Newton's law of gravitation

Until the middle of the 17th century, there were two completely different and seemingly unrelated doctrines of motion side by side: terrestrial mechanics dealing with the motion of ordinary bodies and celestial mechanics studying the motion of planets and stars.
I. Newton followed Galileo's kinematics and built the dynamics of motion of bodies summarized in three well-known Newton's laws (see §1.6 "Four- dimensional spacetime and special theory of relativity ", passage " Newton's classical mechanics ") . The resulting classical mechanics was (and is still to this day) able to explain all the movements of bodies that we encounter in everyday life.
Newton's predecessor, J. Kepler, summarized a large number of astronomical observations to date and traced the general laws governing the motion of planets in the solar system :

These empirical Kepler's laws served I. Newton as a starting point for establishing an even more fundamental law that governs not only the planets, but all bodies "in heaven and on Earth."
Above all, Newton realized that the motion of the planets did not conform to the law of inertia. The planets move in curved orbits (ellipses) around the Sun, so they must be subjected to a force directed at the Sun - an attractive force emanating from the Sun. Kepler's third law applied to the special case of circular orbit says that the squares of periods 4 p ² . r ² / v ² are proportional to the squares of the radius r . Then centripetal acceleration v²/r must be proportional to 1/r² . A similar lead can be shown for elliptical movement. Due to Newton's second law on force causing the planet mass m such centripetal acceleration must be therefore proportional to the m / r ² . According to the law of action and reaction, however, the planet must act with a force of the same magnitude as the Sun acts on the planet, and this force will be proportional to the mass M of the Sun. The mutual attractive force between the planet and the Sun will therefore be proportional to m.M / r ² .

The law of general gravitation
By analyzing Kepler's laws, Newton found that the motion of planets in the solar system can be easily explained by the hypothesis that each two bodies are attracted to each other by a force directly proportional to the mass m_l and m₂of each and indirectly proportional to the square of the distance r among them - Newton's law of gravitation :

where r ° is a unit vector indicating the direction from the body m _l to the body m ₂ . The coefficient of proportionality G - the gravitational constant - must be determined empirically (from observation or experiment), see below.
In addition to deriving the form of the law of attraction between celestial bodies, Newton showed that this force has the same nature as the gravity of the earth, forcing all free bodies to fall to the ground with acceleration. Newton, by comparing the acceleration of the Moon's motion as it orbits at the appropriate distance around earth and by the acceleration of free-moving bodies at the earth's surface, he found that the magnitudes of these accelerations correspond to the law of inverted squares and therefore agree with the law (1.1), provided that both the fall of bodies and the orbit of the Moon around the Earth are conditioned by the Earth's gravity.
The finding that the force that compels the orbit of the planet around the sun or moon around the planet is the same force that causes the fall of bodies to the ground and called weight , or gravity , united on a common basis before completely different phenomena and areas: mechanics, the force of gravity and "heavenly "mechanics. Equation (1.1) is therefore called Newton's law of general gravity .
Gravity - an irremovable all-pervading force
The electrical action between charged bodies (which can be attractive or repulsive according to the signs of charges) depends on the material environment and can in principle be shielded (even by an alternating electromagnetic field using a Faraday cage ) . Gravity, which is always attractive, however, is an all-pervading force that cannot be removed in any way. Gravity cannot be "blocked the way" by anything, it cannot be shielded . If we inserted some massive material between two gravitational bodies, we would only achieve an amplification of the total gravitational action: the gravitational force of the inserted material would add up in vector to the original gravity.
Universal action of gravity
Since the attractive force acting on each body is proportional to the mass of that body, Newton's law is also consistent with Galileo's law of free fall, according to which all bodies fall to the ground with equal acceleration regardless of their mass and composition . 2.2) .
According to 2.Newton's law of classical mechanics - the law of force and acceleration - the force F needed to set a body in motion is proportional to its mass m and acceleration a : F = m . a. If this force is gravity, then for a test body of mass m moving under the influence of gravity of the central body of mass M (eg Earth) at a distance r , we can write Newton's law of gravitation on the other side of the equation: m . a = G. Mm / r ² . On the left and right side of the equation there is a mass m of the test body, which can be shortened , so the acceleration no longer depends on the mass of the observed body - it depends only on the mass of the central gravitational body and the mutual distance: a = G. M / r ² . As a result, the gravitational force gives all bodies the same gravitational acceleration , which explains Galileo 's law of free fall. Although a heavier body is subjected to a greater gravitational force, which at the same time must do more "work" to accelerate it - this heavier body puts more "resistance" to acceleration.
Further physical knowledge has shown that this property of the same - universal - action of gravity applies not only to ordinary matter , but also to their building particles such as electrons, protons, neutrons, as well as to electromagnetic fields and waves such as light and its quantum. photons (undoubtedly also applies to other particles and antiparticles ) . It was this fact that proved to be the most important property of gravity and the key to its new understanding withingeneral theory of relativity (§2.1-2.4 of Chapter 2, especially §2.2 " Universality - a basic property and the key to understanding the nature of gravity ") .
Gravitational bodies

Fig.1.0. Newton's law of general gravity.
Above: Basic situation of gravitational action between two (idealized) point bodies with masses m ₁ and m ₂ .
Middle: Newton's law of gravitation also applies to the gravitational action between spherically symmetric bodies of finite dimensions.
Bottom: A spherically symmetric hollow body (spherical layer or shell) has zero gravitational force inside the cavity, outside the body the gravitational force is given by the standard Newton's law.

