Download complete lab manual

DEPARTMENT OF
ASTROPHYSICAL AND PLANETARY SCIENCES
APS 1010 Laboratory
Introduction to Astronomy
Fall 1999
University of Colorado at Boulder
Acknowledgements
The content of this laboratory manual is based to a large extent upon experiments
compiled and developed over the course of many years by Dr. Roy Garstang for
the APAS 121/122 labs. In 1987 Dr. Garstang published the Sommers-Bausch
Observatory Manual of General Astronomy Projects; a number of his 88 exercises
were adapted for use in the new APAS 101 (now APS 1010) course that was
initiated in 1988.
Since that time the manual has undergone considerable evolution and expansion
under the guidance of faculty lab coordinators Drs. Raul Stern, Scott Roberston,
Larry Esposito, and Fran Bagenal. Many other faculty members have contributed
new exercises or major suggestions for improvements, including in particular Drs.
John Stocke, Larry Esposito, J. McKim Malville, Carl Hansen, and the late Ned
Benton.
Virtually every graduate teaching assistant who has conducted a lab in the
APAS/APS 121/122/101/1010/1030 series has contributed to this manual by
identifying areas of difficulty, made suggestions for improvements, or provided
ideas that germinated into new experiments. Although the contributors from this
quarter are too numerous to list, special mention should be made of those who
developed a new lab exercise or assisted in the coordination and production of
earlier editions: Alexis Hayden, Marsha Allen, Josh Colwell, Ron Wurtz, Joel
Parker, and Eric Perlman - and most recently and significantly to the current version
of the manual, Marc DeRosa, KaChun Yu, Mary Urquhart, and Travis
Rector.
Finally, unfailing educational support and promotion for these laboratories has been
provided by the past and present Directors of Sommers-Bausch Observatory and
Fiske Planetarium: Drs. Bruce Bohannan and Katy Garmany.
Keith Gleason, Editor
SBO Manager/Lab Coordinator
Summer, 1999
Front Cover: Comet Hale-Bopp over the Flatirons
courtesy Niescja Turner and Carter Emmart
Back Cover: CCD Mosaic of the Full Moon
contributed by Keith Gleason
APS 1010 Astronomy Lab
3
Table of Contents
TABLE OF CONTENTS
Celestial Calendar - Fall 1999......................................................
General Information.................................................................
Units and Conversions..............................................................
Scientific Notation...................................................................
Math Review .........................................................................
Using Your Calculator ..............................................................
Celestial Coordinates................................................................
Telescopes & Observing............................................................
4
6
11
15
18
22
27
35
DAYTIME EXPERIMENTS ..................................................
43
Colorado Model Solar System ✩ ..................................................
The Seasons ★ .......................................................................
Motions of the Sun ..................................................................
Motions of the Moon ✩ .............................................................
Gravity and the Mass of the Earth .................................................
Kepler's Laws........................................................................
Temperature of the Sun ★ ..........................................................
Spectroscopy ✩ .....................................................................
Telescope Optics.....................................................................
Planetary Temperatures ✩ ..........................................................
Encounter with the Galilean Moons ...............................................
45
51
59
63
69
75
79
87
93
101
109
O B S E R V I N G P R O J E C T S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Constellation and Bright Star Identification ★ ...................................
Telescope Observing ★ .............................................................
Observing Lunar Features ★ .......................................................
The Messier Catalog (CCD Imaging) ★ ..........................................
★
- Clear skies are required for the exercise
✩ - Clear skies are needed for a portion of the exercise
117
121
129
135
APS 1010 Astronomy Lab
4
Celestial Calendar – Fall, 1999
CELESTIAL CALENDAR - Fall 1999
THE SUN
The Sun reached its highest declination (+23.5°) on June 21st, marking the Summer Solstice.
On September 23rd, it will be directly over the Earth’s equator at 5:31 a.m. Mountain Daylight
Time; the arrangement marks the Autumnal Equinox, the moment when fall officially begins.
Finally, on December 22nd , the Sun will reach its lowest declination (-23.5°), marking the Winter
Solstice and the smallest number of daylight hours of the year.
Our time-keeping method, by which we keep track (roughly) of the Sun’s location in the sky,
makes a one-hour jump backward at 2:00:00 a.m. Mountain Daylight Time on Sunday, October
31st - the next second of time will be referred to as 1:00:01 a.m. Mountain Standard Time.
The last total solar eclipse of the millenium happens on August 11th ; unfortunately for us, the
eclipse will only be visible from Europe and Asia, not from North America.
THE MOON
Month
August
September
October
November
December
Last
New
First
Full
Last
4
2
1
11
9
9
7
7
18
17
17
16
15
26
25
24
23
22
31
29
29
MERCURY
The innermost planet won't be visible as an "evening star" this fall until early December.
However, a rare grazing transit of the Sun by Mercury will occur on the late afternoon of
November 15th , when the tiny black disk of the planet will skim along the edge of the Sun for
about a half-hour. Transits of Mercury occur only 13 times each century, while a grazing transit is
far more rare. If skies are clear, Sommers-Bausch Observatory will be holding a special observing
session for viewing this once-in-a-lifetime opportunity - watch the website lyra.colorado.edu/sbo/
for further details.
APS 1010 Astronomy Lab
5
Celestial Calendar – Fall, 1999
VENUS
Venus appears as a brilliant "morning star" by mid-September, and will remain in the pre-dawn
eastern sky throughout the remainder of the semester.
EARTH
The third planet from the Sun is visible throughout the year, any day or night. Look down.
MARS
Mars will remain very low in the southwest throughout the fall - its prograde motion through the
skies just keeping it always above the horizon just after sunset. However, the planet will be a
disappointing telescope sight, shrinking from 10 arc-seconds in diameter in August to only 5 arcseconds by December, while becoming a full magnitude dimmer.
JUPITER
The King of the Planets will become a prominent evening object after September, when it rises as
twilight fades. October through December, the giant planet's disk will exceed 45 arc-seconds
across, permitting spectacular views of its cloud bands and subtle atmospheric features. The four
Galilean moons (Io, Europa, Ganymede, and Callisto) change their positions around Jupiter
nightly, and are easily seen with a small telescope.
SATURN
Saturn can be observed late at night early in the
semester, or shortly after sunset in November and
December. This fall and winter offers the best viewing
of the ringed planet since 1978! It is higher in the
northern sky (+13 degrees declination), closer to Earth
(disk 20 arc-seconds across and rings 46 arc-seconds
across), with rings highly-tipped (over 20 degrees).
The Ringed Planet is guaranteed to be a glorious jewel
for telescope observation!
URANUS, NEPTUNE
The two outermost gas giants are only several degrees apart in the constellation of Capricornus,
where they may be found by telescope low in the southern skies late evening throughout the
semester. Mars passes only 1.7 degrees from Neptune on November 28th , and 0.7 degrees from
Uranus on December 14th .
PLUTO
Always difficult to see at best, Pluto might be glimpsed in Ophiuchus (with good finder charts)
early evening early in the fall semester.
METEOR SHOWERS
Fall is prime time for good meteor showers, and this fall they should be better than ever. Watch
for the Perseids on the evening of August 12 th , the Leonids on the night of November 17th /18th ,
and the Geminids the night of December 14th . The Leonids in particular may be especially
spectacular this year, perhaps increasing to "storm" levels visible in some locales of the Earth.
Moonlight will not be a problem for midnight observers of any of these showers.
APS 1010 Astronomy Lab
6
General Information
GENERAL INFORMATION
You must enroll for both the lecture section and a laboratory section.
Your lecture section will generally be held in the classroom in Duane Physics Building (just south
of Folsom Stadium). An occasional lecture may be held instead at the Fiske Planetarium (at the
intersection of Regent Drive and Kittridge Loop).
Your laboratory section will meet once per week in the daytime in Room S175 at SommersBausch Observatory (just east of the Fiske Planetarium); follow the walkway around the south
side of Fiske and up the hill to the Observatory.
You will also have nighttime observing sessions using the Observatory telescopes to view
and study the constellations, the moon, planets, stars, and other celestial objects.
Folsom
Street
Folsom
Stadium
Engineering
Center
Events
Center
Colorado
Avenue
Business
DUANE PHYSICS
(CLASSROOMS)
Colorado Scale
Model Solar System
Regent
Drive
CDSS
GAMOW TOWER
(APAS OFFICE)
SOMMERS-BAUSCH
OBSERVATORY
JILA
Tower
18th
Street
Hallet
Willard
FISKE
PLANETARIUM
Kittridge
Regent
MATERIALS
The following materials are needed:
•
APS 1010 Astronomy Lab Manual, Fall 1999 (this document). Available from
the CU Bookstore. Replacement copies are available in Acrobat PDF format
downloadable from the SBO website (http://lyra.colorado.edu/sbo/).
•
Calculator. All students should have access to a scientific calculator which can perform
scientific notation, exponentials, and trig functions (sines, cosines, etc.).
•
A 3-ring binder to hold this lab manual, your lab notes, and lab write-ups.
•
Highly recommended: a planisphere, or rotating star map (available at Fiske
Planetarium, the CU bookstore, other bookshops, etc.).
APS 1010 Astronomy Lab
7
General Information
THE LABORATORY SECTIONS
Your laboratory session will meet for one hour and fifty minutes in the daytime once each week in
the Sommers-Bausch Observatory (SBO) Classroom S175. Each lab section will be run by a lab
instructor who will also grade your lab exercises and assign you a score for the work you hand in.
Your lab instructor will give you organizational details and information about grading at the first
lab session.
The lab exercises do not exactly follow the lectures or the textbook. Rather, we concentrate on
how we know what we know, and thus spend more time making and interpreting observations.
Modern astronomers, in practice, spend almost no time at the eyepiece of a telescope. They work
with photographs, with satellite data, or with computer images. In our laboratory we will explore
both traditional and more modern techniques.
You are expected to attend all lab sessions. The lab exercises can only be done using the
equipment and facilities in the SBO classroom; thus, if you do not attend the daytime lab sessions,
you cannot complete those experiments and cannot get credit. The observational exercises can only
be done at night using the Observatory telescopes; if you do not attend the nighttime sessions, you
cannot complete these either.
NIGHTTIME OBSERVING
You are expected to attend nighttime observing labs. These are held approximately
every third week at the Sommers-Bausch Observatory. Your lab instructor will tell you the dates
and times. Write the dates and times of the nighttime sessions on your calendar so you do not miss
them. If you have a job that conflicts with the nighttime sessions, it is your responsibility to make
arrangements with your instructor to attend at different times. Nighttime sessions are not
necessarily cancelled if it is cloudy; an indoor exercise may be done rather than observing - check
with your instructor. The telescopes are not in a heated area, so dress warmly for the night
observing sessions!
THE LABORATORY WRITE-UPS
You are expected to turn in lab assignments of collegiate quality: neat and easy to read, wellorganized, and demonstrating a mastery of the English language (e.g. grammar, spelling) as well
as mastery of the subject matter.
Your presentation of neat, well-organised, readable work will help you prepare for other courses
and jobs where appearance certainly counts (it also makes your work much easier to grade!). By
requiring you to explain your work clearly, we can ensure that you fully understand it. If you find
writing clear explanations to be difficult, it may mean that you aren't understanding the concepts
well.
Some of the labs are designed to be written up as full reports. A few are of the "fill in the blank"
type. In either case, the following guidelines should be followed.
APS 1010 Astronomy Lab
8
General Information
1.
Write the title and date at the top of the first page of the exercise.
2.
Write an introductory sentence or two describing what you are doing.
"Today we used the heliostat to observe and draw sunspots."
3.
Number the parts of your lab the same as the numbers in the lab manual: I.1, I.2, II.1,
II.2, etc.; this is particularly useful to help the instructor find your responses to the
questions.
4.
Put titles on your figures and label the columns in your tables.
5.
Put units after your numerical answers. One hundred millimeters is a bit different from
one hundred kilometers!
6.
Use complete sentences to answer questions. For example: "The largest planet is Jupiter"
is much better than just "Jupiter".
7.
Be sure to write your results in your own words and not the words of your lab partner.
8.
It is better to neatly cross though long mistakes than to erase them. Use a ruler and a
pencil to make neat lines through mistakes.
For instance:
ACADEMIC DISHONESTY
We expect students in our classes to hold to a high ethical standard. In general we expect you, as
college students, to be able to differentiate on your own between what is honest and dishonest.
Nevertheless, we point out the following guideline regarding laboratory assignments:
All work turned in must be your own. You should understand all work that you write on your
paper. While we encourage you to work in groups if it is helpful, you must not copy the work of
someone else. We have no objection to your consulting friends for help in understanding
problems. If you copy answers without understanding, however, it will be considered academic
dishonesty.
HELPING YOU HELP YOURSELF
Astronomy uses the laws of nature that govern the universe: i.e., physics. We will cover the
physics that is needed as we go along. To move beyond just the "ooooh and aaaah" of stargazing,
we also need to use mathematics. We are also aware that most of you may not be particularly fond
of math. You are expected, however, to be able to apply certain aspects of high school algebra and
trigonometry. This manual includes a summary of the mathematics you will need;
we recommend that you review it as necessary.
The "rule of thumb" for the amount of time you should be spending on any college class is 2-3
hours per week outside lectures and labs for each unit of credit. Thus, for a 4-unit class, you
should expect to spend an additional 8 to 12 hours per week studying. In general, if you are
spending less time than this on a class, then it is either too easy for you, or you are not learning as
much as you could. If you are spending more time than this, you may be studying inefficiently.
APS 1010 Astronomy Lab
9
General Information
We recommend the following strategies for efficient use of your lab time:
•
Review the accompanying material on units and conversions, scientific notation, math,
and calculator usage. Discuss with your lab instructor any problems you have with these
concepts before you need to use them. Use these sections as a reference in case you are
having difficulties.
•
Read the appropriate lab manual exercise before you come to lab.
•
Read any background material in the textbook before you come to lab.
•
Complete the whole of the lab exercise before the end of the lab period; avoid the
temptation to "wrap it up later".
If you have difficulty with a specific lab exercise, get help before the assignment is due! We will
attempt to give you as much help as needed to overcome your difficulties, provided you make the
effort to seek assistance. Do not procrastinate on assignments until the night before they are due it will be too late to get help!
If you find that you are having difficulties of any kind, talk to your lab instructor, your lecture
professor, or your lecture teaching assistant. We can help you only if we are aware that there is a
problem.
SOMMERS-BAUSCH OBSERVATORY
Sommers-Bausch Observatory (SBO), on the University of Colorado campus, is operated by the
Department of Astrophysical and Planetary Sciences to provide hands-on observational experience
for CU undergraduate students, and research opportunities for University of Colorado astronomy
graduate students and faculty. Telescopes include 16, 18, and 24-inch Cassegrain reflectors and a
10.5-inch aperture heliostat.
In its teaching role, the Observatory is used by approximately 1500 undergraduate students each
year to view celestial objects that would otherwise only be seen on the pages of a textbook or
discussed in classroom lectures. The 16- and 18-inch telescopes on the observing deck are both
under computer control, with objects selected from a library that includes double stars, star
clusters, nebulae, and galaxies.
In addition to the standard laboratory room, the Observatory recently added a computer lab
consisting of one instructor and twelve student computers, each capable of running either
Macintosh or PC software. We’re currently developing new computer-based lab exercises, and
have added planetarium-style and image-processing software to further enhance the student’s
hands-on astronomy experience.
