Matrix Method for Coordinates Transformation

Transcription

1 atrix tod for Coordinats Transformation Tosimi Taki January, Rvision A: Fruary 7, Rvision B: Dcmr, Rvision C: January, Rvision D: ovmr, Rvision E: Fruary 9, 4 Tal of Contnts. Introduction.... Rfrncs.... otations.... ot.... Symols Basic Equations of Coordinats Transformation in atrix tod Polar Coordinats and Rctangular Coordinats Coordinat Transformation w Coordinat Systm Rotatd around Z-axis w Coordinat Systm Rotatd around X-axis w Coordinat Systm Rotatd around Y-axis Otaining Polar Coordinats from Dirction Co ots on Approximation Approximation of Trigonomtric Functions Approximation of Otr Functions Applications Transformation from Equatorial Coordinats to Altazimut Coordinats Transformation Equations Exampl Calculation Angular Sparation Equations Exampl Calculation Compnsation of ounting Farication Errors Tlscop Coordinats... /7

2 5.. Farication Errors of ount Drivation of Equations Apparnt Tlscop Coordinat witout Approximation Exampl Calculations Equations for Pointing Tlscop Introduction Transformation atrix Drivation of Transformation atrix Exampl Calculation Commnt on Accuracy of t Pointing tod Polar Axis isalignmnt Dtrmination Drivation of Equations Exampl Calculations Dom Slit Syncronization Ojct in First Quadrant Ojct in Scond Quadrant Ojct in Tird Quadrant Ojct in Fourt Quadrant Intrsction... 7 /7

3 . Introduction Coordinats transformation is a asic part of astronomical calculation and sprical trigonomtry as n long usd for astronomical calculation in amatur astronomy. Sprical trigonomtry quations can a littl it difficult for amaturs to undrstand. In t last two dcads, dvlopmnt of prsonal computrs as rougt aout a cang in t way astronomical calculations ar carrid out. In my opinion, sprical trigonomtry is not appropriat to astronomical calculation ug prsonal computrs. I rcommnd t matrix mtod for coordinats transformation, caus of its simplicity and as of gnralization in writing computr programs. In tis monograp, I dscri coordinats transformation ug t matrix mtod. I also xtnd t mtod to som spcific applications, suc as polar axis misalignmnt dtrmination of quatorial mount (Callis mtod and a tlscop pointing algoritm.. Rfrncs [] Jan us, Astronomical Formula for Calculators, 985, Willmann-Bll, Inc. [] Jan us, Astronomical Algoritms, 99, Willmann-Bll, Inc. [] Ko agasawa, Calculation of Position of Astronomical Ojcts, 985, Cijin-Sokan Co., in Japans [4] W. R. Vzin, Polar Axis Alignmnt of Equatorial Instrumnt [5] Rv. Jams Callis, cturs on Practical Astronomy and Astronomical Instrumnts, 879. [6] Tosimi Taki, A w Concpt in Computr-Aidd Tlscops," Sky & Tlscop, Fruary 989, pp otations. ot In tis monograp, angls ar xprssd in radian, caus all computr languags for prsonal computrs us radian for trigonomtric functions.. Symols Following symols ar usd in tis monograp. X-Y-Z : Gnral rctangular coordinat systm /7

4 X -Y -Z : X -Y -Z : X t -Y t -Z t : Rctangular quatorial coordinat systm Rctangular altazimut coordinat systm Rctangular tlscop coordinat systm α : Rigt Ascnsion (in radian : Dclination (in radian A : Azimut, masurd wstward from t Sout. (in radian : Altitud (in radian ξ : Gnral polar coordinat masurd countrclockwis from X-axis in XY-plan (in radian ζ : Gnral polar coordinat, masurd upward from XY-plan. (in radian ϕ, θ : Tlscop polar coordinats (in radian,, " : Tlscop mount farication rrors (in radian : X-componnt of dirction in of clstial ojct in X-Y-Z coordinats : Y-componnt of dirction in of clstial ojct in X-Y-Z coordinats : Z-componnt of dirction in of clstial ojct in X-Y-Z coordinats : our angl : Osrvr s latitud θ : Sidral tim at Grnwic t : Tim u, v : Tlscop polar axis misalignmnt (in radian JD : Julian day numr d : Angular distanc twn two ojcts [T] : Transformation matrix twn coordinat systms R : Atmospric rfraction (in radian 4/7

5 4. Basic Equations of Coordinats Transformation in atrix tod 4. Polar Coordinats and Rctangular Coordinats In astronomical calculations, polar coordinat systms ar usually usd. S figur 4-. Point O is t osrvation point. Vctor OR sows unit vctor dircting to a clstial ojct. T position of t clstial ojct is xprss in polar coordinats (ξ, ζ. ormally, angl ξ is masurd countrclockwis from X-axis (viwing from positiv Z and angl ζ is masurd upward (toward Z-axis from XY-plan. Z Clstial Ojct R (,, O ζ X ξ Y Spr wit radius Figur 4- Polar Coordinats and Rctangular Coordinats An xampl of polar coordinats is rigt ascnsion and dclination, (α,. S figur 4-. T otr xampl is azimut and altitud, (A,. But azimut is masurd wstward (clockwis from t Sout wic is t opposit dirction to t normal polar coordinat systm. S figur 4-. 5/7

6 Z (Clstial ort Pol Clstial Ojct Clstial Equator R (,, O α Y X (Vrnal Equinox Figur 4- Equatorial Coordinats Z (Znit Clstial Ojct orizon R (,, A O Y (East X (Sout Figur 4- Altazimut Coordinats 6/7