Fig. 1.0 shows three situations in which this law of gravity applies exactly . The basic theoretical situation according to the upper part of the figure consists in the gravitational action between two " point " bodies , in practice bodies whose dimensions are negligibly small in comparison with their mutual distance. However, Newton's law of gravitation also applies exactly to the gravitational action between bodies of finite dimensions with a spherically symmetric distribution of mass density (middle part of the figure). And not only in the outer space around the body, but also inside the spherically symmetric body - the gravitational force F (r) at a distance r from the center is given by the amount of mass m (r) = 4 p ₀n ^{^r} r ( r ). r ² dr contained in an imaginary sphere of radius r : F (r) = Gm (r) / r ² ; matter in the outer shells is not applied, its gravity acting from different directions is canceled (this situation is also discussed in §2.4, passage " Time dilation inside gravitational bodies ") . In the lower part of Fig. 1.0 there is a curious theoretical case of a spherically symmetric hollow body - a spherical layer or shell (it does not occur in nature) . The test specimens located inside the cavity are not subjected to any gravitational force (the gravitational field inside has zero intensity , there is a constant gravitational potential, gravitational forces from different directions are canceled) , but outside the body the gravitational force is exactly given by Newton's law.
In all other configurations of asymmetric inhomogeneous mass density distribution in bodies of finite dimensions, the law of inverted squares applies only approximately , these bodies cannot be completely replaced by their center of gravity (due to the quadratic nonlinearity of the law of gravitation) . In the general case of a source of volume V , in which the mass distribution is given by a function of the local density r ( r ), the gravitational force acting on the test particle mgiven by the vector volume integral F = -Gm ò _V r ( r´ ). I r - r´ I ^-2 ( r - r ´) ⁰ dV ´ of the vector sum of contributions from individual material elements of different density r ( r ) in the source.
Deviations from homogeneity and spherical symmetry
For large material bodies in space (with a diameter greater than about 1000 km), strong gravity automatically ensures an approximately spherical shape with an almost spherically symmetrical density distribution. However, if such a body rotates , it will not be spherical, but only axial symmetry (cf. §3.6 " Kerr and Kerr-Newman geometry " and §4.4 " Rotating and electrically charged Kerr-Newman black holes ").
However, within terrestrial-type planets, there are certain local inhomogeneities in the mass density distribution that cause minor anomalies and fluctuations in the surrounding gravitational field. This is manifested by minor irregularities and deviations from the Kepler orbits(see below " Motion in Newton's gravitational field ")orbiting satellites. These irregularities are more pronounced in the low orbits of the satellites, decreasing at greater distances until they practically disappear. Gravimetric measurements on the Earth's surface and analysis of deviations in the orbits of satellites can be used in geology to detect the internal inhomogeneities of the structure of our planet - and thus to identify possible mineral deposits.
Gravitational constant
For the universal constant G , acting as a coefficient of proportionality in the law of gravity (1.1) - Newton's gravitational constant - the value was experimentally determined

A very small value of the gravitational constant G indicates that the gravitational action is relatively weak and can only be more pronounced if at least one of the interacting bodies has a considerably large mass. Therefore, the mutual gravitational attraction of ordinary macroscopic bodies is not applied in practice, we do not observe it in everyday life and it can be demonstrated and measured only by highly sensitive methods (such as Cavendish and Eötvös measurements) . Gravity is the domain of very material bodies in the universe.
The weakness of the gravitational interaction, among other things, makes it very difficult to accurately measure the gravitational constant. The first measurement of the gravitational constant was made as early as 1798 by H. Cavendish, who used two lead balls weighing 730 g, mounted on a horizontal arm suspended on a thin fiber. This mechanical system functioned as a torsion balance , which was fitted with a mirror from which a beam of light was reflected indicating the deflection of the arm. Cavendish approached the bullets from both sides with a pair of larger lead balls (160kg) and measured the torsional weight arm deflection using a mirror. Cavendish's experiment was originally designed to determine the mass and density of the Earth; later analyzes of its results determined the value of the gravitational constant G » 6,74.10 ^-11 kg ^-1 m ³ s ^-2 ,which differs by only 1% from the current value.
Despite the fact that the gravitational constant G is one of the most important and most basic natural constants, in comparison with other physical constants we know its value with only a relatively small accuracy of about 10 ^-4 (to 4 decimal places). The reason for this "lousy" accuracy is the above-mentioned weakness of the gravitational interaction. Perhaps future planned highly sensitive quantum-radiation measurements using excited atoms in various quantum states, subjected to the gravitational action of test masses, can provide some hope for refining the value of the gravitational constant .

Gradients of gravitational forces - tidal forces
The gravitational field, excited around bodies according to the law of gravity, is inhomogeneous , in the vicinity of a gravitational body it is significantly stronger than at greater distances. Thus, when a finite-sized body is affected by the gravity of another body, the gravitational force on the near and far sides may be slightly different - a gravitational force gradient is created , given by the vector difference in gravitational force (acceleration) between the two locations. If this body is not absolutely rigid (in §1.6 we will see that the theory of relativity "forbids" absolute rigidity) , it leads to deformation body shape (eg, originally a spherical shape extends into an ellipsoid). Gravitational gradients causing this effect are called tidal forces ( tide = rapids, slapping, swelling of water ) , or ebb forces . Thus, tidal forces are created due to the non-zero dimensions of mutually attractive objects: when these two bodies are attracted, a greater attractive force will act on each of them on the side closest to the other body than on the opposite (averted) side.
The name comes from the fact that these forces are well known from our ordinary earthly life - they cause sea tide and outflow. This is the gravitational action of the Moon (to a lesser extent the Sun). During the daily rotation of the Earth around its axis, its individual regions move to the Earth-Moon line and tidal forces cause deformations in this direction. The Earth's crust deforms relatively little (but see below) , but large masses of ocean water can respond to the Moon's attractive gravitational forces almost instantly - they can move around the Earth's surface and ocean levels periodically deform , increase and decrease several times a day even a few meters. The Moon attracts water more strongly in the oceans that are closest to it - a small "bulge" forms on the surface of the ocean, which manifests itself as a tide at the coast.
The magnitude of tidal forces can be determined from the law of gravity (1.1) asvector difference of gravitational force between two given places around a gravitational body *). If we have a (spherical) gravitational body of mass M , in whose gravitational field there is another (test) spherical body of diameter r and mass m at a distance R , then a tidal force F_grad will act on the edges of this body in the direction of the joint (with the basic body M ) :
      ^« F _grad = F (R + r) - F (Rr) = G. [mM / (R + r) ² ] - G. [mM / (Rr) ² ]    »   - 2 GmMr / R ³ .
This tidal force acts outwardly and stretched body in the direction towards the connector gravitated body M . In an analogous manner, the tidal force acting on a body of diameter r in the direction perpendicular to the junction of the two bodies can be determined as a vector difference:
                _á^â F _grad = G. [mM / R ² ] .sin (2r / R)    »    GmMr / R ³ .
In the direction perpendicular, the tidal force is about half and points inwards - in this direction it compresses the body. Thus, tidal forces " stretch " the bodies in the direction of their joint and at the same time " compress " them in the perpendicular direction.
*) The resulting simplified expressions ( » ) are created by linearization - in algebraic expressions or in Taylor power expansion only members with first powers r , R are left , while members with higher powers are neglected.
Tidal forces are more pronounced in principle in three basic situations (and their combinations) :
1. Close circulation of two bodies, such as tight binary stars. Tidal forces periodically deform the partially elastic material of both bodies, with viscous friction generating heat therein . As a result, part of the kinetic energy of the orbital motion is converted into thermal energy, the orbits approaching, the circulation time is shortened ...
2. A rotating body orbiting near another gravitational body. This is the case of the rotating Earth, around which the Moon orbits, mentioned above. In addition to the deformation of ocean levels (tides), there are also periodic deformations of the earth's crust and mantle, which are transformed by viscous friction into heat heating the planet's interior. It is estimated that approximately 40% of geothermal energy comes from this gravitational-tidal mechanism inside the Earth (most geothermal energy comes from the radioactive decay of natural isotopes - see §1.4 " Radionuclides ", passage " Geological significance of natural radioactivity " in the book " Nuclear Physics and Physics ionizing radiation "). This tidal "warm-up" mechanism also takes place on some moons around the outer planets of the solar system, which would otherwise be very cold. An example is Jupiter's moon Io , in which heating by intense tidal influences leads to strong volcanic activity.
  Interestingly, the tidal dissipation of rotational energy has apparently slowed and stopped the earlier rotation of the Moon , which is now facing the same side of the Earth. Furthermore, since the Moon orbits the Earth in the same direction as the Earth's rotation about its axis, tidal forces in the Earth's material cause a gradual transmission.the rotational momentum of the Earth to the orbital momentum of the Moon, whose orbit, as a result, moves away from the Earth by about 40 mm per year. And the Earth's rotation slows down slightly. In the hypothetical case of a counter-rotating orbit, the Moon would approach the Earth.
3. Compact gravitationally collapsed bodies with an extremely strong gravitational field - neutron stars and especially black holes - cause such strong gravitational gradients and enormous tidal forces in their vicinity that even free-falling bodies are intensely stretched in the radial direction and compressed in the vertical direction ( there is a kind of "spaghetti") so that in the end they end up as a "string" of atoms or elementary particles (§4.2, passage "An observer falling into a black hole ").
In tidal phenomena, the laws of conservation of energy and momentum apply . The kinetic energy of the rotation of bodies and their circulation is partially converted into thermal energy in the material of bodies. The momentum is "redistributed" between the rotational motion around its own axis and the orbital motion of bodies around a common center of gravity. The intensity and character of these transformations depend on the individual geometric, kinetic and material configuration of the bodies involved.

