Orbits and Gravity

Orbits and Gravity AST 2010: Chapter 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) AST 2010: Chapter 2

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 AST 2010: Chapter 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 AST 2010: Chapter 2

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 AST 2010: Chapter 2

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 planetmoves 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 AST 2010: Chapter 2

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 AST 2010: Chapter 2

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 AST 2010: Chapter 2

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! AST 2010: Chapter 2

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 AST 2010: Chapter 2

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 AST 2010: Chapter 2

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 AST 2010: Chapter 2

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 AST 2010: Chapter 2

3 S 1 2 Kepler's Second Law (2) 4 • Physicists found that the 2nd law is a consequence of the conservation ofangular 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 The surfaces S-1-2 and S-3-4 are equal AST 2010: Chapter 2

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 AST 2010: Chapter 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 = 3.51 • 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 AST 2010: Chapter 2

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 … AST 2010: Chapter 2

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 AST 2010: Chapter 2

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 AST 2010: Chapter 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 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 AST 2010: Chapter 2

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 AST 2010: Chapter 2

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 AST 2010: Chapter 2

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 AST 2010: Chapter 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” 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 AST 2010: Chapter 2

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 AST 2010: Chapter 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 AST 2010: Chapter 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 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 AST 2010: Chapter 2

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 AST 2010: Chapter 2

Newton’s Las of Gravity (4) • This lawof 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!!! AST 2010: Chapter 2

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 AST 2010: Chapter 2

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 AST 2010: Chapter 2

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 AST 2010: Chapter 2

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 AST 2010: Chapter 2

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 AST 2010: Chapter 2

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 AST 2010: Chapter 2

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 AST 2010: Chapter 2

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) AST 2010: Chapter 2

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 AST 2010: Chapter 2

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 AST 2010: Chapter 2

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) AST 2010: Chapter 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 AST 2010: Chapter 2

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 AST 2010: Chapter 2

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! AST 2010: Chapter 2

Orbits and Gravity