astronomical surveying h.c. king, history of the telescope

ASTRONOMICAL SURVEYING

H.C. King, History of the Telescope

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UNIT IV ASTRONOMICAL SURVEYING

CONTENTS• Celestial sphere• Astronomical terms and definitions • Motion of sun and stars • Apparent altitude and corrections • Celestial co-ordinate systems • Different time systems • Use of Nautical almanac • Star constellations • Calculations for azimuth of a line

Sivapriya Vijayasimhan

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Shape and Size of EarthShape of the earth is sphere regular figure for simplified calculationEquatorial radius of earth (a) = 6378.388 kmPolar radius of earth (b)= 6356.912 km

Survey of India gives a =6377.3097 km and b=6356.1087 km

Ellipticity factor =

India : 1/300-80

Mean radius of earth is 6367.272km


a

ba 22

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Celestial sphere


The celestial sphere is an imaginary sphere of arbitrarily large radius, concentric with the observer•All objects in the observer's sky can be thought of as projected upon the inside surface of the celestial sphere, as if it were the underside of a dome or a hemispherical screen. •The celestial sphere is a practical tool for spherical astronomy, allowing observers to plot positions of objects in the sky when their distances are unknown or unimportant.

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Astronomical terms and definitionsZeinth (z) : It is a point on the upper portion of celestial sphere immediately above

the overhead of an observerNadir (z’) : It is the intersection of a vertical line through the observer’s station to the

lower portion of the celestial sphereCelestial or Rotational Horizon (Geocentric or true horizon): It is a great circle traced

upon the celestial sphere by that plane which is perpendicular to zeinth-nadir line and which passes through the centre of the earth

Sensible Horizon : It is a circle in which a plane passes through the point of observation and tangential to earth’s surface intersects with celestial sphere. The line of sight of an accurately levelled telescope lies in this plane


Visible Horizon:

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Sivapriya Vijayasimhan Fig 1 Fig 2

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Motion of SUN and STARSunLocated at a distance 93,005 km from earthDia of Sun = 109 Dia of earthMass of Sun = 3,32,000 of earthTemperature of earth = 20 million degrees

Motions: Two apparent motion of earth1. With respect to earth east to west2. With respect to fixed stars in celestial sphere

Motion of sun is along the great circle – ecliptic

Obliquity of Ecliptic – angle between the plane of equator and the ecliptic ( 23027’)Equinoctial Point : Point of intersection of ecliptic with equator. Here declination of sun is zeroVernal Equinox : First point of aeries in which the sun’s declination changes from south to

northAutumnal Equinox : First point of libra in which the sun’s declination changes from north to

southSolstices : Sun’s declination is maximumSummer solstices : north declination is maximum at a pointWinter Solstices : South declination is maximumSivapriya Vijayasimhan

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StarMoon rerates the earth in elliptical orbit(average angle 508’)inclined to the plane of ecliptic,

which is intersected at points called NodesMotions:

1. With respect to earth east to west2. With respect to fixed stars it is from west to east

Moon rotates about its polar axis. New and full occur when sun, earth and moon lie in same vertical plane (not necessarily in same straight line)

Conjunction : in new moon ,moon lies between sun and earth and has same latitude as sunOpposition : In full moon , earth lies between sun and moonWaxing : Illuminated limb increase in size of moon between the time interval of new and full

moon periodWaning : Illuminated limb decrease in size of moon between the time interval of full and new

moon periodLunar Month : Time taken between two successive new moons (27 ⅓ days)Siderial Month : One complete revolution relative to stars (29.5 days)Solar Eclipse: moon passes in front of sun’s discLunar Eclipse: Shadow of earth passing over the moon


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SPHERICAL TRIGONOMETRYProperties of spherical and astronomical triangle are studied1.SphereEvery point on the surface of the sphere is equidistant from a certain point called centre of sphere. Every section of sphere is circle.The section through the centre of the sphere is called a great circle The section not passing through the centre is called a small circleCDEFG – great circleMON – diameter of sphereM and N – poles of great circleR- radius of sphereφ – angle subtended by great circleDE = R φ (if R=1), DE = φ In right angle Δ dO1O,

= =sin dOO1 =sin Md

(Md+ dD =90 deg)

