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HOMEWORK assignment 2 due Feb 16
Assignment contains two tutorials; eclipses, phases of the moon
Things like unopened hint bonus, allowable attempts per answer all changes as the semester progresses, these are written on the first page of the homework assignment
First Mid-Term Exam will be Feb 19 (Mon)
Second Mid-Term Exam will be Mar 14 (Weds)
The Birth of "Scientific" Cosmology
Aristotle & Ptolemy
The "State of the Universe" for the Greeks
The "Dark" Millennium
The Recovery of W. European Science
The word Cosmology is from the Greek kosmos (world)
and logia (from legein: to speak).
All civilizations (probably) developed some form of "cosmology". Indeed all civilizations (we know about) seem to have "Creation myths" of some sorts.
In the earliest civilizations (& up to very recent times) cosmology was primarily (often totally) a branch of religion/myth.
Certainly many (all ?) ancient civilizations performed astronomical observations of various levels of sophistication - from the neolithic observatories (eg.Stonehenge), to written reports (eg. The Chinese, Egyptians, etc).
Many earlier beliefs & models may seem "silly" or "absurd" now.
Our beliefs and our scientific methods are "clearly far superior" - aren't they ?
However it worth reminding ourselves from the very start that we have not
figured everything out yet !! (dark matter, cosmic acceln, WIMPs, MACHOS etc)
Many variations on Mythological/Cosmological Ideas, but many with themes
not so different from our own:
AND we should not underestimate the struggles/difficulties early thinkers had attempting to
explain the universe within their various cultural, religious & technological environments….
The Greeks are generally credited with promoting the idea that the
universe was understandable using logic, and could be described
This is a huge leap forward, and the basis of all science today
By ~400BCE, generally thought that the universe worked & evolved
through "natural" processes that can be observed on Earth.
...Divine intervention is not required (at least in the "running" of the universe).
Consensus - there are no limits to what can be observed & understood
(again, at least not concerning the "running" of the universe),
theories could be postulated,
theories revised as necessary
Cosmology was really a branch of philosophy at the time,
but the Empirical Scientific Method was developing.
TheGreeks understood that the Earth is a sphere:-
Observation of ships sailing over the horizon
Observation of Earths shadow on the moon during lunar eclipses
Only five planets were known to the Early Greeks
(Mercury, Venus, Mars, Saturn, Jupiter).
Thus early Greek cosmologists believed they had to account for
8 celestial entities -the 5 planets, the Sun, the Moon, and the "Stars".
Cosmologies were all "naturally"
(centered on the Earth)
Cosmologies generally included
(Sphere count: Pythagoras of Samos,(c.530BCE)8)
The physical universe was finite - beyond the outer
sphere of the stars was the (non-material) spiritual realm.
Reasoned that that this "universe" must be unique
(& have a single center), and have existed for
eternity (& in a steady-state).
Believed celestial bodies move in
Forced to increase the number of spheres due
to refined observations of planetary motion
(Sphere count: Aristotle,(c.350BCE)55)
Constructed a catalog of close to 1000 stars.
Discovered precession (1 degree/century)
- the change of the position of the stars with time
(now known to be due to the precession of the Earth's axis).
In 134 BCE he discovered a new star (a nova),
...in direct contradiction to the paradigm that the "heavens" were unchanging.
Refined distances to (& hence size of) the Moon (via Parallax)
...made the first step determining the scale of the "cosmos"
Suggested that the Sun appeared to be much larger than the Earth.
...some aesthetic concerns for a geocentric universe
… but these were generally ignored…
...the apparent change of position
of a (closer) object as measured
against the positions of more
distant object(s) due to the
movement of the observer.
A Parsec is defined as the distance of an object that exhibits
parallax of 1 arcsec
(Easy to remember since the word parsec is a
construction from parallax and arcsec)
1 parsec = 3.085678 x 1016 m = 3.26 light years
Larger parallax = smaller distance
Hipparchus of Rhodes estimated the distance to the Moon from
measurements taken during a solar eclipse in189BCE.
