Higher multipoles of the cow

Abstract

The spherical cow approximation is widely used in the literature, but is rarely justified. Here, I propose several schemes for extending the spherical cow approximation to a full multipole expansion, in which the spherical cow is simply the first term. This allows for the computation of bovine potentials and interactions beyond spherical symmetry, and also provides a scheme for defining the geometry of the cow itself at higher multipole moments. This is especially important for the treatment of physical processes that are suppressed by spherical symmetry, such as the spindown of a rotating cow due to the emission of gravitational waves. I demonstrate the computation of multipole coefficients for a benchmark cow, and illustrate the applicability of the multipolar cow to several important problems.

Figures (6)

• Figure 1: Illustration of the choice of $\bm{\mathrm{f}}$ used in this work (projected). The point $(\theta, \phi)\in S^2$ is mapped by the homeomorphism $\mathcal{F}$ to a point on the surface of the cow. This point, $\mathcal{F}(\theta, \phi)\in\partial\mathcal{C}$, has a radial coordinate $f_r(\theta, \phi)$, and also has different angular coordinates from the original point. The differences in the $\theta$ and $\phi$ coordinates are given by $\bar{f}_{\Delta\theta}$ and $\bar{f}_{\Delta\phi}$, respectively. The means of these angular differences are then subtracted by rotation of the coordinate system.
• Figure 2: Increasing level sets of $d_{\mathcal{C}}$. The function $\bm{\mathrm{\varphi}}(\bm{\mathrm{x}}, t)$ maps between these level sets from left to right with increasing $t$. This has the effect of "inflating" the cow to a star-shaped domain, from which the boundary can be projected onto the sphere smoothly and bijectively.
• Figure 3: Distance gradient flow as a mapping to the unit sphere ($\mathcal{F}^{-1}$), shown here for points within 0.001 benchmark units of the $yz$ plane. The original points are shown in the interior (blue). Points on the circle (orange) are obtained by integrating along the gradient shown by the gray streamlines. Note the sharply different behavior of the streamlines above and below the cow: below, the presence of the legs (out of the plane shown) pushes the flow away from the front and back of the cow.
• Figure 4: Cow geometry $\partial\mathcal{C}$ reconstructed via the distance gradient flow method including progressively higher multipole contributions. Top: monopole, dipole, and quadrupole. Middle: octupole, hexadecapole, and dotriacontapole. Bottom: tetrahexacontapole, octacosahectapole, and full cow for comparison.
• Figure 5: Homeomorphism $\mathcal{F}$ between the cow and the unit sphere obtained by the method of harmonic maps. In each column, colors correspond to the value of one component of $\bm{\mathrm{f}}$. The unit sphere is represented via Mollweide projection in the bottom row.