Questions tagged [multipole-expansion]
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320 questions
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Difference between the pear-shaped Earth, ellipsoid, and the geoid
About 32 years ago, when I was a university student, I took a course called Advanced Mechanics that also covered some astronomy topics.
I vaguely remember a derivation showing that the shape of the ...
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How to obtain the third gravitational potential term from this series?
In this paper the gravitational potential between a point mass and an extended rigid body is
$$
U = -\mathbb{G}m_1\int_B \frac{\mathrm{d}^3 \mathbf{Q}'\rho(\mathbf{Q}')}{|\mathbf{r} + \mathrm{C}\...
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What are the Euler-Lagrange equations of an approximated rigid body and a point planet interacting via gravity?
The potential due to a rigid body will be approximated using a Lagrange polynomial expansion as
$$
U(x,y,z)=-\frac{\mathbb{G}Mm_p}{r} - \frac{\mathbb{G}m_p(I_1 + I_2 + I_3)}{2r^3} + \frac{3\mathbb{G}...
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Why does one quadrupole operator have a different spectrum from the rest?
Consider a spin-$S$ system, with $2S+1$ eigenstates $\mid s \rangle$, $s\in -S,-S+1,\ldots +S$ and spin operators $S^x$, $S^y$, $S^z$. Consider the quadrupole moment,
$$Q^{\alpha\beta} = S^\alpha S^\...
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What is the relationship between "spectroscopic" and "intrinsic" nuclear quadrupole moments?
Several Physics SE questions/answers note the difference between intrinsic and spectroscopic electric quadrupole moments:
Why do spin-1/2 nuclei have zero electric quadrupole moment?
Measurement of (...
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Measurement of (intrinsic) electric quadrupole moment of neutron?
While there are amazing experimental boundaries for electric dipole moment of electron and neutron, for electric quadrupole moments I could only find for large nuclei.
It seems especially interesting ...
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Why is quadrupolar relaxation so much stronger than dipolar?
In the book "Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications" it is stated that
"For S ≥ 1 spins, the quadrupolar interaction is usually much stronger than ...
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What is meaning of exact in context of series?
In Griffith's book, while deriving the multipole expansion, binomial expansion was used. Now he claimed that the multipole expansion is "exact". Obviously I am aware that it is for large ...
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Expansion of electric field into Spherical Harmonics
A solution of of the Laplace equation in spherical coordinates that is regular at origin can according to Zangwill can be written as
$$
\varphi(r,\theta,\phi) = \sum_{lm} A_{lm} r^l Y_{lm}(\theta,\phi)...
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How to solve the Poisson equation for the source terms as the double derivative of a spherically symmetric scalar function ϕ?
I need help understanding how to solve the Poisson equation in 3D for $g_{ij}$, which incorporates the double derivative of a spherically symmetric scalar function $\phi(\textbf{x})$. Where $\textbf{x}...
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Physical interpretation of selection rules for different multipole orders
While going through this resource, I came across the following statement:
The interaction $eE(0, t) · x$
in (2.8) describes the coupling of the external field to the electric dipole moment $d = ex$ ...
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Getting a CMB map from the Cls using CAMB
I'm trying to get a temperature anisotropies map using CAMB, and I getting a trouble. In first place, I'm doing this with the following code:
...
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Is the quadrupole tensor a $(2,0)$ tensor?
In classical electromagnetism, the quadrupole moment is usually written as
$$Q_{ij}=\int d^3r \rho(\vec{r})(3r_ir_j-r^2\delta_{ij}). \tag{1}$$
However, this represents the quadrupole tensor as a ...
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Where does the factor $1/3$ in the wave metric come from?
In P.T. Chrusciel's "Elements of General Relativity" (Birkhäuser 2020) there is the following calculation:
The linearized Einstein Field equations can, after a suitable choice of coordinates,...
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Quadrupole radiation formula for gravity
I derived the formula for the quadrupole radiation power emitted by a system of masses:
$$P=\frac{1}{45}\dddot{Q}_{kl}\dddot{Q}_{kl} .\quad\quad (*)$$
Note here that: (1) I am using geometrized units, ...
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