Orbital Mechanics and Kepler’s Laws
A Simple Intelligence Test
What is the best baseball team in Southern California?
A
The Angels
B
The Dodgers
x
An Angstrom is the most suitable unit for describing a measurement of
A
B
The distance from the earth to the sun
A small angle
C
The wavelength of visible light
D
The speed of light
Multiplying together two numbers N × 10X and M × 10Y in scientific notation yields:
A
(N × X) × 10M × Y
B
(N + M) × 10X + Y
C
(N × M) × 10X × Y
D
(N × M) × 10X + Y
Orbits and Kepler
•
One of the first challenges to astronomy was the explanation of retrograde motion (the apparent “backtracking” of planets) •
Ptolemy, working with an earth-­
centered model, used epicycles (circles on circles)
Interactive Figure 3.15
•
Copernicus proposed a sun-­
centered system, but still used circles
•
Kepler solved the problem in the early 17th century
Kepler’s Laws
• 1st Law -­ The Planets move in elliptical orbits with the Sun at one focus
• 2nd Law -­ Orbits sweep out equal areas in equal times
• 3rd Law -­ More distant planets orbit at slower average speeds, obeying a simple mathematical relationship
p 2 ∝ a3
where p is the period of an orbit, and a is the semi-­major axis of the planet’s orbit
Solar system data
Halley’s comet
17.8
0.967
75.3
1.0054
Kepler’s 2nd Law
“Equal areas in equal times”
•
•
In a given time, a line connecting the Sun to the planet will sweep out an area that is the same in all parts of the orbit (Int. Fig. 3.20)
Thus, the planet moves faster when it is closer to the Sun (Int. Fig. Kepler II)
Kepler’s 2nd Law Quiz I The planet shown in the drawing obeys Kepler’s Second Law. At which lettered position is the planet speeding up?
B
A
C
D
E
Kepler’s 2nd Law Quiz II The planet shown in the drawing obeys Kepler’s Second Law. Each lettered position represents a particular day during the year. During which day (at which lettered position) will the planet move the shortest distance?
B
A
C
D
E
Kepler’s 2nd Law Quiz III Which of the three orbits shown below would you say most closely matches the shape of Earth’s orbit around the Sun? A
B
C
Kepler’s 2nd Law Quiz IV Which of the three orbits shown below would you say most closely matches the shape of Halley’s comet’s orbit around the Sun?
A
B
C
Kepler’s 3rd Law
•
Kepler’s 3rd Law concerns the relationship between the semi-­major axis (a) of a planet from the Sun and its orbital period (p)
•
The smaller the orbit (a), the shorter its period (p)
•
This is both because the distance traveled each period is shorter and because the closer planets move faster (Interactive Figure 7.1)
•
Kepler’s 3rd Law can be expressed mathematically as p 2 ∝ a3
•
In “solar system units” (p in yrs, a in AU) this becomes
p 2 = a3
•
Note that the period (and speed) do not
depend on the planet’s mass
Kepler’s 3nd Law
Period 2 ∝ Distance3
Solar system data
Halley’s comet
17.8
0.967
75.3
1.0054
Kepler’s 3rd Law Quiz I If a small chunk of rock and the large International Space Station are orbiting Earth at the same altitude above Earth’s surface, which object takes longer to orbit once around Earth?
A
The large space station
B
The small chunk of rock
C
They would take the same amount of time
Kepler’s 3rd Law Quiz II Consider a planet orbiting the Sun. If the orbital distance of the planet doubled, then the planet would take
A
More than twice as long to orbit the Sun
B
Exactly twice as long to orbit the Sun
C
The same amount of time to orbit the Sun
D
Exactly half as long to orbit the Sun
E
Less than half as long to orbit the Sun
Kepler’s 3rd Law Quiz III Consider a planet orbiting the Sun. If the mass of the planet doubled, then the planet would take
A
More than twice as long to orbit the Sun
B
Exactly twice as long to orbit the Sun
C
The same amount of time to orbit the Sun
D
Exactly half as long to orbit the Sun
E
Less than half as long to orbit the Sun
A comet’s orbit has an eccentricity e=0.8. How many times faster is the comet moving at perihelion versus aphelion?
A
3
B
4
C
5
D
9
A new planet is discovered orbiting a 2 M¤ star with a period of 16 years. What is the semi-­major axis of that planet’s orbit?
A
6 AU
B
8 AU
C
16 AU
D
22 AU
Center-­of-­mass