How to find the mass of a planet formula. Masses of celestial bodies (methods of determination). Mass of the Earth and other planets
(1)
Where F- the force of mutual attraction of masses and, proportional to their product and inversely proportional to the square of the distance r between their centers. In astronomy, it is often (but not always) possible to neglect the size of the celestial bodies themselves in comparison with the distances separating them, the difference in their shape from an exact sphere, and to liken celestial bodies to material points in which all their mass is concentrated.
Proportionality factor G = called or the constant of gravity. It is found from a physical experiment with torsion balances, which make it possible to determine the force 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 gravity g. Acceleration g can be determined, for example, by period T oscillations of a vertical pendulum: , where l- length of the pendulum. At latitude 45 o and at sea level g= 9.806 m/s 2 .
Substituting the expression for the forces of gravity into formula (1) leads to the dependence 
, where is the mass of the Earth, and is the radius of the globe. This is how the mass of the Earth was determined 
g. Determination of the mass of the Earth. the first link in the chain of determining the masses of other celestial bodies (Sun, Moon, planets, and then stars). The masses of these bodies are found based either on Kepler's 3rd law (see), or on the rule: distances of k.-l. masses from the general 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 the inverse of 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 (along with the Moon) around the Sun and the motion of the Moon around the Earth: 
, (2)
Where A- semimajor axes of orbits, T- periods (stellar or sidereal) of revolution. Neglecting in comparison with , we obtain a ratio equal to 329390. Hence 
g, or approx. .
The masses of planets with satellites are determined in a similar way. The masses of planets that do not have satellites are determined by the disturbances they exert on the motion of their neighboring planets. The theory of perturbed planetary motion 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 (besides the Sun) can be determined with relatively high reliability only if it is physical component of a visual double star (see), the distance to the cut is known. Kepler's third law in this case gives the sum of the masses of the components (in units): 
,
Where A"" is the semimajor axis (in arcseconds) of the true orbit of the satellite around the main (usually brighter) star, which in this case is considered stationary, R- period of revolution in years, - system (in arcseconds). The value gives the semimajor 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 masses and their ratio make it possible to obtain the mass of each star separately. If the components of a binary have approximately the same brightness and similar spectra, then the half-sum of masses gives a correct estimate of the mass of each component without addition. determining their relationship.
For other types of double stars (eclipsing binaries and spectroscopic binaries), there are a number of possibilities to approximately determine the masses of stars or estimate their lower limit (i.e., the values below which their masses cannot be).
The totality of data on the masses of the components of approximately one hundred binary stars of different types made it possible to discover important statistical data. the relationship between their masses and luminosities (see). It makes it possible to estimate the masses of single stars by their (in other words, by their absolute values). Abs. magnitudes M are determined by the following formula: M = m+ 5 + 5 lg - A(r), (3) where m- apparent magnitude in the selected optical lens. range (in a certain photometric system, e.g. U, V or V; see ), - parallax and A(r)- the magnitude of light in the same optical range in a given direction to a distance.
If the parallax of the star is not measured, then the approximate value of abs. stellar magnitude can be determined by its spectrum. To do this, it is necessary that the spectrogram allows not only to recognize the stars, but also to estimate the relative intensities of certain pairs of the spectrum. lines sensitive to the "absolute magnitude effect". In other words, you first need to determine the luminosity class of a star - whether it belongs to one of the sequences on the spectrum-luminosity diagram (see), and by luminosity class - its absolute value. size. According to the abs. obtained in this way. magnitude, you can find the mass of the star using the mass-luminosity relationship (only and do not obey this relationship).
Another method for estimating the mass of a star involves measuring 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. mass of the Sun, R- radius of the star in units. radius of the Sun, and is expressed in km/s. This relationship was verified using those white dwarfs that are part of binary systems. For them the radii, masses and true v r, which are projections of orbital velocity.
Invisible (dark) satellites, discovered near certain stars from observed fluctuations in the position of the star associated with its movement around the common center of mass (see), have masses less than 0.02. They probably didn't show up. self-luminous bodies and are more like planets.
From determinations of the masses of stars, it turned out that they range from approximately 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. 10 33 g. The difference in the masses of stars turns out to be much smaller than their difference in luminosity (the latter can reach tens of millions). The radii of stars are also very different. This leads to a striking difference between them. densities: from to g/cm 3 (cf. solar density 1.4 g/cm 3).
The mass of the Sun can be found from the condition that the Earth’s gravity towards the Sun manifests itself as a centripetal force that holds the Earth in its orbit (for simplicity, we will consider the Earth’s orbit to be a circle)
Here is the mass of the Earth, the average distance of the Earth from the Sun. Denoting the length of the year in seconds through we have. Thus
from where, 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, the mass of the Earth can be determined using the indicated method.
The mass of the Earth can also be determined by equating the weight of a body to the gravitation of this body towards the Earth, minus that component of gravity that manifests itself dynamically, imparting to a given body participating in the daily rotation of the Earth a 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 by the average radius of the Earth and by the mass of the Earth, we have:
where does the earth's mass come from?
If the average density of the globe is denoted by then, obviously, Hence the average density of the globe is equal to
The average density of mineral rocks in the upper layers of the Earth is approximately Therefore, the core of the globe must have a density significantly exceeding
The study 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 Haalck (1924), approximately the following values of the Earth's density occur at various depths:
The pressure inside the globe, at great depths, is apparently enormous. Many geophysicists believe that already at depth the pressure should reach atmospheres per square centimeter. In the Earth's core, at a depth of about 3000 kilometers or more, the pressure may reach 1-2 million atmospheres.
As for the temperature in the depths of the globe, it is certain that it is higher (the temperature of lava). In mines and boreholes, the temperature rises on average by one degree for every one. It is assumed that at a depth of about 1500-2000 ° and then remains constant.
Rice. 50. Relative sizes of the Sun and planets.
The complete theory of planetary motion, set forth 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, and Uranus were known. It was observed that the motion of Uranus exhibited some "irregularities" which indicated that there was an unobserved planet behind Uranus influencing the motion of Uranus. In 1845, the French scientist Le Verrier and, independently of him, the Englishman Adams, having studied the movement of Uranus, calculated the mass and location of the planet, which no one had yet observed. Only after this the planet was found in the sky exactly in the place indicated by the calculations; this planet was named Neptune.
In 1914, 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 table below 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 1,300 very small planets, so-called asteroids (or planetoids), are known. Their orbits are mainly located between the orbits of Mars and Jupiter.
Newton's law of universal gravitation allows us 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 Kepler’s third refined law,
c) from the analysis of observed disturbances 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 relationship 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 ac- the distances of the planet from the Sun and the satellite from the planet, respectively.
From the equation it follows
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, you must first determine the mass of the Moon. This is very difficult to do. The problem is solved by analyzing the disturbances in the movement of the Earth that the Moon causes.
3. By precise 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 small planet Eros in 1930 - 1931.
Based on 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 for the Sun a mass = 2.1033g, which is 333,000 times greater than that of the Earth.
The masses of planets that do not have satellites are determined by the disturbances they cause in the movement of the Earth, Mars, asteroids, comets, and by the disturbances they produce on each other.
