lec 01 - introduction & brief history of astronomy - mohan apte
DESCRIPTION
By Prof. ApteTRANSCRIPT
PROMINENT ANCIENT INDIAN ASTRONOMERS
1> Acharya Lagadha Vedang Jyotish 1500 BC
2> Aryabhat Aryabhatiya 500 AD
3> Varahamihir Panchsiddhantika 505 AD
4> Bhaskara 1 Mahabhaskriya 600 AD
5> Lallacharya Shishadhivridhidatantra 638 AD
6> Brahmagupta Brahmasphutsiddhanta 628 AD
7> Vateshwar Vateshwarsiddhanta 880 AD
8> Munjala Laghumanasam 932 AD
9> Aryabhat 2 Mahasiddhanta 950 AD
10> Bhaskaracharya 2 Siddhantashiromani 1150 AD
11> Ganesh Daivadna Grahalaghav 1520 AD
PaaOilaSa raomak vaisaYz saaOr pOtamahastu pMcaisaQdaMta:
AaQauinak saUya-isaQdaMt i#a`stabd 1000
PROMINANT ASTRONOMERS FROM KERALA
1> Madhavan of Sangamagram (1340-1425) Karanapaddhati(1375)
2> Narayana Pandit (1340-1400) Ganita Kaumudi
3> Parameshvara (1370-1460) Drigganita
4> Nilakantha Somayaji (1444-1544) Tantrasangraha
5> Jyesthadeva (1500-1610) Yuktibhasa
6> Sankara Variar (1500-1560) Kriya-kramakari
7> Sankar Varman (1774-1839) Sadratnamala
CONCEPT OF YUGAS
Kali 432000 Years
Dwapar 864000 Years
Treta 129600 Years
Krita 1728000 Years
Mahayuga = Kali + Dwapar + Treta + Krita
= 4320000 Years
Manavantar = 71 Mahayugas + 1 Krita
= 71 x 4320000 + 1728000
= 308448000 Years
Kalpa = 14 Manvantaras + 1 Krita
= 14 x 308448000 + 1728000
= 4320000000 Years
Kali Yuga started on 17/18 Feb 3102 BC
CONCEPT OF NAKSHTRAS
Period of revolution of Moon = 27 Days
Total angle covered in this period = 360 degrees
Speed of revolution per day = 360/27 = 13.33 deg/day
One Nakshtra = 13.33 degrees
There are 27 Nakshtras
Moon crosses one Nakshtra per day
Names of 27 Nakshtras
1> Ashvini 10> Magha 19> Mula
2> Bharani 11> Purva Phalguni 20> Purvashadha
3> Krittaka 12> Uttara Phalguni 21> Uttarashadha
4> Rohini 13> Hasta 22> Shravana
5> Mrigashirsha 14> Chitra 23> Dhanishta
6> Aardra 15> Swati 24> Shatataraka
7> Punarvasu 16> Vishakha 25> Purvabhadrapada
8> Pushya 17> Anuradha 26> Uttarabhadrapada
9> Ashlesha 18> Jyeshta 27> Revati
tO%trIya saMihtotIla maihnyaaMcaI naavao
maQau maaQava vasaMtSau@r Sauica g`aIYmanaBasa\ naBasya vaYaa-[Ya }ja- Sardsahs\a\ sahsya homaMttpsa\ tpsya iSaiSar
BaartIya maihnao1Å caO~ 7Å AiSvana2Å vaOSaaK 8Å kait-k3Å jyaoYz 9Å maaga-SaIYa-4Å AaYaaZ 10Å PaaOYa5Å EaavaNa 11Å maaGa6Å Baad`pd 12Å falgauna
caMd` paOiNa-maolaa jyaa naxa~acyaa javaL Asatao %yaa naxa~acao naava maihnyaalaa doNyaat Aalao.PaaOiNa-maolaa ra~Bar to to naxa~ idsato.
