Presentation on theme: "Orbit of Mercury: Following Kepler’s steps"— Presentation transcript:
1Orbit of Mercury: Following Kepler’s steps NATS 1745 BOrbit of Mercury: Following Kepler’s steps
2ObjectiveYou will use a set of simple observations, which you could have made yourself, to discover the size and shape of the orbit of Mercury.
3TerminologySuperior planet - a planet with an orbit greater than Earth’s (e.g. Mars, Neptune)Inferior planet - a planet with an orbit smallerthan Earth’s (Mercury and Venus)Conjunction - planet is directly lined up with the Sun and EarthOpposition - Sun and planet in line with Earth, but in opposite directions (180o apart) on the sky (as seen from Earth)
4Terminology Cont’d Elongation: The angular separation of a planet from the Sun (as seen from the Earth)ElongationEarthSunPlanet
5RE = radius of Earth’s orbit = 1 AU RP = radius of planets orbit Definition:The Astronomical Unit (AU) is the averagedistance between the Earth and the Sun1 AU = x 108 kmRE = radius of Earth’s orbit = 1 AU RP = radius of planets orbitRPSunLine of Sight(LOS)EarthGreatest elongation(from observations)PlanetRERight-angle
7The Motion of the Planets The planets are orbiting the sun almost exactly in the plane of the ecliptic.JupiterVenusMarsEarthMercuryThe moon is orbiting Earth in almost the same plane (ecliptic).Saturn
8Apparent Motion of the Inner Planets Mercury appears at most ~28º from the sun.It can occasionally be seen shortly after sunset in the west or before sunrise in the east.Venus appears at most ~ 48º 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.
9The ellipse F1 F2 Semi-major axis (a) O r1 r2 P Major axis Two focal pointsSemi-minoraxis (b)DistanceOF1 = OF2Definition:Eccentricity (e)
10First Kepler’s lawPlanets have elliptical orbits, with the Sun at one focusperihelionAphelionSunPlanetary orbit - exaggeratedcenter“empty” focus
11time (A to B) = time (C to D) = time (E to G) Second Kepler’s lawThe planet-Sun line sweeps out equal areas in equal timeABCDEGFTime T2nd law says:ifarea AFB = area CFD = area EFGthentime (A to B) = time (C to D) = time (E to G)
12Second Kepler’s law cont’d Perihelion - closest point to SunNear perihelion planet moves fasterAphelion - greatest distance from SunNear aphelion planet moves slowerPlanet at 1/4 oforbital periodPlanet 1/4 of wayaround orbital pathPAphelionPerihelionSunArea (Sun, P, Perihelion) = Area(Sun, P, Aphelion) = 1/4 area of ellipse
13Kepler’s third law P2 = K a3 The square of a planet’s orbital period (P) is proportional to the cube of its orbital semi-major axis (a)a3P2MercuryPlutoSlope = KP2 = K a3where,P = planet orbital perioda = orbit’s semi-major axisK = a constantif P(years) and a(AU) then K = 1 and P2(yr) = a3(AU)
14Observational Evidence Planetsidereal period (years)semi major axis (AU’s)a3/P2Mercury0.2410.3870.998Venus0.6150.7230.999Earth1.000Mars1.8811.524Jupiter11.865.2031.001Saturn29.469.54Uranus84.8119.18Neptune164.830.06Pluto248.639.440.993The above data confirm Kepler’s 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.
16You will havean scale drawing of the Earth's orbit and the Earth's positions on its orbit on some dates, marked of at ten day intervals.a list, similar to this oneMonthDayYearElongationDirectionFeb6158826°WApr1820°EJun524°…Dec1121°Jan1589119°Nov2322°159023°
17PROCEDURE For each elongation: 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.
19PROCEDURECenter 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 2nd 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.
20as seen from Earth, the 2nd line will be From the first line of the example table:Feb 6, elongation = 26° W26º2nd lineas seen from Earth, the 2nd line will beto the left of the Sun if the elongation is to the East,to the right of Sun if the elongation is to the West.FebFeb 6
21PROCEDUREAfter 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.
22After 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.
23After 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 Earth’s 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 bymultiply your measurements in centimetres by the scale to convert them to AUs.Scale = ( 1.5A.U. / l )in (AU/cm)
24Report Plot of Mercury orbit Semi major axis Eccentricity of the orbit Verify Kepler’s second lawDueon Friday Nov 3, 5 pmat Prof. Caldwell’s office (332 Petrie Building)