"full" in Hellespont
partial in Alexandria
20% of the Sun's disk remained visible in Alexandria
Since 20% of the Sun's disk corresponds to 6 arcmins,
then by estimating the distance between the cities
one can derive the distance of the Moon.
Hipparchus estimated distance (4.5 to 5.2) x 108 m
(c.f. modern value of 3.8 x 108 m)
First attempt to scale the cosmos
His (13 volume) master work
Megale Syntaxis("Great Compilation")
Is usually known as by the arab translation
Extended the system of Aristotle,
sticking "religiously" to the ideas of
a geocentric cosmology,
the perfection of spheres
a finite universe.
(There is an on-going debate whether he stole/plagiarized Hipparachus' data !)
[e.g. see Schaefer, Sky & Telescope 2002 Feb issue, p39]
The key elements are
the epicycle to account for the retrograde
motion of the planets
the deferent (main circle) to
account for the brightening &
speeding-up of the planets at
relative tilts between the various planes,
Large number of parameters
(for 8 celestial objects)
The “equant” is the point from which ang. vel
of epicycle ~const)
The model was able to make accurate predictions
Remained the "standard" cosmological model
for 1400 years
Contrary to the Greek ideas that Nature is
simple, perfect, beautiful
In the middle ages some thinkers started suggesting that
the Ptolemaic system of eccentrics/epicycles
do not actually exist,
...but are merely convenient mathematical descriptions
of celestial motion & reality (not reality itself)
Note: The concept of Nature being simple, perfect, beautiful
can be argued to be back with some modern theories (!)
Alan W. Hirshfeld, in
"Parallax - The Race to Measure the Cosmos"
“In a sense, Ptolemy was the Bill Gates of his day.
His Ptolemaic "operating system", despite its known
deficiencies, grew to dominate - in fact, monopolize -
the astronomical market place."
The modern Scientific Method
observations, theory, predictions, tests/revisions...
Roger Bacon (c.1250) helps popularize the scientific method
William of Ockham (or Occam; c.1300) suggests entities should not be
“multiplied unneccessarily”, leading to (interpreted as)
the simplest (most succinct) theory is more likely
to be correct, and certainly a better working model
(to attempt) to disprove first.
It should be remembered that our knowledge of history
is solely dependent us on having written records
Also few (if any) of the original works survive,
so we must rely on later works (true & complete reporting ?)
Who knows what ideas have been lost...
One (radical) idea that was not developed (apparently ignored)
is due to Aristarchus (c280BCE - between Aristotle & Ptolemy)
a Heliocentric universe - the Earth orbiting the Sun (!)
The following should be remembered:
Cosmology one of the oldest philosophies/sciences
Many ancient cosmologies grappled with some of the same deep
philosophical questions we still ponder with today.
The Greeks first (we think)
reasoned that Universe was formed by natural processes
which could be observed, understood/explained by mathematics
Developed the Empirical Scientific Method
Developed a geocentric system (Pythagoras of Samos, c.550BCE;
Aristotle,c.350BCE) culminating with that of Ptolemy,c.150
involving a complex arrangement of spheres & epicycles.
Reason and beauty/perfection were a strong influence of their
The universe was reasoned to be finite but eternal/unchanging
Greeks thought Earth was stationary, if it were moving, wouldn’t we
feel a sense of motion (great winds, loose objects whizzing by us etc)
Mathematics (ie. the Ptolemaic system) seen as a parameterization,
By c.1400, the Ptolemaic (geocentric) system had remained
essentially unchallenged as the cosmology for 1300 years
You should be familiar with
the concept of Parallax
the basics of how the Ptolemaic system works
(how epicycles, deferent etc account for retrograde
the concept of Ockham's Razor
Again, a detailed knowledge of names, dates and places is not required
However, you should be familiar with at least the names & approximate
dates of Aristotle (c.350BCE) and Ptolemy (c.150).