Small arc = arc DE cos dD


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1. Spherical TriangleIt is formed by surface of the sphere by interaction of three arcs of great circleThe angle subtended by the axes at the vertices of the triangle is called spherical anglesABC – spherical triangleAB and AC are great circles with subtended angle BAC = A0

1.1Properties of spherical triangle1. Any angle is less than two right angles or π2. Sum of three angles is less than six right angles or 3 π and greater than two right angles or

π3. Sum of any two sides is greater than the third4. If the sum of any two angles, is equal to two right angles to π, the sum of the angles

opposite them is equal to two right angles or π5. The smaller angle is opposite the smaller side and vice-versaSivapriya Vijayasimhan

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UNIT IV ASTRONOMICAL SURVEYING1.2 Angles and Sides of spherical triangles1. Sin formula : 2. Cosine Formula:

3. = : = : = = : = : =

1.3 Area of spherical triangle Area of spherical triangle = = R- radius of sphere E =A + B +C -


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2. Right angled Spherical TriangleThe relationships of right angled spherical triangle may be obtained from “Napier’s circle of Circular Parts”

The circle is divided into five parts. Consider any part as middle part, the part adjacent to it as adjacent parts, we have Napier’s rule as follows,Sine of middle part = product of tangent of adjacent parts : -A)Sine of middle part = product of cosines of opposite parts : 3. Spherical ExcessSpherical excess of spherical triangle exceeds 180 deg

sseconds r –radius of earth


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Astronomical TrianglesFormed by joining the pole, zenith and any star M on sphere by arcs of great circles

α –altitude of celestial body (M)δ – declination of the celestial body (M)θ – latitude of observerZP = co-latitude of observer = MP = co-declination of the polar distance M= ZM = zenith distance or co – altitude of the body

= Angle Z = MZP = azimuth (A) of the bodyAngle P = ZPM = hour angle (H) of the bodyAngle M = ZMP = parallactic angleIf three sides of the triangles are known, A and H are computed by spherical trigonometry formula


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= =

= (z + c + p)

= =

=


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1.Position of StarsStar of Elongation :When it is at greater distance east or west of meridian. Under this

condition azimuth of star is maximum.Eastern or western elongation of a star is at its greatest distance to west or east of meridian respectively

Star at Prime Vertical: When the observer, at zenith , the angle is right angled in the astronomical triangle. A = 90 deg

Star of Horizon : Its altitude is zero and the zenith distance is equal to 90 degStar at Culmination : When the star crosses an observer meridian the star is said to be culminate or transit. In one revolution, each star crosses a meridian twice.

Upper culmination : altitude is maximumLower culmination : altitude is minimum


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UNIT IV ASTRONOMICAL SURVEYINGCircumpolar Star : Stars which are always above the horizon and which evidently do not set. For an observer it is an circle above the poleDeclination of such stars is always greater than the co-altitude of the place of observationM1 – circumpolar star having circular path A1A2 (path above horizon)

M2 – circumpolar star having circular path B1B2 (path below horizon)


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Co-ordinate SystemsPosition of heavenly body can be located by two-spherical co-ordinates, two angular distances

measured along arcs of two great circles which cut each other at right anglesOne of great circle passing through the heavenly body is called Primary circle of reference, whereas the other is called as Secondary circle of reference

Point M represents heavenly body with reference to a plane OAB O –origin of the co-ordinatesA plane passing through OM shall cut a perpendicular plane OAB in line OBTwo spherical co-ordinates of the point M are angles AOB and BOM at centre O

Systems :1. Altitude and Azimuth , 2.Declination and right ascension system and 3. Declination and hour angle system Sivapriya Vijayasimhan

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1.Altitude and Azimuth SystemAlso called as horizon system which is dependent on the position of the observer Horizon is a plane of reference and the co-ordinates of a heavenly body (azimuth and altitude)

- It is the primary and secondary reference great circle in observer’s meridian- Horizontal and vertical angles are measured - theodolite- The heavenly body can be in eastern or western part of the celestial sphere

Heavenly body in eastern part of celestial sphere. Let Z be the observer’s zenith and P be the celestial pole

Great circle is passing through Z and M is drawn to cut the horizon plane at M’The azimuth (A) angle between the observer’s meridian and the vertical circle through the

body is the first co-ordinateAzimuth is equal to the angle at zenith between the meridian and the vertical circle through

