Spherical Trigonometry of the Projected Baseline Angle

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 defined 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 (diffraction limited to the ratio of the wavelength over the projected baseline length) in the two directions along this line segment, and low (diffraction 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 define the well and poorly resolved by the interferometer, and plane that contains the baseline of a stellar interfer- (ii) a definition 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 defines 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 defines and computes the baseline ties the baseline to sky coordinates, which defines 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 interferometer are related to the geographic longitude East λi, geographic latitude φi and net Earth radius ρ (sum of the Earth radius and altitude above the sea level), A>0 T1   cos λi cos φi Ti ≡ ρ  sin λi cos φi  , i = 1, 2. (1) star sin φ 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 < π < Ab and them apart from other Cartesian coordinates that D < 0. will be used further below. A great circle of radius ρ ≈ 6380 km, centered at the Earth center, joins T1 Straight forward algebra establishes τ from ˆ 1 and T2. The vector J ≡ 2 T1 × T2 is perpen- ρ sin Z cos φ sin φ − sin φ cos φ cos(λ − λ ) dicular to this circle, where a baseline aperture angle cos τ = 1 2 1 2 1 2 ; Z is defined as the apparent size of baseline vector sin Z cos φ sin(λ − λ ) sin τ = 2 2 1 . sin Z b = T2 − T1 (2) There is a caveat of this spherical approxima- tion: The angle τ does not transform exactly into as seen from the Earth center: τ ± π if the roles of the two telescopes are swapped, because the great circle, which has been used to de- T · T = |T | |T | cos Z; fine the direction, is not a loxodrome. This asym- 1 2 1 2 metry within the definition indicates that a more cos Z = cos φ1 cos φ2 cos(λ2 − λ1) + sin φ1 sin φ2. generic framework to represent the geometry is use- ful. If we consider (i) geodetic rather than geocen- Jˆ 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: define a tangent plane to the Earth at T1. rugged landscapes, the five parameters above (ρ, λ , Two orthonormal vectors that span this plane to the i φi) are too constrained to handle the six degrees of North and East are freedom of two earth-fixed telescope positions. Nev- ertheless, a telescope array ”platform” is helpful to   define a zenith, a horizon, and associated coordinates − cos λ1 sin φ1 ˆ   ˆ like the zenith distance z or the star azimuth A. This N1 ≡ − sin λ1 sin φ1 ∼ ∂T1/∂φ1; |N1| = 1. leads to the OIFITS concept, an array center C, for cos φ 1 g example   Cx C =  Cy  , and Cz g

