virgil.orbits

The orbit with longitude of periastron varpi = Ω + ω.

Give two_Omega (2Ω, degrees) for positions alone, which fix Ω only modulo 180°, or Omega when other data (RVs) fix the node; see orientation_priors, which samples them as angle vectors. omega is varpi - Omega, the secondary's argument of periastron, as everywhere in virgil.

The orbit whose position angle at t_ref is theta.

An alternative to dt_peri for short arcs, after Thompson et al. (2023, AJ 166, 164): astrometry measures the position angle at an epoch directly. With φ = θ - Ω, the argument of latitude u = ω + f at t_ref follows from (cos φ, sin φ) ∝ (cos u, sin u cos i): u = atan2(sin φ / cos i, cos φ); then the true anomaly f = u - ω, the eccentric and mean anomalies, and dt_peri = -M P / 2π (in [-P/2, P/2)). Sample theta as an AngleVector, and add position_angle_prior to the likelihood terms, so that the prior stays uniform in the time of periastron.

Singular at i = 90°, where the position angle takes only two values (Ω and Ω + 180°) and does not fix the phase; a concrete inc of 90° is rejected. Near edge-on the map is badly conditioned: keep dt_peri, or use StateVectorOrbit.

Parameters:

The Thiele–Innes constants (A, B, F, G, C, H) (mas).

ddec = A X + F Y, dra = B X + G Y and dz = C X + H Y, with X = cos E - e and Y = √(1 - e²) sin E.

Position of the secondary relative to the primary (mas).

Parameters:

Returns:

d(dra, ddec, dz)/dt (mas per day), exactly.

Angles of the binary frame at mjd (degrees), by name.

• line_pa: position angle of the line of centres, primary to secondary; towards_primary is line_pa + 180.
• line_tilt: the line of centres' elevation out of the sky, arcsin(dz / |r|), positive when the secondary is farther.
• node_pa: Omega, the line of nodes of the orbital plane.
• inc: the orbit's inclination (0–180); apparent_inc: arccos|cos inc| (0–90), the projected tilt, for components whose inc is an apparent inclination.

Separation (mas) and position angle (degrees, North through East, in [0, 360)) of the secondary from the primary.

The same sky orbit as a :class:ThieleInnesOrbit.

A jaxoplanet OrbitalBody for this orbit, and the factor that turns its relative positions into mas.

jaxoplanet's times are days since t_ref, its axes are (X, Y, Z) = (North, East, toward the observer), and its ω is the primary's: (dra, ddec, dz) = (Y, X, -Z) * scale.

The orbit of a jaxoplanet OrbitalBody whose times are days since t_ref, with angular semimajor axis a_mas.

(dra, ddec) of the secondary from the primary (mas).

The :class:KeplerOrbit with these sky positions.

Positions fix Omega only modulo 180°: the result has 0 <= Omega < 180, and (Omega + 180, omega + 180) is the other solution, with dz reversed.

The state of orbit at its t_ref.

The KeplerOrbit of this state.

The periastron and the direction of motion there give the Thiele–Innes constants directly, which fixes the angles in virgil's conventions; the line-of-sight components then pick the node, which positions alone leave ambiguous by 180°. A circular orbit (e = 0) has no periastron: the position at t_ref is used instead.

(dra, ddec, dz) (mas), as for KeplerOrbit.relative.

Positions given as separation (mas) and position angle (degrees, North through East), with independent errors on each.

The positions (dra, ddec) these data would measure (mas).

The orbit's sky positions, seen through this dataset's astrometric calibration: m R(δ) (dra, ddec) with δ = north_angle (degrees), m = plate_scale and R(δ) = [[cos δ, sin δ], [-sin δ, cos δ]]. So a companion at true PA θ and separation ρ is measured at PA θ + δ and separation m ρ: north_angle is the error added to every measured position angle, as in OIData.with_north_angle. None (the default) leaves the positions untouched.

L⁻¹ (data - model) for every epoch, flattened (2n,).

north_angle and plate_scale are this dataset's calibration terms (see model).

Gaussian log-likelihood of the positions under orbit.

A likelihood term for fit's likelihoods.

Parameters:

Predicted radial velocities (km/s).

Parameters:

Effective errors sqrt(d_rv² + jitter²) (km/s).

(rv - model) / sqrt(d_rv² + jitter²) for every epoch.

Gaussian log-likelihood of the velocities.

jitter (km/s) adds an extra scatter in quadrature to every error; the normalization then depends on it.

Whitened residuals of the zero-point-marginalized Gaussian.

The data are d ~ N(m + Aμ, C + AΛAᵀ) with m the Keplerian model (gamma included), A the indicator matrix of the instruments and w ~ N(μ, Λ) the zero points. Returns u of length N with uᵀu = rᵀ(C + AΛAᵀ)⁻¹r, r = d - m - Aμ, by the dense Woodbury whitening of virgil._linear (an exact square root, with no eigendecomposition).

prior is (mean, sd) per instrument, as term builds it.

½ log det(C + AΛAᵀ).

This is the normalization that depends on the jitter, in the same convention as the plain Σ log σ_eff.

Normalized log density of the zero-point-marginalized Gaussian.

Equal to a dense N(m + Aμ, C + AΛAᵀ) log density, computed in O(N k²). A flat prior is the limit Λ → ∞ only up to a constant (-½ Σ log Λ); the finite prior width is part of the model.

Mean and covariance of the zero points w given the orbit.

w | d, θ ~ N(S⁻¹(Λ⁻¹μ + AᵀC⁻¹(d - m)), S⁻¹) with S = Λ⁻¹ + AᵀC⁻¹A. Entry j is the velocity zero point of self.instruments[j]; m includes gamma, so with gamma = 0 they are the systemic velocity seen by each instrument, and differences between them are the offsets.

A likelihood term for fit's likelihoods.

Parameters: