How to find a mass knowing acceleration and radius. How much earth weighs? How to calculate the mass of the planet? Earth mass and other planets

Earth is a unique planet of the solar system. It is not the smallest, but not the largest: ranks fifth in size. Among the planets of the earth group, it is the largest mass, diameter, density. The planet is located in outer space, and find out how much earth weighs is difficult. It is impossible to put it on the scales and weighed, so they say about her weight, summing up the mass of all the substances from which it consists. Approximately this indicator is 5.9 sextillion tons. To understand what kind of figure it is, you can simply write it to mathematically: 5 900 000 000 000 000 000 000. From this number of zeros, somehow ripples in the eyes.

History of attempts to determine the size of the planet

Scientists of all centuries and peoples tried to find an answer to the question of how much Earth weighs. In ancient times, people assumed that the planet is a flat plate, which is kept whales and a turtle. In some nations, instead of whales were elephants. In any case, different peoples of the world represented the planet flat and having their land.

At the time of the Middle Ages, the idea of \u200b\u200bform and weight has changed. The first who spoke about spherical form 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 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 are weights. This discovery made it possible to start a variety of studies. In physical theory, the weight is the strength of the body action on the support. Considering that the Earth has no support, it can be concluded that she has no weight, but there is a lot of mass, and big.

Earth weight

For the first time, the size of the planet tried Eratosthenes - an ancient Greek scientist. In different cities of Greece, he performed the shadows measurements, and after compared the data obtained. So he tried to calculate the volume of the planet. After him, the Italian Galilee tried to hold the calculations. It was he who opened the law of free gravity. The relay to determine how much the earth weighs, was adopted by I. Newton. Thanks to attempts to make measurements, he opened the law of gravity.

For the first time, determine how much the earth weighs, it was possible to the Scottish scientist N. Mekhelin. According to its calculations, the mass of the planet is 5.9 sextillion tons. Now this figure has increased. The differences in weight is associated with sedimentation on the surface of the planet of cosmic dust. Approximately thirty tons of dust remain on the planet every year, making it harder.

Earth weight

To find out how much earth weighs, you need to know the composition and weight of the substances from which the planet consists.

1. Mantle. The mass of this shell is approximately 4.05 x 10 24 kg.
2. Core. This shell weighs less mantle - only 1.94 x 10 24 kg.
3. Earth bark. This part is very thin and weighs only 0.027 x 10 24 kg.
4. Hydrosphere and atmosphere. These shells weigh 0.0015 x 10 24 and 0.0000051 x 10 24 kg, respectively.

Folding all these data, we get the weight of the Earth. However, in different sources, the mass of the planet is different. So how many weighs the planet earth in tons, and how many other planets weigh? The weight of the planet is 5.972 x 10 21 tons. Radius - 6370 kilometers.

Based on the principle of gravity, it is easily determined by the weight of the Earth. For this, it takes a thread, and a small cargo is suspended. Its location is defined for sure. Next place a ton of lead. There is an attraction between the two bodies, due to which the cargo defects aside for a slight 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 strength of attraction in relation to the weight and the strength of the attraction of small cargo to the large. The data obtained allow to carry out calculations of the mass of the Earth.

Earth mass and other planets

The Earth is the largest planet of the earth group. In relation to it, the mass of Mars is about 0.1 terrestrial weight, and Venus is 0.8. It is about 0.05 of the earth. Gas giants are many times larger than the Earth. If you compare the Jupiter and our planet, then the giant is more than 317 times, and Saturn is heavier 95 times, uranium - at 14. There are planets that weigh more land 500 times or more. These are huge gas bodies located outside our solar system.

The basis of the determination of the masses of the heavenly bodies is the law of global gravity expressed by F-Loi:
(1)
Where F. - the power of mutual attraction of the masses and, proportional to their work and inversely proportional to the square of the distance r. between their centers. In astronomy, often (but not always) can be neglected by the sizes of the heavenly bodies themselves compared with the distances separating their distances, distinguish their shapes from the exact sphere and then focus the heavenly bodies in which their entire weight is concentrated.

The ratio of the proportionality G \u003d Naz. or constant. It is found from a physical experiment with twisted scales, allowing to determine the force of gravitats. The interaction of the bodies of the known mass.

In case of free fall, bodies F.acting on the body is equal to the product of body weight to accelerate free fall g.. Acceleration g. may be determined, for example, in the period T. Vertical pendulum oscillations: where l. - The length of the pendulum. On the breadth of 45 o and at sea level g.\u003d 9.806 m / s 2.

