Gravity and revolved

Interest of the field of gravity

The importance of the field of gravity of the Earth for the geodesists is conceived easily when one realizes that its direction in each point, which corresponds to the vertical of the place provided by the plumbline, is used as reference at the time of the erection of any geodetic measuring instrument. In a more detailed way, one includes/understands the interest of the field of gravity for the following reasons:

- Its values on the surface and outside the Earth are used as reference to the majority of the quantities measured in geodesy. In fact, the field of gravity must be known in order to reduce the observable geodetic ones in geometrically defined systems.

- The distribution of the values of gravity on the terrestrial surface allows, in combination with other geodetic measurements, to determine the form of this surface.

- The surface of the most important reference for measurements heights - that one calls the Geoid - is a Equipotential surface field of gravity.

- The analysis of the external field of gravity provides information on the structure and the properties of the interior of the Earth. While making this information available, geodesy becomes an auxiliary science of geophysics. It is what occurred in a way accelerated during the last decades, with the advent of the space Géodésie.

Components of the field of gravity

A body interdependent of the earth's crust is subjected to the gravific attraction of the Earth and other cosmic bodies, like with the force axifuge (or centrifuges) caused by terrestrial rotation. The resultant of these forces is gravity. It depends on the geographical location of the body, but also of time. The study of the space and temporal variations of gravity is one of the main aims of physical geodesy. One can notice right now that the general study of the field of gravity of the Earth is based mainly on the use of artificial satellites turning around the Earth. The rotation of these satellites is uncoupled from terrestrial rotation, and the satellites undergo consequently the only gravific component of planet. The latter implicitly depends however itself on the component axifuge, by the fact that rotation affects the distribution of the masses by more or less flattening the various layers of the Earth according to whether it is more or less fast.

Gravity is a force communicating to a unit of mass an acceleration G , which is variable in space and time. In the international system, the unit of acceleration is the meter a second a second (m/s ²). The intensity of the vector gravity G , i.e. G, in the vicinity of terrestrial surface is close to 10 m/s ², with maximum variations reaching approximately 0,2%. In general, the variations ∆g of G are more important for the geodesist and the geophysicist who absolute intensities - was this only because of the fact that one can make differential measurements with more precision than of absolute measurements. Consequently, a practical unit for gravimetry is the cm/s ². One gave to the latter the name “Gall” in the honor of the large Italian physicist Galileo Galilei.

The maximum variation of G on the surface of the Earth thus reaches about 5 Gall, and is ascribable to the variation of G with the latitude. Variations with shorter wavelengths, known like gravimetric anomalies geoid, are typically of some dizièmes to a few tens of milligals (mgal). In certain geodynamic phenomena whose observation became possible recently thanks to progress of the geodetic instrumentation, one is interested in variations of G according to the time whose amplitude reaches only some microgals (µgal). In fact, theoretical studies (Modes of the core, secular variation of G) currently consider variations of G being at the level of the nanogal (ngal).

In gravimetric prospection and civil engineering, the significant anomalies of G generally are included/understood between some microgals and some dizièmes of milligal. To fix the ideas, let us note that when on the surface of the Earth we rise of three meters, gravity varies approximately 1mgal. Let us retain que
1 Gall =10^-2 m/s ² ≅ 10^-3g,
1 mgal = 10^-3 Gall = 10^-5 m/s ² ≅ 10^-6g,
1 µgal = 10^-6 Gall = 10^-8 m/s ² 10^-9g,
1 ngal = 10^-9 Gall = 10^-11 m/s ² 10^-12g.

Gravific attraction

According to Isaac Newton, which formulated the law of gravitation in its work “Principles mathematical of Natural Philosophy” appeared in 1687, two material points P and Q, of respective masses M_P and M_Q, attract mutually with a force whose intensity is worth

F (P, Q) = GM_PM_Q/d ² (P, Q),

D (P, Q) indicating the distance separating the points P and Q, and G the Constant of gravitation of Newton being worth (6,67259 ± 0,00085) X 10^-11 m ³ kg^-1s-2. This force is carried by the line uniting the points P and Q. Out of P it is directed towards Q: F (P, Q) = F (P, Q) text-decoration: overline > PQ /d (P, Q). In Q, it is directed towards P: F (Q, P) = F (P, Q) text-decoration: overline > QP /d (P, Q).

In the continuation, P will in general indicate the attracted point (one says also the potentié ) and Q the attracting point (or not potentiant ). We will admit that P is a unit particle-test of mass (M_P = 1). The attracting force being exerted out of P is then a force per unit of mass, or forces specific, called gravity or gravific force . We will indicate it here X (P, Q). One thus has:

X (P, Q) = F (P, Q) /M_P = GM_Q text-decoration: overline > PQ d^-3 (P, Q).

Dimensions of this gravific force, which according to the first law of Newton communicates an acceleration G (called gravific acceleration ) with the unit mass out of P, are thus those of an acceleration, i.e. LT^-2. The size X is thus expressed in m/s ².

A body such as the Earth is composed of an quasi-infinite number of mass points, by implying by mass point a discrete point a such atom or a molecule. Earth in its entirety - or very part of this one - armature on the particle-test out of P an attraction force X resulting from the vectorial summation of the forces exerted individually by all the mass points.