The 18-inch telescope has an additional computer interface for planetarium-style "click-and-go"
pointing, plus a charge-coupled device (CCD) electronic camera so that students can image celestial
objects through an 8-inch piggyback telescope.
The 10.5-inch aperture heliostat is equipped for viewing sunspots, measuring the solar rotation,
solar photography, and for studies of the solar spectrum. A unique optical system called
SCRIBES permits simultaneous observations of the photosphere (using white light) and the solar
chromosphere (using red light from hydrogen and ultraviolet light from calcium in the sun).
APS 1010 Astronomy Lab
10
General Information
The 24-inch telescope is primarily used for graduate student training and for research projects not
feasible with larger telescopes because of time constraints or scheduling limitations. A Texas
Instruments 3-phase, 800x800-pixel CCD camera is used for both spectroscopic and direct
imaging applications. CU is the only facility in the nation with a telescope of under 90-inches
aperture to be awarded this advanced technology detector by the National Science Foundation. The
ultra-high efficiency CCD gives the 24-inch telescope at SBO a light-detecting capacity comparable
to the Palomar 200-inch telescope with conventional photographic plates. Images are analyzed and
processed using IDL and the National Optical Astronomical Observatory's IRAF software running
on a Sun Sparcstation 10 microcomputer.
An easy-to-operate, large-format SBIG ST-8 CCD camera with focal reducer assembly is also now
available for graduate and advanced undergraduate use on the 24-inch telescope. In addition, this
fall the 24-inch is being upgraded to include computerized control and automatic pointing
capabilities. These and other recent improvements and additions have all been made possible by
the funding provided by the student laboratory fees.
Free Open Houses for public viewing through the 16-and 18-inch telescopes are held every Friday
evening that school is in session. Students are welcome to attend. Call 492-5002 for starting
times and reservations. Call 492-6732 for general astronomical information.
For additional information about the Observatory, including schedules, information on how to
contact your lab instructor, and examples of images taken by other students in the introductory
astronomy classes, see our website located at
http://lyra.colorado.edu/sbo/
FISKE PLANETARIUM
The Fiske Planetarium and Science Center is used as a teaching facility for classes in astronomy,
planetary science and other courses that can take advantage of this unique audiovisual environment.
The star theater seats 210 under a 62-foot dome that serves as a projection screen, making it the
largest planetarium between Chicago and California. The Zeiss Mark VI star projector is one of
only five in the United States. Two hundred additional slide and special effect projectors are
controlled by a SPICE (Specialized Projector Interface Electronics) automation system.
Astronomy programs designed to entertain and to inform are presented to the public on Fridays and
Saturdays and to schoolchildren on weekdays. Laser-light shows rock the theater late Friday
nights as well. Following the Friday evening starshow presentations, visitors are invited next door
to view the stars at Sommers-Bausch Observatory, weather permitting.
The Planetarium provides students with employment opportunities to work on show production,
presentation, and in the daily operation of the facility.
Fiske is located west of the Events Center on Regent Drive on the Boulder campus of the
University of Colorado. For recorded program information call 492-5001 and to reach our
business office call 492-5002.
Alternatively, you can check out the upcoming schedules and events on the Fiske website at
http://www.colorado.edu/fiske/
APS 1010 Astronomy Lab
11
Units & Conversions
UNITS AND CONVERSIONS
You are probably familiar with the fundamental units of length, mass and time in the American
system: the yard, the pound, and the second. The other common units of the American system are
often strange multiples of these fundamental units such as the ton (2000 lbs), the mile (1760 yds),
the inch (1/36 yd) and the ounce (1/16 lb). Most of these units arose from accidental conventions,
and so have few logical relationships.
Most of the world uses a much more rational system known as the metric system (the SI,
Systeme International d'Unites, internationally agreed upon system of units) with the following
fundamental units:
•
The meter for length. Abbreviated "m".
•
The kilogram for mass.
standard.)
•
The second for time. Abbreviated "s".
Abbreviated "kg".
(Note: kilogram, not gram, is the
Since the primary units are meters, kilograms and seconds, this is sometimes called the ' mks
system'. Some people also use another metric system based on centimeters, grams and seconds,
called the 'cgs system' .
All of the unit relationships in the metric system are based on multiples of 10, so it is very easy to
multiply and divide. The SI system uses prefixes to make multiples of the units. All of the
prefixes represent powers of 10. The table below gives prefixes used in the metric system, along
with their abbreviations and values.
Metric Prefixes
Prefix
deci
centi
milli
micro
nano
pico
femto
atto
Abbreviation
d
c
m
m
n
p
f
a
Value
10-1
10-2
10-3
10-6
10-9
10-12
10-15
10-18
Prefix
decka
hecto
kilo
mega
giga
tera
Abbreviation
da
h
k
M
G
T
Value
101
102
103
106
109
1012
The United States, unfortunately, is one the few countries in the world which has not yet made a
complete conversion to the metric system. (Even Great Britain has adopted the SI system; so what
we used to call "English" units are no more - they are strictly "American"!) As a result, you are
forced to learn conversions between American and SI units, since all science and international
commerce is transacted in SI units. Fortunately, converting units is not difficult. Although you
can find tables listing seemingly endless conversions, you can do most of the lab exercises (as well
as most conversions you will ever need in science, business, etc.) by using just the four
conversions between American and SI units listed below (along with your own recollection of the
relationships between various American units).
APS 1010 Astronomy Lab
12
Units & Conversions
Units Conversion Table
American to SI
1 inch
= 2.54 cm
1 mile
= 1.609 km
1 lb
= 0.4536 kg
1 gal
= 3.785 liters
SI to American
1m
= 39.37 inches
1 km
= 0.6214 mile
1 kg
= 2.205 pound
1 liter
= 0.2642 gal
Strictly speaking, the conversion between kilograms and pounds is valid only on the Earth since
kilograms measure mass while pounds measure weight. However, since most of you will be
remaining on the Earth for the foreseeable future, we will not yet worry about such details. (If
you're interested, the unit of weight in the SI system is the newton, and the unit of mass in the
American system is the slug.)
Using the "Well-Chosen 1"
Many people have trouble converting between units because, even with the conversion factor at
hand, they aren't sure whether they should multiply or divide by that number. The problem
becomes even more confusing if there are multiple units to be converted, or if you need to use
intermediate conversions to bridge between two sets of units. We offer a simple and foolproof
method for handling the problem, which will always work if you don't take shortcuts!
We all know that any number multiplied by 1 equals itself, and also that the reciprocal of 1 equals
1. We can exploit these rather trivial properties by choosing our 1's carefully so that they will
perform a unit conversion for us.
Suppose we wish to know how many kilograms a 170 pound person weighs. We know that 1 kg
= 2.205 pounds , and can express this fact in the form of 1's:
1 kg
1 = 2.205 pounds
or its reciprocal
1=
2.205 pounds
1 kg
.
Note that the 1's are dimensionless; the quantity (number with units) in the numerator is exactly
equal to the quantity (number with units) in the denominator. If we took a shortcut and omitted the
units, we would be writing nonsense: neither 1 divided by 2.205, nor 2.205 divided by 1, equals
"1"! Now we can multiply any other quantity by these 1's, and the quantity will remain unchanged
(even though it will look considerably different).
In particular, we want to multiply the quantity "170 pounds" by 1 so that it will still be equivalent
to 170 pounds, but will be expressed in kg units. But which "1" do we choose? Very simply, if
the unit we want to "get rid of" is in the numerator, we choose the "1" that has that same unit
appearing in the denominator (and vice versa) so that the undesired units will cancel. Hence we
have
170 lbs x 1
=
1 kg
170 lbs x 2.205 lbs
=
170 x 1 lbs x kg
2.205 x lbs
=
77.1 kg .
Note that you do not omit the units, but multiply and divide them just like ordinary numbers. If
you have selected a "well-chosen" 1 for your conversion your units will nicely cancel, which will
assure you that the numbers themselves will also have been multiplied or divided properly. That's
what makes this method foolproof: if you used a "poorly-chosen" 1, the expression itself will
immediately let you know about it:
APS 1010 Astronomy Lab
170 lbs x 1
=
13
170 lbs x
2.205 lbs
1 kg
=
Units & Conversions
170 x 2.205 lbs x lbs
x
1
kg
=
lbs2
375 x kg
!
Strictly speaking, this is not really incorrect: 375 lbs2/kg is the same as 170 lbs, but it's not a very
useful way of expressing it, and it's certainly not what you were trying to do!
Example: As a passenger on the Space Shuttle, you note that the inertial navigation system shows
your orbital velocity at 7,000 meters per second. You remember from your astronomy course that
a speed of 17,500 miles per hour is the minimum needed to maintain an orbit around the Earth.
Should you be worried?
m
7000 s
=
7000 m
1 km
1 mile
60 s
60 min
1 s x 1000 m x 1.609 km x 1 min x 1 hr
7000 x 1 x 1 x 6 0 x 60
= 1 x 1000 x 1.609 x 1 x 1
x
m x km x mile x s x min
s x m x km x min x hr
miles
= 15,662 hour .
Because of your careful analysis using "well-chosen 1's", you can conclude that you will probably
not survive long enough to have to do any more unit conversions.
Temperature Scales
Scales of temperature measurement are tagged by the freezing point and boiling point of water. In
the U.S., the Fahrenheit (F) system is the one commonly used; water freezes at 32 °F and boils at
212 °F (180° hotter). In Europe, the Celsius system is usually used: water freezes at 0 °C and
boils at 100 °C. In scientific work, it is common to use the Kelvin temperature scale. The Kelvin
degree is exactly the same "size" as the Celsius degree, but is based on the idea of absolute zero,
the temperature at which all random molecular motions cease. 0 K is absolute zero, water freezes
at 273 K and boils at 373 K. Note that the degree mark is not used with Kelvin temperatures, and
the word "degree" is often not even mentioned: we say that "water boils at 373 kelvins".
To convert between these three systems, recognize that 0 K = -273 °C = -459 °F and that the
Celsius and Kelvin degree is larger than the Fahrenheit degree by a factor of 180/100 = 9/5. The
relationships between the systems are:
K = °C + 273
°C = 5/9 (°F - 32)
°F = 9/5 K - 459
.
Energy and Power: Joules and Watts
The SI unit of energy is called the joule. Although you may not have heard of joules before, they
are simply related to other units of energy with which you probably are familiar. For example, 1
food Calorie (which actually is 1000 “normal'' calories) is 4,186 joules. House furnaces are rated
in btus (British thermal units), indicating how much heat energy they can produce: 1 btu = 1,054
joules. Thus, a single potato chip (having an energy content of about 9 Calories) could also be
said to possess 37,674 joules or 35.7 btu's of energy.
The SI unit of power is called the watt. Power is defined to be the rate at which energy is used or
produced, and is measured as energy per unit time. The relationship between joules and watts is:
APS 1010 Astronomy Lab
14
joule
1 watt = 1 second
Units & Conversions
.
For example, a 100-watt light bulb uses 100 joules of energy (about 1/42 of a Calorie or 1/10 of a
btu) each second it is turned on. Weight-watchers might be more motivated to stick to their diet if
they realized that one potato chip contains enough energy to operate a 100-watt light bulb for over
6 minutes!
Another common unit of power is the horsepower; one hp equals 746 watts, which means that
energy is consumed or produced at the rate of 746 joules per second. (In case you're curious, you
can calculate (using unit conversions) that if your car has fifty "horses" under the hood, they need
to be fed 37,300 joules, or the equivalent energy of one potato chip, every second in order to pull
you down the road.)
To give you a better sense of the joule as a unit of energy (and of the convenience of scientific
notation, our next topic), here are some comparative energy outputs:
Energy Source
Big Bang
Radio galaxy
Supernova
Sun’s radiation for 1 year
Volcanic explosion
H-bomb
Thunderstorm
Lightening flash
Baseball pitch
Hitting keyboard key
Hop of a flea
Energy
(joules)
1068
1055
1043
1034
1019
1017
1015
1010
102
10-2
10-7
APS 1010 Astronomy Lab
15
Scientific Notation
SCIENTIFIC NOTATION
Scientific Notation: What is it?
Astronomers deal with quantities ranging from the truly microcosmic to the macrocosmic. It is
very inconvenient to always have to write out the age of the universe as 15,000,000,000 years or
the distance to the Sun as 149,600,000,000 meters. To save effort, powers-of-ten notation is
used. For example, 10 = 101; the exponent tells you how many times to multiply by 10. As
another example, 10-2 = 1/100; in this case the exponent is negative, so it tells you how many
times to divide by 10. The only trick is to remember that 100 = 1. Using powers-of-ten notation,
the age of the universe is 1.5 x 1010 years and the distance to the Sun is 1.496 x 1011 meters.
•
The general form of a number in scientific notation is a x 10 n, where a must be between
1 and 10, and n must be an integer. (Thus, for example, these are not in scientific
notation: 34 x 105; 4.8 x 100.5 .)
•
If the number is between 1 and 10, so that it would be multiplied by 100 (=1), then it is
not necessary to write the power of 10. For example, the number 4.56 already is in
scientific notation (it is not necessary to write it as 4.56 x 10 0, but you may write it this
way if you wish).
•
If the number is a power of 10, then it is not necessary to write that it is multiplied by 1.
For example, the number 100 can be written in scientific notation either as 102 or as 1 x
102. (Note, however, that the latter form should be used when entering numbers on a
calculator.)
The use of scientific notation has several advantages, even for use outside of the sciences:
•
Scientific notation makes the expression of very large or very small numbers much
simpler. For example, it is easier to express the U.S. federal debt as $3 x 10 12 rather
than as $3,000,000,000,000.
•
Because it is so easy to multiply powers of ten in your head (by adding the exponents),
scientific notation makes it easy to do "in your head" estimates of answers.
•
Use of scientific notation makes it easier to keep track of significant figures; that is, does
your answer really need all of those digits that pop up on your calculator?
Converting from "Normal" to Scientific Notation:
Place the decimal point after the first non-zero digit, and count the number of places the decimal
point has moved. If the decimal place has moved to the left then multiply by a positive power of
10; to the right will result in a negative power of 10.
Example: To write 3040 in scientific notation we must move the decimal point 3 places to the left,
so it becomes 3.04 x 103.
Example: To write 0.00012 in scientific notation we must move the decimal point 4 places to the
right: 1.2 x 10-4.
APS 1010 Astronomy Lab
16
Scientific Notation
Converting from Scientific to "Normal'" Notation:
If the power of 10 is positive, then move the decimal point to the right; if it is negative, then move
it to the left.
Example: Convert 4.01 x 102. We move the decimal point two places to the right making 401.
Example: Convert 5.7 x 10-3. We move the decimal point three places to the left making 0.0057.
Addition and Subtraction with Scientific Notation:
When adding or subtracting numbers in scientific notation, their powers of 10 must be equal. If
the powers are not equal, then you must first convert the numbers so that they all have the same
power of 10.
Example: (6.7 x 109) + (4.2 x 109) = (6.7 + 4.2) x 10 9 = 10.9 x 10 9 = 1.09 x 10 10. (Note
that the last step is necessary in order to put the answer in scientific notation.)
Example: (4 x 108) - (3 x 106) = (4 x 108) - (0.03 x 108) = (4 - 0.03) x 108 = 3.97 x 108.
Multiplication and Division with Scientific Notation:
It is very easy to multiply or divide just by rearranging so that the powers of 10 are multiplied
together.
Example: (6 x 10 2) x (4 x 10 -5) = (6 x 4) x (102 x 10 -5) = 24 x 10 2-5 = 24 x 10 -3 = 2.4 x 10 -2.
(Note that the last step is necessary in order to put the answer in scientific notation.)
Example: (9 x 108) ÷ (3 x 106) =
9 x 10 8
= (9/3) x (108/106) = 3 x 108-6 = 3 x 102.
3 x 10 6
Approximation with Scientific Notation:
Because working with powers of 10 is so simple, use of scientific notation makes it easy to
estimate approximate answers. This is especially important when using a calculator since, by
doing mental calculations, you can verify whether your answers are reasonable. To make
approximations, simply round the numbers in scientific notation to the nearest integer, then do the
operations in your head.
Example: Estimate 5,795 x 326. In scientific notation the problem becomes (5.795 x 10 3) x (3.26
x 102). Rounding each to the nearest integer makes the approximation (6 x 103) x (3 x 10 2), which
is 18 x 105, or 1.8 x 106 (the exact answer is 1.88917 x 106).
Example: Estimate (5 x 10 15) + (2.1 x 10 9). Rounding to the nearest integer this becomes (5 x
1015) + (2 x 109). We see immediately that the second number is nearly 1015/109, or one million,
times smaller than the first. Thus, it can be ignored in the addition problem and our approximate
answer is 5 x 1015. (The exact answer is 5.0000021 x 1015).
APS 1010 Astronomy Lab
17
Scientific Notation
Significant Figures:
Numbers should be given only to the accuracy that they are known with certainty, or to the extent
that they are important to the topic at hand. For example, your doctor may say that you weigh 130
pounds, when in fact at that instant you might weigh 130.16479 pounds. The discrepancy is
unimportant and will change anyway as soon as a blood sample has been drawn.
If numbers are given to the greatest accuracy that they are known, then the result of a multiplication
or division with those numbers can't be determined any better than to the number of digits in the
least accurate number.
Example: Find the circumference of a circle measured to have a radius of 5.23 cm using the
formula: C = (2 x π x R). Since the value of pi stored in your calculator is probably 3.141592654,
the numerical solution will be
(2 x 3.141592654 x 5.23 cm) = 32.86105916 = 3.286105916 x 101 cm.
If you simply write down all 10 digits as your answer, you are implying that you know, with
absolute certainty, the circle’s circumference to an accuracy of one part in 10 billion! That would
mean that your measurement of the radius was in error by no more than 0.000000001 cm; that is,
its true value was at least 5.229999999 cm, but no more than 5.230000001 cm (otherwise, your
calculator would have shown a different number for the circumference).
In reality, since your measurement of the radius was known to only three decimal places, the
circle’s circumference is also known to only (at best) three decimal places as well: you round the
fourth digit and give the result as 32.9 cm or 3.29 x 10 1 cm. It may not look as impressive, but
it’s an honest representation of what you know about the figure.
Since the value of "2" was used in the formula, you may wonder why we're allowed to give the
answer to three decimal places rather than just one: 3 x 101 cm. The reason is because the number
"2" is exact - it expresses the fact that a diameter is exactly twice the radius of a circle - no
uncertainty about it at all. Without any exaggeration, we could have represented the number as
2.0000000000000000000, but merely used the shorthand "2" for simplicity - so we really didn't
violate the rule of using the least accurately-known number.
APS 1010 Astronomy Lab
18
Math Review
MATH REVIEW
Dimensions of Circles and Spheres
•
The circumference of a circle of radius R is 2π R.
•
The area of a circle of radius R equals π R2.
•
The surface area of a sphere of radius R is given by 4π R2.
•
The volume of a sphere of radius R is 4/3 π R3.
Measuring Angles - Degrees and Radians
•
There are 360° in a full circle.
•
There are 60 minutes of arc in one degree. (The shorthand for arcminute is the single
prime (‘): we can write 3 arcminutes as 3’.) Therefore there are 360 x 60 = 21,600
arcminutes in a full circle.
•
There are 60 seconds of arc in one arcminute. (The shorthand for arcsecond is the double
prime (“): we can write 3 arcseconds as 3”.) Therefore there are 21,600 x 60 =
1,296,000 arcseconds in a full circle.
We sometimes express angles in units of radians instead of degrees. If we were to take the radius
(length R) of a circle and bend it so that it conformed to a portion of the circumference of the same
circle, the angle subtended by that radius is defined to be an angle of one radian.
Since the circumference of a circle has a total length of 2π R, we can fit exactly 2π radii (6 full
lengths plus a little over 1/4 of an additional) along the circumference; thus, a full 360° circle is
equal to an angle of 2 π radians. In other words, an angle in radians equals the subtended
arclength of a circle divided by the radius of that circle. If we imagine a unit circle (where the
radius = 1 unit in length), then an angle in radians numerically equals the actual curved distance
along the portion of its circumference “cut” by the angle.
The conversion between radians and degrees is
360
1 radian = 2 π degrees = 57.3°
2π
1° = 360 radians = 0.017453 radians
Trigonometric Functions
In this course we will make occasional use of the basic trigonometric functions: sine, cosine, and
tangent. Here is a quick review of the basic concepts:
APS 1010 Astronomy Lab
19
Math Review
In any right triangle (one angle is 90°), the longest side is called the hypotenuse and is the side that
is opposite the right angle. The trigonometric functions relate the lengths of the sides of the
triangle to the other (i.e. not 90°) enclosed angles. In the right triangle figure below, the side
adjacent to the angle α is labeled “adj”, the side opposite the angle α is labeled “opp”. The
hypotenuse is labeled “hyp”.
•
The Pythagorean theorem relates the lengths of the sides to each other:
(opp)2 + (adj)2 = (hyp)2
•
.
The trig functions are just ratios of the lengths of the different sides:
(opp)
sin α = (hyp)
(adj)
cos α = (hyp)
(opp)
tan α = (adj)
.
Angular Size, Physical Size and Distance
The angular size of an object (the angle it subtends, or appears to occupy, from our vantage point)
depends on both its true physical size and its distance from us. For example, if you stand with
your nose up against a building, it will occupy your entire field of view; as you back away from
the building it will occupy a smaller and smaller angular size, even though the building’s physical
size is unchanged. Because of the relations between the three quantities (angular size, physical
size, and distance), we need know only two in order to calculate the third.
Suppose a tall building has an angular size of 1° (that is, from our location its height appears to
span one degree of angle), and we know from a map that the building is located 10 km away.
How can we determine the actual physical size (height) of the building?
We imagine we are standing with our eye at the apex of a triangle, from which point the building
subtends an angle α = 1° (greatly exaggerated in the drawing). The building itself forms the
opposite side of the triangle, which has an unknown height that we will call h. The distance d to
the building is 10 km, corresponding to the adjacent side of the triangle.
APS 1010 Astronomy Lab
20
Math Review
Since we want to know the opposite side, and already know the adjacent side of the triangle, we
need only concern ourselves with the trigonometric tangent relationship:
(opp)
tan α = (adj)
or
h
tan 1° = d
which we can reorganize to give
h = d x tan 1°
or
h = 10 km x 0.017455 = 0.17455 km = 174.55 meters.
Small Angle Approximation
We used the adjacent side of the triangle for the distance instead of the hypotenuse because it
represented the smallest separation between us and the building. It should be apparent, however,
that since we are 10 km away, the distance to the top of the building will only be very slightly
farther than the distance to the base of the building; a little trigonometry shows that the hypotenuse
in this case equals 10.0015 km, or less than 2 meters longer than the adjacent side of the triangle.
In fact, the hypotenuse and adjacent sides of a triangle are always of similar lengths whenever we
are dealing with angles that are “not very large”. Thus, we can substitute one for the other
whenever the angle between the two sides is small.
Now imagine that the apex of a small angle α is located at the center of a circle whose radius is
equal to the hypotenuse of the triangle, as shown above. The arclength of the circumference
subtended by that small angle is only very slightly longer than the length of the corresponding
straight (“opposite”) side. In general, then, the opposite side of a triangle and its corresponding
arclength are always of nearly equal lengths whenever we are dealing with angles that are “not very
large”. We can substitute one for the other whenever the subtending angle is small.
Now let’s go back to our equation for the physical height of our building:
h
= d x tan α
(opp)
= d x (adj)
.
Since the angle α is small, the opposite side is approximately equal to the “arclength” subtended by
the building. Likewise, the adjacent side is approximately equal to the hypotenuse, which is in
turn equivalent to the radius of the inscribed circle. Making these substitutions, the above (exact)
equation can be replaced by the following (approximate) equation:
h
= d x
(arclength)
.
(radius)
But remember that the ratio (arclength)/(radius) is the definition of an angle expressed in radian
units rather than degrees - so we have the very useful small angle approximation:
For small angles, the physical size h of an object can be determined directly from its
distance d and angular size in radians by
APS 1010 Astronomy Lab
21
h
Math Review
= d x (angular size in radians)
.
Or, for small angles the physical size h of an object can be determined from its distance d
and its angular size α in degrees by
2π
= d x 360° x α
h
.
Using the small angle approximation, the height of our building 10 km away is calculated to be
174.53 meters high, an error of only about 2 cm (less than 1 inch)! ... and best of all, the
calculation didn’t require trigonometry, just multiplication and division!
When can the approximation be used? Surprisingly, the angles don’t really have to be very small.
For an angle of 1°, the small angle approximation leads to an error of only 0.01%. Even for an
angle as great as 10°, the error in your answer will only be about 1%, and you avoided using trig!
Powers and Roots
We can express any power or root of a number in exponential notation: We say that bn is the “nth
power of b”, or “b to the n (power)”. The number represented here as b is called the base, and n is
called the power or exponent.
The basic definition of a number written in exponential notation states, as you know, that the base
should be multiplied by itself the number of times indicated by the exponent. That is, bn means b
multiplied by itself n times. For example: 52 = 5 x 5; b4 = b x b x b x b.
From the basic definition, certain properties automatically follow:
•
Zero Exponent: Any nonzero number raised to the zero power is 1. That is, b 0 = 1.
Examples: 20 = 100 = -30 = (1/2)0 = 1.
•
Negative Exponent: A negative exponent indicates that a reciprocal is to be taken.
That is,
1
b n
b -n =
1
b
-n
a
=bn
= a x b n.
b -n
Examples: 4-2 = 1/42 = 1/16; 10-3 = 1/103 = 1/1000; 3/2-2 = 3 x 22 = 12.
•
Fractional Exponent: A fractional exponent indicates that a root is to be taken.
n
b1/n =
Examples:
81/3
=
b;
bm/n =
n
m
n 
bm =  b .
2
3
3 
=  8 = 22 = 4
8 =2
82/3
24 = 16 = 4
x1/4 = ( x1/2)
1/2
24/2 =
=
x
APS 1010 Astronomy Lab
22
Using Your Calculator
USING YOUR CALCULATOR
Calculator Rule #1:
Don't switch off your brain when you switch your calculator on! Your calculator is intended to
eliminate tedious arithmetic chores, not to do your thinking for you. It's very easy to accidentally
press the wrong buttons. Before you write down any answer, ask yourself if the answer seems
reasonable. Intelligent use of your calculator will make your success in this course far more likely!
Example: Suppose you punch in the following: (3.28 x 104) x (2.4 x 10-7). Since 3 x 2 = 6, and
(104) (10-7) = 10-3, you know that your answer should be near 6 x 10 -3. If your calculator gives
you the answer 2.3 x 1015, then you would know you have done something wrong and should try
again.
Example: Suppose you punch in: 47 . Since 47 is between 36 and 49, and you know that 36
= 6 and 49 = 7, you know that your answer should be between 6 and 7. If your calculator
gives you the answer 8.3, then you would know you have made an entry error and should try
again.
Variations among Calculators:
Most scientific calculators have keys to perform all of the operations described below. However,
the labeling of the keys on your calculator, and the sequence in which they must be pushed, may
differ slightly from the examples given here. If you cannot get your calculator to do all the
operations in these examples, ask your instructor for assistance.
Scientific Notation:
Your calculator should have a key for entering scientific notation. It probably is labeled either EXP
or EE (for "exponent" or "enter exponent") .
This key is used to enter the power of 10
multiplying the number you have entered previously. For example, to enter the number 3.5 x 10 6
you would use the key sequence:
3.5
EE 6
Note that the EE key can be thought of as meaning "times 10 to the power that I enter next".
Thus, you must be especially careful in entering powers of 10, since there must be a preceding
number for you to multiply "times 10 to the power of ..." with. For example, to enter the number
108 you would press:
1 EE 8
.
A common mistake is to enter: 10 EE 8 . Note that this means 10 x 108 or 109.
APS 1010 Astronomy Lab
23
Using Your Calculator
Most calculators are a bit tricky in the way you must enter a negative exponent. If you want to
enter 5 x 10 -3, for example, you do not press 5 EE -3 = ; the calculator will probably
perform the operation (5 x 100) - 3 = 2. Instead, you must use the "change sign" key, +/- or
CHS , which toggles the sign between positive and negative values (check the display to see if
the exponent is correct):
5 EE +/- 3 =
or
5 EE 3 +/- =
Switching Between "Normal" and Scientific Notation:
Most calculators have a single key which switches between scientific and "normal" notation. The
labeling of this key varies significantly among calculators, but often includes the letters "F" and
"E" (which stand for "floating point notation" and "exponential notation", respectively). A
common label for this key is F-E . On some calculators a sequence of keys must be used instead
of a single key. On Sharp brand calculators, for example, the key is labeled FSE ; you may have
to press this key more than once. On the Casio fx-250, notation is changed by using the MODE
key:
MODE 82
MODE 84
gives you scientific notation with 2 significant figures.
gives you scientific notation with 4 significant figures.
To convert to normal notation, press MODE 9 .
Angles and Trigonometry
Your calculator should have keys for all of the standard trigonometric functions including sin, cos,
tan, and their inverse operations: arcsin, arccos, and arctan.
Using the keys for the trigonometric functions is straightforward, except for one complication: the
trig functions are derived from work with angles, and there is more than one system for measuring
angles. Besides degrees, angles can also be measured in radians (common in math and science)
and in grads (common in engineering). Unless you tell your calculator in advance which type of
angle measurement you are using, your answer may come out incorrect.
Most calculators have a key labeled DRG which switches the calculator between degrees,
radians, and grads; usually, the numerical display has a label showing which mode is in effect.
Before using any trigonometric function, always put your calculator into the correct mode! In this
lab course, if you see the ° symbol, you will know to work with degrees, otherwise you should
assume we are working with radians.
For example, to calculate the sine of 45°, first make sure your calculator is in degree mode, and
then enter: 45 sin = . You will get the answer 0.707 (to 3 decimal places). Now switch your
calculator to radian mode, and find the sine of 45 radians with the same sequence of keys. You
should find that sin 45 = 0.851, demonstrating the importance of setting the mode before starting
APS 1010 Astronomy Lab
24
Using Your Calculator
a problem! (Note: it is not necessary to distinguish between these types of angle measure if you
are not using any trigonometric functions.)
If you know that the tangent of an unknown angle equals 0.235, for example, and you want to
know what the angle itself must be, you should press the inverse function key (usually labeled arc
or INV ) before selecting the trigonometric function:
.235
arc tan =
to get the answer 13.22° or 0.2308 radians (depending upon the angular mode the calculator was
in). Some calculators combine the inverse trigonometric function into a single key, such as tan-1
; you may have to press a "shift key" on your calculator to access it.