Enterprise for the forces of earth attraction in F-Lo (1) leads to dependence , Where - the mass of the Earth, and is the radius of the globe. This way was determined by the mass of the Earth G. Determination of the mass of the Earth of the Yavl. The first link in the mass definitions chains dr. Heavenly bodies (sun, moon, planets, and then stars). The masses of these bodies are found, relying or on the 3rd law of Kepler (see), or on the rule: distances K.L. The masses from the total center of masses are inversely proportional to the masses themselves. This rule allows you to determine the mass of the moon. From the measurements of the exact coordinates of the planets and the Sun found that the Earth and the Moon with a period of one month move around the barcasecenter - the center of the mass system of the Earth - Moon. The distance of the center of the Earth from the Barycenter is 0,730 (it is located inside the globe). Cf. The distance of the center of the moon from the center of the Earth is 60.08. Hence the attitude of the distances of the centers of the Moon and the Earth from the Barcenter is 1/81.3. Because this attitude back is 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 the 3rd law of Kepler to the movement of the Earth (together with the moon) around the Sun and the movement of the moon around the Earth:
, (2)
Where but - large semi-axes orbits, T. - Periods (Star or Sideric) appeals. Neglecting compared to, we get a relationship, equal to 329390. Hence r, or ok. .

Similar to the mass of the planets with satellites determine the masses. The masses of the planets that do not have satellites are determined by indignation, which they put on the movement of the planets neighboring with them. The theory of indignant movement of the planets allowed to suspect the existence of the unknown planets of Neptune and Pluto, to find their masses, predict their position in the sky.

A lot of stars (besides the Sun) can be determined with a relatively high reliability only if it is ivl. Phys. A component of a visual-dual star (see), the distance to to-ro is known. The third law of Kepler in this case gives the amount of mass components (in units):
,
Where but"" - the best semi-axis (in the second arc) of the true orbit of the satellite around the main (usually brighter) stars, in this case, are considered motionless, R - The period of circulation in years, - systems (in the seconds of the arc). The value gives a large feather orbit in a. e. If you can measure the angular distances of the components from the total center of the masses, then their ratio will give the magnitude, the inverse ratio of the masses :. The resulting amount of the masses and their relationship allow us to get a lot of each star separately. If the double components have about the same gloss and similar spectra, then the mass of the masses gives the faithful assessment of the mass of each component and without additional. Definitions of their relationship.

For other types of double stars (elaborate-double and spectral-dual), there are a number of possibilities to approximately determine the masses of the stars or estimate their lower limit (i.e., the values \u200b\u200bless than which they cannot have their masses).

The set of data on the masses of the components about a hundred double stars of different types made it possible to detect important statistics. Dependence between their masses and luminosities (see). It makes it possible to estimate the masses of single stars according to them (in other words, according to their abs.). Abs. Star values M. Defined by F-le: M \u003d M. + 5 + 5 LG - A (R) , (3) where m. - Visible Star Value in the selected optical. Range (in a specific photometric. System, eg U, B. or V.; See), - Pararallax and A (R) - The magnitude of light in the same optical. The range in this direction is up to the distance.

If the parallax stars are not measured, then the approximate value of the ABS. Star values \u200b\u200bcan be determined by its spectrum. For this it is necessary that the spectrogram allows not only to learn the stars, but also to estimate the relative intensities of some couples spectrum. Lines sensitive to "ABS effects". In other words, you must first determine the class of star luminosity - belonging to one of the sequences in the chart spectrum-luminosity (see), and by the class of luminosity - its abs. magnitude. According to the ABS thus obtained. The magnitude can be found a lot of stars using the dependence of the mass-luminosity (this dependence is not obeyed only and).

Another method of assessing the mass of the star is associated with the measurement of gravitats. Red shift spectrum. lines in her field. In a spherical symmetric field, it is equivalent to Doppler red displacement, where - the mass of the star in the unit. Sun masses, R. - Radius star in units. Sun radius, and expressed in km / s. This ratio was verified by those white dwarfs, which are part of the double systems. For them, radii, masses and true v R.which are projections of orbital speed.

Invisible (dark) satellites found near some stars through observed oscillations of the star position associated with its movement near the total mass center (see), have a mass less than 0.02. They probably did not yavl. Self-driving bodies and more similar to the planet.

Of the mass definitions, the stars it turned out that they were concluded from about 0.03 to 60. The greatest number of stars have a mass from 0.3 to 3. Cf. Mass of stars in the nearest surroundings of the Sun, i.e. 10 33. The difference in the masses of the stars is much less than their difference in the luminosities (the latter can reach tens million). The radii stars are very different. This leads to a split difference for their cf. Densities: from to r / cm 3 (cf. Sun density 1.4 g / cm 3).

The law of Worldwide Newton allows you to measure one of the most important physical characteristics of the celestial body - its mass.

Mass can be determined:

a) from the measurements of gravity on the surface of this body (gravimetric method),

b) according to the third refined law of Kepler,

c) from the analysis of the observed perturbations produced by the celestial body in the movements of other celestial bodies.

1. The first method is applied on Earth.

Based on the law, acceleration G on the surface of the Earth:

where m is the mass of the Earth, and R is its radius.

g and R are measured on the ground surface. G \u003d const.

