arXiv:astro-ph/0608273v3 2 May 2009 ln oteotclpt ea ocuetemanuscript the conclude IV. delay Section path tangent in optical the in the (star) to parallactic relating axis plane on optical the comments the Some if from away way these. distances bridge hori- hybrid to or used a is equatorial in angle position in or baseline purely systems. coordinates the done zontal coordinate computes be polar can and angle—which familiar defines III the Section in variables of Pole Celestial North star. the the from to pivot relocated (NCP) the ce- been where of system has definition coordinate point a a re- (ii) in poorly longitudes and and lestial interferometer, well the are by which solved directions detailing orthogonal Inter- 16], Optical [15, the the (OIFITS) Format of Exchange tables ferometry visibility the with associated at.Ti s()teitouto faplrcoordi- polar a of introduction the in the angle (i) polar on system’s is location nate observatory’s This the of Earth. coordinates coordi- independent sky equatorial Ce- are to in the which baseline directions onto the defines ties projection which this this nates, that use is base- to Sphere projected lestial incentive oriented The an meets defines line. baseline prolongated Sphere the the Celestial between where section the point its the and This and circle, star Sphere. great Celestial the a the is and with intersection interferometer intersection stellar its a and star, of baseline the contains ah OtclItreoer:AnwMto o tde fE of Studies for Method new Planets.” A trasolar Interferometry: “Optical bach, † URL: ∗ upre yteNOVC rn 3.4.0 oA Quirren- A. to 639.043.201 grant VICI NWO the by Supported lcrncades [email protected]; address: Electronic noeve,ScinI nrdcstesadr set standard the introduces II Section overview, In h ae ecie tnadt en h ln that plane the define to standard a describes paper The http://www.strw.leidenuniv.nl/ trpstoswihaeaindwt rotooa otegr the orthog to two orthogonal represents or angle with This aligned are tangentia star. which manus plane the positions celest The star a of the in position light. onto measured the outwards stellar defined, at vecto is further the delay baseline baseline of the projected wavefronts the are the vectors projecting of derived of Two plane the star. a in to direction a ewrs rjce aeie aalci nl;positio angle; interferometer parallactic baseline; projected Keywords: 95.75.Kk 95.10.Jk, angle. numbers: “parallactic” PACS di the reference of the definition as common chosen the is to Pole adapted Celestial North The plane. h ai emtyo tla nefrmtrwt w tele two with interferometer stellar a of geometry basic The ednOsraoy ednUiest,PO o 53 2300 9513, Box P.O. University, Leiden Observatory, Leiden peia rgnmtyo h rjce aeieAngle Baseline Projected the of Trigonometry Spherical .SCOPE I. u − ~ mathar v oriaesystem coordinate Dtd coe 9 2018) 29, October (Dated: ihr .Mathar J. Richard x- nl;clsilshr;shrclatooy stellar astronomy; spherical sphere; celestial angle; n epniua oti iceis circle this to perpendicular ee tteErhcne,connects center, Earth the at tered ute on ra iceo radius of used circle great be A will that down. coordinates further Cartesian other from apart iae ftetotlsoe faselrinterferometer longitudes stellar geographic a the of to telescopes related two are the of dinates eada“ a add We liueaoesalevel), sea above altitude radius Earth effective and aeieaetr angle aperture baseline a h aeievector baseline the h atcldrcinfrom direction nautical The olw:dfieatnetpaet h at at Earth the to plane tangent a define follows: sse rmteErhcenter: Earth the from seen is rhnra etr htsa hspaeare plane this span that vectors orthonormal J ˆ steai frtto when rotation of axis the is nagoeti oriaesse,teCreincoor- Cartesian the system, coordinate geocentric a In N ˆ 1 T cos ≡ deto h ea eoddi the in recorded delay the of adient ∗ 1 a pee h oiinageo the of angle position The sphere. ial eto ftepoetdbsln angle, baseline projected the of rection oteclsilshr,tnetpoint tangent sphere, celestial the to l cpscnit fabsln etrand vector baseline a of consists scopes nldrcin ntesy differential sky, the on directions onal   · Z T T ,adtepoetdbsln vector baseline projected the and r, cos = − − g i 2 rp el ihtetrigonometry the with deals cript otegoeti oriae ostthem set to coordinates geocentric the to ” ≡ ALie,TeNetherlands The Leiden, RA = cos sin .ShrclApproximation Spherical 1. I IEGEOMETRY SITE II. .TlsoePositions Telescope A. cos ρ | T λ λ   φ 1 1 1 1 φ | | sin cos sin sin cos 1 T b sin φ 2 φ λ λ φ = | 1 1 i i 2 cos Z cos cos ρ φ   T cos( i a endfie ne which under defined been has smo h at aisand radius Earth the of (sum 2 g Z φ φ − T ∼ J ˆ λ (3) ; i i T 2 1 ≡ T   ∂ − 1 T 1 to g ρ smvdtoward moved is λ 1 i , 2 T /∂φ sin 1 T 1 sin + ) 1 ρ 2 Z u and ≈ 1 = 1 † − scmue as computed is T λ ; i 30k,cen- km, 6380 v 1 , , T × | φ 2 N ˆ 2 . 1 T vector a ; 1 sin T 2 | where , 1 1 = Two . φ 2 T . . (1) (2) (4) (5) φ 2 i . 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 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

