Transcript Kepler`s Laws of Planetary Motion

In This Lesson:
Planetary Motion
(Lesson 2 of 2)
Today is Thursday,
th
April 27 , 2016
Pre-Class:
In your notebooks, draw a rough sketch of the Earth orbiting the Sun.
Importantly, you should show it from above (as though you’re above
the Sun looking down at the orbit path). Finally, indicate the Sun and
where in your drawing the Earth’s orbit is fastest.
Also, get a small piece of a paper towel and your calculator.
 That’s Earth.
http://solarsystem.nasa.gov/images/PIA17171_708a.jpg
Saturn from the Cassini spacecraft
Today’s Agenda
• Kepler’s Laws.
– Look out! Physics!
• Newton’s Laws.
– Look out! Physics again!
• The movements of the planets.
– Look out! Mercury is in retrograde!
• Escape velocity.
– Look out! Gravity!
• Where is this in my book?
– Pages 51-52, 75-88.
By the end of this lesson…
• You should be able to calculate major features
of a planet’s elliptical orbit using mathematics.
• You should be able to explain the apparent
retrograde motion of a celestial body.
• You should be to able to determine the
necessary escape velocity for a projectile
leaving a massive object.
Kepler’s Laws
General Information
• The planets orbit the Sun, yes, but they don’t orbit
in a perfect circle with the Sun at the center.
– It’s more of an ellipse, the eccentricity of which varies
for each planet.
• Kepler’s Laws seek to explain these ellipses and
some of the unifying themes for each.
– I will give you the official law and a “plain English”
translation for each.
– Just like Newton’s Laws, the Laws of Thermodynamics,
and Kardashians to keep up with, there are three of
them.
Kepler’s Laws
• When last we left our early astronomer
friends, the general population of Earth was
just getting over yet another ego-trip.
– “What? I’m not the center of the solar system?”
• In the midst of what was a rather long-lasting
little controversy, Johannes Kepler put forth
what we now call Kepler’s Laws of Planetary
Motion.
Ellipse Review
• Eccentricity is the deviation of an ellipse from
a perfect circle.
Zero
Eccentricity
High Eccentricity
• Major axis is the “long distance” from the
ends of an ellipse.
• Semi-major axis is half the major axis.
Ellipse Details
• The semi-major axis is the distance from the ellipse’s
center to its farthest edge (given by “a”).
• The semi-minor axis (less important to astronomy) is
the distance from the ellipse’s center to its closest edge
(given by “b”).
• The foci are the two points around which the ellipse is
generated (given by “f”).
b
a
f
f
Kepler’s First Law
• The orbit of a planet is an
ellipse with the Sun at one
of the two foci (plural of
focus).
• Plain English: A planet’s
orbit isn’t a perfect circle
and the Sun is at one “end”
of the oval or the other.
• Note: the “radius” of the
orbit is known as the semimajor axis.
Kepler’s Second Law
• A line segment joining a planet and the Sun sweeps out
equal areas during equal time intervals.
• Plain English: Planets move at different speeds through
their orbits, so they each cover equal “ground” in equal
time frames within each of their orbits.
Area 1
=
Area 2
1
Going from P1 to P2
takes as much time as
going from P3 to P4.
2
Kepler’s Second Law
• By the way, do you see aphelion and perihelion?
– As a reminder, for Earth, aphelion (furthest from Sun) is
in July and perihelion (closest to Sun) is in January.
– Remember, Earth’s orbit does not explain the seasons.
Kepler’s Second Law
• Kepler’s Second Law Interactive
Kepler’s Third Law
AKA the Harmonic Law
• The square of the orbital
period of a planet orbiting
the Sun is proportional to
the cube of the semi-major
axis of its orbit.
• Plain English: The more
eccentric (oval) a planet’s
orbit, the longer it will take
to complete a revolution
around the focus/Sun.
• P is the orbital period (year).
• a is the semi-major axis (au).
• P2 = a3
Kepler’s Third Law Example
• Suppose Uranus has a semi-major axis of 19.18 au.
• How long is Uranus’s orbital period? In other words,
how many Earth-years does it take Uranus to make one
orbit around the Sun?
