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MSU AST 207 - 09-25

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1To Frame the World—25 Sept• Kepler found orbit of Mars relative to earth’s orbit.• Goal was to measure the absolute distance (in miles or km) of the solar system• Cassini & Richer 1672Giovanni Domenico Cassini, (1625 - 1712)engraving by N. Dupuis www.sil.si.edu/digitalcollections/hst/scientific-identity/fullsize/SIL14-C1-18a.jpgHipparchus measures the moon’s distance~200BC • At the Hellespont, the solar eclipse of 189BC was total. (Sparta defeated Athens there in 405 BC.)• In Alexandria, the moon covered ¼ of the sun.• Draw a picture to show the relationship between the sun, the moon, the two locations, and the difference between a total & ¼ eclipse. (The diameter of the sun is ½ degree.)http://mkatz.web.wesleyan.edu/medea_lecture/hellespont.gifView @ HellespontView in Alexandria. Moon is offset by 1/4 diameter of sunhttp://www.livius.org2What triangle did Hipparchus use?• Parts of triangle• Angle is due to parallax: moon in foreground shifts with respect to sun in the background.• One leg of triangle is the baseline.• Other leg is distance to moon• Method of parallax.• Angle is the “parallactic shift.” MoonSunfrom HSunfrom ADistance MoonangangbaselineHellespontAlexandriaSmall angle approximation• Measure angles in radians• Arc = radius × angle• For entire circle,Length of arc = circumference = 2πR.• Therefore angle = 2π for the whole circle.•2π radians = 360°• For small angles, arc is approximately equal to the chord.Chord = radius × angle• Application• Baseline = Distance × angleArcChordArcChordangle3Method of parallax• Parts of triangle• Angle is due to parallax: moon in foreground shifts with respect to sun in the background.• One leg of triangle is the baseline.• Other leg is distance to moon• Method of parallax.• Angle is the “parallacticshift.”• Distance = Baseline/ angleMoonSunfrom HSunfrom ADistance MoonangangbaselineDistance = Baseline/ angleHellespontAlexandriaDifficulties• Small angles are hard to measure• Naked eye 1/30° = 1/1700 rad = 6×10-4rad = 600 μrad• Modern telescope used under ideal conditions: 5 μrad• Modern telescope with correction for atmospheric turbulence: 0.5μrad• Moon using Hellespont & AlexandriaAngle = baseline/distance = 1000km / 400,000km= 1/400 rad = 2500 μrad (1/7°)•MarsAngle = baseline/distance = 1000km / 80,000,000km= 13 μrad using Hellespont & Alexandria• Need a reference nearby in the sky• Measuring with a reference on the ground is impossible.4Cassini & Richer 1672• Angle=baseline/distance• What baseline should C&R use to measure distance to Mars?Cassini & Richer 1672• Angle=baseline/distance• What baseline should C&R use to measure distance to Mars?• Cayenne-Paris baseline is 7000km.– Angle=baseline/distance=7000km/60,000,000km= 120μrad– Shift is 25 times width of a star seen with modern telescope.5Cassini & Richer• We are pretending to be Jean Richer and Giovanni Cassini in 1672. We are measuring the distance to Mars by making observations from Paris and Cayenne.• Facing the screen, hold a pencil at arms length. Without moving the pencil, look at it with your left and then your right eye. The pencil tip shifts with respect to something on the screen.1. What corresponds to Mars?A. Left eye or right eyeB. Tip of pencilC. Something in the screenD. The shift of the pencil tip with respect to the screen.2. What corresponds to Paris?3. What corresponds to the star?4. What is proportional to the parallax angle?ParisCayenneMarsStarfrom PStarfrom CDistance MarsangangbaselineDistance = Baseline/ angleStarViewed from ParisView from CayenneMarsCassini & Richer1. What corresponds to Mars? A. Left eye or right eyeB. Tip of pencilC. Something in the screenD. The shift of the pencil tip with respect to the screen.2. What corresponds to Paris? A3. What corresponds to the star? C4. What is proportional to the parallax angle? D5. If Mars were closer, the shift isA. larger.B. same.C. smaller.6. If observing stations were farther apart, the shift is ___.largerParisCayenneMarsStarfrom PStarfrom CDistance MarsangangbaselineDistance = Baseline/ angleStarViewed from ParisView from CayenneMars6To Frame the World• We pretend to be Jean (Giovanni) Cassini, Director of the Paris Observatory.• We propose a grand plan to “Frame the World” to the Louis XIV. The expedition to Cayenne will determine the distance to the outermost planet in the solar system!Giovanni Domenico Cassini, (1625 - 1712)engraving by N. Dupuis www.sil.si.edu/digitalcollections/hst/scientific-identity/fullsize/SIL14-C1-18a.jpgProposal to Louis XIV• I draw the orbit of Mars & Earth at the best opposition.– Mars is 0.38AU from Earth at the closest.– Jean (in Cayenne) & I (in Paris) will measure the distance between Earth & Mars by triangulation.– We will then know the length of an AU.•Kepler’s3rdLaw– Known to Kepler, period of Saturn is 29.5 years–P2 = R3⇒ R = 9.54AU– Because we will have measured the length an AU, we will know the size of the solar


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MSU AST 207 - 09-25

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