caMd` naxa~maasa duNaa itqaI gaNaa tIna ]Naa naxa~ jaaNaa
Aa.sa.kbaro
SIGNS OF ZODIAC
1> MESH ARIES 14 APRIL
2> VRISHABHA TAURUS 14 MAY
3> MITHUN GEMINI 14 JUNE
4> KARKA CANCER 14 JULY
5> SIMHA LEO 14 AUGUST
6> KANYA VIRGO 14 SEPTEMBER
7> TULA LIBRA 14 OCTOBER
8> VRISHIKA SCORPIUS 14 NOVEMBER
9> DHANU SAGITTARIUS 14 DECEMBER
10> MAKAR CAPRICORNUS 14 JANUARY
11> KUMBHA AQUARIUS 14 FEBRUARY
12> MEENA PICES 14 MARCH
INDIAN WESTERN ENTRY OF SUN
DECLINATION OF THE SUN THROUGHOUT AN YEAR
itqaI
Angular speed of Sun = 1 degree per day
Angular speed of Moon = 13.33 degrees per day
Angular speed of Moon with respect Sun = 13.33 – 1
= 12 deg/day
One Tithi = 12 degrees nearly
Tithi is also angular difference between Sun and Moon
Angular difference in white fortnight = 12 x Tithi
Let Tithi = 4, Angular difference = 4 x 12 = 48 degrees
Angular difference in black fortnight = 180 – 12 x Tithi
Let Tithi = 3, Angular difference = 180 – 12 x 3 = 144 degrees
Tithi also indicates the Phase of the Moon
From the Tithi, time of Moon rise can also be estimated.
For example, in the third Tithi of dark fortnight Moon rise
Will be about 2.5 ( 36/15) hours after Sunset.
In the fifth Tithi of white fortnight Moonrise
Will be about 4 (60/15) after Sunrise.
From the Tithi time of high tide can also be estimated.
itqaI vaRiQd AaiNa xaya • At the time of sunrise the current tirhi is the tithi of the day
• The tithi which exists at the consecutive sunrise is the case of
tithi vriddhi or interval of tithi > 24 hours.
• The tithi which falls between the two consecutive sunrises is the
case of tithi kshya or interval of tithi < 24 hours
Tithi Cycle ( Metonic Cycle)
• 19 tropical years = 19 x 365.2422 = 6939.602 days
• 235 synodic months of Moon = 235 x 29.3 = 6939.689 days
• Hence after 19 years sequence of tithis are repeated
A difference of 1 day after 218.5 years.
INTERCALARY MONTH (AiQak maasa )
• Lunar month is an interval between two successive new moons.
Which is equal to 29.530589 days.
• one lunar year = 29.530589 x 12 = 354.36706 days
• one solar year = 365.25636 days.
• Difference between solar and lunar year = 10.8893 days.
• If the difference is allowed to accumulate, Lunar months
will be out of steps with the seasons. Hence idea of intercalary
month was introduced.
• When the difference adds up to a full lunar month, intercalary
month is added in that particular lunar month.
• Intercalary month occurs after 10.8893 x 3 = 32.6679 months,
or after about 33 months.
• Normal lunar year consists of 12 lunar months of 354 0r 355 days.
In a lunar year in which there is intercalary month will have 13
lunar months of 383 or 384 days.
AiQak maasa
xaya maasa
FORTNIGHTS AND ELONGATION
(pxa AaiNa [naaMtr)• Two Fortnights White (Sau@la) and Dark (kRYNa)• In White fortnight angular distance between Sun and Moon increases.
• In Dark fortnight angular distance between Sun and Moon decreases.
• In White fortnight Moon rises after sunrise and sets after sunset.
• In Dark fortnight Moon rises before sunrise and sets before sunset.
• Elongation is angular difference between longitudes of Sun and Planet/star.
• If longitude of a star or planet is greater than longitude of Sun,
elongation is east.
• If longitude of a star or planet is less than longitude of Sun,
elongation is west.
• If elongation is east star or planet will rise after sunrise.