  plus local telescope coordinates ti, − sin λ1 Eˆ 1 ≡  cos λ1  ∼ ∂T1/∂λ1; |Eˆ 1| = 1. Ti ≡ C + ti, i = 1, 2. 0 g Geodetic longitude λ, geodetic latitude φ and site altitude H above the geoid are defined 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 Jˆ × 1 T . The compass rose angle τ is computed ρ 1     ˆ 1 ˆ ˆ C (N + H) cos φ cos λ by the partition J × ρ T1 = cos τN1 + sin τE1 within x    (N + H) cos φ sin λ  the tangent plane. τ is zero if T2 is North of T1, and Cy = £ ¤ , C N(1 − e2) + H sin φ is π/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 ≡ ρe/ 1 − e sin φ 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   B 2.2. Sky coordinates x b =  By  = T2 − T1 B In a plane tangential to the geoid at C we de- z g fine a star azimuth A, a zenith angle z, and a star elevation a = π/2 − z, (defined to stretch from T1 to T2—the opposite sign is also in use (Pearson 1991)) could be converted with     − cos A sin z − cos A cos a       s =  sin A sin z  =  sin A cos a  . (3) bx t2x t1x       cos z sin a b = by = t2y − t1y t t bz t2z t1z t   t t We label this coordinate system t as ”topocentric,” Bx with the first component North, the second compo- = Utg(φ, λ) ·  By  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 = +π/2. This conventional definition of the horizontal means the star to the topocentric system via the rotation matrix coordinates can be transformed to the equatorial pa-   rameters, hour angle h = l − α, declination δ, and − sin φ cos λ − sin φ sin λ cos φ right ascension α (Lang 1998 eq. (5.45), Karttunen Utg(φ, λ) =  sin λ − cos λ 0  . 1987 Eq. (2.13)), cos φ cos λ cos φ sin λ sin φ cos a sin A = cos δ sin h; (4) Just like the star direction (3), the unit vector bˆ cos a cos A = − sin δ cos φ + cos δ cos h sin φ; (5) along the baseline direction defines a baseline az- sin a = sin δ sin φ + cos δ cos h cos φ; (6) imuth Ab and a baseline elevation ab   cos δ cos h = sin a cos φ + cos a cos A sin φ; (7) − cos A cos a b b b sin δ = sin a sin φ − cos a cos A cos φ. (8) bˆ ≡ ≡  sin A cos a  . (12) b b b sin ab t These convert (3) into   The standard definition of the delay vector D cos φ sin δ − sin φ cos δ cos h and projected baseline vector P is s =  cos δ sin h  . (9) b = D + P; D k s; P ⊥ D. sin φ sin δ + cos φ cos δ 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   heads and tails of the vectors are associated with the sin(δ − φ) same portion of the wavefront: Fig. 2. s =  0  , (h = 0), s D cos(δ − φ) t P north of the zenith at A = π if δ − φ > 0, south at θ A = 0 if δ −φ < 0. A reference point on the Celestial T b T Sphere is the North Celestial Pole (NCP), given by 1 2 insertion of δ = π/2 into (9), Fig. 2. Sign conventions of the coplanar vectors b,   cos φ D and P. s =  0  . (10) + We use the term ”projected baseline” both sin φ t ways: for the straight vector of length P , units me- ter, that connects the tails of D and b, or the line The cosine of the angular distance between s and s+ segment of length θ, units radian, on the Celestial is Sphere. s · s+ = sin δ. (11) The dot product of (3) by (12) is

D = s · b = b cos θ, (0 ≤ θ ≤ π), (13) 117 R. J. MATHAR where the angular distance θ between the baseline An overview of the relevant geometry is given and star directions is introduced as in Fig. 3 which looks at the celestial sphere from outside. cos θ = cos ab cos a cos(Ab − A) + sin ab sin a. (14) In Fig. 3, the baseline has been infinitely ex- tended straight outwards in both directions which The star circles around the NCP in 24 hours, which define telescope coordinates T1 and T2, also where changes the distance to the baseline in a centric pe- the baseline meets the Celestial Sphere. For an ob- riodic way (Appendix B). server at the mid-point of the baseline, telescope 1 then is at azimuth τ, telescope 2 at azimuth Ab = τ + π, see Fig. 1. Let the object be at azimuth A and elevation a. 3. PROJECTED BASELINE ANGLES The projection of the baseline occurs in a 3.1. Definition plane including the object and the baseline, thus defining the great circle labeled θ in Fig. 3. If θ is the angle on the sky between the object and the point We define position angles of points on the Ce- on the horizon with azimuth Ab, then the length P lestial Sphere as the bearing angle by which the NCP of the projected baseline is given by must be rotated around the star direction s (the axis) until the NCP and the point appear aligned in the P = b sin θ, same direction (along a celestial circle) from the star. The sign convention is left-handed placing the head where θ is calculated from the relation (14). With of s at the center of the Celestial Sphere. This is this auxiliary quantity, there are various ways of ob- equivalent to drawing a line between the star and taining the position angle pb of the projected baseline the NCP, looking at it from inside the sphere, and on the sky, some of which are detailed in Sections defining position angles of points as the angles in po- 3.2–3.4. The common theme is that the baseline ori- lar coordinates in the mathematical sign convention, entation (Ab, ab) in the local horizontal polar coor- using this line as the abscissa and the star as the dinate system is transferred to a rotated polar coor- center. Another, fully equivalent definition uses a dinate system with the star defining the new polar right-handed turn of the object around the star until axis; the position angle is the difference of azimuths it is North of the star. And finally, as an aid to mem- between T2 and the NCP in this rotated polar coor- ory, it is also the value of the nautical course while dinate system. navigating a ship sailing the outer hull of the Celes- tial Sphere, which is currently poised at the star’s coordinates. 3.2. In Horizontal Coordinates This defines values modulo 2π; whether these are finally represented as numbers in the interval We use a vector algebraic method to span a [0, 2π) or in the interval (−π, +π]—as the SLA BEAR plane tangential to the celestial sphere; tangent point and the SLA PA routines of the SLA library (Wallace is the position s of the star. Within this orthographic 2003) do—is largely a matter of taste. zenithal projection (Calabretta and Greisen 2002) we consider the two directions from the origin, toward ˆ the NCP, s+, on one hand and toward T2, b, on the zenith other. The tangential plane is fixed by any two unit vectors perpendicular to the vector s of (9). Rather arbitrarily they are chosen along ∂s/∂A and ∂s/∂a, z explicitly   sin A eA =  cos A  , (15) 0 t NCP T E star   1 cos A sin a ea =  − sin A sin a  . (16) S cos a θ a t N (The equivalent exercise with a different choice of axes follows in Section 3.4.) These are orthonormal, A -A W T2 b s · eA = s · ea = eA · ea = 0; |eA| = |ea| = 1; s × ea = eA. Fig. 3. Celestial sphere, seen from the outside. The north direction through the object is given by the great Partitioning of the direction from s to the NCP de- circle passing through the celestial poles and the ob- fines three coefficients c0,1,2, ject. The angle θ is the vector P = b − D projected on the sphere. s+ = c0s + c1eA + c2ea. (17) 118 SPHERICAL TRIGONOMETRY OF THE PROJECTED BASELINE ANGLE

