Today is Tuesday, April 30th, 2019
←
→
Page content transcription
If your browser does not render page correctly, please read the page content below
In This Lesson:
Planetary Motion
(Lesson 2 of 2)
Today is Tuesday,
th
April 30 , 2019
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 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.
Key Ellipse Vocabulary
• Eccentricity is the deviation of an ellipse from
a perfect circle, equal to the distance between
the foci divided by major axis.
Zero
High Eccentricity
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.
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
=
1 Area 2
Going from P1 to P2 2
takes as much time as
going from P3 to P4.
Really Nerdy Science Joke https://xkcd.com/21/
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 = a3Kepler’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
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
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.
• You can use AU or km for “a” here.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.Unit 2 Quiz • At this point we’ve covered everything we need for the Unit 2 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.gifRetrograde 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, Mars?
– Fear not. Kepler nailed
this one down too.
http://upload.wikimedia.org/wikipedia/commons/7/70/Apparent_retrograde_motion_of_Mars_in_2003.gifRetrograde 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
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.jpgOpposition 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.jpgNewton’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, technically, the resistance to acceleration an
object has (its inertia).
– You can think of it, for just our class, as “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
DENSITY.
• Angular momentum is momentum caused by
rotation/revolution around a massive object.
– Rotational movement, in other words.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.
In this case, the
force of the bat
on the ball is
given by F = maNewton’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 wayNewton + Kepler = Awesome
• Newton could explain Draw these
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)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.
– G is a constant, so in a way it has no direct effect on gravity
itself.
G=
6.67408 × 10-11 m3 kg-1 s-2
…but we won’t do much
with it in Astronomy.As a result…
• Newton’s Law of Gravitation
explains the whole “long-
distance 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:
2m
m = GM Vescape =
R
G = Gravitational Constant μ = Gravitational Parameter
M = Mass R = Radius of planet
μ = Gravitational Parameter Vescape = Escape Velocity
How do we increase μ? How do we increase Vescape?Combined Escape Velocity Formula
2GM
Vescape =
R
• Let’s imagine that:
• G is 1 (just for simplicity)
• M is 10
• R is 5
• Vescape = 2
• How can 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
2m
Vescape =
R
R = 71, 569.26 km
2(126, 686, 534)
59.5 = D = 2R
R
253,373, 068 D = 143,138.52 km
59.5 =
R
NASA lists Jupiter’s
253,373,068
3540.25 = diameter at 142,984
R km, so it’s pretty close
to our calculations.Escape Velocity Practice
Body μ (km3s−2)
• Escape Velocity Interactive Sun 132,712,440,018
Activity Mercury 22,032
Venus 324,859
– #9-10. Earth 398,600.4418
– Note: For #10, the Moon’s Moon 4,902.8000
radius is 1737.4 km. 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 1108Closure • Smarter Every Day – How to Fly a Spaceship to a Space Station
You can also read
