Astronomi

Beregning af området for synlig højre opstigning og deklination fra bestemt sted + tid

Beregning af området for synlig højre opstigning og deklination fra bestemt sted + tid


We are searching data for your request:

Forums and discussions:
Manuals and reference books:
Data from registers:
Wait the end of the search in all databases.
Upon completion, a link will appear to access the found materials.

I betragtning af en bestemt dato og et klokkeslæt og koordination af et sted på jorden, hvordan kan jeg beregne (selv eller ved hjælp af en pythonpakke) række af synlig RA og dec?

For eksempel (men jeg leder efter en generel løsning og formel):

input:

  1. dato og klokkeslæt: 01-05-2019 23:00:00

  2. jordkoordinater: 32,0853 ° N, 34,7818 ° E

produktion:

  1. RA min:?

  2. RA max:?

  3. Dec min:?

  4. Dec max:?


I princippet passerer alle RA-linjer hver dag gennem din placering. Det er deklinationen, der er begrænset af din breddegrad. Hvis du var ved ækvator, skulle du være i stand til at se alle deklinationer (igen, det er i princippet). Forudsat at du befinder dig på den nordlige halvkugle, vil den nordlige himmelpol, dvs. + 90 ° deklination, være A ° over din nordlige horisont, hvor A angiver din breddegrad. Så himmelkuglen 'vippes' og begrænser den sydligste deklination, der er synlig til - (90-A) °.

Deklinationslinjerne forbliver faste til Jorden, mens RA ser ud til at gå rundt. Situationen ændres, hvis du var på en af ​​de to poler. Begge linjer i RA og Decliantion ser ud til at gå rundt, men med den forskel, at al RA, i det mindste en del af dem, altid vil være over horisonten og mødes i højdepunktet.

Dette kan være et dumt spørgsmål og det hele, men jeg spekulerede på, hvordan ville du vide, hvad højre opstigning og deklinationskoordinater er synlige fra et bestemt sted? Jeg mener, at jeg kiggede på en gammel almanak, der lå her og ikke rigtig kunne forstå, hvordan du kan se fra disse koordinatværdier, at objektet kan ses fra din placering.

Så neutrino du siger, at virkelig den rigtige opstigning er koordinaten, der bestemmer, hvornår du ser objektet ?? Og hvordan fungerer det? Jeg ved, at den rette opstigning har at gøre med afstanden fra det første punkt i vædderen. Så ville du finde ud af, hvad tid der er synlig fra din placering eller noget?

@ Pighvar
Ja, jeg tror, ​​jeg har set dem før, så jeg vil se på at få fat i en. Jeg synes ikke, det var virkelig for dyrt. Jeg tror, ​​det var noget som $ 5.

Lad os se, om vi kan visualisere dette lidt mere konkret. Se på følgende billede:

Forestil dig, at kuglen i midten er jord, og den ydre kugle er det himmelske koordinatsystem (højre opstigning og deklination). Koordinatsystemet blev grundlæggende defineret således, at det projicerer ud fra jordens koordinatsystem (længdegrad og breddegrad), så du behøver ikke bekymre dig om jordens hældning.

Ok, find nu din breddegrad på jorden i det diagram og vælg et hvilket som helst punkt på den linje med konstant breddegrad. Det bliver sandsynligvis lettere, hvis du vælger et punkt på kanten, men det betyder ikke rigtig. Når du har valgt dette, skal du mentalt (eller grafisk) overveje et plan, der er tangent til det, dvs. et plan, der er vinkelret på linjen, der forbinder jordens centrum og din valgte placering. Det skal groft skære den større kugle i halvdelen. Dette fly afgrænser alt, hvad du kan se på et øjeblik i tiden. Da vi ikke har normaliseret koordinatsystemerne (dvs. sæt numre på diagrammet), kan dette være når som helst, dag eller nat, men det bliver ikke noget. Bemærk også, at for at udføre denne øvelse mere korrekt, burde jorden være meget, meget mindre end den store kugle, så den flyver virkelig klipper den større kugle i halvdelen.

I hvert fald, som jeg sagde, over dette fly ligger alt på himlen på et øjeblik i tiden. At bestemme alt, hvad du kunne se hele året, bare drej det plan 360 grader omkring jordaksen (dvs. linjen, der forbinder nord- og sydpolen). Hver del af den ydre kugle, der er over dette plan på ethvert tidspunkt i rotation, vil være synlig for dig på et eller andet tidspunkt i løbet af året. For at sikre, at du forstår, hvad jeg mener, prøv øvelsen for en observatør ved nordpolen og ved ækvator. Personen på nordpolen vil altid se den samme halvdel af himlen hele året, mens personen på ækvator vil se hele himlen, når året skrider frem.

I betragtning af denne konceptuelle forklaring kan du måske lave en lille geometri for at bestemme, at rækkevidden af ​​deklinationer, som du aldrig kan se, er givet af:


Baner og navigation

Stanley Q. Kidder, Thomas H. Vonder Haar, i satellitmeteorologi, 1995

2.5.1.1 Vektorrotationsmetoden

Figur 2.11 illustrerer, hvad vi kalder vektor rotation metode. Det diskuteres i en noget anden form af Escobal (1965) og andre. I det første trin er satellitten placeret i planet for sin bane, det vil sige den sande anomali θ og radius r beregnes. Dette gøres ved (1) at løse for e ved hjælp af ligning 2,8, (2) beregning θ ved anvendelse af ækv. 2.10a, og (3) beregning r ved hjælp af ligning 2.7. (For en cirkulær bane forenkles dette trin, fordi den gennemsnitlige anomali, den excentriske anomali og den sande anomali er identiske, og r er konstant.)

FIGUR 2.11. Rotationer, der bruges til at placere en satellit i sin bane: (a) satellitten i planet for sin bane, (b) rotation omkring z akse gennem argumentet om perigee (ω), (c) rotation omkring x akse gennem hældningsvinklen (jeg) og (d) rotation omkring z akse gennem højre opstigning af stigende knude (Ω).

I det andet trin dannes en vektor, der peger fra Jordens centrum til satellitten i det rigtige opstigningskoordinatsystem. De kartesiske koordinater for denne vektor er

På dette tidspunkt antages den orbitale ellipse at ligge i xy fly med perigee på det positive x akse (figur 2.11a).

I de sidste tre trin roteres vektoren således, at orbitalplanet er korrekt orienteret i rummet.

I det tredje trin roteres vektoren omkring z akse gennem argumentet om perigee (fig. 2.11b). Denne rotation udføres bekvemt ved at multiplicere vektoren med en rotationsmatrix, i dette tilfælde

I det fjerde trin roteres vektoren omkring x akse gennem hældningsvinklen (fig. 2.11c).

I det femte og sidste trin roteres vektoren omkring z akse gennem den høje opstigning af den stigende knude (fig. 2.11d).

Vektoren (x″′, y″′, z″ ′) Er placeringen af ​​satellitten i koordinatsystemet til højre opstigning – deklination t. Denne vektor kan konverteres til radius, deklination og højre opstigning af satellitten ved

Når man har beregnet satellitens højre opstigning, deklination og radius, er det nyttigt at beregne breddegrad og længdegrad for undersatellitpunktet. Hvis vi antager, at Jorden er en kugle, er breddegraden (kendt som geocentrisk breddegrad) er simpelthen lig med deklinationen. Længden af ​​subsatellitpunktet er forskellen mellem satellitens højre opstigning og den primære meridian (0 ° længdegrad) højre opstigning, der passerer gennem Greenwich, England (fig. 2.12). Den højre opstigning af Greenwich kan beregnes ved at kende dens højre opstigning på et givet tidspunkt og jordens rotationshastighed. 11 Da rotationshastigheden ændres meget let på grund af vind- og havstrømmens handlinger, kræver meget nøjagtig viden om Greenwichs højre opstigning observationer. Nogle satellitbulletiner giver den rigtige opstigning af Greenwich ud over satellitkredsløbselementerne.

FIGUR 2.12. Forholdet mellem Jordens længdegrad og højre opstigning.

Det omvendte problem med at finde, når en satellit passerer over (eller tæt på) et bestemt punkt, løses iterativt ved (1) at estimere tiden, (2) at beregne satellitens position og (3) at korrigere tidsestimatet. Trin 2 og 3 gentages, indtil en tilfredsstillende løsning er fundet.


