aryabhata and axial rotation of earth-1--p51-68

8/3/2019 Aryabhata and Axial Rotation of Earth-1--p51-68

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51RESONANCE March 2006

GENERAL ARTICLE

KeywordsAryabhatiya; axial rotation; ce-

lestial sphere, celestial poles,

celestial equator, Earths equa-

tor, latitude.

Aryabhata and Axial Rotation of Earth

1. Khagola (The Celestial Sphere)

Amartya Kumar Dutta

.

Amartya Kumar Dutta is

in the Stat-Math Unit of

Indian Statistical Instiutte,

Kolkata. His research

interest is in commutative

algebra.

A ry a b h a t.a (b o rn 4 7 6 C E ) is reg a rd ed a s a p io -

n e e r o f m a th e m a tic a l a str o n o m y in a n cie n t In d ia .

In th is th ree -p a rt a rtic le , I sh a ll d isc u ss o n e im -

p o rta n t c o n trib u tio n o f A ry a b h a t.a in a stro n o m y

in v o lv in g th e le a st te ch n ic a l b a ck g ro u n d { th e

p rin cip le o f a x ia l ro ta tio n (P a rt 1 ) a n d c o m p u -

ta tio n s o n th e sid e re a l d a y (P a rt 2 ). P o rtio n s o f4 v e rse s, p e rta in in g to E a rt h 's ro ta tio n , fro m h is

fa m o u s tre a tise A ry a b h a t.y a w ill b e q u o te d . In

P a rt 3 , I sh a ll g iv e a b rie f h isto rica l a c c o u n t o n

th e c o n ce p t o f a x ia l r o ta tio n o f E a rth .

T h e A ry ab h at.y a ( co m p o sitio n of A ryab h at.a') is d e-sign ed as a con cise text of 12 1 verses b rie y p resen t-in g im p ortan t p rin cip les in a stron o m y a n d m a th em a ticsin th e sty le o f terse a p h orism s. T h is h ig h ly co n d en sedtrea tise is n ot m ea n t to p rov id e c om p lete o r d etailed ex -

p ositio n s. V a rio u s fa cts o r m eth o d s, w h ich w e re p erh a p sw ell-kn ow n in h is tim e, are taken for granted in th e text.T h erefore, a m a jo r p o rtio n of th is article w ill in tro d u cecertain b a sic fea tu res o f sp h erica l a stro n om y in m o d ernla n g u a ge to fac ilitate d iscu ssion s o n th e q u o ted v erses.T h ere w ill b e in cid en tal referen ces to oth er statem en tso f A rya b h at.a .

T h e A rya b h at.y a is d iv id ed in to 4 section s (pad a). Inth e rst an d in tro d u ctory section G tika (13 verses),th ere are ten ve rses called D asa gtika-sutra (ten ap h o -

rism s in th e gtik a m etre) su m m arisin g , fo r th e b eg in n er,th e essen tial p ara m eters o f A rya b h at.a's sy stem . T h erem a in in g th ree section s (total 10 8 v erses), th a t c o n sti-tu te th e m ain tex t,a reG an.ita (m ath em atics): 33 verses,


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52 RESONANCE March 2006

GENERAL ARTICLE

K alakriya (recko n in g o f tim e): 25 verses an d G ola (ce-lestial sp h ere): 5 0 v erses.

T h e origin al v erses o f A ryab h at.ya (alo n g w ith E n g -lish tra n slation s) a re g iv en in ([1 ]). O n e ca n see ([2 ])

fo r a g en eral stu d en t-frien d ly g u id an ce a n d ([3 ],[4 ]) forov erview articles o n In d ian astron om y, ([5]) fo r a com -p ila tio n of a n cien t S a n sk rit v erses (w ith E n g lish tran s-latio n s) o n va rio u s co n cep ts, th eo ries, tech n iq u es a n din stru m en ts in a stro n o m y, a n d ([6 ]) fo r a stan d a rd m o d -ern treatm en t o n sp h erical astron om y.

T h e P rin cip le o f R o ta tio n

T h e great d iscovery th a t th e E arth rotates arou n d itsow n a x is from w est to ea st is reco rd ed in th e A ry ab h at.ya(G tik a 3,6; K alakriya 5; G ola 9,10). T h e you n g as-tron o m er b old ly d eclared th a t th e a p p aren t m otion ofth e h eav en ly b o d ies rou n d th e E a rth is o n ly an illu sio n(G o la 1 0 ). H e ex p lain ed it u sin g th e fo llow in g sim ile(G ola 9):

B o x 1 . A ry a b h a t.a o rA ry a b h a t.t.a ?

The celebrated astronomer-mathematician is popularly known as ` Aryabhat. t. a' in Indianlanguages. Early Indologists like H T Colebrooke (1817) too spelt the name with double`t. '. However, as Bhau Daji (1 865) pointed out ([7], p 518) , the Sanskrit spelling of thename, as found in the manuscript s of Brahmagupta and other ancient Indian astronomers,corresponds t o ` Aryabhat. a'. The situation was aptly summarised by W E Clark (1 930)in ([8], vii-viii):

\There has been much discussion as to whether the name of the author should be spelledAryabhat. a or Aryabhat. t. a. B h a t.a means `hireling,' `mercenary,' `warrior,' and b h a t.t.ameans `learned man,' `scholar'. ` Aryabhat. t. a' is the spelling which would naturally beexpected. However, all the metrical evidence seems to favour the spelling with onet. . It is claimed by some that the metrical evidence is inclusive, that b h a t.a has beensubstituted for b h a t.t.a for purely metrical reasons, and does not prove that Aryabhat. ais the correct spelling.... However, until more denite historical or metrical evidencefavoring the spelling Aryabhat. t. a is produced I prefer to keep the form Aryabhat. a. "

Historians of Indian science have adopted ` Aryabhat. a' as the ocial spelling.


