Transcript Orbits and Gravity - Wayne State University

Orbits and 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 pretelescope 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
From his study of Mars and also the other planets,
Kepler discovered 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
This discovery was a significant
departure from the prevailing
thinking at the time, rooted in
ancient Greek philosophy, that
planetary orbits must be circles
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Kepler’s Ellipse (1)
• An ellipse is the simplest (next
to the circle) kind of closed
curve, belonging to a family of
curves known as conic sections
• It has two different diameters,
and the larger of the two is
called its major axis
• The semi-major axis is
• one half of the major axis
• equal to the distance from the
center of the ellipse to one end of
the ellipse
• also the average distance of a
planet from the Sun at one focus
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Kepler’s Ellipse (2)
• The minor axis of an ellipse is
the length of its shorter
diameter
• 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 orbit that is
farthest from the Sun
• The aphelion is thus on the major axis directly opposite
the perihelion
• The line connecting the aphelion and the
perihelion is none other than the major axis
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Kepler’s Ellipse (3)


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An ellipse has two special
points, called its foci
(singular: focus), along its
major axis
The sum of the distances
from any point on the
ellipse to the foci is always the same
The Sun is at one of the two foci (nothing
is at the other one) of each planet's
elliptical orbit, NOT at the center of the
orbit!
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Kepler’s Ellipse (4)

The eccentricity (e) of an ellipse is defined
as the ratio of the distance between its foci
to the length of its major axis
 The eccentricity indicates how elongated the
ellipse is

An ellipse becomes a circle when the 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
 A straight line has an eccentricity of 1
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Orbits of Planets

The orbits of planets 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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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 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 Pythagorean
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 2nd law
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Kepler's Second Law (2)

Physicists found that the 2nd law is
a consequence of the conservation
of angular momentum
3
1
S
2
The surfaces S-1-2
and S-3-4 are equal
 The angular momentum of a planet
is a measure of the amount of its
orbital motion and does NOT change
as the planet orbits the Sun
 The angular momentum of a planet equals (its mass) ×
(its transverse speed) × (its distance from the Sun)
•
The transverse speed of a planet is the amount of its
orbital velocity that is in the direction perpendicular to
the line joining the planet and the Sun
 Thus, for example, if the distance decreases, then the
speed must increase to compensate
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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 3rd 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 Copernical
(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

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Newton (1643-1727), who was born
to a family of farmers in Lincolnshire,
England, in the year after Galileo's
death, went to college at Cambridge
and was later 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
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
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 counter-

intuitive 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
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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 statets 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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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)

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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” 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
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Examples of density

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 as small as 0.1 g/cm3
 Gold, on the other hand, is "heavy" and has a
density of 19 g/cm3
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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
•
•
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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 Las 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
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Kepler’s Third Law Revisited (2)
 In the Newton’s formulation above
 D is distance to the Sun, expressed in
astronomical units (AU)
 P is the period, expressed in years
 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
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Kepler’s Third Law Revisited (3)
 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 the 3rd law
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Kepler’s 3rd Law Revisited (4)


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
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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
 Provided that 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
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Artificial Satellites and Space Flight (2)
 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
17500 miles per hour)
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Artificial Satellites and Space Flight (3)
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

Sputnik, the first artificial Earth satellite,
was launched by what wast the 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 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 burn 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 spacecrafts, are designed to make the
re-entry possible by adding a heat shield below
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 complicated and are best done
today with large computers
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Discovery of Neptune

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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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