellipticity is about 1/300. The deviations of the geoid from the ellipsoid of revolution do not exceed 100 m. The difference between the two semi-axes of the equatorial ellipse in the case of a triaxial ellipsoid model of the Earth is only about 80 m.
Geoid concept
Sea level, even in the absence of the effects of waves, winds, currents and tides, does not form a simple mathematical figure. The undisturbed surface of the ocean should be the equipotential surface of the gravitational field, and since the latter reflects density inhomogeneities inside the Earth, the same applies to equipotentials. Part of the geoid is the equipotential surface of the oceans, which coincides with the undisturbed mean sea level. Beneath the continents, the geoid is not directly accessible. Rather, it represents the level to which water will rise if narrow channels are made across the continents from ocean to ocean. The local direction of gravity is perpendicular to the surface of the geoid, and the angle between this direction and the normal to the ellipsoid is called the deviation from the vertical.
Deviations
It may seem that the geoid is a theoretical concept with little practical value, especially in relation to points on the land surface of continents, but this is not the case. The heights of points on the ground are determined by geodetic alignment, in which a tangent to the equipotential surface is set with a spirit level, and calibrated poles are aligned with a plumb line. Therefore, the differences in height are determined with respect to the equipotential and therefore very close to the geoid. Thus, the determination of 3 coordinates of a point on the continental surface by classical methods required the knowledge of 4 values: latitude, longitude, height above the Earth's geoid and deviation from the ellipsoid at this place. The vertical deviation played a large role, since its components in orthogonal directions introduced the same errors as in the astronomical determinations of latitude and longitude.
Although geodetic triangulation provided relative horizontal positions with high accuracy, the triangulation networks in each country or continent started from points with assumed astronomical positions. The only way to combine these networks into a global system was to calculate the deviations at all starting points. Modern methods of geodetic positioning have changed this approach, but the geoid remains an important concept with some practical utility.
Form definition
The geoid is essentially the equipotential surface of a real gravitational field. In the vicinity of a local excess of mass, which adds the potential ΔU to the normal potential of the Earth at the point, in order to maintain a constant potential, the surface must deform outwardly. The wave is given by the formula N= ΔU/g, where g is the local value of the acceleration of gravity. The effect of mass over the geoid complicates a simple picture. This can be solved in practice, but it is convenient to consider a point at sea level. The first problem is to determine N not in terms of ΔU, which is not measured, but in terms of the deviation of g from the normal value. The difference between local and theoretical gravity at the same latitude of an ellipsoidal Earth free of density changes is Δg. This anomaly occurs for two reasons. Firstly, due to the attraction of excess mass, the effect of which on gravity is determined by the negative radial derivative -∂(ΔU) / ∂r. Secondly, due to the effect of height N, since gravity is measured on the geoid, and the theoretical value refers to the ellipsoid. The vertical gradient g at sea level is -2g/a, where a is the radius of the Earth, so the height effect is given by (-2g/a) N = -2 ΔU/a. Thus, combining both expressions, Δg = -∂/∂r(ΔU) - 2ΔU/a.
Formally, the equation establishes a relationship between ΔU and the measurable value Δg, and after determining ΔU, the equation N= ΔU/g will give the height. However, since Δg and ΔU contain the effects of mass anomalies throughout an undefined region of the Earth, and not just under the station, the latter equation cannot be solved at one point without reference to others.
The problem of the relationship between N and Δg was solved by the British physicist and mathematician Sir George Gabriel Stokes in 1849. He obtained an integral equation for N containing the values of Δg as a function of their spherical distance from the station. Until the launch of satellites in 1957, the Stokes formula was the main method for determining the shape of the geoid, but its application presented great difficulties. The spherical distance function contained in the integrand converges very slowly, and when trying to calculate N at any point (even in countries where g has been measured on a large scale), uncertainty arises due to the presence of unexplored areas that may be at considerable distances from stations.
Satellite contribution
The advent of artificial satellites, whose orbits can be observed from Earth, completely revolutionized the calculation of the shape of the planet and its gravitational field. A few weeks after the launch of the first Soviet satellite in 1957, an ellipticity value was obtained that supplanted all previous ones. Since that time, scientists have repeatedly refined the geoid with observation programs from near-Earth orbit.
The first geodetic satellite was Lageos, launched by the United States on May 4, 1976, into an almost circular orbit at an altitude of about 6,000 km. It was an aluminum sphere with a diameter of 60 cm with 426 reflectors of laser beams.
The shape of the Earth was established through a combination of Lageos observations and surface measurements of gravity. Deviations of the geoid from the ellipsoid reach 100 m, and the most pronounced internal deformation is located south of India. There is no obvious direct correlation between continents and oceans, but there is a connection with some basic features of global tectonics.
Radar altimetry
The geoid of the Earth over the oceans coincides with the mean sea level, provided that there are no dynamic effects of the action of winds, tides and currents. Water reflects radar waves, so a satellite equipped with a radar altimeter can be used to measure the distance to the surface of the seas and oceans. The first such satellite was Seasat 1 launched by the United States on June 26, 1978. Based on the data obtained, a map was compiled. Deviations from the result of calculations made by the previous method do not exceed 1 m.