Coordinate Transformations¶

We briefly introduce how to extract the off-center angle $\beta$, cf. Fig. 1 in BDM Physics, from two widely used astronomical coordinate systems: galactic and equatorial coordinates. This enables the input of arbitrary supernova (SN) locations from astrophysical databases, e.g.
SRcat, into snorer, allowing users to evaluate its corresponding BDM signature.

Galactic coordinate¶

The galactic coordinate system is shown in Fig. 1, with longitude $\ell$ and latitude $b$. Once $(\ell, b)$ for a SN is specified, its position on the celestial sphere is determined. Given that the distance from Earth to the SN, $R_s$, is typically provided by the database, the final task is to extract $\beta$, the deviation of SN from galactic center (GC) in polar direction.

Figure 1. The galactic coordinate system, including longitude $\ell$ and latitude $b$.

Before proceeding, note that there are two conventions within this coordinate system. In the Cartesian representation, Earth is placed at the origin, and the GC is at $(x, y, z) = (0, R_e, 0)$. See the side view of this geometric configuration in Fig. 2.

Figure 2. The Cartesian representation of galactic coordinate with distance included.

With two unit vectors—one pointing from Earth to the SN, $\hat{\mathbf{s}} = (x_s, y_s, z_s) / R_s$, and the other pointing from Earth to the GC, $\hat{\mathbf{g}} = (0, 1, 0)$—we immediately find

\[\begin{equation} y_s = R_s \cos b \cos\ell. \end{equation}\]

The reason why $x_s$ and $z_s$ are not needed becomes clear when we evaluate $\beta$,

\[\begin{equation}\label{eq:cos_beta_MW} \cos\beta = \hat{\mathbf{g}} \cdot \hat{\mathbf{s}} = \cos b \cos\ell, \end{equation}\]

which involves only the $y$-component of the SN position in galactic coordinates.

Note that when we extract $\beta$ from galactic coordinates, the inverse transformation is not possible. It is evident that the map from $(\ell, b)$ to $\beta$ is many-to-one: $\beta$ is a scalar while $(\ell, b)$ contains two degrees of freedom. Without an additional constraint, the inverse problem is ill-posed. Therefore, reconstructing the SN location relative to the galactic plane using only $\beta$ is implausible.

For completeness, to fully recover the spatial location, one would need the missing azimuthal angle around the Earth-GC axis, which depends on both $x_s$ and $z_s$. We omit further detail here, as it is irrelevant to our current purpose.

Supernova in arbitrary distant galaxy¶

We now consider a more general case where the SN lies in an arbitrary distant galaxy, and we wish to evaluate the corresponding BDM signature from that SN. Again, regardless of how the scene changes, the underlying task remains the same: we must determine three quantities, $R_g$, $R_s$, and $\beta$. The latter two are already familiar, while $R_g$ refers to the distance between Earth and the center of the distant galaxy. Effectively, this means we replace $R_e$ by $R_g$ in the calculation. See Fig. 3.

Figure 3. SN in an arbitrary distant galaxy.

Suppose the galactic coordinates for the two stellar objects are $(\ell_s, b_s)$ for the SN and $(\ell_g, b_g)$ for the distant GC. We can use the same procedure as before to obtain their Cartesian representations: $(x_s, y_s, z_s)$ for the SN and $(x_g, y_g, z_g)$ for the galaxy center. Thus, for the SN:

\[\begin{align*} x_s &= R_s \cos b_s \sin(2\pi - \ell_s), \\ y_s &= R_s \cos b_s \cos \ell_s, \\ z_s &= R_s \sin b_s, \end{align*}\]

and for the distant galaxy:

\[\begin{align*} x_g &= R_g \cos b_g \sin(2\pi - \ell_g), \\ y_g &= R_g \cos b_g \cos \ell_g, \\ z_g &= R_g \sin b_g. \end{align*}\]

We define two unit vectors for the two stellar objects: $$ \hat{\mathbf{s}} = (-\cos b_s \sin \ell_s, \cos b_s \cos \ell_s, \sin b_s), $$ for pointing to the SN direction and $$ \hat{\mathbf{g}} = (-\cos b_g \sin \ell_g, \cos b_g \cos \ell_g, \sin b_g). $$ for pointing to the GC. The angle $\beta$ can then be retrieved using the same dot product formula: \begin{equation} \cos \beta = \hat{\mathbf{g}} \cdot \hat{\mathbf{s}} = \cos b_s \cos b_g \cos(\ell_s - \ell_g) + \sin b_s \sin b_g. \end{equation} In the case of our MW, where $\ell_g = b_g = 0$, this expression reduces to Eq. \eqref{eq:cos_beta_MW}.

Equatorial coordinate¶

Another commonly used coordinate system is the equatorial coordinate system, which is specified by right ascension $\alpha$ (RA) and declination $\delta$ (DEC). Due to its complexity, we do not provide the detailed mathematical conversion from $(\alpha, \delta)$ to $\beta$ here.

To handle this task, we make use of Astropy to convert $(\alpha, \delta)$ into $(\ell, b)$. Once this conversion is done, we can follow the discussion in the previous section to compute $\beta$. Note that in snorer, we assume the input $(\alpha, \delta)$ values are expressed in the ICRS J2000.0 frame.