7 T vctor OR is also xprssd in rctangular coordinats, (,,. (,, is calld dirction in. In t matrix mtod, dirction ins ar usd to xprss coordinat transformation. Rlationsip twn rctangular coordinats and polar coordinats can xprssd in matrix form as follows. ζ ξ ζ ξ ζ. Equation (4- For quatorial coordinats, α α. Equation (4- For orizontal coordinats, ( A ( A. Equation (4- ot tat (-A is usd in t quation (4- instad of A, caus azimut A is masurd clockwis. 7/7

8 4. Coordinat Transformation 4.. w Coordinat Systm Rotatd around Z-axis w coordinat systm, X -Y -Z is gnratd rotating X-Y-Z coordinats around Z-axis as sown in figur 4-4. T polar coordinats in X -Y -Z coordinat systm is (ξ, ζ and t dirction in in X -Y -Z coordinat systm is (,,. T rlationsip twn t dirction ins in ot coordinat systms is xprssd as follows. ζ ξ ζ ξ ζ. Equation (4-4 θ z θ z θ θ z z. Equation (4-5 θ θ z z θ θ z z. Equation (4-6 Y Y O OR : unit vctor R X θ z (rotat countrclockwis around Z-axis X ooking ormal to XY-plan Figur 4-4 Coordinats Rotation around Z-axis 8/7

9 4.. w Coordinat Systm Rotatd around X-axis w coordinat systm, X -Y -Z is gnratd rotating X-Y-Z coordinats around X-axis as sown in figur 4-5. T polar coordinats in X -Y -Z coordinat systm is (ξ, ζ and t dirction in in X -Y -Z coordinat systm is (,,. Tn t rlationsip twn t dirction ins in ot coordinat systms is xprssd as follows. ζ ξ ζ ξ ζ. Equation (4-7 θ x θ x θ θ x x. Equation (4-8 θ x θ x. Equation (4-9 θ x θ x Z Z O OR : unit vctor R Y θ x (rotat countrclockwis around X-axis Y ooking ormal to YZ-plan Figur 4-5 Coordinats Rotation around X-axis 9/7

10 4.. w Coordinat Systm Rotatd around Y-axis w coordinat systm, X -Y -Z is gnratd rotating X-Y-Z coordinats around Y-axis as sown in figur 4-6. T polar coordinats in X -Y -Z coordinat systm is (ξ, ζ and t dirction in in X -Y -Z coordinat systm is (,,. Tn t rlationsip twn t dirction ins in ot coordinat systms is xprssd as follows. ζ ξ ζ ξ ζ. Equation (4- θ θ y y θ θ y y. Equation (4- θ y θ y θ θ y x. Equation (4- X X O OR : unit vctor R Z θ y (rotat countrclockwis around Y-axis Z ooking ormal to ZX-plan Figur 4-6 Coordinats Rotation around Y-axis /7

11 4. Otaining Polar Coordinats from Dirction Co Aftr coordinat transformation ug t matrix mtod it is ncssary to otain t polar coordinats (ξ, ζ from t dirction ins. Ug quation (4-4, ξ and ζ ar otaind from dirction ins as sown low. tanξ. Equation (4- Wn >, ξ is in t st quadrant or t 4t quadrant. Wn <, ξ is in t nd quadrant or t rd quadrant. ζ. Equation (4-4 -/ (-9 o < ζ < / (9 o /7

12 4.4 ots on Approximation 4.4. Approximation of Trigonomtric Functions Wn w procss small angls in trigonomtry, approximation of trigonomtric functions is oftn usd. In t following approximations, θ is vry small angl and xprssd in radian. θ θ. Equation (4-5 θ. Equation (4-6 For igr ordr approximation, θ θ. Equation ( Approximation of Otr Functions For otr functions, following approximation can usd wn x is vry small compard to. x x. Equation (4-8 ( x x. Equation (4-9 /7

13 /7 5. Applications 5. Transformation from Equatorial Coordinats to Altazimut Coordinats 5.. Transformation Equations Altazimut coordinat systm, X -Y -Z is rotatd (/ - around Y -axis to quatorial coordinat systm, X -Y -Z. is osrvr s latitud. S figur 5.-. T dirction ins ar xprssd in angls as follows. A A ( (. Equation (5.- Wr A is azimut masurd wstward from t Sout and is altitud. ( (. Equation (5.- Wr is local our angl masur wstward from t Sout and is dclination. Rlationsip twn t coordinats is xprssd in matrix form as sown low.. Equation (5.-. Equation (5.-4 A tan(. Equation (5.-5 Wn >, (-A is in t st quadrant or t 4t quadrant. Wn <, (-A is in t nd quadrant or t rd quadrant.. Equation (5.-6

14 -/ (-9 o < < / (9 o X (ridian Z (Znit Clstial Ojct Z (ort Pol Equator - orizon O Y, Y (East X (Sout ridian Figur 5.- Altazimut Coordinats and Equatorial Coordinats Comparison wit sprical trigonomtric quations (rf. [] is prformd low. From quations (5.-, (5.-4 and (5.-5, w otain t following quations. tan( A ( (. Equation (5.-7 ( ( tan ( tan 4/7

15 (. Equation (5.-8 Ts quations ar t sam as quations (8.5 and (8.6 in rf. []. 5.. Exampl Calculation Exampl 8. in rf. []: Find t azimut and t altitud of Saturn on 978 ovmr at 44ms UT at t Uccl Osrvatory (longitud 7m5.94s, latitud 5 o (radian; t plant s apparnt quatorial coordinats, intrpolatd from t A.E., ing α 57m5.68s (radian 8 o o.478 (radian T apparnt sidral tim at Grnwic, θ 8m46.5s. ocal our angl, is, θ - - α 8m46.5s 7m5.94s 57m5.68s -8m.66s x 5 / (8/ (radian (radian From quation (5.-, (.769 ( From quation (5.-4, 5/7

16 6/ (.6849 (.6849 (.6849 ( From quations (5.-5 and (5.-6, tan( A -A.966 (radian A (radian o (radian o

