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Welcome to astronomy at ISM
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About me and astronomy
About you and astronomy
About this class: handout, website, text, me
Notes in class: write what I write
Our toolbox: eyes, Stellarium, physlets, models,
telescopes
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Is astronomy class fun?
FUN
This class can be interesting, mindblowing,
awesome, gratifying, and rewarding.
But you have to do your work in class and outside
of class.
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ISM Lab Safety Rules
No horseplay in the lab
No students may stand on chairs or tables
Students may not move chemicals from the lab
Eye goggles or safety glasses are required by all people in the
lab for the following conditions
–Any heat source is in use
–Anyone is using any material (liquid, dry, or gas) that would not be harmful if
put directly into an eye
–A pressure source (higher than a blown up balloon) is in use
–Activities are in the lab that could result in chips, shards, or any flying objects,
such as hammering, launching projectiles, etc.
– Food in the labs
•Food is not allowed in the lab (except water in bottles)
•Food can be allowed for planned special activities and with teacher
permission, but all table tops must be disinfected and wiped down
beforehand.
•Food or water are not allowed if a lab activity is ongoing
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23 Aug: Resources for the class
• Ismscience.org
– Note assignment
– See the points for this week
• Astronomy: A Beginner’s Guide to the
Universe, 7th ed. Chaisson and McMillan
• Powerpoint slides
• Stellarium software
• Apps such as Solar Walk and Star Walk
• “Yakko’s Universe” song from Animaniacs
• Monty Python’s “Galaxy Song”
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Neil Armstrong died this week, 2012
Show video for curiosity launch
https://www.youtube.com/watch?v=P4boyXQuUIw
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Andromeda galaxy M31
Find on stellarium
2.5 x 106 light years away from us
Nearly one trillion stars
Nearest spiral galaxy to the milky way
Overall is 6x as wide as the full moon!
the great thing about astronomy is how
beautiful the subjects are.
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• https://www.youtube.com/watch?v
=EIxwTNp38W8&feature=youtu.be
• Look at video; the stars and planets
will go on, oblivious to the collision!
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You are responsible to learn and
use Stellarium
• Go to stellarium.org and download it
• Find each and report magnitude
– Andromeda galaxy
– Polaris
– The moon
• You will demonstrate in two weeks that you
know how to use it.
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Preseason quiz
1.
2.
3.
4.
Define Astronomy.
How many planets in our solar system?
Name as many planets as you can.
Arrange these in order of size, smallest first:
Saturn, Earth, galaxy, solar system, star, nebula
5. What country was first to put a satellite in
space, a dog in space, a man in space, and a
human to orbit the earth?
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Lecture Outline
Chapter 0
Charting the
Heavens
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Chapter 0
Charting the Heavens
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Units of Chapter 0
The “Obvious” View
Earth’s Orbital Motion
The Motion of the Moon
The Measurement of Distance
Science and the Scientific Method
Summary of Chapter 0
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How to read/preview a chapter
• FIRST, in 20 minutes
– Read learning goals and intro page carefully
– Major titles and pictures
– Highlighted words
– Skip “discovery”, and Skip “more precisely”
– Skip equations
– Read summary
• Next, in 60-120 minutes
– Now read (by skimming) all sections
– Keep thinking about major points, skip equations
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QUIZ FOR 24 Aug
• Write down 10 of the sources of information
in each chapter of your text.
• Write down 10 additional resources in the
book outside of the chapters.
• Work as a group, each contributing….
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2 sept. 0.1 The “Obvious” View
• Earth is average – we don’t occupy
any special place in the universe
• Universe: Totality of all space, time,
matter, and energy
• Astronomy: Study of the universe
• Scales are very large, measured in
light-years, the distance light travels
in a year – about 10 trillion miles
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Part Opener 1.1
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Part Opener 1.2
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Part Opener 1.3
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Part Opener 1.4
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Part Opener 1.5
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Part Opener 2.1
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Part Opener 2.2
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Part Opener 2.3
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Part Opener 2.4
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Part Opener 2.5
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Part Opener 3.1
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Part Opener 3.2
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Part Opener 3.3
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Part Opener 3.4
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Part Opener 3.5
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Part Opener 4.1
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Part Opener 4.2
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Part Opener 4.3
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Part Opener 4.4
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Part Opener 4.5
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Once in a blue moon (tonight)
A blue moon can refer to either the third full moon in a
season with four full moons, or the second full moon in a
month.[1] Most years have twelve full moons that occur
approximately monthly. In addition to those twelve full lunar
cycles, each solar calendar year contains roughly eleven days
more. The extra days accumulate, so every two or three years
there is an extra full moon. The term "blue moon" comes from
folklore. Different traditions and conventions place the extra
"blue" full moon at different times in the year.