Body motion in Newton's gravitational field
Within classical physics motion of bodies in Newtonian gravitational field controls the basic second Newton's law of force F and acceleration a : a o d ²y / dt ² = F , where after the force F is substituted gravitational force by inverse square law (1.1). Newton's law of gravitation naturally follows not only the original Kepler's laws , but also other observed properties and possibilities of planetary motion, star motion in binary and multiple systems and within galaxies, hydrostatic equilibrium in stars (except for the final stages). and other effects.

Motion of a body in a centrally symmetric gravitational field . Escape velocity.
In addition to the idealized and simple case of a homogeneous gravitational field (such as free fall and throw of bodies in the Earth's gravitational field) , the simplest and basic task for moving a body in a gravitational field is when we have one point (or centrally symmetric) gravitational body of mass M and investigate the motion of a small test particle in its gravitational field. It is therefore important to investigate the movement of the test particle mass m in a centrally symmetric field excited Newtonian gravitujícím body M . On the test particlem at a distance r from the center of the body M , according to (1.1), a gravitational force F = GmM / r ² will act , which according to the law of force and acceleration will give it acceleration and according to the equation m . a = G. mM / r ² . The mass of the test body m is shortened here, so that the gravitational acceleration will be

has a centripetal direction and does not depend on the mass m - all bodies fall with the same acceleration . At the point source r = 0, the gravitational acceleration of the test particle would be theoretically infinite.
If the spherical gravitational body M is of finite radius R (situation according to Fig. 1.0 in the middle) , the simple dependence (1.3a) will apply only outside the body for r> R, while inside for r <R the dependence will be more complex, depending on the density r (r ) mass distribution. The gravitational force F (r) at a distance r from the center is given by the amount of mass m (r) = 4 p ₀ò ^{^r} r ( r ). r ² dr contained in an imaginary sphere of radius r : F (r) = Gm (r) / r ² (the mass in the outer shells does not apply, its gravity acting from different directions is canceled) . The gravitational acceleration of the test particle m for a spherical gravitational body M of radius R is therefore:

The gravitational acceleration for an extended body M is finite everywhere, in the center r = 0 it is zero (the gravitational force is zero there) .
It is also useful to determine the potential energy U (r) of the test particle m in the gravitational field at r . It is the work done against the gravitational force F (r) = GmM / r ² , necessary to "pull" a particle of mass m from the point r of the gravitational field to an infinite distance r ® ¥ , where gravity no longer acts and the potential energy is there already zero. It is therefore given by the integral U (r) = Gm _rò ^{^¥} (M / r ²) dr = - GmM / r. The gravitational potential j (r), which is the potential energy of the body m per unit of its mass U(r) / m , is

This simple relation applies for all distances r only for the point source M ; in its place r = 0 the potential acquires an infinite value (similar to the gravitational force F) . If the spherical body M is of finite radius R (situation according to Fig. 1.0 in the middle) , a simple dependence will apply only outside the body for r> R, while inside for r <R the dependence will be more complex, depending on the density profile r (r) of mass distribution:
The amount of mass contained in an imaginary sphere of radius r is m _r = 4 p ₀ò ^{^r} r ( r ). r ² dr , so the gravitational force per unit mass of the test particle will be F (r) = G. ( _₀ ò ^{^r} 4pr (r) r ² dr ) / r ² and the potential will be the integral of this force j (r) = - _rò ^{^¥} F (r) dr. If we divide the space around the body M into the outside r> R and the inside r <R, the gravitational potential will be formed by two components:

The gravitational potential for an expanded body M is finite everywhere (if, of course, the mass density is finite) , at the center r = 0 it has a maximum value (even if the gravitational force is zero there) ; somewhat lower on the surface r = R: j (R) = -GM / R.
The gravitational potential j plays an important role in the general theory of relativity , as the physics of gravity and spacetime (starting with §2.4 " Physical laws in curved spacetime " and others) .
(Please do not confuse the designation "phi" with the coordinate "j" in the polar coordinate system!)
The simplest motion of a body in the central field is radial motion in the direction of the body M , or in the opposite direction, "away" from the gravitational body. The result of this movement will depend on the radial velocity v of movement of the body in the place of the distance r from the center of the central body M .
As it moves outwards from the body M , the velocity of the body v gradually decreases with increasing r due to gravitational attraction . According to the law of conservation occurs converting the kinetic energy of motion mv ² /2 potential energy G.m.M / R in gravitational field. If m.v²/2 = G.m.M / r, ie the sum of the kinetic and potential energy of the particle is zero, the particle continues to move to infinity (where there is zero potential energy) and stops there.
Thus, the so-called escape velocity v₂ is very important , which is the smallest radial velocity needed for the body m to leave the gravitational field of the central body M from a distance r forever and move to infinity :