17 X (ridian Clstial Ojct : our Angl θ : ocal Sidral Tim O Y Y α: Rigt Ascnsion X (Vrnal Equinox ooking from ort Pol ormal to Equatorial Plan Figur 5.- our Angl and Sidral Tim 7/7

18 5. Angular Sparation 5.. Equations T angular distanc d twn two clstial ojcts, P and P is drivd ug t matrix mtod. Position of ojct, P : (ξ, ζ Position of ojct, P : (ξ, ζ Dirction ins of t two ojcts ar, ζ ξ ζ ξ ζ ζ ξ ζ ξ ζ. Equation (5.-. Equation (5.- Ug scalar product of t two unit vctors, OP and OP, angular sparation d is otaind as follows. S figur 5.-. d ξ ς ξ ς ς ς ς ς ξ ( ξ ξ ξ ς. Equation (5.- ς ς ς Tis quation is idntical to quation (9. in rf. []. P d O P Figur 5.- Angular Sparation 8/7

19 Wn angular sparation is vry small, (ξ -ξ and (ζ -ζ ar narly zro and quation (5.- can not usd. Equation (5.- is transformd to a nw quation as follows. d ς d ς ς ς ς ς ς ς ς ( ξ ς ( ς ( ξ Wr ς ς ς ς ς ς ς ( d ( ( ξ ς ξ ς ( ξ ( ξ ς ς ς ( d ( ς ( ξ ς ( ξ ς ( ς d. Equation (5.-4 ot: Wn θ is vry small, t following approximation can usd. θ θ (in radian 5.. Exampl Calculation Exampl 9.a in rf. []: Calculat t angular distanc, d twn Arcturus and Spica. T 95 coordinats of ts stars ar, Arcturus : α 4m.8s.45 o o Spica : α m.s.688 o o /7

20 / α α α α ( ( ( ( d d (radian.87 o

21 5. Compnsation of ounting Farication Errors 5.. Tlscop Coordinats A tlscop as tlscop coordinat systm as sown in figur 5.-. Tru tlscop polar coordinats is (ϕ, θ. Tru mans tat w considr ypottical prfct tlscop mount witout farication rror. If t X t -axis points to t Sout and Z t -axis points to znit, tis mount is an alt-azimut mount. ϕ A θ If t Z t -axis points to clstial nort pol and X t -axis points to mridian, tis mount is an quatorial mount. ϕ θ Z t θ: Elvation Angl X t Y t ϕ: orizontal Angl Figur 5.- Tlscop Coordinats 5.. Farication Errors of ount In t ral world all mountings av farication rrors. Tr ar tr diffrnt farication rrors to considrd as sown in figur 5.-. ( : Error in prpndicularity twn orizontal axis and vrtical axis, or polar axis /7

22 and dclination axis ( : Collimation rror twn vrtical or polar axis and tlscop optical axis ( : Sift of zro point in apparnt lvation angl or dclination angl Tlscop Optical Axis θ Tlscop Vrtical Axis Tlscop orizontal Axis Figur 5.- Tlscop ount Farication Error 5.. Drivation of Equations T apparnt tlscop coordinats (ϕ, θ is t coordinats masurd wit stting circls of t tlscop mount. Rlationsip twn t tru tlscop coordinats and t apparnt tlscop coordinats ar drivd as follows. S figurs 5.- to ( Tlscop optical axis, R -axis points to a clstial ojct of tru tlscop coordinats (ϕ, θ. R S -plan is t plan dfind y tlscop optical axis and tlscop vrtical axis. Tis mans tat dirction ins of t clstial ojct in R -S -T coordinats is. /7

23 ( T coordinat systm R -S -T is rotatd countrclockwis around T -axis and coms a nw coordinat systm R -S -T. ( T coordinat systm R -S -T is rotatd (θ countrclockwis around S -axis and coms R -S -T coordinat systm. (4 T coordinat systm R -S -T is rotatd - countrclockwis around R -axis and coms R-S-T coordinat systm. (5 Finally, t coordinat systm R-S-T is rotatd ϕ countrclockwis around T-axis and coms X-Y-Z coordinat systm wic is t tru tlscop coordinats. θ ϕ ϕ ϕ θ ϕ ϕ ϕ θ ( θ ( θ ( θ ( θ ( θ ϕ ϕ ( θ ϕ ( θ ϕ ϕ ( θ ϕ ( θ. Equation (5.- Equation (5.- is an xact solution to otain tru tlscop coordinat from apparnt tlscop coordinat. Ug t following approximation, ϕ ϕ, ( θ ϕ θ ϕ ϕ ϕ, ( θ ϕ θ ϕ Equation (5.- is drivd from quation (5.-. ( θ ϕ (θ ϕ ϕ θ ϕ / ( θ ϕ (θ ϕ ϕ θ ϕ / ( θ ( θ /. Equation (5.- /7

24 Ug quation (5.-, an xact solution of θ and an approximat solution of ϕ ar otaind. Furtr approximation can mad as follows. Assuming tat t rrors ar vry small, quations (5.- and (5.- ar simplifid as follows. θ ϕ ( θ ϕ ϕ ( θ ϕ θ ϕ ( θ ϕ ϕ ( θ ϕ θ ( θ ( θ ϕ θ ϕ ϕ θ ϕ ( θ ϕ θ ϕ ϕ θ ϕ ( θ θ. Equation (5.-. Equation ( Apparnt Tlscop Coordinat witout Approximation ( Exact Solution Exact solution of apparnt tlscop coordinat ϕ and θ is otaind as follows. From quation (5.-, θ ( θ Tn, θ θ. Equation (5.-5 ϕ ( θ θ ϕ { ( θ } { ( θ } { ( θ }. Equation (5.-6 θ ϕ ( Itration Anotr way to otain an xact solution of apparnt tlscop coordinat ϕ and θ is an itration mtod. 4/7