Beregning af Polaris position

Jeg forsøger at finde ud af en måde, givet en kendt breddegrad, længdegrad og tid. du kan matematisk beregne placeringen af ​​Polaris.

Jeg ved, at højden vil være lig med bredden.

Jeg prøver at finde ud af, hvordan jeg skriver en formel til at beregne antallet af grader fra magnetisk nord for at rotere.

# 2 Pharquart

Formlen vil omfatte to komponenter. Den ene varierer over mange år, og den anden varierer i løbet af en dag. Den første er placeringen af ​​den himmelske pol. Den nordlige himmelpol er over en flad horisont med et beløb svarende til lokalitetens breddegrad, og det er en fast afstand øst eller vest for magnetisk nord afhængigt af placering. Beløbet er kendt som stedets & quotmagnetic deklination. & Quot Jeg er sikker på, at der er en matematisk formel, der kan beregne det, ellers ville de have svært ved at producere de diagrammer, der viser det. Google & quotmagnetic deklination & quot, og du får kortet.

Den anden komponent ville forudsige Polaris position, når den roterer omkring himmelpolen. Polaris er mindre end 1 grad væk fra himmelpolen i løbet af 24 timer og trækker en lille cirkel omkring polen. Der plejede at være papirhjulslommeregne (som en cirkulær glidereegel), der viste stangens placering i forhold til Polaris givet en dato og lokal tid. Folk brugte dem (og kan stadig) til at bestemme, hvor de skulle placere Polaris på deres polære justeringsomfang, så deres GEM pegede lige mod himmelpolen. Denne formel til at skabe dette er sandsynligvis lettere end at beregne magnetisk deklination. Du bliver nødt til at finde en bestemt dato / tid, hvor Polaris var direkte & quotabove & quot (i forhold til horisonten) polen, og derefter figur en fuld cirkel mod uret plus ca. 1 grad (1 / 365,25 af en cirkel) omkring polen hver dag .

Så din formel ville være sådan:

Polaris placering = placering af himmelpolen (højde = bredde, øst / vest placering = magnetisk nord forskudt ved magnetisk deklination baseret på nøjagtig placering) + afvigelse af Polaris fra himmelpolen (ca. 0,7 grader i en retning bestemt af dato og lokal standardtid )


Hor2eq: Konverterer lokale horisontkoordinater (alt-az) til ækvatorial.

Denne funktion beregner ækvatoriale (ra, dec) koordinater fra horisont (alt, az) koordinater. Det er typisk nøjagtigt til ca. 1 buesekund eller bedre og udfører korrektioner med præcession, nutation, aberration og refraktion. Indgange kan være vektorer undtagen observatørens breddegrad, længdegrad og højde. ra, dec, alt og az skal være vektorer af samme længde, men jd kan være en skalar eller en vektor med samme længde.

Trin i beregningen:
Forløber Ra-Dec til nuværende equinox
Ernæringskorrektion til Ra-Dec
Korrektion af aberration til Ra-Dec
Beregn lokal gennemsnitlig sidetidstid
Beregn lokal tilsyneladende sidetid
Beregn timevinkel
Anvend sfærisk trigonometri for at finde tilsyneladende Alt-Az
Anvend brydningskorrektion for at finde observeret Alt

Brugeren kan tilføje specifikationer for temperatur og tryk bruges af funktion co_refract at beregne atmosfærens brydningseffekt. Se co_refract for flere detaljer.


Parsing af konstellationsdata

Konstellationer er et andet sæt data, der skal være relativt til et bestemt katalog. Jeg fandt dette websted, der præsenterer konstellationslinjerne i et tekstformat:

Problemet er, at filen indeholder konstellationer som en række linier (linjestrimler), som du har brug for kontinuerligt at tegne som en pen, der ikke forlader papiret. Jeg besluttede at konvertere dette til en liste over segmenter, så jeg er nødt til at analysere tekstfilen og foretage konverteringen.

Der er et par forbehold her:

  • Konstellationer kan være til stede mere end én gang, de har 2 ikke sammenhængende linjer.
  • Parsing udføres to gange, først for at beregne forskydninger af endelige segmenter (især for konstellationer med flere linjer), for det andet for faktisk at analysere dataene.

Dette er også en interessant brug af hydra_lexer - rygraden i HFX-sproget. I denne demo er den allerede brugt, men i de følgende opdaterer jeg & rsquoll den mere og mere.

Den mest interessante del for mig er at se parsingsløkken og have et lexer / tokenizer som et personligt værktøj er et MUST! Konstellationer indeholder et kort fra navnet til posten. Den anden parse læser bare de faktiske stjernetal og sætter dem på det rigtige sted. Ikke sikker på, at det er interessant kode at læse her. Vi har nu en liste over segmenter og dermed 2 point for hver konstellation i en sammenhængende hukommelsesblok.


Beregning af rækkevidden af ​​synlig højre opstigning og deklination fra bestemt sted + tid - Astronomi

Feltrotation, polarjustering og afdækning
Beregninger, formler og relation til astrofotografi

Del I: Højde-azimuth-monteringer

Når du bruger et alt-azimuth-beslag til at se på tingene på himlen, selv med et beslag, der er i stand til at spore stjernerne (såsom Meade LX200-serien), ser billederne ud til at rotere inden for okularfeltet. Dette fænomen er kendt som feltrotation.

Det er let at forstå, hvordan dette opstår ved at overveje et specifikt eksempel. Forestil dig, at vi er ved ækvator og ser på et par stjerner meget nær nordpolen - dvs. meget tæt på horisonten - sådan at de to stjerner tilfældigvis har den samme højre opstigning (RA). Lad os trække en linje gennem dem, faktisk vil denne linje falde sammen med deres RA-linje. Når stjernerne stiger over horisonten, vil denne linje være næsten parallel med horisonten. Når stjernerne passerer gennem meridianen, vil denne linje være lodret med horisonten, faktisk falder den sammen med meridianen. Endelig, når stjernerne indstiller, vil linjen igen være parallel med horisonten. Forestil dig nu, at vi ser på stjernerne med et teleskop og - til vores nuværende formål - et okular med et rektangulært synsfelt, således at den ene side af rektanglet er parallel med horisonten. Den måde, hvorpå alt-azimuth-monteringer fungerer, sikrer, at denne side af rektanglet altid vil forblive parallel med horisonten (husk at linjer med konstant højde er parallelle med horisonten). Imidlertid har jeg netop nævnt, at linjen trukket gennem vores par stjerner ser ud til at rotere i forhold til horisonten, hvilket betyder, at når de ses gennem vores rektangulære teleskop synsfelt, vil de helt sikkert også rotere. Med lidt ekstra tanke er det let at se, at denne feltrotation virkelig er uafhængig af okularets synsfelt.

Feltrotation er ikke rigtig et problem for visuel observation, men når man prøver at udføre astrofotografering, betyder feltrotation, at der er en vis maksimal eksponeringstid, som du kan bruge til et alt-azimuth-monteret fotografisk instrument, før dine stjerner bliver optaget som buer.

Denne side beskriver mine forsøg på at beregne den nøjagtige størrelse af feltrotation som en funktion af astronomens breddegrad. Derefter vil jeg også forsøge at udlede et udtryk for den maksimale tilladte eksponeringstid, før der opstår mærkbar billedudstrygning, specifikt i sammenhæng med planetarisk fotografering med høj effekt. Jeg antager, at læseren forstår multivariabel og vektorberegning samt nogle meget elementære ideer fra differentieret geometri. Giv mig besked, hvis du opdager beregnings- eller andre matematiske fejl.

For nemheds skyld lad os forestille os, at vores stjerner alle sidder fast i en enhedssfære, og vi som observatører er matematiske punkter lige i midten af ​​sfæren. For et givet objekt på himlen (dvs. på enhedskuglen) kan vi specificere dets koordinater ved hjælp af et kartesisk system (x, y, z). Dette kan også relateres til dets højre opstigning og deklination eller til dets højde og azimut. Lad os bruge lidt tid på at opsætte disse to forskellige sæt koordinater på enhedssfæren. (Du undrer dig måske over, hvorfor vi har brug for to sæt koordinater - læs videre for at finde ud af det.)