•
•
•
•
•
P2 = a 3
P2 = 19.183 au
P2 = 7055.79 au
P = 7055.79 au
P = 83.998 years
• (which is true – Uranus’s orbital period is 84 years, which means in
combination with its axial tilt means each seasons is 20+ years long)
Kepler’s Third Law
• Kepler’s Third Law Interactive
Practice
• Kepler’s Laws Practice worksheet
Extensions of Kepler
• Kepler’s Laws allow us a couple other useful
equations, still using P (orbital period) and a
(semi-major axis).
– We have to throw in one variable: e (eccentricity).
• The distance at aphelion; Q = a (1 + e)
• The distance at perihelion; q = a (1 – e)
– You can remember which is which since aphelion
will always work out to be longer than perihelion.
Aphelion/Perihelion Example
• Calculate the distance between Mercury and
the Sun during Mercury’s closest pass to the
Sun. Mercury’s semi-major axis is 0.387 au
and its orbit’s eccentricity is 0.2056.
• Distance at perihelion = a (1 – e)
• Distance at perihelion = 0.387 (1 – 0.2056)
• Distance at perihelion = 0.387 (0.7944)
• Distance at perihelion = 0.307 au.
– Sure enough, NASA lists Mercury’s perihelion
distance as 4.6 x 107 km, or 0.3075 au.
Kepler Practice Quiz
• To see how you’re doing, we’re going to take a
practice quiz on Kepler’s laws.
• Keep in mind you will be graded on accuracy.
– Kepler Practice Quiz
Backward Planets
• So those are Kepler’s Laws. Pretty logical.
• However, early stargazers noticed that some
of the planets appear to move backward (!) in
the sky during certain times of the year.
– Instead of going East to West (relative to Earth, or
West to East relative to the celestial sphere), they
reverse direction momentarily before continuing
on their normal way.
• Uh…what?
– Did you miss something, Kepler?
Backward Planets
• Let’s get a couple terms down:
– Prograde motion is when an object
moves in the same direction relative
to another.
• Like how the Sun rotates
counterclockwise and Earth orbits
counterclockwise.
– Retrograde motion is when an object
moves in the opposite direction
relative to another.
• Venus has a retrograde rotation.
• Some planets appear to have retrograde
motion.
https://upload.wikimedia.org/wikipedia/commons/8/82/RetrogradeBaan.gif
Retrograde Motion
• To explain retrograde motion,
in ~150 AD, Ptolemy
(remember him?) put forth
the idea of epicycles, which
he said were smaller orbits
within larger orbits called
deferents.
• This was his way of explaining
retrograde motion.
– Recall that Earth is at the
center of Ptolemy’s solar
system.
– This model, geocentric but
complete with epicycles, is
called the Ptolemaic model.
Retrograde Motion
• As wrong as Ptolemy
was, his idea was
accepted for 1300 years.
• Planets don’t do that
epicycle thing, but then
how do you explain
movement as shown in
multiple-exposure shots
of, let’s say, Mercury?
– Fear not. Kepler nailed
this one down too.
http://upload.wikimedia.org/wikipedia/commons/7/70/Apparent_retrograde_motion_of_Mars_in_2003.gif
Retrograde Motion
• The answer lies in the “overtaking” of one planet by
another.
– It’s much like being in a faster car as you pass a slower
one.
• Retrograde Motion Interactive
– Even with this being a completely natural phenomenon,
a lot of people cite “Mercury being in retrograde” as a
reason for technology malfunctioning and warn against
signing contracts and other commitments during those
time periods.
Retrograde Motion
Practice
• Retrograde Motion Activity
Now to put it all together…
• Unit 2 Quiz
Opposition and Conjunction
• Let’s go back in time…to when we defined the
terms sidereal and synodic.
• Sidereal means relating to…?
– Background stars.
• Synodic means relating to…?
– Conjunctions between two celestial bodies.
• Wait…what?
Opposition and Conjunction
• Before we define that, another thing to
consider is this example:
– How far away is Mars?
