towards modern astronomy - astrophysicsfverbunt/iac2011/modern.pdf · satellites of jupiter: not...

Towards modern astronomy

Outlinelongitude and RoyalObservatories

I star cataloguesI the Moon, JupiterI size of the solar system

Jupiter, SaturnI perturbation theoryI error theory

towards the deep universe:Herschelstellar astronomy

I first parallaxI spectra

LiteraturePannekoek

Van Helden, Measuring theUniverse

Stigler, The history ofstatistics

Sobel, Longitude

Gillispie, Science and Polityin France (2 books)

Holmes, The age of wonder(esp. chapters on Herschel)

Frank Verbunt (Astronomical Institute Utrecht) Towards modern astronomy July 15, 2011 1 / 30

Longitude and Royal Observatories

Longitude and clocksposition at Sea:

I latitude can bemeasured

I longitude requiresclock

several large shippingdisasters: search for goodclock

astronomical clocks

mechanical clocks: JohnHarrison (∼1730)

Astronomical clockssatellites of Jupiter: not possiblefrom ship; may be used forlongitude of main land⇒ Römervelocity of lightMoon passing stars: requires

I accurate positions of starsI good theory of motion of

Moon

1667 Observatoire Royal deParis

1676 Royal ObservatoryGreenwich


Accurate star positions: catalogues and aberration

Star catalogues

year astronomer stars acc.150 Ptolemaios 1028 ∼ 1◦

1437 Ulugh Beg 1016 ∼ 1◦

1598 Tycho Brahe 1004 ∼ 2′

1602 Tycho Brahe 777 ∼ 2′

1627 Brahe/Kepler 1004 ∼ 2′

1690 J. Hevelius 1690 ∼ 2′

1725 Flamsteed 2554 ∼ 10′′

1818 Bradley ∼ 1′′

Johannes Hevelius & ElisabethKoopman 1690


First new measurements of star positions: Ulugh Beg

Ulugh Beg 1393-1449Ulugh Beg, the grandson ofTamarlane (Timur Leng), wasking of Samarkand, and had aninterest in astronomy

He built a large quadrant(R = 40 m) to measure starpositions

He was murdered by themullahs, his court astronomerescaped to Istambul and carriedseveral copies of the starcatalogue. One copy now inBodleian Oxford

Ulugh Beg Quadrant


New accuracy: James Bradley Astronomer Royal

James Bradley Resulting discoveriesproper motion Aldebaran,Sirius, Arcturus (Halley 1718)

Aberration (Bradley 1728)annual shift varying from+20′′ to −20′′

Nutation (Bradley 1748) 18 yrshift in declination from+18′′ to −18′′

parallax (in solar system) forSun: R⊕/AU corresponds to8.8′′

refraction from 0 in zenit to31′ at horizon


Perturbing effects: parallax and refraction

When an object P isobserved from O , its position must be corrected for two effects:

1 parallactic correction: to obtain the angle at which the object is seenfrom the center E of the earth, the correction is given by angle OPE,and depends on the location of O . O depends on time of day, and onlongitude, latitude on Earth. The parallax of P is defined as the angleEPL , i.e. radius of the earth divided by distance to P.

2 refraction: due to refraction of the light in the earth atmosphere, anobject is seen higher above the horizon. The correction is angleO ′OP (exaggerated in the figure).


A wrong parallax leads to a wrong derived refraction

Brahe and CassiniTycho Brahe: stars, Sun andMoon each have differentrefractions; values for stars wrongby up to 2′. Cassini hadremarkably good values for therefraction. See upper frame. Thelower frame shows the differenceswith the modern values. Braheassumed Rm = 0 for h > 21◦,thus his errors at these h equalthe modern value. The errors inCassini’s values directly (•) andmultiplied by 10 (o).

Comparison with modern value


Solution of recfraction by Richer & Cassini

From the measurements by Richer of the maximum height of the Sun inCayenne (summer and winter solstice), Cassini could show that hisrefraction and solar parallax were consistent, whereas those of Brahewere not.

