Earth model calculation using the results of the Struve geodetic arc

How measurements of meridian length allow us to determine the size of the Earth

Since the time of Eratosthenes, attempts to estimate the size and shape of the Earth have been made to realize that the shape of the Earth essentially corresponds to a mathematical model – an ellipsoid or spheroid, consisting of a semi-major axis a and a semi-minor axis b (Figure 1).

1.1
1.2

Figure 1. The Earth's ellipsoid.

Eratosthenes calculated that if we could consider the Earth's ellipsoid as an irregular circle of 360 degrees and measure the length of one degree in the circle, we would get the entire circumference of the Earth. However, the task is not so simple, because, as we can see from Fig. 1, the radius of curvature of the Earth's ellipsoid is variable, i.e. the Earth varies unevenly at the equator and in the North or South axes. Therefore, it is necessary to determine the angle of curvature α of the ellipsoid as close as possible to the real Earth's surface by ground and astronomical measurements, and once it is determined, we can also obtain the flattening of the Earth, which is the difference between these semi-axes divided by the angle of curvature

Also used in mathematics is the term "eccentricity", a positive real number that expresses the characteristics of inclination:

The dependence of meridional curvature can be expressed through eccentricity

The length of the fixed meridianS, or changedm = M(φ) dφfrom the equator to the latitudefapskaičiuojamas

Expression of the specified meridian segment

It should be appreciated that due to the shape of the Earth, the flatness is critically different between the Earth's axes and the equator. If we were to try to measure the distance along the meridian obtained at the same angle of change between the Earth's ellipsoid axes and the equator (Fig. 2), the distanceS1would be greater than the distanceS2.

 

 

 

 

 

 

 

 

 

Figure 2. Differences in the flatness of the Earth.

From the given example it is clear that if we could determine the distance on the Earth's meridian and find out how many degrees of latitude this distance covers on the Earth's ellipsoid, we would calculate the distance of 1 degree on the meridian, and multiplying it by 360, we would get the entire circumference of the Earth. The longer we determine the length of the meridian S, the closer we could apply the change in curvature to the flatness of the Earth (Fig. 3). Having determined the length of 1 degree of the meridian S, we can easily calculate the circumference of the Earth, and also the radius of the Earth

 

Figure 3. Meridian length in an ellipsoid.

 

 

 

 

 

 

 

 

 

 

Measuring the length of a meridian with a triangulation chain

The triangulation method, which was introduced in the 17th century, made it possible to measure large distances more accurately by eliminating irregularities in the earth's surface. A modified method developed by the Dutch scientist was used for measurements of the Struve geodetic arc.Vilebordas Fast from Rojenas(Willebrord Snel van Royen)triangulation method. Triangle segments were designed in the Struve geodetic arc circuit, the side (base) of one triangle of which was very accurately measured on the ground at the beginning and end of the segment, as shown by the red double line in Fig. 4.

Figure 4. Triangulation chain along the meridian

 

 

 

 

 

 

 

 

 

 

During the triangulation measurements, all angles of the triangle chain between points were measured in the area, and after applying the formulas of the trigonometric theorem of sines (Fig. 5), the lengths of each side were calculated. In order to ensure control of the calculations, precise measurements of the length of one side were again made in the area at the end of each triangle segment (Fig. 4).

 

 

 

 

 

 

 

Figure 5. Applications of the principles of the trigonometric theorem of sines.

In total, 10 sides (bases) were precisely measured in the Struve geodetic arc, and in order to determine the position of points in space and calculate the radius of curvature of the earth, astronomical observations were made at 13 points in the Struve geodetic arc, determining astronomical latitudes and directions to adjacent points - astronomical azimuths. Astronomical measurements were carried out using high-precision telescopes, observing the position of stars and recording the time of measurements, according to which the parameters of the position of stars were adjusted from astronomical reference books.

