the earth is spherical

Image: Harrison [Hagan] Schmitt
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By FELIPE APL COSTA

From space, Earth looks like a blue marble.

Viewed from space, the Earth appears to be shaped like a perfect sphere. The same impression we have in front of the Sun and the Moon, as well as in front of images of the other planets of the Solar System or even of some of its satellites.

Depending on the distance, the image of Earth makes us think of a blue marble[1] – see the image that accompanies this article. The predominance of the bluish coloration has to do with the fact that the oceans cover most of the earth's surface (~71%).[2]

Shape and size of the Earth.

Speculation about the shape of the planet is old. The Greeks, for example, based on observations of the Earth's shadow on the Moon during eclipses, already assumed that the planet was a gigantic sphere. And the most impressive: the Greeks were able to calculate the dimensions of such a sphere.

This is what Eratosthenes of Cyrene (276-194 BC) did [3].

The Greek philosopher and astronomer developed a method of calculation with which he obtained a very accurate estimate for the circumference of the Earth – 250 stadia, or 46.250 km [4].

Here is Singh's comment (2006, p. 20-1):

“In the library [of Alexandria] Eratosthenes learned of the existence of a well with remarkable properties, located near the city of Siena, in southern Egypt, near present-day Aswan. Every year, at noon on June 21st, the day of the summer solstice, the sun shone directly into the well and illuminated everything to the bottom. Eratosthenes realized that, on that particular day, the sun must be directly overhead, something that never happened in Alexandria, which was several hundred kilometers north of Siena. Today we know that Siena is close to the Tropic of Cancer, the northernmost latitude where the Sun can appear right at the zenith.”

Aware that the Earth's curvature was the reason the Sun didn't shine the same above Syena and Alexandria at the same time, Eratosthenes wondered if he couldn't use this to measure the Earth's circumference. He didn't think of the problem in the same way as we would, since his interpretation of geometry and his annotation were different, but here is a modern explanation of his approach. [Consider] how the parallel rays of light from the Sun reached the Earth at noon on June 21st. At the same moment that sunlight was sinking vertically into the bottom of the well at Syena, Eratosthenes stuck a stick vertically into the ground at Alexandria and measured the angle between the stick and the sun's rays. And what's crucial to the problem is that this angle equals the angle between two radial lines drawn from Alexandria and Siena to the center of the Earth. He measured the angle to be 7,2°.

Now imagine someone in Siena who decides to walk a straight line to Alexandria, and then keeps walking until they circle the world and return to Siena. When it circles the Earth completely, it would describe a complete circle covering 360°. Thus, if the angle between Siena and Alexandria is only 7,2°, then the distance between Siena and Alexandria represents 7,2/360 or 1/50 of the circumference of the Earth. The rest of the calculation is simple. Eratosthenes measured the distance between the two cities, which turned out to be 5.000 stadia. If this represents 1/50 of the Earth's circumference then the total circumference must be 250.000 stadia.

The same method would later be used to calculate other astronomical quantities, such as Earth-Sun and Earth-Moon distances. [5].

2 – “I saw the Earth! She is so beautiful."

The first human being to see Earth as a gigantic sphere was Soviet cosmonaut Yuri [Alekseyevich] Gagarin (1934-1968).

On 12/4/1961, aboard the spacecraft Vostok 1 and orbiting Earth at an average altitude of 322 km [6], Gagarin made a single loop around the planet.

The flight lasted only 108 minutes [7], but it was enough to turn the episode into an epic and historic feat.

While in orbit, in addition to delivering an 'official speech' addressed to humanity as a whole, Gagarin told his Soviet colleagues: “I see the Earth! She is so beautiful" [8].

3 – Oblate spheroid.

It turns out that the Earth is not perfectly spherical. Rigorous measurements indicate that the equatorial radius (6.378 km) is slightly larger than the polar radius (6.357 km) [9]. The terrestrial globe is then said to be a spheroid – ie, an approximately spherical object.

This deviation was perhaps a surprise to the Greeks, but not to the person who predicted and explained it: the English mathematician and naturalist Isaac Newton (1643-1727). [10].

