H1 - rotation of the milky way (tangent method)

most stars and gas in MW are flattened into an exponentially-decaying disk surrounding a bulge
if any object with mass emits light, light will trace the mass
it can be concluded that most of the mass is concentrated within a few kpc from the centre
defining an encircling radius $r_{e n c}$ above with the galactic mass is constant with radius, $M (r > r_{e n c}) = const$
so, Keplerian rotation can be used to estimate the velocity profile of the galaxy at large radii
equating the gravitational and centrifugal acceleration, so at $r > r_{e n c}$ , the rotational curve of the galaxy is:

\frac{G M}{r^{2}} = \frac{v^{2}}{r} ⟹ v (r) \propto r^{- 1 / 2}

measuring the rotation of MW is harder than for other galaxies as earth's own motion needs to be accounted for
to reconstruct the rotation profile, Doppler shift in the spectra of stars, HII regions, HI, CO, etc can be measured
any measured Doppler shift can be converted into a radial component along the line of sight of the velocity
these then need to be converted into orbital velocities
using geometry to solve for all the orbital components internal to the orbit of the sun is called the tangent method
geometrically:

v_{r} = v \cos θ - v_{⊙} \sin l

using the sine rule on the triangle defined by the sun, the object, and the galactic centre:

\frac{\sin l}{R} = \frac{\sin (α + π / 2)}{R_{⊙}} = \frac{\cos α}{R_{⊙}}

∴ v_{r} = R_{⊙} \sin l [\frac{v}{R} - \frac{v_{⊙}}{R_{⊙}}]

where, $R_{⊙}$ and $l$ are known, and $v$ and $R$ are unknown

moving along the line of sight to the point where $α = 90 °$ , ie. the line of sight is tangent to the orbit:

R = R_{⊙} \sin l

unless the rotational velocity increases with radius, the maximum radial velocity along the line of sight is measured at the tangent:

v (R) = v_{r, m a x} - v_{⊙} \sin l

note that the tangent point is only defined for $R < R_{⊙}$ , limiting the method to galactic longitudes to between $- π / 2$ and $π / 2$
so, beyond sun's orbit, estimates of the rotational curve rely on different distance methods, and are more subject to uncertainty