The measurements of the chain were not necessarily carried out exactly according to the meridian, because often the terrain and vegetation of the area did not provide such an opportunity. After processing the calculation results and calculating the astronomical latitudes of each point of the chain, these were mathematically projected onto the meridian line, thus fixing the established length of the meridian. The length of the Struvė geodetic arc is 2822 km, fixing the northernmost point of the arcThe birds(Norway) and the southernmost –Staro NekrasovkaThe chain points in Professor FGW Struve's report were projected into the so-calledTartu meridian– 26 degrees and 43 minutes East of Greenwich.

The first to practically apply the parameters of the Struve geodetic arc wasFriedrich Wilhelm Bessel(Friedrich Wilhelm Bessel), who calculated the parameters of the so-called Bessel ellipsoid: radius a, from the center of the Earth to the equator a = 6337397, Earth's oblateness α = 1 / 299.15. The Bessel ellipsoid served for more than 100 years as a reliable basis for the Earth model in many applicable geodetic coordinate systems.

Figure 6. Determination of the meridian segment AB on the ellipsoid.

 

 

 

 

 

 

 

 

 

 

When considering the spherical transformation of the latitudes and longitudes of the Struve geodetic arc, several aspects of mathematical analysis should be emphasized. Let us evaluate two points on the spherical surface -A, whose latitudeφ1is of durationλ1andB, whose latitudeφ2is of durationλ2(Fig. 6). The connecting spherical segment (from A to B) is AB, whose length iss12and which have azimuths at both endsa1ir a2. Once we have determined the sizesA, a1, ands12when solving a geodetic problem, we can determine the parameters of another point B and its azimuth in the north directiona2.

According to the given example, when solving a spherical trigonometry problem, when the north direction of the meridian is clear, we can calculate the triangle NAB by applying mathematical trigonometric (triangle formulas.where the sum of the angles of the triangles is always

Figure 7. Elements of the meridian projection.

 

 

 

 

 

 

 

 

 

 

Evaluating the change in curvature ρ on the surface of an ellipsoidF.W. Beselisderives the relationship between azimuthα, change in lengthds, and the change in latitudedφ (Figure 7).

Expression of the meridian segment:

Prepared by Dr. Saulius Urbanas.

Sources / References

Bessel, F. W. (2010) [1825]. Translated by Karney, C. F. F.; Deakin, R. E. „The calculation of longitude and latitude from geodesic measurements”. Astronomische Nachrichten. 331 (8): 852–861. arXiv:0908.1824

Karney, C. F. F. (2013). „Algorithms for geodesics”. Journal of Geodesy. 87 (1): 43–55. arXiv:1109.4448. Bibcode:2013 Journal of Geodesy, Volume 87.

Smith J. R: The Struve Geodetic Arc. International Institution for History of Surveying & Measurement. FIG, 2005.

Struve W. 1957. Arc of Meridian. Moscow

Struve, W.: The arc of the meridian of 2520′ between the Danube and the Arctic Sea, measured from 1816 to 1855 under the direction of K. Tenner, N. H. Selander, Ch. Hansteen, and W. Struve.

Struve, W.: Geodesy. On the junction of two measurements of degrees carried out in Russia, Universal Library of Sciences, Belles-Lettres and Arts. Sciences and arts. Following the British Library, Written in Geneva

https://en.wikipedia.org/wiki/Geodesics_on_an_ellipsoid, 2023.

This information was prepared within the framework of the 2014–2020 Interreg VA Latvia-Lithuania Cross-Border Cooperation Programme project LLI-477 "Creation of the International Tourist Route "Struve Geodetic Arc"" / STRUVE, which aims to strengthen the development of cognitive tourism, increase the number of visitors and extend the duration of their visits to the regions by providing a variety of tourism opportunities.

This information has been prepared with the financial support of the European Union. Total project funding: 850.5 thousand EUR (including ERDF funding – 723 thousand EUR).

The content of this information is the sole responsibility of the Aukštaitija Protected Areas Directorate and can under no circumstances be taken to reflect the official opinion of the European Union.