In calculating the dates for Lent and Easter, the Clergy identify
the Lent Moon. It is thought that historically when the moon's
timing was too early, they named an earlier moon as a
"betrayer moon" (belewe moon), thus the Lent moon came at
its expected time.[2]
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0.1 The “Obvious” View
Stars that appear close in the sky may not
actually be close in space.
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**0.1 The “Obvious” View
The celestial sphere:
• Stars seem to be on the
inner surface of a sphere
surrounding the Earth.
• They aren’t, but we can
use two-dimensional
spherical coordinates
(similar to latitude and
longitude) to locate sky
objects.
•What is the celestial
equator?
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More Precisely 0-1: Angular Measure
• Full circle contains
360° (degrees).
• Each degree contains
60′ (arc-minutes).
• Each arc-minute
contains 60″ (arcseconds).
• Angular size of an
object depends on
actual size and
distance away.
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*** 0.1 The “Obvious” View
• Declination: Degrees north or south of celestial
equator
• Right ascension: Measured in hours, minutes,
and seconds eastward from position of the Sun at
vernal equinox
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30 aug. 0.2 Earth’s Orbital Motion
•Daily cycle, noon to noon,
is diurnal motion – solar
day.
• Stars aren’t in quite the
same place 24 hours later,
though, due to Earth’s orbit
around the Sun; when they
are in the same place
again, one sidereal day has
passed.
•Show this on Stellarium
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A synodic day is the period of time it takes for a planet to rotate
once in relation to the body it is orbiting (as opposed to a
sidereal day which is one complete rotation in relation to the
stars).
Thus, a synodic day may be "sunrise to sunrise'" whereas a
sidereal day may be "star-rise to star-rise".
Diurnal motion, meaning daily motion, describes the synodic
day.
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Orbit vs. rotation
• Be careful with the words
• Orbit = to go around something else
– Earth’s orbit around the sun requires how many
days?
– Moon around earth?
• Rotation (usually) means to spin on its own
axis.
– Earth’s rotation requires how many days?
– The sun’s “spin” requires how many days?
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0.2 Earth’s Orbital Motion
The 12 constellations the Sun moves through
during the year are called the zodiac; the path is
called the ecliptic. (see fig 0.9, find vernal equinox)
What you
see at night
is opposite
to the sign’s
season!
See solar walk movie
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***0.2 Earth’s Orbital Motion
• Ecliptic is plane of Earth’s path around the Sun; at 23.5o
to celestial equator.
• Northernmost point (above celestial equator) is summer
solstice; southernmost is winter solstice; points where
path crosses celestial equator are vernal and autumnal
equinoxes.
• Combination of
day length and
sunlight angle gives
seasons.
• Time from one
vernal equinox to
next is tropical year
(“year”).
=Solar year. Is there a siderial year?
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Quiz for Wednesday
1. Out of 3000 stars in the visible sky, western civilization
created over 80 constellations. Why do we feature only 12
constellations in the zodiac? a: there are 12 months in the
year b: there are 12 eggs in a dozen c: ancient people had
12 fingers d: 12 constellations fit along the ecliptic
2. If you look at the stars every night at midnight, they appear
at slightly different positions. a: the earth spins on its axis
b: the moon moves c: the earth moves around the sun d:
the sun moves through the sky
3. a: diurnal b: declination c: ecliptic d: angle
is the word used to describe position in the sky and is
measured in degrees north or south of the Earth’s equator
4. Describe what the ecliptic is: _____________
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A model of the earth-moon system
Sketch this in your notes
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•
Globe, diameter: 30cm (this sets the scale)
Moon, diameter: 8cm to scale
Earth-moon distance: 9.0 m to scale
Sun, diameter 33 m to scale! (we can’t do
this to scale)
• Earth-sun distance: 3500 m to scale!
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News you can use.
