1AST 2010: Chapter 25 Jul 2005

Orbits & Gravity

5 Jul 2005 AST 2010: Chapter 2 2

Laws of Planetary MotionLaws of Planetary MotionTwo of Galileo’s contemporaries made dramatic advances in understanding the motions of the planets

Tycho Brahe (1546-1601)

Johannes Kepler (1571-1630)

5 Jul 2005 AST 2010: Chapter 2 3

Tycho Brahe (1)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 IIThis enabled Tycho to establish, at age 30, an observatory on the North sea island of HvenHe was the last and greatest of the pre-telescope observers in Europe

5 Jul 2005 AST 2010: Chapter 2 4

Tycho Brahe (2)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

5 Jul 2005 AST 2010: Chapter 2 5

Johannes Kepler 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

5 Jul 2005 AST 2010: Chapter 2 6

Kepler's First LawKepler's First LawKepler’s most detailed study was of MarsFollowing 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 ideaHe 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

7AST 2010: Chapter 25 Jul 2005

Introduction to the Ellipse 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 axisOne half of the major axis is called the semi-major axis, and that of the minor axis the semi-minor axis

8AST 2010: Chapter 25 Jul 2005

The Ellipse (1)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 fociAn 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

5 Jul 2005 AST 2010: Chapter 2 9

The Ellipse (2)The Ellipse (2)The roundness of an ellipse depends on how close together the two foci are, compared with the major axisThe 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

5 Jul 2005 AST 2010: Chapter 2 10

Kepler’s EllipseKepler’s EllipseThe 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

5 Jul 2005 AST 2010: Chapter 2 11

Orbits of PlanetsOrbits of PlanetsThe 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

5 Jul 2005 AST 2010: Chapter 2 12

Orbital Data for the PlanetsOrbital Data for the Planets

5 Jul 2005 AST 2010: Chapter 2 13

Orbits of CometsOrbits of CometsA 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 ellipsesThe comets, therefore, spend most of their time far away from the Sun, moving very slowly

5 Jul 2005 AST 2010: Chapter 2 14

Kepler's Second Law (1)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

5 Jul 2005 AST 2010: Chapter 2 15

Kepler's Second Law (2)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

5 Jul 2005 AST 2010: Chapter 2 16

Conservation of Angular Conservation of Angular MomentumMomentumAn 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 sameSimilarly, when the planet is moving farther from the Sun, it moves more slowly

Another example is a figure skater spinning on ice

5 Jul 2005 AST 2010: Chapter 2 17

Kepler's Third Law (1)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

5 Jul 2005 AST 2010: Chapter 2 18

Kepler’s Third Law (2)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 = 3.51Mars 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

5 Jul 2005 AST 2010: Chapter 2 19

Sir Isaac Newton 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 MathematicsHe 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 problemsHis work on mechanics led to his famous three laws of motion …

5 Jul 2005 AST 2010: Chapter 2 20

Newton's Laws of MotionNewton's Laws of MotionThe 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 actingThe 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

5 Jul 2005 AST 2010: Chapter 2 21

Newton's First Law (1)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 movesThe 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

22AST 2010: Chapter 25 Jul 2005

Newton's First Law (2)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 bodyThe 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

5 Jul 2005 AST 2010: Chapter 2 23

Newton's Second LawNewton's Second LawThe 2nd law defines force in terms of its ability to change momentumThus, 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

5 Jul 2005 AST 2010: Chapter 2 24

Newton's Third LawNewton's Third LawThe 3rd law states that to every action there is an equal and opposite reactionConsider 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

5 Jul 2005 AST 2010: Chapter 2 25

Momentous QuestionMomentous Question

If a spaceship is moving at If a spaceship is moving at constant speed along a constant speed along a straight line, is there an straight line, is there an outside force acting on outside force acting on

the ship? the ship?

5 Jul 2005 AST 2010: Chapter 2 26

Mass, Volume, and Density (1)Mass, Volume, and Density (1)The mass of an object is a measure of the amount of material in the objectThe 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

5 Jul 2005 AST 2010: Chapter 2 27

Mass, Volume, and Density (2)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 isTo summarize, mass is “how much”, volume is “how big”, and density is “how tightly packed”

5 Jul 2005 AST 2010: Chapter 2 28

Examples of density (1)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 

Examples of density (2)Examples of density (2)

5 Jul 2005 AST 2010: Chapter 2 30

Newton’s Law of Gravity (1)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

5 Jul 2005 AST 2010: Chapter 2 31

Newton’s Law of Gravity (2)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 calculusEventually, 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

5 Jul 2005 AST 2010: Chapter 2 32

Newton’s Law of Gravity (3)Newton’s Law of Gravity (3)Newton provided the formula for this gravitational attraction between any two bodies:

Force = G M1 M2 / R2

whereG is called the constant of gravitationM1 is the mass of the first body

M2 is the mass of the second body

R is the distance between the two bodies

5 Jul 2005 AST 2010: Chapter 2 33

Newton’s Law of Gravity (4)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!

5 Jul 2005 AST 2010: Chapter 2 34

Newton’s Law of Gravity (5)Newton’s Law of Gravity (5)Everything with a mass is subject to this law of universal attractionFor most pairs of objects, this attraction is rather smallIt takes a huge body such as the Earth, or the Sun, to exert a large force of gravity

5 Jul 2005 AST 2010: Chapter 2 35

Kepler’s Third Law Revisited (1)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 

5 Jul 2005 AST 2010: Chapter 2 36

Kepler’s Third Law Revisited (2)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

5 Jul 2005 AST 2010: Chapter 2 37

Kepler’s Third Law Revisited (3)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

5 Jul 2005 AST 2010: Chapter 2 38

Artificial Satellites and Space Artificial Satellites and Space Flight (1)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

39AST 2010: Chapter 25 Jul 2005

Launching a Satellite into OrbitLaunching a Satellite into OrbitTo 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

5 Jul 2005 AST 2010: Chapter 2 40

Artificial Satellites and Space Artificial Satellites and Space Flight (2)Flight (2)

Sputnik, the first artificial Earth satellite, was launched by what was called the Soviet Union on October 4, 1957Since 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

5 Jul 2005 AST 2010: Chapter 2 41

Artificial Artificial SatelliteSatellite

s and s and Space Space

Flight (3)Flight (3)Satellites in Earth orbit

5 Jul 2005 AST 2010: Chapter 2 42

Artificial Satellites and Space Artificial Satellites and Space Flights (4)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 atmosphereSpacecraft 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

5 Jul 2005 AST 2010: Chapter 2 43

Interplanetary Spacecraft (1)Interplanetary Spacecraft (1)The exploration of our solar system has been carried out mostly by automated spacecraft or robotsTo 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)

5 Jul 2005 AST 2010: Chapter 2 44

Interplanetary Spacecraft (2)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

5 Jul 2005 AST 2010: Chapter 2 45

Gravity with More than Two Gravity with More than Two BodiesBodies

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)

5 Jul 2005 AST 2010: Chapter 2 46

Discovery of NeptuneDiscovery of NeptuneUranus was discovered by William Herschel in 1781The orbit of Uranus was calculated and “known” by 1790, but it did not appear to be regular, namely, to agree with Newton’s lawsIn 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!