Newton's law of universal gravitation makes it possible to measure one of the most important physical characteristics of a celestial body - its mass.
Mass can be determined:
a) from measurements of gravity on the surface of a given body (gravimetric method),
b) according to the third refined Kepler's law,
c) from an analysis of the observed perturbations produced by a celestial body in the movements of other celestial bodies.
1. The first method is used on Earth.
Based on the law of gravity, the acceleration g on the Earth's surface is:
where m is the mass of the Earth and R is its radius.
g and R are measured at the Earth's surface. G = const.
With the currently accepted values of g, R, G, the mass of the Earth is obtained:
m = 5.976.1027g = 6.1024kg.
Knowing the mass and volume, you can find the average density. It is equal to 5.5 g/cm3.
2. According to Kepler's third law, it is possible to determine the ratio between the mass of the planet and the mass of the Sun, if the planet has at least one satellite and its distance from the planet and the period of revolution around it are known.
where M, m, mc are the masses of the Sun, the planet and its satellite, T and tc are the periods of revolution of the planet around the Sun and the satellite around the planet, a and ace are the distances of the planet from the Sun and the satellite from the planet, respectively.
It follows from the equation
The M/m ratio for all planets is very high; the ratio m/mc is very small (except for the Earth and the Moon, Pluto and Charon) and can be neglected.
The M/m ratio can be easily found from the equation.
For the case of the Earth and the Moon, one must first determine the mass of the Moon. This is very difficult to do. The problem is solved by analyzing the perturbations in the motion of the Earth, which are caused by the Moon.
3. By exact determinations of the apparent positions of the Sun in its longitude, changes with a monthly period, called "lunar inequality", were discovered. The presence of this fact in the apparent motion of the Sun indicates that the center of the Earth describes a small ellipse during the month around the common center of mass "Earth - Moon", located inside the Earth, at a distance of 4650 km. from the center of the earth.
The position of the Earth-Moon center of mass was also found from observations of the minor planet Eros in 1930-1931.
According to disturbances in the movements of artificial Earth satellites, the ratio of the masses of the Moon and the Earth turned out to be 1/81.30.
In 1964, the International Astronomical Union adopted it as const.
From the Kepler equation, we obtain the mass for the Sun = 2.1033 g, which is 333,000 times greater than the earth's.
The masses of planets that do not have satellites are determined by the perturbations they cause in the motion of the Earth, Mars, asteroids, comets, by the perturbations they produce on each other.
The mass of the Sun can be found from the condition that the gravitation of the Earth towards the Sun manifests itself as a centripetal force holding the Earth in its orbit (for simplicity, we will consider the orbit of the Earth a circle)
Here is the mass of the Earth, the average distance of the Earth from the Sun. Denoting the duration of the year in seconds through we have. In this way
whence, substituting numerical values , we find the mass of the Sun:
The same formula can be applied to calculate the mass of any planet that has a satellite. In this case, the average distance of the satellite from the planet, the time of its revolution around the planet, the mass of the planet. In particular, by the distance of the Moon from the Earth and the number of seconds in a month in this way, it is possible to determine the mass of the Earth.
The mass of the Earth can also be determined by equating the weight of a body to the gravitation of this body to the Earth, minus that component of gravity, which manifests itself dynamically, informing this body, participating in the daily rotation of the Earth, the corresponding centripetal acceleration (§ 30). The need for this correction disappears if, for such a calculation of the mass of the Earth, we use the acceleration of gravity that is observed at the poles of the Earth. Then, denoting through the average radius of the Earth and through the mass of the Earth, we have:
where does the mass of the earth come from
If the average density of the globe is denoted by then, obviously, From here the average density of the globe turns out to be equal to
The average density of the mineral rocks of the upper layers of the Earth is approximately equal to
The study of the question of the density of the earth at various depths was undertaken by Legendre and continued by many scientists. According to the conclusions of Gutenberg and Gaalck (1924), at various depths, approximately the following values of the density of the Earth take place:
The pressure inside the globe, at great depths, seems to be enormous. Many geophysicists believe that already at a depth the pressure should reach atmospheres per square centimeter. In the core of the Earth, at a depth of about 3000 kilometers or more, the pressure may reach 1-2 million atmospheres.
As for the temperature at the depth of the globe, it is certain that it is higher (the temperature of the lava). In mines and boreholes, the temperature rises by an average of one degree per each. It is assumed that at a depth of about 1500-2000 ° and then remains constant.
Rice. 50. Relative sizes of the Sun and planets.
A complete theory of the motion of the planets, expounded in celestial mechanics, makes it possible to calculate the mass of a planet from observations of the influence that a given planet has on the motion of some other planet. At the beginning of the last century, the planets Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus were known. It was observed that the movement of Uranus exhibited some "irregularities" which indicated that there was an unobserved planet behind Uranus affecting the movement of Uranus. In 1845, the French scientist Le Verrier and, independently of him, the Englishman Adams, having studied the motion of Uranus, calculated the mass and location of the planet, which no one had yet observed. Only after that the planet was found in the sky just in the place that was indicated by the calculations; this planet was named Neptune.
