= Sky projection of the Galactic-centre orbit model
{title2=$\boldsymbol\theta=(x/D,y/D)$}
Let $z$ increase away from the observer, with $x,y$ spanning the sky plane. A rotation $R=R_z(\Omega)R_x(i)R_z(\omega)$ sends the orbital-plane coordinates to $\mathbf r=R\mathbf r'$. Here $i$ is <orbital inclination>, $\Omega$ the <longitude of ascending node>, and $\omega$ the <argument of periapsis>. For distance $D\gg a$, the small-angle model gives <angular positions> $\theta_x=x/D$, $\theta_y=y/D$ in <radians>. <Doppler spectroscopy> instead supplies the line-of-sight <velocity> $v_{\rm los}=\dot z+v_{\rm sys}$, with a fitted systemic offset $v_{\rm sys}$.
Define $\alpha=a/D$, the angular scale of the \b[deprojected] fitted <orbit>. It is not generally the semi-major axis of the apparent ellipse on the sky. Projection also need not send a true focus to a focus of the apparent ellipse. Fitting the rotated orbit preserves that distinction; measuring a bounding ellipse alone loses it.
\Image[/galactic-centre-stellar-orbit-projection.png]
{title=A true stellar orbit and its apparent path on the sky}
{description=Original Newtonian orbit with chosen $a=825\,\mathrm{AU}$, $P=18$ years, $e=0.70$, $i=55$ degrees, $\omega=35$ degrees and $\Omega=20$ degrees, viewed from $D=7.5\,\mathrm{kpc}$. Markers are equally spaced in model time. This is not a reconstruction of S2 or historical astrometry.}
{source=galactic-centre-stellar-orbit-projection.py}
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