Building the square on both sides yields the famil- Building the square on both sides yields iar formula for the sum of squares of the projected 02 02 02 cosines, c0 + c1 + c2 = 1. (20) 2 2 2 c0 + c1 + c2 = 1. (18) Dot products of (19) solve for the expansion coeffi- Two dot products of (17) using (10) and (15)–(16) cients, with (12) and (15)–(16): solve for two coefficients, 0 ˆ c1 = b · eA = − cos A cos a sin A + sin A cos a cos A c1 = s+ · eA = cos φ sin A, b b b b = cos a sin(A − A), (21) c2 = s+ · ea = cos φ cos A sin a + sin φ cos a. b b 0 ˆ c2 = b · ea An angle ϕ between eA and s+ is defined via Fig. 4, = − cos ab sin a cos(Ab − A) + sin ab cos a. (22) This defines an angle ϕ0 in the polar coordinates of c cos ϕ = p 1 , the tangent plane, 2 2 c1 + c2 0 c1 c2 cos ϕ0 = p , sin ϕ = p . 02 02 2 2 c1 + c2 c1 + c2 c0 sin ϕ0 = p 2 . From (18), and since (11) equals c , 02 02 0 c1 + c2 q Since c0 equals bˆ · s = cos θ in (14), c2 + c2 = cos δ. 0 1 2 q 02 02 c1 + c2 = sin θ. The equivalent splitting of bˆ into components within and perpendicular to the tangent plane is: The baseline position angle is the difference between the two angles, as to redefine the reference direction ˆ 0 0 0 from eA to the direction of the NCP (see Fig. 4): b = c0s + c1eA + c2ea. (19) 0 pb = ϕ − ϕ (mod 2π). ea Sine and cosine of this imply NCP 0 0 sin pb = sin ϕ cos ϕ − cos ϕ sin ϕ c 2 c c0 − c0 c p = 1 2 1 2 ; (23) cos δ sin θ 0 0 cos pb = cos ϕ cos ϕ + sin ϕ sin ϕ c c0 + c c0 = 1 1 2 2 , (24) cos δ sin θ p where b ϕ 0 0 c1c2 − c1c2 = cos φ(sin A sin ab cos a − cos ab sin a sin Ab) − sin φ cos a cos ab sin(Ab − A), (25) b c' 2 and 0 0 ϕ' c1c1 + c2c2 = cos φ(cos A cos a cos θ pb − cos ab cos Ab) 0 star +c2 sin φ cos a. (26) eA c 1 c'1 ψ The computational strategy is to build c c0 − c0 c tan p = 1 2 1 2 . (27) b c c0 + c c0 Fig. 4. In the tangent plane spanned by the unit 1 1 2 2 ˆ vectors (15) and (16), the components of b and s+ The (positive) values of cos δ and sin θ in the denom- 0 0 inators of (23) and (24) need not to be calculated. are c1, c1, c2 and c2 defined by (17) and (19). The view is from the outside onto the plane, so eA, the The branch ambiguity of the arctan is typically han- vector into the direction of increasing A, points to dled by use of the atan2() functionality in the li- the left. braries of higher programming languages. 119 R. J. MATHAR

