Gravitational acceleration

A gravitational field is a force field caused by gravity and possibly certain inertial forces. The field strength of the gravity field is the gravity, formula sign . As a weight force of a specimen related to its mass, it has the SI unit N/kg = m/s² and is also called gravitational acceleration or acceleration due to gravity. This acceleration sets a freely falling body in motion.

is a vector quantity with magnitude and direction. The direction is called the perpendicular direction. The magnitude is also called location factor to emphasize that and thus also the weight of a body depends on the location. In Germany, the gravitational acceleration is about 9.81 m/s² = 981 Gal. The variation over the earth's surface is a few Gal.

In a narrower sense - especially in the geosciences - the gravitational field of a celestial body is composed of its gravitational field ("gravity") and the centrifugal acceleration in the reference frame rotating with the body and possibly falling freely with it in the gravitational field of other celestial bodies.

In celestial mechanics, non-rotating reference frames are often used. The gravitational field of one or more celestial bodies is then based only on gravity.

In a broader sense, one speaks of the gravitational field in arbitrarily accelerated reference systems. In the gravitational field of a centrifuge, the centrifugal force dominates. In freely falling reference systems (e.g. space station) weightlessness prevails.

Measurement

In addition to directly measuring the acceleration of a freely falling body, one can calculate the amount of gravitational acceleration from the period of oscillation of a pendulum. A modern gravimeter is a special spring balance and achieves a precision of one microgal, approx. 10-9 g. One could use it to register a change in altitude on Earth of less than a centimeter. Fluctuations in atmospheric pressure cause changes of the same order of magnitude, mountains or different rock densities in the Earth's crust affect g even by up to 100 milligal, somewhat weaker also tidal forces due to the inhomogeneity of external gravitational fields, especially from the Moon and the Sun.

Sum of gravitational and centrifugal acceleration

The gravitational acceleration is the vector sum of a gravitational and a centrifugal component:

• The gravitational acceleration is caused by the gravitational field. Provided that the celestial body can be considered spherically symmetrical, the gravitational acceleration is calculated according to the law of gravity:

Here is the gravitational constant, is the mass of the celestial body, is the distance between the center of gravity of the celestial body and the sample, and is a unit vector directed from the center of gravity of the celestial body to the sample. If the mass distribution of the celestial body is not isotropic, as is usually the case, this results in gravity anomalies.

• The centrifugal acceleration affects because one is on the surface of the celestial body in a co-rotating reference frame.

• Tidal forces arise from the influence of other celestial bodies (e.g. the moon or the sun). Whether these forces are considered part of the gravitational field is a matter of definition. In this article they are not counted as part of the gravitational field.

For the gravitational field at a planetary surface it follows: The gravitational acceleration depends on the altitude, because according to the law of gravity . It also follows from this relationship that due to the oblateness of the planet, the distance to the center of the planet is smallest at the poles, and therefore the gravitational effect is greatest. In addition, at the poles of the celestial body the centrifugal acceleration disappears because the distance from the axis of rotation is zero. The gravitational field is thus weakest at the equator: there the centrifugal acceleration is at a maximum and counter to the gravitational effect, and the distance from the centre of the planet is at its greatest.

The direction of the gravitational acceleration is called the perpendicular direction. This perpendicular direction points approximately towards the gravitational centre of the celestial body. Deviations occur (apart from gravity anomalies) because the centrifugal acceleration at mid-latitudes is at an oblique angle to the gravitational acceleration. Lines that follow the perpendicular direction are called perpendiculars. They are the field lines of the gravitational field. If a body moves in the gravitational field, the direction of the effective acceleration deviates from the perpendicular direction with increasing velocity. This can be interpreted as the effect of the Coriolis force.