# Orbit of Mercury: Following Kepler’s steps

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

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.

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)

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

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

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

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

Apparent 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.

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

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

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

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 at 1/4 of orbital period Planet 1/4 of way around orbital path P Aphelion Perihelion Sun Area (Sun, P, Perihelion) = Area(Sun, P, Aphelion) = 1/4 area of ellipse

Kepler’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) a3 P2 Mercury Pluto Slope = K P2 = K a3 where, P = planet orbital period a = orbit’s semi-major axis K = a constant if P(years) and a(AU) then K = 1 and P2(yr) = a3(AU)

Observational Evidence
Planet sidereal period (years) semi major axis (AU’s) a3/P2 Mercury 0.241 0.387 0.998 Venus 0.615 0.723 0.999 Earth 1.000 Mars 1.881 1.524 Jupiter 11.86 5.203 1.001 Saturn 29.46 9.54 Uranus 84.81 19.18 Neptune 164.8 30.06 Pluto 248.6 39.44 0.993 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.

The assignment

You will have 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. a list, similar to this one Month Day Year Elongation Direction Feb 6 1588 26° W Apr 18 20° E Jun 5 24° Dec 11 21° Jan 1589 1 19° Nov 23 22° 1590 23°

PROCEDURE 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.

From the first line of the example table: Feb 6

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.

as seen from Earth, the 2nd line will be
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

PROCEDURE 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.

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.

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. Scale = ( 1.5A.U. / l ) in (AU/cm)

Report 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)