3.3. Via Parallactic Angle the directions along ∂s/∂h and ∂s/∂δ:   If the position angle of the zenith p sin φ sin h eh =  cos h  ; cos φ sin A tan p = − cos φ sin h t cos φ cos A sin a + sin φ cos a   cos φ cos δ + sin φ sin δ cos h sin h = eδ =  − sin δ sin h  . cos δ tan φ − sin δ cos h sin φ cos δ − cos φ sin δ cos h t is known from any other source, a quicker approach to the calculation of pb employs an auxiliary angle ψ The further calculation follows the scheme of Section (Ch. Leinert 2006, priv. commun.) 3.2: The axes are orthonormal,

s · eh = s · eδ = eh · eδ = 0; pb ≡ p + π + ψ (mod 2π). (28) |eh| = |eδ| = 1; s × eδ = eh. The calculation of ψ is delegated to the calculation of its sines and cosines The decomposition of s+ in this system deﬁnes three expansion coeﬃcients d0,1,2, cos ψ = − cos pb cos p − sin pb sin p;

sin ψ = − sin pb cos p + cos pb sin p. s+ = d0s + d1eδ + d2eh. In the right hand sides we insert (23)–(24) Multiplying this equation in turn with eh and eδ yields cos φ sin A sin p = ; (29) d2 = 0; d1 = s+ · eδ = cos δ. (33) cos δ cos φ cos A sin a + sin φ cos a ˆ cos p = , (30) The direction b to T2 is projected into the same cos δ tangent plane, defining three expansion coefficients d0 , and after some standard manipulations 0,1,2 ˆ 0 0 0 b = d0s + d1eδ + d2eh. (34) cos a sin(A − A) sin ψ = b b ; (31) sin θ e sin a cos θ − sin a δ cos ψ = b sin θ cos a cos a sin a cos(A − A) − sin a cos a = b b b .(32) sin θ pb NCPd1 0 0 [these are c1 and −c2 of (21) and (22) divided by sin θ.] In Fig. 4, ψ can therefore be identified as the d' angle between −e and bˆ, and this is consistent with 1 a b (28) which decomposes the rotation into the angle p, a rotation by the angle π (which would join the end of p with the start of ψ in Fig. 4), and finally a rotation by ψ. In summary, the disadvantage of this approach star is that p must be known by other means, and the ad- d'2 eh vantage is that computation of ψ via the arctan of pb 0 0 (31) and (32) only requires c1 and c2, but not c1 or c2.

3.4. In Equatorial Coordinates Fig. 5. Fig. 4 after switching from the (eA, ea) ˆ axes to (eh, eδ). Axis components of s+ and b are Section 3.2 provides pb, given star coordinates indicated. (A, a). Subsequently we derive the same value in terms of δ and h. The straightforward approach is Building the dot product of this equation with the substitution of (4)–(8) in (25) and (26). A less e using (12) yields exhaustive alternative works as follows (R. K¨ohler h 2006, priv. commun.): the axes eA and ea in the tan- 0 gential plane are replaced by two diﬀerent orthonor- d2 = − cos Ab cos ab sin φ sin h + sin Ab cos ab cos h mal directions, better adapted to δ and h, namely − sin ab cos φ sin h. (35) 120 SPHERICAL TRIGONOMETRY OF THE PROJECTED BASELINE ANGLE