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GENERAL ARTICLEa n u lo m a ga tirn a u sth a h. p asy atya ca la _m v ilom a ga _myad vat

acalan i b h an itad va t sam ap ascim a g an ila _n k a y a m

[an ulom a : in th e n atu ral d irection ; gati : m ov in g; n au: b oa t; stha : b ein g in ; pasyati : sees; acala : station a ry,m ou ntain ; vilom a : a ga in st th e n atu ral d irectio n ; ga :m ov in g; yadvat : ju st a s; acalan i : statio n ary (p lu ral);bh an i : sta rs; tadvat : lik ew ise; sam a : sam e, eq u al,exact, sim ilar (also com p lete, w h ole, entire); pascim a :w est; gan i : m ovin g (p lu ral); la _n kayam : a t L a _n k a .]

Ju st as a passenger in a boat m ovin g dow n-stream sees the station ary (trees on the riverbanks) as traversing upstream , just so doesan observer at La _n ka see the xed stars asm oving tow ards the w est at exactly the sam espeed (at w hich the E arth m ov es from w estto east).

F u rth er, as w e sh all see in th e n ex t p a rt of th is a rticle,h e m ad e an a ccu rate estim ate of th e tim e taken by th eE a rth fo r o n e co m p lete rotatio n .

N ote th a t L a _n k a , in th e q u oted verse, d o es n ot refer tom od ern S ri L an ka. It d en oted th e p oin t o n th e equ a-tor sou th o f U jjay in (23 .0 9 N , 75.43 E ), i.e., th e p o in tat w h ich th e m erid ia n th rou gh U jjay in (7 5 .43 E ) in ter-sects th e eq u a tor. (T h u s th e p oin t is n ow on th e In d ia nO cea n n ea r th e M a ld ive isla n d s.) B u t w h y th e referen ceto L a _n k a? W e rst d e n e th e celestia l sp h ere a n d its rel-eva n t circles m en tio n in g th e an cien t In d ian a n a lo g ou es.

T h e C e le stia l S p h e re

T h e sky ap p ea rs to u s as th e u p p er h a lf of a larg e h ol-low sp h ere. T h e sta rs a n d p la n ets a p p ea r a s lu m in o u sp o in ts o n th e in n er su rfa ce o f th is u p p er h em isp h ere.T h e co m p lete sp h ere seem s to m eet th e E a rth 's su rfa ce

The great

discovery that the

Earth rotatesaround its own axis

from west to east

is recorded in the

Aryabhatiya.

Aryabhata made

an accurate

estimate of thetime taken by the

Earth for one

complete rotation.

.


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GENERAL ARTICLE

in a circle w h ich w e ca ll th e (visib le) h orizo n . T h is im ag -in a ry h o llow sp h ere o f a n a rb itrarily large rad iu s w ith

th e ob server at th e cen tre is ca lled th e celestial sphere.T h e rad iu s o f th e celestia l sp h ere is u su a lly co n ceiv edto b e su cien tly large so th at th e en tire E a rth ca n b eregard ed a s a p o in t a t th e cen tre of th e h u g e sp h ere.

A lth o u gh th e sta rs are scattered in sp a ce at d i erentd istan ces from th e E arth , th ey all a p p ear eq u ally re-m o te to a n o b server { th e d i eren ces in th eir d istan cesare n o t p ercep tib le to o rd in ary o b servation . T h u s th eap p aren t p osition o f a star o n th e im ag in a ry celestia lsp h ere sign i es on ly th e direction of th e star from th e

ob serv er; n o th in g else is in d ica ted ab ou t its p osition insp a ce. T w o stars w ill b e v ery close o n th is sp h ere w h enth ey h av e n ea rly th e sam e d irectio n eve n if o n e o f th esta rs is p h y sica lly m u ch m o re rem ote th a n th e o th er.

T h e a p p a ren t d istan ce b etw een tw o celestia l b od ies issim p ly th e d i eren ce in th eir d irection s a n d is m ea su redb y th e an g le su b ten d ed at th e o b serve r b y th e tw o o b -jects. S p h erica l a stro n om y ' ex a m in es th is a n g u la r d is-tan ce b etw een ob jects on th e celestia l sp h ere. S in ceth e a ctu a l lin ea r d istan ces b etw e en celestial ob jects are

n o t of m u ch relev a n ce h ere, th e ex p ressio n a n gu lar d is-tan ce' is o ften ab b rev ia ted to d istan ce' w h en th ere is n oscop e fo r co n fu sio n . T h e a n gu lar d ista n ce is m ea su redin d eg rees, m in u tes an d seco n d s. R ec all: 10 = 6 00a n d10= 6 000. T h u s, \ star X is a t a d ista n ce of 300from starY " m ea n s th a t th e d irection s o f th e tw o sta rs X an d Yfrom a terrestrial ob serv er m a ke an an g le o f 1

2d eg ree.

T h e geo cen tric m o d el of celestial sp h ere, p resen tin g th eu n iverse as it appears to th e ob server, p rov id es a co n ve-n ien t fram ew ork for th e c om p u tatio n al stu d y o f th ose a s-

tron o m ica l p h en o m en a w h ich essen tially in vo lv e th e d i-rection s of celestia l ob jects w h en v iew e d from th e E a rth .M o d ern tex ts o n b a sic a stro n om y, th erefore, b eg in w itha ch a p ter o n th e celestial sp h ere. In sta n d ard p lan e-

The apparent

distance between

two celestial bodies

is simply the

difference in their

directions and is

measured by theangle subtended at

the observer by the

two objects.


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GENERAL ARTICLE

Figure 1. Geocentric paral-

lax.

OX: Observed direction

CX: Geocentric direction

CXO: Geocentric parallax.