The escape velocity does not depend on the mass m or the composition of the escaping body (universality of gravity) , it depends only on the mass M of the gravitational body and the radius r from which the escaping body starts (for a body starting from the Earth's surface the escape velocity is about 11.2 km / s - the so-called second cosmic velocity , which is why we mark it v₂ , the escape velocity from the "surface" of the Sun is significantly higher, about 617 km / s) . In §4.2 (section " Complete gravitational collapse. Black hole. ") We will see that in the gravitational collapse of massive stars, the escape velocity can reach the speed of light - thus creatingblack hole .
Another important "cosmic" velocity in the central gravitational field is the velocity of the circular orbit v₁ , which a body of mass m must reach in the azimuthal direction in order to balance the centrifugal force F _rot = mv² /r with the attractive force F _g = GmM / r²of the gravitational of a body M so that the body can orbit in a circular path of radius r :

Again, it does not depend on the mass m or the composition of the orbiting body (universality of gravity) , it depends only on the mass M of the gravitational body and the radius r along which the body orbits (for satellites orbiting the Earth in low orbit near the planet 7.9 km / s - the so - called first cosmic velocity , that's why we mark it v₁ ) .
By comparing equations (1.5) and (1.6) we see that the escape velocity v₂ is Ö2 (= 1.414) - multiple of the circular orbital velocity v₁ .
Note: The additional cosmic velocities mentioned in cosmonautics (3-6) are given by the specific situation of the Earth and its orbit around the Sun. They are therefore not of universal significance and will not be considered in our general treatise on astrophysics and gravity.
We will analyze more complex types of movements below in the reduced motion of two bodies :

Gravitational motion of two bodies
In the case of Newton's gravitational field, it is shown (will be shown below) that the motion of two bodies can be converted to the problem of the motion of one test body . So let's have two bodies with masses m ₁ and m ₂ , which act on each other by gravity according to Newton's law (and are otherwise free). The equations of motion of these bodies will then be

where r ₁ , r ₂ are the position vectors of the bodies m ₁ and m ₂ with respect to the given reference point O (origin of the reference system), r ₁₂ = r ₂ - r ₁ is the position vector of the body m ₂ with respect to the body m ₁ (ie r ₂₁ = - r ₁₂ ) ar = | r ₁₂ | = | r ₂₁ | is the distance between the two bodies. Subtracting both equations gives an equation describing the relative motion of the body m ₂relative to the body m ₁ :

where r = r ₁₂ is the position vector of the body m ₂ with respect to the body m ₁ .

It is advantageous to place the origin of the O coordinates in the common center of gravity of both bodies. Then m ₁ applies . r ₁ + m ₂ . r ₂ = 0, so that the position vectors of the individual particles are to their distance vector r associated relationships

called reduced mass , the equation of motion (1.10) can be written in the form

This relationship has the form of equation of motion of a single particle of mass m moving in an external field F (r) which is formed by the m ₁ and m ₂ and is symmetrical relative to the origin of the coordinate r = 0. The determination of the motion of two interacting bodies is thus reduced to the problem of the movement of a single imaginary body m in a centrally symmetric field around a fixed center of gravity. If this problem has a solution r = r (t), it is easy to determine the individual trajectories r ₁ = r ₁ (t) and r ₂ = r ₂ (t) on the basis of relations (1.11a) of the original bodies m ₁ and m ₂ .

If we multiplies equation (1.12) the scalar vector v = r^., we can write them in the form after editing

here ¹/₂ mv² is the kinetic energy and U (r) is the potential energy of the body m , related to the field strength in the relation

It follows that the total energy E of a particle is a constant independent of time:

which expresses the law of conservation of energy during the motion of a body of mass m in Newton's central field.

Trajectories of bodies
If we introduce the polar coordinates r, j in the path plane (motion in the central field is planar) , it is possible to divide the equation of motion (1.12) into radial and tangent components :

expresses the law of conservation of the momentum when moving in the central field and at the same time Kepler's law of surfaces (area velocity (1/2) r ² j^. is constant over time), where the quantity L is the momentum per unit mass of the particle ("specific momentum") L = J / m.

However, it is easier to follow the law of conservation of energy (1.13 ) rewritten in polar coordinates

which is an equation describing the radial component of motion. From the law of conservation of momentum (1.15) rewritten in the form d j = (J / mr ² ) dt and from equation (1.16b) the equation between r and j

Equation (1.16a), resp. (1.16b) shows that the radial part of the motion corresponds to a one-dimensional motion in a central field with an " effective potential "

composed of both the gravitational potential energy U (r) and the centrifugal potential energy J ² / 2mr ² . Values of r at which is V_ef(r) = E, and thus according to (1.16b) r ^.= 0 , correspond to the turning points of the path, at which the distance function from the center r (t) transitions from increase (distance) to decrease (approach) or vice versa. These turning points determine the range of distances from the center at which the particle m can move. A graphical representation of the course of the effective potential for the motion of a body in the Newton gravitational field of a central body is shown in Fig. 1.1a. If there is only one turning point r = r _min , it is the movement of unrestricted (infinite), begining and terminating at infinity. If there are two turning points r = r _min (r ^.. <0) ar = r _max (r ^.. > 0), then the whole trajectory lies inside the annulus r _min <r < r _max - this is a limited movement (finite) , in this case after the ellipse . The idea of the effective potential is a very useful tool in the study of the motion of bodies in centrally (or axially) symmetrical fields, as we will see in §3.4,4.3 and §4.4 in the analysis of the motion of bodies in the gravitational fields of black holes.

The shape of the path is obtained by solving equation (1.17). For a Newtonian gravitational field with U (r) = -Gm ₁ m ₂ / r (as well as for any central field with a potential proportional to 1 / r or 1 / r ² ) the integration can be performed analytically :

where the integration constant can be canceled by a suitable choice of the beginning of the reading of the angle j ( j = 0 in perihelion). The trajectory equation can then be rewritten in the form

This is the equation of a conic section with a focus at the origin of the O coordinates , ie in the common center of gravity of both bodies, p is a parameter and e is the eccentricity of the orbit determining what kind of conic :

Fig.1.1. Motion of bodies under the influence of Newton's gravitational field.
a) The course of the "effective potential" V _ef (r) controlling the radial component of the motion in the central Newtonian gravitational field. Intersections with the energy line E = const. they determine the turning points at which the radial component of the motion changes direction. Lines E = const. <0 (if they are permissible, ie E> min (V _ef )) correspond to finite motion either elliptical (radial component oscillates between r = r _A and r = r _B ) or circular (constantly r = r _C ) motion. If E> = 0, the motion is infinite - it starts and ends at infinity; point D corresponds to the closest approach of the body to the center.
b) Finite movement (E <0) of the two bodies interacting gravitationally m ₁ am ₂ happens ellipses having the same eccentricity with the focus in a common centroid C . The problem of two bodies is equivalent to the problem of motion of a body with a reduced mass m in the gravitational field centrally symmetrically to the common center of gravity C .
c) Analogously infinite motion (E> 0) occurs after hyperbolách around the center C .
d) Cross section of some equipotential surfaces of a system of two bodies M ₁and M ₂ orbiting a common center of gravity. The bold line indicates the first common equipotential of both bodies - the Roche limit. Furthermore, the intersections of equipotential surfaces are marked - Lagrange libration points L ₁ , L ₂ , ..., L ₅ .