Lad os starte med højde-azimuth-koordinaterne. Hvis vi lader x-y-planet repræsentere det tilsyneladende plan (horisont), vi observerer fra, ville z-aksen pege på zenitten. Givet en stjerne på sfæren, lad vinklen undertrykt af z-aksen og den radiale linje, der forbinder stjernen og vores sfærecenter (også centrum af vores x-y-z-akser) være e. Projicér stjernen vinkelret på x-y-planet og træk en linje fra det projicerede punkt til (0,0,0). Angiv vinklen, der er undertrykt af denne linje, og x-aksen som en. Disse koordinater ligner meget højde- og azimutkoordinater, bortset fra at jeg har valgt dem på denne måde for at gøre beregningerne lettere.

Vi har således koordinaterne for et givet punkt på sfæren & mikro [e, a] = (x, y, z) = (sin [e] cos [a], sin [e] sin [a], cos [e ]).

Lad os oprette de rigtige koordinationer til opstigning-deklination. Fordi vi ikke rigtig vil være interesseret i den nøjagtige højre opstigning af de pågældende objekter, foreslår jeg at bruge følgende system til at gøre beregningerne lettere. Forestil dig det i stedet for ovenstående koordinater, ændrer jeg deres navne til i stedet. Derefter roterer jeg hele kuglen omkring y-aksen med en vinkel w (uden at rotere akserne selv - ellers er vi tilbage til den samme situation). Bemærk dette sæt nyt roteret koordinater svarer til et RA-Dec-system for en breddegrad på (90 grader - w grader). Matematisk betyder det, at vi erstatter ovenstående med og multiplicer ovenstående vektor med en rotationsmatrix.

Efter nogle trigonometri og lineær algebra har vi & micro [R, D] = (x, y, z) = (cos [R] cos [w] sin [D] + cos [D] sin [w], sin [D] sin [R], cos [D] cos [w] - cos [R] sin [D] sin [w]).

Men i sidste ende vil jeg gerne have mine formler i form af R, D, så det ville være mere bekvemt at skifte koordinater, således at sines og cosinus for w vises med e og a. Men da rotationsmatricer er lette at invertere, opnår vi straks

& mikro [R, D] = (x, y, z) = (sin [R] cos [D], sin [R] sin [D], cos [R]) = & mikro [e, a] = (x, y, z) = (cos [e] cos [w] sin [a] - cos [a] sin [w], sin [a] sin [e], cos [a] cos [w] + cos [e] synd [a] synd [w]).

En alt-azimuth-montering, der sporer stjernerne, er en, der bevæger sig på en sådan måde, at et omfang monteret på det altid peger på den samme konstant-deklinationslinie på himlen (dvs. på enhedssfæren). Forestil dig, at vi kan se to sæt fint placerede gitre oven på vores himmel, når vi kigger gennem et teleskop monteret på en sådan montering: det ene sæt er det for højdeazimut og det andet RA-dec. Fordi vores er en alt-azimuth-montering, ser alt-azimuth-gitteret ikke ud til at rotere, når vi følger stjernerne, men RA-Dec-linjerne ville af de grunde, der allerede var diskuteret tidligere, mens orienteringen af ​​objekter på himlen (f.eks. retningen af ​​linjen trukket gennem vores hypotetiske stjernepar) forbliver fast med hensyn til RA-Dec-gitteret (f.eks. linjen gennem de 2 stjerner forbliver oven på en bestemt konstant-RA-linje). Dette får os til at indse, at feltrotation på et givet punkt på himlen virkelig er hastigheden for ændring af vinklen dannet af enten RA-linjen eller Dec-linjen med azimut- eller højdelinjen, der passerer gennem den, når vi bevæger os langs i RA, mens vi sporer stjernerne. På et mere teknisk sprog er vi nødt til at opnå to rammer på hvert punkt på sfæren: en med enhedsvektorer parallelt med RA- og Dec-linjer, der passerer gennem det givne punkt, og den anden med enhedsvektorer parallelt med azimut- og højdelinjer, der passerer det samme. Når monteringen sporer, bevæger den sig i RA, men fordi det er en alt-azimuth-montering, "forbliver" den i det sidstnævnte rammefelt, mens det tidligere RA-Dec-rammefelt roterer i forhold til sidstnævnte.

Da vi kun er interesseret i vinkler, behøver vi kun at få en vektor til hver ramme.

Til RA-Dec-rammen fik jeg vektoren, der pegede parallelt med den konstante Dec-linje ved at differentiere & micro [R, D] med hensyn til R, for at opnå d & micro / dR = (-Sin [D] Sin [R], Cos [R] Sin [D], 0). Ved at tage prikprodukt med sig selv ser vi længden af ​​vektoren er (Sin [D]) ^ 2, og enhedsvektoren er således r_hat = (1 / Sin [D) (-Sin [D] Sin [R], Cos [R] Sin [D], 0).

For højde-azimut-rammen fik jeg vektoren, der pegede parallelt med den konstante højdelinje ved at differentiere & micro [R, D] med hensyn til e, for at opnå d & micro / de = (Cos [D] Cos [R] dD / de - dR / de Sin [D] Sin [R], Cos [R] dR / de Sin [D] + Cos [D] dD / de Sin [R], dD / de Sin [D]). Hvis vi tager prikprodukter med sig selv, ser vi, at længden er (dD / de ^ 2 + dR / de ^ 2 Sin ^ 2 [D]) ^ 0,5. Den tilsvarende enhedsvektor er således E_hat = (dD / de ^ 2 + dR / de ^ 2 Sin ^ 2 [D]) ^ 0,5 (Cos [D] Cos [R] dD / de - dR / de Sin [D] Sin [R], Cos [R] dR / de Sin [D] + Cos [D] dD / de Sin [R], dD / de Sin [D]).

Vi skal finde ud af, hvad dD / de og dR / de er. For at gøre det henviser vi til det faktum, at & micro [R, D] = & micro [e, a], da de virkelig er den samme vektor. Se på z-komponenten i matrixligningen: cos [e] = cos [D] cos [w] - cos [R] sin [D] sin [w]. Ved at differentiere begge sider med hensyn til R, med hensyn til D, ved hjælp af forholdet sin ^ 2 + cos ^ 2 = 1, og ved hjælp af det faktum, at dx / dy = 1 / (dy / dx), kan vi derefter opnå dR / de og dD / de.


En "zoomet ind" af de to sæt koordinatsystemer omkring et givet punkt på himmelsfæren. De to rammer, der er brugt i vores beregninger, vises med de ikke-normaliserede vektorer. Det er svært at tegne buede linjer, så jeg tegnede lige linjer - i virkeligheden er koordinatlinjer buede. Med en vis fantasi kan man se, hvordan feltrotation finder sted: det alt-azimuth-monterede teleskop leverer en visning, hvori ramme (angivet med d & micro / de og d & micro / da-vektorer) roterer ikke, og på samme tænder forbliver orienteringen af ​​himmelobjekterne faste i forhold til RA-Dec-gitteret (og dermed med hensyn til ramme, angivet med d & micro / dR og d & micro / dD-vektorer) . Nu bevæger teleskopet sig, hvis det spores, i RA (eller i vores system med variabler simpelthen R). Da koordinatlinierne generelt er buede, vil vinklen G mellem rammevektorerne d & micro / dR og d & micro / de ændre sig, når R ændres, hvilket giver anledning til feltrotation. Denne måde at tænke på feltrotation danner også grundlaget for vores beregninger. (Bemærk, at jeg ikke har gjort noget for at gøre R-, Dec-, e- og skalaerne ensartede, fordi de kun er der til illustrative formål. For eksempel = <90d, 16d> er bestemt ikke det samme punkt som = <31d, + 14d>.)

Fra vektoranalyse har vi r_hat. E_hat = || r_hat || || E_hat || cos [G] = cos [G], hvor G er vinklen mellem de to enhedsvektorer fra RA-Dec- og alt-azimuth-rammefelterne. Vi ønsker at finde ud af, hvad ændringshastigheden for denne vinkel G er, når vores monteringsspor i RA, så vi adskiller nu begge sider med hensyn til R. (d / dR) (r_hat. E_hat) = -Sin [G] dG / dR. Ved at bruge forholdet sin ^ 2 + cos ^ 2 = 1 igen får vi derfor et udtryk for dG / dR. Men vi har allerede udtryk for r_hat og E_hat, så alt hvad jeg havde brug for var at sætte dem i Mathematica. Desuden, hvad vi virkelig ønsker er dG / dt, rotationshastigheden med tiden, så vi gør dG / dt = dG / dR x dR / dt, hvor dR / dt simpelthen er jordens rotationshastighed, som er 2 pi (dvs. 360 grader) divideret med en siderisk dag (23,93446965 x 60 x 60 sekunder).

hvor enhederne er i radianer pr. sekund, R er vores pseudo-RA-koordinat, D er vores pseudo-dec-koordinat, og L er vores observatørs breddegrad. For at konvertere D til den sædvanlige deklination d skal du bemærke, at for D = 0 grader, d = +90 grader D = +90 grader, d = 0 D = +180 grader, d = -90 grader.)