1 For observers in distinct posi-

tions O1

and O2, even though

the directions O1Xand O

2Xof a

star Xcoincide, the celestial co-

ordinates ofXare different. Fig-

ure 2 shows a star X to be si-

multaneously at the horizon of

O1, zenith ofO2 and nadir ofO3.

tariu m sh ow s, th e rela tiv e p osition s (i.e., d irection s) ofcelestia l o b jects a re d ep icted on a d o m e rep resen tin g th e

celestia l sp h ere.O b servers a t d i eren t p la ces on E a rth g et d i eren t p ic-tu res o f th e celestial sp h ere. T h is is b eca u se th e h o rizo n -tal p la n es a t tw o d istin ct p o in ts on th e E a rth 's su rfa ceare d i eren t. T w o o b servers at d i eren t sp ots see d if-ferent p o rtion s of th e sky. M o reover, even if a sta r isseen b y b oth ob serv ers, its orien tatio n s tow ard s th e tw oh orizo n s a re d i eren t.

F or a sim u ltan eo u s a n aly sis o f th e situ a tio n s a t d i eren tp la ces, it w o u ld b e d esirab le to en v isag e a co m m o n cen -

tre o f ob serva tio n . T h e cen tre of th e E a rth is a n a tu ralch o ice . T h erefore, w h ile it is u n d o u b ted ly co n ven ien t tocon sid er th e sp h ere cen tred at th e ob serv er at a p la ceO , it m igh t a lso b e n ecessary to red u ce th e o b served d i-rection s of celestia l b o d ies (as o b served from O ) to th eco rresp o n d in g geocen tric d irectio n s { th e d irectio n s ofth e celestia l ob jects th at w o u ld b e ob serv ed b y a h y p o -th etica l o b server a t th e cen tre of th e E a rth C p osition edalo n g C O (i.e., p a rallel to th e o b serve r a t O ). T h e a n g leb etw een th e ob served d irection of a h eaven ly b o d y X

an d its g eo cen tric d irectio n , i.e., th e a n g le su b ten d ed b yC O at X , is ca lled th e geocen tric pa rallax o f X (F igu re1).

N o w each star is at su ch a n en o rm o u s d ista n ce from th eE a rth th a t its ge o cen tric p ara llax is p ractica lly zero .T h erefore, a s m en tion ed in th e rst p ara grap h of th issection ,C a n d O m ay b e trea ted a s th e sam e p oin t w h ilem a p p in g th e stars.1 H ow ever, w h ile con sid erin g ob jectsw h ich are relatively n ear { like th e M oo n or even th eS u n { o n e can n ot b e so d ism issive ab ou t th e E arth 's

rad iu s. A t th e h o rizo n , th e g eo cen tric p a rallax o f th eM oo n is a b ou t 5 7 0 w h ile th a t of th e S u n is 800. T hu s,w h ile stu d y in g th e o rb its o f th e M o on , th e S u n a n d th ep la n ets, at d i erent p laces of th e E a rth , co rrectio n s for


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GENERAL ARTICLE

Figure 2. Coordinates of a

star X for different observ-

ers O1, O

2, O

3.

Z: Zenith of O1

H: At horizon of O1

(north

point/south point)

N: zenith of O2and nadir of

O3.

th e geo centric p arallax h av e to b e em p loy ed fo r accu -racy.

T h e In d ian astron o m ers co n ceiv ed o f th e c elestial sp h erean d an a ly sed it in d etail. R o tatin g sp h eres w ere co n -stru cted as co n crete m o d els for th e celestia l sp h ere. T h eS an sk rit term s for th e celestia l sp h ere a re bhagola, i.e.,sp h ere o f th e sta rs [bh a : sta r o r p la n et; gola : sp h ere],an d khagola, i.e., h ollow sp h ere o r sp h ere o f th e sk y orcelestial sp h ere [kh a : h ollow , sk y, h eav en ]. T h e termbhagola w a s u sed fo r th e celestia l sp h ere cen tred a t th eE arth 's centre,w h ilekhagolad en oted th e sp h ere c en tredat th e ob server. A ryab h at.a b ega n th e section \G o la"w ith a b rief d escrip tio n o f th e bhagola a n d th e khagolaan d u sed th em to d em on strate th e m o tion of th e ce-lestial b o d ies { th e bhagola w a s u sed fo r d escrib in g th e


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GENERAL ARTICLEFigure 3. Bhagola.

O: Observer

C: Centre of the Earth as

well as Bhagola

CO: radius of the Earth

__: Visible portion of

Bhagola

- - - : Invisible portion of

Bhagola.

At a given place,

the picture of the

celestial sphere

changes slowly

but continuously

with time. The

Earth rotates from

west to east

around the linejoining the

geographical north

and south poles.

m o tio n of th e S u n , th e M o on , a n d th e p la n ets in th eirorb its; th e khagola for th e a p p a ren t d a ily m o tio n o f th eh eav en ly b o d ies d u e to E a rth 's rota tio n ([1 ], p 1 13 ). InG ola 15, A ry ab h a t.a ex p la in ed : \ O n e h alf of th ebhagola,d im in ish ed b y th e E a rth 's ra d iu s, is v isib le fro m a lev elsu rface. T h e (v iew of) oth er h alf, in creased b y th eE a rth 's rad iu s, is cu t o b y th e E a rth ." (see F igure 3 ).T h e focu s on bhagola for th e S u n , th e M o on , an d th ep la n ets, is p erh a p s a re ectio n o f th e co n cern fo r a ccu -

rac y (recall g eo cen tric p a ralla x ). F or sim p licity, w e sh a lln ot d istin g u ish b etw een th e tw o sp h eres.

