Transcript Orbits and Gravity - Wayne State University

Orbits & Gravity
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Laws of Planetary Motion
Two of Galileo’s contemporaries made
dramatic advances in understanding the
motions of the planets
Tycho Brahe (1546-1601)
Johannes Kepler (1571-1630)
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Tycho Brahe (1)
Born to a familiy of Danish nobility, Tycho
developed an early interest in astronomy and as a
young man made significant astronomical
observations
Among these was a careful study of the explosion
of a star (a nova)
Thus he acquired the patronage of Danish King
Frederick II
This enabled Tycho to establish, at age 30, an
observatory on the North sea island of Hven
He was the last and greatest of the pre-telescope
observers in Europe
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Tycho Brahe (2)
He made a continuous record of the positions of
the Sun, Moon, and planets for almost 20 years
This enabled him to note that the actual positions
of the planets differed from those in published
tables based on Ptolemy’s work
After the death of his patron, King Frederick II,
Tycho moved to Prague and became court
astronomer for the Emperor Rudolf of Bohemia
There, before his death, Tycho met Johannes
Kepler, a bright young mathematician who
eventually inherited all of Tycho’s data
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Johannes Kepler
Kepler served as an assistant to Tycho Brahe, who set
him to work trying to find a satisfactory theory of
planetary motion — one that was compatible with the
detailed observations Tycho made at Hven
For fear that Kepler would discover the secrets of the
planetary motions by himself, thereby robbing Tycho of
some of the glory, Tycho was reluctant to provide Kepler
with much material at any one time
Only after Tycho’s death did Kepler get full possession
of Tycho’s priceless records
Their study occupied most of the following 20 years of
Kepler’s time
Using Tycho's data, Kepler derived his famous three
laws of planetary motion
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Kepler's First Law
Kepler’s most detailed study was of Mars
Following the prevailing thinking at the time,
rooted in ancient Greek philosophy, Kepler had
initially thought that the orbits of planets had to
be circles, but his study of Mars and also the
other planets contradicted this idea
He discovered instead
that each planet moves
about the Sun in an
orbit that is an ellipse,
with the Sun at one
focus of the ellipse
This is known as
Kepler's First Law
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Introduction to the Ellipse
The ellipse is the simplest (next to the
circle) kind of closed curve belonging
to a family of curves known as conic
sections, which are formed by the
intersection of a plane with a cone
Animation showing various conic sections
Unlike a circle, an ellipse has two
special points inside it called its foci
(plural for focus), and also two
different diameters
The larger diameter is called the major
axis, and the smaller one the minor axis
One half of the major axis is called the
semi-major axis, and that of the minor
axis the semi-minor axis
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The Ellipse (1)
The sum of the distances from the foci of an
ellipse to any point on the ellipse is always the
same
Animation showing elliptical motion and the foci
An ellipse can be drawn using a pencil, two tacks,
and a loop of string
The locations of the tacks
become the two foci
Since the length of the
string remains the same,
at any point where the
pencil may be, the sum
of the distances from the
pencil to the two tacks is
a constant length
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The Ellipse (2)
The roundness of an ellipse depends on how close together
the two foci are, compared with the major axis
The eccentricity (e) of an ellipse is defined as the ratio of
the distance between its foci to the length of its major axis
Thus, the eccentricity indicates how elongated the ellipse is
An ellipse becomes a circle when the two foci are at the
same place
Thus the eccentricity of a circle
is zero, e = 0
A very-long and skinny ellipse
has an eccentricity close to 1
and is said to be very eccentric
Thus, a straight line has an
eccentricity of 1
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Kepler’s Ellipse
The Sun is at one of the two
foci (nothing is at the other
one) of each planet's elliptical
orbit, NOT at its center!
The perihelion is the point on
a planet's orbit that is closest
to the Sun
Thus, the perihelion is on the
major axis
The aphelion is the point on a planet’s orbit that is
farthest from the Sun
The aphelion is thus on the major axis
directly opposite the perihelion
The orbits of the different planets
have different eccentricities
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Orbits of Planets
The orbits of the planets around
the Sun have small eccentricities
In other words, the orbits are
nearly circular
This is why astronomers before
Kepler thought the orbits were exactly circular
This slight error in the orbital shape accumulated into a
large error in a planet’s positions after a few hundred
years
Only very accurate and precise observations can show
the elliptical character of the orbits
Tycho's meticulous observations, therefore, played a key
role in Kepler's discovery
This is an excellent example of a fundamental
breakthrough in our understanding of the universe
being possible only from greatly improved
observations
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Orbital Data for the Planets
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Orbits of Comets
A comet is a small body of icy and dusty
matter that revolves around the Sun
When it comes near the Sun, some of its material
vaporizes, forming a large head of gas and often a
tail
The orbits of most comets have large
eccentricities
In other words, the orbits
look much like flattened
ellipses
The comets, therefore,
spend most of their time
far away from the Sun,
moving very slowly
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Kepler's Second Law (1)
From Tycho’s observations of the
planets’ motion (particularly Mars'),
Kepler found that the planets speed
up as they come near the Sun and
slow down as they move away from it
This is yet another break with the ancient Greek
paradigm of uniform circular motion!