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 , 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 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. The sign convention is left- handed placing the head of s at the center of the Celestial − sin φ cos λ − sin φ sin λ cos φ Sphere. This is equivalent to drawing a line between the   Utg(φ, λ)= sin λ − cos λ 0 . (24) star and the NCP, looking at it from inside the sphere,  cos φ cos λ cos φ sin λ sin φ  and defining position angles of points as the angles in polar coordinates in the mathematical sign convention, Just like the star direction (12), the unit vector bˆ along using this line as the abscissa and the star as the center. the baseline direction defines a baseline azimuth Ab and Another, fully equivalent definition uses a right-handed a baseline elevation ab turn of the object around the star until it is North of the star. And finally, as an aid to memory, it is also b − cos Ab cos ab the nautical course of a ship sailing the outer hull of the ˆ   b ≡ ≡ sin Ab cos ab . (25) Celestial Sphere, which is currently poised at the star’s b sin a  b t coordinates and is navigated toward the point. This defines values modulo 2π; whether these are fi- D The standard definition of the delay vector and pro- nally represented as numbers in the interval [0, 2π) or in P jected baseline vector is the interval (−π, +π]—as the SLA BEAR and the SLA PA routines of the SLA library [23] do—is largely a matter b = D + P; D k s; P ⊥ D. (26) of taste. P—and later its circle segment projected on the Celestial An overview on the relevant geometry is given in Fig- Sphere—inherits its direction from b such that heads and ure 3 which looks at the celestial sphere from the outside. tails of the vectors are associated with the same portion In Figure 3, the baseline has been infinitely extended of the wavefront: Fig. 2. We use the term “projected straight outwards in both directions which defines tele- baseline” both ways: for the straight vector (of length P , scope coordinates T1 and T2 also where the baseline pen- units of meter) that connects the tails of D and b, or the etrates the Celestial Sphere. Standing at the mid-point line segment (length θ, units of radian) on the Celestial of the baseline, telescope 1 then is at azimuth τ, telescope Sphere. 2 at azimuth Ab = τ + π, see Fig. 1. Let the object be The dot product of (12) by (25) is at azimuth A and elevation a. The projection of the baseline occurs in a plane in- D = s · b = b cos θ, (0 ≤ θ ≤ π), (27) cluding the object and the baseline and thus defining the 4

cos A sin a e =  − sin A sin a  . (31) zenith a cos a  t

z (The equivalent exercise with a different choice of axes follows in Section III D.) These are orthonormal,

