Serb. Astron. J. } 177 (2008), 115 - 124 UDC 520{872 : 521{181 DOI: 10.2298/SAJ0877115M Professional paper SPHERICAL TRIGONOMETRY OF THE PROJECTED BASELINE ANGLE R. J. Mathar Leiden Observatory, Leiden University, P.O. Box 9513, 2300RA Leiden, The Netherlands E{mail: [email protected] (Received: March 11, 2008; Accepted: May 16, 2008) SUMMARY: The basic vector geometry of a stellar interferometer with two telescopes is de¯ned by the right triangle of (i) the baseline vector between the telescopes, of (ii) the delay vector which points to the star, and of (iii) the projected baseline vector in the plane of the wavefront of the stellar light. The plane of this triangle intersects the celestial sphere at the position of the star; the intersection is a circular line segment. The interferometric angular resolution is high (di®raction limited to the ratio of the wavelength over the projected baseline length) in the two directions along this line segment, and low (di®raction limited to the ratio of the wavelength over the telescope diameter) perpendicular to these. The position angle of these characteristic directions in the sky is calculated here, given either local horizontal coordinates, or celestial equatorial coordinates. Key words. Methods: analytical { Techniques: interferometric { Reference systems 1. SCOPE with the visibility tables of the Optical Interferome- try Exchange Format (OIFITS) (Pauls et al. 2004, 2005), detailing the orthogonal directions which are The paper describes a standard to de¯ne the well and poorly resolved by the interferometer, and plane that contains the baseline of a stellar interfer- (ii) a de¯nition of celestial longitudes in a coordinate ometer and the star, and its intersection with the Ce- system where the pivot point has been relocated from lestial Sphere. This intersection is a great circle, and the North Celestial Pole (NCP) to the star. its section between the star and the point where the prolonged baseline meets the Celestial Sphere de¯nes In overview, Section 2 introduces the standard an oriented projected baseline. The incentive to use set of variables in the familiar polar coordinate sys- this projection onto the Celestial Sphere is that this tems. Section 3 de¯nes and computes the baseline ties the baseline to sky coordinates, which de¯nes position angle|which can be done purely in equa- directions in equatorial coordinates independent of torial or horizontal coordinates or in a hybrid way if the parallactic angle is used to bridge these. Some the observatory's location on the Earth. This is (i) comments on relating distances away from the op- the introduction of a polar coordinate system's po- tical axis (star) in the tangent plane to the optical lar angle in the u ¡ v coordinate system associated path delay conclude the manuscript in Section 4. 115 R. J. MATHAR 2. SITE GEOMETRY North A −A T2 2.1. Telescope Positions b In a geocentric coordinate system, the Carte- Ab τ>0 sian coordinates of the two telescopes of a stellar in- terferometer are related to the geographic longitude East ¸i, geographic latitude Ái and net Earth radius ½ (sum of the Earth radius and altitude above the sea level), A>0 T1 0 1 cos ¸i cos Ái Ti ´ ½ @ sin ¸i cos Ái A ; i = 1; 2: (1) star sin Á i g Fig. 1. In the horizontal coordinate system, the baseline direction is characterized by the azimuth an- We add a "g" to the geocentric coordinates to set gle Ab. In the case shown, 0 < A < ¼ < Ab and them apart from other Cartesian coordinates that D < 0. will be used further below. A great circle of radius ½ ¼ 6380 km, centered at the Earth center, joins T1 Straight forward algebra establishes ¿ from ^ 1 and T2. The vector J ´ 2 T1 £ T2 is perpen- ½ sin Z cos Á sin Á ¡ sin Á cos Á cos(¸ ¡ ¸ ) dicular to this circle, where a baseline aperture angle cos ¿ = 1 2 1 2 1 2 ; Z is de¯ned as the apparent size of baseline vector sin Z cos Á sin(¸ ¡ ¸ ) sin ¿ = 2 2 1 : sin Z b = T2 ¡ T1 (2) There is a caveat of this spherical approxima- tion: The angle ¿ does not transform exactly into as seen from the Earth center: ¿ § ¼ if the roles of the two telescopes are swapped, because the great circle, which has been used to de- T ¢ T = jT j jT j cos