• http://www.nytimes.com/2014/09/03/science
/space/the-v838-monocerotis-star-still-hasastronomers-headsexploding.html?smprod=nytcoreiphone&smid=nytcore-iphone-share
• The eruption of a star 20,000 years ago made
it a million times as luminous as the sun, and
its violent beauty still echoes through space.
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*0.2 Earth’s Orbital Motion
Precession: Rotation of Earth’s axis itself;
makes one complete circle in about 26,000
years (demo with bike wheel or top)
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The north star will shift slowly!
More about precession
• See for example problem 1.
• Precission is the shift in our tilt during time.
– Takes 26000 years for one full rotation
• Recall that seasons (equinox, solstice) depend on
that tilt, so seasons shift
• Therefore, the tropical year shifts during that 26000
years.
• Therefore, siderial and tropical year differ by 365
days/26000 = 20 minutes.
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Precession
• what exactly is precession?
– The shift in our tilt during time.
– Takes 26000 years for one full rotation
• Recall that seasons (equinox, solstice) depend on
that tilt
• Therefore, the tropical year (solar year) shifts
during that 26000 years.
• Therefore, siderial and tropical year differ by 365
days/26000 = 20 minutes.
• The zodiac year is where in the background stars
the vernal equinox points
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0.2 Earth’s Orbital Motion
Time for Earth to orbit once around the Sun,
relative to fixed stars, is sidereal year. This is
20 minutes longer than a tropical year!
Tropical year follows seasons; sidereal year
follows constellations – in 13,000 years July
and August will still be summer, but Orion will
be a summer constellation.
Did you know they just (2011) arranged the
signs of the zodiac to account for this?
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September 1 was the date, in 1859, of a massive solar superstorm. It's
sometimes called the "perfect space storm" or the Carrington Event, after
British astronomer Richard Carrington. He reported witnessing a massive
white-light solar flare: a bright spot suddenly appearing on the surface of the
Sun. At the same time, the Sun produced a coronal mass ejection, or CME: a
large eruption of magnetized plasma. CMEs usually take three to four days to
reach Earth, but the magnetic burst from the superstorm of 1859 reached us
in just under 18 hours.
While Earthlings of 1859 didn't have any cell phones, GPS units, or television
signals to worry about, they were growing accustomed to rapid
communication over the telegraph, which had been in use for 15 years.
Within hours of the CME, telegraph wires began shorti ng out, starting fires
and disrupting communication in North America and Europe. Compasses
were useless because the Earth's magnetic field had gone haywire. The
northern lights were seen as far south as Cuba and Hawaii, and the southern
lights — aurora australis — were seen in Santiago, Chile. People in the
northeastern United States could read the newspaper by the light of the
aurora, and the Sun itself was twice as bright during the event.
See http://www.youtube.com/watch?v=lT3J6a9p_o8 for good video
See the SDO for our response…….
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1 sep. 0.3 The Motion of the Moon
The Moon takes about
29.5 days to go
through whole cycle
of phases – synodic
month.
Phases are due to
different amounts of
sunlit portion being
visible from Earth.
Time to make full 360°
around Earth, sidereal
month, is about 2
days shorter than
synodic month.
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The moon
phases and size
http://www.youtube.com/watch?v=qGwRp38-gzE
• Why does it get bigger and smaller?
• Is it at the same phase when it is biggest?
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Play Movie: See next page for detail.
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A NASA camera aboard the Deep Space Climate Observatory
(DSCOVR) satellite captured on 16 July 2015 a unique view of
the moon as it moved in front of the sunlit side of Earth last
month. The series of images shows the fully illuminated “dark
side” of the moon that is never visible from Earth.
The images were captured by NASA’s Earth Polychromatic
Imaging Camera (EPIC), a four megapixel CCD camera and
telescope on the DSCOVR satellite orbiting 1 million miles from
Earth. From its position between the sun and Earth, DSCOVR
conducts its primary mission of real-time solar wind monitoring for
the National Oceanic and Atmospheric Administration (NOAA).
NASA will post daily color images of Earth to a dedicated public
website. These images, showing different views of the planet as it
rotates through the day, will be available 12 to 36 hours after they
are acquired.