s·eA = s·ea = eA·ea = 0; |eA| = |ea| = 1; s×ea = eA. (32) Decomposition of the direction from s to the NCP defines NCP E T1 star three projection coefficients c0,1,2,

s+ = c0s + c1eA + c2ea. (33) S a θ Building the square on both sides yields the familiar for- N mula for the sum of squares of the projected cosines, 2 2 2 A -A c0 + c1 + c2 =1. (34) W T2 b Two dot products of (33) using (20) and (30)–(31) solve FIG. 3: Celestial sphere, seen from the outside. The north for two coefficients, direction through the object is given by the great circle pass- c1 = s+ · e = cos φ sin A, (35) ing through the celestial poles and the object. The angle θ is A the vector P = b − D projected on the sphere. c2 = s+ · ea = cos φ cos A sin a + sin φ cos a. (36)

ea great circle labeled θ in Figure 3. If θ is the angle on the sky between the object and the point on the horizon with NCP azimuth A , then the length P of the projected baseline b c 2 is given by p P = b sin θ, (29) where θ is calculated from the relation (28). With this auxiliary quantity, there are various ways of obtaining the pb of the projected baseline on the sky, pb some of which are detailed in Sections III B–III D. The ϕ common theme is that the baseline orientation (Ab,ab) in the local horizontal polar coordinate system is trans- ferred to a rotated polar coordinate system with the star defining the new polar axis; the position angle is the dif- b c'2 ference of azimuths between T2 and the NCP in this rotated polar coordinate system. ϕ' pb star B. In Horizontal Coordinates eA c 1 c'1 ψ

We use a vector algebraic method to span a plane tan- gential to the celestial sphere; tangent point is the po- FIG. 4: In the tangent plane spanned by the unit vectors sition s of the star. Within this orthographic zenithal ′ (30) and (31), the components of bˆ and s+ are c1, c1, c2 and projection [3] we consider the two directions from the c′ defined by (33) and (40). The view is from the outside s 2 origin toward the NCP, +, on one hand and toward T2, onto the plane, so eA, the axis vector into the direction of bˆ, on the other. The tangential plane is fixed by any two increasing A, points to the left. unit vectors perpendicular to the vector s of (18). Rather s s arbitrarily they are chosen along ∂ /∂A and ∂ /∂a, ex- An angle ϕ between eA and s+ is defined via Fig. 4, plicitly c1 cos ϕ = , (37) c2 + c2 sin A p 1 2 e   c2 A = cos A , (30) sin ϕ = . (38)  0  c2 + c2 t p 1 2 5

From (34), and since (21) equals c0, The computational strategy is to build

′ ′ c2 + c2 = cos δ. (39) c1c2 − c1c2 q 1 2 tan pb = ′ ′ . (52) c1c1 + c2c2 The equivalent splitting of bˆ into components within and perpendicular to the tangent plane is: The (positive) values of cos δ and sin θ in the denomi- nators of (48) and (49) need not to be calculated. The ˆ ′ ′ ′ b = c0s + c1eA + c2ea. (40) branch ambiguity of the arctan is typically handled by use of the atan2() functionality in the libraries of higher Building the square on both sides yields programming languages.

′2 ′2 ′2 c0 + c1 + c2 =1. (41) Dot products of (40) solve for the expansion coefficients, C. Via with (25) and (30)–(31): If the position angle of the zenith p is known by any ′ ˆ c1 = b · eA other sources—see App. A—a quicker approach to the

= − cos Ab cos ab sin A + sin Ab cos ab cos A calculation of pb employs an auxiliary angle ψ [11],

= cos ab sin(Ab − A), (42) pb ≡ p + π + ψ (mod 2π). (53) ′ ˆ c2 = b · ea

= − cos ab sin a cos(Ab − A) + sin ab cos a. (43) The calculation of ψ is delegated to the calculation of its sines and cosines This defines an angle ϕ′ in the polar coordinates of the tangent plane, cos ψ = − cos pb cos p − sin pb sin p; (54)

