Tangent-point method

Look along a line of sight through the inner Galaxy, at longitude l in the first quadrant. That line passes closest to the Galactic centre at one point, the tangent point, at a distance R = R₀ sin l from the centre. If the gas moves in circular orbits, the gas at the tangent point is moving exactly along your line of sight, so it has the largest radial velocity of anything on that line. In a spectrum, that’s the high-velocity edge of the hydrogen emission, called the terminal velocity.

Measure the terminal velocity at a series of longitudes and each one gives you a point on the rotation curve: a radius, and the orbital speed at that radius.

Worked example

At l = 30°, suppose the high-velocity edge of the spectrum sits at an LSR velocity of 91.2 km/s. (This is an illustration of the arithmetic, the same one used in LAB-013.) With R₀ = 8.5 kpc and V₀ = 220 km/s:

• R = 8.5 × sin 30° = 4.25 kpc
• V = 91.2 + 220 × 0.5 = 201 km/s

Repeat at l = 20°, 40°, 50° and so on and the points start to trace a curve that stays nearly flat, which is the famous result: the Galaxy does not rotate the way its visible mass alone predicts.

What it needs

• LSR-corrected velocities. Without the LSR correction your edges are off by up to tens of km/s, and that error goes straight into V.
• Quadrant I or IV only, inside the solar circle. At l > 90° there is no tangent point; the method only works for R < R₀. Between about 20° and 70° is the comfortable range from mid-northern latitudes.
• A consistent edge definition. Pick a rule (for example, the velocity where the emission falls to a set fraction of its peak, or crosses a few sigma above the noise) and apply it the same way at every longitude.
• State your R₀ and V₀. The IAU 1985 values are still what most amateur comparisons use; newer measurements put R₀ nearer 8.2 kpc. Either is fine if you say which.

The transit planner tells you when each longitude is highest, which is when to take the spectrum. Galactic coordinates explains the quadrants.