Powers and Roots:
Your calculator should have keys for raising numbers to powers or for taking roots. To raise a
number to a power, use the key which is usually labeled as y x or
x y . For example, to do
53 you would use the following sequence of keys:
5 y x 3 =
and get the answer 125.
Many people are reluctant to use the "to the power of" key because they don't remember whether
the base number or the exponent should be entered first. If you find yourself in this situation,
here’s a simple test you can do to figure it out: enter the sequence 2 y x 1 = , and note the
result. If the answer is "2", then the base number comes first, because the calculator performed the
operation 21 = 2. If the answer is "1", then the exponent comes first, since 12 = 1.
To take roots, you use the key usually labeled
x
y or
y
x or x1/y . For example, to take the
4
4th root of 16 ( 16 or 161/4) you would use the keys:
16
x
y 4 =
and get the answer 2. If your calculator doesn't have a "root" key, you can get the same answer
by using the "power" key sequence:
16 y x .25
since 160.25 = 161/4 =
4
16 .
=
APS 1010 Astronomy Lab
25
Using Your Calculator
Reciprocals, Squares, Square Roots:
Reciprocals, squares, and square roots, can all be done using the "power" and "root" keys
described above. For example, to get the reciprocal of 25 you could enter the calculator equivalent
of 25-1:
25 y x +/- 1 =
and get the answer 0.04 (which is 1/25). To square a number, you can use the "power" key and
an exponent of "2"; To get the square root, you can use the power key with an exponent of "0.5",
or the "root" key and a root of "2".
To make your life simpler, however, most calculators have special keys for taking squares (key
labeled x2 ), square roots (key labeled
), and reciprocals (key labeled 1/x or x-1 ).
Some calculators also have special keys for cubes x3 and cube roots
For example, to calculate 38 2 you would enter: 38
x2
=
calculate the decimal equivalent of 1/25, you would enter
0.04.
25
3
.
and get the answer 1,444. To
1/x
=
and get the answer
You can even use the square root key twice to find the fourth root of a number, since the square
root of a square root is the fourth root: for example, you could find that 3 is the fourth root of 81:
4
1/2
81
= (81)
1/4
= (
811/2
=
)
81
by keying in the sequence
81
=
Pi (π):
The key labeled π can be used to automatically enter the numerical value of π. Simply press the
π key to see that π = 3.141592654 (to 9 decimal places).
Logarithms
You probably won't be doing any logarithms in this course, but since scientific calculators come
equipped with this capability, we will mention them very briefly. To take the common logarithm
of a number (to find out what power that 10 must be raised to in order to equal that number), use
the log key; for example, 50 log will produce an answer of 1.699, showing that 101.699 = 50.
To perform the inverse operation (to find out what number you would get if you raised 10 to a
particular power), you use the INV log or 10x keys.
APS 1010 Astronomy Lab
26
Using Your Calculator
Finally, there is the mysterious ln or ln x key, which performs what is known as a "natural"
logarithm. It works like a "common" logarithm, except that the base is the number e =
2.718281828459 instead of 10. Believe it or not, the number e is as important to science and
mathematics as the number π. However, if you aren't overly fond of these subjects, you won't
have to worry about this key or its inverse ex either.
The Parentheses Keys:
All calculators have a priority system for performing operations. For example, multiplication has a
higher priority than addition, so that if you enter the key sequence:
5 + 3 x 2 =
the calculator will first multiply 3 x 2 = 6 and then add the 5 to get 11 (as opposed to first adding 5
+ 3 to get 8 and then multiplying by 2, which would give 16.) When performing calculations
involving multiple steps, you can get incorrect answers if you are not careful to understand your
calculator's priority system. You often can avoid problems, however, by making judicious use of
the keys ( and ) (if your calculator has them) to group your operations. For example, if you
wished to alter the priority of the above calculation, you could enter:
( 5 + 3 ) x 2 =
which will give you the answer 16.
Of course, if you are uncomfortable with long numerical calculations and want to make sure that
your calculator doesn't do something that you didn't intend for it to do, you can simply perform
the mathematical operations one at a time in the correct sequence, write down the intermediate
answers, and re-enter the needed numbers when appropriate. It may not be the most elegant use of
your calculator, but who cares, as long as you get what you wanted!
APS 1010 Astronomy Lab
27
Celestial Coordinates
CELESTIAL COORDINATES
GEOGRAPHIC COORDINATES
The Earth's geographic coodinate system is familiar to everyone - the north and south poles
are defined by the Earth's axis of rotation; equidistant between them is the equator. North-south
latitude is measured in degrees from the equator, ranging from -90° at the south pole, 0° at the
equator, to +90° at the north pole. East-west distances are also measured in degrees, but there is
no "naturally-defined" starting point - all longitudes are equivalent to all others. Humanity has
arbitrarily defined the prime meridian (0° longitude) to be that of the Royal Observatory at
Greenwich, England (alternately called the Greenwich meridian).
Each degree (°) of a 360° circle can be further subdivided into 60 equal minutes of arc ('), and each
arc-minute may be divided into 60 seconds of arc ("). The 24-inch telescope at Sommers-Bausch
Observatory is located at a latitude of 40°0'13" North of the equator and at a longitude 105°15'45"
West of the Greenwich meridian.
North Pole
+90° Latitude
Greenwich
0° Longitude
Boulder
+40° 0' 13" Latitude
+105° 15' 45" Longitude
Equator
0° Latitude
The Earth's
Latitude - Longitude
Coordinate System
Parallels
or
Small Circles
Meridians
ALT-AZIMUTH COORDINATES
The alt-azimuth (altitude - azimuth) coordinate system, also called the horizon system, is a
useful and convenient system for pointing out a celestial object.
One first specifies the azimuth angle, which is the compass heading towards the horizon point
lying directly below the object. Azimuth angles are measured eastwardly from North (0° azimuth)
to East (90°), South (180°), West (270°), and back to North again (360° = 0°). The four principle
directions are called the cardinal points.
APS 1010 Astronomy Lab
28
Celestial Coordinates
Next, the altitude is measured in degrees upward from the horizon to the object. The point
directly overhead at 90° altitude is called the zenith. The nadir is "down", or opposite the zenith.
We sometimes use zenith distance instead of altitude, which is 90° - altitude.
Every observer on Earth has his own separate alt-azimuth system; thus, the coordinates of the same
object will differ for two different observers. Furthermore, because the Earth rotates, the altitude
and azimuth of an object are constantly changing with time as seen from a given location. Hence,
this system can identify celestial objects at a given time and location, but is not useful for
specifying their permanent (more or less) direction in space.
In order to specify a direction by angular measure, you need to know just how “big” angles are.
Here’s a convenient “yardstick” to use that you carry with you at all times: the hand, held at arm's
length, is a convenient tool for estimating angles subtended at the eye:
3°
2°
4°
Index
Finger
8°
1°
18°
Hand
10°
Fist
EQUATORIAL COORDINATES
Standing outside on a clear night, it appears that the sky is a giant celestial sphere of indefinite
radius with us at its center, and upon which stars are affixed to its inner surface. It is extremely
useful for us to treat this imaginary sphere as an actual, tangible surface, and to attach a coordinate
system to it.
APS 1010 Astronomy Lab
29
Celestial Coordinates
The system used is based on an extension of the Earth's axis of rotation, hence the name
equatorial coordinate system. If we extend the Earth's axis outward into space, its
intersection with the celestial sphere defines the north and south celestial poles; equidistant
between them, and lying directly over the Earth's equator, is the celestial equator.
Measurement of "celestial latitude" is given the name declination (DEC), but is otherwise
identical to the measurement of latitude on the Earth: the declination at the celestial equator is 0° and
extends to ±90° at the celestial poles.
The east-west measure is called right ascension (RA) rather than "celestial longitude", and
differs from geographic longitude in two respects. First, the longitude lines, or hour circles,
remain fixed with respect to the sky and do not rotate with the Earth. Second, the right ascension
circle is divided into time units of 24 hours rather than in degrees; each hour of angle is equivalent
to 15° of arc. The following conversions are useful:
24 h =
360°
1m =
15'
1h
=
15°
4s
=
1'
4m =
1°
1s
=
15"
The Earth orbits the Sun in a plane called the ecliptic. From our vantage point, however, it
appears that the Sun circles us once a year in that same plane; hence, the ecliptic may be alternately
defined as "the apparent path of the Sun on the celestial sphere".
The Earth's equator is tilted 23.5° from the plane of its orbital motion, or in terms of the celestial
sphere, the ecliptic is inclined 23.5° from the celestial equator. The ecliptic crosses the equator at
two points; the first, called the vernal (spring) equinox, is crossed by the Sun moving from
south to north on about March 21st, and sets the moment when spring begins. The second
crossing is from north to south, and marks the autumnal equinox six months later. Halfway
between these two points, the ecliptic rises to its maximum declination of +23.5° (summer
solstice), or drops to a minimum declination of -23.5° (winter solstice).
As with longitude, there is no obvious starting point for right ascension, so astronomers have
assigned one: the point of the vernal equinox. Starting from the vernal equinox, right ascension
increases in an eastwardly direction until it returns to the vernal equinox again at 24 h = 0 h.
North Celestial
Pole
+90° DEC
Hour Circles
4h
16h
Celestial
Equator
0° DEC
18h
2h
20h
RA
22h
24h = 0h
Vernal
Equinox
0h RA, 0° DEC
Earth
23.5°
Ecliptic
Winter Solstice
18h RA, -23.5° DEC
Sun
(March 21)
APS 1010 Astronomy Lab
30
Celestial Coordinates
The Earth precesses, or wobbles on its axis, once every 26,000 years. Unfortunately, this means
that the Sun crosses the celestial equator at a slightly different point every year, so that our "fixed"
starting point changes slowly - about 40 arc-seconds per year. Although small, the shift is
cumulative, so that it is important when referring to the right ascension and declination of an object
to also specify the epoch, or year in which the coordinates are valid.
TIME AND HOUR ANGLE
The fundamental purpose of all timekeeping is, very simply, to enable us to keep track of certain
objects in the sky. Our foremost interest, of course, is with the location of the Sun, which is the
basis for the various types of solar time by which we schedule our lives.
Time is determined by the hour angle of the celestial object of interest, which is the angular
distance from the observer's meridian (north-south line passing overhead) to the object,
measured in time units east or west along the equatorial grid. The hour angle is negative if we
measure from the meridian eastward to the object, and positive if the object is west of the meridian.
For example, our local apparent solar time is is determined by the hour angle of the Sun,
which tells us how long it has been since the Sun was last on the meridian (positive hour angle), or
how long we must wait until noon occurs again (negative hour angle).
If solar time gives us the hour angle of the Sun, then sideral time (literally, "star time") must be
related to the hour angles of the stars: the general expression for sidereal time is
Sidereal Time = Right Ascension + Hour Angle
which holds true for any object or point on the celestial sphere. It’s important to realize that if the
hour angle is negative, we add this negative number, which is equivalent to subtracting the positive
number.
For example, the vernal equinox is defined to have a right ascension of 0 hours; thus the equation
becomes
Sidereal time = Hour angle of the vernal equinox
Another special case is that for an object on the meridian, for which the hour angle is zero by
definition. Hence the equation states that
Sidereal time = Right ascension crossing the meridian
Your current sidereal time, coupled with a knowledge of your latitude, uniquely defines the
appearance of the celestial sphere; furthermore, if you know any two of the variables in the
expression ST = RA + HA , you can determine the third.
The following illustration shows the appearance of the southern sky as seen from Boulder at a
particular instant in time. Note how the sky serves as a clock - except that the clockface (celestial
sphere) moves while the clock "hand" (meridian) stays fixed. The clockface numbering increases
towards the east, while the sky rotates towards the west; hence, sidereal time always increases, just
as we would expect. Since the left side of the ST equation increases with time, then so must the
right side; thus, if we follow an object at a given right ascension (such Saturn or Uranus), its hour
angle must constantly increase (or become less negative).
APS 1010 Astronomy Lab
31
Celestial Coordinates
Sidereal Time
= Right Ascension on Meridian
= 21 hrs 43 min
Vernal Equinox
23h 20m
0 h 0m
22h 40m
22h 0m
21h 20m
20h 40m
20h 0m
19h 20m
Ecliptic
Saturn
RA = 21h 59m
Sidereal Time =
Right Ascension +
Hour Angle
Hour Angle
- 0h 16m
Saturn:
21h 59m RA +
(- 0h 16m) HA
= 21h 43m ST
Hour Angle
+ 2h 21m
Uranus
RA = 19h 22m
Hour Angle of
Vernal Equinox
= - 2h 17m
= 21h 43m
= Sidereal Time
Uranus:
19h 22m RA +
2h 21m HA
= 21h 43m ST
1:50 am MDT 20 August 1993
SOLAR VERSUS SIDEREAL TIME
Every year the Earth actually makes 366 1/4 complete rotations with respect to the stars (sidereal
days). Each day the Earth also revolves about 1° about the Sun, so that after one year, it has
"unwound" one of those rotations with respect to the Sun; on the average, we observe 365 1/4
solar passages across the meridian (solar days) in a year. Since both sidereal and solar time use
24-hour days, the two clocks must run at different rates. The following compares (approximate)
time measures in each system:
SOLAR
365.25 days
1 day
0.99727 d
SIDEREAL
366.25 days
1.00274 d
1 day
SOLAR
24 hours
23h 56m 4s
6 minutes
SIDEREAL
24h 3m 56s
24 hours
6m 1s
The difference between solar and sidereal time is one way of expressing the fact that we observe
different stars in the evening sky during the course of a year. The easiest way to predict what the
sky will look like (i.e., determine the sidereal time) at a given date and time is to use a
planisphere, or star wheel. However, it is possible to estimate the sidereal time to within a halfhour or so with just a little mental arithmetic.
At noontime on the date of the vernal equinox, the solar time is 12h (since we begin our solar day
at midnight) while the sidereal time is 0h (since the Sun is at 0h RA, and is on our meridian).
Hence, the two clocks are exactly 12 hours out of syncronization (for the moment, we will ignore
the complication of "daylight savings"). Six months later, on the date of the autumnal equinox
(about September 22) the two clocks will agree exactly for a brief instant before beginning to drift
apart, with sidereal time gaining about 1 second every six minutes.
APS 1010 Astronomy Lab
32
Celestial Coordinates
For every month since the last fall equinox, sidereal time gains 2 hours over solar time. We simply
count the number of elapsed months, multiply by 2, and add the time to our watch (converting to a
24-hour system as needed). If daylight savings is in effect, we subtract 1 hour from the result to
get the sidereal time.
For example, suppose we wish to estimate the sidereal time at 10:50 p.m. Mountain Daylight Time
on August the 19th. About 11 months have elapsed since fall began, so sidereal time is ahead of
standard solar time by 22 hours - or 21 hours ahead of daylight savings time. Equivalently, we can
say that sidereal time lags behind daylight time by 3 hours. 10:50 p.m. on our watch is 22h 50 m
on a 24-hour clock, so the sidereal time is 3 hours less: ST = 19h 50m (approximately).
