Vertical position of the Sun with $\gamma$-rays

We illustrate a method for estimating the vertical position of the Sun above the Galactic plane by $\gamma$-ray observations. Photons of $\gamma$-ray wavelengths are particularly well suited for geometrical and kinematic studies of the Milky Way because they are not subject to extinction by interstellar gas or dust. Here, we use the radioactive decay line of $\mathrm{^{26}Al}$ at $1.809\,\mathrm{MeV}$ to perform maximum likelihood fits to data from the spectrometer SPI on board the INTEGRAL satellite as a proof-of-concept study. Our simple analytic 3D emissivity models are line-of-sight integrated, and varied as a function of the Sun's vertical position, given a known distance to the Galactic centre. We find a vertical position of the Sun of $z_0 = 15 \pm 17\,\mathrm{pc}$ above the Galactic plane, consistent with previous studies, finding $z_0$ in a range between $5$ and $29\,\mathrm{pc}$. Even though the sensitivity of current MeV instruments is several orders of magnitude below that of telescopes for other wavelengths, this result reveals once more the disregarded capability of soft $\gamma$-ray telescopes. We further investigate possible biases in estimating the vertical extent of $\gamma$-ray emission if the Sun's position is set incorrectly, and find that the larger the true extent, the less is it affected by the observer position. In the case of $\mathrm{^{26}Al}$ with an exponential scale height of $150,\mathrm{pc}$ ($700\,\mathrm{pc}$) in the inner (full) Galaxy, this may lead to misestimates of up to $25\,\%$.