Since the position vectors r₁ and r₂of both bodies m ₁ and m ₂ are proportional to the vector r , each of them also describes a conic with a focus in a common center of gravity. As can be seen from (1.11), the ratio r ₁ / r ₂ is the same for any point of the path, so that the bodies move relative to the center of gravity along paths that have the same shape (orbiting, for example, ellipses generally of different sizes but the same eccentricity ) - see Fig.1.1b, c.

The most important case is the gravitationally coupled motion along an ellipse , whose longer half-axis a and shorter half-axis b are given by

It is obvious that the longer semiaxis not depend on the momentum, but only for the energy E . The turning points r _min = a (1-e) and r _max = a (1 + e), ie, the "perihelion" and the "aphelium" of orbit, are also the roots of the equation V _ef (r) = E. The time of one orbit along an elliptical orbit , i.e. the period T , can be easily determined from (1.15) and (1.17) by integration according to time from t = 0 to T and according to j from j = 0 to 2 p . After the adjustment, we get the relationship

which is the exact wording of Kepler's third law. If m ₁ >> m ₂ , as is the case, for example, in the solar system, then 3. Kepler's law has the usual form

where M = m ₁ denotes the mass of the central body (eg the Sun). The ratio of the squares of the orbital periods and the third powers of the large half-axes is thus approximately the same for all planets, but the relation (1.18a) holds exactly. In the special case of circular circulation (e = 0) it is finally possible that 3.Kepler's law (1.18b) can be expressed in the form

If Newton's law (1.1) applies , then a closed trajectory (in a frame of reference firmly connected to the center of gravity) is based on finite motion . To trajectory finite his motion was closed, the angle must be Dj by which the position vector r is rotated for the time in between two has spots turnover r _min and r _max , rational multiple of 2p , i.e. Dj = 2 p. M / n, where m and n are integers. Then, for n periods of the radial component of the motion, the body performs m circulates and returns to the starting position. However, in the case of a deviation from the law of inverted squares in Newton's law, this condition is no longer met and the "elliptical" trajectory is not closed. If the deviation is not too large, such a path can be imagined again as an ellipse, which, however, is no longer fixed, but the whole rotates slowly (performs a precessional movement) around the center of gravity. Such precession causes perihelion and aphelium to be in a slightly different place in each cycle. The elliptical orbits of the planets around the Sun actually perform the above-mentioned precessional motion, while the deviation from the law of inverted squares is due to the fact that it is not exactly the central field (gravitational influence of other planets, the Sun and planets are not point). The general theory of relativity shows that Newton's law is not accurate even for the centrally symmetric case for strong gravitational fields; emerging anomalous precession and slide of perihelion was actually demonstrated in Mercury (see §4.3).

The problem of motion of several bodies
In fact, there are a large number of individual bodies and formations of different masses in outer space, which are affected by gravity. Therefore, the motion of planets, moons, and stars in binary or multiple systems actually differs from the above-derived simple laws of motion of two bodies in a common central gravitational field. The study of the motion of more gravitationally influencing bodies is called the n-body problem . This problem is very difficult even for the case of only 3 bodies, it is generally not analytically solvable. Only in some special cases is it analytically solvable. As early as 1772, J.L.Lagrange showed that for each system of two orbiting bodies, 5 significant points can be found in the coordinate system rotating together with the connection of both bodies, the so-called pound points . If we place a third body in one of them, at suitable speeds the movement of all three bodies will take place again in conic sections.
The problem of movement of the three bodies is well solved if the weight of the third body is negligibly small with respect to the two basic bodies, circulating undisturbed around the common center of gravity. Then, in fact, the general problem of motion of three bodies breaks down into the motion of two bodies discussed above and into a separate problem of motion of one "test" body in the resulting gravitational and centrifugal field of two basic bodies - see the following passage :

Binary system: equipotential surfaces, Roche limit, libration points
So far we have dealt with the mutual motion of two bodies under the influence of their own gravitational field. The most important such case is the so-called binary system - a system of two gravitationally coupled bodies orbiting a common center of gravity. An example is the binary systems that often occur in space. The binary system of bodies M ₁ and M ₂ is schematically shown in Fig. 1.1d. To analyze the motion of a small particle (such as gas atoms in the space of a binary system) in the gravitational and centrifugal field of a binary system, it is useful to determine the shape of surfaces that are places of a certain gravitational potential - equipotential surfaces. For a system of two bodies of masses M ₁ and M ₂ rotating around a common center of gravity at an angular velocity w according to Fig. 1.1d, we choose the coordinate system x, y, z rotating together with the bodies so that the x- axis is identical with the line M ₁ and M ₂ and the origin is in the center of gravity of both bodies. The gravitational potential j for any point P (x, y, z) will then be