• If elongation is west star or planet will rise before sunrise.
• Time of rise of a star or planet = elongation/15.
KNOWING THE NAME OF THE MONTH, THE TYPE OF THE FORTNIGHT
AND THE TITHI, ENTIRE SKY IS KNOWN.
Angle between the Sun, Earth and Planet is called Elongation
Longitude of Sun > Longitude of planet : Elongation is west
Elongation = Longitude of Sun – Longitude of planet
If Elongation is West then planet rises before sunrise and sets before
sunset
Longitude of planet > Longitude of Sun : Elongation is east
Elongation = Longitude of planet – Longitude of Sun
If elongation is East then planet rises after sunrise and sets after sunset
ELONGATION
elongation2
]<arayaNa AaiNa dixaNaayana• After Winter Solstice uttarayan starts.
• At present Uttarayan starts on 22 December every year
• After Summer Solstice Dakshinayan starts
• At present Dakshinayan starts on 22 June every year.
• After onset of Uttarayan day length increases night length decreases.
• After onset of Dakshinayan day length decreases night length increases.
• Lengths of day and night are equal on Vernal Eqinox (vasaMtsaMpat) and
Autumnal Equinox (SardsaMpat).longitude
22 Dec
22 JunSummer Solstice
Winter Solstice
Niryan
AYANANSHA
(ica~a)
vaodaMga jyaaoitYayaqaa iSa#aa mayauraNaama\ naagaanaama\ maNayaao yaqaatqaOva sava- Saas~aNaama\ jyaaoitYama\ maUQa-ina itYzit
Aacaaya- lagaQa
EVENT JULIAN GREGORIAN
WINTER SOLSTICE 03 JAN 1752BC 19 DEC 1753 BCFUL MOON DAY 17 JAN 1752 BC 02 JAN 1752VERNAL EQUINOX 04 APL 1752 BC 20 MAR 1752 BCSUMMER SOLSTICE 05 JUL 1752 BC 20 JUN 1752 BC
DIFFERENCE BETWEEN GREGORIAN AND JULIAN 15 OR 16 DAYS
GREGORIAN AND JULIAN CALENDAR DATES OF VEDANGA JYOTISH
WINTER SOLSTICE
NOW 22 DECEMBER
SUMMER SOLSTICE
NOW 22 JUNE
SUN
MOOMOON
3 JANUARY 1752 BC NEW MOON DAY IN DHANISHTA
17 JANUARY 1752 BC FULL MOON DAY IN MAGHA
4 APRIL 1752 BC VERNAL EQUNOX IN BHARANI
5 JULY 1752 BC DAKSHINAYAN IN MID OF ASHLESHA
SUN
SHIFT OF VERNAL EQUINOX
1 : Earth rotates
2 : PI = 3.1416
3 : Sidereal day = 23hrs 56 min 4.1 sec
4 : Diameter of Earth = 12810 km
5 : Earth is round
6 : length of Earth’s shadow = 1392000km
7 : Solution of first order indeterminate equation
8 : Sidereal year = 365.2563
9 : Revolution of Moon = 27.3216 days
11 : Period revolutions of the planets (days)
planet Aryabhat modern
Sun 365.25868 365.25636
Moon 27.32167 27.32166
Mercury 87.96988 87.9693
Venus 224.69814 224.7008
Mars 686.99974 686.9797
Jupiter 4332.2721 4332.5887
Saturn 10766.06465 10759.201
GLIMPSES ARYABHATIYAARYABHAT : BORN IN 476 AD
Sines in terse verse : Roddam Narasingh : Nature 20/27 DEC 2001
maiK BaiK fiK QaiK naiK HaiK =iK hsJa skik ikYga SGaik ikQvaGlaik ikga` h@ya Qaik ikca sga JaSa =va @la Pt f C klaaQa-jyaa:
arc ABC = radius R
Angle AOC = 57deg 17 min 44 sec = 3438 minutes
All reference circles will have, Radius 3438
For example: sin(3.75deg) = sin(225min) = 225/3438 = 0.0654
radian
R
R
R
saPtOto haoroSaa: SanaOScaraVa yaqaak`maM SaIGa`a:SaIGa`k`maat\ catuqaa- BavaMit saUyaao-dyaat\ idnapa:
weekdaysHOURS OF A DAY
(HORA)
1 2 3 4 5 6 7
8 9 10 11 12 13 14
15 16 17 18 19 20 21
22 23 24
SATURN
SUN
SATURN
JUPITER
MARS
SUN
VENUS
MERCURY
MOON
SLOWEST
FASTEST
THE OF THE WEEK
1: caturiQakM Satma\\ AYTgauNama = 104 x 8 = 832
2: vdaYaiYTstqaa sahs~aNaama = 62000
(1) + (2) = 62000 + 832 = 62832
Ayautvdya = 20000
π = 62832/20000 = 3.146 (approximately)\
The most accurate value among those of ancients.
