18 Jan 2005 AST 2010: Chapter 2 1

Orbits and Orbits and GravityGravity

18 Jan 2005 AST 2010: Chapter 2 2

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)

18 Jan 2005 AST 2010: Chapter 2 3

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

18 Jan 2005 AST 2010: Chapter 2 5

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) 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

18 Jan 2005 AST 2010: Chapter 2 11

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 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

S1

2

34

The surfaces S-1-2 and S-3-4 are equal

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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  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

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) 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 • 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• 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) 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 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!