where r ₁ and r ₂ are the distances of the point P (x, y, z) from the centers of the bodies M ₁ and M ₂ , r _{by the} distance from the center of gravity of both bodies. The gradient of the first two terms indicates the gravitational acceleration that both masses M ₁ and M ₂ cause at the point P (x, y, z) , the third term expresses the centrifugal acceleration caused by the rotation of the system.
Selected equipotential levels are shown in cross section perpendicular to the axis of rotation in Fig. 1.1d. In the vicinity of each of the bodies, the equipotential surfaces have a slightly deformed spherical shape and are enclosed around each of them separately. At greater distances, the deformation increases until these surfaces touch both bodies, and at even greater distances, both bodies already have common equipotential surfaces.
The equipotential surfaces, which touch at one point, in the inner libration point L₁ , form the so-called critical Roche limit - it is the first common equipotential surface of both bodies. Within this limit, each particle moves under the predominant gravitational influence of one or the other body. At the libration point L ₁ , the test particle can pass from the gravitational sphere of influence of one body to the area of gravity of the other body.
An interesting and astrophysically important phenomenon occurs when, for example, the body M ₁of the gaseous state fills (or exceeds) the entire space defined by the Roche boundary. In such a case, the gravitational action of the second body "draws" or "sucks out" the gas from the upper layers M ₁ , which overflows around the inner libration point L ₁ onto the second body M ₂ . This phenomenon often occurs in close binary stars and can lead to dramatic astrophysical processes, as will be shown in Chapter 4 on the evolution of stars and black holes - §4.1, 4.2, 4.8, illustrated in Figure 4.26. In the orbit of solids, the Roche limit can manifest itself when a smaller body (eg the moon) approaches its mass around a more massive body (eg a planet) so much that the libration point L ₁ finds itself inside this smaller body. In this case, opposing gravitational force gradients (additionally having a time-varying tidal force character) can cause this lighter body to rupture (however, for smaller compact bodies, the strength of the body material can prevent this).
In space (in the gravitational and centrifugal field) around a system of two bodies rotating around a common center of gravity, there are a total of 5 significant points L ₁ , L ₂ , ..., L ₅ - the so-called Lagrange libration points (Fig. 1.1d). The libration points are the points where the attractive and centrifugal forces acting on the test particle are balanced. The body located at these points in them can remain at rest against the junction of the two bodies M ₁ and M ₂ . The most important libration point is the already mentioned internal libration point L ₁ , which is located on the connection between the bodies. The outer libration points L2 and L3 lie outside the system on a line passing through both bodies. The exact positions of all these 3 points depend on the specific weights M ₁ and M ₂ , their distance and speed of rotation. The libration points L ₄ and L ₅ lie symmetrically outside the line and form isosceles triangles with the centers of the bodies M ₁ and M ₂ .

We will not deal with a detailed analysis of the motion of bodies under the influence of Newton's gravitational force here, it is a matter of "celestial" mechanics (for a more general case of motion in the gravitational field of a black hole, however, the relevant analysis is performed in §3.4,4.3 and §4.4) .

Astronomical significance of Newton's law of gravitation
Newton's law of gravitation proved to be very successful in elucidating all the motions of planets, moons, comets and other bodies in the solar system. A great triumph was the use of Newton's law of gravitation to accurately analyze some of the anomalies in the motion of planets from Kepler's laws, which at first seemed against the law of gravity. In 1840, astronomers discovered that the last known planet in the solar system at the time, Uranus, deviated slightly from the calculated orbit as it moved. Therefore, there were temporary doubts about the validity of Newton's law at such great distances from the Sun. However, further calculations have shown that the anomalous behavior of Uranus can be fully explained by the gravitational pull of an even more distant, as yet undiscovered planet, which slightly diverts the motion of Uranus from its ideal orbit; the position of a hypothetical planet in the sky was also determined.
The Atom and the Planetary System: Similarities and Differences
After discovering that the atom is a system of positively charged nuclei and negatively charged electrons bound by an electric force, the already well-researched solar system, bound by gravitational force, became the inspiration for clarifying the structure of this system. There is an obvious analogy at three points:
¨ The electric and gravitational forces decrease with the square of the distance;
¨ The attractive gravitational force and the attractive electric force (between charges of opposite sign) can be permanently compensated in a vacuum by centrifugal force during orbital motion;
¨ The same Kepler's laws apply to motion in central gravitational and electric fields.
Based on these analogies, Rutheford's planetary model of the atom was created (see eg "Nuclear physics and physics of ionizing radiation", §1.1 "Atoms and atomic nuclei", section " Atom structure "). However, there are also fundamental differences between the planetary system and the atom :
¨ The difference in the properties and strength of electric and gravitational forces. While the orbits of the planets are stable for a long time *), the orbiting motion of an electron in an atom according to Maxwell's electrodynamics would result in intense radiation of electromagnetic waves, which quickly carry away the kinetic energy of the orbit.
*) According to the general theory of relativity, gravitational waves are emitted even during the orbit of the planets, but their energy is completely negligible and does not affect the orbits for many millions of years.
¨Huge difference in size and weight. The planetary system (masses » 10 ³⁰ kg, diameter » 10 ⁸ km) can be fully described by Newton's classical mechanics, while the atom (diameter » 10 ^-8 cm) is typically a quantum system.
These differences forced Bohr's quantum modification of the planetary model of the atom. Nevertheless, the planetary idea of ??the atom is used in illustrative qualitative considerations.
"Cosmographic Mystery" ?
In addition to Kepler's basic laws, astronomers have tried to explain the specific distances of planets (radii of orbit) and other bodies in the solar system. Already Kepler is at work "Mysterium Cosmographicum" tried to explain the distances of planets from the Sun using "Platon's polyhedra" circumscribed by the spheres of individual planets. More recently, inspired by Bohr's model of the atom (and Balmer's series of spectral lines of the hydrogen atom), Titus, Bode, and Mohorovic (and other authors) have sought to find a "quantum law" for the distances (radii of orbit) of planets in the solar system. We now know that these laws are only apparent - the structure of the solar system is, in addition to the laws of gravity and mechanics, the product of complex and often random processes of its formation (including collisions and various resonant phenomena) and has nothing to do with quantum laws. None "fundamental law" for distances or radii of planetary orbit it does not exist, efforts to find it result in mere "numerology". "Cosmographic mystery" is just fiction ...

Regular and chaotic motion in gravitationally coupled systems
The above analysis of motion in Newton's gravitational field was performed for simplified cases that could be converted to motion in the central field. A high degree of symmetry leads to an integrable problem with a regular solution , the laws of conservation of motion integrals will apply. Two gravitationally bound bodies will orbit around the common center of gravity along stable elliptical orbits (here we neglect the emission of gravitational waves, in the case of the solar system also, for example, the pressure of radiation from the Sun, etc.). In more complex cases of three or more bodies, the orbits will interact with gravitational perturbations- the symmetry is broken. Calculations and computer simulations show that in such complex systems a small change d_oof the initial conditions causes the initially close trajectories to diverge exponentially from time t : d = d _o .e^-^l^.^t . After a long enough time, the system eventually becomes chaotic . The degree of linear stability or instability - "chaotic" of such a system can be characterized by the so-called Lyapunov time T _L = 1 / l , for which the system deviates 2.7 times (this factor increases each initial deviation); parameter l = 1 / T_L is sometimes called Lyapunov's exponent . For the inner planets of the solar system (excluding Mercury), the Lyapunov time is estimated to be T _L » 5.10 ⁶ years. The high value of this time explains the extraordinary accuracy of astronomical predictions of planetary motions over time horizons of hundreds and thousands of years. However, at intervals of hundreds of millions to billions of years, the chaotic nature of the planet's orbits would already be decisive; some of the planets could even leave the bound system of the solar system. From a general point of view, the behavior of chaotic systems is outlined in §3.3, section " Determinism - chance - chaos? ".