A x is o f R o ta tio n a n d C e le stia l P o le s

A t a g iv en p lace, th e p ictu re o f th e ce lestia lsp h ere ch an g-es slow ly b u t continu ou sly w ith tim e. T h e E arth rotatesfrom w est to ea st a rou n d th e lin e jo in in g th e geo g rap h i-ca l n o rth a n d sou th p o les. It is d u e to th is rotatio n th atan ob serve r o n E arth sees th e sy stem o f \ x ed sta rs" (seeB o x 2 for clari ca tio n of th e term \ x ed " ) a s revo lv in gfrom east to w est arou n d a lin e th rou g h th e ob serverp ara llel to th e E a rth 's a x is o f rotatio n (F igu re 4). It isas th o u g h th e celestia l sp h ere is a rota tin g h o llow glob ew ith ea ch of th e x ed stars rm ly stu ck o n its in n er su r-fa ce. S in ce th e rad iu s of E a rth is n eg ligib le co m p a red to


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GENERAL ARTICLE

To an observer on Earth, the relative positions of the stars (excluding the Sun) a p p e a r

to be xed; i.e., the angle subtended (at an observer) by two such stars does not seem tochange with time. Hence, the stars (other than the Sun) are referred to as \xed stars"to distinguish them from other star-like celestial ob jects (e.g., planets). Ancient Indianastronomers used the term n aks.a tra for a xed star and ta ra g r a h a for a planet.

We add a caveat regarding the adj ective \xed" . In reality, the stars t ravel in space indierent directions at dierent velocities. However, all stars (other than the Sun) areseveral \light years" away from the Earth. Due to the enormous distances, the relativeangular displacements between the stars (excluding the Sun) become too minute to benoticed by the unaided human eye. Only around 200 stars are known to have an angularmotion (relative to the rest) at a rate exceeding 1 second (i.e., 1

3 6 0 0 degree) per year!

But such tiny shifts in relative positions can no longer be ignored when one considershuge time intervals. For instance, the celebrated \Saptars. i Man. d. ala" now appears in

the shape of a question mark (?) or a hook (see [2], p 20). An observer is unlikely todiscern any change in this shape during a lifetime; but after, say, a million years, theshape would become considerably dierent from what it is now!

B o x 2 . T h e F ix e d S ta rs


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GENERAL ARTICLE

The Pole Star is a visible star whose direction from a terrestrial observer is so close tothat of the celestial north pole that it appears to play the role of the pole itself. To anobserver in the northern hemisphere, other \xed stars" appear to revolve around the\Pole Star" while this distinguished star appears to remain xed in the same positioneven after a long t ime. In Indian astronomy, the Pole S tar was called d h r u v a n a k s .atra ordh ru vata ra (d h r u v a : xed, rm, immovable, unchangeable, c onstant, lasting, permanent,eternal). At present, Polaris, a moderately bright star belonging to the Ursa Minorconstellation, is called the Pole Star. In reality, Polaris is now about 10 away from thecelestial north pole. There is presently no analogous star near the celestial south polebright enough to catch the eye of an observer in the southern hemisphere.

No individual star remains the \Pole Star" for ever. For, due to a phenomenon called\precession", the celestial north pole executes a slow circular orbit around a certain pointon the celestial sphere { it takes about 25,800 years to complete an orbit. Consequently,

dierent epochs witness dierent Pole Stars. At present, t he pole is approaching Polaris { the current \Pole Star" { and will continue to do so till about the year 2100 CE whenthey will be at their nearest but still 27:50 apart. There will be long periods when thecelestial north pole will be unmarked by any bright star in its vicinity. Around 1 4000CE, Vega (Alpha Lyrae), the brightest star of the northern hemisphere, will have thedistiction of being the Pole Star just as it was around 12000 BCE. However, it will thenbe twice as far from the true pole as our present Pole Star is today. Around 3000 BCE

{ at the time of the early astronomers of the Egyptian, Mesopotamian and Indus valleycivilisations { the star Alpha Draconis was the Pole Star.

From ancient times, navigators and travellers have relied on the Pole Star for determiningdirections during journeys at night. The concept has inspired a literary metaphor for therm, ever-present, shining, eternal Guide whose constant unfailing luminous radiation

alone can be trusted by man for orienting his course through the darkness of his perilousvoyage across the sombre ocean of life.

B o x 3 . T h e P o le S t a r

th e d ista n ce of a n y star from E a rth , th e ax is o f rota tio nof th e celestia l sp h ere a n d th e a x is of E a rth 's rota tio nm ay b e regard ed as co in cid en t.

T h e ax is of rotation , w h en ex ten d ed b oth w ay s, m eetsth e celestia l sp h ere in tw o d iam etrica lly op p o site p oin ts.O f th e tw o p oin ts, th e p o in t in th e d irectio n o f th eE arth 's g eo gra p h ica l n orth p ole is ca lled th e celestial

n orth pole (uttara dhruva); th e oth er is called celestialsouth pole (daks.in.a d hruva). F o r p ractica l p u rp ose s, th e\P o le S tar" (see B o x 3) is id en ti ed w ith th e celestia ln orth p ole an d u sed to lo cate th e ap p rox im ate n orthd irectio n .


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GENERAL ARTICLE

Figure 5. Some great circles

on the Khagola (celestial

sphere); The portions of the

great circles lying below the

horizon (and hence invis-

ible to the observer) are in-

dicated by dotted lines.

O: Observer

Z: Zenith

n: Nadir

CNP: Celestial north pole

CSP: Celestial south pole

N: North point

S: South point

E: East point

W: West point.

S o m e G re a t C irc le s o f th e C e lestia l S p h ere

A great circle o n a sp h ere is a circle o n its su rfa ce (in -

n er or o u ter) o f m a x im u m p o ssib le rad iu s { th e rad iu sof th e sp h ere. It is form ed b y th e in tersection o f th esp h ere w ith a p la n e p a ssin g th rou gh th e cen tre o f th esp h ere. G iven tw o d istin ct p o in ts o n th e sp h ere w h ichare n ot d iam etrica lly op p osite, th ere is a u n iq u e grea tcircle p a ssin g th rou gh th e tw o p o in ts. A n y tw o d istin ctgrea t circles b isect ea ch oth er. T h e lin e, th rou g h th ecen tre o f a g reat circle, a n d p erp en d icu lar to th e p lan eco n tain in g th e g rea t circle, in tersects th e sp h ere in tw op o in ts { th ese p o in ts a re ca lled poles of th e great circle.