M. The co-ordinate of M is the altitude (α), which is measured above or below the horizon on vertical circle


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Heavenly body in western part of celestial sphere. The concerned angles NOM(azimuth) and MOM’ (altitude)In northern hemisphere, the azimuth is always measured from north to east or westIn southern hemisphere, the azimuth is measured from south to east or west

Zenith Distance = ZM-MM’

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2.Declination and Right Ascension System (Independent equatorial system)Two great circles :

1. Equatorial circle – primary circle2. Declination circle – secondary circle

The first co-ordinate of heavenly body is the right ascensionIt is the angle along the arc of celestial equator measured from the first point of aeries and also the angle between the hour circle through (γ)

Declination (δ) is the angle of the body measured from equator along the arc of declination circle

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3.Declination and Hour angle System (Dependent equatorial system)Two great circles :

1. Horizon – primary circle2. Declination circle – secondary circle

The first co-ordinate of M is the hour angle It is the angle subtended at the pole, between observer's meridian and the declination of the bodyIn northern hemisphere the hour angle is measured from south towards the west up to the declination circle. It varies between 00 to 3600 .00 to 1800 – star is in western hemisphere180o to 3600 – star is in eastern hemisphere

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Relationships between co-ordinates1.Relationship between altitude of the pole and latitude of the observer H-H horizon plane E-E equatorial plane O – is the centre of the earth ZO is perpendicular to HH while OP is perpendicular to EE Latitude of place Altitude of pole

Equating both equation

Altitude of the pole is always equal to the latitude of the observer


POZPOZHOPHOZ

ZOPZOPEOZEOP

HOP

EOZ

0

0

90

90

POZZOP

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2.Relationship between latitude of observer and declination an altitude of a point on the meridian

declination meridian altitude of star meridian zenith of star latitude of the observer

If star is below the equator, -ve sign for δ and also if the star is to the north of zenith –ve sign for z

If the star is north of zenith but above the pole as at M2

p= polar distance = M2 P

If the star is north of zenith but below the pole p= polar distance = M3 P


z

ZMEMEZ

EZ

zZM

SM

EMM

11

1

1

11

p

p

PMZMZP

)90()90( 00

22

p

p

PMZPZM

)90()90( 00

33

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3.Relationship between right ascension and hour angleM – position of the star - westerly hour angle - westerly hour angle for first position of aeries position γ - right ascension of star

Hour angle of equinox = Hour angle of star + RA of star


PM

SP

SPM

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Correction to Apparent altitude1.Instrumental correction2. Observational correction1. Instrumental Correctiona. Corrections for altitudes i. Index and ii.Bubble errorb. Corrections for azimuths

i. Index error Small vertical angle between the line of collimation and the horizontal bubble line of the

altitude or azimuthal bubble Procedure1. With telescope normal in face left position any well-defined object such as church spire or a

chimney is bisected and angle is α1

2. The face is changed (Right face) and the telescope is reversed and the same object is bisected and angle is α2

Mean,

If the observations are not possible to take on both sides, correction for index error is applicable

It can be eliminated by taking reading on both faces Sivapriya Vijayasimhan

221'

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ii.Bubble error If the bubble tube is not at the centre while taking reading, correction for bubble error is

applicable Correction fro bubble error, C (seconds) - sum of readings of the object glass end of the bubble - sum of readings of the eye piece end of the bubblen – number of bubble ends readv- angular value of one division of bubble in seconds

The observed altitude when corrected for index error and bubble error is called apparent altitude

b. Corrections for azimuths

c – correction for azimuthsb – inclination of horizontal axis of the transit with respect to horizontal, secα – vertical angle to high point


vn

EO

OE

tanbc

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2.Astronomical Correctioni. Correction for parallax ii. Correction for refractioniii. Correction for dip of the horizon iv. Correction for semi diameter

i. Correction for parallaxWhen Sun and star are viewed from different points, change in direction of the body is

observed due to parallaxParallax in altitude is called diurnal parallaxO – centre of earthA – plane of observationS – position of Sun at time of observationS’ – position of sun at horizonOC – true horizonAB – sensible horizon = observed altitude = true altitude, corrected fro parallax = parallax correction = Sun’s horizontal parallax


h

a

pASO

pASB

SOC

SAB

'