Building the dot product of (34) with eδ yields The segment of the baseline in the tangent plane primarily defines an orientation in the u − v 0 d1 = − cos Ab cos ab cos φ cos δ plane. However, modal decomposition of the ampli- − cos A cos a sin φ sin δ cos h tudes in products of radial and angular basis func- b b tions (akin to Zernike polynomials) means that the − sin Ab cos ab sin δ sin h + sin ab sin φ cos δ Fourier transform to the x − y plane preserves the − sin ab cos φ sin δ cos h. (36) angular basis functions; see the calculation for the 3D case, the Spherical Harmonics. In this sense, the Some simplification in this formula is obtained by angle pb is also ”applicable” in the x − y plane. trading the hour angle h for the angle θ, which might be more readily available since θ is measured via the delay: In the second term we use (5) to substitute 4. DIFFERENTIALS IN THE FIELD-OF-VIEW cos φ sin δ + cos A cos a sin φ cos h → . cos δ 4.1. Sign Convention In the third term we use (4) to substitute The provisions of the previous section, namely sin A cos a – the OIFITS sign convention of the baseline sin h → , vector, Eq. (2) and Fig. 2, cos δ – the conventional formula (13) and in the last term we use (6) to substitute fix the sign of the delay D as follows: if the wavefront hits T2 prior to T1, the values of D and cos θ sin a − sin δ sin φ are positive; if it hits T1 prior to T2, both values are cos φ cos h → . negative. cos δ The transitional case D = 0 (θ = π/2) occurs if the star passes through the ”baseline meridian” Further standard trigonometric identities and (14) plane perpendicular to the baseline; the two lines to yield the Northwest and Southeast in Fig. 1 show (pro- jections of) these directions. The baseline in most 0 sin ab sin φ − cos ab cos Ab cos φ − sin δ cos θ d1 = . optical stellar interferometers is approximately hor- cos δ izontal, a ≈ 0. From (13) we see that this case is (37) b In the numerator we recognize a baseline declination approximately equivalent to cos(Ab − A) ≈ 0. Since cos a > 0, the sign of D coincides with the sign of δb, cos(Ab − A). This is probably the fastest way to obtain the sign of the Optical Path Difference from s · bˆ ≡ sin δ + b FITS (Hanisch et al. 2001) header keywords; the = sin ab sin φ − cos ab cos Ab cos φ. (38) advantage of this recipe is that the formula is independent of which azimuth convention is actually in pb is the angle that rotates the projected vector use. (d2, d1) within the tangent plane into the direction The baseline length b follows immediately (d0 , d0 ) (Fig. 5), from the Euclidean distance of the two telescopes 2 1 entries for column STAXYZ of the OIFITS table d0 OI ARRAY (Pauls et al. 2005). cos p = 1 , b sin θ d0 4.2. External Path Delay sin p = − 2 . b sin θ The total differential of (14) relates a direc- Again, sin θ does not need actually to be calculated, but only tion (∆A, ∆a) away from the star to a change in the 0 angular distance between the star and the baseline, −d2 tan pb = 0 , (39) d1 £ − sin θ∆θ = ∆a − cos ab sin a cos(Ab − A) and again, selection of the correct branch of the arc- ¤ tan is easy with atan2() functions if the negative + sin ab cos a 0 sign is kept attached to d2. +∆A cos a cos ab sin(Ab − A). 0 0 The use of (37) is optional: d2 and d1 are completely defined in terms of the baseline direction (31) and (32) turn this into (Ab, ab), the geographic latitude φ and the star coordinates (h, δ) via (35) and (36). With these, one can immediately proceed to (39). sin θ∆θ = sin θ(cos ψ∆a − sin ψ cos a∆A), (40) OIFITS (Pauls) defines no associated angle in the plane perpendicular to the direction of the phase consistent with the decomposition in Fig. 4. The center. two directions of constant delay are characterized by 121 R. J. MATHAR