25 Rwriting quation (5.-, w gt t following quation. ( θ ϕ (θ ϕ ϕ ( θ ϕ / ( θ ϕ (θ ϕ ϕ ( θ ϕ / ( θ ( θ / (θ ϕ ϕ ( θ ϕ / / (θ ϕ ϕ ( θ ϕ / / ( θ /. Equation (5.-7 Ug quation (5.-, t first approximat solution (ϕ, θ is otaind. T first approximat solution is input into quation (5.-5, ( θ ( ϕ θ ϕ ( θ (θ ϕ ϕ (θ ϕ (θ ϕ ϕ (θ ϕ (θ / / / / /. Equation (5.-8 Solving tis quation, t scond approximat solution (ϕ, θ is otaind. Tis itration will prformd until t solution convrgs. If t mount farication rrors ar aout dgr, two itrations ar noug. 5/7

26 Z θ Y ϕ X Figur 5.- Tru Tlscop Coordinats T, T Z, T R R T θ S, S Y R, R ϕ X S Figur 5.-4 Apparnt Tlscop Coordinat 6/7

27 R T Tlscop Optical Axis T T, Z T, T R θ S, S R, R S S Tlscop Vrtical Axis Tlscop orizontal Axis Figur 5.-5 Apparnt Tlscop Coordinats wit ount Error 7/7

28 5..5 Exampl Calculations Apparnt Tlscop Coordinats Tru Tlscop Coordinats Find tru tlscop coordinats from apparnt tlscop coordinats. ( Data ount rrors ar givn as sown low..5 o.5 / 8 x (radian -.8 o -.8 / 8 x (radian. o. / 8 x (radian asurd position (apparnt tlscop coordinats of a clstial ojct, (θ, ϕ is, θ 6. o 6. / 8 x.874 (radian ϕ 5.5 o 5.5 / 8 x.9755 (radian ( Calculation θ (radian Exact solution is otaind from quation (5.-, θ ϕ θ ϕ θ (.9664 ( ( (.9664 ( ( ( ( From quations (4- and (4-4, 8/7

29 tan ϕ j (radian 5.86 o θ.8877 q.9844 (radian o Approximat solution is otaind from quation (5.-4 as follows. θ ϕ θ ϕ θ ( ( From quations (4- and (4-4, tan ϕ j (radian 5.86 o θ.8878 q.9878 (radian 6.5 o 9/7

30 5..5. Tru Tlscop Coordinats Apparnt Tlscop Coordinats Find apparnt tlscop coordinats from tru tlscop coordinats ( Data ount rrors ar t sam as o.5 / 8 x (radian -.8 o -.8 / 8 x (radian. o. / 8 x (radian Tru tlscop coordinats of a clstial ojct, (θ, ϕ is, ϕ (radian 5.86 o θ.9844 (radian o ( Calculation From quation (5.-, ( θ ϕ ( θ ϕ ( θ ( ( (.9664 / (.9664 ( ( (.9664 / (.9664 ( (.9664 / ( From quations (4- and (4-4,.778 tan ϕ j.9758 (radian 5.5 o (θ.8878 /7

31 θ.9878 (radian 6.5 o q 6.5 o. o 6. o Approximat solution is otaind from quation (5.-5. ( θ ( θ ( θ ϕ ϕ ( ( ( ( From quations (4- and (4-4,.74 tan ϕ j (radian 5.5 o (q D.8877 q D.9844 (radian o q 6.5 o. o 6.99 o /7

32 5.4 Equations for Pointing Tlscop 5.4. Introduction Ug stting circls in tlscop mount, you can point a tlscop to a targt ojct wos quatorial coordinats is known. You don t nd align t tlscop mount. You just av to point your tlscop to two rfrnc stars and masur t stting circl radings of t stars. Input t data to your computr, and t computr will crat transformation quations. Aftr tat, you just input quatorial coordinats of a targt into t computr and t computr will rturn t stting circl numrs for t targt (rf. [6]. T tlscop coordinat systm is dfind as sown in figur T position of a star will spcifid in orizontal angl, ϕ and lvation, θ. ot tat t orizontal angl is masurd from rigt to lft. Tis is t opposit dirction to azimut. T tlscop is not ncssarily lvld or alignd wit any dirctions. Equatorial mounts and altazimut mounts ar t spcial cass. For quatorial mounts, ϕ corrsponds to rigt ascnsion, α and θ corrsponds to dclination,. For altazimut mounts, ϕ corrsponds to (azimut angl and θ corrsponds to altitud. Z Z (Clstial ort Pol X (Vrnal Equinox at t θ: Elvation Angl Y at t X ϕ: orizontal Angl Y Figur 5.4- Tlscop Coordinats and Equatorial Coordinats /7

33 5.4. Transformation atrix T rlationsip twn tlscop coordinats and quatorial coordinats is drivd in tis sction. Transformation from quatorial coordinats to tlscop coordinats is xprssd in matrix form as follows. l m n T T T T T T T T T [ T]... Equation (5.4- Transformation from tlscop coordinats to quatorial coordinats is xprssd as follows. Tis is t invrs form of quation (5.4-. [ T] l m n... Equation (5.4- Wr, l θ ϕ m θ ϕ n θ... Equation (5.4- : Dirction in of an ojct in tlscop coordinat systm ( α k( t t ( α k( t t. Equation (5.4-4 : Dirction in of an ojct in quatorial coordinat systm [ T ], [ ] T : Transformation matrix and its invrs matrix t : Tim t : Initial tim ϕ : orizontal angl of an ojct θ : Elvation angl of an ojct /7