ENVISIONING THE CELESTIAL SPHERE
With time and practice, you will begin to "see" the imaginary grid lines of the alt-azimuth and
equatorial coordinate systems in the sky. Such an ability is very useful in planning observing
sessions and in understanding the apparent motions of the sky. To help you in this quest, we’ve
included four scenes of the celestial sphere showing both alt-azimuth and equatorial coordinates.
Each view is from the same location (Boulder) and at the same time and date used above (10:50 pm
MDT on August 19th, 1993). As we comment on each, we'll mention some important
relationships between the coordinate systems and the observer's latitude.
Looking North
From Boulder, the altitude of the north celestial pole directly above the North cardinal point is 40°,
exactly equal to Boulder's latitude. This is true for all observing locations:
Altitude of the pole = Latitude of observer
The +50° declination circle just touches our northern horizon. Any star more northerly than this
will be circumpolar - that is, it will never set below the horizon.
Declination of northern circumpolar stars > 90° - Latitude
Most of the Big Dipper is circumpolar. The two pointer stars of the dipper are useful in finding
Polaris, which lies only about 1/2° from the north celestial pole. Because these two stars always
point towards the pole, they must both lie approximately on the same hour circle, or equivalently,
both must have approximately the same right ascension (11 hours RA).
APS 1010 Astronomy Lab
33
Celestial Coordinates
Looking South
If you were standing at the north pole, the celestial equator would coincide with your local horizon.
As you travel the 50° southward to Boulder, the celestial equator will appear to tilt up by an
identical angle; that is, the altitude of the celestial equator above the South cardinal point is 50° from
the latitude of Boulder. Your local meridian is the line passing directly overhead from the north
to south celestial poles, and hence coincides with 180° azimuth. Generally speaking, then,
Altitude of the intersection of the celestial equator with the meridian = 90° - Latitude
Since the celestial equator is 50° above our southern horizon, any star with a declination less than
-50° is circumpolar around the south pole, and will never be seen from Boulder.
Declination of southern circumpolar stars < Latitude - 90°
The sidereal time (right ascension on the meridian) is 19h 43m - only 7 minutes different from
our estimate.
Looking East
The celestial equator meets the observer's local horizon exactly at an azimuth of 90°; this is always
true, regardless of the observer's latitude:
The celestial equator always intersects the east and west cardinal points
APS 1010 Astronomy Lab
34
Celestial Coordinates
At the intersection point, the celestial equator makes an angle of 50° with the local horizon; in
general,
The intersection angle between celestial equator and horizon = 90° - Latitude
Our view of the eastern horizon at this particular time includes the Great Square of Pegasus, which
is useful for locating the point of the vernal equinox. The two easternmost stars of the Great
Square both lie on approximately 0 hours of right ascension. The vernal equinox lies in the
constellation of Pisces about 15° south of the Square.
Looking Up
From a latitude of 40°, an object with a declination of +40° will, at some point in time during the
day or night, pass directly overhead through the zenith. In general
Declination at zenith = Latitude of observer
The 24 Ephemeris Stars in the SBO Catalog of Astronomical Objects have Object Numbers ranging
from #401 to #424. Each of these moderately-bright stars passes near the zenith (within 10° or so)
over the course of 24 hours. At any time, the ephemeris star nearest the zenith will usually be the
star whose last two digits equals the sidereal time (rounded to the nearest hour). For example, at
the current sidereal time (19h 43m), the zenith ephemeris star is #420 (δ Cygni).
At this time of the year (late summer) and at this time of night (mid-evening), the three prominent
stars of the Summer Triangle are high in the sky: Vega (in Lyra the lyre), Deneb (at the tail of
Cygnus the swan), and Altair (in Aquila the eagle). However, the Summer Triangle is not only
high in the sky in summer, but at any period during the year when the sidereal time equals roughly
20 hours: just after sunset in October, just before sunrise in May, and even around noontime in
January (though it won't be visible because of the Sun).
35
APS 1010 Astronomy Lab
Telescopes & Observing
TELESCOPES & OBSERVING
Telescope Types
Telescopes come in two basic types - the refractor, which uses a lens as its primary or
objective optical element, and the reflector, which uses a mirror. In either case, light
originating from an object (usually at infinity) is brought to a focus within the telescope to form
an image of the object. The size of a telescope refers to the diameter its primary lens or mirror,
rather than its length.
By placing photographic film at the focal plane, the objective lens or mirror forms a camera
system. If instead we position an eyepiece lens at an appropriate distance behind the focal plane,
we form an optical telescope.
The optical arrangement of a refracting telescope is shown below. The image is formed by the
refraction of light through the lens. The refractor has an advantage over reflectors in that there is
no central obscuration to produce diffraction patterns, and therefore yields crisper images.
However, refraction introduces chromatic abberation, which is corrected by using two-element
(doublet or achromat) or three-element (apocromat) lenses. The lens complexity makes the
refractor very expensive; hence, refractors are much smaller in diameter than comparably-priced
reflectors. All of the Sommers-Bausch Observatory (SBO) finder telescopes are of the refractor
type.
Objective Lens
Light from
Object
at Infinity
Eyepiece Lens
Eye
Focal
Plane
A reflecting telescope focusses and re-directs light back towards the incident direction, and
therefore requires additional optics to get the image "out of the way". This central obscuration
reduces the amount of light reaching the primary, and adds diffraction patterns which degrade the
resolution. However, the telescope does not suffer from chromatic abberation, since only mirrors
are used. Reflectors can be built much larger since the mirror is supported from behind (while a
lens is mounted only at its edges). Furthermore, only one objective surface must be ground and
polished, making the reflector much less expensive. As a result, virtually all large telescopes are of
the reflector type.
The Newtonian (invented by Sir Isaac Newton) is the simplest form of reflector. It uses a
diagonal mirror (a plane mirror tilted at a 45° angle) to re-direct the light out the side of the
telescope to the eyepiece. The placement of the eyepiece at the "wrong" end of the telescope limits
it practical size, and its asymmetric design precludes the use of heavy instrumentation. The
newtonian is used principly by amateur astronomers since it is the "cleanest" as well as least
expensive reflector.
36
APS 1010 Astronomy Lab
Light from
Object
at Infinity
Telescopes & Observing
Diagonal
Mirror
Prime Focal Plane
Primary
Mirror
Eyepiece
Eye
In the Cassegrain reflector, a convex secondary mirror intercepts the light from the primary and
reflects it back again, reducing the convergence angle in the process. The light passes through a
central hole in the primary and comes to a focus at the back of the telescope. The location of the
Cassegrain focus makes it easy to mount instrumentation, and the folded optical design permits
larger diameter telescopes to be houses within smaller domes; hence this design is most widelyused at professional observatories. All three of the permanently-mounted SBO telescopes (16",
18", 24") are Cassegrain reflectors.
Eyepiece
Light from
Object
at Infinity
Eye
Prime
Focus
(not used)
Secondary
Mirror
Primary
Mirror
Telescopes that use a combination of lenses and mirrors to form images are called catadioptric.
One form is the Schmidt camera, which uses a weak lens-like corrector plate at the entrance to
the telescope to correct for off-axis image abberations; this specialized telescope can't be used for
viewing but does produce panoramic sky photographs.
SBO has an 8" Schmidt for
astrophotography.
Corrector
Lens
Light from
Object
at Infinity
Focal Plane
Concave
Mirror
Film
One of the most popular, albeit expensive, telescope designs is the Schmidt-Cassegrain. As
the name suggests, it looks like the Cassegrain telescope but with the addition of a Schmidt
corrector plate. SBO has six of these smaller, portable telescopes: an 8" Meade, two 5"
Celestrons, and three 3.5" Questars.
37
APS 1010 Astronomy Lab
Telescopes & Observing
The fundamental optical properties of any telescope are described by three parameters: the focal
length, the aperture, and the focal ratio (f/ratio). The focal length (f) is the distance from
the objective where the image of an infinitely-distant object is formed. The aperture (A) of a
telescope is simply the diameter of its light-collecting lens (or mirror). The f/ratio is defined to be
the ratio of the focal length of the lens or mirror to its aperture:
f
f/ratio = A
.
(1)
Obviously, if you know any two of the three fundamental parameters, you can calculate the third.
Focal Length
As shown in the diagram below, an object that subtends an angular size θ in the sky will form
an image of linear size h given by θ ≅ tan(θ) = h/f. Therefore
h = θ radians f
Object
at
Infinity
Parallel
Incident
Rays
θ
57.3° f
=
.
(2)
Image
at
Focal
Plane
f
θ
θ
h
The image scale is the ratio of linear image size to its actual angular size:
Image Scale =
h
θ
=
f
57.3°
.
(3)
Thus, the scale of the image of any object depends only upon the focal length of the telescope, not
on any other property: two telescopes with identical focal lengths will produce identically-sized
images of the same object, regardless of any other physical differences. Furthermore, the image
scale is directly proportional to f: doubling the focal length will produce an image twice the linear
size (and four times the areal size).
The plate scale (used in photography) is usually expressed as the inverse of the above - the
angular size of the object (usually in arc-seconds) corresponding to a linear size (usually in
millimeters) in the focal plane.
For a compound telescope (with more than one active optical element), we refer to the effective
focal length (EFL) of the entire optical system, and treat it as a simple telescope with single lens
or mirror of focal length f = EFL.
Aperture
The aperture A of a telescope is the diameter of the principle light-gathering optical element. The
light-gathering ability, or light grasp, of the telescope is proportional to the area of the objective
element, or A2. For point sources such as stars, all of the collected light will (to a first
38
APS 1010 Astronomy Lab
Telescopes & Observing
approximation) converge to form a point-like image; hence, stars appear brighter, and fainter stars
can be detected, with telescopes of larger aperture. Telescope aperture is the principle criteria for
determining the limiting visible stellar magnitude.
The resolving power (ability to resolve adjacent features in an image) of an optical telescope is
also chiefly a function of aperture; the larger the aperture, the smaller the diffraction pattern formed
by each point source, and therefore the better the resolution. The theoretical diffraction-limited
resolution of a telescope is given by
θ diff ≥
1.2 λ
A
5 arc-sec
≅ Inches of Aperture
(4)
where λ is the wavelength of the light used, assumed to be 5500 Å.
Focal Ratio
Although the amount of light gathered by a telescope is proportional to A2, the collected light is
spread out over an area at the focal plane that is proportional to h2, and therefore propertional to f2;
hence, the image brightness of an extended (non-pointlike) object will scale linearly with the ratio
(A/f)2, or inversely with the square of the f/ratio = f/A of the telescope:
Brightness ∝
1
f/ratio2
.
(5)
The f/ratio of a telescope is therefore the only factor that determines the image brightness of an
extended object. For example, if one telescope has twice the aperture and twice the focal length of
another, both telescopes are geometrically similar and would use exactly the same photographic
exposure to produce identically-bright images of, say, the Moon. Of course, the first telescope
would produce an image twice the linear size (and four times the area) as the latter, but both would
have the same photographic "speed". Note that the smaller the f/ratio, the brighter the image and
the faster the speed: a 35mm camera using an f/ratio of f/2 will photograph the Milky Way in less
than 5 minutes, while the same film on an f/15 telescope will require an hour or more to capture
the diffuse glow! On the other hand, individual stars will record much better using the telescope.
The optical speed of a compound telescope is simply the ratio of its effective focal length to its
aperture. Both effective or actual f/ratios are a measure of the final angle of convergence angle of
the cone of light before it forms the image; the smaller the f/ratio, the greater the convergence, and
the more critical the focus; this is why "slow" optics, with a slowly-converging beam, exhibit a
larger "depth-of-field".
Seeing and Clarity
Equation (4) gives the theoretical limit to the angular size that can be resolved by a telescope. This
limit is almost never achieved in practice, since atmospheric turbulence is always present to blur
our image of the object. The quality of the seeing is measured by the angular separation needed
between two stars for their images to just be resolved. Average seeing in the Boulder area is about
2 arc-seconds, making it a marginal task to "split" the 2 arc-sec pairs of the "double-double" star
Epsilon Lyrae. Good seeing in Boulder is when the atmosphere is stable enough to resolve 1"
separations. Poor seeing conditions can be as bad as 5" or even worse. By comparison, half-arcsecond seeing occurs routinely at several optimally-located major observatories, but it is rare for
any ground-based telescope to experience 0.1" conditions.
One can estimate the seeing of a night simply by glancing up. If the stars glow solidly, the seeing
is probably "good"; if they twinkle, the seeing is "average"; if bright stars dance and planets flash
with color, the seeing is "poor". Another measure is to look towards Denver - if there are lots of
39
APS 1010 Astronomy Lab
Telescopes & Observing
particulates in the air, and Denver is on smog alert, you will see a bright horizon glow; in this
situation, you can usually count on good seeing! These conditions are created by an inversion
layer of stagnant air, which permits rock-solid astronomical viewing.
The best conditions for good seeing are the worst for sky clarity. Good clarity implies dark skies
due to a lack of light-scattering dust particles, and an absence of water-vapor haze. Clarity usually
improves in Boulder after the passage of a thunderstorm, which clears out the dust. Of course, the
conditions that improve clarity usually destroy seeing.
Since ideal nights of good clarity and seeing are rare, an observer must be prepared to take
advantage of the best properties of the night, if any. Lunar and planetary observing requires good
seeing conditions, since planetary disks are small, only 2-40 arc-seconds in diameter. Poor clarity
is not a major problem for bright solar system objects, although low-contrast features may be
washed out. On the other hand, good clarity (and the absence of strong moonlight) is essential to
observe galaxies and diffuse nebulae, since the light from these objects is faint and dark skies are
needed for contrast. Seeing is not critical, since these objects are fuzzy patches to begin with.
When both seeing and clarity are poor, it's best to focus your attention on bright star clusters and
widely-spaced double stars.
Eyepieces and Magnification
An eyepiece or ocular is simply a magnifier that allows you to inspect the image formed by the
objective from a very short distance away. The eyepiece is placed so that its focal plane coincides
with the focal plane of the objective, so that the rays from the image emerge parallel from the
eyepiece. As a result, the telescope never forms a final image - the lens of your eye does that.
By selecting appropriate rays for both the objective lens (obj) and the eyepiece lens (eye), we can
see that rays incident at the telescope at an angle θ obj will emerge from the eyepiece at a larger
angle θ eye; the observer perceives that the object subtends a much larger angle than is actually the
case. A little geometry gives the angular magnification Mθ produced by the arrangement:
Mθ ≡
θ eye
≅
θ obj
fobj
feye
.
(6)
That is, magnification is the ratio of the objective focal length divided by the eyepiece focal length.
For example, the 16-inch f/12 telescope has a focal length of 192 inches, or 4877 mm. If we use
an eyepiece with a focal length of 45 mm (engraved on its barrel), the "power" of the telescope will
be about 108 X; by switching to an 18 mm eyepiece, we will have 271 X. The answer to the
question "what is the power of this telescope?" is "whatever you want - within the range of the
available eyepiece assortment".
40
APS 1010 Astronomy Lab
Telescopes & Observing
Eyepieces come in a variety of designs, each representing a different trade-off between
performance and cost, ranging from the inexpensive short-focal-length two-element Ramsden (R)
to the 6-element triple-doublet Erfle (Er). Other types include: the Kellner (K), an improved
Ramsden; orthoscopic (Or), good at moderate focal lengths at reasonable cost; and the expensive
low-power wide-field designs - the Plossl (including Clave), the Konig, and the Televue.