From this finding, he discovered another rule of
planetary orbits: the straight
line joining a planet and the
Sun sweeps out equal areas in
space in equal intervals of time
This is now known as Kepler's
Second Law
The surfaces S-1-2 and S-3-4 are equal
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Kepler's Second Law (2)
Physicists found that the 2nd law is a consequence of
the conservation of angular momentum
Angular momentum is a measure of the rotational
motion of an object about some fixed point
Whenever we study rotating or spinning objects, from
planets to galaxies, we have to consider angular
momentum
The angular momentum of an object equals (its mass)
× (its speed) × (its distance from the fixed point
around which it turns)
Generally, in any rotating system in which no external
forces act, angular momentum is constant
This implies that if the distance decreases, for example,
then the speed must increase to compensate
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Conservation of Angular Momentum
An example of such a system (where angular
momentum is conserved) is a planet moving
around the Sun on an elliptical orbit
When the planet approaches the Sun, the distance
to the orbital center decreases, and the planet
speeds up to keep angular momentum the same
Similarly, when the
planet is moving
farther from the Sun,
it moves more slowly
Another example is a
figure skater spinning
on ice
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Kepler's Third Law (1)
Finally, after several more years of calculations, Kepler
found a simple and elegant relationship between the
distance of a planet from the Sun and the time the
planet took to go around the Sun
The relationship is that the squares of the planets’
periods of revolution about the Sun are in direct
proportion to the cubes of the planets’ average
distances from the Sun
This is now known as Kepler's Third Law
For each planet in the solar system, if the period is
expressed in years and the distances is expressed in
AU (the Earth’s average distance from the Sun),
Kepler’s 3rd law takes the very simple form
(period)2 = (average distance)3
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Kepler’s Third Law (2)
As an example, Kepler’s third law is satisfied by Mars'
orbit
The length of Mars’ semi-major axis (the same as Mars’
average distance from the Sun) is 1.52 AU, and so
1.523 = 3.51
Mars takes 1.87 years to go around the Sun, and so
1.872 = 3.51
Kepler’s third law, as well as the other two, provided a
precise description of planetary motion within the
framework of the Copernican (heliocentric) system
Despite the successes of Kepler’s results, they are
purely descriptive and do not explain why the planets
follow this set of rules
The explanation would be provided by Newton
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Sir Isaac Newton
Newton (1643-1727) was born to a
family of farmers in Lincolnshire,
England, in the year after Galileo's
death and went to college at
Cambridge, where he would later be
appointed Professor of Mathematics
He worked on a large number of science topics,
establishing the foundation of mechanics and
optics, and even created new mathematical tools
to enable him to deal with the complexity of the
physics problems
His work on mechanics led to his famous three
laws of motion …
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Newton's Laws of Motion
The 1st law states that every body continues
doing what it is already doing — being in a
state of rest, or moving uniformly in a straight
line — unless it is compelled to change by an
outside force
The 2nd law states that the change of motion
of a body is proportional to the force acting on
it, and is made in the direction in which that
force is acting
The 3rd law states that to every action there is
an equal and opposite reaction, or the mutual
actions of two bodies on each other are always
equal and act in opposite directions
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Newton's First Law (1)
This is basically a restatement of one of
Galileo's discoveries, called the conservation
of momentum
Momentum is a measure of a body's motion and
depends on 3 factors:
The body’s speed — how fast it moves
The direction in which the body moves
The body’s mass, which is a measure of the amount of
matter in the body
The momentum of the body is then its mass times
its velocity (velocity is a term physicists use to
describe both speed and direction)
Thus, a restatement of the 1st law is that in
the absence of any outside influence (force), a
body's momentum remains unchanged
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Newton's First Law (2)
At the onset, the 1st law is rather counterintuitive because in the everyday world forces
(such as friction, which slows things down) are
always present that change the state of
motion of a body
The 1st law is also called the law of inertia
Inertia is the natural tendency of objects to keep
doing what they are already doing
Thus, the 1st law implies that, in the absence
of outside influence, an object that is already
moving tends to stay moving
This contradicts the Aristotelian idea that every
moving object is always subject to an outside force
Animations illustrating the 1st law
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Newton's Second Law
The 2nd law defines force in terms of its ability to
change momentum
Thus, a restatement of the 2nd law is that the
momentum of a body can change only under the
action of an outside force
In other words, a force is required to change the
speed of a body, its direction, or both
The rate of change in the velocity of a body (its
change in speed, direction, or both) is called
acceleration
Newton showed that the acceleration of a body
was proportional to the force applied to it
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Newton's Third Law
The 3rd law states that to every action there is an
equal and opposite reaction
Consider a system of two bodies completely isolated
from influences outside the system
The 1st law then implies that the momentum of the
entire system should remain constant
Consequently, according to the 3rd law, if one of the
bodies exerts a force (such as pull or push) on the other,
then both bodies will start moving with equal and
opposite momenta, so that the momentum of the entire
system is not changed
The 3rd law implies that forces in nature always occur
in pairs: if a force is exerted on an object by a second
object, the second object will exert an equal and
opposite force on the first object
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Momentous Question
If a spaceship is moving at constant
speed along a straight line, is there
an outside force acting on the ship?