En af de praktiske anvendelser af ovenstående formel er følgende. Jeg har tænkt på at kassere mit tunge ækvatoriale beslag og lille teleskop og erstatte det med en stor Dob med et drevet alt-azimuth-beslag. Jeg vil dog bruge den til at fotografere planeterne i høj opløsning. Spørgsmålet er, hvor lang en eksponering kan jeg lave uden at mit billede er udtværet og sløret af feltrotation?

For at overveje dette, lad os antage, at radiusen på vores planet er sigma i buesekunder. Vi antager også, at vores planet er placeret lige i midten af ​​vores synsfelt, man skal huske dette, når man laver planetarisk fotografering. Efter en tid på tau sekunder ville planeten som set i min Dob have roteret med nogenlunde dG / dt x tau radianer, hvilket betyder, at et givet punkt på omkredsen ville have bevæget sig med en vinkelafstand på dG / dt x tau x sigma. (Det nøjagtige udtryk kan opnås på mindst to måder. Man ville kræve at erstatte R med et passende tidsafhængigt udtryk - f.eks. R = 23,93446965 x 60 x 60 t - i ovenstående formel og derefter integrere det med hensyn til t over passende grænser. Den anden ville være at erstatte R med det tidsafhængige udtryk i det netop udarbejdede r_hat. E_hat = Cos [G] og derefter tilslutte de passende start- og sluttider for at opnå to separate sæt ligninger, tage de inverse cosinus fra begge sider og trække.) Fordi omkredsen af ​​planeten ville være mest udtværet, er det tilstrækkeligt at sikre, at et givet punkt på omkredsen ikke ser ud til at være udtværet efter en given tids tau. For at få et udtværet billede skal den vinklede afstand, der tilbagelægges af et givet punkt, være større end omtrent halvdelen af ​​den mindste vinkelafstand, der kan løses med teleskopet. Sidstnævnte (i radianer) er givet med en velkendt formel: 1,22 x lambda / diameter af omfang, hvor lambda er lysets bølgelængde - da vi leder efter den maksimalt tilladte tid, kan vi lige så godt tilslutte den korteste bølgelængde ( dvs. det mest følsomme) af synligt lys: 4 x 10 ^ -7 m.

Dette betyder dG / dt x tau x sigma & lt (1/2) (1,22 x 4 x 10 ^ -7) / Diameter, hvilket fører til den maksimalt tilladte tid tau er

hvor R, D og L er som før, er Gamma teleskopets diameter i tommer, og Sigma er planetens radius i buesekunder.

Som et konkret eksempel for Mars-modstand i august 2003 ville den røde planet have en radius (sigma) på cirka 12 '' og deklination på ca. +14 grader (D = 90 grader - 14 grader). Hvis jeg observerer fra Singapore (breddegrad på 1 grad, som jeg afrunder til nul) med en 10 tommer Dob, er det følgende plot af den maksimalt tilladte eksponeringstid over hele bevægelsen over himlen (dvs. over området af R fra 0 til 2 x pi).

Som du kan se, er minimumet større end 135 sekunder eller 1 min 15 s. Dette er meget tilstrækkeligt til planetfotografering, da eksponeringer sjældent overstiger flere sekunder (ikke tilrådeligt alligevel på grund af turbulens). Selv hvis planeten placeres uden for centrum med en afstand på f.eks. 10 radier, ville vi stadig have mindst 13 sekunder, tilstrækkelig tid til fotografering med webkameraer og film med høj hastighed. Dette illustrerer imidlertid, hvorfor alt-azimuth-monteringer uden feltrotatorer er upraktiske for deepsky-astrofotografi, da den relevante radius (sigma) i så fald meget større, og derfor er den maksimalt tilladte tid meget kortere, mens den nødvendige eksponeringstid normalt er mindst ti minutter.

Del II: Ækvatoriale monteringer

En lille tanke afslører, at vi kan bruge vores resultater ovenfor til at løse et relateret problem inden for dybsky-astrofotografi for ækvatoriale monteringsbrugere.

Det er umuligt at opnå perfekt polar tilpasning. Denne fejl i polarjustering fører til feltrotationsproblemer svarende til den ovenfor beskrevne. Det naturlige spørgsmål er så, hvor tæt på den sande pol skal man justere sin ækvatoriale monteringsakse for ikke at lide af feltrotationseffekter i et guidet fotografi i en given ønsket eksponeringslængde?

Tidligere brugte vi koordinatsystem til at repræsentere bevægelse af alt-azimuth-mount - billeder set gennem et omfang placeret på en sådan driven mount ville ikke rotere i forhold til det lokale gitter. Nu kan vi bruge det samme gitter til at repræsentere bevægelse ved hjælp af en ækvatorial montering, bortset fra at gitteret nu skal drejes således, at e = 0-punktet (dvs. monteringens nordpol) vil være placeret tæt på D = 0-punktet (dvs. den nordlige himmelpol), hvor vinkelafstanden mellem e = 0 punkt og D = 0 punkt målt fra kuglens centrum (0, 0, 0) derefter ville være fejlen i polær justering. Men dette svarer simpelthen til ovenstående problem, hvis vi observerer, at variablen w = (90 - L) grader, vi brugte ovenfor, er nøjagtigt den polære justeringsfejl, vi i øjeblikket forfølger.

Ved at erstatte L med (90 - w) grader fortsatte jeg med at finde ud af, i en given eksponeringstid, hvilke områder af himlen der ikke ville lide af feltrotationseffekter, når de blev fotograferet. Ved at antage, at jeg bruger en perfekt autoguider, der styrer en stjerne lige i midten af ​​mit fotografi, lavede jeg plot til min 5 "refraktor (770 mm brændvidde), hvilket giver cirka et 2,68 grader bredt felt på 36 mm siden af ​​35 mm filmen format.

Først kiggede jeg på sagen i 20 minutters eksponering.

Polær justeringsfejl: 1,5 grader (lodret akse: D, vandret akse: R)

Polær justeringsfejl: 1 grad (lodret akse: D, vandret akse: R)



Polær justeringsfejl: 0,5 grad (lodret akse: D, vandret akse: R)

Polær justeringsfejl: 0,15 grad (lodret akse: D, vandret akse: R)


Disse plot viser de områder af himlen, der kan fotograferes med min 5 "refraktor uden at lide af feltrotationseffekter i en periode på 20 minutter. Den lodrette akse er vores pseudo-dec-koordinat (i radianer), hvilket betyder toppen af ​​plottet er den himmelske nordpol, midten er midterste bredde og bunden er sydpolen. Den vandrette akse er vores pseudo-RA-koordinat (i radianer), hvilket betyder, at hvis du ser på plottet fra venstre mod højre, er du ser på himlen fra meridianen til den østlige halvkugle til den meridian vestlige halvkugle og derefter tilbage til meridianen på den østlige halvkugle (faktisk er øst og vest afhængig af din definition, men jeg håber man får ideen - hvis ikke se på mit første diagram på dette De hvide områder er de områder på himlen, der kan fotograferes uden at lide af feltrotationseffekter, hvorimod de farvede områder ikke kan. (Disse diagrammer er virkelig 3D-tegninger af (x, y, z) = (R, D, Vi ser dem 'face-on' fra 'bunden' for at opnå dette tilsyneladende 2D-perspektiv, idet man ignorerer tau-aksen.)

Vi observerer straks to ting. Den første er, at polarområderne lider mest af felterotationseffekter eller for at sige det på en anden måde. Hvis du ønsker at fotografere polarområderne, skal du polarere din montering meget mere præcist, end du har brug for fotografering af ækvatoriale regioner. Det andet er, at polar tilpasning til deepsky-astrofotografi virkelig skal være nøjagtig - vi ser, at selv med en fejl på kun 1 grad og en relativt kort eksponering på 20 minutter, kan cirka 35 grader himmel omkring hver pol ikke fotograferes med succes.

Her er to eksempler til det samme teleskop. Den første er til 1 times eksponering.