0 0 ∆θ = 0. Equating the right hand side of the previous = b(∆δ d1 + ∆h cos δ d2) equation with zero, this means = b sin θ(∆δ cos pb − ∆h cos δ sin pb). (42) ∆a = tan ψ; (∆θ = ∆D = 0). (41) This translates Calabretta’s (Calabretta and Greisen cos a∆A 2002) Appendix C to our variables. When ∆θ = 0, For the direction of maximum change in ∆D, we have the directions of zero change in ∆D, per- which runs perpendicular to (41), pendicular to the projected baseline, ∆a 1 ∆δ = − , (pb : max ∆D) = tan p , (∆θ = ∆D = 0). (43) cos a ∆A tan ψ cos δ∆h b which leads to The direction of maximum change is cos a ∆A ∆D = −b sin θ∆θ = b sin θ ∆δ 1 sin ψ = − , (pb : max ∆D), cos δ∆h tan p ∆a b = −b sin θ , (max ∆D). cos ψ and along this gradient

Example: a ﬁeld of view of ∆θ = ±1” at a cos δ∆h ∆δ baseline of b = 100 m at a position θ = 45◦ scans ∆D = −b sin θ = b sin θ , (max ∆D). ∆D = ±340 µm. The details of the optics determine sin pb cos pb how far this contributes to the instrumental visibility (loss). zenith 5. SUMMARY

To standardize the nomenclature, we propose star 1 ∆θ to choose the North Celestial Pole as the ”reference” direction (direction of position angle zero) and star 2 a handedness to deﬁne the sign of the position angles (mathematically positive if looking at the Celestial Sphere from the inside). The set of position angles discussed here in- cludes – the direction of the zenith, the ”parallactic” angle, – the direction toward the point where the base- A line pinpoints the Celestial Sphere, the ”pro- b jected baseline angle,” – position angles of ”secondary” stars in diﬀer- Fig. 6. Example of three concentric circles on the ential astrometry. Celestial Sphere, centered at the baseline, which unite Computation of the angle proceeds via (27) if star directions of D = const. Two star positions are the object coordinates are given in the local altitude- connected by the short diagonal dash. A quarter of azimuth system, or via (39) if they are given in the each of the two projected baselines, which intersect equatorial system. at (Ab, ab) near the horizon, is also shown. The mathematics involved is applicable to ob- servatories at the Northern and the Southern hemi- sphere: positions (A, a) or (δ, h) are mapped onto a Fig. 6 sketches this geometry: the change ∆D 2D zenithal coordinate system. The position angles while re-pointing from one star by (∆A, ∆a) to an- play the role of the longitude (the North Celestial other star along some segment of the Celestial Sphere Pole the role of Greenwich or Aries). A distance be- can be split into a change associated with the radial tween the object and the star—in the range from 0 direction toward/away from T2 along a great circle, to π along great circles through the Celestial Pole in plus no change moving along a circle of radius cos θ an ARC projection—might take the role of the polar centered on the baseline azimuth. distance—although an orthographic SIN projection We may repeat the calculation in the equato- had been used for the calculations in this script. rial system and split ∆θ into ∆δ and ∆h, ∆D = −b sin θ∆θ Acknowledgements – This work is supported by the ˆ ˆ = b(∆δeδ · b + ∆h cos δeh · b) NWO VICI grant 639.043.201.