In the words of Nussenzveig (2013, p. 249):

Newton calculated the effect of the Earth's rotation on its shape: in the absence of rotation, that is, only under the effect of gravity, the planets should have a spherical shape; however, the 'centrifugal forces' produced by rotation lead to a flattening at the poles and widening at the equator, leading to an oblate spheroid shape […] [11].

According to Newton's calculation, the Earth's polar diameter must be to the equatorial as 229/230, leading to an ellipticity of 1/230 [12].

4- Gravity and the shape of celestial bodies.

But, after all, why are the Earth, the Sun, the Moon and so many other celestial bodies spherical?

The answer has to do with the following: each and every astronomical object whose diameter is above a certain minimum value tends to become spherical for the simple reason that its shape starts to be molded by the gravity.

In the words of Luminet (1996, p. 53-4):

“Earth is effectively almost spherical because it is an astronomical object and, as such, its shape is governed by gravitation. In very general terms, all forms in the universe are governed by the four fundamental forces. Among these fundamental forms, there are two nuclear interactions that govern the structure of atomic nuclei – although this is not our purpose today – electromagnetism and gravity.

A good example of fairly massive bodies, but not so massive as to prevent electromagnetic forces and gravitational forces from acting at the same time, is that of asteroids and cometary nuclei. These objects can be between a few kilometers and a few hundred kilometers in diameter and have completely bizarre shapes, as varied as those of the pebbles we find on a beach: they do not have a spherical shape, because they are not sculpted by gravitation. In fact, it can be demonstrated that gravitation only becomes the dominant organizing force from bodies that have diameters of the order of 500 kilometers. It is the reason why all bodies in the solar system, over 500 kilometers in diameter – that is to say, all planets and most satellites of planets – have spherical shapes. Why? Because it is the very nature of gravitation that imposes it. The force of gravitation attracts each material particle of a body to what we call the center of mass (or center of gravity) of the body. [13]. It acts the same way in all directions, with an intensity that depends only on the mass of the particles and their distance from the center. So, if a body is homogeneous, gravitation inevitably 'sculpts' it into a spherical shape. This goes for the planets and a fortiori for stars, which are much more massive.”

Tail

In summary: (1) The Earth is spherical because it is an astronomical object large enough (> 500 km in diameter) to the point that its shape is governed by gravity. By becoming the dominant force, gravity tends to make celestial bodies assume a spherical shape. (2) But the Earth is not a perfect sphere. The deviation (imperceptible in a photograph – see the figure that accompanies this article) is the result of the planet's rotation. Generated by such movement, the centrifugal force tends to make the accumulation of matter a little greater along the equatorial axis of the planet.

*Felipe APL Costa is a biologist. Author, among others, of books, The Flying Evolutionist & Other Inventors of Modern Biology.

This article was extracted and adapted from the book The power of knowledge & other essays: An invitation to science (in press).

References


Boorstin, DJ. 1989 [1983]. the discoverers. RJ, Civilization.

Boyer, CB & Merzbach, UC. 2012 [2011]. history of mathematics, 3rd ed. SP, Blucher.

Comins, NF & Kaufmann, WJ, III. 2010 [2008]. discovering the universe, 8th ed. Porto Alegre, Bookman.

Christie, T. 2015. Calendrical confusion or just when did Newton die? The Renaissance Mathematicus, on 20/3/2015. [The author's blog is here.]

Garrison, T. 2010 [2006]. fundamentals of oceanography, 4th ed. SP, Cengage.

Luzum, B & more 11. 2011. The IAU 2009 system of astronomical constants: the report of the IAU working group on numerical standards for Fundamental Astronomy. Celestial Mechanics and Dynamical Astronomy 100: 293-304.

Nussenzveig, HM. 2013. Basic physics course, v. 1: Mechanics, 5rd ed. SP, Blucher.

Ronan, CA. 1987 [1983]. Illustrated history of science, vol. 1: From the origins to Greece. RJ, J Zahar.

