orbit of mercury: following keplers steps nats 1745 b
TRANSCRIPT
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)