ASTA01 at UTSC – Lecture 10

Chapter 3 The Origin of Modern Astronomy

- The Copernican revolution - Brahe and Kepler - Galileo

- Newton (and Hooke)

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Isaac Newton, Gravity, and Orbits • The problem of the place of Earth was resolved by the Copernican Revolution.

• The problem of planetary motion, though, was only partly solved by Kepler’s laws. 2

Isaac Newton, Gravity, and Orbits • For the last 10 years of his life, Galileo studied the nature of motion, especially the accelerated motion of falling bodies.

• Although Galileo made extraordinary progress in observations and formulated what is now known as the first law of Newton’s dynamics, he was not able to relate his discoveries about motion to the heavens.

• That final step was taken by Isaac Newton.

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Isaac Newton • Galileo died in January 1642. • Almost a year later Isaac Newton was born in the English village of Woolsthorpe.

Isaac Newton • Newton was a quiet child from a farming family.

• However, his work at school was so impressive & he seemed so inept at agriculture that his parents decided that Isaac should become a priest, not a farmer.

• His uncle financed his education at Trinity College, Cambridge – where he studied mathematics and physics. 5

Isaac Newton • In 1665, plague epidemics swept through England, and the colleges were closed.

• During 1665 and 1666, Newton spent his time back home in Woolsthorpe —thinking and studying. It was during these years that he made most of his scientific discoveries. • Among other things, he studied optics, developed three laws of motion, probed the nature of gravity, and invented differential calculus. 6

• • Isaac Newton and Robert Hooke But he was slow to publish most of his discoveries, which later led to conflict with those who made them independently (most famously, G. W. Leibniz discovered calculus, and R. Hooke formulated and published the idea of universal gravitation

before

Newton in 1670-1674) Robert Hooke (1635-1703) Physcist and inventor, discovered Hooke’s law of ideal spring, first identified and named “cells” in plants using his improved design of a microscope, etc.

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Isaac Newton • The publication of his work in his book

Principia Mathematica Philosophiae Naturalis

in 1687 placed the fields of physics and astronomy on a new firm base.

• But it was only though prodding by the famous astronomer Edmund Halley and the work of Newton’s arch-enemy Hook, that Newton was persuaded to write

Principia

. 8

Isaac Newton • • Edmund Halley, architect Christopher Wren, and physicist and the secretary of the Royal Society Robert Hooke had meetings and discussions at coffeehouses of London.

They were interested in physical causes of elliptic orbits discovered by Kepler and confirmed in comets by Halley. • Hooke had an exchange of letters 1679-1680 about it with Newton, which probably gave Newton both a motivation (

which he admitted later

) and crucial ideas to study orbits, like combining inertia and gravity to produce orbits, and the Universal Gravitation (

all of which Newton later denied

) 9

Robert Hooke • Hook suspected that the cause is universal gravitation between massive bodies, whose strength follows the inverse square law of the form

F = const/r 2

. He constructed funnels for balls, which simulated the motion in a gravity field, new that a pendulum simulates

F=const*r

. He also knew how the orbits follow from “compunding” the inertial staight-line motion and gravity that bends the trajectory by providing radial additions to velocity. • For those interested in history of science: an article by • M. Nauenberg “Robert Hooke’s Seminal Contributions to Orbital Dynamics” – link 25 on the course web page At the time, some of Newton’s ideas on that problem were clearly incorrect (spiral trajectory of a body in some cases). 10

Isaac Newton – the story of Principia • In 1684 Wren offered a reward for showing that 1/r 2 force  elliptical orbit (1 st Kepler’s law). Robert Hooke said he has a demonstration of this but said that he

can’t find it

! He said he’ll provide it

later

… (In fact, he did have a demonstration explained to his friends in letters in 1685, but it was based on mechanical analogs, mechanical experiments, not mathematical physics). Halley traveled in 1684 to Cambridge to ask Newton the same question. Newton ALSO claimed he had a proof of elliptical orbits following from inverse-square law of gravity.

“Sir Isaac looked among his papers but

could not find it

, but he promised him to renew it and

then to send it

him…” 11

Isaac Newton • Halley started doubting in both Hooke’s and Newton’s claims… Yet Newton afterwards made good on his promise!

De motu corporum in gyrum

(1984) &

Principia Mathematica Philosophiae Naturalis

(1987) were the result of E. Halley’s interest and private financing. 12

1687 Newton’s

Principia

are today considered the most important book in physical sciences.

13

Isaac Newton • In order to understand Newton’s work, we must begin with a general framework for describing the motion of any object.

• Position and time specify where and when an object is.

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Isaac Newton • Speed is the rate at which an object moves (changes position).