′ sin ψ = − sin pb cos p + cos pb sin p. (55) ′ c1 cos ϕ = ′2 ′2 , (44) c1 + c2 In the right hand sides we insert (48)–(49) and (A1)– p ′ ′ c (A2), and after some standard manipulations sin ϕ = 2 . (45) c′2 + c′2 p 1 2 cos a sin(A − A) sin ψ = b b ; (56) ′ ˆ Since c0 equals b · s = cos θ in (28), sin θ sin a cos θ − sin a cos ψ = b ′2 ′2 sin θ cos a qc1 + c2 = sin θ. (46) cos a sin a cos(A − A) − sin a cos a = b b b . (57) The baseline position angle is the difference between the sin θ e two angles, as to redefine the reference direction from A ′ ′ to the direction of the NCP (see Fig. 4): These are c1 and −c2 of (42) and (43) divided by sin θ. In Fig. 4, ψ can therefore be identified as the angle be- ′ pb = ϕ − ϕ (mod 2π). (47) tween −ea and bˆ, and this is consistent with (53) which decomposes the rotation into the angle p, a rotation by Sine and cosine of this imply [1, 4.3.16, 4.3.17] the angle π (which would join the end of p with the start ′ ′ of ψ in Fig. 4), and finally a rotation by ψ. sin pb = sin ϕ cos ϕ − cos ϕ sin ϕ ′ ′ In summary, the disadvantage of this approach is that c1c − c c2 = 2 1 ; (48) p must be known by other means, and the advantage is cos δ sin θ that computation of ψ via the arctan of (56) and (57) ′ ′ ′ ′ cos pb = cos ϕ cos ϕ + sin ϕ sin ϕ only needs c1 and c2, but not c1 or c2. ′ ′ c1c + c2c = 1 2 , (49) cos δ sin θ D. In Equatorial Coordinates where ′ ′ c1c2 − c1c2 = cos φ(sin A sin ab cos a − cos ab sin a sin Ab) Section III B provides pb, given star coordinates (A, a). − sin φ cos a cos ab sin(Ab − A), (50) Subsequently we derive the same value in terms of δ and h. The straight forward approach is the substitution of and (13)–(17) in (50) and (51). A less exhaustive alternative e e ′ ′ works as follows [9]: the axes A and a in the tangential c1c1 + c2c2 = cos φ(cos A cos a cos θ − cos ab cos Ab) plane are replaced by two different orthonormal direc- ′ +c2 sin φ cos a. (51) tions, better adapted to δ and h, namely the directions 6 along along ∂s/∂h and ∂s/∂δ: Building the dot product of (63) with eδ yields

′ sin φ sin h d1 = − cos Ab cos ab cos φ cos δ e  cos h  h = ; (58) − cos Ab cos ab sin φ sin δ cos h  − cos φ sin h  t − sin Ab cos ab sin δ sin h + sin ab sin φ cos δ cos φ cos δ + sin φ sin δ cos h   − sin ab cos φ sin δ cos h. (65) eδ = − sin δ sin h . (59) sin φ cos δ − cos φ sin δ cos h  t Some simplification in this formula is obtained by trading the hour angle h for the angle θ, which might be more The further calculation follows the scheme of Section readily available since θ is measured via the delay: In the IIIB: The axes are orthonormal, second term we use (14) to substitute s e s e e e e e s e e · h = · δ = h· δ = 0; | h| = | δ| = 1; × δ = h. cos φ sin δ + cos A cos a (60) sin φ cos h → . (66) cos δ The decomposition of s+ in this system defines three ex- pansion coefficients d0,1,2, In the third term we use (13) to substitute

s+ = d0s + d1eδ + d2eh. (61) sin A cos a sin h → , (67) cos δ Multiplying this equation in turn with eh and eδ yields and in the last term we use (15) to substitute d2 = 0; d1 = s+ · eδ = cos δ. (62) sin a − sin δ sin φ The direction bˆ to T2 is projected into the same tangent cos φ cos h → . (68) ′ cos δ plane, defining three expansion coefficients d0,1,2, Further standard trigonometric identities [1, ˆ ′ ′ ′ b = d0s + d1eδ + d2eh. (63) 4.3.10,4.3.17] and (28) yield