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0.3 The Motion of the Moon
Lunar eclipse:
• Earth is between the Moon and Sun
• Partial when only part of the Moon is in shadow
• Total when all is in shadow
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Umbra and Penumbra
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0.3 The Motion of the Moon
Solar eclipse: the Moon is between Earth and Sun
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Solar walk movie:
https://www.youtube.com
/watch?v=hTucwKxXxSg
http://www.nytimes.com/2016/09/01/science/ring-of-fire-solar-eclipse.html?smprod=nytcore-iphone&smid=nytcore-iphone-share&_r=0
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6 sept: 0.3 The Motion of the Moon
Solar eclipse is
partial when only
part of the Sun is
blocked, total when
all is blocked, and
annular when the
Moon is too far
from Earth for total.
What direction does
the moon rotate as it
orbits the earth?
And the earth around
the sun? ….
Copyright © 2010 Pearson Education, Inc.
Do eclipses occur during the same
months?
•
•
•
•
eclipses can only occur when the Moon is close to a node and it is also either full or new.
With no other forces acting, the line of nodes would therefore be in line with the Sun every
six months.
However, also like the Earth's axis, the gravitational force of the Sun on the Moon causes the
line of nodes to precess.
This precession is a much more noticeable effect than the Earth's precession, with one every
19 years.
As a result, the time between alignments is decreased to about 5.4 months.
Because of the finite size of the Earth, Moon, and their shadows, multiple eclipses can occur
whenever the line of nodes points near the Sun.
So, eclipses are actually very common!
During a one-year period, there can be between two and five eclipses of each kind (solar and
lunar), with a total of between four and seven.
This includes partial and penumbral lunar eclipses, and partial and annular solar eclipses.
Lunar eclipses are much more likely to be observed, since anyone on the night side of the
Earth can see them.
Solar eclipses, on the other hand, cover only a small fraction of the Earth, and often occur
over unpopulated locations such as the polar regions or the oceans.
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Solar Eclipses: 2011 - 2020
Note that
eclipses
come
during all
times of
the year.
Calendar Date
TD of
Greatest
Eclipse
Eclipse
Type
Saros Series
(Link to Global
Map)
(Link to
Animation)
(Link to
Google
Map)
(Link to
Saros)
Eclipse
Magnitude
Central Duration
Geographic Region of Eclipse Visibility
(Link to Path Table)
2011 Jan 04
08:51:42
Partial 151
0.858
-
Europe, Africa, c Asia
2011 Jun 01
21:17:18
Partial 118
0.601
-
e Asia, n N. America, Iceland
2011 Jul 01
08:39:30
Partial 156
0.097
-
s Indian Ocean
2011 Nov 25
06:21:24
Partial 123
0.905
-
s Africa, Antarctica, Tasmania, N.Z.
2012 May 20 23:53:53
Annula 128
r
0.944
05m46s
2012 Nov 13
Total
133
1.050
04m02s
2013 May 10 00:26:20
Annula 138
r
0.954
06m03s
2013 Nov 03
12:47:36
Hybrid 143
1.016
01m40s
2014 Apr 29
06:04:32
Annula 148
r
0.987
-
2014 Oct 23
21:45:39
Partial 153
0.811
-
Asia, Pacific, N. America
[Annular: China, Japan, Pacific, w U.S.]
Australia, N.Z., s Pacific, s S. America
[Total: n Australia, s Pacific]
Australia, N.Z., c Pacific
[Annular: n Australia, Solomon Is., c Pacific]
e Americas, s Europe, Africa
[Hybrid: Atlantic, c Africa]
s Indian, Australia, Antarctica
[Annular: Antarctica]
n Pacific, N. America
2015 Mar 20
09:46:47
Total
120
1.045
02m47s
2015 Sep 13
06:55:19
Partial 125
0.788
-
2016 Mar 09
01:58:19
Total
130
1.045
04m09s
2016 Sep 01
09:08:02
Annula 135
r
0.974
03m06s
2017 Feb 26
14:54:32
Annula 140
r
0.992
00m44s
s S. America, Atlantic, Africa, Antarctica
[Annular: Pacific, Chile, Argentina, Atlantic, Africa]
2017 Aug 21
18:26:40
Total
1.031
02m40s
N. America, n S. America
[Total: n Pacific, U.S., s Atlantic]
22:12:55
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145
Iceland, Europe, n Africa, n Asia
[Total: n Atlantic, Faeroe Is, Svalbard]
s Africa, s Indian, Antarctica
e Asia, Australia, Pacific
[Total: Sumatra, Borneo, Sulawesi, Pacific]
Africa, Indian Ocean
[Annular: Atlantic, c Africa, Madagascar, Indian]
0.3 The Motion of the Moon
Eclipse tracks, 2010 - 2030
How long does an eclipse last
at any one point on earth?