Polær justeringsfejl: 0,5 grader (lodret akse: D, vandret akse: R)

Polær justeringsfejl: 0,15 grader (lodret akse: D, vandret akse: R)

Polær justeringsfejl: 0,05 grader (lodret akse: D, vandret akse: R)

Det andet eksempel er en 2 timers eksponering. Læg mærke til, hvor nøjagtig polarjustering skal være i denne sag - selv en 0,5 graders fejl i polarjustering efterlader meget af himlen uegnet til fotografering.

Polær justeringsfejl: 0,5 grader (lodret akse: D, vandret akse: R)

Polær justeringsfejl: 0,15 grader (lodret akse: D, vandret akse: R)


Vi kan sammenfatte resultaterne som følger:

1) The larger the aperture of your photographic instrument (the larger the Gamma), the higher its resolution, and hence the more sensitive your photograph will be to mis-polar alignment.

2) The wider the field of view of your photographic instrument - and hence the larger the radius of rotation (sigma) - the more susceptible your photograph is to mis-polar alignment errors.

3) To minimize field rotation problems, always try to guide near the center of your photograph. The key is one needs to minimize the angular distance between the guide star and the farthest point from it in your photograph.

4) Be especially careful in polar alignment when photographing near the poles.

Part III: Declination Drift

We can solve yet another problem quite easily, with the work already done above. This is the issue of declination drift. When performing polar alignment, astrophotographers usually use what is called the drift alignment method for polar alignment. After roughly aligning the polar axis of her/his mount to the north, the astrophotographer usually proceeds to point her/his telescope to a star low in the east (or west), center it on usually a crosshair and watch which way it drifts in declination as the mount tracks the star with its motor drive on. There is a standard procedure to determine the corresponding adjustments needed. (With a little extra effort, one can also derive the procedures from the current discussion.) Next the astronomer looks at a star on the meridian and performs similar observations and adjustments. The question that then naturally arises is, how much declination drift within a given time interval is acceptable if one desires to obtain a non-field rotated photograph - i.e. the field rotation effects do not smear the stars sufficiently to be noticeable?

First I recall a result I obtained above, but did not display. I calculated the cosine of the angle G (Cos[G]), which in the present case is really the angle between the instantaneous direction of movement of the telescope's center of view and that of the star at the center of its view at a given moment. Now the earth and the mount rotate at a rate of u = (2 Pi) / ( 23.93446965 x 60 x 60) per second. That means in a small time interval, the apparent movements of the center of the telescope's view and the star would appear to each trace out a line, which when taken together forms a small wedge. This small wedge can be taken as a flat triangle (in Euclidean space) as a first approximation - even though they actually sweep out arcs on the unit sphere defined above. Then, assuming the angle of the wedge is small, we can further assume that the declination drift can also be approximated by the circular arc subtended by the two lines forming the wedge, even though to be precise we'd need to resolve it into perpendicular (aka declination drift) and parallel (aka RA drift) components and also take into account the curvature of our unit sphere.

Let t be the time that has passed since our astronomer began watching the star. Both the telescope's center of view and the star would have swept out an angular distance of (u t). That means the distance between the ends of the two lines is, roughly G x (u t). That is our desired declination drift, in radians:

For the same formula in arc seconds - a more useful unit for practical declination drift purposes, we divide by 2 Pi and multiply by (360 x 60 x 60) arc seconds:

[Dt]

where t is the total amount of time in seconds star has been allowed to drift from the center of view, w is the angular distance the true celestial pole is displaced from the mount's polar axis (aka polar misalignment), D and R are the pseudo-RA and Dec coordinates defined above in Part I.

Suppose we have figured out - say from the considerations described in Part II above - that we need to get the mount's polar axis within w radians (or degrees - whichever you wish to plug in) of the true pole. And suppose we're only patient enough to watch the star drift for t seconds (e.g. a common time used is 300 seconds, or 5 minutes). Then how much can the star drift within this t time without giving smeared pictures?

The answer depends on what direction the mount's pole is displaced from the true pole. Let's first note that the arc on the unit sphere R = 0 / R = Pi is also the direction in which the poles are displaced from each other on the coordinate system defined. The horizon can of course be oriented in any way with respect to the coordinate system I defined above. That is why we need to perform drift alignment on stars at the meridian and low in the east or west. To see this, let's observe that if our mount's polar axis is displaced along the true meridian, then on the true meridian our mount's constant declination lines (lines that our telescope's center of view will sweep out on the celestial sphere when the mount tracks an object) will be parallel to the true constant declination lines on the celestial sphere, and the two sets of declination lines will intersect at the largest angle 90 degrees away from the meridian. That means in such a case there will be no declination drift if we do drift alignment on a star on the meridian and the most severe declination drift can be observed 90 degrees away from the meridian this also can be seen by putting R = 0 or R = Pi in the above formula, since ArcCos vanishes when its argument is unity. On the other hand, if the mount's pole is displaced 'east' or 'west' of the true pole, then the mount declination lines and the true declination lines will be parallel roughly (I say roughly, because 'east' and 'west' is not orthogonal to the direction of the meridian, and so this is an approximation for small displacements) 90 degrees at the meridian and intersect at the largest angle on the meridian. (When I have more time, I'd draw a picture to illustrate this perhaps some kind reader can contribute to this?)

To summarize, here is how we do drift alignment. After we do our rough alignment, we look at stars 90 degrees away from the meridian (so the above statement regarding low in the east or west is not entirely accurate) to watch for drift. This is to check for magnitude of the component along the meridian of the displacement of the mount's pole from the true pole. Then we watch for drift on a star on the meridian. This checks how far the mount's pole is from the true pole in the 'east' or 'west' direction. (Remember these statements are approximations, good for small displacements - this is reasonable since we have already assumed that rough polar alignment has already been done.) We repeat this procedure until the corresponding adjustments we make reduce the drift to a level that is low enough, say alpha arc seconds in t time, for both stars on the meridian and 90 degrees away from it.

Where does our formula come in? As discussed above, w, D and t are known quantities - in fact quantities determined by the astrophotographer. (D is the pseudo- declination of the star that one is doing drift alignment on - to convert to the usual declination, please look at Part I.) We know the maximum declination drift is produced when R = 0 in our above formula. Therefore a necessary (but alas not sufficient) condition that we have met our required standards for drift alignment is when the total amount of drift (alpha) observed in the telescope is less than the answer one obtains when plugging our values for w, D, R = 0, and t into the above formula [Dt].

July 2005 : Putting R = 0 (see below) completely removes dependance on w and D, since sin[0] = 0. Because of this I suspect the necessary condition imposed by [Dt] is too weak to be truly useful.

(Footnote: In order to derive the sufficient condition for the accuracy needed, one would have to do further analysis. Specifically, here is one possibility: w would have to be resolved into the displacement along the meridian and also along the 'east' or 'west' directions (say p and q respectively). Given a particular mount setup, obtaining two measurements on the declination drift of stars on the meridian and 90 degrees away from the meridian would give us, after plugging them into [Dt] a pair of simultaneous equations that could be solved - by software! - for the values of p and q. Then, provided you could make accurate adjustments on the altitude and azimuth axes of the mount (an unfortunately non-existent feature, as far as I know), it would be possible to obtain very quick polar alignment - no need to perform tedious repetitions of the drift alignment procedure - to a degree of accuracy limited only by the accuracy of the mount's altitude and azimuth adjustment system and of course the accuracy of the drift measurements.)


Calculating the range of visible Right Ascension and Declination from specific location + time - Astronomy

My main goal here is to be able to to develop a simple program that, given the lon/lat/time of a location on the earth, use that to calculate the ascension and declination of the sun at that time, from that position.

While this is not a trivial calculation, most of the heavy lifting is done already and is provided in the form of tables of positions of heavenly bodies at specific time intervals.

Overview of Ephemeris Data

Ephemerides files provide data used in the calculation of the positions of astronomical objects in the sky for specific time periods. NASA’s Jet Propulsion Lab provides a development ephemeris which can be downloaded from their anonymous FTP site.

The ephemeris data is constructed by combining calculations of the motions of the planets, taking into consideration effects that force their motion to vary unexpectedly (e.g. gravitation, tidal, libration, nutation, etc). It is provided in their files as numerical coefficients (Chebyshev polynomials). These coefficients are used instead of the actual data they are much more compact than the tabular ephemerides data and because positions and velocities can be interpolated from them for an arbitrary date.