A circle o n th e sp h ere w h ich is n o t a grea t circle is ca lleda sm all circle.

In a p lan e, th e p osition of a p oint is referred to in term sof tw o x ed lin es a n d a p oin t ca lled o rig in . A n a log o u sly,in a stron o m y, th e co -ord in a te o f a h ea ve n ly b o d y o n th ecelestia l sp h ere is d escrib ed w ith referen ce to x ed grea tcircles an d an origin . W e n ow d e n e a few im p ortan tgrea t circles o n th e (in n er su rfa ce of) celestia l sp h ere(F igure 5 ).


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GENERAL ARTICLEFigure 6. Dip of the horizon.

C: Centre of the Earth

E: Exact location of the

observers eye.

2 To avoid a possible confusion,

we clarify here that if the dip

can be neglected, it would be

not because of the smallness

of the Earths radius relative to

stellar distances but rather the

largeness of the Earth s radius

relative to the height of the ob-

server. While the Earths radius

can be regarded as negligible

for mapping the stars, the dip

of the visible horizon has noth-

ing to do with the distances of

the stars. It depends solely on

the height of the observer s eye

from the Earth s surface and the

Earths radius.

The great circle of the

celestial sphere

whose plane is

perpendicular to theaxis of the celestial

sphere is called the

celestial equator.

T h e g rea t circle o f th e celestial sp h ere in w h ich th e (h ori-zo n tal) tan gen t p la n e to th e E a rth 's su rfa ce, at th e p osi-tio n o f th e ob server, m eets th e celestial sp h ere, is ca lledth e celestial ho rizon (ks.itija). A s th e ob server's eye is atsom e d istan ce a b ov e th e sea -lev el, th e \ v isib le h orizo n "a p p ea rs so m ew h a t b elow th e celestia l h o rizo n a s a sm a llcircle (F igure 6). H ow ever, to m in im ise tech n icality, w esh all ig n ore th e d i eren ce2 in th is a rticle a n d reg a rd th ecelestia l h o rizo n itself to b e th e v isib le h o rizo n a n d ca llit h orizo n . T h e ob serv er sees on ly th e p ortion of th e

celestia l sp h ere ab ov e th e h o rizo n . In F igu res 3 a n d 5 ,th e in v isib le p ortion of th e celestia l sp h ere a n d its circlesa re in d ica ted b y d o tted lin es. T h e risin g a n d settin g o fth e sta rs a n d p la n ets tak e p la ce o n th e h o rizo n tow a rd sth e ea st a n d w est resp ectiv ely.

T h e g reat circle o f th e celestial sp h ere w h o se p lan e isp erp en d icu la r to th e ax is o f th e ce lestial sp h ere (i.e., toth e lin e jo in in g th e ce lestia l p oles) is ca lled th e celes-tial equ ator (vis.uvadvr.tta o r n ad.ivr.tta). T h u s, as w ithth e a x es o f rotatio n (F igure 4 ), th e p lan e of th e celestia leq u a tor is p a rallel to th e p lan e o f th e E a rth 's eq u a tor(niraks.a) { th e tw o p la n es p ractica lly co in cid e. H alf th ecelestia l eq u ator is a b ov e th e celestia l h orizo n (b o th b e-in g great circles). T h e celestial p oles are th e \p oles" of


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GENERAL ARTICLE

th is \great circle". T hu s each xed star traverses a cir-cu la r orb it p a rallel to th e celestia l eq u a tor m a in tain in g

a co n sta n t a n g u la r d istan ce from th e celestial p o les. W em en tio n h ere th at th e celestial eq u a tor is th e fu n d am en -tal g reat circle u sed fo r d e n in g m ea su res o f tim e.

T h e zen ith is th e p oint on th e celestial sp h ere verticallyov erh ea d . T h u s th e lin e join in g th e zen ith a n d th e o b -serv er is a v ertica l lin e p erp en d icu lar to th e p lan e of th eh orizo n . A n y g reat circle th rou g h th e zen ith is ca lleda vertical great circle. T h e vertical great circle th rou ghth e celestia l p oles is ca lled th e observer's m eridian orcelestial m eridian (yam yottara m an.d.ala).

T h e p o in t b elow th e celestia l n orth p ole w h ere th e o b -serve r's m erid ia n m eets th e h o rizo n is c alled th e N o rthp oin t. T h e p o in t on th e h orizon op p o site th e N o rthp o in t is ca lled th e S o u th p o in t. T h e m erid ian is thu sth e ve rtica l circle th rou g h th e N o rth an d S o u th p o in ts.T h e p o in ts of in tersectio n o f th e celestia l eq u ato r a n dth e h orizon are th e E ast an d W est p oin ts { th e E astp o in t lies to th e righ t of th e N o rth -S o u th lin e fo r a no b server fa cin g N o rth ; th e W est p o in t lies to th e left.T h ese fo u r p oin ts a re ca lled ca rd in al p oin ts. T h e v er-

tica l grea t circle th rou gh th e E ast an d W est p o in ts iscalled th e prim e vertical (sam am an.d.ala). T h e g reat cir-cle th rou gh th e celestial p o les an d th e E ast an d W estp o in ts is ca lled th e six o'clock circle (u n m a n .d.ala).