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Mass, Volume, and Density (1)
The mass of an object is a measure of the
amount of material in the object
The volume of an object is a measure of the
physical size or space occupied by the object
Volume is often measured in units of cubic
(centi)meters or liters
Thus, the volume indicates the size of an
object and has nothing to do with its mass
A cup of water and a cup of mercury may have
the same volume, but they have very different
masses
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Mass, Volume, and Density (2)
The density of an object is its mass divided by
its volume
Density is thus a measure of how much mass an
object has per unit volume
One of the common units of density is gram per
cubic centimeter (gm/cm3)
In everyday language, we often use “heavy”
and “light” as indications of density
Strictly speaking, the density of an object is
primarily determined by its chemical
composition — the stuff it is made of — and
how tightly pack that stuff is
To summarize, mass is “how much”, volume is
“how big”, and density is “how tightly packed”
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Examples of density (1)
An example of calculating density
If a block of some material has a mass of
600 g and a volume of 200 cm3, then its
density is (600 g)/(200 cm3) = 3 g/cm3
Familiar materials around us span a
large range of density
Artificial materials such as plastic insulating
foam can have densities less than 0.1 g/cm3
Gold, on the other hand, is "heavy" and has
a density of 19 g/cm3
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Examples of density (2)
Newton’s Law of Gravity (1)
Newton's 1st law tells us that an object at rest
remains at rest, and that an object in uniform
motion (with fixed speed and direction)
continues with this same motion
Thus, it is the straight line, not the circle, that
defines the most natural state of motion of an
object
So why are planets revolving around the Sun,
instead of moving in a straight line?
The answer is simple: some force must be bending
their paths
Newton proposed that this force is gravity
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Newton’s Law of Gravity (2)
To handle the difficult calculations of planetary
orbits, Newton needed mathematical tools that
had not been developed, and so he then
invented what we today call calculus
Eventually, he formulated the hypothesis of
universal attraction among all bodies
He showed that the force of gravity between
any two bodies
drops off with increasing distance between the
two in proportion to the inverse square of their
separation
is proportional to the product of their masses
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Newton’s Law of Gravity (3)
Newton provided the formula for this
gravitational attraction between any two
bodies:
Force = G M1 M2 / R2
where
G is called the constant of gravitation
M1 is the mass of the first body
M2 is the mass of the second body
R is the distance between the two bodies
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Newton’s Law of Gravity (4)
This law of gravity not only works for the
planets and the Sun, but also is universal
Therefore, this law should also work for, say, the
Earth and the Moon
Objects on the surface of the Earth — at R =
Earth’s radius — are observed to accelerate
downward at 9.8 m/s2
The Moon is at a distance of 60 Earth-radii
from Earth’s center
Thus the Moon should experience an acceleration
toward the Earth that is 1/602 or 3,600 times
less — that’s 0.00272 m/s2
This is precisely the observed acceleration of the
Moon in its orbit!
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Newton’s Law of Gravity (5)
Everything with a mass is subject to this
law of universal attraction
For most pairs of objects, this attraction
is rather small
It takes a huge body such as the Earth,
or the Sun, to exert a large force of
gravity
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Kepler’s Third Law Revisited (1)
Kepler's three laws of planetary motion are just
descriptions of the orbits of objects moving
according to Newton's laws of motion and law of
gravity
The knowledge that gravity is the force that
attracts the planets towards the Sun, however, led
to a new perspective on Kepler's third law
Newton's law of gravity can be used to show
mathematically that the relationship between the
period (P) of a planet’s revolution and its distance
(D) from the Sun is actually
D3 = (M1+M2) x P2
D is distance to the Sun, expressed in astronomical
units (AU)
P is the period, expressed in years
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Kepler’s Third Law Revisited (2)
Newton's formulation introduces a factor
which depends on the sum of the masses
(M1+M2) of the two celestial bodies (say, the
Sun and a planet)
Both masses are expressed in units of the Sun’s
mass
How come Kepler missed the mass factor?