• Answering that depends on whether Mars is
on our side of the Sun or not…since it could be
very far away.
• Enter the terms opposition and conjunction.
Opposition and Conjunction
• Opposition is when the
lines of sight between two
celestial bodies are
completely opposite one
another.
– If Earth is between two
celestial bodies, those
bodies are at opposition as
viewed from Earth.
• If another celestial object
is along the same line of
sight as another, that
object is in conjunction
with Earth.
http://darkerview.com/darkview/uploads/Astronomy/ElongationOppositionConjunction.jpg
Opposition and Conjunction
• Conjunction goes further:
– If the object in conjunction
is between the Sun and
Earth, it’s at inferior
conjunction.
• It follows that only the
inferior planets – Mercury
and Venus – can reach
inferior conjunction.
– If the object in conjunction
is on the other side of the
Sun, it’s at superior
conjunction.
http://darkerview.com/darkview/uploads/Astronomy/ElongationOppositionConjunction.jpg
Newton’s Laws
• It turns out we also need to investigate
Newton’s Laws of Motion, since combined
with Kepler’s Laws we get a nice view of the
solar system.
• Let’s take a look at Newton’s Laws, then a
combination of Kepler and Newton to
interpret the motion of the planets.
• To help us understand Newton, here’s your
brief physics lesson/reminder.
Intro to Physics
• Mass is the amount of matter in an object.
• Weight is the force on an object due to gravity.
– Mass and weight are not the same: Your Weight on
Other Worlds
• Gravity is the attractive force between physical
bodies; gravity generally increases with increased
mass.
• Angular momentum is momentum caused by
rotation/revolution around a massive object.
Newton’s First Law of Motion
• An object at rest will stay at rest until some force acts on it.
• An object in motion will stay in uniform motion until another
force acts on it.
• In one word? Inertia.
• If the object’s velocity changes, it is a change in acceleration.
Newton’s Second Law of Motion
• The relationship between an object’s mass
(m), its acceleration (a), the force (F) applied
to get that mass accelerating is F = ma.
Newton’s Third Law of Motion
• For every action, there is an equal and
opposite reaction.
– Like stepping off a skateboard:
• You move forward.
• The skateboard moves backward.
– Like rockets:
Rocket is pushed this way
Fuel is pushed this way
Newton + Kepler = Awesome
• Newton could explain
Kepler’s 2nd and 3rd laws
using gravity:
– Planets traveling in ellipses at
constant speeds.
– The more oval-shaped, the
longer the orbit.
• Newton found that three
possible orbits could come
from Kepler’s 2nd/3rd laws:
– Elliptical (Bound)
– Parabolic
– Hyperbolic (straight line)
 Draw these 
Explaining Planetary Motion
• Here on Earth, a ball
thrown upward comes
down in an arc due to
the ever-present force of
gravity.
– The gravity force is
shown as the downward
black arrows in the
diagram.
• Gravity is the force
preventing the ball from
continuing in a straight
line.
Explaining Planetary Motion
• Were it not for the force of
gravity acting on planets,
they would continue in a
straight line away from the
Sun.
• The Sun’s gravity, however,
is constantly pulling the
planet inward, resulting in
a circular-ish path.
Newton + Kepler = Awesome
• The complicated part is
how to explain gravity
acting over a long
distance.
– For that, we need Newton’s
Law of Gravitation.
• Dude had a law for
everything.
Newton’s Law of Gravitation
• The gravitational force (Fg) on an object is proportional
to the mass of the first object (M1) times the mass of
the second object (M2) divided by the square of the
distance between them (d), all multiplied by the
gravitational constant (G).
– So generally, the smaller the distance or greater the mass,
the greater the force of gravity.
As a result…
• Newton’s Law of Gravitation
explains the whole “longdistance gravity” thing because
the Sun is so much bigger than
any of the planets.
• As a result, the center of mass
between the two (Sun + a
planet) is relatively close to the
Sun.
– In the same way, when you
throw a ball, the center of mass
is pretty much Earth, negating
the distance.
Misconception Alert!
• A lot of people misinterpret Newton’s Law of
Gravitation to suggest that the outer planets have
less gravity than the inner planets.