The Earth moves from Center Stage
And Then the Apple Dropped…
Summary at the beginning of the C20th
By 1400, the geocentric cosmology of Aristotle & Ptolemy
(based on concentric spheres, epicycles etc) had been essentially unchallenged for well over a thousand years.
However, in the 15th & 16th centuries, following the years of the
"Black Death" & centuries of strife, the start of the Renaissance in W.Europe finally allowed scientific & technological progress.
In c.1430, Nicholas de Cusa published
On Learned Ignorance
In which he suggested
the universe is infinite
(the universe does not have a center,
the pattern of stars would look the
same at all locations.
all motion is relative,
& that the Earth might not be stationary
Homogeneity & Relativity
The suggestion by Nicholas de Cusa (c.1430) that
the Earth might not be stationary,
was supported by Leonardo da Vinci (c1490),
who amongst many (!) other things also suggested
the Earth moves (rather than the Sun).
However it was not until 1543 when
Nicholas Copernicus publishes his
Revolutionibus Orbium Coelestium
(The Revolution of the Celestial Spheres)
that this idea was put of a more rigorous footing.
the planets rotate (on circles)
around a central Sun
….with "slower" planets being
further from the Sun.
Copernicus also acknowledged
the Earth rotates on its axis
The heliocentric model of Copernicus obviously could be used
to make predictions, that could be compared to observations.
It was simpler than the model of Ptolemy that it replaced.
it’s predictions were not any better than those of Ptolemy’s model
unless (much smaller) epicycles were added
One important aspect of Copernicus’ work - he took his heliocentric model, went further and made a model for the cosmos by saying, lets assume several things, then use observations to test whether this is a good model
Cosmological principles are the assumptions which allow us to deduce the whole of nature on the basis of the observable to the unobservable. Not surprisingly, any study of cosmological principles must combine elements of astronomy, physics and philosophy.
The Copernican Cosmological Principle
This is sometimes simply referred to as simply
“The Cosmological Principle”
The Copernican Cosmological Principle is a logical
extension of the the Copernican theory that the Earth is not the center of the universe.
Thus the Earth is not "special", thus the "laws of nature" on (or around) Earth are not special.
It is essentially a philosophical requirement/simplification necessary/assumed for all modern cosmologies:
- our laws of physics are otherwise "irrelevant"
The Copernican Cosmological Principle is that
On a large scale, the universe is both
homogeneous and isotropic(in 3-D space),
and has/will always be so.
Note that the statement "has/will always be so" refers to the universe continuing to display the properties of homogeneity & isotropy.
The CCP does not imply that any actual observable parameter (e.g. the density of matter in the universe) will remain constant with time. Indeed, the CCP allows the properties of the universe to evolve with time, but states that at any given time the universe will be both homogeneous and isotropic (in 3-D space).
Another way of expressing
the Copernican Cosmological Principle is that ...
will see identical properties & laws everywhere- homogeneity
will NOT see any preferred direction - isotropy
i.e. this was the suggestion that we do not occupy a special place in the universe
homogeneous - same properties everywhere
isotropic - no special direction, uniform in all directions
homogeneous but not isotropic
isotropic but not homogeneous
A (small) sentient being living in the center of a "perfect" loaf of bread…!
There may be obvious structure on small scales (air bubbles etc), but on the large scale the loaf can be considered
uniform and isotropic
The laws of physics (e.g. which caused the dough to rise)
are the same throughout the loaf.
The loaf might still be rising - but (in this perfect loaf) this happens
uniformly & following then same lawsthroughout the loaf
The best support for the Copernican Cosmological Principle is the Cosmic Microwave Background (CMB), which is isotropic to 1 part in 105
The obvious observational evidence against the Copernican Cosmological Principle seems to be the structure seen in the universe on a variety of scales
(stars, galaxies, clusters, super-clusters..the cosmic web)
This is why the qualifier "On a large scale.." is required to be added to the principle.