Answer:
Expressed in units of the Sun’s mass, the mass of
each of the planets is much much smaller than one
This means that the factor M1+M2 is essentially one
(unity) and is, therefore, difficult to identify as
being different from one in the approach taken by
Kepler to derive his 3rd law
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Kepler’s Third Law Revisited (3)
Is this factor significant anywhere ?
Answer:
In the solar system, the Sun dominates the show
and all other objects have negligible masses
compared to the Sun’s mass and, therefore, the
factor is essentially equal to one
There are many cases in astronomy, however,
where this factor differs drastically from unity and,
therefore, the two mass terms have to be included
This is the case, for instance, when two stars, or two
galaxies, orbit around one another
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Artificial Satellites and Space Flight (1)
Kepler's laws apply not only to the motions of planets,
but also to the motions of artificial (man-made)
satellites around the Earth and of interplanetary
spacecraft
Once an artificial satellite is in orbit, its behavior is no
different from that of a natural satellite, such as the
Moon
As long as it is at sufficient altitude to avoid friction with
the atmosphere, the artificial satellite will "fly" or orbit
the Earth indefinitely following Kepler's laws
Maintaining an artificial satellite once it is in orbit is
thus easy, but launching it from the ground is an
arduous task
A very large amount of energy is required to lift the
spacecraft (which carries the satellite) into orbit
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Launching a Satellite into Orbit
To launch a bullet (or any other
object) into orbit, a sufficiently
large horizontal velocity is
needed
The speed required for a
circular orbit happens to be
independent of the size and
mass of the object (bullet or
anything else) and amounts to
approximately 8 km/s (or
17,500 miles per hour)
Newton’s cannon simulation
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Artificial Satellites and Space Flight (2)
Sputnik, the first artificial Earth satellite, was
launched by what was called the Soviet Union
on October 4, 1957
Since then, about 50 new satellites each year
have been launched into orbit by such nations
as the United States, Russia, China, Japan,
India, and Israel, as well as the European
Space Agency (ESA)
At an orbital speed of 8 km/s, objects circle
the Earth in about 90 minutes
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Artificial
Satellites
and Space
Flight (3)
Satellites
in Earth
orbit
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Artificial Satellites and Space Flights (4)
Low orbits are not stable indefinitely because
of drag forces generated by friction with the
upper atmosphere of the planet
The friction eventually leads to a “decay” of the
orbit
Upon re-entry in the atmosphere, most
satellites are burned or vaporized as a result
of the intense heat produced by the friction
with the atmosphere
Spacecraft such as the Space Shuttle, and
other recoverable spacecraft, are designed to
make the re-entry possible by adding a heat
shield around the spacecraft
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Interplanetary Spacecraft (1)
The exploration of our solar system has been
carried out mostly by automated spacecraft or
robots
To escape the Earth’s gravitational attraction,
these craft must achieve escape velocity, which is
the minimum velocity required to break away
from the Earth's gravity forever
The escape velocity is independent of the mass and
size of craft, and is solely determined by the mass
and radius of the Earth
This velocity amounts to approximately 11 km/s
(about 25,000 miles per hour)
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Interplanetary Spacecraft (2)
Once the spacecraft have broken away from Earth’s
gravity forever, they coast to their targets, subject
only to minor trajectory adjustments provided by
small thruster rockets on board
The craft’s paths obey Kepler’s laws
As a spacecraft approach a planet, it is possible by
carefully controlling the approach path to use the
planet’s gravitational field to redirect a flyby to a
second target
Voyager 2 used a series of gravity-assisted encounters to yield
successive flybys of Jupiter (1979), Saturn (1980), Uranus
(1986), and Neptune (1989)
The Galileo spacecraft, launched in 1989, flew past Venus
once and Earth twice to gain the speed required to reach its
ultimate goal of orbiting Jupiter
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Gravity with More than Two Bodies
The calculations of planetary motions
involving more than two bodies tend to
be very complicated and are best done
today with very
powerful computers
(supercomputers)
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Discovery of Neptune
Uranus was discovered by William Herschel in 1781
The orbit of Uranus was calculated and “known” by
1790, but it did not appear to be regular, namely, to
agree with Newton’s laws
In 1843, John Couch Adams made a detailed analysis
of the motion of Uranus, concluding that its motion
was influenced by a planet and predicted the position
of that planet
A prediction was also made independently by Urbain J.J.
Leverrier
The predictions by Adams and Leverrier were
confirmed by Johann Galle, who on September 23,
1846, found the planet, now known as Neptune
This was a major triumph for Newton’s theory of
gravity and the scientific method!
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