Z; ¯ne the direction, is not a loxodrome. This asym- 1 2 1 2 metry within the de¯nition indicates that a more cos Z = cos Á1 cos Á2 cos(¸2 ¡ ¸1) + sin Á1 sin Á2: generic framework to represent the geometry is use- ful. If we consider (i) geodetic rather than geocen- J^ is the axis of rotation when T1 is turned toward T2. tric representations of geographic latitudes (NIMA The nautical direction from T1 to T2 is computed as 2000) or (ii) long baseline interferometers build into follows: de¯ne a tangent plane to the Earth at T1. rugged landscapes, the ¯ve parameters above (½, ¸ , Two orthonormal vectors that span this plane to the i Ái) are too constrained to handle the six degrees of North and East are freedom of two earth-¯xed telescope positions. Nev- ertheless, a telescope array "platform" is helpful to 0 1 de¯ne a zenith, a horizon, and associated coordinates ¡ cos ¸1 sin Á1 ^ @ A ^ like the zenith distance z or the star azimuth A. This N1 ´ ¡ sin ¸1 sin Á1 » @T1=@Á1; jN1j = 1: leads to the OIFITS concept, an array center C, for cos Á 1 g example 0 1 Cx C = @ Cy A ; and Cz g 0 1 plus local telescope coordinates ti, ¡ sin ¸1 E^ 1 ´ @ cos ¸1 A » @T1=@¸1; jE^ 1j = 1: Ti ´ C + ti; i = 1; 2: 0 g Geodetic longitude ¸, geodetic latitude Á and site altitude H above the geoid are de¯ned with the array center (Jones 2004, Vermeille 2004, Wood The unit vector from T to T along the great circle 1 2 1996, Zhang et al. 2005), is J^ £ 1 T . The compass rose angle ¿ is computed ½ 1 0 1 0 1 ^ 1 ^ ^ C (N + H) cos Á cos ¸ by the partition J £ ½ T1 = cos ¿N1 + sin ¿E1 within x @ A @ (N + H) cos Á sin ¸ A the tangent plane. ¿ is zero if T2 is North of T1, and Cy = £ ¤ ; C N(1 ¡ e2) + H sin Á is ¼=2 if T2 is East of T1: Fig. 1. z g g 116 SPHERICAL TRIGONOMETRY OF THE PROJECTED BASELINE ANGLE where e isp the eccentricity of the Earth ellipsoid, and 2.3. Baseline: Generic Position 2 N ´ ½e= 1 ¡ e sin Á the distance from the array center to the Earth axis measured along the local The baseline vector coordinates of the geocen- vertical. tric OIFITS system 0 1 B 2.2. Sky coordinates x b = @ By A = T2 ¡ T1 B In a plane tangential to the geoid at C we de- z g ¯ne a star azimuth A, a zenith angle z, and a star elevation a = ¼=2 ¡ z, (de¯ned to stretch from T1 to T2|the opposite sign is also in use (Pearson 1991)) could be converted with 0 1 0 1 ¡ cos A sin z ¡ cos A cos a 0 1 0 1 0 1 s = @ sin A sin z A = @ sin A cos a A : (3) bx t2x t1x @ A @ A @ A cos z sin a b = by = t2y ¡ t1y t t bz t2z t1z t 0 1 t t We label this coordinate system t as "topocentric," Bx with the ¯rst component North, the second compo- = Utg(Á; ¸) ¢ @ By A nent West and the third component up. The value of B A used in this script picks the convention that South z g means A = 0 and West means A = +¼=2. This con- ventional de¯nition of the horizontal means the star to the topocentric system via the rotation matrix coordinates can be transformed to the equatorial pa- 0 1 rameters, hour angle h = l ¡ ®, declination ±, and ¡ sin Á cos ¸ ¡ sin Á sin ¸ cos Á right ascension ® (Lang 1998 eq. (5.45), Karttunen Utg(Á; ¸) = @ sin ¸ ¡ cos ¸ 0 A : 1987 Eq. (2.13)), cos Á cos ¸ cos Á sin ¸ sin Á cos a sin A = cos ± sin h; (4) Just like the star direction (3), the unit vector b^ cos a cos A = ¡ sin ± cos Á + cos ± cos h sin Á; (5) along the baseline direction de¯nes a baseline az- sin a = sin ± sin Á + cos ± cos h cos Á; (6) imuth Ab and a baseline elevation ab 0 1 cos ± cos h = sin a cos Á + cos a cos A sin Á; (7) ¡ cos A cos a b b b sin ± = sin a sin Á ¡ cos a cos A cos Á: (8) b^ ´ ´ @ sin A cos a A : (12) b b b sin ab t These convert (3) into 0 1 The standard de¯nition of the delay vector D cos Á sin ± ¡ sin Á cos ± cos h and projected baseline vector P is s = @ cos ± sin h A : (9) b = D + P; D k s; P ? D: sin Á sin ± + cos Á cos ± cos h t P|and later its circle segment projected on the Ce- A star at hour angle h = 0 is on the Meridian, lestial Sphere|inherits its direction from b such that 0 1 heads and tails of the vectors are associated with the sin(± ¡ Á) same portion of the wavefront: Fig. 2. s = @ 0 A ; (h = 0); s D cos(± ¡ Á) t P north of the zenith at A = ¼ if ± ¡ Á > 0, south at θ A = 0 if ± ¡Á < 0.