‘

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When Sun is on horizon (apparent altitude is zero),Sun’s horizontal parallax varies from 8.95” from Jan to 8.66” during early JulyTrue altitudeParallax correctionFrom

But,

pa and ph are very small,

Correction for parallax = (horizontal parallax) x cos (apparent altitude) = 8.8” cos

- Correction is additive- Correction is maximum when the Sun is at horizon Sivapriya Vijayasimhan

'OS

Rph

apASBSABSBSSOC ''

ap '

'cossinsin

sin'

'cos)'90sin(sin

sinsin

,

0

ha

h

a

pp

pOS

OA

OS

OAOS

OA

OS

OAp

OS

OAOASSO

AOS

'cosha pp

'

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ii. Correction for refractionAs the distance from surface increases, the layers of atmospheric air

surrounding the earth becomes thinner Due to variation in atmospheric density, the ray of light passes

through the atmosphere bentsBecause of this, body appears to be nearer to zenith than it actual

Refraction angle of correction: Deviation of angle of ray from its direction on entering the earth’s atmosphere to its direction at the surface of earth

At a pressure of 76 cm of mercury and 10 o C , Correction for refraction ( in sec) = 58 “ cot α = 58” tan z α – apparent altitude of heavenly body : z – apparent zenith distance of heavenly body

Correction is subtractiveFactors influencing1. Density of air 2. Temperature 3. Barometric pressure and 4. Altitude


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iii. Correction for dip of the horizonAngle of Dip : angle between the true and visible horizonDue to curvature of earth, visible horizon is below the true horizonAngle of dip is angle between the two horizons and this has to be

subtracted from the observed altitude of the body

A – position of observerAB – h – height of observer above sea-levelS – position of Sun or StarAD – visible horizonAC – true horizon - observed altitude of sun or star - true altitude of sun or star - angle of dipR - radius of earthThen,BO = R and AO = (R + h)


CAD

SAC

SAD '

22 RhRAD

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R

hradians

eapproximatR

h

exactR

hRh

R

RhR

OD

AD

AODCAD

2)(tan

)(2

tan

)()2(

tan

tan

2

2

22

Β= small, then

Correction for dip is subtractive

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iv. Correction for semi-diameterHalf of angle subtended at centre of earth by sun and star is the semi-diameter of earthSemi diameter of earth varies from 15’46” (July) to

16’18” (January)

Mean distance value is 16’1.18”

Sun’s diameter is the tangent sight of sun’s image by cross hair


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OA – ray corresponding to lower limb of Sun - observed altitudeα - corrected altitudeOB – ray corresponding to upper limb of Sun - observed altitudeγ/2 is semi diameter,

When horizontal angle is measured to Sun’s right or left limb correction is equal to sun’s semi-diameter times the second of altitude is applied.

Correction for semi – diameter in azimuth = semi diameter x secant α


1

2

22 21

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Time SystemEarth moves from west to eastMeasurements of time depends on the apparent motion of heavenly bodies by earth’s

rotation on its axis

Four kinds of time1. Sidereal time 3. Mean solar time 2. Apparent solar time 4. Standard time

1.Sidereal TimeSidereal Day : Time interval between two successive upper transits of first point of aeries over same meridianSidereal noon : instant of crossingTime : 1 Day ( 0 to 24 hrs) 1 hrs = 60 min 1 min = 60 secondsLocal sidereal time (LST) : Right Ascension (RA) of meridian of place

LST = RA of star + westerly hour angle of star If LST > 24 hrs, 24 hrs has to be deducted: If LST < 24 hrs, 24 hrs has to be added

LST = RA of mean sun ± 12 hr + (mean time of that place)

Sidereal time of transit of star = RA of star


Astronomers Relevant every day

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UNIT IV ASTRONOMICAL SURVEYING2. Apparent Solar timeApparent solar Day : Time interval between two successive lower transit of center of sun over the same meridianApparent solar day(24 hrs) 60 min 60 secCalculated on the basis of “motion of Sun”

3.Mean Solar TimeMean sun(imaginary body) is assumed to move at a uniform rate along the equator in order to make solar day of uniform period.Mean Solar Time : Time when measured by diurnal motion of mean sun (clock time)Mean solar day or civil day : time interval between two successive lower transits of mean sun

over same meridian

Astronomical day : zero hr to midnightCivil day : 1. midnight to noon - anti meridian (am)