Physical significance of Newton 's law of gravitation
Before Newton's law, physics encountered the force acting between bodies only during their mechanical contact - impact or friction. With his law of gravity, Newton first introduced into physics the concept of direct action of bodies at a distance ("actio in dis tans") in empty space *). However, both Newton himself and his followers were not satisfied with this idea and tried to find an "environment" transmitting gravitational force effects and thus explain the nature of gravity (§1.3). Later developments in physics have shown that the idea of ??direct action at a distance through empty space is correct, no environment is needed, but it cannot be an immediate action (as Newton's law assumes), but always properlyretarded (see §2.1).
*) The development of the concept of the physical field has shown that even though two bodies do not physically touch, they "touch" or even intersect their fields. And that causes their force interaction. In addition to its immediate practical contribution, Newton's law of gravitation is also of great unitarianism . The law of gravity also describes the fall of a stone to the earth, the motion of a planet around the Sun, or perhaps the motion of a star in a galaxy. This, for the first time, bridged the gap that (in the understanding of humans) previously existed between the Earth and the universe. The same physical laws have been shown to apply in the solar system, and probably throughout the universe
. Newton synthesis A Kepler's kinematics of planetary motions with her and Galileo dynamics of movement earthly bodies is so in the history of the first m case process, which later in the development of science many times repeated, and that continues today: the unification of formerly independent branches of physics showing that the laws of nature are consistent and mutually articulated - see Appendix B "Unitary field theory and quantum gravity".
Despite all the successes of Newton's theory, however, one anomaly remained unexplained under Newton's law of gravitation. It was a peculiarity of the orbit of the planet closest to the Sun, Mercury. The orbit of this planet, whose orbit is quite eccentric (and therefore perih é is well established)lium), is markedly different from Kepler's laws. If it were a motion of a single body in Newton's centrally symmetric gravitational field, Mercury would have to orbit along an ideal constant ellipse with the Sun in focus. The observed rate of precession and perihelia amounts of even 5600 "for 100 years although crucial part (approximately 5026") having a kinematic origin - is caused by the movement of the reference system. The remaining 575 "/ 100 years actual precession movement perihelia showing the elliptical orbit of Mercury slowly rotates. Almost all the feed can be explained disruptive influence of other planets, p øedevším Venus. After subtracting the gravitational influence of planets known from observed of Mercury, however, it does not get an ideal ellipse, but some very small anomalous displacement remains perihelion about 43 "/ 100 years. This anomalous shift of perihelion remained unexplained in Newton 's theory (efforts to explain it, for example, by the influence of another unknown planet between Mercury and the Sun were unsuccessful).

Distribution of kinetic and potential energy. Viral theorem.
If a system of bodies or particles moves in a force field , the kinetic energy E_kinof motion and the potential energy E_pot in the force field acquire certain values ; the internal energy U is given by the sum of the kinetic and potential energy of the system. If this force field is a gravitational attraction , where the interacting force between the individual particles is inversely proportional to the square of their distance F ~ r ^–2 , then between the mean value of the total kinetic energy <E _kin > And the mean value of the potential energy <E _pot > the relation :
2 <E _kin > + <E _pot > = 0, or <U> = <E _kin > + <E _pot > = ¹/₂ <E _pot > = - <E _kin > .
Thus, the sum of the potential energy and twice the kinetic energy of the stationary system of bodies is equal to zero . Or the total energy of the gravitationally coupled system (at equilibrium) is equal to half the mean value of the potential energy of the system (or negatively taken total mean kinetic energy). These important laws, which can be derived in the framework of classical mechanics [165] is called theorem virial or virial theorem (lat. vires, virium = power, energy ) . This knowledge plays an important role in astrophysics - the formation and evolution of stars (§4.1, passage " Star formation ") , the behavior of galaxies and galaxy clusters (§5.6, the passage " Hidden-dark matter in galaxies and galaxy clusters ") .

Modifications of Newton's Law of Gravity
Minor difficulties in celestial mechanics (whether real or apparent) raised various doubts about the accuracy of the law of inverted squares in Newton's law of gravitation (1.1). Therefore, during the 18th and 19th centuries, attempts were made to "refine" and modify Newton's law of gravitation by introducing various small corrections in the law of inverted squares, e.g.

      m. M          a                                                 m. M
F = - G -------- (1 + ----) r ^o , n = 1 or 2;          F = - G -------- r ^o ,
        r ²              r ⁿ                                                    r ²⁺^b
(Clairaut's law)                         (Hall's law)

where a and b are small constants (corrections) correspondingly modifying the original law of inverted squares to correspond to the observed anomalies. Another modification of the law of gravitation has its origin in Seeliger's well-known "gravitational cosmological paradox" arising from an attempt to use Newton's law of gravitation in an infinite Euclidean space (universe) filled with matter with a constant non-zero density. The law of gravity in the form (1.1) in such a case gives an infinite value of the gravitational potential and infinite gravitational force (a satisfactory solution is obtained only if the mass distribution density in all directions from a given point decreases faster than r ^-2 ). In order to make the law of gravity compatible with the idea of an infinite space homogeneously filled with cosmic matter, a modification of Newton's law was proposed using an additional exponential factor :