• It is the total distance moved divided by the total time taken to move that distance.

• For example, if it took you 2 hours to travel 100 km, then your speed was 50 km/hr.

Although we are used to thinking of speeds in km/hr, in science the Standard International (SI) units are metres/second.

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Isaac Newton • Velocity specifies both the speed and direction of travel of an object.

• For example, if car A moves 60 km east in 2 hours and car B moves 60 km south in 2 hours, they have the same speed of 30 km/hour, but their velocities are different because they are travelling in different directions.

• Thus, velocity can change if: (i) the speed changes, (ii) the direction changes, or (iii) both speed and direction change.

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Isaac Newton • Acceleration is the rate of change of velocity with time.

• It is thus the change in velocity divided by the time taken for the change to occur.

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Isaac Newton • Since velocity changes if speed changes, speeding up is an example of acceleration and slowing down is negative acceleration (in a direction opposing the direction of travel), or deceleration.

a = dv /dt.

• On the other hand, velocity also changes if there is a change of direction, so turning is also an example of acceleration a = dv /dt .

v 0 dv v v 0 18

Isaac Newton • Newton knew that the motion of all objects is a result of the forces (pulls or pushes) acting on them.

• He was able to find three universal laws of motion that made it possible to predict exactly how a body would move if the forces acting on it were known.

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Isaac Newton • Newton’s first law of motion [actually formulated earlier by Hook on the basis of Galileo’s idea] states that an object remains at rest or at constant velocity unless a net force acts to change its speed or direction.

• When a car is at rest or travelling at a constant speed and direction, the forces exerted by the wheels to drive it forward are balanced by the air drag and other drag forces in such a way that the net (total) force is zero.

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Isaac Newton • “If you wanted to speed up or slow down or change direction , then the engine would have to cause an additional force.

• The effect of this additional force is described by Newton’s second law .” • [Here, our textbook is a bit wrong. Do you know why?] 21

Isaac Newton • If the mass m (amount of matter) of the object does not change, then the acceleration is proportional to the force exerted.

• Hence, if you want to double your acceleration the applied force must be doubled.

• • F = m a a = F/m or 22

Isaac Newton • An example of acceleration that we are all familiar with is the acceleration due to gravity.

• All falling objects on Earth have a constant acceleration downwards toward the centre of the Earth.

• This acceleration was first pointed out and measured by Galileo.

• Thus, after the first second its speed will be roughly 10 m/s; after two seconds its speed will be 20 m/s, and so on until it crashes into the ground 23

Isaac Newton • • The acceleration due to gravity, g, is 9.8 metres per second per second, g = 9.8 m/s 2 .

• • This means that if you drop

any

object, say an apple, from rest, its speed will increase by roughly 10 m/s with each second of falling,

if one ignores air resistance.

Watch Galileo’s experiment on the Moon

http://www.youtube.com/watch?v=GdHlWp9k_sY&f eature=fvsr 24

The Universal Theory of Gravitation • Newton realized that some force must pull the Moon toward Earth’s centre. • If there were no such force altering the Moon’s motion, it would continue moving in a straight line and leave Earth forever. • It can circle Earth only if Earth attracts it.

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The Universal Theory of Gravitation • Hook’s and Newton’s insight was to recognize that the force that holds the Moon in its orbit is the same as the force that makes apples fall from trees.

• Gravitation is sometimes called universal

The Universal Theory of Gravitation • Newton’s third law occur in pairs. points out that forces • If one body attracts another, the second body must also attract the first. • Thus, gravitation must also be mutual, two bodies attract each other with the same force F, only oppositely directed.

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The Universal Theory of Gravitation • The mass of an object is a measure of the amount of matter in the object.

• This is usually expressed in kilograms. 29

The Universal Theory of Gravitation • You may be used to thinking of “massive” objects as very large objects.

• However, in science massive objects are those that contain a lot of matter.

• They may or may not be large.

• For example, a two-centimetre ball of lead is more massive than a large balloon full of air.

• In everyday life, the terms mass and weight are used interchangeably.

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The Universal Theory of Gravitation • • In science, mass is not the same as weight. In fact, weight is the force W that results from mass m and acceleration of gravity g: W = m g • Mass is an intrinsic property of an object and is the same no matter what forces are acting on an object.

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The Universal Theory of Gravitation • An object’s weight is the force that gravity exerts on the object. • Thus, an object in space far from Earth might have no weight.

• However, it would contain the same amount of matter and would thus have the same mass that it has on Earth.

• W = m g 32

The Universal Theory of Gravitation • Gravitational force of attraction between two objects depends on the product of the masses of the two objects.

