Gravity & Orbits

That’s the trouble with science. You’ve

got a bunch of empiricists trying to describe things of

unimaginable wonder.

Gravity & Orbits

Solar System AstronomyChapter 3

But first… history

Solar System Astronomy

Some History Ancient cultures & astronomical

knowledge Already in the stone and bronze ages, human

cultures realized the cyclic nature of motions in the sky.

Monuments dating back to ~ 3000 B.C. show alignments with astronomical significance.

Those monuments were probably used as calendars or even to predict eclipses


knowledge Nebra sky disk

~1600 BC bronze disk ~30cm diameter sun or full moon, a lunar crescent, and stars

(including Pleiades) two golden arcs along the sides marking angle between the solstices arc at the bottom perhaps

Milky Way or Rainbow

Some History

Some History Ancient cultures & astronomical knowledge

Often timekeeping methods Some objects ‘wandered’

Greek πλανήτης, derivative of the word πλάνης "moving”

7 in the sky Sun Moon Mars Mercury Jupiter Venus Saturn

Some History Ancient cultures & astronomical knowledge

Often timekeeping methods Some objects ‘wandered’

Greek πλανήτης, derivative of the word πλάνης "moving”

7 in the sky Sun Dimanche Moon Lundi Mars Mardi Mercury Mercredi Jupiter Jeudi Venus Vendredi Saturn Samedi


knowledge Often timekeeping methods Lunar cycles gave rise to months


knowledge Often timekeeping methods When is Easter?

observed on the Sunday after the first full moon on or after the day of the vernal equinox

full moon involved is not an astronomical full moon, but an ecclesiastical moon – though it coincides more or less with the astronomical full moon

Easter falls on the first Sunday following the first ecclesiastical full moon that occurs on or after March 21 (the day of the ecclesiastical vernal equinox)


knowledge Often timekeeping methods Sirius, the Dog Star

seiros σείριος, “scorcher” Rising just before Sun marks the beginning of

summer

Some History Ancient Greece

Astronomy was a branch of mathematics Astronomers sought to create geometrical

(mathematical) models that could imitate the appearances of celestial motions

tradition began with Pythagoreans, who placed astronomy among the four mathematical arts (along with arithmetic, geometry, and music)

Our conception of astronomy (and many sciences) inherited from Ancient Greek


Eudoxus Aristotle Eratosthenes Hipparchus Ptolemy

All sought to describe motion of heavenly bodies

Mathematical behavior of motion


Eudoxus (4009 – 356 BC) assigned to each planet a set of 27 concentric

spheres first to attempt a mathematical description of the

motions of the planets


Aristotle (384 – 322 B.C.) major authority of philosophy until late Middle Ages divided universe into two parts

imperfect, changeable Earth perfect heavens (described by spheres)

expanded Eudoxus’ model to 55 spheres Aristotle taught the Earth is round

Why would he think that?


Aristotle (384 – 322 B.C.) perhaps the philosopher most respected by

European thinkers during and after the Renaissance erroneous positions taken as a given, which held

back science analysis of procreation presupposes:

active, ensouling masculine element bringing life to

inert, passive, female element assertion that objects of different mass fall

at different speeds under gravity refuted by Philoponus


Aristotle (384 – 322 B.C.) held back science for 2000 years

refuted Democritus's claim Milky Way was multitude of stars

refuted stars of night sky were just like Sun calculated they would have to be

millions of times farther away than Sun thus dismissed for hundreds of years

theory of the natural slave was used to justify European domination of Native Americans

On Animals…


Eratosthenes (276 BC - 194 BC) Greek mathematician, geographer and astronomer

contemporaries nicknamed him Beta (2nd letter of alphabet) –supposedly proved himself to be the second in the Mediterranean world in many fields

first known to have calculated the size of the Earth


Eratosthenes (276 BC - 194 BC) Angular distance between Syene and

Alexandria is ~ 7° Linear distance ~ 5,000 stadia Earth Radius ~ 40,000 stadia

probably ~ 14 % too large


Hipparchus (2nd century BC) access to Babylonian observations & predictions

used to create better geometrical models, constructed accurate models for movement of Sun & Moon

moved Earth from center of spheres compiled star catalogue

recorded the position and brightness of the stars Considered among most important Greek astronomers