A ryab h a t.a d e n ed th e p rim e v ertica l, m erid ia n a n d h o ri-zo n in G o la 1 8, th e six o'clo ck circle in G ola 1 9 ; w h ileth e eq u a tor is im p licit in G o la 1. T h e h o rizo n is p ic-tu red a s an en circlin g cord a rou n d th e p rim e vertica la n d th e m erid ian (as if fa sten in g th em ). T h e su g g estiv eim a ge ry is in terestin g . A m id st th e a p p a ren t ro tatio n of

th e celestia l sp h ere, th e h orizo n rem a in s rm ly x ed inth e m id d le o f th e sp h ere a t rig h t a n gles to th e p rim ev ertica l an d th e m erid ia n .


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GENERAL ARTICLE

Figure 7. Altitude of CNP =

Geographical Latitutde of

O.

GNP: Geographical North

Pole

GSP: Geographical South

Pole

C: Centre of Earth

CNP: Celestial North Pole

N: North point

S: South point

O: Observer

Z: Zenith.

3 Thus the latitude of a place in

the northern hemisphere would

be approximately the altitude

of the Pole Star at that place.

For an observer at Bangalore

(12.58N, 77.38E) or Chennai

(13.04N, 80.17E), the Pole Star

appears about 130 above thenorthern horizon, while for an

observer in Kolkata (22.34N,

88.24E), the elevation of the

Pole Star is about 22 1/2 0.

G e o g ra p h ic a l L a titu d e a n d A ltitu d e o f C e le stia l

N o rth P o le

T h e a n gu lar d ista n ce b etw een th e celestia l n orth p o lea n d th e N o rth p o in t (i.e., th e elev a tio n o f th e celestia l

n o rth p ole fro m th e h o rizo n ) is p recisely th e g eo g rap h i-ca l la titu d e o f th e p la ce o f th e o b serve r (see F igu re 7 ).3

It is a lso th e d ep ressio n o f th e celestia l so u th p o le fromth e h o rizo n a s a lso th e a n gle b etw een th e celestial eq u a -tor a n d th e p rim e m erid ia n . T h e eq u a lity o f th e la titu d e

w ith th e a ltitu d e o f th e celestia l n o rth p o le (o r th e d e-p ressio n o f th e ce lestia l so u th p o le) is im p licit in G o la1 9 . In p a rticu la r, fo r a n o b serve r o n th e E a rth 's eq u a tor,th e celestia l n o rth p o le co in cid es w ith th e N o rth p oin t,th e celestial sou th p o le w ith th e S ou th p oin t an d th e

celestia l eq u a tor w ith th e p rim e v ertica l; w h ile fo r a no b serv er in th e N o rth P o le, th e celestia l n o rth p o le is a tth e z en ith a n d th e c elestia l eq u a to r b eco m es th e h o rizo n !

A t th e eq u a tor, th e six o 'clo ck circle co in cid es w ith th ecelestia l h orizo n an d h en ce u n m a n .d.ala w as som etim esreferred to as th e \h o rizo n a t L a _n k a" ([8], p 69).


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GENERAL ARTICLE

4 We are restricting ourselves to

theoretical explanations based

essentially on the geometrical

aspects of the celestial sphere.But, in reality, one has to take

into account the effects of the

Earths atmosphere which con-

siderably modify our simplified

picture. For instance, due to

refraction, a star can appear

higher in the sky than it s actual

geometric position. The effect

o f re f ra c t ion i s mos t p ro-

nounced at the horizon. As a

result, the actual duration of the

day is several minutes longer

than the theoretical duration.

At a place on the equator,

where day and night should be

of equal duration from geomet-

ric consideration, the actual

durations are 12 hours 5 min-

utes and 11 hours 55 minutes

respectively. In this article we

shall not discuss the visibility

corrections employed in ancient

India.

C h o ic e o f L a _n k a

D u e to E a rth 's rota tio n arou n d its ax is, th e o b serve r sees

ea ch ob ject on th e celestial sp h ere a s revo lv in g aro u n dth e a x is o f th e celestia l sp h ere. S o m e o f th ese o b jects(lik e th e S u n , th e M o o n a n d th e p la n ets) a lso h av e a d -d itio n al m o tio n s d u e to o th er p h en om en a . H ow ev er, fo ra \ xe d sta r", th e on ly m otion o n th e celestia l sp h ere isth e o n e ca u sed b y E a rth 's ro tatio n .

A s m en tion ed earlier, a x ed star is seen to d escrib ea circle o n th e celestial sp h ere parallel to th e celestia leq u a tor. T h erefo re, an o b server a t th e E arth 's eq u a tor(w h ere th e celestial eq u a tor co in cid es w ith th e p rim e

ve rtica l an d th e a x is o f rotation lies in th e p lan e of th eh orizo n ) sees th e x ed star as trav ellin g from east tow est in a vertical circle (p a rallel to th e p rim e vertica lan d p erp en d icu lar to th e h o rizo n ). A t a n in term ed ia telatitu d e b etw een th e eq u ato r an d a p o le, th e orb it ofan y x ed sta r is o b liq u e an d th u s th e fa ct th at th e staris m ov in g from ea st to w est co u ld b e less o b v iou s to acasu a l ob server. In th e ex trem e case of th e p oles, th esta rs a re seen to trav erse in h o rizo n tal circles (p a rallelto th e h o rizo n ).

It is o n ly a t th e eq u a tor th a t all x ed sta rs rise a n d set.T h eir o rb its cross th e h orizon a t righ t an g les an d areb isected by it, so th a t every x ed star rem ain s a b oveth e h orizo n fo r 1 2 h o u rs d aily (if w e n eg lect refraction4

an d oth er su b tleties).