Concept of potential of gravity

To simplify these calculations, it is advantageous to pass from the vector Field X to a scalar Champ V, by using the theorem of representation of Helmholtz. This last provides a method of decomposition for a general vector field v , with the help of the relation

v = ∇ϕ + ∇ X ψ

provided with a condition of gauge

∇ ・ ψ = 0

to reduce the component count independent of the vector potential ψ to 2. Here, we will suppose the quantities v , ϕ and ψ expressed in terms of Cartesian coordinates x₁, x₂, x₃. The Cartesian axes are located by means of the unit vectors E ₁, E ₂, E ₃ parallel respectively with the Ox₁ directions, Ox₂, Ox₃, O being the origin of the system of axes. Then, the preceding equations can be written in the form of components respectively

v₁ = ∂₁ϕ + ∂₂ψ₃ - ∂₃ψ₂,
v₂ = ∂₂ϕ + ∂₃ψ₁ - ∂₁ψ₃,
v₃ = ∂₃ϕ + ∂₁ψ₂ - ∂₂ψ₁,

and

∂₁ψ₁ + ∂₂ψ₂ + ∂₃ψ₃ = 0.

According to a rather current habit, we indicate by ∂_i the IE component of the gradient ∇, i.e. the partial derivative compared to x_i:

∂_i = ∇_i = ∂/∂x_i.

It is well-known that the rotational one of a gradient just as the divergence of rotational are null. That is proven by the following identities:

∇ X (∇ϕ) ≡ 0,

∇ ⋅ (∇ X ψ ) ≡ 0.

Like the field of gravific force X (x₁, x₂, x₃) is irrotational, in other words ∇ X X = 0 one sees that X can be written like gradient of a scalar potential. Instead of the general notation ϕ, we will indicate in the continuation this potential gravific by V, therefore X (P, Q) = ∇V (P, Q) or, in notation of components (i.e. in indicielle notation): X_i (P, Q) = ∂_iV (P, Q).

Let us consider the quantity now

d^-1 = + (y₂-x₂) ² + (y₃-x₃) ² ^-1/2.

If we interpret, as we will systematically do it in the continuation, the y₁, y₂, y₃ like the components of the vector-position there associated with a point potentiant Q, and the x₁, x₂, x₃ like the components of the vector-position X associated with a point potentié P, then d^-1 represents the reverse of the distance between the points P and Q: d^-1 = d^-1 (P, Q). Its partial derivative evaluated at the point P,

∂_id^-1 (P, Q) = (y_i-x_i) + (y₂-x₂) ² + (y₃-x₃) ² ^-3/2

represent the IE component of the gradient of the distance reverses d^-1. By comparing this expression with the expression providing produced gravity out of P by a mass point in Q, we see that

X (P, Q) = GM_Q ∇d^-1 (P, Q).

Consequently, the potential V produces out of P by Q can be written

V (P, Q) = GM_Qd^-1 (P, Q) + V_∞.

The additive constant V_∞ is not essential since it does not intervene when the gradient is taken. Nevertheless, one can cancel it by a judicious definition of the potential, as indicated below.

Work to carry out to bring a particle-test in the field gravific X produces by a mass point Q of a P_∞ point located ad infinitum at the point P is provided by the expression

T (P_∞→P, Q) = ∫_{P_∞→P} X ⋅ of = ∫_{P_∞→P} (X₁dx₁ + X₂dx₂ + X₃dx₃),

of being itself an arbitrary element of the trajectory. Taking into account the fact that X_i (P, Q) = GM_Q ∂d^-1 (P, Q)/∂x_i, the curvilinear integral above is evaluated easily and one finds simply

T (P_∞→P, Q) = GM_Q - d^-1 (P_∞, Q) = GM_Q d^-1 (P, Q)

since d^-1 (P_∞, Q) = ∞.

By conventionally defining the potential gravific as work to be carried out to bring a particle-test in a field gravific of an infinite distance in a point P, the relations above imply that V_∞ = 0. The potential gravific produces out of P by a specific mass M_Q located in Q is thus

V (P, Q) = GM_Q d^-1 (P, Q).

By using the principle of superposition, the potential gravific created out of P by NR isolated specific masses located at the Q₁ points, Q₂, ⋯, Q_N can be written

V (P) = ∑_{i=1→N GM_i d^-1 (P, Q_i)}

if we indicate by M_i the mass of the material point located in Q_i.

The terrestrial material is generally designed like a continuous accumulation of mass points rather than like a system of discrete mass points. One passes from discrete description to continuous description via the concept of density improperly but universally called density as well in sciences of the Earth, in particular in geodesy, as in sciences of the Universe.

The density in a point Q is defined so that: by the thought let us surround the point Q of a small volume Δτ (Q), and is ΔM (Q) the mass of the quantity of matter contained in this volume. One then defines the density ρ (Q) in Q by the limit of report/ratio ΔM (Q)/Δτ (Q) when Δτ (Q) becomes increasingly small, provided that this limit exists. In macroscopic description that we make here of Nature, we will admit that this limit always exists. Consequently, we will suppose the Earth subdivided in elements of mass DM (Q) associated with elements of volume dτ (Q) by the relation

DM (Q) = ρ (Q) dτ (Q).

Mathematically, the elements of volume Δτ (Q) must be conceived like infinitely smalls volumes, but from a physical point of view they must be sufficiently large to contain enough structural entities (atoms, molecules, microcrystals, minerals, rocks, according to the smoothness of desired description), so as to be able to define significant averages for the various physical properties. With the help of these concepts, one can pass from discrete description to a description continues: one replaces the mass point Q of mass M (Q) by the material element dτ (Q) recovering the point potentiant Q, of elementary mass DM (Q) provided by DM (Q) = ρ (Q) dτ (Q), and one replaces the sum on all the points isolated Q_i by the Intégrale from Stieltjes

V (P) = ∫_M GdM (Q) /d (P, Q), integration being extended to all the mass M of the Earth.

The potential function V was explicitly introduced by Laplace and was used by Legendre at the end of the XVIIIe century, and the concept of potential was implicit in writings of Laplace and Gauss dating from first half of the XIXe century. However, the potential term itself was forged by Stokes only in second half of the XIXe century.

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