• For example, doubling one of the masses ( m or M ) would double the gravitational force F, and doubling both masses would quadruple the force. • F = G M m / r 2 where G is a gravitational constant 33

The Universal Theory of Gravitation • Gravitational force of attraction between two objects depends on the product of the masses of the two objects.

• Keeping the masses constant but doubling the distance r would decrease the force 4 times. • This relationship is known as the inverse square relation.

• F = G M m / r 2 where G is a gravitational constant 34

The Universal Theory of Gravitation • Gravity is universal.

• Your mass affects Neptune, the galaxy M31, and every other object in the universe.

• Their masses affect you – although not much, because they are so far away and your mass is relatively very small.

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Orbital Motion • Newton’s laws of motion and gravitation make it possible for you to both understand why and how the Moon orbits Earth and the planets orbit the Sun and discover why Kepler’s laws work.

• It explains why the pendulum swings, and also why galaxies look as they look, and why they sometimes interact and merge.

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Orbital Motion around a point mass • There are three important ideas to note about orbiting Earth.

Orbital Motion • One, an object orbiting Earth is actually falling (being accelerated due to the gravitational force) toward Earth’s centre. • An object in a stable orbit continuously misses Earth because of its horizontal velocity .

Orbital Motion • Two, objects orbiting each other actually revolve around their mutual centre of mass.

• We can detect unseen planets by looking at stars

Orbital Motion • Three, note the difference between closed orbits and open orbits.

Orbital Motion • If you want to leave Earth, never to return, you must give your spaceship a high enough velocity (the so called

escape speed

) to put it in an open orbit.

Orbital Motion • • How to find a speed that puts a spaceship into a circular orbit? Scientists call it • • • “Keplerian speed” or V K That circular velocity depends only on the mass of the planet and the distance from the centre of the planet.

V K 2 = G M / r [you will see that V K 2 ~ 1/r on a later slide] 42

Orbital Motion • Once the engines fire and the ship reaches circular velocity, the engines can shut down. • The ship is in orbit and will fall around the planet forever – so long as it is above the atmosphere’s friction. • No further effort is needed to maintain orbit, according to the laws Galileo, Hook and Newton described.

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Orbital Motion: Newton’s vs. Kepler’s laws • Newton’s laws of motion and theory of universal gravitation enabled Newton to explain mathematically Kepler’s laws of planetary motion.

• Kepler’s first law, that the planets move in elliptical orbits, is a direct result of the inverse square law of gravitation.

• Newton proved that any object moving in a closed orbit according to the inverse square law of attraction must follow an elliptical path. That includes a special case – the circle.

• Open orbits are: parabola and hyperbola.

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Orbital Motion • Furthermore, just like the spaceship in stable orbit around the Earth, the planets, the Moon, and all objects in the universe will remain on their respective paths forever unless an external force (such as a collision with another) acts on them .

• [Actually, the textbook may be a little misleading. They’ll not, because gravity IS such an external force. If there are more than 2 bodies, the problem is mathematically unsolvable (Newton said “..and if I am not mistaken, exceeds the capabilities of human mind”). The orbits of planets are NOT constant ellipses then and in general slowly change their orientation and shape, making escape or collision possible.] 45

Orbital Motion • Newton’s inverse square law of gravitation also explains Kepler’s second law, which states that planets move faster when they are closer to the Sun.

• As the planets continuously fall toward the Sun in their orbits, they go faster when they approach the Sun thanks to the inverse relationship between force and distance.

[This explanation is also misleading. In fact, the 2 nd Kepler’s law follows entirely from the fact that the force points toward the center of attraction, and is thus true of ANY force law, be it F~1/r 2 , F=const., or F~r, or any other F(r).] 46

Orbital Motion L =

m v t r

=

const

• A measure of a planet’s rotational motion is its angular momentum L, which is proportional to its velocity [ velocity perpendicular to radius = tangential = v t • ] times its distance from the Sun.

As a result of Newton’s laws , in the absence of additional rotational forces the total angular momentum of a planet is conserved (remains constant).

• Thus, when its distance from the Sun increases, its velocity must decrease to balance out the increased distance and vice versa.

• [this is true of sideways velocity = transversal = tangential velocity, but false of radial velocity] 47

Orbital Motion • You can observe the conservation of angular momentum for yourself by watching an ice skater spinning on the ice.

