Spherical Trigonometry of the Projected Baseline Angle Richard J. Mathar∗ Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, The Netherlands† (Dated: October 29, 2018) The basic geometry of a stellar interferometer with two telescopes consists of a baseline vector and a direction to a star. Two derived vectors are the delay vector, and the projected baseline vector in the plane of the wavefronts of the stellar light. The manuscript deals with the trigonometry of projecting the baseline further outwards onto the celestial sphere. The position angle of the projected baseline is defined, measured in a plane tangential to the celestial sphere, tangent point at the position of the star. This angle represents two orthogonal directions on the sky, differential star positions which are aligned with or orthogonal to the gradient of the delay recorded in the u−v plane. The North Celestial Pole is chosen as the reference direction of the projected baseline angle, adapted to the common definition of the “parallactic” angle. PACS numbers: 95.10.Jk, 95.75.Kk Keywords: projected baseline; parallactic angle; position angle; celestial sphere; spherical astronomy; stellar interferometer I. SCOPE II. SITE GEOMETRY A. Telescope Positions The paper describes a standard to define the plane that contains the baseline of a stellar interferometer and the 1. Spherical Approximation star, and its intersection with the Celestial Sphere. This intersection is a great circle, and its section between the In a geocentric coordinate system, the Cartesian coor- star and the point where the prolongated baseline meets dinates of the two telescopes of a stellar interferometer the Celestial Sphere defines an oriented projected base- are related to the geographic longitudes λi, latitudes φi line. The incentive to use this projection onto the Ce- and effective Earth radius ρ (sum of the Earth radius and lestial Sphere is that this ties the baseline to sky coordi- altitude above sea level), nates, which defines directions in equatorial coordinates which are independent of the observatory’s location on cos λi cos φi Earth. This is (i) the introduction of a polar coordi- Ti ≡ ρ sin λi cos φi , i =1, 2. (1) sin φ nate system’s polar angle in the u − v coordinate system i g associated with the visibility tables of the Optical Inter- ferometry Exchange Format (OIFITS) [15, 16], detailing We add a “g” to the geocentric coordinates to set them the orthogonal directions which are well and poorly re- apart from other Cartesian coordinates that will be used solved by the interferometer, and (ii) a definition of ce- further down. A great circle of radius ρ ≈ 6380 km, cen- lestial longitudes in a coordinate system where the pivot tered at the Earth center, connects T1 and T2; a vector Jˆ 1 T T point has been relocated from the North Celestial Pole perpendicular to this circle is ≡ ρ2 sin Z 1 × 2, where arXiv:astro-ph/0608273v3 2 May 2009 (NCP) to the star. a baseline aperture angle Z has been defined under which the baseline vector In overview, Section II introduces the standard set b = T2 − T1 (2) of variables in the familiar polar coordinate systems. Section III defines and computes the baseline position is seen from the Earth center: angle—which can be done purely in equatorial or hori- zontal coordinates or in a hybrid way if the parallactic T1 · T2 = |T1| |T2| cos Z; (3) angle is used to bridge these. Some comments on relating cos Z = cos φ1 cos φ2 cos(λ2 − λ1) + sin φ1 sin φ2.(4) distances away from the optical axis (star) in the tangent plane to the optical path delay conclude the manuscript Jˆ is the axis of rotation when T1 is moved toward T2. in Section IV. The nautical direction from T1 to T2 is computed as follows: define a tangent plane to the Earth at T1. Two orthonormal vectors that span this plane are ∗ Electronic address: [email protected]; − cos λ1 sin φ1 URL: http://www.strw.leidenuniv.nl/ mathar ~ Nˆ 1 ≡ − sin λ1 sin φ1 ∼ ∂T1/∂φ1; |Nˆ 1| =1. †Supported by the NWO VICI grant 639.043.201 to A. Quirren- cos φ1 bach, “Optical Interferometry: A new Method for Studies of Ex- g trasolar Planets.” (5) 2 to the North and example − sin λ1 C x Eˆ 1 ≡ cos λ1 ∼ ∂T1/∂λ1; |Eˆ 1| =1. (6) C = Cy , (9) 0 C g z g to the East. The unit vector from T1 to T2 along the plus local telescope coordinates 1 great circle