caturiQakM Satma\\ AYTgauNama\ vdaYaiYTstqaa sahs~aNaama\Ayautvdya ivaYkmBasya Aasanna: vaR<apirNa:
VALUE OF π
The gods living at north pole and demons living at south pole
See the Sun for six months when it rises.
The ancestors living on Moon see the Sun for half of the lunar
month i.e. for 15 days.
The people living on Earth see the Sun for half of the day i.e.
for 12 hours.
The meaning of this verse is,
On north and south pole the day and night cycle is of six months.
On Moon the day and night cycle if of 15 days.
On Earth the day and night cycle is of 12 hours.
Day and night cycle
On north and south pole, on Moon and the Earth
rivavaYaaQa-M dovaa: pSyaMit ]idtM rivMa tqaa pota:SaiSamaasaaQa-M iptr: SaiSagaa: kuidnaaQa-M [h manaujaa:
Planet Bhagana Sidereal time of revolution (days)
Aryabhata Modern
Sun 4320000 365.25868 365.25636
Moon 57753336 27.32167 27.32166
Mercury 17937020 87.96988 87.9693
Venus 7022388 224.69814 224.7008
Mars 2296824 686.99974 686.9797
Jupiter 364224 4332.27217 4332.5887
Saturn 146564 10766.06465 10759.201
Ascending node 232226 6794.74951 6793.39108
BHAGANANumber revolutions around the Earth in Mahayuga
SYNODIC TIME = NUMBER OF YEARS IN YUGA/NUMBER OF CONJUNCTIONSNUMBER OF YEARS IN YUGA = 4320000NUMBER OF CONJUNCTIONSIN YUGA = N1 – N2 N= REVOLUTIONSSYNODIC TIME OF SUN AND MOON = 4320000/(57753336 – 4320000)
= 4320000/5343336 = .0808484 X 365.25868= 29.530582 days
Planet Aryabhatiya Modern
Mercury 0.375 0.387
Venus 0.725 0.723
Mars 1.538 1.523
Jupiter 5.16 5.2
Saturn 9.41 9.54
Heliocentric distances of the planets
in terms of Earth to Sun distance
Planet Aryabhatiya(days) Modern(days)
Mercury 115.88 115.975
Venus 583.89 585.92
Mars 779.92 779.93
Jupiter 398.89 398.75
Saturn 378.08 378.00
SYNODIC TIMES OF THE PLANETS
PLANET AVERAGE DAILY SPEED(DEG)
ARYABHAT MODERN
Moon 13.176333 13.178029
Mercury 4.092029 4.0953717
Venus 1.6019858 1.6013164
Sun 0.9856377 0.9861002
Mars 0.524014 0.5234908
Jupiter 0.0830973 0.0833221
Saturn 0.0334383 0.03330
AVERAGE DAILY SPEEDS OF THE PLANETS
Epycycle
MV = Equation of Center ( maMdfla )
At A and C
Sun : 840 min
Moon : 1920 min
At B and D
Sun : 820 min
Moon ; 1900 min
]<ar QaRvalanka-p23
]jaO[ina
ivaYauvavaR<alaMka
dixaNa QaRva (vaDvaanala)
]jaO[ina laMkayaa: tt\ caturMSao samaao<art:]jaO[ina AxaaMSa 22.5 ],<ar (vaastva 23.1)
LANKA AND FOUR CARDINAL CITIES
]dyaao laMkayaama\ sa: Astmaya: saivatu: eva isaQdpuromaQyaa*nao yavakaoTyaaM raomakivaYaya: AQa-ra~: syaat\
four-cities
isaQdpUr
raomak
laMka
yavakaoiT
saUya-ikrNa
saUya-ikrNa
Brahmagupta occupies an important place in the history
of Oriental culture. Brahmagupta taught astronomy to
the Arabs before they came to know Ptolemy’s work,
since, references to the works, ‘Sindhind’ , and ‘Al-
Arkand frequently occur in Arabic literature. These are the translations of Brahmagupta’s works,
‘BrahmaSiddhanta’, and ‘Khandakhadyka’ .