• She can increase or decrease her velocity of rotation by pulling her arms in or spreading them out and thus increasing or decreasing her “distance” from her axis of rotation. •

L = m v t r

48

Orbital Motion – derivation of 3 rd Kepler’s law • • • • • • • Newton was also able to combine his laws of motion with the law of gravitation to derive a relationship between a planet’s orbital period and average distance from the Sun, which was identical to Kepler’s third law. Newton did it for

elliptic

orbits. Earlier done for

circular

orbits by Hook and Halley: F~ 1/r 2 (inverse-square force law) F centrifugal =mV which V =V K 2 /r (C. Huyghens centrifugal force formula, in is the constant speed of the orbiting body) F=F centrifugal <=> mV 2 /r = GmM/r 2 <=> V 2 = GM/r Since V = 2π r /P, where P is the orbital period, the force balance results in the relationship: P 2 ~ r 3 (3 rd Kepler’s law, when ‘r’ is replaced by ‘a’) 49

Orbital Motion & tides • You now understand the power of Newton’s work.

• He was able to explain all the patterns of planetary motion observed by Kepler by using very simple and universal rules.

• But this was not all. Gravity is also the key to understanding another critical phenomenon on Earth: ocean tides.

50

Tides: Gravity in Action • Tides are caused by small differences in gravitational forces. As the Earth and Moon orbit around each other, they attract • • each other gravitationally .

Because the side of Earth toward the Moon is a bit closer, the Moon pulls on it more strongly than on the rest that pulls up a bulge. 51

Tides: Gravity in Action • Because the far side of Earth toward the Moon is a bit further, the Moon pulls on it less strongly than on the rest that pulls up a bulge, too, because the rest is pulled strongly away from the bulge. Because there are two bulges, Any point rotating on a sphere encounters two tides a day.

52

Tides: Gravity in Action • The Sun also produces tides on Earth.

• However, they are smaller than lunar tides.

• At new and full moons, the lunar and solar tides add together to produce extra high and extra low tides that are called spring tides.

53

Tides: Gravity in Action • Although the oceans flow easily into tidal bulges, the nearly rigid bulk of Earth flexes into tidal bulges and the plains and mountains rise and fall a few centimetres twice a day. • Friction of a constantly flexed body of Earth is gradually slowing Earth’s rotation • Fossil evidence shows that Earth used to rotate faster. The days were once shorter than now. 54

Tides: Gravity in Action • In the same way, Earth’s gravity produces tidal bulges in the Moon.

• Although the Moon used to rotate faster, friction has slowed it down and it now keeps the same side facing Earth. It is not slowing down further but keeps its spin (rotation) and orbit (revolution) in 1:1 synch.

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Tides: Gravity in Action • Tides can also affect orbits. • The rotation of Earth drags the tidal bulges slightly ahead of the Moon, and the gravitation of the bulges of water pull the Moon forward in its orbit. • This makes the Moon’s orbit grow larger by about 3.8 cm a year, an effect that astronomers measure by bouncing lasers off reflectors left on the Moon by the Apollo astronauts . 56

Tides: Gravity in Action 57

Newton’s Universe • Newton’s insights gave the world a new conception of nature.

• His laws of motion are general laws that described the motions of all bodies under the action of external forces.

• Just imagine! A few simple laws that can explain how your car accelerates, how the Canadian hockey team manoeuvres on ice, how apples fall from trees, and even how the planets move 58

Newton’s Universe • Furthermore, Newton’s laws and the theory of gravitation allow us to break the bonds of Earth and the solar system and understand the motion of all objects in the universe. • [In the neighborhood of the highly compact ones, like black holes we need Einstein’s theory of gravity: general relativity] As you will see in later chapters, we can detect planets around other stars by observing the motion of the star as it gravitationally interacts with any planets orbiting it.

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Newton’s Universe 60

Newton’s Universe • We can calculate the mass of these new planets using the law of gravitation.

• Indeed, this has been first used to calculate the mass of Earth and all the other planets, and the Sun. (This required first finding the gravitational constant G from experiment, the so-called Cavendish experiment.) • We can even detect and “weigh” black holes at the centres of galaxies by observing the motion of objects around it.

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Newton’s Universe • The story of the development of astronomy that you have just learned is also the story of the development of the scientific method.

• Ancient astronomers began the process by carefully gathering and recording data.

• Gradually, models were developed that best fit the data and over time they were tested against observations and discarded if necessary.

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Newton’s Universe • Good scientific theories are those that can make a broad range of predictions confirmed against observations, and that can provide new insight into nature.

• Science and in particular astronomy today progress through the careful application of this scientific method of studying nature: • Openness & publication, verifiability & repeatability by other scientists, confronting theory with empirical studies, are the hallmarks of the scientific method. It differs much from the business model, politics or religion. 63

Newton’s Universe • The shift from the geocentric to the heliocentric viewpoint was a harsh lesson in humility for humanity.

• Earth became just another planet orbiting the Sun. • But this first revolution of thought started us on a fantastic journey of scientific discovery. And now the Earth and other Solar System planets are just another solar system… connected with the galaxy and the universe.

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