is Jˆ × T1. The compass rose angle τ is ρ T ≡ C + t , i =1, 2. (10) Jˆ 1 T Nˆ i i computed by its decomposition × ρ 1 = cos τ 1 + sin τEˆ 1 within the tangent plane. It becomes zero if T2 is North of T1, and becomes π/2 if T2 is East of T1: Fig. B. Sky coordinates 1. Geodetic longitude λ, geodetic latitude φ and altitude North H above the geoid are defined with the array center [7, T2 21, 24, 25], Ab−A C (N + H)cos φ cos λ x Cy = (N + H)cos φ sin λ , (11) Ab τ>0 2 Cz N(1 − e )+ H sin φ g g East where e is the eccentricity of the Earth ellipsoid, and 2 2 N ≡ ρe/ 1 − e sin φ the distance from the array center A>0 p T to the Earth axis measured along the local vertical. 1 In a plane tangential to the geoid at C we define a star azimuth A, a zenith angle z, and a star elevation a = π/2 − z, star − cos A sin z − cos A cos a FIG. 1: In the horizontal coordinate system, the baseline di- s = sin A sin z = sin A cos a . (12) rection is characterized by the azimuth angle Ab. In the case cos z sin a shown, 0 <A<π<Ab and D < 0. t t We label this coordinate system t as “topocentric,” with (Note that the tables in [5] use a different definition.) the first component North, the second component West Straight forward algebra establishes τ from and the third component up. The value of A used in this script picks one of the (countably many) conventions: cos φ1 sin φ2 − sin φ1 cos φ2 cos(λ1 − λ2) cos τ = ; (7) South means A = 0 and West means A = +π/2. This sin Z conventional definition of the horizontal means the star cos φ2 sin(λ2 − λ1) coordinates can be transformed to the equatorial param- sin τ = . (8) sin Z eters of hour angle h = l − α, declination δ, and right ascension α [10, (5.45)][8, (2.13)], 2. Caveats cos a sin A = cos δ sin h; (13) cos a cos A = − sin δ cos φ + cos δ cos h sin φ; (14) The angle τ of the previous section does not transform sin a = sin δ sin φ + cos δ cos h cos φ; (15) exactly into τ ± π if the roles of the two telescopes are cos δ cos h = sin a cos φ + cos a cos A sin φ; (16) swapped, because the great circle, which has been used to define the direction, is not a loxodrome. This asym- sin δ = sin a sin φ − cos a cos A cos φ. (17) metry within the definition indicates that a more generic These convert (12) into framework to represent the geometry is useful. If we consider (i) geodetic rather than geocentric rep- cos φ sin δ − sin φ cos δ cos h resentations of geographic latitudes [14] or (ii) long base- s = cos δ sin h . (18) line interferometers build into rugged landscapes, the 5 sin φ sin δ + cos φ cos δ cos h t parameters above (ρ, λi, φi) are too constrained to han- dle the 6 degrees of freedom of two earth-fixed telescope A star at hour angle h = 0 is on the Meridian, positions. Nevertheless, a telescope array “platform” is helpful to define a zenith, a horizon, and associated co- sin(δ − φ) ordinates like the zenith distance or the star azimuth. s = 0 , (h = 0), (19) C cos(δ − φ) This leads to the OIFITS concept, an array center , for t 3 north of the zenith at A = π if δ − φ> 0, south at A =0 s if δ − φ < 0. A reference point on the Celestial Sphere D P is the North Celestial Pole (NCP), given by insertion of δ = π/2 into (18), θ T b T cos φ 1 2 s+ = 0 . (20) sin φ FIG. 2: Sign conventions of the co-planar vectors b, D and t P. The angular distance between s and s+ is where the angular distance θ between the baseline and s · s+ = sin δ. (21) star directions is introduced as C. Baseline: Generic Position cos θ = cos ab cos a cos(Ab − A) + sin ab sin a. (28) The star circles around the NCP in 24 hours, which The baseline vector coordinates of the geocentric changes the distance to the baseline in a centric periodic OIFITS system way (App. B). B x b = By = T2 − T1 (22) B III. PROJECTED BASELINE ANGLES z g A. Definition (defined to stretch from T1 to T2—the opposite sign is also in use [17]) could be converted with We define position angles of points on the Celestial bx t2x t1x Bx Sphere relative to a star’s position as the bearing angle b = by = t2y − t1y = Utg(φ, λ)· By by which the NCP must be rotated around the star direc- b t2 t1 B s z t z t z t z g tion (the axis) until the NCP and the particular point (23) of the object are aligned to the same direction (along a to the topocentric system via the rotation matrix celestial circle) from the star.