The question them becomes a question of scale (now large is "large" ?), and whether the observed structures on large scales are indeed representative of the universe on these scales (or are "perturbations" which "happen” to be visible to us).
We are up to 1400
We briefly mentioned the ideas of Nicholas de Cusa
Homogeneity & Relativity
We discussed the Heliocentric system of Nicholas Copernicus
We then discussed the Copernican Cosmological Principle
On a large scale, the universe is both
homogeneous and isotropic (in 3-D space)
Also famous for having lost his nose in a swordfight
Tycho Brahe (c.1570)
did accept that the (other) Planets
move around the Sun
but did not accept that the Earth & Stars
move around the Sun
… Falling bodies fall towards the Earth
... The lack of Stellar Parallax
Falling Bodies fall towards the Earth
Indeed if you throw something vertically upwards,
it falls vertically downwards (to the same spot)
Tycho Brahe reasoned this surely meant
the Earth was the center of the universe
Tycho Brahe was unable to detect (by naked-eye) Stellar Parallax
and reasoned that in a Copernican system this would require
the Stars to be so far away they would have to be
Tycho Brahe was primarily an observer
position did not change (so it was not a comet or meteor),
- lack of Parallax must be in one of the “outer spheres”
-therefore the outer sphere of stars does change!
position did not change significantly throughout the night.
- lack of Parallax, must lay beyond the orbit of the Venus
Observed positions of Mars twice-daily
which implied its orbit intersects that of the Sun.
- apparent crossing/smashing of the “crystalline spheres”
“there are no solid spheres "holding" the celestial bodies”
- observations that strengthened the rejection of the cosmology of Aristotle & Ptolemy
So, even though Tycho Brahe never found the “right” model…
… his observations did play a major role in the final
of the notions of Aristotle/Ptolemy that
the celestial bodies are carried by crystalline spheres,
with everything beyond the Moon eternal & unchanging.
In addition Tycho Brahe also actually published his data !
In particular his twice-daily measurements of the position of Mars
provided Johannes Kepler with a crucial database a few years later.
a + b = constant
The amount of "flattening" of the ellipse is termed the eccentricity
All ellipses have eccentricities lying between zero and one.
e=ratio; dist. between foci compared to major axis
Kepler's Second Law: Line joining planet and the Sun sweeps out equal areas in equal times
Kepler's Third Law:The squares of the periods of the planets are proportional to the cubes of their semi-major axes:
Kepler’s 3rd Law: version 2
This equation may then be solved for the period P of the planet, given the length of the semi-major axis axis,
or for length of the semi-major axis, given the period of the planet
Second example, let us calculate the orbital period for Pluto, given that its observed average separation from the Sun is 39.44 astronomical units. From Kepler's 3rd Law
Galileo saw shadows cast by the mountains on the Moon.
He observed craters.
The Moon had a landscape; it was a “place”, not a perfect heavenly body.
Galileo discovered that Jupiter had four moons of its own.
Jupiter was the center of its own system.
Heavenly bodies existed which did not orbit the earth.
No gibbous or full phases!
All phases are seen!
His astronomical work got him in trouble with the Church!
Galileo observed all phases!
This version of 3rd Law can use *any* units for P and R (hours, years, weeks, km, feet etc etc
This version of 3rd Law is simpler, because we are constrained to use years and AU
Rearranged versions ‘isolate’ P or R
All you need is to be able to use the the squared and cubed buttonson your calculator, plus the square-root or cube-root 3 buttons
Rearranged versions ‘isolate’ P or R
All you need is to be able to use the the “squared” and “cubed” buttonson your calculator, plus the square-root or cube-root 3 buttons
Calculate R cubed (RxRxR) then take the square-root of that number
Calculate R squared (RxR) then take the cube-root of that number