# Newton's law of universal gravitation

The Newtonian law of gravitation is a physical law of classical physics, according to which each point mass acting on any other ground point with an attractive gravitational force directed along the line connecting the two mass points and the amount is proportional to the product of their masses and inversely proportional to the square of their distance. The cause of the force is the gravitational field.

- 3.1 Theoretical limits
- 3.2 contradictions for observation

## History

The Newtonian law of gravitation is one of the fundamental laws of classical physics. It was formulated by Isaac Newton in 1686 in his work Philosophiae Naturalis Principia Mathematica. Thus, Newton managed under the system established by him Newtonian mechanics, the first common explanation for the gravity on the earth, for the moon to orbit the Earth and the planets move around the sun. The Newtonian theory of gravitation explains this and more with the gravity -related phenomena as the tides on earth and perturbations of the planets with great accuracy. It was not until the early 20th century it has been refined by the general theory of relativity by Albert Einstein.

## Mathematical formulation

### Mass points

The magnitude of the force between two point masses and at a distance

The size is the gravitational constant. The forces acting on the two masses forces are equal and opposite and always have the direction to each other point mass; the Newtonian law of gravitation describes so unlike the mathematically similar Coulomb law an increasingly attractive force. In vector form is the force acting on the mass point 1 force

Where and are the positions ( position vectors ) are the two mass points. The modulus signs in the denominator of the expression represent the magnitude of the vector. If the ground point one of several mass points 2, 3, ..., n tightened, then the individual forces acting on the mass point to add one total force

The amounts of acceleration ( acceleration due to gravity or gravitational field strength, see gravitational field ) and who experienced two point masses and at a distance in the absence of other forces by the Newtonian law of gravitation, results after the second Newtonian axiom:

The mass attracts mass and vice versa. The two individual accelerations relate to the common center of gravity. The total acceleration of the body to one another is the sum of the accelerations, and their amount is

Now, if one of the masses is much smaller than the other, it is sufficient approximation to consider only the greater mass. So the Earth has much more mass than an apple, a man or a truck, so that it is enough for all of these objects to use the mass of the Earth in the equation for the acceleration. All three objects if they are in the same place, the same accelerated in the direction of center of the earth. They fall at the same speed and in the same direction. However, if one considers a binary star system, one must consider both stellar masses, because they are about the same.

If only very slightly changed during the movement of an object, the gravitational acceleration is practically constant, such as an object near the Earth's surface, which falls only a few feet deep, so vanishingly small compared to the Earth's radius of r = about 6370 km. In a sufficiently small area so the gravitational field can be considered homogeneous.

### Extensive bodies

Real bodies are not point masses, but have a spatial extent. Since the law of gravity is linear in the masses, one can decompose the body mentally into small pieces and add their contributions as shown in the previous section ( vectorial). Finally Traces the border crossing with infinitely small parts, resulting instead of a conventional sum of an integral.

In this way it can be shown, among other things, that an object with a spherically symmetric mass distribution in the outer space has the same gravitational effect as if its entire mass united in its focus. Therefore, you must treat extensive ( approximately spherically symmetric ) celestial bodies as mass points. Since a spherically symmetric mass distribution (eg, a hollow sphere ) produces no gravitational force in its interior, the gravitational force stirs inside a ball exactly on the proportion of the total mass here, the inner than the observed distance r. Newton proved this theorem in Philosophiae Naturalis Principia Mathematica. It does not apply in general for non-spherical symmetric body.

## Limits of the theory

Although it is sufficiently accurate for practical purposes, the Newtonian law of gravitation is only an approximation for weak and time-independent gravitational fields. For strong fields one uses the more accurate description by means of the general theory of relativity, from which the Poisson equation of the classical theory of gravity and thus the Newtonian law of gravitation can be derived directly, if you just assume that it is in the gravitation to a conservative field. This is called the law, therefore, today often referred to as the limit of small fields. The general theory of relativity in particular solves the problems described here, the Newtonian theory of gravitation.

### Theoretical limits

- The Newtonian theory is an effective theory, that is, they are not a cause for the gravitational force to, nor does it explain how gravity can act over the distance. This distance effect was also unsatisfactory for Newton. To fill this explanatory gap, the so-called Le Sage gravitation was developed as a model, but this could never really prevail.
- The Newtonian theory assumes that the gravitational effect of infinitely fast spreading, so that Kepler 's laws are satisfied. This led to conflict with the special theory of relativity. This calls namely, that the gravity propagates with the speed of light.
- The equivalence of inertial and gravitational mass is not explained in the Newtonian mechanics.

### Contradictions for observation

- The Newtonian theory does not explain fully the perihelion of the planet orbits, especially of Mercury. In this, the difference between the after -Newtonian theory calculated and the observed perihelion is 43 arc seconds per century.
- Whether light is deflected in a gravitational field or not depends on the Newtonian theory on which nature is attributed to the light. Will it be construed as an electromagnetic wave, then there is no distraction. However, when it be construed as a mass -affected particles according to the corpuscular theory, is then obtained according to the Newtonian law of gravitation, a light deflection, which from the equation of motion, a prediction can be made that is independent of the mass and thus also in the limit of vanishing mass remains valid. This value is, however, only half of the actual observed distraction. The measured value is calculated correctly from the equations of general relativity.