In 1914, the astronomer Lovell similarly predicted the existence of another planet even further from the Sun than Neptune. Only in 1930 this planet was found and named Pluto.
Basic information about the major planets
(see scan)
The following table contains basic information about the nine major planets of the solar system. Rice. 50 illustrates the relative sizes of the sun and planets.
In addition to the listed large planets, about 1300 very small planets are known, the so-called asteroids (or planetoids). Their orbits are mainly located between the orbits of Mars and Jupiter.
Earth is a unique planet in the solar system. It is not the smallest, but not the largest either: it ranks fifth in size. Among the terrestrial planets, it is the largest in terms of mass, diameter, and density. The planet is located in outer space, and it is difficult to find out how much the Earth weighs. It cannot be put on a scale and weighed, so they talk about its weight by summing up the mass of all the substances of which it consists. Approximately this figure is equal to 5.9 sextillion tons. To understand what this figure is, you can simply write it down mathematically: 5,900,000,000,000,000,000,000. This number of zeros somehow dazzles your eyes.
History of attempts to determine the size of the planet
Scientists of all ages and peoples have tried to find an answer to the question of how much the Earth weighs. In ancient times, people assumed that the planet was a flat plate held by whales and a turtle. Some nations had elephants instead of whales. In any case, different peoples of the world represented the planet as flat and having its edge.
During the Middle Ages, ideas about shape and weight changed. The first to talk about a spherical view was J. Bruno, however, the Inquisition executed him for his beliefs. Another contribution to science, which shows the radius and mass of the Earth, was made by the traveler Magellan. It was he who suggested that the planet is round.
First discoveries
Earth is a physical body that has certain properties, among which there is weight. This discovery led to a variety of studies. According to physical theory, weight is the force of a body acting on a support. Given that the Earth has no support, we can conclude that it has no weight, but there is a mass, and a large one.
Earth weight
For the first time, Eratosthenes, an ancient Greek scientist, tried to determine the size of the planet. In different cities of Greece, he measured the shadow, and then compared the data obtained. Thus he tried to calculate the volume of the planet. After him, the Italian G. Galilei tried to make calculations. It was he who discovered the law of free gravity. The relay race to determine how much the Earth weighs was adopted by I. Newton. Through trying to take measurements, he discovered the law of gravity.
For the first time, the Scottish scientist N. Makelin managed to determine how much the Earth weighs. According to his calculations, the mass of the planet is 5.9 sextillion tons. Now this figure has increased. Differences in weight are due to the settling of cosmic dust on the surface of the planet. Approximately thirty tons of dust are left on the planet every year, making it heavier.
Mass of the Earth
To know exactly how much the Earth weighs, you need to know the composition and weight of the substances that make up the planet.
- Mantle. The mass of this shell is approximately 4.05 X 10 24 kg.
- Nucleus. This shell weighs less than the mantle - only 1.94 X 10 24 kg.
- Earth's crust. This part is very thin and weighs only 0.027 X 1024 kg.
- Hydrosphere and atmosphere. These shells weigh 0.0015 X 10 24 and 0.0000051 X 10 24 kg, respectively.
Adding all these data, we get the weight of the Earth. However, according to different sources, the mass of the planet is different. So how much does planet Earth weigh in tons, and how much do other planets weigh? The weight of the planet is 5.972 X 10 21 tons. The radius is 6370 kilometers.
Based on the principle of gravity, one can easily determine the weight of the Earth. To do this, a thread is taken, and a small load is hung on it. Its location is determined accurately. A ton of lead is placed nearby. An attraction arises between two bodies, due to which the load deviates to the side by an insignificant distance. However, even a deviation of 0.00003 mm makes it possible to calculate the mass of the planet. To do this, it is enough to measure the force of attraction in relation to the weight and the force of attraction of a small load to a large one. The data obtained allow us to calculate the mass of the Earth.
Mass of the Earth and other planets
Earth is the largest terrestrial planet. In relation to it, the mass of Mars is about 0.1 of the Earth's weight, and Venus is 0.8. is about 0.05 of the earth. Gas giants are many times larger than the Earth. If we compare Jupiter and our planet, then the giant is 317 times larger, and Saturn is 95 times heavier, Uranus is 14 times heavier. There are planets that weigh 500 times more than the Earth or more. These are huge gaseous bodies located outside our solar system.
The determination of the masses of celestial bodies is based on the law of universal gravitation, expressed by f-loy:(1)
where F- the force of mutual attraction of the masses and , proportional to their product and inversely proportional to the square of the distance r between their centers. In astronomy, one can often (but not always) neglect the dimensions of the celestial bodies themselves in comparison with the distances separating them, the difference between their shape and the exact sphere, and liken the celestial bodies to material points, in which all their mass is concentrated.
Proportionality factor G = nam. or constant gravity. It is found from a physical experiment with torsion balances, which allow determining the strength of gravity. interactions of bodies of known mass.