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Is the moon’s orbit circular?
• NO. No orbit is perfectly circular.
• Check: how elliptical is it?
• Know the phases of the moon
– Fig 0.13
– Know days and names
• If the moon moved twice as fast, what would
change above?
• Use SolarWalk to see this:
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*0.3 The Motion of the Moon
Eclipses don’t occur every month because
Earth’s and the Moon’s orbits are not in the
o
same plane. Angle is 5.2 . Where did that
arise?
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See solar walk on lunar
eclipses
Practice
• In groups of 3, show sun-earth-moon system
for:
– Lunar eclipse
– Solar eclipse
– First quarter moon
– Waning gibbous moon
– High-high tide
Copyright © 2010 Pearson Education, Inc.
0.4 The Measurement of Distance
Triangulation:
Measure baseline
and angles, and
you can calculate
distance.
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0.4 The Measurement of Distance
Parallax: Similar to
triangulation, but looking at
apparent motion of object
against distant background
from two vantage points
The measure of parallax
allows us to measure the
distance to remote objects in
space.
(note, we can always find a
further star for reference)
A good applet
http://www.astro.ubc.ca/~scharein/applets/Sim/
new-parallax/Parallax.html
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Sizing up planet Earth
• Discovery 0.1
• Please know
– The size of the earth: r = 6380km
– That 2300 years ago, learned people knew the
earth was round
• Was Columbus learned only 500 years ago?
– That 2200 years ago, the size of the earth was
calculated within 1%.
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0.5 Science and the Scientific Method
Scientific theories:
• Must be testable
• Must be continually tested
• Should be simple
• Should be elegant
Scientific theories can be proven wrong, but
they can never be proven right with 100%
certainty.
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0.5 Science and the Scientific Method
• Observation leads to theory explaining it.
• Theory leads to predictions consistent with previous
observations.
• Predictions of new
phenomena are
observed. If the
observations agree
with the prediction,
more predictions can
be made. If not, a new
theory can be made.
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Scientific method and the
discovery of the first Galaxy
• http://scienceblogs.com/startswithabang/2013/09/0
2/messier-monday-andromeda-the-object-thatopened-up-the-universe-m31/
• Observe: Fuzzy nebula, thought to be a typical
nebula (nova or dust cloud or cluster of stars nearby)
• Then, new data, astrophotography, plus variable star
to give distance, meant it was far away and very large
• Much farther than any other star
• Thus, a new “universe”, called a galaxy
• All the subsequent data supported this
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What is the practical value of
astronomy to humans?
1.
2.
3.
4.
5.
6.
7.
8.
9.
tides
Energy from sun
weather
seasons
navigation
tourism
Asteroid avoidance
minerals and materials in space
solar storms affect us
10. Sun affects earth’s weather and warming, solar storms dangerious, future
Copyright © 2010 Pearson Education, Inc.
of humankind is out there, asteroid risk
Practical, for example:
• How do these effects all interact on earth?
– Axial Tilt (varies on 40,000 yr cycle)
– Orbital Eccentricity (100,000 yr cycle)
– Axial Precession (26,000 yr cycle)
• astronomer Milutin Milanković (1920’s) linked
them to insolation (the amt of sunlight striking
earth)
• Prof Larry Edwards (2016) at U of M
Geosciences linked this to cave stalactites and
weather
Copyright © 2010 Pearson Education, Inc.
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**Summary of Chapter 0
• Astronomy: Study of the universe
• Stars can be imagined to be on inside of
celestial sphere; useful for describing location.
• Plane of Earth’s orbit around Sun is ecliptic; at
23.5° to celestial equator.
• Angle of Earth’s axis causes seasons.
• Moon shines by reflected light, has phases.
• Solar day ≠ sidereal day, due to Earth’s rotation
around Sun.
Copyright © 2010 Pearson Education, Inc.
Summary of Chapter 0, cont.
• Synodic month ≠ sidereal month, also due to
Earth’s rotation around Sun
• Tropical year ≠ sidereal year, due to
precession of Earth’s axis
• Distances can be measured through
triangulation and parallax.
• Eclipses of Sun and Moon occur due to
alignment; only occur occasionally as orbits
are not in same plane.