The ephemeris data is versioned, with revisions based on accuracy for given time ranges and optimizations according to different variables. Explanations for the different ephemerides is available on the JPL website, under the section “AVAILABLE EPHEMERIDES“. The data is further subdivided into blocks of time (20 years) for which you’re interested in doing the calculations.

For this exercise, I’ll only be dealing with version DE405 which is accurate for the date range December 9, 1599 through February 20, 2201. This version was created in June 1997, using observations made by the Magellan spacecraft orbiting Venus.

Ephemeris Coordinate System

The data in the ephemerides file plots the positions of various celestial bodies on an XYZ coordinate system. This coordinate system is parallel to the Earth’s equatorial plane, and where the X-axis runs in the direction of the vernal equinox, the Y-axis runs through the celestial equator at 6h right ascension, 0 o declination, and the Z-axis runs through the north celestial pole.

Here’s an image that illustrates the celestial sphere the coordinate system of the ephemerides is analogous to the coordinate system illustrated here.


Step 3: Ingredients

Here's the list of components I used for this project (note that the above image only pictures a few of them, that's just a shot of my GPS test setup). With the exception of the major components, I haven't specified a particular product or supplier. This is because that might not be helpful to people who like in a different country to me, and also because for many of these components (switches etc.) it doesn't hugely matter what you use.

  • Skywatcher Heritage 130p Flextube Dobsonian Telescope
  • Adafruit ADS1015 12-bit ADC (analogue to digital converter)
  • 16x2 LCD Display (I used Winstar WH1602B-RTI-JT Red Characters)
  • Arduino Mega 2560 Microcontroller Board
  • GPS Module (I used GY-GPS6MV2)
  • Battery holder for AA batteries (6 cells in series to produce 9V)
  • Small breadboard to fit inside the control box
  • SPST Rocker Switch with red LED
  • SPST Momentary Pushbutton Switch with red LED
  • Four red LEDs
  • SPDT Rocker Switch, Centre Off
  • Rubber tube to use as hose to connect the handset to the telescope mount.
  • Arduino Uno (not part of the finished device, but useful to have a second Arduino for testing purposes)
  • Suitable resistors to use with each of the LEDs.
  • One continuous rotation potentiometer for measuring the Azimuth (Heading) of the telescope.
  • One potentiometer for measuring the Altitude (Inclination) of the telescope.

You'll also need jumper leads of different lengths and types, multicore wire, solder, insulation tape, filament for your 3D printer, and a computer with the Arduino IDE software installed to allow you to program the Arduino.


The Palomar Sky Survey

Between 1949 and 1956, a 1.2-meter diameter Schmidt camera on Palomar Mountain was used to survey the entire sky visible from a low latitude in the continental United States. Sponsored by Palomar Observatory and the National Geographic Society, the resulting set of 900 pairs of photographs are still a widely used reference in astronomical research. Each image covers a useful field about 6° across allowing for overlap so that the edges are photographed twice. The sky was photographed once in blue light to capture hot stars and dusty nebulae, and again in red light to show cool stars and gaseous nebulae. Originally, paper copies of the original glass plates were distributed to observatories around the world where astronomers used them to identify objects for more detailed study. Most large telescopes have such a small field of view that they can look only at one thing of special interest at a time, whereas the Schmidt camera has a large field of view, almost big enough to capture the entire Virgo cluster in one image. With the development of digital image technology and the capacity to store enormous quantities of data, it is also possible to automate mapping the entire sky, and to reconstruct the needed wide field views from many small individual images. Now, imaging databases recorded in visible light, x-rays, and in the infrared, are available in digital form over the Internet, providing astronomers with access to a virtual telescope from their desktop. The background images of the sky in Aladin-Lite and in [ https://www.google.com/sky/ Google-Sky] are from the Palomar Sky Survey and its successors. Recent images such as the one below taken by an amateur astronomer are nearly as good.

Let's start with M87. It is one of the largest galaxies in the cluster, and it is fairly easy to find on the images. Near the center of this picture it seems to be an isolated round object. That's the galaxy you want, south and west of the similar looking pair of galaxies M84 and M86. In Aladin you can enter a name in the target window and the cursor will center on it. Try it with M84, then zoom out to see all the other nearby galaxies, and ask for M86.

Using the + and - buttons on the map, you can zoom out a little from the close view that a search will provide by default to see where M87 is in the cluster. Or, you can zoom in to see a more detailed image. A seach on Google will bring up images from the Hubble Space Telescope and other sources, although you have to be cautious about misidentified photos in web searches. A source of some of the best images is "Astronomy Picture of the Day", known as APOD. The page shows a new image each day. At the bottom, click on Search and enter a name to see if what you are interested in has been highlighted by them. Do that for M87, and you may find this link to the Virgo Cluster on APOD.

The M in these names stands for Messier, after Charles Messier, the French comet hunter nicknamed Cometferret, who made a comprehensive list of the brightest non-stellar objects in the sky. There are 103 objects on this list that he discovered himself, and 109 altogether including others he confirmed. His work on the catalog was largely complete by 1789 and the time of the French Revolution. Messier, who died in 1817 at age 86, was a contemporary of William Herschel, the discoverer of Uranus.

The first object on Messier's list is M1, the Crab Nebula. It is a remnant of a supernova in Taurus that he found in 1758. The galaxies in Virgo were all found much later, probably because observing faint objects is difficult when they are low in the sky, as this cluster of galaxies is when seen from Paris.

Look closely at these images of M87. Compare them with various types of galaxies illustrated in books and on the web. Try to see how this galaxy fits in the classification scheme devised by Edwin Hubble.

Now go back and look again at the large image of the Virgo Cluster above, or explore the region with Aladin, and you'll see lots of other similar galaxies. The most common kind of galaxy in a cluster like Virgo is elliptical.

In the Hubble classification, the number following E indicates just how elliptical the galaxy appears. It should be obvious that M87 is an elliptical galaxy. The ratio of the long dimension a to the short dimension b is used to tell just how elliptical the galaxy is, according to 10(a-b)/a, so if b and a are the same it is E0, and if b is 30 percent of a it is E7.


3. Estimate the large and small dimensions for M87 and calculate 10(a-b)/a. How elliptical is it? This is the number after "E" in the classification.

Hints: A simple way to do this is with a millimeter scale on a ruler. Hold it up to the screen and measure across the galaxy's long (a) and short (b) dimensions. Calculate the ratio (a-b)/a and multiply by 10. You can also do this in Sky-Map, reading the coordinates and converting to seconds of arc. Since the galaxy is not oriented north-south or east-west, finding the long and short dimensions this way requires more math skills than simply measuring with a ruler.

If you look at the detailed images of M87 again, you should be able to see many apparently tiny fuzzy dots all around it. Each of these "dots" is a spherical cluster of hundreds of thousands of stars. Similar globular star clusters surround our Milky Way.

There are about 2000 such globular clusters with M87, but only about 100 around the Milky Way. The halo of clusters around this galaxy help to define its size. Return to M87 on Sky-Map and measure the diameter of the halo in seconds of arc. You can do this by using the coordinate readout that changes as you move the cursor around the image. Declination (north and south) is indicated in degrees, minutes, and seconds of arc. Right Ascension (east and west) is measured in hours, minutes, and seconds of time. Each second of time is 15 seconds of arc.


4. How large in seconds of arc is the image of M87 and its halo of globular clusters?

Hints: Look at the declination reading in Sky-Map for the center of the galaxy and for a point just beyond the halo north of the galaxy center so that right ascension is the same as at the center. The difference in declination readings will give you the radius of the halo. Remember that 1 degree is 60 arcminutes, and 1 arcminute is 60 arcseconds. The diameter is twice the radius.


To give you a sense of its true size, let's convert this measurement into a diameter in light years. The angle that you see in the sky is determined by the distance to the galaxy, and its real size. Something quite large (like this galaxy) will appear to be very small simply because it is far away. Mathematically, we say that


where S is the real size, R is the distance, and A is the angle in seconds of arc. In this formula both S and R have the same units. The 206,000 converts from seconds of arc to radians. Assume that the galaxy is 60,000,000 light years away (R=60,000,000) and calculate how large it must be in light years to appear to be the size you measured.


5. How large is M87 in light years? Compare to the Milky Way which with its halo is about 130,000 light years in diameter.