Richer & Brahehmax (North) hmax (South)71◦27′40′′ 61◦35′16′′ measured Richer

0′′ 0′′ refraction correction Brahe55′′ 1′28′′ parallax correction Brahe

71◦28′35′′ 61◦36′44′′ corrected values18◦31′25′′ 28◦23′16′′ 90◦ − hmax ≡ hcomp

ε = 0.5[hcomp(N) + hcomp(S)] = 0.5(46◦54′41′′) = 23◦27′21′′

This is significantly different from Brahes value at Hveen: 23◦31′30′′.


Solution of refraction by Richer & Cassini

From the measurements by Richer of the maximum height of the Sun inCayenne (summer and winter solstice), Cassini could show that hisrefraction and solar parallax were consistent, whereas those of Brahewere not.

Richer & Cassinihmax (North) hmax (South)71◦27′40′′ 61◦35′16′′ measured Richer

-20′′ -32′′ refraction correction Cassini3′′ 4′′ parallax correction Cassini

71◦27′23′′ 61◦34′48′′ corrected values18◦32′37′′ 28◦25′12′′ 90◦ − hmax ≡ hcomp

ε = 0.5[hcomp(N) + hcomp(S)] = 0.5(46◦57′49′′) = 23◦28′55′′

This is virtually equal to Cassini’s value in Paris: 23◦29′.


Distance to Sun: from antiquity to modern

Aristarchos Ratio of distances to Sun and Moonfrom the angle Sun-Earth-Moonwhen the Moon is exactly half dark:see Figure (not to scale; the threedots above are to scale withAristarchos’ numbers).He then determined the distance tothe Moon in Earth radii from its sizecompared to the size of the earthshadow in eclipse. The figure aboveis simplified: Aristarchos uses a finitedistance to the Sun, hence an Earthshadow smaller than the Earthdiameter, and finds radius of theMoon is about 0.35 R⊕


’All planets have same angular size as seen from Sun’

For simplicity assume circularorbits. From Kepler’s third law:(

2πP

)2

=G(M� + Mp)

a3'

GM�a3

Thus for Venus and Earth:

aE

aV=

(365.26224.70

)2/3

= 1.3825

The assumption of equal angularsizes from the Sun then givesRV = R⊕/1.3825 (' 4600 km).

Horrocks & Venus transitHorrocks observed diameter ofVenus during transit in 1639: 1′16′′

(compare Brahe’s value 12′18′′).The derived distance Earth to Venusis aEV = 4600 km/tan 38′′ =2.5 107 km and to the SunaE = aEV 1.382/0.382 = 9.0 107 km '15000 R⊕.Note: R⊕/RV = 1.054 (modernvalue).


The parallax of Mars

Modern valueThe minimum (maximum)distance of Mars to the Sun is 206(249) million km. The distance ofthe Earth to the Sun is 150 millionkm.Thus, the parallax of Mars at agood opposition is 6378 km / 56million km = 0.00011 = 23.5′′.For comparison: at a distance of2 meter, 1◦=3.5cm, 1′=0.058cm,and 1′′=0.001cm.

Richer in Cayenne (1672)The analysis of the data obtainedin Cayenne were inconclusive;and included a possible parallaxof 0, i.e. infinite distance of Mars.Cassini and Richer then selectedthe observations which gave aparallax in agreement with earlierestimates.


Distance to Sun: overview

Archimedes: methodunknown; and large-numberplay in the ‘Sand Reckoner’

Ptolemaios to Brahe:touching spheres: maximumdistance of planet n isminimum distance of planetn + 1

Kepler: in/out-scribed regularsolids or third law(period-distance relation)plus relation distance andsize, surface or volume.

Remus to Flamsteed: planetshave same angular size as seenfrom Sun

Cassini/Richer select from Marsparallax measurements thosethat agree with earlier estimates

Newton takes value ofFlamsteed

‘Venus’ various publishedresults of transits 1761, 1769

Encke 1835: re-analysis ofVenus transition of 1761, 1769.