(& greatest astronomer of antiquity by some) introduced the concept of exact prediction into

astronomy last innovative astronomer before Ptolemy


Ptolemy (c. 90 – c. 168) compiled and extended the astronomical knowledge

and theories of the ancient Greek & Babylonian world

star catalog in Almagest lists forty-eight constellations

ancestral to the modern system of constellations geocentric model widely accepted until Copernicus


Ptolemy (c. 90 – c. 168) Retrograde motion…


Ptolemy (c. 90 – c. 168) Epicycles…


Ptolemy (c. 90 – c. 168) Epicycles Ptolemaic system was considered the “standard

model” of the Universe until the Copernican Revolution

Some History Copernican Revolution & beyond

99 years that changed our view of the Universe

Nicolaus Copernicus Galileo Galilei Tycho Brahe Johannes Kepler Isaac Newton

Some History Copernican Revolution

Copernicus (1473-1543) one of the great Renaissance polymaths mathematician, astronomer, jurist,

physician, classical scholar, governor, administrator, diplomat, economist, and soldier


Copernican Model No center to universe Center of Solar System near sun

heliocentric Distance to Sun imperceptible vs distance to stars Earth’s motion accounts for apparent motion:

rotation accounts for daily rotation of stars annual cycle of Sun’s movements caused by Earth’s

orbit apparent retrograde motion of planets is caused by

orbit


Copernican Model arrived at correct order of the known planets Explained precession of the equinoxes correctly

slow change in the position of the Earth's rotational axis

gave a clear account of the cause of the seasons Earth's axis not perpendicular to plane of orbit

added motion to Earth, keeping axis pointed throughout the year at the same place in the heavens


Copernican Model not experimentally better than Ptolemy’s from 1543 publication until about 1700, few

astronomers were convinced by the Copernican system, though the book was relatively widely circulated

influenced Galileo & Kepler, others

ironically, this was at the behest of the Catholic Church as part of the Catholic Reformation efforts for a means of creating a more accurate calendar for its activities


Galileo (1564 – 1642) Italian

physicist astronomer astrologer philosopher

closely associated with the Scientific Revolution


Galileo (1564 – 1642) improved (but did not invent) telescope invented first known microscope observed Galilean moons

motion indicated orbits about Jupiter observed rings of Saturn observed full set of phases with Venus


Galileo (1564 – 1642) one of first Europeans to observe sunspots

Existence contradicted unchanging perfection of the heavens (Aristotle)

First to report lunar mountains & craters Concluded “rough and uneven, and just like the

surface of the Earth Not Aristotle’s perfect sphere

Observed the Milky Way Found to be multitude of distant stars


Galileo (1564 – 1642) Demonstrated equal acceleration of unequal masses

Contrary to Aristotle Determined rate objects fell Demonstrated velocity constant (not zero) without

outside force Contrary to Aristotle

Basic principle of relativity


Galileo (1564 – 1642) Early example of science conflicting with

Church/authority 1633 Heresy

required to recant heliocentric ideas idea that Sun is stationary condemned as “formally

heretical” heliocentrism was never formally or officially

condemned ordered imprisoned

sentence later commuted to house arrest Dialogue Concerning the Two Chief World Systems banned

and all past & future books


Tycho Brahe (1546 – 1602) preeminent observational astronomer of pre-telescopic

period said to own 1% of Denmark’s entire wealth at one observations of stellar/planetary positions unparalleled

accuracy observations of nova contradicted immutable heavens

Lost nose in duel 1566 Fair amount of drinking involved Replacement of silver & gold