34 α : Rigt Ascnsion of an ojct : Dclination of an ojct k Drivation of Transformation atrix Suppos tat data st of quatorial coordinats and tlscop coordinats for tr rfrnc stars ar otaind as follows. Rfrnc Star Osrvation Tim Equatorial Coordinats Rigt Dclination Ascnsion Tlscop Coordinats orizontal Elvation Angl Angl Star t α ϕ θ Star t α ϕ θ Star t α ϕ θ Ug t data aov, dirction in of ac star is xprssd in ot tlscop coordinats and quatorial coordinats. l m n θ ϕ θ ϕ θ... Equation (5.4-5 ( α k( t ( α k( t t t. Equation (5.4-6 l θ ϕ m θ ϕ n θ. Equation (5.4-7 ( α ( α k( t k( t t t. Equation ( /7

35 5/7 θ ϕ θ ϕ θ n m l. Equation (5.4-9 ( ( ( ( α α t t k t t k. Equation (5.4- Rlationsip twn tlscop coordinats and quatorial coordinats ar, [ ] T n m l [ ] T n m l [ ] T n m l Comining t tr quations aov, w otain t following quation. [ ] T n n n m m m l l l ultiplying to t ot sid of t quation aov, t transformation matrix is drivd as follows. [ ] n n n m m m l l l T. Equation (5.4-

36 6/7 Altoug w us tr rfrnc stars in quation (5.4-, two stars ar noug. An indpndnt vctor (dirction in is cratd from rfrnc star and star ug vctor product. T nw dirction ins will rplac dirction ins for rfrnc star. T dfinition of vctor product is sown in figur 5.4- and quation (5.4-. m l m l n l l n m n m n OP OP OP. Equation (5.4- Wr, n m l OP, n m l OP Figur 5.4- Vctor Product w dirction ins ar cratd from t coordinats of t rfrnc star and t rfrnc star ug quation (5.4.. ot tat t vctor products ar dividd y t lngt of t vctor caus dirction ins sould unit lngt. ( ( ( m l m l n l l n m n m n m l m l n l l n m n m n n m l. Equation (5.4- O P P P O P P P

37 ( ( (. Equation (5.4-4 Us quations (5.4- and (5.4-4 in quation (5.4- instad of (5.4-9 and ( Exampl Calculation T following data was masurd ug my.5 inc Dosonian wit stting circls. Calculat t transformation matrix from tlscop coordinats and quatorial coordinats assuming t mount dos not av farication rrors. Rfrnc Star Osrvation Tim Equatorial Coordinats Rigt Dclination Ascnsion Tlscop Coordinats orizontal Elvation Angl Angl Initial Tim Star : α And t ms t 7m56s α ϕ 7m54s 9.8 o 99.5 o θ 8.87 o.4688 Star : α Umi t 7ms α m45s o.5578 ϕ.98 o θ 5.4 o.656 From quation (5.4-5, l m n From quation (5.4-6, 7/7

38 8/ ( ( ( (.447 From quation (5.4-7, n m l From quation (5.4-8, ( ( ( (.685 From quation (5.4-,

39 9/ ( (.7648 ( ( ( ( (( (.5694 ( ( (.596 n m l From quation (5.4-4, ( ( ( ( ( (( T invrs matrix is,

40 From quation (5.4-, w otain t following transform matrix. [ T ] l l l m m m n n n If you want to aim t tlscop at β Ct (α 4m7s, -8.8 o at 5ms, from quation (5.4-, α 4m7s. 88radian o radian t 5ms radian From quation (5.4-, 4/7

41 ( α k( t t ( α k( t t (.48 ( ( (.48 ( ( ( From quation (5.4-, l m n [ T ] From quations (4- and (4-4, tlscop coordinats ar calculatd as follows. tanϕ ϕ.7546radian. o θ.68 θ radian 7.6 o Tis calculatd tlscop coordinats is vry clos to t masurd tlscop coordinats, ϕ.46 o, θ 7.67 o. 4/7

42 5.4.5 Commnt on Accuracy of t Pointing tod T accuracy of t pointing mtod is affctd y t following lmnts. ( ount rrors dscrid in 5.. ( ount dformation du to flxiility of mount ( Atmospric rfraction (4 Prcssion (5 Accuracy of angular masurmnt y ncodr or stting circls Tortically, all t ffcts xcpt t last on can takn into account. I will includ tm in t monograp in t nar futur. 4/7

43 5.5 Polar Axis isalignmnt Dtrmination T matrix mtod is applid to driv t quations of t dclination drift mtod for polar axis misalignmnt dtrmination in tis sction. T dclination drift mtod was proposd y Callis [5] to dtrmin t polar axis misalignmnt of quatorial mount. T advantag of t dclination drift mtod is its simplicity of t masurmnt Drivation of Equations Rlationsip twn Coordinat Systms Following Coordinat systms ar usd. S figur Equatorial coordinat systm is X -Y -Z. Z -axis dircts to t nort pol. Y -axis is in t orizontal plan and dircts to t ast. X -axis is on t mridian. Polar coordinats of a clstial ojct in t quatorial coordinat systm is (-,, wr is our angl and is dclination. Tlscop coordinat systm is X-Y-Z and its polar coordinats is (ξ, ζ. isalignmnt of t tlscop polar axis Z from t clstial polar axis Z is dfind as follows. First, quatorial coordinat systm X -Y -Z is rotatd θ clockwis around Z -axis (polar axis, tn t nw coordinat systm is rotatd γ clockwis around t nw X-axis. 4/7

44 Z (Znit Z (ort Pol Z (Polar Axis of Tlscop ridian θ (rotat clockwis around Z -axis v u X O X (ridian X (Sout Equator orizon Y Y, Y (East Figur 5.5- isalignmnt of Tlscop Polar Axis Dirction in of a star in quatorial coordinats is, ( ( Equation (5.5.- Dirction in of t sam star in tlscop coordinats is, ς ξ ς ξ ς Equation (5.5.- Rlationsip twn t coordinat systms is drivd as follows ug quations sown in sction /7