Prof.Sachau, in the translation of ‘ Al-Biruni’s India
BRAHMAGUPTA (BORN IN 598 AD)
BRAHMAGUPTA (BORN IN 598 AD)
First to give the formula for the area of Cyclic Quadrilateral
In trigonometry,
the Brahmagupta interpolation formula
is a special case of the Newton-Stirling interpolation formula,
which calculates the values of sine at different intervals.
The formula was developed by Brahmagupta in 665,
which was later expanded by Newton and Stirling
around a thousand years later to develop
the more general Newton-Stirling interpolation formula.
The Brahmagupta interpolation formula is defined as:
Angle Indian sine First difference Second difference
0 0
15 39 39
30 75 36 -3
45 106 31 -5
60 130 24 -7
75 145 15 -9
90 150 5 -10
150Sine (67) = sine (60) + sine(7) = 130 + {(7/15)(15 + 24)/2 + (7/15)(15 – 24)/2}
= 130 + 8.12 = 138.12
Sine(67) = 138.12/150 = 0.9208 (=9205 actual)
Glimpses of Siddhantshiromani
1 : Idea of infinity : Khahara rashi
2 : Numbers from one to 10^18
3 : PI = 3.1416
4 : Idea of calculus d(sin x) = cos x dx
5 : Khandameru = Pascal trtiangle
6 : Two line proof of so called
Pythagoras hypothesis
7 : General solution of Indeterminate
equation of second order
61x^2 + 1 = Y^2
X=226153980 Y = 1766319049
8 : Circumference of Earth = 40000km
SIDDHANTASHIROMANI
1 : Lilavati
2 : Beejaganit
3 : Ganiytadhyaya
4 : Goladhyaya
BHASKARACHARYA (1114 -1183 AD)
BIRTH AND DATE OF WRITING SIDDHANTASHIROMANI
RASA = 6 GUNA = 3 PURNA = 0 MAHI = 1
BHASKARACHARYA WAS BORN IN SHAKE 1O36
THAT IS 1036 + 78 = 1114 AD
SIDDHANTASHIROMANI WAS WRITTEN AT THE AGE
RASA = 6 GUNA = 3 THAT IS
WHEN HE WAS 36 YEARS OLD
VALUE OF PI
bha = 27 nanda = 9 agni = 3 kha = 0 bana = 5 suryai = 12
PI = 3927/1250 = 3.1416
PI = 22/7
OR
EDUCATION OF BHASKARACHRYA
NAMES OF THE NUMBERS
THE PLACE AND THE TEACHER
BHASKARACHARYA
PERIODIC TIME OF THE PLANETS
Planet Revolutions in Kalpa Periodic Time (years)
Sun 4,320,000,000 1
Moon 57,753,300,000 27.3 days
Mercury 17,936,998,984 87.96 days
Venus 7,022,389,492 224.76 days
Mars 2,296,828,522 1.88
Jupiter 365,226,455 11.86
Saturn 146,567,298 29.47
Moon’s Node 2,323,111,698 18.56
Moon’s Perigee 488,105,858 8.85
TIME UNITES
1 Muhurta = 24/30 hours = 48 minutes
1 Ghatika = ½ Muhurta = 24 minutes
1 Kala = 1/30 Ghatika = 48 seconds
1 Kashtha = 1/30 Kala = 8/5 seconds
1 Nimesh = 1/18 Kashtha = 4/45 seconds
1 Tatpar = 1/30 Nimesh = 2/675 seconds
1 Truti = 1/100 Tatpar = 2/67500 seconds