Kept a jester (Jepp) beneath the table during dinner Believed to be clairvoyant

Tame elk (or moose)…


Kepler (1571 – 1630) German Mathematician, astronomer, astrologer,

sci-fi writer


Kepler (1571 – 1630) Used the precise observational tables of Tycho Brahe

to study planetary motion mathematically Consistent description by abandoning both

Circular motion Uniform motion

Planets move around the sun on elliptical paths, with non-uniform velocities


Kepler (1571 – 1630) Kepler’s Laws

1) The orbits of the planets are ellipses with the sun at one focus

2) A line from a planet to the sun sweeps over equal areas in equal intervals of time

3) A planet’s orbital period (P) squared is proportional to its average distance from the sun (a) cubed


Kepler’s Law

#1 & #2


Kepler’s Law

#3

Kepler’s Laws

Some History 99 years changed our universe


Where do Kepler’s Laws come from?

Why are they true?

We don’t know…Until

Newton.

Sir Isaac Newton, FRS

Newton’s Laws Newton discovered laws applying to all

objects Universal Describe forces that act on all objects

Like gravity

Newton’s Laws of Motion Stationary objects stay stationary, moving

objects stay in constant motion Unless a force acts upon it “constant” refers to speed AND direction Forces change motion

No change in motion means no forces OR balanced forces

#1

Newton’s Laws of Motion Force = mass × acceleration

F = ma a = F/m

More mass » less acceleration Larger force » larger acceleration

#2

Newton’s Laws of Motion Forces always come in pairs

Same size Opposite direction

Applies to ALL forces, including gravity

#3

Newton’s Law of Gravitation The story of the apple…


Plague years 1665-1666, Newton sat in his garden at Woolsthorpe

Wondered at the fall of an apple


The force that drew the apple to Earth certainly extended to far greater heights

What if it reached as far as the Moon? The circling Moon and falling apple are fellow

captives of the Earth And perhaps the Sun likewise holds all the planets in

thrall

A Law of Universal Gravitation!


The force that drew the apple to Earth certainly extended to far greater heights


What did Newton start with? Galileo – equal acceleration of unequal masses Kepler – P2 = a3

Circular motion F = ma What is a?

Δv/Δt


We start at circular motion


Kepler’s Law also tells us something


So we put them together And what else is important?

Mass(es)


They say he derived Kepler’s rules from his law of gravity…

…but it was the other way around

Newton’s Law of Gravitation Gravity is a force between any two

objects. It depends on the objects’ masses It depends on the distance between them

Important! Gravity is a mutual force, acting on both

bodies The force on each body is the same size (but

has the opposite direction)

Newton’s Law of Gravitation Gravity is a force between any two

objects. It depends on the objects’ masses It depends on the distance between them

Variables M & m, or m1 & m2, are the two masses The force on each body is the same size (but

has the opposite direction)

Newton’s Law of Gravitation Orbital velocity Escape velocity

Ideas… Why doesn’t the Space Shuttle fall from

space? Is gravity still acting on it?

What does weightless mean?

Einstein & Relativity Special Relativity General Relativity

Confirmation

Albert Einstein (1879 – 1955) Annalen der Physic 1905

“On a Heuristic Viewpoint Concerning the Production and Transformation of Light”

“On the Motion — Required by the Molecular Kinetic Theory of Heat — of Small Particles Suspended in a Stationary Liquid”

“On the Electrodynamics of Moving Bodies” “Zur Elektrodynamik bewegter Körper”

Annus Mirabilis papers

Albert Einstein (1879 – 1955)

Who cares about Einstein Anybody who wants to pass this course… Anyone who want to use GPS…

Moving clocks slow (-7 ms/day) Orbiting clocks fast (+45 ms/day) Net 38 ms fast each day

GPS would stop working in 2 minutes 10 km error adds up each day

Albert Einstein (1879 – 1955)

Gravity & Orbits

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