122 SPHERICAL TRIGONOMETRY OF THE PROJECTED BASELINE ANGLE

REFERENCES A time-independent offset b sin δ sin δb and an ampli- tude b cos δ cos δb, defined by products of the polar and equatorial components of star and baseline, re- Calabretta, M. R., Greisen, E. W.: 2002, Astron. spectively, constitute the regular motion of the delay. Astrophys., 395, 1077. The phase lag hb is determined by the arctan of Hanisch, R. J., Farris, A., Greisen, E. W., Pence, W. D., Schlesinger, B. M., Teuben, P. J., Thomp- son, R. W., Warnock III, A.: 2001, Astron. cos δb sin hb = sin Ab cos ab; Astrophys., 376, 359. cos δb cos hb = cos Ab cos ab sin φ + sin ab cos φ, Jones, G. C.: 2004, J. Geod., 76, 437. Karttunen, H., Kröger,P., Oja, H., Poutanen, M., Donner, K. J. (eds.): 1987, Fundamental As- and plays the role of an interferometric hour angle. tronomy, Springer, Berlin, Heidelberg. Lang, K. R.: 1998, Astrophysical Formulae, vol. II of Astronomy and Astrophysics Library, (3rd A.2. Closure Relations edn.), Springer, Berlin, Heidelberg. E: In the second line of (5.45), sin A should read cos A. National Imagery and Mapping Agency (NIMA): An array T1,T2 and T3 of telescopes forms a 2000, Department of defense world geodetic baseline triangle system 1984, Tech. Rep. TR8350.2. Pauls, T. A., Young, J. S., Cotton, W. D., Monnier, b + b + b = 0. J. D.: 2004, In: New Frontiers in Stellar In- 12 23 31 terferometry, Traub, W. A. (ed.), Proc. SPIE 5491, Int. Soc. Optical Engineering, 1231. If all point to a common direction s, some phase clo- Pauls, T. A., Young, J. S., Cotton, W. D., Monnier, sure sum rules result in: J. D.: 2005, Publ. Astron. Soc. Pacific, 117, 1255. Pearson, T. J.: 1991, Introduction to the Caltech D12 + D23 + D31 = 0; VLBI programs, Tech. rep., Astronomy De- D12 + D23 + D31 = 0; partment, California Institute of Technology. b12 cos θ12 + b23 cos θ23 + b31 cos θ31 = 0. http://www.astro.caltech.edu/~tjp/citvlb/ Vermeille, H.: 2004, J. Geod., 76, 451. Wallace, P. T.: 2003, SLALIB—Positional Astron- The triangle of projected baselines in the tangent omy Library 2.4–12 Programmer’s Manual. plane is Wood, T. E.: 1996, SIAM Review, 38, 637. Zhang, C.-D., Tsu, H. T., Wu, X. P., Li, S. S., Wang, P12 + P23 + P31 = 0. Q. B., Chai, H. Z., Du, L.: 2005, J. Geod., 79, 413. These vectors can be represented as numbers in a complex plane by introduction of three moduli P and three orientation angles p : APENDICES b

ipb12 ipb23 ipb31 P12e + P23e + P31e ipb12 ipb23 ipb31 A.1. Diurnal motion = b12 sin θ12e + b23 sin θ23e + b31 sin θ31e = 0.

The diurnal motion of the external path difference is described by separating terms proportional These equations remain correct if transformed by complex conjugation or multiplied by a complex to sin h and proportional to cos h. We rewrite (13) in phase factor; therefore these closure relations of the equatorial coordinates by multiplying (9) with (12), projected baseline angles are correct for (i) both and collect terms with the aid of (38): senses of deﬁning their orientation, and (ii) any ref- D = s · b = b[sin δ sin δb + cos δ cos δb cos(h − hb)]. erence direction of the zero angle.

123 R. J. MATHAR

SFERNA TRIGONOMETRIJA PROJEKCIJE UGLA OSNOVE

R. J. Mathar

Leiden Observatory, Leiden University, P.O. Box 9513, 2300RA Leiden, The Netherlands E–mail: [email protected]

UDK 520–872 : 521–181 Struqni qlanak

Osnovna geometrija zvezdanog interfe- nije. Razdvojna mo interferometra je ve- rometra koga qine dva teleskopa odreena lika u dva pravca du ovog linijskog seg- je pravouglim trouglom koji grade (i) os- menta (ograniqena difrakcijom na odnos ta- nova, odosno vektor koji spaja dva teleskopa, lasne duine i duine projektovane osnove (ii) vektor ”kaxǌeǌa” u pravcu ka zvezdi, interferometra), a mala normalno na ove i (iii) projekcija vektora osnove interfe- pravce (ograniqena difrakcijom na odnos ta- rometra na ravan talasnog fronta upadnog lasne duine i preqnika teleskopa). U ovom zraqeǌa. Ravan trougla preseca nebesku qlanku izraqunati su pozicioni uglovi ovih sferu po krugu koji prolazi kroz zvezdu; karakteristiqnih pravaca u horizontskim i sam presek predstavǉa segment krune li- nebeskim ekvatorskim koordinatama.

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