2. noon to midnight – post meridian (pm)

i. Conversionii. Relationshipsiii. Local mean time


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i. Conversionsa. If civil time is am, the astronomical time is sameb. If civil time is pm, the astronomical time = civil time + 12 hrsc. If astronomical time is less than 12 hrs, civil time is samed. If astronomical time is greater than 12 hrs, civil time = astronomical time+12 hrs

ii. Relationships Between hour angle, right ascension and timeApparent solar time = hour angle + 12 hrsMean solar time = hour angle of mean sun + 12 hrLocal sidereal time = RA of mean sun + hour angle of mean sunSidereal time of apparent moon(sun crosses the meridian of any place) = RA of SunSidereal time of mean noon = RA of mean sun

iii. Local Mean TimeMean time at meridian of observerAll places along the same meridian shall have same local time.Mean time - Greenwich mean time


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UNIT IV ASTRONOMICAL SURVEYING4.Standard TimeMean time on meridian as the standard time for the whole of country Standard meridianMeridian passing Greenwich – Greenwich mean time (GMT)

Time : 0 to 24 hrsMean time associated with standard meridian - Standard time

India : 82030’ E or 5 hrs 30 m east

Standard time = LMT ± difference of longitude in time between the given place and standard meridian

+ sign – standard meridian to west- sign - standard meridian to east


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Equation of TimeDifference between apparent solar time

and mean solar time+ sign – Sun after clock- Sign – Sun before clockEquation of time = RA of mean Sun – RA of Sun varies between 0 to 16 minApril 15, June 14 , September 1 and December 25 - mean time and apparent time are sameThe difference is due to obliquity of real sun and mean sunLST = RA of mean sun + hour angle of mean sunLST = RA of sun + hour angle of sunRA of mean sun – RA of sun = hour angle of sun – hour angle of mean sunEquation of time

= hour angle of sun – hour angle of mean sun Equation of time

= apparent time – mean time


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Azimuth of a Survey LineAngle between observer's meridian and vertical circle passing through the body

Azimuth Observation1. Measuring the horizontal angle between a reference mark and heavenly body2. Determine the azimuth of the celestial body

Reference mark – azimuth of star or heavenly body- Triangulation station lantern or electric light- Line of sight should be well above ground to minimum the error due to lateral deflection

Azimuth of reference mark is calculated from measured angle and known azimuth of celestial body

Azimuth of survey line may be obtained measuring the horizontal angle between the reference mark and line and combining with azimuth of the reference


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Determination of Azimuth of a Survey Line- Extra meridian observation of the Sun- Extra meridian observation of circumpolar star or of a star near Prime vertical- Observation of a circumpolar star at elongation

1.Extra meridian observation of SunAstronomical triangle ZPS is used to compute azimuth Sun

Azimuth OB =Observation of Sun’s time = 8 am to 10 am or between 2 and 4 pm


coscos

sinsinsincos

2

)sin(sin

)sin()sin(

2tan

A

PSZSZPs

PSss

ZSsZPsA

BODABODNODNOB

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2.Extra meridian observation of circumpolar starObservation of Star is taken when it is on or near the prime vertical as it move slowly in

azimuthRefraction will be greater if the star is too low

3. Observation of a circumpolar star at elongationPlane of declination and plane of vertical circle is right anglesProcedure to calculate star elongation1.Hour angle of star is calculated by knowing latitude of the place and declination of star

2.Hour angle is converted into time and added to RA of star (west elongation) or subtracted to RA of star(east elongation)

3.Time is converted into mean time

Azimuth of Star,


)tan(

)tan()cos(

ndeclinatio

latitudehourangle

cos

cos

)cos(

)cos(sinsin

latitude

ndeclinatioPZSA

45


Nautical Almanac(NA) Astronomical data availableSalient Features1. Greenwich hour angle of Sun and declination are given for every angle of GMT to 0.1’

Tables for increments and corrections for every minute and second2. Equation of time(ET) is given to nearest second for intervals of 12 hours and time of

meridian passage every day3. ET is the quantity to be added to mean solar time to get apparent solar time4 . Semi-diameter of sun is given to 0.1’ for every 3 day period5. Sidereal hour angle and declinations are given for 173 stars including 57 selected stars

(accuracy = 0.1’)6. Polar star table are given


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