Positioning

To position the BDM signals, cf. Fig. 1 in BDM Physics, we need to analyze the geometry of propagation in order to determine how large the emissivity is at point B. Once the geometrical relations are properly understood, we can proceed to construct a class that solves this problem on-the-fly. See also Ref. [1], and note that we slightly modify the notations in this document for improved clarity.

Geometry

The geometry for BDM propagation is depicted in Fig. 1. The three coplanar points S, G, and E remain the same as before. Repeated quantities retain their original definitions, while we also introduce additional auxiliary terms whose purposes will become clear shortly.

geometry1
Figure 1. The geometry of BDM propagation.

By examining the blue and green triangles, we have

\[ \begin{align} h &= \ell \sin\theta \\ b &= \ell \cos\theta. \end{align} \]

Moreover, the blue and brown triangles are identical, with the brown one being a rotation of the blue triangle around the axis \(\overline{\mathsf{SE}}\) by angle \(\varphi\). See Fig. 2.

geometry1
Figure 2. Blue and brown triangles are identical.

Suppose point B is at a distance \(r\) from G, then its rotated counterpart B′ is at distance \(r^\prime\). Clearly, \(r\) is a special case of \(r^\prime\) when \(\varphi = 0\). This is crucial because when the SN is not located at the GC, the DM number density \(n_\chi\) is no longer spherically symmetric with respect to the SN. To correctly evaluate \(j_\chi\) at the boost point B, one must use the DM density evaluated at the rotated distance: \(n_\chi(r^\prime(\varphi))\).

geometry1
Figure 3. Another set of auxiliary triangles.

We draw another set of auxiliary triangles in Fig. 3, also cf. Fig. 1, and immediately observe that, in the left figure,

\[\begin{equation}\label{eq:rprime} r^{\prime 2} = a^2 + h^2 \cos^2\varphi, \end{equation}\]

while

\[\begin{equation} a^2 = (\rho \sin\delta - h \sin\varphi)^2 + \rho^2 \cos^2\delta, \end{equation}\]

and from the law of cosines, the right figure gives

\[\begin{equation} \rho = \sqrt{b^2 + R_e^2 - 2b R_e \cos\beta}. \end{equation}\]

To determine \(\delta\), we again apply the law of cosines (since we need to know whether \(\delta > \pi/2\)):

\[ R_e^2 = \rho^2 + b^2 - 2\rho b \cos(\pi - \delta), \]

which yields

\[\begin{equation} \cos\delta = \frac{R_e^2 - \rho^2 - b^2}{2\rho b}. \end{equation}\]

One can check that we have already determined \(r^\prime\) (Eq. \eqref{eq:rprime}) in terms of known quantities \((d, \ell, \theta, \varphi)\) and \((R_s, R_e, \beta)\). The first set is specified during the evaluation of BDM signatures, and the second set defines the SN location.

The last quantity to compute is the scattering angle \(\psi\), which can be obtained via the law of cosines:

\[ R_s^2 = d^2 + \ell^2 - 2d \ell \cos(\pi - \psi), \]

so that

\[\begin{equation} \cos\psi = \frac{R_s^2 - d^2 - \ell^2}{2d\ell}, \end{equation}\]

where

\[\begin{equation}\label{eq:d} d = \sqrt{\ell^2 + R_s^2 - 2\ell R_s \cos\theta}. \end{equation}\]

This indicates that \(d\) is not an independent quantity, but is instead determined by \(\ell\) and \(\theta\).

Static to time-dependent

From Eq. (15) in BDM Physics and offsetting by \(t_\nu = R_s / c\), we have

\[ t = \frac{d}{c} + \frac{\ell}{v_\chi} - t_\nu, \]

which leads to \begin{equation}\label{eq:t_dependent} d + \frac{\ell}{\beta_\chi} = R_s + ct, \end{equation} where \(\beta_\chi = v_\chi / c\). For convenience, we define \begin{equation} \zeta(t) = R_s + ct, \end{equation} and plug Eq. \eqref{eq:d} into Eq. \eqref{eq:t_dependent}, then solve for \(\ell\) \begin{equation}\label{eq:ell_t} \ell(t) = -\frac{\beta_\chi}{1 - \beta_\chi^2} \left( \alpha + \gamma - \zeta \right), \end{equation} where

\[\begin{align*} \alpha &= \sqrt{(R_s^2 - \zeta^2)(1 - \beta_\chi^2) + (R_s \beta_\chi \cos\theta - \zeta)^2}, \\ \gamma &= R_s \beta_\chi \cos\theta. \end{align*}\]

We can now determine \(\ell\) at any time \(t\) using Eq. \eqref{eq:ell_t}, and since \(d\) depends on \(\ell\) when \(\theta\) is specified (via Eq. \eqref{eq:d}), the geometry of BDM propagation becomes time-dependent, transitioning from a static configuration.

References

  1. Y.-H. Lin et al., Phys. Rev. D. 108, 083013 (2023)