Besides focal length (and image quality), the other important characteristic of an eyepiece is its
field-of-view - the size of the solid angle viewable through the ocular, or when used on a particular
telescope, the actual angular size of the observable sky field. The field available with a given
telescope-eyepiece combination can be measured directly by positioning a bright star just at the
northern edge of the field; after noting the declination, you move the telescope northward until the
star is at the southern boundary of the field, note the declination again, and calculate the difference
in angle. The field can also be calculated from the image scale of the telescope (equation (3)):
measure the diameter of the field stop (the ring installed within the open end of the eyepiece) ,
equate that to the linear image size h, and calculate the corresponding angle θ.
The Barlow is a telecompressor (concave or negative) lens that can be installed in front of the
eyepiece. It reduces the angle of convergence of the objective light cone and hence increases the
f/ratio and the EFL of the telescope, thus increasing the magnification (by a factor of 2 to 3) from a
given eyepiece. It helps achieve high magnification without sacrificing eye relief (see below).
Selecting an Eyepiece
The choice of eyepiece is one of the few factors that an observer may control to enhance the
visibility of astronomical objects.
The Observatory's eyepieces range from focal lengths of 70 mm down to 4 mm. The longer focal
lengths are in the 2"-diameter format, which fit directly into the large telescope tubes; the shorter
(higher magnification) eyepieces have 1-1/4" diameter barrels, but can be used in the 2" tube using
an eyepiece adapter plug.
Although equation (6) implies that it's possible to have any magnification one desires, there are
practical limits. The entrance pupil of the human eye (the diameter of the iris opening) is about
7 mm for a fully dark-adapted eye (5 mm for older individuals). The exit pupil of an telescope
(diameter of the bundle of light rays exiting the eyepiece) can be shown to be
feye
A
Exit Pupil = Mθ = f/ratio
(7)
If the exit pupil is larger than the entrance pupil of the eye, the eye can't intercept it all and some of
the light is wasted. This occurs if the magnification is too low. If the exit pupil just matches the
eye pupil, all of the light is utilized and we have the brightest-possible, or "richest-field"
arrangement. This is why 7x50 (7 power, 50 mm aperture) and 11x80 (11 power, 80 mm
aperture) binoculars are excellent for night observing - their exit pupils optimally match the darkadapted human eye. At higher magnifications, all of the light enters the eye but is spread over a
larger solid angle which dims the field.
The average human eye can just barely resolve two objects separated by about 1 arc-minute,
although a separation of about 4' (240") is much more comfortably perceived (about 50 such 4 arcminute "pixels" comprise the face of "the man in the Moon"). Hence, one criterion for a
telescope's maximum useful power Mmax is "that magnification that matches a 250 arc-second
visual separation to the diffraction limit of the telescope"; from equation (4) this equates to
Mmax ≅ 50 x Aperture of Telescope in Inches
(8)
APS 1010 Astronomy Lab
41
Telescopes & Observing
By this "rule of thumb", the 16-inch telescope has a maximum useful magnification of 800 X on a
night of perfect seeing, which would be achieved with a 6 mm eyepiece. Any greater
magnification will be "empty" - that is, the image will become larger, but the enlargement will
contain only "blur", not additional detail. (Note: poor-quality 2.4"-refractors are provided with 4
mm eyepieces plus a cheap 2.5X barlow lens so that the manufacturer can advertise "600 power" 5 times beyond any useful application!)
The above magnification limit is somewhat optimistic for telescopes of moderate-to-large aperture,
since detail is almost invariably limited by atmospheric seeing rather than by telescope resolution.
For example, if the seeing is about 1", a 250" perceived separation implies a useful maximum
magnification of about 250 X - or an eyepiece in the 18 - 24 mm range for the 16-inch telescope.
For 2" seeing, a reasonable choice is about 125 power, or eyepieces in the 32 - 45 mm range.
There are several additional considerations related to high magnification. On the negative side,
short-focal-length eyepieces require critical focussing and eye placement, making them difficult for
inexperienced observers to use. In addition, they exhibit short eye relief - the distance behind the
lens where the eye is positioned to see the entire field-of-view. In particular, eyepieces shorter
than about 12 mm focal length are problematic for eye-glass wearers, since glasses prevent the eye
from being placed close enough to avoid "tunnel vision".
On the positive side, high magnification reveals fainter stars. Remember that stars are unresolved
point sources whose light is concentrated (to a first approximation) at a single point in the image
regardless of magnification. The glow of the background sky is a diffuse source, which will be
spread out by higher magnification, reducing its brightness. Although high magnification doesn't
increase the brightness of faint stars, it improves their contrast against the sky, making them more
visible! By this same token, high magnification helps diminish the intense glare from bright solar
system objects, making the Moon less painful to look at, and the bands of Jupiter easier to see.
Finally, there is the consideration of the type and angular extent of the object being viewed. High
magnification certainly helps resolve fine planetary detail and separates close double stars,
provided that seeing conditions permit. Open clusters usually extend over a large patch of the sky,
and hence need low power (a wide field-of-view) to encompass them - in fact, the small finder
scopes may present a more pleasing view than the main telescope. High power won't help the
contrast between sky background and diffuse nebulae (since both are extended sources), and in
fact may keep the observer from seeing the glow since all aspects of the image are dimmer. Low
power is essential for extremely large nebulae, since these objects can fill the field-of-view creating a situation where "you can't see the forest for the trees". Globular clusters can be
appreciated at a variety of magnifications: low power creates an impression of a "cotton-ball"
floating in space, while high magnification shows a magnificent sparkling array of bright points
reminiscent of distant city lights. In the end, your personal experience and judgement should be
your guide.
Observing Technique
Observing through an eyepiece is an unnatural experience; good observing techniques come with
time and experience. Here are some hints to help you get the most out of your telescope time:
•
Let your eyes become fully dark-adapted before doing "serious" observing. This can
take 20-30 minutes.
•
Keep both eyes open; this will help you relax your eyes to infinity focus.
•
"Gaze", don't "glimpse". It takes a a few moments to mentally and optically adjust to the
scene in an eyepiece. Continued observation also improves the chances for brief periods
of atmospheric stability when new details may appear.
APS 1010 Astronomy Lab
42
Telescopes & Observing
•
Don't stare fixedly through the eyepiece; let your eye wander. The human eye is much
more sensitive to changes in brightness than to constant light levels. By looking around,
you'll see fainter detail.
•
Use averted vision to help bring out dim features. The color-sensitive cone cells
concentrated near the center of the eye's retina don't respond to low light levels. If you
look out of the "corner of your eye", you'll utilize the much more sensitive black-andwhite rod cell receptors that are used for peripheral vision.
•
Finally, don't expect to see the colors that appear in photographs; for the reason just
mentioned, most faint objects appear greenish to whitish because they are too faint to
stimulate the color receptors in your eye.
APS 1010 Astronomy Lab
43
DAYTIME
EXPERIMENTS
Daytime Experiments
APS 1010 Astronomy Lab
44
The SBO Heliostat ...
... and a television image of a solar flare observed with it
Daytime Experiments
APS 1010 Astronomy Lab
45
Colorado Model Solar System
THE COLORADO MODEL SOLAR SYSTEM
SYNOPSIS: A walk through a model of our own solar system will give you an appreciation of
the immense size of our own local neighborhood and a “feel” for astronomical distances.
EQUIPMENT: This lab write-up, a pencil, perhaps a calculator, and walking shoes.
Astronomy students and faculty have worked with CU to lay out a scale model solar system along
the walkway from Fiske Planetarium northward to the Engineering complex (see figure below).
The model is a memorial to astronaut Ellison Onizuka, a CU graduate who died in the explosion of
the space shuttle Challenger in January 1986.
ve
Regent Dr.
A
do
ra
olo
C
Folsom St.
o
ut
Pl
Ne
u
pt
ne
Ur
u
an
s
S
u
at
rn
r
te
J
i
up
s
ar
M
th
r
Ea
s
ry
V
u
en
M
e
u
rc
Su
n
The Colorado Scale Model Solar System is on a scale of 1 to 10 billion (1010 )!!! That is, for every
meter (or foot) in the scale model, there are 10 billion meters (or feet) in the real solar system.
Note: A review of scientific notation can be found on page 15 of this manual.
All of the sizes of the objects within the solar system (where possible), as well as the distances
between them, have been reduced by this same scale factor. As a result, the apparent angular sizes
and separations of objects in the model are accurate representations of how things truly appear in
the real solar system.
The model is unrealistic in one respect, however. All of the planets have been arranged roughly in
a straight line on the same side of the Sun, and hence the separation from one planet to the next is
as small as it can possibly be. The last time all nine planets were lined up this well in the real solar
system, the year was 1596 BC!
In a more accurate representation, the planets would be scattered in all different directions (but still
at their properly-scaled distances) from the Sun. For example, rather than along the sidewalk to
our north, Jupiter could be placed in Kittridge Commons to the south; Uranus might be found on
the steps of Regent Hall, Neptune in the Police Building, and Pluto in Folsom Stadium. Of
course, the inner planets (Mercury, Venus, Earth, and Mars) will still be in the vicinity of Fiske
Planetarium, but could be in any direction from the model of the Sun.
APS 1010 Astronomy Lab
46
Colorado Model Solar System
I. The Inner Solar System
Read the information on the pyramid holding the model Sun in front of Fiske Planetarium. Note
that the actual Sun is 1.4 million km (840,000 miles) in diameter - but on a one-ten-billionth scale,
it’s only 14 cm (5.5 inches) across.
I.1
Step off the size of the orbit of each of the four inner planets.
walking stride, count the number of paces from the model Sun to
Using your normal
(a) Mercury: __________ paces
(b) Venus:
__________ paces
(c) Earth:
__________ paces
(d) Mars:
__________ paces
Contemplate the vast amount of empty space that lies between these planets and the Sun. As the
saying goes, “you ain’t seen nothin’ yet” (There's a whole lot more of "nothingness" to come)!
I.2
Since the Sun is the size of a grapefruit, what appropriately-sized objects might you
choose to represent the scaled size of each of the four inner planets?
Read the information at each of the four innermost planets, and answer the following:
1.3
(a) Which planet is most like the Earth in temperature?
(b) Which has the greatest range of temperature extremes?
(c) Which planet is most similar to the Earth in size?
(d) Which is the smallest?
(e) Which planet has a period of rotation (a day) very much like the Earth’s?
(f) Which planet has an extremely long period of rotation?
The real Earth orbits about 150 million km (93 million miles) from the Sun - a distance known as
an astronomical unit, or AU for short. The AU is very convenient for comparing relative distances
in the solar system by using the Earth-Sun separation as a “yardstick”.
I.4
What fraction of an AU does one of your paces correspond to in the model?
I.5
(a) Using your pace measurements from I.1, what is the distance in AU between the Sun
and Mercury?
(b) Between the Sun and Mars?
(c) Between the Sun and Venus?
(d) Which of these planets comes closest to the Earth?
Another way to describe distance is to use “light-time” - the time it takes light, travelling at 300,000
km/second (186,000 miles/sec, or 670 million mph!) to get from one object to another. Since light
APS 1010 Astronomy Lab
47
Colorado Model Solar System
travels at the fastest speed possible in the universe, light-time represents the shortest time-interval
in which information can be sent from one location to another.
Sunlight takes 500 seconds, or 8 1/3 minutes, to reach the Earth; we say that the Earth is 8.33
light-minutes from the Sun. Remember, the light-minute is a measure of distance!
I.6
(a) Again, walk at a normal pace from the Sun to the Earth, and time how many seconds
it takes for you to travel that distance:
(b) Since light takes 500 seconds to travel that same distance in the real solar system,
how many times “faster than light” do you walk through the model?
I.7
(a) Mars is depicted in the model at its closest approach to Earth (that is, both the Earth
and Mars are on the same side of the Sun). In this configuration, what is the distance, in
astronomical units, between Earth and Mars?
(b) What is this distance expressed in light-minutes?
On July 4th, 1997, the Pathfinder spacecraft landed a rover named Sojourner on the surface of
Mars. The vehicle was equipped with a camera and an on-board computer that enables it to
recognize obstacles and to drive around them.
I.8
Explain why Sojourner was not steered remotely by an operator back on the Earth.
Finally, take a closer look at Earth's own satellite, the Moon - the furthest object to which mankind
has traveled "in person". It took three days for Apollo astronauts to cross this "vast" gulf of empty
space between the Earth and the Moon.
I.9
(a) How many years has it been since mankind first walked there?
(b) At the scale of this model, estimate how far (in cm or inches) mankind has ventured
into space.
I.10
Assuming that we could make the trip in a straight line over the smallest possible
separation (neither of which are feasible), roughly how many times farther would
astronauts have to travel to reach even the nearest planet to the Earth?
II. The View From Earth
Stand next to the model Earth and take a look at how the rest of the solar system appears from our
vantage point (remember, since everything is scaled identically, the apparent sizes of things in the
model are the same as they appear in the real solar system).
II.1
(a) Stretch out your hand at arm’s length, close one eye, and see if you can cover the
model Sun with your index finger. Are you able to completely block it from your view?
(b) Estimate the angle, in degrees, of the diameter of the model Sun as seen from Earth.
Note: see page 28 of this manual for a “handy” way to estimate angles.
APS 1010 Astronomy Lab
II.2
48
Colorado Model Solar System
If it’s not cloudy, use the same technique to cover the real Sun with your outstretched
index finger. Verify that the apparent size of the real Sun as seen from the real Earth is
the same as the apparent size of the model Sun as seen from the model Earth.
Caution! Staring at the Sun can injure your eyes - be sure the disk of the Sun is covered!
Now look towards Mars. This Sun-Earth-Mars arrangement is called opposition, since Mars lies
in the opposite direction from the Sun as seen from Earth.
II.3
(a) When is a good time of day (or night) for humans to observe a planet in opposition?
(Hint: if you can see Mars, is the Sun “up” in the sky at the same time, or not?)
(b) When Mars is in opposition, will it appear larger or smaller to Earthlings than at some
other point in its orbit?
II.4
Now turn to face Venus. What time of day are you simulating now? (Once again,
consider the direction of the Sun!)
This configuration is called a conjunction, meaning that a planet appears lined up with (i.e., in
conjunction with) the Sun as seen from Earth.
II.5
Although Venus is as close to Earth as possible in this configuration, it would not be a
good time to observe the planet. Why not?
II.6
Neither Mercury nor Venus can ever be viewed (from Earth) in a direction opposite the
Sun. Explain why not. (Hint: If you’re having trouble answering this question, have a
classmate “orbit” at Mercury’s distance of 6 meters (20 feet) from the Sun while you
observe from Earth - and note which directions you look to follow Mercury’s motion.)
II.7
Mars can appear in conjunction with the Sun as well as in the opposite direction. Explain
or sketch how this is possible. (Hint: have a classmate “orbit” the Sun at the distance of
Mars).
Off in the distance to the north, along the walkway leading to the Engineering Building, you can
just see the pedestals for the next two planets, Jupiter and Saturn.
II.8
Before launching off on your journey to these planets, use your hand to measure the
angle between Jupiter and Saturn, as seen from the model Earth. (Remember, if this
were the real solar system, 10 billion times bigger in all directions, the angle between
them would still be the same!)
III. Journey to the Outer Planets
As you cross Regent Drive heading for Jupiter, you’ll also be crossing the region of the asteroid
belt, where thousands of “planetoids” or “minor planets” can be found crossing your path. The
very largest of these is Ceres, which is 760 km (450 miles) in diameter.
