Orbit of Mercury: Following Kepler’s

steps

Orbit of Mercury: Following Kepler’s

steps

NATS 1745 BNATS 1745 B

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 smallerthan 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 planetfrom the Sun (as seen from the Earth)

Elongation

Earth

Sun

Planet

RE = radius of Earth’s orbit = 1 AURP = radius of planets orbit

RP

Sun

Line of Sight(LOS)

Earth

Greatest elongation(from observations)

Planet

RE

Right-angle

Definition: The Astronomical Unit (AU) is the averagedistance between the Earth and the Sun

1 AU = 1.496 x 108 km

Standard Planetary Configurations

E

Opposition

QuadratureQuadrature

Conjunction

Superior conjunction

Greatest westernelongation

Greatest easternelongation

Inferior Conjunction

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

MarsEarth

Venus

Mercury

Saturn

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.

Apparent Motion of the Inner Planets

The ellipse

Definition:Eccentricity (e)

DistanceOF1 = OF2

F1F2

Semi-major axis (a)

O

r1 r2

P

Major axis

Two focal points

Semi-minor axis (b)

2

1 1

a

b

a

OFe

First Kepler’s law

Planets have elliptical orbits, with the Sun at one focus

perihelionAphelionSun

Planetary orbit - exaggerated

center

“empty” focus

Second Kepler’s law

The planet-Sun line sweeps out equal areas in equal time

A

B

CD

E

G FTime T Time T

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:

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

Perihelion

P

SunAphelion

Planet 1/4 of wayaround orbital path

Planet at 1/4 oforbital period

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 = Kwhere, 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)

Planetsidereal 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 1.000 1.000

Mars 1.881 1.524 1.000

Jupiter 11.86 5.203 1.001

Saturn 29.46 9.54 1.000

Uranus 84.81 19.18 1.000

Neptune 164.8 30.06 1.000

Pluto 248.6 39.44 0.993

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.

The assignment

Month Day Year Elongation Direction

Feb 6 1588 26° W

Apr 18 1588 20° E

Jun 5 1588 24° W

… … … … …

Dec 11 21° E

Jan 18 1589 24° W

Apr 1 19° E

… … … … …

Nov 23 … 22° E

Jan 1 1590 23° W

… … … … …

Nov 5 1590 23° 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.

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:

Feb 6

Feb

From the first line of the example table:

Feb 6

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 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.

PROCEDURE

Feb 6

Feb

2nd line26º

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.

From the first line of the example table:

Feb 6, elongation = 26° W

• 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

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)