A n ob server in th e n o rth ern h em isp h ere sees th at n o rth -w a rd from th e celestia l eq u a tor, th e d aily orb its o f th esta rs com e u p m o re a n d m o re ab ov e th e h o rizo n till th eyare en tirely a b ov e it; sou thw a rd from th e celestia l eq u a -tor, th ey a re d ep ressed m ore an d m ore u n til th ey d isap -p ea r co m p letely from th e ob serv er's v iew . T h u s w h ilesom e of th e stars clo se to th e c elestia l n o rth p o le rem a inv isib le all th e tim e, som e o f th e stars close to th e celes-

tia l sou th p ole h av e o rb its co m p letely b elow th e h o rizo n


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GENERAL ARTICLE

Figure 8. Star-orbits at 3

locations.

N: North point

S: South point

E: East point

W: West point

Z: Zenith

__ : Horizon

___ : Portions of star-orbits

above horizon

---- : Portions of star-orbits

below horizon__> : Direction of an orbit.

an d rem a in in v isib le a ll th e tim e. F or in sta n ce, a t ap la ce o f latitu d e 2 3N , th ere is a circu la r a rea arou n d th e

celestia l n o rth p ole o f ra d iu s 2 30

co n tain in g sta rs th a tn ev er set a n d a sim ila r a rea aro u n d th e celestia l sou thp o le co n tain in g sta rs th at n eve r com e in to v iew . T h erem a in in g b a n d of th e sk y co n tain s th e stars th a t risean d set. T h is b a n d is sy m m etrica l w ith resp ect to th ecelestia l eq u a tor an d ex ten d s 670 on eith er sid e o f it.(S ee F igure 8 .) T h e situ atio n is e x a ctly reversed in th esou th ern h em isp h ere.

A s d escrib ed b y A rya b h at.a in G ola 16 , an ob server atth e N orth P ole sees o n e h a lf o f th e celestia l sp h ere a s

rev olv in g from left to righ t (i.e., clo ck w ise) w h ile a n o b -server a t th e S o u th P o le sees th e o th er h alf as revo lv in gfrom rig h t to left (i.e., a n ticlo ck w ise).

T h u s th e eq u a tor is th e on ly p art of th e E arth w h ereeach a n d ev ery p ortion of th e en tire sky is b rou gh t in tov iew b y th e d aily rotatio n fo r so m e tim e d u rin g 2 4 h o u rs.A t an y oth er p lace, som e p ortion of th e sky rem ain sp erm a n en tly h id d en from th e ob serve r. T h e p rop ortio nof th e in v isib le p o rtio n in crea ses a s o n e a p p roa ch es th ep o les; a t ea ch p o le, o n e h alf o f th e sky rem a in s v isib le

all th e tim e w h ile th e oth er h a lf rem a in s p erm an en tlyin v isib le.

T h e v isib ility o f th e w h o le sky an d th e v ertica l n a tu reof th e stella r o rb its m u st h av e b een a m o n g th e reaso n s


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GENERAL ARTICLE

In Gola 14, Arybhata

conveyed a rough

idea of the distance

of Ujjayini from

Lanka (i.e., the

latitude of Ujjayini)

by mentioning that it

is one-sixteenth of

the Earths

circumference.

for A ry ab h a t.a sp ecify in g a p o in t on th e ge og rap h ica leq u a tor (L a _n ka) in th e q u oted verse (G ola 9). F igu re

8 sh ow s h ow th e orb its o f th e stars ap p ear b efore ano b serv er a t (i) L a _n k a (on eq u a tor), (ii) U jjay in (23.09 N )an d (iii) U ttara M eru (N orth P ole).

P rim e M er id ia n in In d ia n A stro n o m y

T h e v isu al im p act of ve rtical o rb its w ou ld b e sam e atan y p oin t o n E a rth 's eq u ato r. H o w ev er L a _n k a w a s situ -a ted o n a sp ecial g eo g rap h ica l m erid ia n { th e m erid ianth rou g h U jjay in. T h is m erid ian w a s ch osen b y In d iana stro n o m ers as th e \ P rim e M erid ia n " (ju st as th e m erid -ian th rou gh G reenw ich is n ow tak en to b e th e p rim em erid ian ). T h e accep tan ce of on e p rim e m erid ian by allIn d ia n a stro n o m ers re ects th e cu ltu ral u n ity o f an cien tIn d ia . U jjay in w a s a grea t cen tre o f lea rn in g , esp eciallyin astron o m y. T h e ac a d em ic, cu ltu ral a n d p o litica l em -in en ce o f th e city, as w ell a s its cen tra l lo ca tion , m u sth av e in u en ced th e ch o ice of th e p rim e m erid ia n .

It w a s th en n atu ral for A rya b h at.a to select L a _n ka torep resen t an eq u ato ria l p o in t. In fact, L a _n k a u sed tob e a sta n d a rd referen ce p o in t in In d ia n a stron o m y. F orin stan ce, in A ryab h at

.ya (G tika 4), th e co m m en cem ent

o f th e cu rren t \yuga" is recko n ed from su n rise a t L a _n k a .

T h e d eterm in a tio n of ge og rap h ica l latitu d e a n d lo n g i-tu d e of a p lace on E arth from ob servation s of celes-tia l b o d ies is o n e of th e m ost im p o rtan t ap p lica tio n s ofsp h erica l a stron o m y. In G ola 1 4 , A ryab h at.a conveyed a rou g h id ea of th e d istan ce of U jjay in from L a _n ka (i.e.,th e la titu d e of U jjay in) b y m en tion in g th a t it is on e-six teen th of th e E arth 's circu m feren ce. A s th e read ercan see, 1

1 6(=

2 21

2

36 0) is a convenient fractio n fairly close

to th e ac tu al ratio of th e d ista n ce of a p lace a t 2 3.09 Nfrom th e eq u ato r to th e len g th o f th e circu m feren ce.