III.1
If the asteroids are scaled like the rest of the solar system model, would you be able to
see Ceres as you passed by it?
APS 1010 Astronomy Lab
49
Colorado Model Solar System
Jupiter contains over 70% of all the mass in the solar system outside of the Sun - but still “weighs”
less that one-tenth of one percent of the Sun itself.
III.2
(a) What object would you choose to represent this, the solar system’s largest planet?
(b) How many times more massive is it than the Earth?
(c) What moon orbits Jupiter at about the same distance as our Moon orbits the Earth?
Besides Io, there are three additional large moons of Jupiter that are easily seen with a telescope
from Earth: Europa, Ganymede, and Callisto. These moons orbit Jupiter at a distance of roughly
2.5, 4, and 7.5 inches, respectively at our scale. Ganymede is slightly larger than the planet
Mercury - making it the 8th largest object in the solar system after the Sun!
III.3
(a) Use your hand measurement to compare the apparent size of the Sun as seen from
Jupiter with your earlier measurement (in II.1) from the Earth. (You’ll have to walk a
few paces to the west to avoid the intervening evergreen, which plays no useful role in
our model.)
(b) How does the surface temperature of Jupiter compare with the temperatures of the
inner planets? (The answer shouldn’t surprise you, considering what you just noted
about the apparent size of the Sun.)
Next stop, the beautiful Ringed Planet - the most distant object in the solar system that you can see
from Earth without a telescope.
III.4
On your way, count the number of steps between Jupiter and Saturn: ________ paces.
III.5
(a) Earlier (in II.8) you found that Jupiter and Saturn appeared to be very close to each
other as seen from Earth. How far apart are they really, expressed in astronomical units?
(b) Compare the size of the entire inner solar system (from the Sun out to Mars) with the
empty space between the orbits of Jupiter and Saturn.
III.6
Compare the diameter of the rings of Saturn to the diameter of planet Jupiter. Which do
you think would appear wider (through a telescope) from Earth: the ball of Jupiter, or the
rings of Saturn? (Hint - consider distance from Earth as well as actual size).
Now continue the journey onward to Uranus. On your way:
III.7
(a) Time how long it takes for you to walk from the orbit of Saturn out to the orbit of
Uranus: _______________ seconds.
(b) Since you already have figured how many times “faster-than-light” that you walk (in
I.6), calculate how long would it take a beam of light to cover this same distance.
As you walk to Neptune, ponder the vast amount of “nothingness” through which you are passing.
When you arrive, answer the following:
III.8
(a) Which of the planets you’ve just explored most resembles Neptune?
(b) How many complete orbits has Neptune made around the Sun since its discovery?
(c) What is the name of the only spacecraft to visit this planet, and when did it pass?
APS 1010 Astronomy Lab
50
Colorado Model Solar System
Now begin the final walk to Pluto. When you get there, you’ll be standing three-tenths of a mile,
or one-half of a kilometer, from where you started your journey. You’ll also be standing about 33
times further away from the Sun than when you left the Earth.
III.9
You may have been surprised by how soon you arrived at Pluto after leaving Neptune.
Read the plaque, and then explain why the walk was so short.
III.10
(a) Although you can’t actually see the model Sun from this location, describe how you
imagine it would appear from Pluto.
(b) How does the temperature compare out here in the depths of space with that in the
inner solar system?
IV. Beyond Pluto
Although we’ve reached the edge of the solar system visible through a telescope, that doesn’t mean
that the solar system actually ends here - nor does it mean that mankind’s exploration of the solar
system ends here, either.
Down by Boulder Creek to your north, at 58 AU from the Sun, Voyager II is still travelling
outwards towards the stars, and still sending back data to Earth.
IV.1
Voyager II uses a nuclear power cell instead of solar panels to provide electricity for its
instruments. Why do you think this is necessary?
In 2000 years, Comet Hale-Bopp will reach its furthest distance from the Sun (aphelion), just
north of the city of Boulder at our scale. Comet Hyakutake, the Great Comet of 1996, will require
23,000 years more to reach its aphelion distance - 15 miles to the north near the town of Lyons.
Beyond Hyakutake’s orbit is a great repository of comets-yet-to-be: the Oort cloud, a collection of
a billion or more microscopic (at our scale) “dirty snowballs” scattered over the space between
Wyoming and the Canadian border. Each of these is slowly orbiting our grapefruit-sized model of
the Sun, waiting for a passing star to jog it into a million-year plunge into the inner solar system.
And there is where our solar system really ends. Beyond that, you’ll find nothing but empty space
until you encounter Proxima Centauri, a tiny star the size of a cherry, 4,000 km (2,400 miles) from
our model Sun! At this scale, Proxima orbits 160 kilometers (100 miles) around two other stars
collectively called Alpha Centauri - one the size and brightness as the Sun, the other only half as
big (the size of an orange) and one-fourth as bright. The two scale model stars of Alpha Centauri
orbit each other at a distance of only 1000 feet (0.3 km).
IV.2
Suppose there's an Earth-like planet orbiting the Sun-like star of Alpha Centauri, at the
same distance as the Earth is from the Sun. Use your imagination to describe what the
nighttime and/or daytime sky might look like to a resident of than planet.
APS 1010 Astronomy Lab
51
Seasons
THE SEASONS
SYNOPSIS This exercise involves making measurements of the Sun every week throughout the
semester, and then analysing your results at semester’s end. You’ll learn first-hand what factors
are important in producing the seasonal changes in temperature, and which are not.
EQUIPMENT: Gnomon, sunlight meter, heliostat, tape measure, calculator, and a scale.
REVIEW: Angles and trigonometry (page 19), using your calculator (page 22).
Everybody knows that it’s colder in December than in July - but why? Is it because of a change in
the number of daylight hours? The height of the Sun above the horizon? The “intensity” of the
sunlight? Or are we simply closer to the Sun in summer than in winter?
We can measure each of these factors relatively easily, and will show you how to do so. But
seasonal changes occur rather slowly, so we’ll need to monitor the Sun over a long period of time
before the important factors become apparent. We’ll also need to collect a considerable amount of
data from all of the lab classes - to gather information at different times of the day, and to make up
for missing data on cloudy days.
Today you’ll learn how to take the solar measurements. Then, every week during the semester,
you or your classmates will collect additional observations. At semester’s end, you’ll return to this
exercise to analyze your findings to find out just what factors are responsible for the Earth’s
heating and cooling.
Part I. Start of the Semester: Learning to Make Solar Measurements
TIME OF DAY
As the Sun moves daily across the sky, the direction of the shadows cast by the Sun move as well.
By noting the direction of the shadow cast by a vertical object (called a gnomon), we can
determine the time-of-day as defined by the position of the Sun. Such a device, of course, is called
a sundial.
I.1
Note the time shown by the sundial on the deck of the Observatory. Officially, this is
known as “local apparent solar time”, but we’ll just refer to it as “sundial time”.
Determine the time indicated by the shadow to the nearest quarter-hour: Sundial Time =
__________. (Don’t be surprised if your watch and the sundial disagree - this is normal
and is to be expected due to a number of factors which we won’t go into here!)
ALTITUDE OF THE SUN
When the Sun first appears on the horizon at sunrise, shadows are extremely long; as it rises
higher in the sky, the lengths of shadows become shorter. Hence, the length of the shadow cast
by a gnomon can also be used to measure the altitude of the Sun (the angle, in degrees, between
the Sun and the point on the horizon directly below it).
The figure on the next page shows that the shadow cast by a gnomon forms a right-triangle. The
Sun’s altitude is the angle from the horizontal ground to the top of the gnomon, as seen from the
tip of the shadow. The tangent of that angle is the opposite side of the right-triangle (the height of
the gnomon, H) divided by the adjacent side of the triangle (the length of the shadow, S).
APS 1010 Astronomy Lab
52
Seasons
Mathematically, this relationship is: tan(altitude) = H/S. We’ve prepared a 1-meter high gnomon
(H = 1,000 mm) on a stand to help you make the measurement.
I.2
Carry the gnomon and a metric tape-measure to the
Observatory deck. Carefully measure the shadow
length S from the base of the gnomon to the tip of the
shadow: S = ____________ mm.
I.3
Calculate the tangent value for the solar altitude:
tan(altitude) =
I.4
H
1000 mm
=
= ________
S _____ mm
Now use the “arc-tangent” function on your calculator
to determine what angle, in degrees, has a tangent equal
to the number you obtained in I.3:
Solar altitude = ________ °
THE SUNLIGHT METER
The “sunlight meter” is a device that enables you to deduce the relative intensity of the sunlight
striking the flat ground here at the latitude of Boulder, compared to some other place on the Earth’s
surface where the Sun is, at this moment, directly overhead at the zenith.
I.5
On the observing deck, aim the “meter” by rotating the base and tilting the upper plate
until its gnomon (the perpendicular stick) casts no shadow. When properly aligned, the
upper surface of the apparatus will directly face the Sun.
The opening in the upper plate is a square 10 cm x 10 cm on a side, so that a total area of 100 cm2
of sunlight passes through it. The beam passing through the opening and striking the horizontal
base covers a larger, rectangular area. This is the area on the ground here at Boulder that receives
solar energy from 100-square-centimeter’s “worth” of sunlight.
I.6
Place a piece of white paper on the horizontal base, and draw an outline of the patch of
sunlight that falls onto it.
I.7
Measure the width of the rectangular region; is it still 10 cm? _______ Measure the
length ( ______ cm), and compute the area of the patch of sunlight (width x length):
Area = ___________ cm2.
I.8
Now calculate the relative solar intensity - the fraction of sunlight we’re receiving here in
Boulder compared to how much we would receive if the Sun were directly overhead:
Relative Solar Intensity =
100 cm 2
= _____________
Area
THE RELATIVE SIZE OF THE SUN AS SEEN FROM EARTH
Finally, in the lab room your instructor will have an image of the Sun projected onto the wall using
the Observatory’s heliostat, or solar telescope. As you know, objects appear bigger (we say they
“subtend a larger angle”) when they are close, and appear smaller at a distance. By measuring the
projected size of the Sun using the heliostat throughout the semester, you’ll be able to determine
whether or not the distance to the Sun is changing - and if so, whether the Earth is getting closer to
it, or farther away, and by how much.
APS 1010 Astronomy Lab
I.9
53
Seasons
Use a meter stick to measure the diameter, to the nearest millimeter, of the solar image
that is projected onto the wall. (Note: since the wall isn’t perfectly perpendicular to the
beam of light, a horizontal measurement will be slightly distorted; so always measure
vertically, between the top and bottom of the image). Solar Diameter = _________ mm.
PLOTTING YOUR RESULTS
You’ll use three weekly group charts to record your measurements, which will always be posted
on the bulletin board at the front of the lab room: the Solar Altitude Chart, the Solar Intensity
Chart, and the Solar Diameter Chart. This first week, your instructor will take a representative
average of everyone’s measurements and show you how to plot a datum point on each graph.
After this week, it will be the responsibility of assigned individuals to measure and plot new data
each class period. You yourself will be called upon at least once during the semester to perform
these measurements, so it’s important for you to understand what you’re doing, and to make your
measurements and plots as accurately and as neatly as possible!
I.10
On the weekly Solar Altitude Chart, carefully plot a symbol showing the altitude (I.4)
of the Sun in the vertical direction and the sundial time (I.1) along the horizontal
direction, showing when the measurement was made. Use a pencil (to make it easy to
correct a mistake) and use the symbol appropriate for your day-of-the-week (M-F) as
indicated on the chart. Other classes will have added, or will be adding, their own
measurements to the chart as well.
I.11
On the weekly Solar Intensity Chart, carefully plot a point that shows the relative
solar intensity (I.8) that was measured at the corresponding sundial time (I.1). Again,
use the appropriate symbol.
I.12
On the weekly Solar Diameter Chart, plot your measurement of the diameter in mm
(I.9) vertically for the current date (horizontal axis). (There may be as many as three
points plotted in a single day from three different classes.)
Part II. During the Semester: Graphing the Behavior of the Sun
At the end of each week, the Observatory staff will collect the three weekly charts and construct a
best-fit curve through the set of datum points. In addition, the curves will be extrapolated to earlier
and later times of the day so that the entire motion of the Sun, from sunrise to sunset, will be
represented (adding a few additional early-morning and late-day measurements of our own if
necessary). Although the data represent readings over a 5-day period, the curve will represent the
best fit for the mid-point of the week, Wednesday. These summaries of everyone’s measurements
will be available for analysis the following week and throughout the remainder of the semester.
Every week take a few moments to analyze the previous week’s graphs, and record in the table on
the next page:
II.1
(a) The date of the mid-point of the week (Wednesday).
(b) The greatest altitude above the horizon the Sun reached that week.
(c) The number of hours the Sun was above the horizon, to the nearest quarter-hour.
(d) The maximum value of the intensity of sunlight received here in Boulder, relative to
(on a scale of 0 to 1) the intensity of the Sun if it had been directly overhead.
(e) The average measured value of the apparent diameter of the Sun as determined using
the heliostat.
APS 1010 Astronomy Lab
(a)
Week
#
Date
54
Seasons
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Maximum
Solar
Altitude
(deg)
Number
of
Daylight
Hours
Maximum
Solar
Intensity
Solar
Diam.
(mm)
“Weight”
of
Sunlight
(grams)
SolarConst.
Hours
KWH/
Meter2
per
Day
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
II.2
Also each week, transfer your new data from columns (a) through (e) in the table above
to your own personal semester summary graphs on the next two pages: Maximum Solar
Altitude, Hours of Daylight, Maximum Solar Intensity, and Solar Diameter. Be sure to
include the appropriate date at the bottom of each chart.
APS 1010 students using a primitive sundial.
APS 1010 Astronomy Lab
55
Seasons
Maximum Solar Altitude (Semester Summary)
Week 1
90°
2
3
2
3
4
5
6
7
8
9
10
11
12
13
14
15
14
15
Maximum Solar Altitude (degrees)
80°
70°
60°
50°
40°
30°
20°
10°
0°
Date:
Hours of Daylight (Semester Summary)
Week 1
Number of Hours of Daylight
14
12
10
8
6
4
2
0
Date:
4
5
6
7
8
9
10
11
12
13
APS 1010 Astronomy Lab
56
Seasons
Maximum Solar Intensity (Semester Summary)
Week 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
13
14
15
Maximum Solar Intensity (relative to sun at zenith)
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Date:
Solar Diameter (Semester Summary)
Week 1
785
Apparent Diameter of the Sun (mm)
780
775
770
765
760
755
750
745
740
735
Date:
2
3
4
5
6
7
8
9
10
11
12
APS 1010 Astronomy Lab
57
Seasons
Part III. End of the Semester: Analyzing Your Results
By now, at the end of the semester, your collected data should provide ample evidence to explain
the cause of the seasonal change in temperatures.
III.1
Draw best-fit curves through your graphed datum points on the previous two pages. The
lines should reflect the actual trend of the data, but should smooth out the effects of
random errors or bad measurements.
Do you expect these downward or upward trends to continue indefinitely, or will they
eventually flatten out and then reverse direction? Explain your reasoning.
Now, use your graphs to review the trends you've observed:
III.2
Measured at noontime, did the Sun’s altitude become higher or lower in the sky during
the course of the semester?