Sagan, C. 1996 [1994]. pale blue dot. SP, Companhia das Letras.

Singh, S. 2006 [2004]. Big Bang. RJ, Record.

Stephenson FR; Morrison LV & Hohenkerk CY. 2016. Measurement of the Earth's rotation: 720 BC to AD 2015. Proceedings of the Royal Society A 472: 20160404 (http://dx.doi.org/10.1098/rspa.2016.0404).

Notes


[1] Ball or Blue Marble (eng., The Blue Marble) was how one of the first color images of Earth became known. Dated 7/12/1972, the photograph was taken by American geologist and astronaut Harrison [Hagan] Schmitt (born 1935). (There is an earlier photographic image, dated 1967. But it was taken by a satellite and is relatively little known – see here.) Schmitt was one of three crew on Apollo 17 (7-19/12/1972), the last manned mission to land on the Moon.

[2] The rest of the earth's surface (29%) exhibits other colors, especially greenish tones (closed forests), brownish tones (deserts, deforested areas or sparse vegetation) or whitish tones (polar caps and mountain tops, threatened today by an accelerated melting process). A combination of physical and chemical properties gives ocean water its bluish tint – see Garrison (2010).

[3] On the historical relevance of Eratosthenes' work, see Ronan (1987) and Boyer & Merzbach (2012).

[4] In Ancient Greece, stadium was the standard distance (185 m) at which races were held. The result obtained by Eratosthenes (46.250 km) is a slight overestimation of the value adopted today for the equatorial circumference of the planet. Let's see. The circumference length (C) measures 2πr, where π is a constant and r it's the ray. Making π = 3,14 and r = 6,378x106 m (see note 8), we get C = 4,0054x107 m (or 40.054 km), equivalent to 87% of the value obtained by the Greek philosopher.

[5] Like the Greeks, European navigators who arrived in the New World, such as Christopher Columbus (1451-1506) and Pedro Álvares Cabral (1467-1520), were aware that we live on a spherical planet. In the words of Boorstin (1989, p. 214): “By this time [1484], educated Europeans no longer had any doubts about the sphericity of the planet”. The disagreement resided in the value of the dimensions. The Earth globe model adopted by Columbus, for example, was significantly smaller than predicted by Eratosthenes' calculations. Which is why his trip to the New World took longer than expected.

[6] Orbiting so close to the planet, Gagarin did not come to see Earth as a blue marble. When Schmitt took his famous photograph (see note 1), Apollo 17 was about 45 km away from Earth. As we move further away, the image of the planet changes and evokes other analogies and metaphors. In early February 1990, for example, the Voyager I space probe (launched on 5/9/1977) was about 6 billion km from Earth. And it kept moving away out of the Solar System. At this distance, our planet is practically imperceptible in a photographic image – it becomes a speck against a background sprinkled with countless other specks or a pale blue dot. pale blue dot), to use the literary expression adopted by Sagan (1996).

[7] For a real-time recreation of Gagarin's flight, including images and original audio clips, see the film first orbit (2011), by Christopher Riley.

[8] For details, see here e here.

[9] Equatorial radius: ~6,37814 x 106 m (Luzum et al. 2011). Earth's current rotational speed is 1.670 km/h. In 24 hours, therefore, a fixed point on the equator describes a circumference of 40.080 km in length (= 24 h x 1.670 km/h). It is noteworthy that as the rotation decreases, the length of the day increases – for details, see Comins & Kaufmann (2010). The speed of rotation was once greater, which implies that the length of the Earth's day was once shorter than it is today. It is estimated that the day is gaining 1,8 ms (thousandths of a second) every century (Stephenson et al. 2016).

[10] For a discussion of Newton's birth and death years, see Christie (2015).

[11] Oblate spheroid is one whose equatorial axis is greater than the polar axis. When the polar axis is longer, the spheroid is said to be prolate.

[12] Also Nussenzveig (2013, p. 249): “The most recent experimental determinations give an ellipticity of ≈1/297.”

[13] For technical details, see Comins & Kaufmann (2010) and Nussenzveig (2013).

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