Question #f66fb

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Answer 1 · 2016-02-21T10:54:05

A gravitational quadrupole is a mathematical concept to explain the gravitational effect of a mass which is not just a point but which has an extension in the space

Explanation:

The most easy way to show its effects is considering 4 equal masses (m each of them) placed in every corner of a square of side $2 a$ $2a$ and calculate their interaction with a distant mass (M). The distance from the center of square to this mass is $r$ $r$ . It is very important that $r$ $r$ is huge in comparison with $a$ $a$ .

By using the Newton's gravitational law

$F = G \cdot \frac{M \cdot m}{R^{2}}$ $F=G*(M*m)/R^2$

$r$ $r$ , the distance between the mass M and the center of square, can be written in its components $x$ $x$ and $y$ $y$

$r^{2} = x^{2} + y^{2}$ $r^2 = x^2+y^2$

the same for the distance from mass M to every corner of square

$r_{1}^{2} = {(x - a)}^{2} + y^{2}$ $r_1^2=(x-a)^2+y^2$
$r_{2}^{2} = x^{2} + {(y - a)}^{2}$ $r_2^2=x^2+(y-a)^2$
$r_{3}^{2} = {(x + a)}^{2} + y^{2}$ $r_3^2=(x+a)^2+y^2$
$r_{4}^{2} = x^{2} + {(y + a)}^{2}$ $r_4^2=x^2+(y+a)^2$

the total gravitational force in the system is

$F = G \cdot M \cdot m \cdot [\frac{1}{r_{1}^{2}} + \frac{1}{r_{2}^{2}} + \frac{1}{r_{3}^{2}} + \frac{1}{r_{4}^{2}}]$ $F=G*M*m*[1/r_1^2+1/r_2^2+1/r_3^2+1/r_4^2]$

Taken in account the approximation formula

$\frac{1}{x + ε} \approx \frac{1}{x} \cdot (1 - \frac{ε}{x})$ $1/(x+epsilon) ~~ 1/x *(1-epsilon/x)$

$F \approx G \cdot \frac{M \cdot 4 m}{r^{2}} \cdot [1 - \frac{a^{2}}{r^{2}}] =$ $F~~G*(M*4m)/r^2*[1-a^2/r^2] =$

$= G \cdot \frac{M \cdot 4 m}{r^{2}} - G \cdot \frac{M \cdot 4 m \cdot a^{2}}{r^{4}}$ $= G*(M*4m)/r^2 -G*(M*4m*a^2)/r^4$

In this expression, we have 2 elements. The first one

$G \cdot \frac{M \cdot 4 m}{r^{2}}$ $G*(M*4m)/r^2$ is the interaction of 4 masses (m) as if they were

concentrated in a single point, just in the center of the square. The

second one $Q = - G \cdot \frac{M \cdot 4 m \cdot a^{2}}{r^{4}}$ $Q =−G⋅(M⋅4m⋅a^2)/r^4$ is the gravitational

quadropole for this example. See that this element is negative and decrease faster with distances.

This concept can be generalised to any geometrical configuration with 4 masses and also to larger number of masses (octopole, hexapole, etc.). When we have a extensive mass, its gravitational effects can be approximated in this way (point mass plus gravitational dipole plus gravitational quadrupole plus ...).

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