• Scientific method: Observation, theory,
prediction, observation …
Copyright © 2010 Pearson Education, Inc.
quiz
1. Describe precession in a sentence or two.
2. Which is longer: solar day or siderial day?
1. By how many minutes?
3. During what phase of the moon does an
eclipse occur? Why?
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**I love space probes
• Voyager 1, good bye and good luck!
– Wikipedia and pale blue dot photo
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Super close Supernova
• http://latimesblogs.latimes.com/nationnow/2
011/08/new-supernova-discovered-closestexploding-star-to-earth-in-25-years.html
• Hard to see with small telescopes, but
possible.
• Supernovae are key to 2011 Nobel Prize
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What is this?
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2015 solar system model
• Scale to a sun of 30mm.
• Entire model fits on a foamcore board and
allows orbital motion of each planet
• Distance scale is for convenience, but the true
scale distance is shown on the connecting
spoke.
• We will show 19 objects total
•
•
•
Sun
Saturn
Oort cloud
Mercury
Venus
Saturn moons
Uranus
Hubble Space Telescope
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Earth
Nepturn
moon
Kuyper belt
Mars
Pluto
Asteroid belt
Jupiter
Comet with New Horizons
Jupiter moons
Astronomy lab #1: build solar
system model in the school
• http://www.exploratorium.edu/ronh/solar_sy
stem/ is one tool, I did work on Excel.
• New challenge this year: scale distance AND
size
• Start with earth of 1mm
• Show limits of manned flight
• Show limits of robotic flight
Copyright © 2010 Pearson Education, Inc.
Why build a solar system model?
• https://www.youtube.com/watch?v=zR3Igc3R
hfg 7 minutes
Copyright © 2010 Pearson Education, Inc.
2014 Steps to build solar system
1. Set the size of one object
Our choice: Earth = 1.0mm in diameter
2.
3.
4.
5.
Scale all other sizes from this
Scale all distances from this
Assign planets to students
Find materials and construct, using black
backgrounds for visibility.
6. Post in school, using Google Earth distances
7. Admire and be amazed
8. Use for rest of the year
Copyright © 2010 Pearson Education, Inc.
2013 model
• Scale all planets to the sun, 1.0 m in diameter
– Use printed sun pic
• Distance scale is different, so that all 8 will fit
in the room.
– Very compressed
Copyright © 2010 Pearson Education, Inc.
2014 solar system
Object
Actual
scale
Actual
scale distance
distance from diameter in
assignment
diameter km
from
sun
in
m
sun 106 km
mm
scale factor
12.8*109
Sun
1.40E+06
0
Mercury
4880
Venus
12756
109.752
0.000
adam
57.9
0.383
4.539
kamin
12104
108.2
0.949
8.482
luyu
Earth
12756
149.6
1.000
11.728
emma
Mars
6788
227.9
0.532
17.866
brad
Jupiter
142984
778.4
11.209
61.022
trahern
Saturn
120537
1427
9.449
111.869
jamie
Uranus
51118
2871
4.007
225.071
xinzhu
Neptune
49532
4498
3.883
352.618
ryan
moon
3475
0.384
0.272
0.030
dr. f
Copyright © 2010 Pearson Education, Inc.
from moon to
earth
Google Earth distances
Copyright © 2010 Pearson Education, Inc.
2013 Steps to build solar system
1. Calculate the correct distance and size scales
2. Decide which of these we can model in the
lab
3. set final scale reference (sun)
4. Assign planets
5. Do research
6. Find materials and construct durable object
7. Hang where convenient and visible.
8. Admire
Copyright © 2010 Pearson Education, Inc.
Setting the scale 2013
Object
Actual
diameter
km
Actual
distance
from sun
106 km
Sun
14E5
0
Mercury
4880
57.9
Venus
12104
Earth
Scaled
Scaled
diameter for distance for
model cm
model cm
assignment
0
Fisher
0.35
0.14
Ashley
108.2
0.86
0.26
Georgie
12756
149.6
0.91
0.37
Sophia
Mars
6788
227.9
0.48
0.56
Leo
Jupiter
142984
778.4
10.21
1.90
Ashley
Saturn
120537
1427
8.61
3.49
Georgie
Uranus
51118
2871
3.65
7.02
Lydia
Neptune
49532
4498
3.54
11.00
Kirsten
Copyright © 2010 Pearson Education, Inc.
100