Conventions

This page documents the coordinate systems, angle definitions, and sign conventions used throughout ROTIR. All conventions are consistent across the geometry, orbital, radial-velocity, and plotting code.

Sky plane

ROTIR adopts the standard astronomical sky-plane convention:

Internally, projected vertex coordinates are stored as proj_west and proj_north (both in mas). Plots negate the West component so that East points left, matching the standard on-sky orientation for optical interferometry.

Stellar surface coordinates

The unit sphere is parameterized in geometric spherical coordinates:

Conversion to Cartesian: x = sin(θ)cos(φ), y = sin(θ)sin(φ), z = cos(θ).

Spin-axis angles

Inclination

Angle between the stellar spin axis and the line of sight (z-axis toward observer).

Position angle (PA)

Angle of the projected spin axis on the sky, measured from North through East (counterclockwise on plots where East is left).

The projected spin-axis direction on the sky is:

spin_x = -sin(inc) sin(PA)   (West component)
spin_y =  sin(inc) cos(PA)   (North component)
spin_z =  cos(inc)           (toward observer)

Rotation matrix

ROTIR uses a ZXZ Euler convention. The three angles are applied as a right-multiply on vertex coordinates:

xyz_rotated = xyz · R(ψ, inc, PA)

where R = Rz(-ψ) · Rx(inc) · Rz(-PA) and the angles are:

Rotation sense: prograde (counterclockwise on sky plots when viewed from the visible pole). A surface feature moves West → North → East on the sky as time increases.

Rotation sense consistency

Stellar spin and orbital motion use the same sense: both are counterclockwise on the sky when their angular momentum vector points toward the observer.

This was verified numerically: a spin feature at (West=1, North=0) at t=0 moves to (West=0, North=1) at t=P/4; an orbiting secondary at (North=1, East=0) at ν=0 moves to (North=0, East=1) at ν=π/2. Both trace counterclockwise arcs on the standard sky plot.

Visibility

A surface element is visible when its outward normal has a positive z-component (pointing toward the observer). ROTIR uses a soft visibility model:

weight = sigmoid(κ · normal_z)

with κ controlling the sharpness of the limb transition (default 50). Pixels with weight < 0.01 are treated as invisible.

Binary orbital elements

Orbital elements follow standard astrometric/spectroscopic conventions:

The Thiele-Innes coefficients (Laplace coefficients in the code) are computed from (Ω, i, ω) in compute_coeff() and used to project the orbit into the observer frame.

Orbital output frame

binary_orbit_rel and binary_orbit_abs return positions in the observer frame:

Note that the orbital z-axis points away from the observer (positive = receding), while the stellar geometry z-axis points toward the observer. The orbit_to_rotir_offset function handles this conversion when placing binary components on the sky plane:

offset_west  = -(East component)    # West = -East
offset_north =   North component

Radial velocities

binary_RV returns radial velocities in km/s following the spectroscopic convention:

Vrad = γ + K [cos(ν + ω) + e cos(ω)]

where ν is the true anomaly and ω the argument of periapsis (with a π offset between primary and secondary).

Annotated example

The following figure shows a sphere at inclination 45°, position angle 45°, with the spin axis, rotation arrow, graticules, and E/N compass all displayed: