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03Presentazionedef.Pdf CosmicCosmic DistancesDistances 1010 ly 107 ly 105 ly 102 ly 10-4 ly Radar Ranging ItIt is is based based on on the the measurement measurement ofof the the round-trip round-trip time time (“t”) (“t”) ofof a a powerful powerful radio radio signal signal (with(with speed speed “v”) “v”) that that is is reflected reflected onon the the atmosphere atmosphere of of Venus Venus DD = = v v (t/2) (t/2) = = E-VE-V 6 == 42 42 x x 10 106km km (average) (average) UsingUsing this this method method we we can can obtainobtain the the value value of of AstronomicalAstronomical Unit Unit AstronomicalAstronomical Unit Unit is is the the averageaverage Earth-Sun Earth-Sun distance distance SinceSince D D is is known, known, so so it's it's possible possible to to E-VE-V determinedetermine the the distance distance Earth Earth - - Sun Sun trigonometricallytrigonometrically TheThe maximum maximum apparent apparent distance distance of of Venus Venus from from the the Sun Sun (as (as seen seen from from Earth) Earth) varies varies between between 45° 45° and and 48 48 °. °. For For simplicity, simplicity, we we take take an an average average value value betweenbetween these these : : 46 46 ° ° D /D = sin 46° = 0,72 DS-V /DS-E = sin 46° = 0,72 D D = = 1 1 AU AU D D = = 0,72 0,72 AU AU D D = = 0,28 0,28 AU AU S-V S-E S-ES-E S-VS-V E-VE-V 8 D = 0,72 D 8 DS-V = 0,72 DS-E 1 1 AU AU = = D D / /0,28 0,28 = = 1,5 1,5 x x 10 10 km km S-V S-E E-VE-V Parallax WhenWhen we we want want to to determine determine the the weight weight with with a a scale, scale, we we may may note note that that the the needleneedle varies varies position position each each time time we we move move our our head: head: if if we we move move it it toto the the right, right, we we will will project project the the needle needle to to the the left left toto the the left, left, we we will will see see the the needle needle projectedprojected to to the the right. right. The shift of the object is only apparent and it is This effect is called The shift of the object is only apparent and it is This effect is called causedcaused by by a a change change of of the the position position of of the the observer. observer. PARALLAX PARALLAX Another example TheThe stars stars have have a a parallax parallax effect, effect, annual annual parallax, parallax, caused caused by by the the motion motion of of the the Earth Earth around around the the Sun. Sun. TheThe Earth Earth revolves revolves around around the the Sun Sun drawing drawing an an ellipse ellipse whose whose major major axis axis is is about about 300 300 million million km km.. TheThe position, position, from from which which we we observe observe the the sky, sky, changes changes over over the the course course of of the the year. year. LetLet us us fix fix the the position position of of a a star star and and let let repeat repeat the the observation observation from from the the same same place place on on Earth Earth after after six months when the Earth is in the opposite point in the orbit around the Sun, six months when the Earth is in the opposite point in the orbit around the Sun, wewe will will note note a a small small change change in in the the star's star's position position.. InIn the the course course of of a a year, year, a a star star describes describes a a small small ellipse ellipse in in the the sky. sky. S S 2 1 TheThe parallax parallax method method is is a a method method similar similar toto the the triangulation. triangulation. In In fact, fact, if if we we join join thethe point point E E with with S S and and S S with with E E , , 11 11 22 22 wewe define define a a triangle, triangle, which which the the major major axis axis asas base base and and the the distances distances S-E S-E S-E S-E asas sides. sides. 11 22 S By By convention, convention, the the angle angle of of the the annual annual parallaxparallax (p) (p) is is equal equal to to half half the the apparent apparent angularangular shif shif of of a a star star in in 1 1 year, year, and and correspondscorresponds to to the the vertex vertex of of the the triangle triangle whichwhich is is based based on on the the semi-axis semi-axis ofof Earth's Earth's orbit orbit around around the the Sun Sun (R(R = = 150000000 150000000 km) km) R andand has has vertex vertex the the star. star. E E 1 2 TheThe parallax parallax of of a a star star decreases decreases with with distance distance and and it it has has always always very very small small values. values. TheThe nearest nearest stars stars to to Earth Earth have have a a value value of of the the parallax parallax a a little little less less than than 1 1 second second of of arc. arc. AsAs the the distance distance increases, increases, the the measurement measurement of of the the parallax parallax angle angle becomes becomes more more difficult, difficult, as as thethe annual annual change change in in position position decreases decreases significantly. significantly. For For this this reason reason the the more more distant distant stars stars maintainmaintain a a fixed fixed position position and and do do not not exhibit exhibit apparent apparent shifts. shifts. ToTo derive derive the the distance distance to to a a star star directly directly from from the the value value of of the the parallax, parallax, itit is is convenient convenient to to adopt adopt a a particular particular unit unit of of measurement: measurement: parsec parsec (pc). (pc). 11 parsec parsec is is the the distance distance at at which which a a body body subtend subtend a a parallactic parallactic angle angle of of 1” 1” of of arc. arc. IfIf we we express express the the distance distance in in parsecs parsecs and and assign assign the the value value 1 1 to to the the distance distance Sun-Earth Sun-Earth = = “R”, “R”, “d” “d” is is d=1/p d=1/p To obtain it we use trigometric p ratios : d = R / sin p = 1/ sin p d Since the angles with which we deal are very small, we use the approximation sin p ~ p dd (pc) (pc) = = 1/p 1/p R By studying the annual shifts in the positions of the stars, we are able to measure parallaxes corresponding to a maximum distance of 50 pc. TheThe measurement measurement of of parallax parallax seems seems very very easy, easy, but but in in reality reality it it is is necessary necessary to to use use tools tools that that reach reach accuraciesaccuracies much much greater greater than than those those of of a a "normal" "normal" telescope, telescope, that that does does not not reach reach the the required required accuracy. accuracy. WithWith the the Spacecraft Spacecraft Hypparcos Hypparcos we we can can measure measure parallaxes parallaxes up up to to 300 300 pc. pc. 300 pc ByBy the the end end of of 2013 2013 the the Spacecraft Spacecraft Gaia Gaia will will be be launched launched and and it it will will be be able able to to measuremeasure parallaxes parallaxes up up to to 10000 10000 parsec parsec ExercisesExercises Proxima Centauri Proxima Proxima Centauri Centauri is the nearest star to the SolarSystem. It has a parallax of 0,775"; so its distance is 1 /0,775 = 1,295 pc Sirius Sirius SiriusSirius is is the the brightest brightest starstar in in the the night night sky. sky. ItIt has has a a parallax parallax of of 0,380”;0,380”; so so its its distance distance is is 1/0,3801/0,380 = = 2,632 2,632 parsec parsec UnitUnit ofof MeasuramentMeasurament ConversionConversion FactorsFactors AstronomicalAstronomical Unit Unit ( (auau):): average average 1 au ≈ 4.85×10-6 pc ≈ 15.81×10-6 ly Earth-SunEarth-Sun distance distance (149 (149 600 600 000 000 km) km) ParsecParsec (pc): (pc): the the distance distance from from the the Sun Sun toto an an astronomical astronomical object object which which 1 light-year ≈ 63,24x103 au ≈ 0,31 pc hashas a a parallax parallax angle angle of of oneone arcsecond arcsecond (1/3,600 (1/3,600 of of a a degree). degree). LightLight Year Year (l.y.): (l.y.): the the distance distance that that 4 lightlight travels travels in in a a vacuum vacuum in in one one year year 1 parsec ≈ 20,62x10 au ≈ 3,26 ly withwith a a speed speed of of ~ ~ 300 300 000 000 km/s km/s CAUTION! LIGHT YEAR is NOT a unit of measurament of time. It is correct to say that the image of celestial body, which is far a certain number of light years, shows us that celestial body as it was the same number of years ago, and not at this time. The distance Earth - Andromeda is 2 million ly, so if an observer could see the Earth from Andromeda, he would see the Earth today as it was 2 million years ago.
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