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

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

5. 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

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

7. 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
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. 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)

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

11. 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)

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 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

13. 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)

14. 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.

15. The assignment

16. 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°

17. 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.

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

21. 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.

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. 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)

24. 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)