Building the dot product of this equation with e using ′ sin ab sin φ − cos ab cos Ab cos φ − sin δ cos θ h d = . (69) 1 cos δ

In the numerator we recognize a baseline declination δb, eδ

s+ · bˆ ≡ sin δb = sin ab sin φ − cos ab cos Ab cos φ. (70)

pb is the angle that rotates the projected vector (d2, d1) within the tangent plane into the direction (d′ , d′ ) (Fig. pb 2 1 NCPd1 5),

′ d1 cos pb = , (71) d'1 sin θ ′ b d2 sin p = − . (72) b sin θ Again, the sin θ does not need actually to be calculated, but only star ′ d' −d2 2 tan pb = , (73) eh d′ pb 1 and again, selection of the correct branch of the arctan is easy with atan2() functions if the negative sign is kept ′ attached to d2. FIG. 5: Fig. 4 after switching from the (eA, ea) axes to Note also that the use of (69) is optional: d′ and d′ ˆ 2 1 (eh, eδ). Axis components of s+ and b are indicated. are completely defined in terms of the baseline direction (Ab,ab), the geographic latitude φ and the star coordi- (25) yields nates (h,δ) via (64) and (65). With these one can imme- ′ diately proceed to (73). d2 = − cos Ab cos ab sin φ sin h + sin Ab cos ab cos h OIFITS [15, 16] defines no associated angle in the plane − sin ab cos φ sin h. (64) perpendicular to the direction of the phase center. 7

The segment of the baseline in the tangent plane pri- Equating the right hand side of the previous equation marily defines an orientation in the u−v plane. However, with zero, this means modal decomposition of the amplitudes in products of ∆a radial and angular basis functions (akin to Zernike poly- = tan ψ (∆θ = ∆D = 0). (76) nomials) means that the Fourier transform to the x − y cos a∆A plane preserves the angular basis functions (see the cal- culation for the 3D case, Spherical Harmonics, in [13]). For the direction of maximum change in ∆D, which runs perpendicular to (76), In this sense, the angle pb is also “applicable” in the x−y plane. ∆a 1 = − , (p : max ∆D) (77) cos a ∆A tan ψ b

IV. DIFFERENTIALS IN THE FIELD-OF-VIEW which leads to

A. Sign Convention ∆D = −b sin θ∆θ cos a ∆A ∆a = b sin θ = −b sin θ , (max ∆D(78)). The provisions of the previous section, namely sin ψ cos ψ

• the OIFITS sign convention of the baseline vector, Example: a field of view of ∆θ = ±1′′ at a baseline Eq. (2) and Fig. 2, of b = 100 m at a “random” position θ = 45◦ scans ∆D = ±340 µm. The details of the optics determine how • the conventional formula (27) far this contributes to the instrumental visibility (loss). Fig. 6 sketches this geometry: the change ∆D when re- fix the sign of the delay D as follows: if the wavefront pointing from one star by (∆A, ∆a) to another star along hits T2 prior to T1, the values of D and cos θ are positive; some segment of the Celestial Sphere can be split into a if it hits T1 prior to T2, both values are negative. change associated with the radial direction toward/away The transitional case D = 0 (θ = π/2) occurs if the from T2 along a great circle, and a zero change moving star passes through the “baseline meridian” plane per- along a circle of radius cos θ centered on the baseline pendicular to the baseline; the two lines to the North- azimuth. west and Southeast in Fig. 1 show (projections of) these directions. The baseline in most optical stellar inter- ferometers is approximately horizontal, ab ≈ 0. From zenith (27) we see that this case is approximately equivalent to cos(Ab − A) ≈ 0. Since cos a > 0, the sign of D coin- star 1 cides with the sign of cos(Ab − A). This is probably the ∆θ fastest way to obtain the sign of the Optical Path Dif- ference from FITS [6] header keywords; the advantage of star 2 this recipe is that the formula is independent of which azimuth convention is actually in use. The baseline length b follows immediately from the Eu- clidean distance of the two telescopes entries for column STAXYZ of the OIFITS table OI ARRAY [16].