45 γ γ θ γ θ γ θ θ θ γ θ θ γ θ γ θ γ γ θ γ Equation (5.5.- ς ξ θ ς ξ γ θ ς γ θ θ γ θ γ θ ( γ ( γ Equation ( Basic Equations for Dclination Drift tod Two stars ar slctd for masurmnt. Point t tlscop to t first star and driv t quatorial mount around polar axis only. asur t drift of t star in dclination in a crtain tim intrval. Sam masurmnt is don for t scond star. Assum tat dclination drifts of two stars ar otaind from osrvation as sown in tal Atmospric rfraction is nglctd in tis sction. Effct of atmospric rfraction will discussd in sction Star Tal Osrvd Data Position of Star Tim Rigt Dclination, Start End Ascnsion, α Drift of Dclination (Rfraction is nglctd. Star α t a t ζ - ζ a Star α t a t ζ - ζ a From quation (5.5.-4, ς γ θ ( γ θ ( γ Equation ( /7

46 Ug t data from star in tal 5.5.-, ς a γ θ ( a γ θ ( a γ Equation ( Wr, a : our angl of t first star at tim t a : our angl of t first star at tim t ζ a : Dclination in Tlscop Coordinats at tim t a : Dclination of t first star in Equatorial Coordinats Assuming tat γ is vry small, quation ( is xprssd as, ς γ θ ( γ θ ( a γ a ( θ ( a θ ( Equation ( Considring t following rlationsip (assuming tat is vry small, ( a a W gt t following quation from quation ( ς a γ ( θ ( γ θ ( a a θ ( a γ θ ( a Equation ( Sam quation is drivd for tim t. ς γ θ ( γ θ ( Equation ( Rtracting quation ( from (5.5.-9, ς ς γ θ (( ( a γ θ (( ( a a Equation (5.5.- Putting ς ς u γ θ, v γ θ in quation (5.5.-, u(( ( a v(( ( a a Equation (5.5.- Ug t data of star in tal 5.5.-, ς ς u(( ( a v(( ( a a Equation (5.5.- Equations (5.5.- and (5.5.- ar t asic quations of dclination drift mtod. 46/7

47 Wn you otain t data in tal from osrvation, you can calculat t polar axis misalignmnt u and v from quations (5.5.- and ( Two important commnts can mad asd on t quations. ( In ordr to maximiz t accuracy of t mtod, it is dsiral to tak two stars wic locations at osrvation ar narly 9 dgr apart. ( It is not ncssary to slct stars nar quator. Stars far from clstial quator can work. Tis conclusion is drivd from t fact tat t dclinations of t stars do not appar in quations (5.5.- and ( Callis tod Callis original dclination drift mtod rquirs tr masurmnts wit on star. S tal for t rquird data. Tal Osrvd Data Callis tod Position of Star Tim Drift of Dclination Star Rigt Ascnsion, α Dclination, Start End (Rfraction is nglctd. Star α t a t ζ - ζ a t a t c ζ c - ζ a Basd on quations (5.5.- and (5.5.-, ς ς c ς ς a a u(( u(( c ( ( a a v(( v(( c ( ( a Equation (5.5.- a Equation ( /7

48 Compnsation of Atmospric Rfraction Effct of atmospric rfraction is includd in t masurd data. Tal Osrvd Data Position of Star Tim Drift of Dclination Star Rigt Dclination, (Rfraction is Start End Ascnsion, α includd. Star α t a t ζ - ζ a Star α t a t ζ - ζ a Altitud of t star at osrvd instant is ncssary to calculat t atmospric rfraction. Rlationsip twn quatorial coordinats and altazimut coordinats is (s sction 5.., Equation ( Wr, is osrvr s latitud. ( A ( A Equation ( A is azimut masurd wstward from t Sout and is airlss altitud. ( A ( ( A ( ( tan( A Equation ( From quation (5. in rf. [] (pag, atmospric rfraction R is xprssd as follows. R is addd to airlss altitud to otain apparnt altitud. ot tat quation ( is valid for t altitud largr tan 5 dgr. 48/7

49 R tan tan (in radian 6 8 / 6 8 / Equation ( Ug t following rlationsip, tan Equation ( Equation ( is xprssd as follows R 6 8 / / Equation (5.5.- Apparnt altitud is xprssd as follows. R Equation (5.5.- Rfractd position of t star in altazimut coordinats is, ( R( A R ( A R( A ( R( A R( A R( A ( R R R R ( A A R ( A R A R Equation (5.5.- Tn, rfractd position of t star in quatorial coordinats is, 49/7

50 5/7 A A A R A A R A A R Equation (5.5.- Rfractd position of t star in tlscop coordinats is drivd from quations (5.5.- and ( A A A R γ θ γ θ γ γ θ γ θ γ θ θ Equation ( Putting osrvd data of star at t in tal to quation (5.5.-, ( ( ( R A R A R R A R ( ( γ θ γ θ γ ς Considring γ and R ar vry small,

51 ς γ θ ( γ θ R ( A A γ θ ( γ θ ( R R Considring t following rlationsip (assuming tat is vry small, ( ς γ θ ( γ θ ( R A Equation ( Sam quation is drivd for data of star at t a in tal ς a γ θ ( a γ θ ( a R a a A a Equation ( a Rtracting quation ( from (5.5.-6, ς ς a Aγ θ Bγ θ C Equation ( Wr, A ( ( a B ( ( a A C R R tan R R a a a a tan A a a a a Sam quation is drivd for data of star. ς ς a Dγ θ Eγ θ F Equation ( Wr, D ( ( a E ( ( a 5/7

52 F R R A tan R R a a a a A a tan a a a Putting Au Du u γ θ, v γ θ in quations (5.5.- and (5.5.-4, Bv C ( a Ev F ( ς a ς ς Equation ( ς Equation (5.5.- Equations (5.5.-9, (5.5.-, ( and (5.5.- ar t quations for dclination drift mtod wit atmospric rfraction. 5/7