where e is a small positive constant. This modification can be related to the hypothesis of " absorption " of gravity by the environment lying between gravitational bodies. The law of gravitation considering the absorption of gravity would indeed have the form F = -G (mM / r ² ). e ^-^mr^r , where r is the density of the environment between the bodies M and m (homogeneous is assumed for simplicity) and mis the absorption coefficient. Attempts to demonstrate gravity absorption have not yielded convincing results. In addition, the absorption of gravity would lead to a violation of the proportionality between inertial and gravitational mass, which would lead to an inadmissible violation of the 3rd Kepler's law. Furthermore, the effect of gravity absorption on the Earth's surface would cause appropriate variations in gravity and acceleration (with a period of 24 hours), caused by shielding the gravitational action of the Sun and the Earth's Moon. In experiments with pendulums no similar effect was observed, as well as properties of the marine tide caused by the Moon and Sun e show no observable abnormalities which could be attributed to changes in tidal acceleration stemming from absorbing gravity.
All similar attempts to modify Newton's law were in the nature of formal ad hoc hypotheses , were not substantiated by deeper physical reasons, and ultimately failed to satisfactorily explain one of the observed anomalies without other side effects and anomalies contradicting the results of the observations.
The difficulty in shifting Mercury's perihelion was not serious enough to threaten Newton's theory more seriously; some hypotheses, such as that the Sun is slightly flattened and the gravitational field is therefore not exactly spherically symmetrical, could explain similar effects. However, Newton's theory has some more serious conceptual shortcomings , which manifested themselves in confrontation with a newer knowledge of the laws of nature. From the point of view of the depth of knowledge, it can be considered a shortcoming that Newton's law of gravitation does not explain the exact equality (proportionality) of gravity and inertia. This equality is purely empirical here and has the character of chance (for more details see §2.1).
Velocity of gravitation
The main weakness of Newton's theory of gravitation, however, is the already mentioned assumption of immediate and immediate gravitational action " at a distance ". In Newton's law of gravitation (1.1) time does not appear in any way; according to him, a change in the position of one body is gravitationally reflected immediately on other bodies, even very distant - gravity has infinite speed .
This assumption has proved to be incompatible with the knowledge obtained during research into the phenomena of electromagnetic characterized ch and generalized in Einstein's special relativity (see §1.6 and 2.1). The need arose to modify Newton's law by introducing a time factor - retardation reflecting the final velocity of the gravitational interaction *). Indeed, this approach leads to a satisfactory and consistent theory of gravity - Einstein's general theory of relativity (Chapter 2) - which not only incorporated gravity into the context of modern physics, but even came up with the idea of the determining role of gravity for all the laws of physics, to identify gravity with the properties of space and time. In addition to its profound conceptual significance, the general theory of relativity quite naturally explains the equality of inertia and gravity, the anomalous shift of Mercury's perihelion, the curvature of light rays in the gravitational field, and other phenomena and facts beyond Newton's theory. According to the general theory of relativity, the commotion in a gravitational field propagates at the speed of light c , in the form of gravitational waves ( §2.7 " Gravitational waves " ) .
*) The velocity of the gravitational interaction will be dealt with in §2.5 " Einstein's equation of the gravitational field " and mainly in §2.7 " Gravitational waves ",where in the passage " How fast is gravity? ", general questions of the speed of propagation of changes in the gravitational field and the possibilities of its experimental determination will be discussed.

Galactic Modifications of Newton's Law of Gravitation - MOND
Newton's law of gravitation, in co-production with the other 3 laws of mechanics, is phenomenally successful in analyzing all mechanical processes here on Earth, planetary motions and other bodies in the solar system, star formation dynamics and evolution (except final stages). ), stellar motions in binary and multiple systems. It fails only in extreme situations of strong gravitational fields in gravitationally collapsed compact objects in the final stages of massive star evolution (§4.2 " Final stages of stellar evolution. Gravitational collapse. Black hole formation. ") And in analyzing the structure and evolution of the universe on cosmological scales (§5.4 "Standard cosmological model. Big Bang. Forming the structure of the universe. "). There must be generalized Einstein's law of gravitation in the general theory of relativity (§2.5 ' Einstein's gravitational field equation ").
The surprising incompatibility but appeared at astronomical measuring the speed of movement - circulation - stars and gas in the peripheryof galaxies and speed of galaxies in galactic clusters: these velocities appeared to be higher than expected based on Newtonian mechanics and gravity (the velocities of motion and intensity of the gravitational field are relatively small here - non-relativistic ). The visible matter in galaxies and their clusters is insufficient for the dynamics of motion in Newton's law analysis. This paradox was solved in astrophysics by the hypothesis that the dynamics of galaxies is determined by an additional massive invisible - hidden dark matter (§5.6, section " Future evolution of the universe. Hidden-dark matter. ") .
However, the hypothetical dark matter has not yet been explicitly discovered - to clarify what it is composed of, to detect its particles. An alternative hypothesis to explain the observed dynamics of motion in galaxies is a suitable modification of Newton's theory (and possibly its relativistic generalization) - the so-called MOND (Modified Newtonian Dynamics).). The observed astronomical measurements of the dynamics of the motion of stars in galaxies and galaxies in galaxy clusters, which are generally attributed to the gravitational effect of dark matter, try to explain with the modified form of the law of gravitation . Such a modification of MOND was proposed in 1983 by M. Milgrom, in which Newton's law of gravitation F = GM .m / r²is modified by the hypothesis that in addition to mass and distance the gravitational force F_MONDalso depends on the value of acceleration :
F _MOND = G.M.m / [f (a/a_o) r ² ] ,
where G is the gravitational constant, M is the gravitational mass, m is the mass of the test piece, r is the distance. The modification of the classical Newton's law is given by the empirical function f depending on the value a acceleration of the test body, in relation to a certain empirical constant a _o (indicating the scale transition between Newtonian and MOND dynamics) . The function f is usually chosen in the form f(a / a _o ) = 1 / (1 + a _o / a ), or f(a / a _o ) = 1 / Ö [1+ (a _o / a ) ²]. To the thus modified law of gravitation explained the observed velocity curves in galaxies, for the constant a_o by the fitation were determined value a_o » 1.2 × 10^-10 m ^-2 .
Note: In MOND, this is a different (and otherwise motivated) modification of the law of gravity than the "Modifications of Newton's law of gravitation" mentioned above!
Experts differ on these efforts - both the concept of dark matter and MOND are ad hoc hypotheses . Milgrom's MOND with its acceleration dependence is an artificial and not very convincing theory of gravity . Whyshould the gravitational force depend on the acceleration - and even on a very small acceleration? It has no physical logic... Most astrophysicists are more inclined to the concept of dark matter, in which the dynamics of galaxies can be explained in more physically justified ways (it is not even excluded that it may be just a mistake to model the distribution of standard baryonic matter in galaxies and galaxy clusters..?..) .

Entropic hypothesis of the origin of gravity - Verlinde
Originally, this is another newer attempt at an alternative clarification - exclusion - of dark matter. In this hypothesis (dating from 2016), E. Verlinde declared entropy to be the primary very basic quantity, the growth of which causes the "entropic force" that is the essence of gravity. So there is no gravity! - we observe it only as an apparent force, a macroscopic manifestation of entropy growth with random statistical behavior of microscopic particle-quantum field systems. This entropic force causes the bodies to move and performs mechanical work. This rather bizarre and definitely very unlikely hypothesis is described in §B-5, passage "Entropic hypothesis of the nature of gravity".

Mechanistic hypothesis of the origin of gravity - LeSage
is the oldest attempt to simply explain the origin of gravity, long abandoned - see the following §1.3 "LeSage's hypothesis".

Gravity, black holes and space-time physics :
Gravity in physics	General theory of relativity	Geometry and topology
Black holes	Relativistic cosmology	Unitary field theory
Anthropic principle or cosmic God
Nuclear physics and physics of ionizing radiation
AstroNuclPhysics ® Nuclear Physics - Astrophysics - Cosmology - Philosophy


1.1.Historical development of knowledge about gravity		1.3.LeSage's hypothesis

The law of general gravity