The solar parallax from antiquity to modern


Distance to Sun: transit of Venus

GeometryFrom Kepler’s third law (see above)

aE = 1.3825aV

The distance h of the center ofthe Earth E to the orbital plane ofVenus is not known, but one canmeasure the distance dh betweentwo locations on earth (in thedirection perpendicular to theorbit of Venus). One then seesthat the difference dz in distanceof the solar center S to the imageof Venus on the Sun is given bydz = dh/0.3825.



Difficult measurement. . .dz is not easily measured! Fordh=6000 km on has dz=15700 km.This was expected to be (and indeedis) such a small fraction of the solardiameter, that its directmeasurement as an angle is notaccurate. Hence one converts thedifference to a time difference.For Mercury (PM=88 days, henceaE = 2.56aM) it is worse, asdz = dh/1.56

Angular velocityAngular velocity of Venus, asseen from Earth:

ωV =2πav/Pv − 2πaR/PE

aE − aV

' 0.6352πPE≡ 0.635ω�

Angular velocity of Venus withrespect to the Sun is

ωV + ω� = 1.635ω�



Geometry The time difference between thetracks at z and z + dz is found bydividing dl expressed in an angle(by dividing by aE) by the relativeangular velocity:

∆T =dl/aE

ωV + ω�

By measuring ∆T and computingdl from dh and the (alsomeasured) α, one finds aE .Example: for dh=6000 km, andα = 45◦ one has ∆T ' 300 s.


Distance to Sun: end of 19th century

Venus & Marsin 1874 and 1882 two moreVenus transitions wereobserved, which led to better,but still disappointinglyinaccurate, values for thesolar parallax. The range ofdifferent published valueswas larger than the accuracyEncke had claimed for the1761-1769 transits!

a more accurate value wasderived from the firstsignificant measurement ofthe parallax of Mars

Aberration and Erosthe orbital velocity of the Earthis v = 2πa/P. The aberrationangle is θ = v/c. At the end ofthe 19th century themeasurement of c in thelaboratory was very accurate,and a was determined from c!

the asteroid Eros, discovered in1898, is in a highly eccentricorbit, and comes closer to theEarth than Mars. Its parallaxcould be measured accuratelyand through this the solarparallax


Distance to Sun: end of 19th century


Towards error analysis: France and Saturn

‘Cassini has taken more land from methan I conquered on my enemies’ (LouisXIV)

Triangulationmethod described bySnellius

applied to France byCassini for Louis XIV

N triangles, each witherror d: total errorD = Nd ?

repeated under Turgot(Louis XVI), continuedduring revolution

error propagationstudied by Laplace


Towards the least squares method

Euler 1749Halley (1676) had shown thatthe orbit of Jupiter is shrinking,that of Saturn expanding

Does this imply that Newton iswrong? and/or that the solarsystem is unstable?

Euler worked out a theory with 6parameters. He had 75observations. He could not finda way to solve the parameters,since each set of 6 observationsgave a different answer. . .

Mayer 1750The lunar orbit was studiedbecause it might serve as aclock

Mayer measured the positionof the crater Manilius on theMoon-disk as observed fromEarth, and tried to determinethe libration of the Moon

He had 27 observations,which he divided in threesets of 9. By adding theequations in each set, heended with 3 equations, andsolved the 3 unknowns


Towards the least squares method

Laplace 1788Laplace analysed the problemof Saturns orbit in terms ofperturbation by Jupiter

He used 24 observations tosolve for 4 unknowns, bycleverly combining them into 4equations.