In the case of free falling bodies, the force F, acting on the body, is equal to the product of the body mass and the acceleration of free fall g. Acceleration g can be determined, for example, by the period T oscillations of the vertical pendulum: , where l is the length of the pendulum. At latitude 45 o and at sea level g\u003d 9.806 m / s 2.
Substitution of the expression for the forces of gravity in f-lu (1) leads to the dependence 
, where is the mass of the Earth, and is the radius of the globe. In this way, the mass of the Earth was determined 
d. Determination of the mass of the Earth yavl. the first link in the chain of determining the masses of other celestial bodies (the sun, moon, planets, and then stars). The masses of these bodies are found, relying either on Kepler's 3rd law (see), or on the rule: distances to. masses from the common center of mass are inversely proportional to the masses themselves. This rule allows you to determine the mass of the moon. From measurements of the exact coordinates of the planets and the Sun, it was found that the Earth and the Moon, with a period of one month, move around the barycenter - the center of mass of the Earth-Moon system. The distance of the center of the Earth from the barycenter is 0.730 (it is located inside the globe). Wed the distance of the center of the moon from the center of the earth is 60.08 . Hence the ratio of the distances of the centers of the Moon and the Earth from the barycenter is 1/81.3. Since this ratio is inverse to the ratio of the masses of the Earth and the Moon, the mass of the Moon
G.
The mass of the Sun can be determined by applying Kepler's 3rd law to the motion of the Earth (together with the Moon) around the Sun and the motion of the Moon around the Earth: 
, (2)
where a- semi-major axes of the orbits, T- periods (stellar or sidereal) circulation. Neglecting in comparison with , we obtain the ratio equal to 329390. Hence 
g or ok. .
In a similar way, the masses of planets with satellites are determined. The masses of planets that do not have satellites are determined by the perturbations they have on the motion of their neighboring planets. The theory of the perturbed motion of the planets made it possible to suspect the existence of the then unknown planets Neptune and Pluto, to find their masses, and to predict their position in the sky.
The mass of a star (other than the Sun) can be determined with relatively high reliability only if it is yavl. physical component of a visual double star (see), the distance to which is known. Kepler's third law in this case gives the sum of the masses of the components (in units): 
,
where a"" - the semi-major axis (in arc seconds) of the true orbit of the satellite around the main (usually brighter) star, which in this case is considered fixed, R- period of revolution in years, - systems (in seconds of arc). The value gives the semi-major axis of the orbit in a. e. If it is possible to measure the angular distances of the components from the common center of mass, then their ratio will give the reciprocal of the mass ratio: . The found sum of the masses and their ratio allow us to obtain the mass of each star separately. If the binary components have approximately the same brightness and similar spectra, then the half-sum of the masses gives a correct estimate of the mass of each component and without additional. determining their relationship.
For other types of binary stars (eclipsing binaries and spectroscopic binaries) there are a number of possibilities to approximately determine the masses of stars or to estimate their lower limit (i.e., values below which their masses cannot be).
The totality of data on the masses of the components of about a hundred binary stars of various types made it possible to discover an important statistical the relationship between their masses and luminosities (see ). It makes it possible to estimate the masses of single stars from their (in other words, from their abs.). Abs. magnitudes M determined by f-le: M=m+ 5 + 5 lg - A(r), (3) where m- apparent stellar magnitude in the selected optical. range (in a certain photometric system, e.g. U, V or V; see ), - parallax and A(r)- the amount of light in the same optical. range in a given direction up to a distance of .
If the parallax of the star is not measured, then the approximate value of abs. stellar magnitude can be determined by its spectrum. For this it is necessary that the spectrogram should not only allow one to recognize the stars, but also to estimate the relative intensities of certain pairs of spectra. lines sensitive to the "abs. magnitude effect". In other words, first you need to determine the luminosity class of the star - belonging to one of the sequences on the spectrum-luminosity diagram (see), and according to the luminosity class - its abs. size. According to the thus obtained abs. value, you can find the mass of the star using the mass-luminosity dependence (only and do not obey this dependence).
Another method for estimating the mass of a star is related to the measurement of gravity. redshift spectrum. lines in its gravitational field. In a spherically symmetric gravitational field, it is equivalent to the Doppler redshift , where is the mass of the star in units. sun mass, R- radius of the star in units. radius of the Sun, and expressed in km/s. This relationship has been verified for those white dwarfs that are part of binary systems. For them, the radii, masses and true vr, which are projections of the orbital velocity.
Invisible (dark) satellites, discovered near certain stars from the observed fluctuations in the position of the star associated with its movement around a common center of mass (see ), have masses less than 0.02. They are probably not yavl. self-luminous bodies and are more like planets.
From the definitions of the masses of stars, it turned out that they are approximately in the range from 0.03 to 60. The largest number of stars have masses from 0.3 to 3. Wed mass of stars in the immediate vicinity of the Sun, i.e. 1033. The difference in the masses of the stars turns out to be much smaller than their difference in luminosities (the latter can reach tens of millions). The radii of the stars also differ greatly. This leads to a striking difference between their cf. densities: from to g/cm 3 (compare the density of the Sun 1.4 g/cm 3).
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