53 5.5. Exampl Calculations Two Star Dclination Drift tod wit Atmospric Rfraction glctd ( Osrvd Data Osrvd data is sown in tal Tal Osrvd Data Position of Star Tim Rigt Star Dclination, Drift of Dclination Ascnsion, Start End α α Boo t a t α ay 4, ay 4, 45m49s 9 o 9 ms 5ms ζ - ζ a -4.5 α Boo t a t α ay 4, ay 4, 45m49s 9 o 9 5ms ms ζ - ζ a Osrvation ocation: atitud, 5 o 9., ongitud, o 8.6E All t data is convrtd to radian. Tal Osrvd Data (in radian Position of Star Tim Rigt Star Dclination, Drift of Dclination Ascnsion, Start End α t a t α ζ - ζ a α Boo ay 4, ay 4, t a t α ζ - ζ a α Boo ay 4, ay 4, Osrvation ocation: atitud,.9877, ongitud, /7

54 ( Sidral Tim Julian Day numr JD corrsponding to ay 4, UT is (captr 7 in rf. []. T sidral tim at Grnwic at ay 4, UT is (captr of rf. [], T JD θ T at UT T.879T dg 4.664dg m8.95s ( our Angl From captr of rf. [], our angls ar calculatd as follows. θ α a a ( ( ( ( (4 Basic Equations From quation (5.5.-,.676 u(( ( v(( ( u. 4995v Equation (5.5.- From quation (5.5.-,.94 u(( ( v(( ( /7

55 u. 4457v Equation (5.5.- Solving quations (5.5.- and (5.5.-, u.796 radian.454 o 6 v.79 radian.5 o /7

56 5.5.. Callis tod wit Atmospric Rfraction glctd ( Osrvd Data Osrvd data is sown in tal Tal Osrvd Data Position of Star Tim Rigt Star Dclination, Drift of Dclination Ascnsion, Start End α t a ay 4, t ay 4, ζ - ζ a -4.5 α Boo α ms 5ms 45m49s 9 o 9 t a t c ay 4, ms ay 4, 5ms ζ c - ζ a -.4 Osrvation ocation: atitud, 5 o 9., ongitud, o 8.6E All t data is convrtd to radian. Tal Osrvd Data (in radian Position of Star Tim Rigt Star Dclination, Drift of Dclination Ascnsion, Start End α t a t ζ - ζ a ay 4, ay 4, α α Boo t a t c ζ c - ζ a ay 4, ay 4, Osrvation ocation: atitud,.9877, ongitud, ( Sidral Tim Julian Day numr JD corrsponding to ay 4, UT is (captr 7 in rf. 56/7

57 []. T sidral tim at Grnwic at ay 4, UT is (captr of rf. [], T JD θ T at UT T.879T dg 4.664dg m8.95s ( our Angl From captr of rf. [], our angls ar calculatd as follows. θ α a c ( ( ( (4 Basic Equations From quation (5.5.-,.676 u(( ( v(( ( u. 4995v Equation (5.5.- From quation (5.5.-4,.94 u(( ( v(( ( u v Equation ( Solving quations (5.5.- and (5.5.-4, u.85 radian.447 o 66 v.4 radian.5 o /7

58 5.5.. Two Star Drift tod wit Atmospric Rfraction Compnsatd ( Osrvd Data Sam data as sction is usd. Tal Osrvd Data Position of Star Tim Rigt Star Dclination, Drift of Dclination Ascnsion, Start End α α Boo t a t α ay 4, ay 4, 45m49s 9 o 9 ms 5ms ζ - ζ a -4.5 α Boo t a t α ay 4, ay 4, 45m49s 9 o 9 5ms ms ζ - ζ a Osrvation ocation: atitud, 5 o 9., ongitud, o 8.6E All t data is convrtd to radian. Tal Osrvd Data (in radian Position of Star Tim Rigt Star Dclination, Drift of Dclination Ascnsion, Start End α t a t α ζ - ζ a α Boo ay 4, ay 4, t a t α ζ - ζ a α Boo ay 4, ay 4, Osrvation ocation: atitud,.9877, ongitud, ( Sidral Tim 58/7

59 Julian Day numr JD corrsponding to ay 4, UT is (captr 7 in rf. []. T sidral tim at Grnwic at ay 4, UT is (captr of rf. [], T JD θ T at UT T.879T dg 4.664dg m8.95s ( our Angl From captr of rf. [], our angls ar calculatd as follows. θ α a a ( ( ( ( (4 Basic Equations Atmospric rfraction is calculatd ug quations ( and ( a a (.9877 (.4664* ( (.9877( (.9877(.4664 * ( (.9877( (.9877(.4664 * ( (.9877( /7

60 (.9877(.4664* ( (.9877( a a ( ( ( ( a a a a R a / radian 4.7" / R / radian 8." / R a / radian 8." / /7

61 R / radian 8.5" / From quation (5.5.-7, A B a ( ( ( ( ( ( ( ( a C R tan R a a tan a a tan tan D ( ( a E ( (.4457 a ( ( ( ( /7

62 F R tan R a a tan a a tan tan From quations ( and (5.5.-,.84847u.4995v u.4457v..94 Solving t quations, u.85 radian.459 o 65 v.78 radian.5 o 449 Comparing ts valus wit t rsult in sction (rfraction nglctd, ffct of atmospric rfraction is small in tis xampl. 6/7