– That makes no sense.
•
•
•
•
Neptune has a greater force of gravity than us.
So does Jupiter.
Pluto doesn’t.
Uranus doesn’t.
• When we talk about distance between objects, we
don’t mean between a planet and the Sun.
– Distance refers to the space between the planet and
an object on the planet.
For Fun, Perspective, and Practice
And then perspective again.
• Family Guy – Gravity
• UniverseToday – Can You Escape the Force of
Gravity?
• Gravity Variations Interactive
• Gravity Exploration activity
• UniverseToday – How Do Gravitational
Slingshots Work?
Escape Velocity
• All this gravity/planet stuff brings up a valid point:
– How can we get a spacecraft off Earth and into orbit (or
beyond)?
• In short: we need to be mindful of what’s called
escape velocity (or escape speed).
– Escape velocity is the speed necessary to escape and
become unbound by Earth’s gravity.
• When Ronald Reagan described “slipping the surly
bonds of Earth” after the Challenger disaster, he
was talking in part about escape velocity.
*Don’t kill me, physics people: Technically “speed” is the more accurate term here.
Launch Videos
• Challenger Disaster
• Ronald Reagan Space Shuttle Challenger Explosion
Speech 1-28-1986
• Apollo 11 Launch
• What do you notice, in both launch videos, about the
angle of the launch? Why launch like that?
– The transition of the launch vehicle from “straight up” to
“kinda sideways-ish” is known as a gravity turn.
– It’s done to provide a more efficient launch either into orbit
or out of Earth’s gravity entirely.
• The turn is after getting through the thickest part of the atmosphere.
• But why does that work? Physics.
– And that escape velocity thing.
Escape Velocity
• As we said, escape velocity is the launch speed
necessary to get an object into space and free from
the gravity of the underlying planet.
– Tie-in: UniverseToday – Why Doesn’t the Sun Steal the
Moon?
• There’s an equation to learn here, but rather than
jump straight into the math, let’s just start with a
conceptual interactive for you:
– What Determines Escape Velocity?
– Escape Velocity Interactive Activity
• Don’t do questions 9 and 10 yet. For reals.
Escape Velocity
• Launch something
slowly and it will
simply come back
to Earth.
• Launch it faster and
it may go into orbit.
– Like “falling around
the planet.”
• Launch it fast
enough and it will
escape Earth’s
gravity.
Escape Velocity
Escape Velocity Formula
• In order to do questions 9 and 10, we need to
know how to mathematically determine the
escape velocity from a celestial body.
• Here are the two formulas:
m = GM
Vescape =
2m
R
G = Gravitational Constant
M = Mass
μ = Gravitational Parameter
μ = Gravitational Parameter
R = Radius of planet
Vescape = Escape Velocity
How do we increase μ?
How do we increase Vescape?
Escape Velocity Example Calculation
• The gravitational parameter of Jupiter is
126,686,534 km3s-2. The escape velocity is
59.5 km/sec. What is the diameter of Jupiter?
– Okay, this one’s tough.
– We need to get the diameter and that’s going to
come from r.
– Let’s fill in the equations…
Escape Velocity Example Calculation
Vescape =
2m
R
59.5 =
2(126, 686, 534)
R
59.5 =
253,373, 068
R
253,373,068
3540.25 =
R
R = 71, 569.26 km
D = 2R
D = 143,138.52 km
NASA lists Jupiter’s
diameter at 142,984
km, so it’s pretty close
to our calculations.
Escape Velocity Practice
• Escape Velocity Interactive
Activity
– #9-10.
– Note: For #10, the Moon’s
radius is 1737.4 km.
Body
μ (km3s−2)
Sun 132,712,440,018
Mercury
22,032
Venus
324,859
Earth
398,600.4418
Moon
4,902.8000
Mars
42,828
Ceres
63.1
Jupiter
126,686,534
Saturn
37,931,187
Uranus
5,793,939
Neptune
6,836,529
Pluto
871
Eris
1108
Closure
• Smarter Every Day – How to Fly a Spaceship to
a Space Station