Ab B. External Path Delay FIG. 6: Example of three concentric circles on the Celestial The total differential of (28) relates a direction Sphere, centered at the baseline, which unite star directions of (∆A, ∆a) away from the star to a change in the angular D = const. Two star positions are connected by the short di- distance between the star and the baseline, agonal dash. A quarter of each of the two projected baselines, which intersect at (Ab,ab) near the horizon, is also shown.

− sin θ∆θ = ∆a [− cos ab sin a cos(Ab − A) + sin ab cos a]

+∆A cos a cos ab sin(Ab − A). (74) We may repeat the calculation in the equatorial system and split ∆θ into ∆δ and ∆h, (56) and (57) turn this into ∆D = −b sin θ∆θ (79) sin θ∆θ = sin θ(cos ψ∆a − sin ψ cos a∆A), (75) = b(∆δeδ · bˆ + ∆h cos δeh · bˆ) (80) ′ ′ consistent with the decomposition in Fig. 4. The two = b(∆δ d1 + ∆h cos δ d2) (81) directions of constant delay are characterized by ∆θ = 0. = b sin θ(∆δ cos pb − ∆h cos δ sin pb). (82) 8

′ This translates [3, App. C] to our variables. When c2 = cos a. This also turns (28) into cos θ = sin a. (48)– ∆θ = 0, we have the directions of zero change in ∆D, (49) become perpendicular to the projected baseline, cos φ sin A sin p = ; (A1) ∆δ cos δ = tan pb, (∆θ = ∆D = 0). (83) cos φ cos A sin a + sin φ cos a cos δ∆h cos p = . (A2) cos δ The direction of maximum change is The ratio of these two equations is ∆δ 1 cos φ sin A = − , (pb : max ∆D), (84) tan p = . (A3) cos δ∆h tan pb cos φ cos A sin a + sin φ cos a and along this gradient This expression can be transformed from the horizontal to equatorial coordinates as follows: Multiply numerator cos δ∆h ∆δ and denominator by cos a and use (13) in the numerator, ∆D = −b sin θ = b sin θ , (max ∆D). (85) sin p cos p b b cos φ cos δ sin h tan p = . (A4) cos φ cos A sin a cos a + sin φ cos2 a V. SUMMARY Insert (17) in the denominator, cos φ cos δ sin h To standardize the nomenclature, we propose to choose tan p = 2 the North Celestial Pole as the “reference” direction (di- sin a(sin a sin φ − sin δ) + sin φ cos a cos φ cos δ sin h rection of position angle zero) and a handedness to define = . (A5) the sign of the position angles (mathematically positive sin φ − sin a sin δ if looking at the Celestial Sphere from the inside). Replace sin a with (15) in the denominator, eventually The set of position angles discussed here includes divide numerator and denominator through cos φ cos δ: • the direction of the zenith, the “parallactic” angle, cos φ cos δ sin h tan p = sin φ − (sin δ sin φ + cos δ cos h cos φ) sin δ • the direction toward the point where the baseline pinpoints the Celestial Sphere, the “projected base- cos φ cos δ sin h = 2 line angle,” cos δ sin φ − cos δ sin δ cos h cos φ sin h = . (A6) • position angles of “secondary” stars in differential cos δ tan φ − sin δ cos h astrometry. These manipulations involved only multiplications with Computation of the angle proceeds via (52) if the ob- positive factors like cos φ, cos δ and cos a; therefore one ject coordinates are given in the local altitude-azimuth can use sin h and cos δ tan φ − sin δ cos h separately as system, or via (73) if they are given in the equatorial the two arguments of the atan2 function to resolve the system. branch ambiguity of the inverse tangent. The mathematics involved is applicable to observato- This definition of the parallactic angle coincides with the dominant one in the literature [2, 18, 19, 20, 22]; ries at the Northern and the Southern hemisphere: posi- ◦ tions (A, a) or (δ, h) are mapped onto a 2D zenithal co- others exist: Equation (1) in [4] generates the 90 - ordinate system. The position angles play the role of the complement of our definition, for example. longitude (the North Celestial Pole the role of Greenwich If we insert (57)–(56) into (75) [1, 4.3.18] and compare or Aries). A distance between the object and the star (in with (82), we find the range from 0 to π along great circles through the Ce- ∆ ∆δ a + tan p lestial Pole in an ARC projection) might take the role of cos a∆A = ∆a , (A7) the polar distance—although an orthographic SIN pro- cos δ∆h 1 − tan p cos a∆A jection had been used for the calculations in this script. which means the parallactic angle mediates between di- rections expressed as axis ratios in the two coordinate systems [12]. APPENDIX A: PARALLACTIC ANGLE

