Table 3: Analytical expression for Alterman et al. (1959) solution of the gravito-elastic deformation equations, generalized for a three-layer Earth.
Earth Core Inner core

y₁ $\frac{C_1}{r^n}+\frac{C_2}{r^{n+2}}+C_3r^{n+1}+C_4r^{n-1}$ y₅/g_n G₃rⁿ⁺¹+G₄ r^n-1

y₂ $\frac{-2\mu C_1(n^2+3n-1)}{(n+1)r^{n+1}} -\frac{2C_2(n+2)\mu}{r^{n+3}}+\frac{2}{n}C_3\mu (n^2-n-3)r^n$ / $\frac{2}{n}\mu_gG_3\left(n^2-n-3\right) r^n+2\mu_gG_4(n-1)r^{n-2}$

$\; \; \; \hspace{1cm} +2\mu C_4(n-1)r^{n-2}-\rho_my_5+g\rho_my_1$ $\; \;\;\hspace{1cm} -\rho_gy_5+g\rho_gy_1$

y₃ $-\frac{(n-2)C_1}{n(n+1)r^n}- \frac{C_2}{(n+1)r^{n+2}}+\frac{C_3(n+3)}{n(n+1)}r^{n+1}+\frac{C_4}{n}r^{n-1}$ / $G_3r^{n+1}\frac{n+3}{n(n+1)}+r^{n-1}\frac{G_4}{n}$

y₄ $\mu C_1\frac{2(n-1)}{nr^{n+1}}+\mu C_2\frac{2(n+2)}{(n+1)r^{n+3}}+\mu C_3 \frac{2(n+2)}{n+1}r^n$ 0 $2\mu_gG_3r^n\frac{n+2}{n+1}+2\mu_gG_4\frac{n-1}{n}r^{n-2}$

$+\frac{2(n-1)}{n}\mu C_4 r^{n-2}$

y₅ $C_5r^n+\frac{C_6}{r^{n+1}}$ $Cr^n+\frac{Dc}{r^{n+1}}$ G₅ rⁿ

y₆ $n C_5 r^{n-1}-C_6\frac{n+1}{r^{n+2}}-\frac{3g_0\rho_my_1}{a\rho_M}$ $nCr^{n-1}-\frac{Dc(n+1)}{r^{n+2}}-\frac{3g_0\rho_{\rm f}y_1}{a\rho_M}$ $G_5 nr^{n-1}-3\frac{g_0\rho_g y_1}{a\rho_M}$

**Table 3:** Analytical expression for Alterman et al. (1959) solution of the gravito-elastic deformation equations, generalized for a three-layer Earth.
	Earth	Core	Inner core
y₁	$\frac{C_1}{r^n}+\frac{C_2}{r^{n+2}}+C_3r^{n+1}+C_4r^{n-1}$	y₅/g_n	G₃rⁿ⁺¹+G₄ r^n-1
y₂	$\frac{-2\mu C_1(n^2+3n-1)}{(n+1)r^{n+1}} -\frac{2C_2(n+2)\mu}{r^{n+3}}+\frac{2}{n}C_3\mu (n^2-n-3)r^n$	/	$\frac{2}{n}\mu_gG_3\left(n^2-n-3\right) r^n+2\mu_gG_4(n-1)r^{n-2}$
	$\; \; \; \hspace{1cm} +2\mu C_4(n-1)r^{n-2}-\rho_my_5+g\rho_my_1$		$\; \;\;\hspace{1cm} -\rho_gy_5+g\rho_gy_1$
y₃	$-\frac{(n-2)C_1}{n(n+1)r^n}- \frac{C_2}{(n+1)r^{n+2}}+\frac{C_3(n+3)}{n(n+1)}r^{n+1}+\frac{C_4}{n}r^{n-1}$	/	$G_3r^{n+1}\frac{n+3}{n(n+1)}+r^{n-1}\frac{G_4}{n}$
y₄	$\mu C_1\frac{2(n-1)}{nr^{n+1}}+\mu C_2\frac{2(n+2)}{(n+1)r^{n+3}}+\mu C_3 \frac{2(n+2)}{n+1}r^n$	0	$2\mu_gG_3r^n\frac{n+2}{n+1}+2\mu_gG_4\frac{n-1}{n}r^{n-2}$
	$+\frac{2(n-1)}{n}\mu C_4 r^{n-2}$
y₅	$C_5r^n+\frac{C_6}{r^{n+1}}$	$Cr^n+\frac{Dc}{r^{n+1}}$	G₅ rⁿ
y₆	$n C_5 r^{n-1}-C_6\frac{n+1}{r^{n+2}}-\frac{3g_0\rho_my_1}{a\rho_M}$	$nCr^{n-1}-\frac{Dc(n+1)}{r^{n+2}}-\frac{3g_0\rho_{\rm f}y_1}{a\rho_M}$	$G_5 nr^{n-1}-3\frac{g_0\rho_g y_1}{a\rho_M}$

Source LaTeX | All tables | In the text

	1 Introduction
	2 Model used and main hypotheses
	3 Theoretical excitation of the free modes
	4 Normal modes and transfer function
	5 Love number for a three-layer Earth
	6 Excitation by a white noise
	7 Conclusions
	References