Orbit of Mercury: Following Kepler’s steps

1. Orbit of Mercury: Following Kepler’s steps NATS 1745 B

2. 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.

3. Terminology Superior planet- a planet with an orbit greater than Earth’s (e.g. Mars, Neptune) Inferior planet- a planet with an orbit smaller than Earth’s (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 (180o apart) on the sky (as seen from Earth)

4. Terminology Cont’d Planet Sun Elongation Earth Elongation: The angular separation of a planet from the Sun (as seen from the Earth)

5. Line of Sight (LOS) Planet Right-angle RP RE Earth Sun Greatest elongation (from observations) The Astronomical Unit (AU) is the average distance between the Earth and the Sun 1 AU = 1.496 x 108 km RE = radius of Earth’s orbit = 1 AURP = radius of planets orbit Definition:

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

7. 0 The Motion of the Planets The planets are orbiting the sun almost exactly in the plane of the ecliptic. Jupiter Venus Mars Earth Mercury The moon is orbiting Earth in almost the same plane (ecliptic). Saturn

8. Apparent Motion of the Inner Planets 0 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.

9. P Semi-minor axis (b) r1 r2 O F2 F1 Two focal points Semi-major axis (a) Major axis The ellipse Distance OF1 = OF2 Definition: Eccentricity (e)

10. Planetary orbit - exaggerated center Aphelion perihelion Sun “empty” focus FirstKepler’s law Planets have elliptical orbits, with the Sun at one focus

11. Time T D C B E Time T Time T F G A Second Kepler’s law The planet-Sun line sweeps out equal areas in equal time 2nd law says: if area AFB = area CFD = area EFG then time (A to B) = time (C to D) = time (E to G)

12. Second Kepler’s law cont’d • Perihelion- closest point to Sun • Near perihelion planet moves faster • Aphelion- greatest distance from Sun • Near aphelion planet moves slower Planet 1/4 of way around orbital path Planet at 1/4 of orbital period P Aphelion Perihelion Sun Area (Sun, P, Perihelion) = Area(Sun, P, Aphelion) = 1/4 area of ellipse

13. P2 Pluto Slope = K Mercury a3 if P(years) and a(AU) then K = 1 and P2(yr) = a3(AU) Kepler’s third law The square of a planet’s orbital period (P) is proportional to the cube of its orbital semi-major axis (a) P2 = K a3 where, P = planetorbital period a = orbit’ssemi-major axis K = a constant

14. Observational Evidence • The 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.

15. The assignment

16. You will have an scaledrawingofthe Earth's orbitand theEarth's positionson its orbit on some dates, marked of atten day intervals. a list, similar to this one

17. PROCEDURE For each elongation: • Locate the date of the maximum elongation on the orbit of the Earth and draw a lightpencil line from this position to the Sun.

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

19. PROCEDURE • 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 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.

20. From the first line of the example table: Feb 6, elongation = 26° W 26º 2nd line • as seen from Earth,the 2nd line will be • to the left of the Sun if the elongation is to the East, • to the right of Sun if the elongation is to the West. Feb Feb 6

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

22. 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.

23. Scale = ( 1.5A.U. / l ) in (AU/cm) 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 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 by • multiply your measurements in centimetres by the scale to convert them to AUs.

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