Unified Astronomical Positioning and Orientation Model Based on Robust Estimation

Astronomical positioning and orientation are important components of astronomical geodesy tasks, and they could also be used for real-time celestial navigation. In traditional methods, astronomical positioning and astronomical orientation can only be implemented separately, and all the observations with different levels of precision are given the same weight. To change this, a unified astronomical positioning and orientation model based on robust estimation was developed. First, the problem that the coordinate rotation parameters are difficult to calculate directly was solved by using the Rodrigues matrix. Consequently, the astronomical longitude, latitude, and azimuth can be calculated at the same time. Second, as a result of the introduction of a robust estimation method, specific weight is matched to observations with corresponding precision when calculating the parameters, and thus more accurate astronomical longitude, latitude, and azimuth valuation can be calculated simultaneously. Experiments indicate that the unified model was able not only to achieve astronomical positioning and orientation simultaneously but also improve the accuracy of final results.

Astronomical geodesy is a technique that uses the sun, moon, stars, and other natural objects as beacons takes the celestial horizon coordinates of such objects as observations and finally determines the geographical position of the station and the azimuth of one direction (Hirt et al. 2010). Because it is a kind of absolute positioning, astrogeodetic points in the geodetic control network are often used to control the accumulation of measurement error and determine vertical deflection (Hirt and Flury 2008 Hirt and Seeber 2008 Müller et al. 2004). Because of their advantages, such as good stealth, not easy to be disturbed, and strong independence, astronomical positioning and orientation could be used as a method of independent autonomous navigation (Kaplan 1999 Vulfovich and Fogilev 2010). In the traditional method, astronomical positioning and orientation are realized by theodolite to observe natural objects one by one the horizontal angle and elevation angle are recorded at the same time and then the astronomical longitude, latitude, and azimuth are solved, respectively, according to algorithms (Robbins 1977). In the celestial sphere, the north celestial pole, zenith, and stars can constitute a positioning triangle, as shown in Fig. 1.

Fig. 1. Astronomical positioning triangle (reprinted from China Satellite Navigation Conference (CSNC) 2013 Proceedings, “A New Celestial Positioning Model Based on Robust Estimation”, 2013, 479–487, Chonghui Li, Yong Zheng, Zhuyang Li, Liang Yu, and Yonghai Wang, © Springer-Verlag Berlin Heidelberg 2013, with permission of Springer)

In Fig. 1, P = north celestial pole Z = zenith W = west point W M Q = horizon circle W N Q ′ = celestial equator and σ = a star in the celestial sphere. According to the spherical triangle cosine formula, the relationship among zenith angle z , astronomical latitude φ , longitude λ , star right ascension α , declination δ , and Greenwich true sidereal time S is

In Eq. (1), right ascension α and declination δ can be obtained through an apparent position calculation procedure (Bangert et al. 2009). Zenith distance z and UTC time T UTC can be obtained through observation. Therefore, only λ and φ remain unknown in Eq. (1) if n stars have been observed, the astronomical longitude and latitude can be calculated according to the least-squares method, and then astronomical positioning can be realized.

When the astronomical longitude and latitude of the observing location are known, there are several kinds of astronomical orientation methods to determine the azimuth of a ground target, but the most frequently used one is the Polaris hour angle method (Adams 1968).

As is shown in Fig. 2, the azimuth ( A B ) of ground target B is the angle between the meridian plane and the vertical plane that goes through the station and the target two points it is measured clockwise from north point N in the range of 0–360°. Although north point N cannot be observed directly like ground targets, it can be determined by means of observing the azimuth of celestial bodies (Zhan et al. 2016 Carter et al. 1978). Let L σ represent the observed horizontal angle of celestial body at Greenwich UTC time T UTC , let L B represent the observed horizontal angle of the ground object, let A σ represent the azimuth of Polaris at the observing epoch, and let L N represent the horizontal angle of north point N therefore:

Fig. 2. Astronomical orientation principle

Obviously, the azimuth A B of the ground target can be determined based on the azimuth A σ of Polaris at the observing epoch. According to the principle of hour angle method, A σ can be calculated according to the following formula:

Then the azimuth A B can be calculated according to Eq. (2). In practice, the final azimuth can be estimated as the average of A B if the Polaris has been observed m times, and then astronomical orientation is realized.

With the development of imaging and digital processing technology, a new kind of celestial navigation method that uses star sensors on the ground and at sea to image multiple stars simultaneously has appeared (Hughes et al. 2009 Li et al. 2014 Parish et al. 2010). Unfortunately, some defects still exist in these methods. On the one hand, astronomical positioning and orientation are implemented separately, leading to low efficiency. On the other hand, although multiple stars can be imaged simultaneously by a charge-coupled device (CCD), some low-precision star observations still exist because of weather or other severe conditions, and calculating with these data at the same weight directly will reduce the accuracy of astronomical positioning and orientation.

To solve these problems, a unified astronomical positioning and orientation model based on robust estimation is proposed. By using the Rodrigues matrix, this model realizes the calculation of astronomical longitude, latitude, and azimuth simultaneously. By introducing robust estimation, the influence of outliers is excluded, the use of low-precision observations is limited, and the high-precision observations are fully utilized thus, more reliable and effective valuations of astronomical longitude, latitude, and azimuth can be obtained.

In the traditional method, astronomical positioning is generally carried out as the first step, and then the calculated longitude and latitude are deemed as known quantities to determine the astronomical azimuth of a ground target thus, errors in longitude and latitude will further increase the error of the azimuth. In the horizontal circle astronomy method, the error equation is established on the difference between the observed and calculated azimuth, and the longitude, latitude, and azimuth can be solved simultaneously (White 1966). However, the elevation-angle observations are underutilized. Therefore, the horizontal-angle and elevation-angle observations are both used in the unified astronomical positioning and orientation model. This method is actually a vector algorithm that can not only improve the calculation efficiency and uniformity but also restrict the error propagation and improve the calculation accuracy.

Suppose at the observing epoch T UTC , the horizontal angle and zenith distance of a celestial body are, respectively, L and z . The rectangular coordinate of the celestial body in the instrument coordinate system is

In the true equatorial coordinate system, the coordinates of the celestial body could be represented by right ascension α and declination δ as

In Eq. (9), S represents the Greenwich true sidereal time of the observing epoch, which can be converted from the UTC time T UTC . Therefore, Eq. (6) could be changed to the following:

Obviously, it is difficult to calculate λ , φ , and A H according to Eqs. (13) and (14) directly. The antisymmetric matrix Q is introduced here as follows:

Because R is an orthogonal rotation matrix with three degrees of freedom, it can be seen as a Rodrigues matrix that is composed of Q and I (Yao et al. 2006), namely:

Then Eq. (13) is changed as follows:

It can be further expressed in the form of rectangular coordinates as

Antisymmetric matrix Q has the following property:

Multiply ( I − Q ) − 1 on the right side:

Then multiply ( I − Q ) − 1 on the left side:

Then substitute Eq. (15) into Eq. (22) and expand it:

Because there are only two independent equations in Eq. (23), the three independent parameters a , b , and c cannot be solved by observing only one celestial body. However, if two or more celestial bodies are observed, then four or more than four independent equations can be determined the three independent parameters a , b , and c can be calculated by the least-squares method and then the matrix Q and the rotation matrix R can be obtained accordingly. Therefore, the astronomical longitude λ , astronomical latitude φ , and azimuth A H can be expressed according to Eq. (14):

Then, in the horizontal plane, the azimuth A B of the observed ground target is

Thus, astronomical positioning and astronomical orientation are realized simultaneously in this unified model.

Because of the influence of severe conditions, such as clouds and background light, some errors and even few outliers inevitably exist in the observations, which will cause the final least-squares estimations of longitude, latitude, and azimuth to deviate from their true values. Considering this problem, robust estimation, which helps to take full advantage of high-precision observations, limit low-precision ones, and exclude outliers, is introduced. This way, more reliable, effective, and meaningful parameter estimations can be obtained (Li et al. 2013).