We define the parallactic angle p as the position angle APPENDIX B: of the zenith. Since the formulas in Section III dealt with the general case of coordinates on the Celestial Sphere, The diurnal motion of the external path difference is ′ one may just set ab = π/2 in (25), which induces c1 = 0, described by separating the terms ∝ sin h and ∝ cos h. 9

We rewrite (27) in equatorial coordinates by multiplying therefore these closure relations of the projected baseline (18) with (25), and collect terms with the aid of (70): angles are correct for (i) both senses of defining their orientation, and (ii) any reference direction of the zero D = s · b = b[sin δ sin δb + cos δ cos δb cos(h − hb)]. (B1) angle.

A time-independent offset b sin δ sin δb and an amplitude b cos δ cos δb, defined by products of the polar and equa- torial components of star and baseline, respectively, con- APPENDIX D: NOTATIONS stitute the regular motion of the delay. The phase lag hb is determined by the arctan of α right ascension cos δb sin hb = sin Ab cos ab; (B2) a, ∆a star elevation and its difference cos δb cos hb = cos Ab cos ab sin φ + sin ab cos φ,(B3) A, ∆A star azimuth angle, and its difference and plays the role of an interferometric hour angle. Ab baseline azimuth b,b baseline vector and length δ declination APPENDIX C: CLOSURE RELATIONS D signed external path difference eh, eδ, eA, ea unit vector in tangent plane An array T1, T2 and T3 of telescopes forms a baseline e eccentricity of geoid triangle φ geographic or geodetic latitude

b12 + b23 + b31 =0. (C1) h hour angle H altitude above geoid If all point to a common direction s, some phase closure l local sum rules result: λ geographic longitude

D12 + D23 + D31 =0; (C2) N distance to the Earth axis measured along local normal D12 + D23 + D31 =0; (C3) p position angle of the zenith, “parallactic b12 cos θ12 + b23 cos θ23 + b31 cos θ31 =0. (C4) angle” The triangle of projected baselines in the tangent plane pb position angle of the projected baseline is P projected baseline length ρ Earth radius P12 + P23 + P31 =0. (C5) ρe Earth equatorial radius s These vectors can be represented as numbers in a com- unit vector into star direction plex plane by introduction of three moduli P and three s+ unit vector into NCP direction orientation angles pb: θ angular separation between star and baseline vector ipb12 ipb23 ipb31 P12e + P23e + P31e ψ angle between projected baseline circle ipb12 ipb23 ipb31 = b12 sin θ12e + b23 sin θ23e + b31 sin θ31e and star meridian =0. (C6) z zenith angle Z baseline angle seen from the Earth cen- These equations remain correct if transformed by com- ter plex conjugation or multiplied by a complex phase factor;

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