# Question f66fb

Feb 21, 2016

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 $2 a$ and calculate their interaction with a distant mass (M). The distance from the center of square to this mass is $r$. It is very important that $r$ is huge in comparison with $a$.

By using the Newton's gravitational law

$F = G \cdot \frac{M \cdot m}{R} ^ 2$

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

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

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

${r}_{1}^{2} = {\left(x - a\right)}^{2} + {y}^{2}$
${r}_{2}^{2} = {x}^{2} + {\left(y - a\right)}^{2}$
${r}_{3}^{2} = {\left(x + a\right)}^{2} + {y}^{2}$
${r}_{4}^{2} = {x}^{2} + {\left(y + a\right)}^{2}$

the total gravitational force in the system is

$F = G \cdot M \cdot m \cdot \left[\frac{1}{r} _ {1}^{2} + \frac{1}{r} _ {2}^{2} + \frac{1}{r} _ {3}^{2} + \frac{1}{r} _ {4}^{2}\right]$

Taken in account the approximation formula

$\frac{1}{x + \epsilon} \approx \frac{1}{x} \cdot \left(1 - \frac{\epsilon}{x}\right)$

$F \approx G \cdot \frac{M \cdot 4 m}{r} ^ 2 \cdot \left[1 - {a}^{2} / {r}^{2}\right] =$

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

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

$G \cdot \frac{M \cdot 4 m}{r} ^ 2$ 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⋅(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 ...).