With the values \u200b\u200bof G, R, G received now, the mass of the Earth turns out:

m \u003d 5,976 .1027g \u003d 6.1024kg.

Knowing the mass and volume, you can find an average density. It is 5.5 g / cm3.

2. According to the third law of Kepler, 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 is known for its distance from the planet and the period of appealing around it.

where m, m, mc masses of the sun, planet and its satellite, t and tc- periods of planet appeals around the sun and satellite around the planet, but and aC- The distances of the planet from the Sun and the satellite from the planet, respectively.

From the equation follows

The ratio M / M for all planets is very large; The attitude of the M / MC, very little (except Earth and the Moon, Pluto and Charon) and they can be neglected.

The M / M ratio can be easily found from the equation.

For the case of land and the moon, you must first determine the mass of the moon. It is very difficult to do it. The task will be solved by analyzing the perturbations in the movement of the Earth, which causes the moon.

3. According to the exact definitions of visible Sun positions in its longitude, changes were discovered with a monthly period, called "lunar inequality". The presence of this fact in the visible movement of the Sun indicates that the center of the Earth describes a small ellipse within a month around the common center of the Mass-Luna mass center, located inside the Earth, at a distance of 4650 km. from the center of the earth.

The situation of the center of the mass of the Earth-Moon was also found from the observations of the Small Planet Eros in 1930-1931.

In the disturbances in the movements of the artificial satellites of the Earth, the ratio of masses of the moon and the land turned out to be 1/81.30.

In 1964, the International Astronomical Union took it as const.

From the Kepler equation, we get a mass for the sun \u003d 2.1033, which is 333,000 times the earthly.

The masses of the planets that do not have satellites are determined by the disturbances they cause in the movement of the Earth, Mars, asteroids, comet, on the disturbances produced by them on each other.

The mass of the Sun can be found from the condition that the land towards the Sun is manifested as the centripetal force holding the land at its orbit (the Earth's orbit for simplification we will consider the circle)

Here is the mass of the Earth, the average distance of the earth from the sun. Destinating the duration of the year in seconds through we. In this way

from where, substituting numeric values, we find the mass of the Sun:

The same formula can be applied to calculate the mass of any planet having a satellite. In this case, the average distance of the satellite from the planet, the time of its appeal 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 the month of the specified way, it is possible to determine the mass of the Earth.

The mass of the Earth can also be determined, equating the weight of any body to the burden of this body to the ground, minus the component, which manifests itself dynamically, informing this body participating in the daily rotation of the Earth, corresponding to the centripetal acceleration (§ 30). The need for this amendment disappears, if for such a calculation of the mass of the Earth, we will use the acceleration of gravity, which is observed on the poles of the Earth then, denoting the average radius of the Earth and through the mass of the Earth, we have:

where did the mass of the Earth

If the average density of the globe designate it, obviously, hence the average density of the globe is obtained equal

The average density of mineral rocks of the upper layers of the Earth is approximately known, the core of the globe must have a density significantly exceeding

The study of the issue of land density at various depths was undertaken by Legeandre and continued by many scientists. According to the conclusions of Gutenberg and Galka (1924) at various depths, approximately the following values \u200b\u200bof the density of the Earth are taken:

The pressure inside the globe, at great depth, apparently enormously. Many geophysicists believe that at the depth of pressure should reach an atmospheres, per square centimeter in the Earth's kernel, at a depth of 3,000 and more kilometers pressure, possibly reaches 1-2 million atmospheres.

As for the temperature and the depth of the globe, it is reliably that it is higher (lava temperature). In mines and drilling wells, the temperature rises an average of one degree to each assume that at a depth of near the temperature reaches 1500-2000 ° and then it remains constant.

Fig. 50. The relative sizes of the sun and the planets.

The full theory of the movement of the planets, set in heavenly mechanics, allows us to calculate the mass of the planet for observations of that effect that this planet has any other planet. At the beginning of the last century, the planet Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus were known. It was observed that the movement of uranium detects some "wrongness", which indicated that the Uranium is unlightened by the planet, affecting the movement of uranium. In 1845, the French scientist Leerier and independently of him, England Adam, examining the movement of uranium, calculated the mass and location of the planet, which no one had not previously watched. Only after that, the planet was found in the sky just in the place that was indicated by calculations; This planet was named neptune.

In 1914, Astronome Lovelle was similar to predicted the existence of another planet, which is even further from the Sun than Neptune. Only in 1930, this planet was found and named Pluto.

Basic planet information

(see scan)

The table contains basic information about nine large planets of the solar system. Fig. 50 illustrates the relative sizes of the sun and planets.

In addition to the listed large planets, about 1,300 is a very small planets, the so-called asteroids (or planet) their orbits are mainly located between the orbits of Mars and Jupiter.