20 Legendre Polynomials
Legendre Polynomials are an important tool for modelling dipole and multi-pole potentials with axial-symmetric symmetry. In this chapter, we introduce some examples of their usefulness.
Key applications include modeling the Earth’s oblateness, gravitational potential, tidal effects, satellite orbit perturbations, global circulation patterns, and large-scale geomagnetic and seismic phenomena. The simplification to axisymmetric phenomena is particularly useful when longitudinal variations are negligible or when studying global-scale effects.
20.1 Gravitational Potential
We depart from the potential \(V(\mathbf r)\) at a point \(\mathbf{r}\) caused by a point mass \(m\) located at \(\mathbf{r}'\).
The potential at \(\mathbf{r}\) due to a point mass \(m\) at \(\mathbf{r}'\) is given by: \[ V(\mathbf{r}) = -f \frac{m}{|\mathbf{r} - \mathbf{r}'|}. \]
Suppose the point mass is slightly displaced to \(\mathbf{r}' = (0, 0, s)^\top\) along the \(z\)-axis.
20.1.1 Expansion of the term \(1/|\mathbf{r} - \mathbf{r}'|\)
The frequently occurring term \(1/|\mathbf{r} - \mathbf{r}'|\) can be expanded into a power series when \(|\mathbf{r}'| \ll |\mathbf{r}|\).
Let \(\mathbf{r} = (x, y, z)^\top\) (observation point) and \(\mathbf{r}' = (0, 0, s)^\top\) (source point of the mass). The Euclidean distance between the observation and source points is \(q = |\mathbf{r} - \mathbf{r}'|\).
Using the magnitudes \(r = |\mathbf{r}|\) and \(s = |\mathbf{r}'|\), and applying the law of cosines, we have: \[ \frac{1}{q} = f(r, \theta) = \frac{1}{r} \left[1 + \left(\frac{s}{r}\right)^2 - 2 \frac{s}{r} \cos \theta \right]^{-1/2}. \]
20.1.2 Binomial Expansion for Small Displacements
For a small displacement of the source point away from the origin, i.e., for \(s \ll r\), the function \(f(r, \theta)\) can be expanded as a binomial series \[ f(r, \theta) = \frac{1}{r} (1 + b)^{-1/2} = \frac{1}{r} \sum_{n=0}^\infty \binom{-1/2}{n} b^n, \] where \[ b = \left(\frac{s}{r}\right)^2 - 2 \frac{s}{r} \cos \theta, \quad |b| < 1. \]
Rearranging the series in powers of \(\frac{s}{r}\) gives \[ f(r, \theta) = \frac{1}{r} \sum_{n=0}^\infty \left(\frac{s}{r} \right)^n P_n(\cos \theta). \]
The \(P_n(\cos \theta)\) are polynomials of degree \(n\) in \(\cos \theta\). They are known as Legendre Polynomials or zonal spherical harmonics. Since they are independent of the azimuthal angle \(\varphi\), they have constant values over spherical zones (\(\theta = \text{const}\)).
The Legendre polynomials can also be defined as the coefficients in a formal expansion in powers of \(t\) of the generating function
\[ \frac{1}{\sqrt{1 - 2tx + t^2}} = \sum\limits_{n=0}^\infty P_n(x)t^n. \] The coefficient of \(t^n\) is a polynomial in \(x\) of degree \(n\) with \(|x| \le 1\).
The first Legendre polynomials are
| \(n\) | \(P_n(x)\) |
|---|---|
| \(0\) | \(1\) |
| \(1\) | \(x\) |
| \(2\) | \(\dfrac{1}{2}(3x^2 -1)\) |
Legendre polynomials can be calculated using Rodrigues’ formula
\[ P_n(x) = \frac{1}{2^n n!} \frac{\mathrm d^n}{\mathrm dx^n}(x^2 - 1)^n \]
20.1.3 Description of Latitudinal Variations in Potential
The superposition of Legendre polynomials can describe the latitude-dependent variations in potential.
