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Orbit of Mercury: Following Keplers steps NATS 1745 B

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Objective You will use a set of simple observations, which you could have made yourself, to discover the size and shape of the orbit of Mercury.

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Terminology Superior planet - a planet with an orbit greater than Earths (e.g. Mars, Neptune) Inferior planet - a planet with an orbit smaller than Earths (Mercury and Venus) Conjunction - planet is directly lined up with the Sun and Earth Opposition - Sun and planet in line with Earth, but in opposite directions (180 o apart) on the sky (as seen from Earth)

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Terminology Contd Elongation: The angular separation of a planet from the Sun (as seen from the Earth) Elongation Earth Sun Planet

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R E = radius of Earths orbit = 1 AU R P = radius of planets orbit RPRP Sun Line of Sight (LOS) Earth Greatest elongation (from observations) Planet RERE Right-angle Definition: The Astronomical Unit (AU) is the average distance between the Earth and the Sun 1 AU = 1.496 x 10 8 km

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Standard Planetary Configurations E Opposition Quadrature Conjunction Superior conjunction Greatest western elongation Greatest eastern elongation Inferior Conjunction

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The Motion of the Planets The planets are orbiting the sun almost exactly in the plane of the ecliptic. The moon is orbiting Earth in almost the same plane (ecliptic). Jupiter Mars Earth Venus Mercury Saturn

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Mercury appears at most ~28 º from the sun. It can occasionally be seen shortly after sunset in the west or before sunrise in the east. Venus appears at most ~ 48 º from the sun. It can occasionally be seen for at most a few hours after sunset in the west or before sunrise in the east. Apparent Motion of the Inner Planets

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The ellipse Definition: Eccentricity (e) Distance OF 1 = OF 2 F1F1 F2F2 Semi-major axis (a) O r1r1 r2r2 P Major axis Two focal points Semi-minor axis (b)

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First Keplers law Planets have elliptical orbits, with the Sun at one focus perihelionAphelion Sun Planetary orbit - exaggerated center empty focus

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Second Keplers law The planet-Sun line sweeps out equal areas in equal time A B C D E G F Time T if area AFB = area CFD = area EFG then time (A to B) = time (C to D) = time (E to G) 2nd law says:

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Second Keplers law contd Perihelion - closest point to Sun –Near perihelion planet moves faster Aphelion - greatest distance from Sun –Near aphelion planet moves slower Perihelion P Sun Aphelion Planet 1/4 of way around orbital path Planet at 1/4 of orbital period Area (Sun, P, Perihelion) = Area(Sun, P, Aphelion) = 1/4 area of ellipse

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Keplers third law P 2 = K a 3 The square of a planets orbital period (P) is proportional to the cube of its orbital semi-major axis (a) a3a3 P2P2 Mercury Pluto Slope = K where, P = planet orbital period a = orbits semi-major axis K = a constant if P(years) and a(AU) then K = 1 and P 2 (yr) = a 3 (AU)

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Planet sidereal period (years) semi major axis (AUs) a 3 /P 2 Mercury0.2410.3870.998 Venus0.6150.7230.999 Earth1.000 Mars1.8811.5241.000 Jupiter11.865.2031.001 Saturn29.469.541.000 Uranus84.8119.181.000 Neptune164.830.061.000 Pluto248.639.440.993 Observational Evidence The above data confirm Keplers third law for the planets of our solar system. The same law is obeyed by the moons that orbit each planet, but the constant k has a different value for each planet-moon system.

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The assignment

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MonthDayYearElongationDirection Feb6158826°W Apr18158820°E Jun5158824°W …………… Dec1121°E Jan18158924°W Apr119°E …………… Nov23…22°E Jan1159023°W …………… Nov5159023°E You will have a list, similar to this one an scale drawing of the Earth's orbit and the Earth's positions on its orbit on some dates, marked of at ten day intervals.

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PROCEDURE 1.Locate the date of the maximum elongation on the orbit of the Earth and draw a light pencil line from this position to the Sun. For each elongation:

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Feb 6 Feb From the first line of the example table: Feb 6

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2.Center a protractor at the position of the Earth and draw a second line so that the angle from the Earth-Sun line to this 2nd line is equal to the maximum elongation on that date. Extend this 2 nd line well past the Sun. Mercury will lie somewhere along this second line. As you draw more lines (dates) you will see the shape of the orbit taking form. PROCEDURE

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Feb 6 Feb 2 nd line 26º as seen fromEarth, the 2nd line will be to the left of the Sun if theelongation isto the East, to the right of Sun if theelongation isto the West. From the first line of the example table: Feb 6, elongation = 26° W

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After you have plotted the data you may sketch the orbit of Mercury. The orbit must be a smooth curve that just touches each of the elongation lines you have drawn. The orbit may not cross any of the lines. PROCEDURE

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After you drew the orbit Through the Sun draw the longest diameter possible in the orbit of Mercury (remember, this is the major axis of the ellipse). Measure the length of the major axis. Draw the minor axis through the center perpendicular to the major axis. –Note that the Sun is NOT at the center of the ellipse.

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After you measured the semi-axis To convert your measurements to A.U.: –measure the length, in centimetres, of the scale at the bottom of the figure of Earths orbit. –call this measurement l. Be sure to measure the full 1.5 A.U. length. –calculate the scale in units of AU/cm. The scale is given by –multiply your measurements in centimetres by the scale to convert them to AUs. Scale = ( 1.5A.U. / l ) in (AU/cm)

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Report Plot of Mercury orbit Semi major axis Eccentricity of the orbit Verify Keplers second law Due on Friday Nov 3, 5 pm at Prof. Caldwells office (332 Petrie Building)

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