Robust estimation stems from the concept of statistical robustness, and it is proposed for least-squares estimation that has no characteristic of anti-interference. In Eq. (23), if order:

Where P ¯ is the equivalent-weight matrix, the success of the robust estimation mainly depends on the accuracy of the initial equivalent weights. The observations in astronomical surveying are independent from each other therefore, P ¯ is a diagonal matrix, and its diagonal element is p ¯ i (Yang et al. 2002). Because the accuracy of each observation is not equal, and observations may even contain gross errors, the least-squares estimations are not suitable as the initial value. According to the characteristics of astronomical measurements, first set the L 1 norm minimum as a criterion to calculate the initial parameters (Yang et al. 2001). Suppose:

Normally, in Eq. (31), the value of k 0 and k 1 can be assigned as 1.5 and 3.0, respectively v ̃ i n − 1 is the (n – 1)th standardized residual of the i th observation, namely:

In Eq. (33), σ 0 is the unit-weight mean square error its theoretical or empirical value is often used. P ¯ n − 1 = (n – 1)th equivalent-weight matrix p i n − 1 = i th diagonal element in P ¯ n − 1 and A i = i th line of matrix A —namely, it is the i th coefficient vector of the error equation (Yang and Wu 2006). Therefore, the final robust valuation of the parameters can be calculated by Eq. (30). The astronomical longitude λ , latitude φ , and azimuth A B can be calculated according to Eqs. (24) and (25).

To verify the effectiveness and accuracy of the proposed unified astronomical positioning and orientation model based on robust estimation, two kinds of observations were used to calculate the longitude, latitude, and azimuth, respectively. The first kind of observation was obtained by a large field of view (FOV) star sensor placed on the ground, which simply needed to capture a star image and eliminate artificial operation when carrying out the experiment. After being processed by such automated steps as star extraction, determination of star image center coordinates, and star pattern matching, a total of 541 stars was identified from the star image [Fig. 3(a)]. The other kind of observation was obtained by a Leica total station (TS 50). During the experiment, 24 stars were manually observed in a period, and each star was measured approximately 10 times [Fig. 3(b)]. The two experiments were carried out at the same position the true value of the instrument station ( λ = 113 ° 6 ′ 13.50 ″ , φ = 34 ° 31 ′ 28.39 ″ ) and azimuth ( A H = 43 ° 19 ′ 56.27 ″ ) of a well-designed ground target were determined by first-class astronomical surveying, and their accuracies were 0.3″ and 0.5″, respectively (General Administration of Quality Supervision, Inspection and Quarantine of the People’s Republic of China 2000). One of the most important details is how to determine the observing time because errors in the observing epoch will directly propagate into astronomical longitude. First, the authors used the pulses per second (1 pps) single and the National Marine Electronics Association-0183 (NMEA-0183) data generated by a timed GNSS receiver to correct a time counter in the measure control unit before starting measurement, and then the time counter was synchronized to UTC scale therefore, the GNSS receiver is not needed in the process of measuring. In the star sensor experiment, the precise time delay of the shutter is necessary. A calibration procedure was developed based on the comparison of two epochs. The first one is a count value recorded as the shutter opening epoch by the control unit, and the other one is imaged by the star sensor from a screen displaying the real-time value of the time counter. The comparison of both count values revealed a significant delay of the shutter motion, and then the time delay could be calculated The accuracy of the implemented timing technique has been verified to be approximately ±5 ms. In the total station experiment, the observing epoch was determined by an electronic pulse from the control unit that was synchronously sent to the corrected time counter and to the total station the accuracy of this method has been demonstrated to be approximately ±3 ms.

Fig. 3. Star observations: (a) obtained by a large FOV star sensor (b) obtained by total station

As can be seen from Fig. 3(a), the stars are mainly uniformly distributed throughout the sky. However, in Fig. 3(b), the stellar azimuths are different from each other, resulting in the elevation angle of each star appearing as a column. It can be seen from Fig. 3(b) that the elevation angle of each star is approximately 60°, and the horizontal-angle intervals are similar to each other, indicating that the 24 stars are substantially distributed near the 60° altitude circle.

According to the star number, the theoretical azimuth and elevation angles of these stars at their observing epoch can be calculated accurately. The observation azimuth of the stars can be obtained by adding the observation horizontal angles on the basis of the azimuth of the star sensor’s x-axis, and then by subtracting it from the true azimuth of these stars, the residual of the horizontal-angle observations can be obtained these are shown in Fig. 4.

Fig. 4. Residuals of horizontal angle: (a) by star sensor (b) by total station

As can be seen in Fig. 4, although the observed precision of the total station was significantly higher than that of the large FOV star sensor, they both obeyed normal distribution. The horizontal angle residuals of most stars in Fig. 4(a) were found to be within the range of ±100″, the observed accuracy of which is relatively high, and a few of the azimuth errors were within the range of ±100–±300″, the observed accuracy of which is relatively low. Similarly, the horizontal-angle residuals in Fig. 4(b) were found to be mostly within the range of ±4″, and the others were within the range of ±4–±10″.

The elevation angles of the two kinds of observations were both influenced by atmospheric refraction, so they should be corrected before being used to calculate the parameters. However, the elevation angles cannot be completely corrected by the refraction correction equation it always remains a quantity in the elevation-angle values. Therefore, the residuals of elevation angles from their true values may still contain some errors, but their regularities of distribution can be used to research the data quality (Fig. 5).

Fig. 5. Residuals of elevation angle: (a) by star sensor (b) by total station

As shown in Figs. 5(a and b), the elevation-angle residuals of most stars were found to be within the range of ±50″ to ±5″, respectively a few of the elevation angle residuals were within the range of ±50″ to ±110″ and ±5″ to ±10″, respectively and individual errors were greater than ±110″ and ± 10″, respectively, which could be considered as outliers.

Three processing schemes were used to deal with the experimental data. Scheme 1 used the least-squares (LS) algorithm to calculate the astronomical longitude and latitude first, then used the hour angle method to determine the azimuth. Scheme 2 used the unified method based on the Rodrigues matrix to solve the astronomical longitude, latitude, and azimuth simultaneously. Scheme 3 further adopted robust estimation based on Scheme 2 to calculate the parameters. The difference between the results of the three schemes and their true values are shown in Table 1.

Table 1. Comparison of Errors of Three Schemes

SchemeLongitude error (a)Latitude error (a)Positioning error (a)Orientation error (a)Longitude error (b)Latitude error (b)Positioning error (b)Orientation error (b)
13.02″−2.85″4.15″−8.75″0.93″–0.59″1.10″4.25″
22.32″−3.59″4.27″−8.13″−0.81″−0.68″1.06″3.37″
30.82″−2.34″2.48″−5.64″−0.31″−0.43″0.50″2.05″

As can be seen from Table 1, the positioning error and orientation error of Scheme 2 were close to those of Scheme 1, which indicates that the proposed model using the Rodrigues matrix in the solving process can realize astronomical positioning and orientation simultaneously. Furthermore, by comparing the positioning error and orientation error of Scheme 3 with those of Scheme 2, it can be determined that the positioning error declined to 2.48″ from 4.27″, and the orientation error declined to –5.64″ from –8.13″ for the star sensor observations. This is because 17 observations were identified as outliers and were given zero weight 164 other observations were identified as abnormal values, and their weights were reduced and another 360 high-precision observations were given greater weight in the robust estimation process. Similarly, 9 total station observations were identified as outliers, 49 other observations were identified as abnormal values, and another 204 observations were identified as high-precision values. Therefore, the astronomical positioning error declined to 0.50″ from 1.06″, and the orientation error changed to 2.05″ from 3.37″ for the total station observations. These results show that the proposed unified model based on robust estimation can effectively take full advantage of high-precision observations, limit utilization of low-precision observations, and exclude outliers, through which more accurate astronomical longitude, latitude, and azimuth estimations can ultimately be obtained.

A unified astronomical positioning and orientation model based on robust estimation is proposed. Through the introduction of the Rodrigues matrix, this model achieves the calculation of astronomical longitude, latitude, and azimuth simultaneously, eliminating the inconvenience of the traditional method in which astronomical positioning and orientation must be carried out separately in. In addition, by the use of a robust estimation method, this model can effectively suppress the influence of outliers, reduce the impact of systematic errors, make full use of high-precision observations, and eventually obtain results with relatively high accuracy. Compared with traditional models, the proposed method has improved efficiency and accuracy. Even in foggy or cloudy weather conditions, the astronomical positioning and orientation problems can be solved simultaneously by just observing several visible bright stars therefore, the method also has important application prospects in the celestial navigation field.

This research is funded by the National Nature Science Foundation of China (Grants 41604011, 11673076, 11373001), the Geographic Information Engineering State Key Laboratory Open Fund of China (Grant SKLGIE2016-M-2-2), and the Outstanding Youth Fund of Information Engineering University (Grant 2016611705). The authors thank two reviewers for their comments on the manuscript and the editor for the handling of the review process.


Se videoen: Arbeitsblatt: Adjektivdeklination im Akkusativ (Oktober 2022).