For \(r \ll s\), a similar result is obtained by factoring out \(1/s\) \[ f(r, \theta) = \begin{cases} \dfrac{1}{s} \sum_{n=0}^\infty \left(\dfrac{r}{s}\right)^n P_n(\cos \theta), & r \leq s, \\ \dfrac{1}{r} \sum_{n=0}^\infty \left(\dfrac{s}{r}\right)^{n+1} P_n(\cos \theta), & r \geq s. \end{cases} \]
20.1.4 Equivalence of Series Expansions
For \(r \geq s\), the series satisfy \[ \frac{1}{r} \sum_{n=0}^\infty \left(\frac{s}{r}\right)^n P_n(\cos \theta) = \frac{1}{s} \sum_{n=0}^\infty \left(\frac{s}{r}\right)^{n+1} P_n(\cos \theta). \]
20.1.5 Special Case for \(s = 0\)
When \(s = 0\), only the term for \(n = 0\) contributes a non-zero value. Thus \[ f(r, \theta) = \frac{1}{r}. \]
20.2 Electrostatic potential
Observation point at \(\mathbf{r} = (x, y, z)^\top\), \(r = |\mathbf{r}|\).
Source points \(\pm \mathbf{r}' = (0, 0, \pm a)^\top\).
Two charges on the \(z\)-axis: \(+Q\) at \(z = a\) and \(-Q\) at \(z = -a\).
The distances between the charges and the observation point are obtained using the law of cosines.
Distance to the charge \(+Q\): \[ | \mathbf{r} - \mathbf{r}'| = \sqrt{r^2 + a^2 - 2ar \cos \theta} = r \sqrt{1 - 2 \frac{a}{r} \cos \theta + \left( \frac{a}{r} \right)^2}. \]
Using \(\cos \theta = -\cos(\pi - \theta)\) (complementary angle), the distance to the charge \(-Q\) is: \[ | \mathbf{r} + \mathbf{r}'| = \sqrt{r^2 + a^2 + 2ar \cos \theta} = r \sqrt{1 + 2 \frac{a}{r} \cos \theta + \left( \frac{a}{r} \right)^2}. \]
20.2.1 Substitution into the Potential
Substituting these distances into the potentials: \[ U(r) = \frac{Q}{4 \pi \varepsilon r} \left\{ \left[1 - 2 \frac{a}{r} \cos \theta + \left( \frac{a}{r} \right)^2 \right]^{-1/2} - \left[1 + 2 \frac{a}{r} \cos \theta + \left( \frac{a}{r} \right)^2 \right]^{-1/2} \right\}. \]
20.2.2 Expansion Using Legendre Polynomials
We apply Legendre polynomials, defined as: \[ P_n(x) = \frac{1}{2^n n!} \frac{\mathrm{d}^n}{\mathrm{d}x^n} \left( x^2 - 1 \right)^n, \] with \(t := \frac{a}{r}\) and \(x := \cos \theta\).
For \(tx := -\frac{a}{r} \cos \theta\), the sign propagates through \(t^n = \left(-\frac{a}{r}\right)^n = (-1)^n \left(\frac{a}{r}\right)^n\): \[ U(r) = \frac{Q}{4 \pi \varepsilon r} \left[ \sum_{n=0}^\infty P_n(\cos \theta) \left( \frac{a}{r} \right)^n - \sum_{n=0}^\infty P_n(\cos \theta) (-1)^n \left( \frac{a}{r} \right)^n \right]. \]
20.2.3 Simplification
Terms with even powers cancel because \((-1)^{2n} = 1\).
Thus, the potential simplifies to: \[ U(r) = \frac{2Q}{4 \pi \varepsilon r} \left[ P_1(\cos \theta) \left( \frac{a}{r} \right) + P_3(\cos \theta) \left( \frac{a}{r} \right)^3 + \ldots \right]. \]
For \(r \gg a\), the term with \(P_1(\cos \theta)\) dominates.
This leads to the electric dipole potential: \[ U(r) = \frac{2aQ}{4 \pi \varepsilon} \cdot \frac{P_1(\cos \theta)}{r^2}. \]
20.2.4 Legendre Polynomial for \(n = 1\)
The Legendre polynomial of degree \(n=1\) is: \[ P_1(\cos \theta) = \cos \theta. \]
This corresponds to the equation (with the electric dipole moment \(p = 2aQ\)) derived using the electric dipole and the superposition principle.