His solution is very good. If yi isSaturn’s observed position in ′

and ym the model,∑24i−1(yi − ym)2 = 21.4.

compare least squares solution:∑24i−1(yi − ym)2 = 20.9

Laplace, Legendre, GaussLaplace & Legendre debatedwhether one should minimize(A)

∑|yi − ym | or

(B)∑

(yi − ym)2

Gauss showed that if (B) iscorrect, the measurementerrors must follow aGaussian distribution

The first application of theleast squares method aspublished by Gauss (1805),is a re-analysis of the Venustransits by Encke (1835):8.57116±0.0371′′


Towards stellar astronomy: John Michell 1767

Binaries are realstatistical argument

separation βCap is 3′20′′

distribute 230 stars randomlyin sky

probability of pair at (small)angular distance x degrees:

(1 − p)n×n ' 1 − n2p

where p = 7.6 × 10−5x2

1 in 80 for βCap

hence: stars have differentluminosities!

Pleiades real clustera fortiori: the Pleiades must bereal cluster


Towards the deep and evolving universe: Herschel

William Herschel Backgroundgrew up in Hannover,Germany, son of militarybandleader

made (poor) living as bandleader & musician/composerin Englanduntil he became director ofpublic concerts in Bath

I buys own houseI makes own telescopesI invites sister Caroline

no education in astronomy

discovered by Royal Societywhile observing on street



Observing programmehis mirror telescopes aresuperior (>1773)

idea of Galilei: faint stars arefurther⇒ have smallerparallax⇒ observe closepairs to detect parallax ofbright = nearby star

Castor A and B

1759 measurement by Bradley



Discovery Uranus1781 in program closebinaries discovers ‘comet’ inGemini

turns out to be new planet

solar system doubles inlinear size! (Saturn 9.5 AU;Uranus 19.2 AU)

others have lessertelescopes: scepticism

obtains pension by KingGeorge

and money for largetelescope

Structure of Universewith sister Caroline countsstars (he at telescope, she inshed below)

many stars: edge far away

few stars: edge nearby

1783



The deep skyasks more money for newtelescope

and asks queen to becomesponsor of Caroline: the firstpaid woman astronomer!

helps William in cataloguingnebulae 1782-1802

in her ‘free time’ discoverscomets

deep evolving universe:evolutionary sequence ofclusters and nebulae ofdifferent density

large telescope unwieldy;many discoveries done withsmaller

40 foot telescope



Appraisal1800 discovers infrared light

other astronomers weredoing ‘boring’ work: accurateposition measurements ofstar

the Herschel’s are roamingthe universe

and speculating on itsnature. . .

thanks to absence ofuniversity training?

both Herschel’sinternationally recognized


Combination of mathematical and telescope techniques

Small planetsplanets between Mars andJupiter:

I 1801 Piazzi discoversCeres

I 1802 Olbers: PallasI 1804 Harding: JunoI 1807 Olbers: Vesta

contra Hegel. . .

Gauss devises method toderive orbit from threeobservations

‘clumps of mud’

NeptuneLaplace: celestial (i.e.planet) mechanics ‘nousn’avons pas besoin de cethypothèse’

Laplace 1821: concludesextraneous influence onUranus

Le Verrier analyses this andcomputes position of newplanet (1845)

confirmed by Galle (1846)

English: we too! i.e. Adams.(not true)


Distance to the stars

in geocentric system touchingspheres (no vacuum!) give thedistance to stars as the outerdistance of Saturn

in heliocentric system the limitto stellar parallax gives lowerlimit to distance

1837-1840 Bessell determinesthe first stellar parallax, of 61Cygni, at 0.31-0.35′′ (themodern, Hipparcos, value is0.287).


Exam questions

Cassini used measurementsby Richer in Cayenne toderive the value of theobliquity, and to show that hisvalue of the refraction andsolar parallax wereconsistent. Explain hisreasoning, and in doing soexplain how refraction andsolar parallax affect thedetermination of the obliquity,and from the numbers givenin the lecture notes derivethe latitude of Cayenne.

Herschel was different fromother astronomers in that hehad no formal education inastronomy. Explain how thismay have helped him inbecoming the most succesfulBritish astronomer of hiscentury. What were otherBritish astronomers doing?


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