G o la -Y a n tra

G ola-Y an tra d en otes a sp h erica l ap p ara tu s rep resen tin g


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GENERAL ARTICLE

Excerpts from a

passage of the

astronomer Lalla:How can the ignorant,

who know neither

mathematics nor the

fundamental principles

of the celestial sphere,

ever detect the motion

of the planets? An

astronomer without

the knowledge of this

sphere is like adisputant without the

knowledge of

grammar, a sacrificer

without the light of the

Vedas and a

physician without the

experience of an

active practitioner.

th ebhagola; it is d esig n ed to rotate u n iform ly a t th e ra teof on e rotatio n p er d ay , i.e., at th e rate of E arth 's ro ta-

tion . In a sin gle verse (G ola 22),A rya b h at.a p rescrib edth e co n stru ctio n of th is rotatin g sp h ere w ith a cry p tic

b rev ity ch a rac teristic o f A rya b h a t.y a :

k a s.t.h am aya _m sa m a v r.tta _m sam antatah . sam -ag u ru _m la g h u _m g o la m p a rad a taila jala ista _m b h ram ay et sva d h iy a cak alasam am

(M ake) a w ooden gola (globe) w hich is per-

fectly sphericala n d u n iform ly d en se alla rou n d(but is) light in w eigh t. U sin g m e rcu ry, o ilan d w ater, an d ap plyin g on e's o w n in tellect,(m ake the globe) rotate (at the required rate)to keep pace w ith tim e.

T h is em p h a sis o n th e creative u se of th e in tellect fo rw ork in g o u t th e d etails from terse a p h o rism ic in d ica -tio n s is a recu rren t fea tu re in th e o rig in a l w o rks o f th ean cien t In d ian M a sters. S o m etim es th e com m en tators

p rov id e m o re elab ora te ex p o sitio n s. In h is co m m en taryon A ryab h at.y a , th e 1 2th cen tu ry a stron o m er S u rya d evad escrib ed in d etail h ow m ercu ry, oil an d w ater ca n b eu sed \in tellig en tly " to rotate th e sp h ere at th e req u iredrate (see [1], p 12 9 -1 30 or [5], p 86 ).

Im p o rta n ce o f th e G o la

In d ia n a stro n o m ers laid sp ecia l em p h a sis o n th e im p o r-tan ce o f th e sp h ere G o la (referrin g to b o th th e co n cep tan d th e p hy sica l m o d el o f th e c elestia l sp h ere). T o q u o te(tran sla tion s of) ex cerp ts from a lon g p a ssag e of th e a s-tron o m er L a lla (arou n d 8th cen tu ry C E ) on th e in d is-p en sa b ility an d p rofu n d ity o f th is stu d y ([5 ], p 74 ): \ N oastron om y trea tise is co m p lete w ith ou t a section o n th esp h ere o f th e u n iv erse. T h e a stro n o m ers stress th a t th is


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GENERAL ARTICLE

Address for Correspondence

Amartya Kumar Dutta

Stat-Math Unit; Indian

Statistical Institute;

203, B.T. Road; Kolkata

700108.

Email: [email protected]

Suggested Reading

[1] K S Shukla and K V Sarma, A r y a b h a t.ya o fA r y a b h a t.a , Indian

National Science Academy, New Delhi , 1976.

[2] S Balachandra Rao,Indian Astronomy An Introduction, Universities

Press, Hyderabad, 2000.

[3] P R Ray and S N Sen (ed),The Cultural Heritage of India Vol VI : Science

and Technology, The Ramakrishna Mission Institute of Culture,

Calcutta, 1986; reprinted, 2002.

[4] D M Bose, S N Sen and B V Subbarayappa (ed), A Concise History of

Science in India, Indian National Science Academy, New Delhi , 1971;

reprinted, 1989.

[5] B V Subbarayappa and K V Sarma (ed),Indian Astronomy; A Source-

Book, Nehru Centre, Bombay, 1985.

[6] W M Smart,Text-Book on Spherical Astronomy, Cambridge University

Press, 1956.

[7] D Chattopadhyaya (ed) , Studies in the History of Science in India,

Vol.II, Editorial Enterprise, New Delhi , 1982.

[8] W E Clark (ed), The A ryabha t.y a o fA r y a b h a t.a , University of

Chicago, Illinois, 1930.

[9] M Monier Williams, Sanskrit-English Dictionary, Clarendon Press,

Oxford (1899); reprinted Munshiram Manoharlal, New Delhi , 2002.

sp h ere is sp eci ca lly n eed ed fo r m a th em atica l c om p u -tation s an d th at th ose w h o w ish to stu d y th e p lan etsm u st b e exp erts o n th e sp h ere. T h e m ean m otion s ofth e p lan ets a n d o th er celestia l o b jects a re clea rly p er-cep tib le o n th is sp h ere, esp ecia lly to th e sch olar w h o h asm a stered th e scien ce o f th eir g eo m etrica l rep resenta tio n .H ow can th e ign oran t, w h o k n ow n eith er m ath em aticsn o r th e fu n d a m en tal p rin cip les o f th e celestial sp h ere,eve r d etect th e m o tion of th e p la n ets? A n a stron o m erw ith ou t th e k n ow led ge of th is sp h ere is like a d isp u tan tw ith ou t th e k n ow led ge of gram m ar, a sacri cer w ith -ou t th e lig h t of th e V ed as an d a p h y sicia n w ith ou t th eex p erien ce o f a n a ctive p ractitio n er. H e w h o ac q u ires

a co m p reh en sive k n ow led g e of th e celestial sp h ere co n -tain in g th e S u n etc. sees, in fron t o f h is ey es a s it w ere,th e w h o le u n iv erse; h e b eh old s it d ec orated b ea u tifu llyw ith a w id e va riety of ex q u isite p h en om en a. H e getssp iritu a lly en rich ed a n d attain s m oks.a (L ib era tio n ); h eb ecom es fam ou s."
mailto:[email protected]

aryabhata and axial rotation of earth-1--p51-68

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