63 5.6 Dom Slit Syncronization Computr control of tlscop as com popular in amatur astronomy, and now t advancmnt xpands furtr. Dom slit control is an xampl of suc advancmnt. In tis sction, quations to prform dom slit control for a tlscop on Grman quatorial mount ar drivd. T quations ar actually usd y Jon Olivr of Univrsity of Florida to dvlop is dom control softwar DomSync. T cntr of t tlscop tu on Grman quatorial mount is offst from t cntr of a dom as sown in figur Bcaus of tis, t azimut of t dom slit is not coincidnt wit t azimut of t ojct wic t tlscop is aimd. T ojctiv is to dvlop quations of t azimut of t dom slit. Tis prolm is a vry good xampl of matrix mtod application. Figur 5.6- sows dfinition of coordinat systms usd in tis sction. Point O is t cntr of t dom. Point P is t intrsction of t polar axis and t dclination axis of t Grman quatorial mount. Point Q is t intrsction of t tlscop tu cntrlin and t dclination axis. Dimnsions of t dom and t mount ar also dfind in t figur. X (ridian Z dom (Znit Ojct Z (ort Pol Dom : atitud Q r Z dom P Azimut X dom O R Y (East Y dom (East X dom (Sout Figur 5.6- Dfinition of Coordinat Systms Dom Slit Position 6/7

64 5.6. Ojct in First Quadrant Equation wn t ojct is in t first quadrant (8 o < < 7 o and tlscop is in ast sid of mount is drivd in tis sction. Point S in figur is on t dom surfac. X (ridian Z dom (Znit Z (ort Pol Dom : atitud S Ojct R P r O Q Y (East Y dom (East X dom (Sout Tlscop is in ast sid of mount. 8 o < < 7 o Figur Ojct in First Quadrant From quation (5.-, unit vctor QS QS is, QS QS ( ( Vctor P Q in X -Y -Z coordinat is, 64/7

65 65/7 ( ( r Z Y X PQ From quation (5.-4, vctor O S in X dom -Y dom -Z dom coordinat is, Fk C Ek B Dk A k r Z k r k r X Z X k r r Z Y X OS dom dom dom dom dom dom dom ( ( ( ( ( ( ( ( ( ( Equation (5.6.- Wr, k: Constant ( ( ( ( ( ( F E D r Z C r B r X A dom dom Equation (5.6.-

66 5.6. Ojct in Scond Quadrant Ojct is in t scond (7 o < < 6 o and tlscop is in wst sid of mount. X (ridian Z dom (Znit Z (ort Pol Dom S Ojct : atitud Q r P R O Y (East Y dom (East X dom (Sout Tlscop is in wst sid of mount. 7 o < < 6 o Figur Ojct in Scond Quadrant From quation (5.-, unit vctor QS QS is, QS QS ( ( Vctor P Q in X -Y -Z coordinat is, 66/7

67 67/7 r Z Y X PQ From quation (5.-4, vctor O S in X dom -Y dom -Z dom coordinat is, Fk C Ek B Dk A k r Z k r k r X Z X k r r Z Y X OS dom dom dom dom dom dom dom ( ( ( ( ( Equation (5.6.- Wr, k: Constant

68 A X dom r B r C Z dom r D ( E ( F ( Equation ( Ojct in Tird Quadrant Ojct is in t tird quadrant ( o < < 9 o and tlscop is in ast sid of mount. X (ridian Z dom (Znit Z (ort Pol Ojct Dom S : atitud R P r O Q Y (East Y dom (East X dom (Sout Tlscop is in ast sid of mount. o < < 9 o Figur Ojct in Tird Quadrant 68/7

69 69/7 From quation (5.-, unit vctor QS QS is, ( ( QS QS Vctor P Q in X -Y -Z coordinat is, r Z Y X PQ From quation (5.-4, vctor O S in X dom -Y dom -Z dom coordinat is, Fk C Ek B Dk A k r Z k r k r X Z X k r r Z Y X OS dom dom dom dom dom dom dom ( ( ( ( ( Equation (5.6.- Wr, k: Constant

70 A X dom r B r C Z dom r D ( E ( F ( Equation ( Ojct in Fourt Quadrant Ojct is in t fourt quadrant (9 o < < 8 o and tlscop is in wst sid of mount. X (ridian Z dom (Znit Ojct Z (ort Pol Dom S : atitud Q r P R O Y (East Y dom (East X dom (Sout Tlscop is in wst sid of mount. 9 o < < 8 o Figur Ojct in Fourt Quadrant 7/7

71 7/7 From quation (5.-, unit vctor QS QS is, ( ( QS QS Vctor P Q in X -Y -Z coordinat is, r Z Y X PQ From quation (5.-4, vctor O S in X dom -Y dom -Z dom coordinat is, Fk C Ek B Dk A k r Z k r k r X Z X k r r Z Y X OS dom dom dom dom dom dom dom ( ( ( ( (

72 Wr, k: Constant A X dom r B r C Z dom r D ( E ( F ( Equation ( Equation ( Intrsction Figur sows vctor xprssion of points on a straigt lin. Ojct S PS P PQ Q OS OP O Points P, Q and S ar on a straigt lin. PS kpq OS OP PS OP k PQ Wr k is a scalar. Figur Vctor Exprssion of Points on a Straigt in 7/7

73 Sinc point S is on t dom surfac, lngt of vctor O S is R. ( A Dk ( B Ek ( C Fk R Tn, ( D E F k ( AD BE CF k ( R A B C Solving tis quation for k, k ( AD BE CF ( AD BE CF ( D E F ( R A B C D E F Equation ( X OS Y Z dom dom dom A Dk B Ek C Fk Equation ( From quations (5.-5 and (5.-6, azimut and lvation of dom slit ar, Y dom tan( A dom Equation ( X dom Wn X dom >, (-A dom is in t first quadrant or t fourt quadrant. Wn X dom <, (-A dom is in t scond quadrant or t tird quadrant. Azimut, A dom is masurd from sout to wstward. Z dom dom Equation ( R -/ (-9 o < dom < / (9 o 7/7