Aqa kismaMiScat puro AxaaMSaana\ &a%vaa tsmaat\ purat\ ]<art: Anyaismana\ puro &oyaa:tt: toYaama\ AMtraMSaO: purantr yaaojanaO: ca Anaupat:yat\ AMtraSaO: puraMtr yaaojanaaina laByanto tda cak`aMSaO: ikma\ [itflaM BaUPairiGa yaaojanaaina
lunar eclipse
D
D1
L
Length of the Earth’s shadow = L = (2Re x D)/(2Rs – 2Re) = 1392000 km
Re = radius of Earth Rs = radius of Sun D = Earth - Sun distance
L = (Earth’s diameter x Earth-Sun distance)/(Sun’s diameter-Earth’s diameter)
Diameter of Earth’s shadow at Moon’s distance=d = (L -D1)2Re/L = 9000 km
D1 = Earth - Moon distance 2Re = diameter of Earth
caMd`ga`hNa
MADHAV OF SANGAMGRAMAM (1350 – 1435)
Greatest of the Indian medieval Astronomer- Mathematician
POWER SERIES
Madhav Gregory Series
Madhav Newton Series
Series for PI
ga`hlaaGava
• Written by Ganesh Daivadna in 1520 AD.
• Accepted 11 year cycle to reduce number of days from the epoch.
• 11 year cycle consists of only 4016 days.
• Removed the use of trigonometry. Still calculations are correct.
• Simplified the calculations.
• Accepted all over India form almanac preparations.
• Epoch Chaitra Shukla Pratipada Shake 1442 (1520 AD)
• Ganesh wrote more than 15 Astronomical books.
• Most brilliant Astronomer of Medieval India.
jyaacaapkma-rihtM saulaGaup`karma\ ktR-ma\ ga`hp`krNaM sfuTma\ ]Vtaosisma
ganesh2
gaNaoSa gauNaakar
JANTARMANTAR JAIPUR
SAVAI JAISINGH
(1686-1743)
SAMRAT YANTRA
SAMRAT YANTRA
ASTRONOMICAL INSTRUMENTS
Instrument Description Use
Chap One half of the Protractor Altitude of celestial body
Turiya One quarter of Protractor Altitude of celestial body
Nadiwalaya Two fixed circular plates Time and meridian passage
parallel to equator of Sun
and Gnomon
Shanku Vertical Sundial N-S line, Latitude,Time
Yashti Straight staff Height, Altitude, Latitude
Chakra A ring mounted between Declination, Hour Angle
two posts along N -S line
Jayprakash A hollow hemisphere with Coordinates of celestial body
number of inscribed arcs
Ram A huge cylinder with its Altitude, Azimuth
height equal to its radius
Bhitti N – S wall with circular Meridian passage of Sun, stars
marking and Gnomon
Samrat Massive Gnomon with Accurate time
equatorial half circles
marked in degrees and minutes
Ghatika A hollow bowl with hole at bottom Time
shanku yantra copy.jpg
shanku yantra2 copy.jpgNORTH SOUTH DIRECTION N
S
shadow 21 march copy.jpg
MARCHSHADOW21
Sin (longitude ) = sin (declination)/sin(obliquity of ecliptic)
22 JUNE SHADOW
DECLINATION MARKINGS