On average, how many degrees per week did the Sun's altitude change?
III.3
Did the number of hours of daylight get greater or less?
On average, by how many minutes did daylight increase or decrease each week?
III.4
What was the maximum altitude of the Sun on the date of the equinox this semester?
(Consult the Celestial Calendar at the beginning of this manual for the exact date.)
On that date, how many hours of daylight did we have?
III.5
Did the “sunlight meter” indicate that we in Boulder received more or less solar intensity
at noon as the semester progressed?
III.6
Did the Sun’s apparent size grown bigger or smaller?
Does this mean that we are now closer to, or farther from, the Sun as compared to the
beginning of the semester?
III.7
Has the weather, in general, become warmer or colder as the semester progressed?
Which factor or factors that you've been plotting (solar altitude, solar intensity, number
of daylight hours, distance from the Sun) appear to be correlated with the change in
temperature?
Which factor or factors seems to be contrary to an explanation of the seasonal change in
temperature?
If you receive a bill from the power company, you’re aware that each kilowatt-hour of electricity
you use costs money. One kilowatt-hour (KWH) is the amount of electricity used by a 1000-watt
appliance (1 kilowatt) in operation for a full hour. For example, four 100-watt light bulbs left on
for 5 hours will consume 2.0 KWH, and will cost you about 15 cents (at a rate of $0.075/KWH).
Every day, the Sun delivers energy to the ground, free of charge, and the amount (and value) of
that energy can be measured in the same units that power companies use. The amount of energy
received by one square meter on the Earth directly facing the Sun is a quantity known as the solar
constant, which has a carefully-measured value of 1,388 watts/m2 . That is, a one-square-meter
solar panel, if aimed constantly towards the Sun, will collect and convert to electricity 1.388 KWH
of energy every hour (worth slightly more than a dime).
APS 1010 Astronomy Lab
58
Seasons
Each weekly Solar Intensity Chart contains all the information you need to find out how much
energy was delivered by the Sun, in KWH, on a typical day that week. Note that one “solarconstant hour” is equivalent to the rectangular area on the chart 1.0 intensity units high (the full
height of the graph, corresponding to a solar panel that always directly faces the Sun) and one hour
wide. The actual number of “solar-constant hours” delivered in a day to flat ground in Boulder is
likewise the total area under the plotted intensity curve, from sunrise to sunset.
A simple trick for measuring an irregularly shaped area is to calibrate and weigh the paper itself!
Use scissors to cut out a rectangular area corresponding to 10 solar-constant hours, and carefully
weigh it on an accurate gram scale. Divide the result by 10 to find out what one solar-constant
hour “weights”: __________ gram. Next, cut out the shape of the area under the curve of a Solar
Intensity Chart, and weigh it as well. Divide the latter weight by the former to determine the
number of of “solar-constant hours”. Finally, multiply that number by 1.388 KWH to obtain the
total energy contribution of the Sun to each square meter of ground during the course of a day.
III.8
As a class group exercise, determine the number of KWH delivered on a sunny day to
every square meter of ground in Boulder, for each week you’ve collected data. Using the
table on the previous page, record the weights in column (f), the calculated solar-constant
hours in column (g), and finally, the equivalent kilowatt-hours in column (h).
III.9
From column (h), what is the ratio of the amount of energy received at the end of the
semester compared to that at the beginning of the semester? (This ignores any effect due
to a change in the distance to the Sun.)
(Final Week' s KWH )
(First Week' sKWH )
= ___________
Fractional Change in SolarEnergy
A ratio greater than 1 implies than an increase in energy was received over the semester, while a
ratio smaller than 1 implies that less solar energy was delivered as the semester progressed.
Now let’s find out just how important was the change in distance from the Sun:
III.10
What is the ratio of the apparent diameter of the Sun between the end and the start of the
semester?
Diameter ratio =
(Final Week' s Diameter)
= __________
(First Week's Diameter)
The energy delivered to the Earth by the Sun varies inversely as the square of its distance from us.
The diameter ratio calculated above is already an inverse relationship (that is, if the diameter
appears bigger, the Sun’s distance is smaller) - so all we have to do is square that ratio to
determine the change in energy from the Sun caused by its changing distance from us.
For example, if the ratio is 1.10 (10% closer), the Sun will deliver 21% more energy (1.102 =
1.21). If the ratio is 0.90 (Sun 10% further away), it will deliver 0.902 = 0.81 = 81% as much
energy (or, if you prefer, 19% less energy).
III.11
How much more or less energy (in percent) does the Sun deliver to us now, compared to
the start of the semester, solely as a result of its changed distance?
If only the distance from the Sun caused the seasonal changes in temperature, would we
be warmer or colder in the wintertime? Explain your reasoning.
III.12
Compare the relative importance of the sunlight intensity-duration effect (III.9) with the
solar-distance effect (III.11) in producing seasonal changes in temperature. Which is, by
far, the most important? Explain clearly how you arrived at your conclusion.
APS 1010 Astronomy Lab
59
Motions of the Sun
MOTIONS OF THE SUN
SYNOPSIS: The objective of this lab is to become familiar with the apparent motions of the Sun
in the Boulder sky.
EQUIPMENT: A globe of the Earth, a light box.
Part I. Daily Motion of the Sun
The daily motion of the Sun is caused by the Earth’s rotation rather than the Sun actually moving
across the sky. (A subtle point! It took civilization thousands of years to figure this out!) The
diagram below show an overhead view of the Earth and Sun:
We will now follow a person (the stick figure) as the Earth rotates. At point ‘A’ our viewer is
rotated into view of the Sun, and the Sun appears to rise (i.e. sunrise!) Noon is defined as the
time when the viewer is located at point ‘B.’ This is the point where the Sun will be at its highest
point in the sky. At point ‘C’ our viewer is being rotated out of view of the Sun, and he or she
sees sunset. Midnight is defined as point ‘D’; the point opposite Noon. The Sun at points ‘A’,
‘B’ and ‘C’ as seen on Earth are shown below:
APS 1010 Astronomy Lab
60
Motions of the Sun
In the next part you will use the globe and light box:
I.1
Put the globe at one end of the table. At the other far end position the light box such that
it is illuminating the globe; the light box will simulate the Sun. Find Boulder on the
globe; and position the globe such that Boulder (latitude 40°, longitude 105°) is facing the
light box (at position ‘B’, i.e. Noon.) Orient the globe such that the Earth’s axis is
pointing 90° (in either direction) away from the light box. In a moment you will see that
this orientation represents the equinoxes; on these days the Northern and Southern
hemispheres each receive 12 hours of sunlight. What fraction of the Earth is illuminated?
I.2
At the time Boulder is at Noon:
• What major cities or small countries are experiencing sunrise and sunset? (pick
one for each.)
• What country is at the same latitude as Boulder but on the other side of the globe?
• What time is it there?
• What are the people of Tibet doing right now?
• Are the Honda Corporation offices in Tokyo likely to be open at this moment?
I.3
Rotate the globe such that Boulder is now at sunset. How much of the Earth is
illuminated now? What time is it now for the country on the other side of the globe?
Part II. The Annual Motion of the Sun
The position of the Sun in the sky also changes as the Earth orbits around the Sun. This motion is
not to be confused with the daily motion of the Sun described above! The Earth’s axis is tilted
23.5° with respect to the perpendicular of its orbital plane (see diagram below). As a consequence
different parts of the Earth receive different amounts of sunlight depending on where the Earth is in
its orbit. Note: the orbit is circular, but it is shown here in projection:
Due to the Earth’s tilt the northern hemisphere receives more light during the summer months; and
less light during the winter. The Sun also rises higher in the sky during the summer; and this
direct sunlight is more warming. These two effects cause the summer to be warmer than the
winter. The summer is not warmer than the winter because the Earth is closer to the Sun; in fact,
the Earth is slightly closer to the Sun during the Northern hemisphere's winter! The Sun’s path
across the summer and winter sky is shown on the next page. Note that the winter Sun does not
rise as high; nor is it up for as long:
APS 1010 Astronomy Lab
61
Motions of the Sun
If you were to measure the position of the Sun every day at Noon over the span of a year, you
would notice that it slowly moves up and down with the seasons. For example, on June 21st the
Sun at Noon is at its highest, northernmost position, while on December 22nd the Sun is at its
lowest, southernmost position. These days are called the summer and winter solstices
respectively. Solstice literally means “Sun stop”; on those two days the Sun appears to stop
moving and change directions (e.g. on the Summer solstice the Sun stops moving to the North and
starts to descend to the South.)
The illustration above shows the Earth at the summer solstice. Position your globe such that
Boulder is at Noon and it is the Summer solstice. From the globe and the illustration above answer
the following questions:
II.1
What direction do the stars rotate around Polaris, the “North Star”, as seen from the
Northern hemisphere. Do they go around clockwise or counter-clockwise?
II.2
Go to the country at the same latitude as Boulder but on the other side of the globe. What
country is this? What season is this part of the world in?
II.3
Now go back to Boulder. Follow the 105° meridian south to a latitude of 40° south.
What is there? You should find Easter Island just to the north. (Did you notice South
America is much farther east than the U.S.?) What season is this part of the world in?
APS 1010 Astronomy Lab
62
Motions of the Sun
II.4
Finally, find the antipode to Boulder - the point directly opposite on the globe, through
the center of the Earth. (Hint: It is not China!)
II.5
The Tropic of Cancer marks the northernmost points where the Sun can be seen at the
zenith (i.e. directly overhead) respectively. On the summer solstice anyone standing on
the Tropic of Cancer will see the Sun overhead at Noon. Is there anywhere in the United
States where you could see the Sun at the zenith?
II.6
The Arctic Circle marks the northern region of the Earth where the Sun is sometimes,
always (or never!) visible. Find the town of Barrow, Alaska. On the summer solstice at
what times will the Sun rise and set here?
Part. III The Southern Hemisphere
Without changing the orientation of the globe study what is happening “down under” in Australia.
III.1
On the Northern hemisphere's Summer solstice is the Sun to the North or South? What
season is Australia experiencing? Would this be a good time to ski the Snowy Mtns. in
New South Wales?
III.2
The Tropic of Capricorn marks the southernmost point at which the Sun can be seen
at zenith. Look at the town of Alice Springs in the center of Australia. Is the sun ever
directly overhead in Alice Springs? If so, when? (Refer to the dot marking the town.)
III.3
The Antarctic Circle is the southern equivalent of the Arctic Circle. On our summer
solstice what time does the Sun rise at the South Pole?
III.4
In Australia, do the stars rise in the east and set in the west? Which way do the stars
revolve around the south celestial pole?
Medieval Noon mark
from Bedos de Celles,
La Gnomonique
pratique (Paris, 1790),
a book on the use of
sundials. If you look
carefully at the shadows
you may notice that the
artist has made a
mistake.
APS 1010 Astronomy Lab
63
Motions of the Moon
MOTIONS OF THE MOON
SYNOPSIS: The objective of this lab is to become familiar with the motion of the Moon and its
relation to the motions of the Sun and Earth.
EQUIPMENT: A globe of the Earth, light box, styrofoam ball, pinhole camera tube.
Part I. The Moon’s Orbit
In the previous lab you learned how the time of day and the position of the Sun are related to the
Earth’s daily rotation. Now we will add the Moon and its orbit around the Earth. The diagram
below shows the overhead view of the Earth and Sun from before, but now the Moon has been
added:
The ‘A’, ‘B’,’C’ and ‘D’ positions on the Earth are the same as before. The Moon orbits the Earth
counter-clockwise (in the same direction the Earth rotates). Note that half of the Moon is always
illuminated (just like the Earth); even though it may not appear to be!
First let’s follow the Moon’s progression from new to full (from position #1 to #3). As the Moon
starts its orbit (at position #1) it is between the Earth and the Sun. This is a new Moon. The half
of the Moon illuminated by sunlight is facing away from us, and we would only see the dark side;
however, since the Moon appears close to the Sun in the sky at this time, we cannot see it because
the Sun is so bright. As the Moon moves in its orbit, it moves away from the Sun, and we start to
see the illuminated half of the Moon in the form of a crescent. At this time the Moon is called a
APS 1010 Astronomy Lab
64
Motions of the Moon
waxing crescent because each night the crescent appears to grow (“waxing” means “to grow”).
When the Moon reaches position #2 in its orbit half of it will be illuminated - this is 1st Quartercalled as such because the Moon has completed the first quarter of its orbit (a bit confusing since
half, not a quarter, of the Moon is illuminated!) As the Moon starts to move behind the Earth
(relative to the Sun; look at its orbit above) we see more and more of the illuminated half of the
Moon. This is the waxing gibbous (the “growing fat” Moon!) phase. Finally the Moon moves
behind the Earth and we see the entire illuminated face - this is full Moon (position #3). As you
can see from the progression above the Moon spends half of its orbit “growing” from the nonilluminated side to the fully illuminated side. As you will see in the next progression it spends the
second half of its orbit doing just the opposite:
After reaching full Moon at position #3 the Moon starts to move back in front of the Earth, and the
illuminated half begins to disappear. This is the waning gibbous phase. Note that right side of
the Moon is dark now; while before it was the left side that was dark. At position #4 only the left
half of the Moon is illuminated. The Moon continues on to complete its orbit, going through the
waning crescent phase, and then back to the new Moon.
You will now experience firsthand the phases of the Moon using the light-box and a ball.
I.1
Imagine that your head is the Earth; and that you live on the end of your nose. Position
the light-box such that your head is illuminated the same way as the Earth was in the
“Motions of the Sun” lab. Slowly turn your head around counter-clockwise to simulate
the daily cycle: sunrise, noon, sunset, midnight and back to sunrise This should give
you a feel of what direction in space you’re looking during the different times of the day.
I.2
Now hold the foam ball in your hand, with your arm extended and the ball at the height
of your head. This will simulate the Moon. Position the Moon (foam ball) such that it
appears next to the Sun (the light box); this is the new Moon. You should only see the
dark side of the Moon. Is it easy to see the Moon next to the bright light?
What time of day is it on your nose?
I.3
While holding the Moon at arm’s length move your arm counter-clockwise (i.e. to your
left). Keep your eyes on the Moon the entire time. Watch the phase of the Moon change
as you move; can you see the illuminated half of the Moon come into view? Stop turning
when you can see exactly half of the Moon illuminated (i.e. 1st quarter). Which side of
the Moon is being illuminated?
What is the angle between the foam ball, your head, and the light-box?
Does this agree with the diagram on the first page?
What time is it for you on Earth?
I.4
Continue to turn slowly to your left until the Moon is almost right behind you. You
should see that the Moon is nearly full - you are now looking at the half of the Moon that
is illuminated by the Sun. What time of day is it? If you move the Moon a little more
you’ll probably move it behind the shadow of your head. This is lunar eclipse - when
the shadow of the Earth covers the Moon.
APS 1010 Astronomy Lab
I.5
65
Motions of the Moon
Again continue to turn to your left, holding the Moon at arm’s length. Which side of the
Moon is now illuminated?
Move until the Moon is half illuminated (3rd quarter). What time is it now? Continue to
slowly move the Moon back to new.
Part II. Moonrise and Moonset
As you observed in the last exercise, there is a relationship between the position of the Moon in its
orbit and the time of day on Earth when you see it. When (i.e. what time of day) you can see the
Moon depends on where it is in its orbit; and as you just saw the position of the Moon in its orbit
also determines the Moon’s phase.
Now use the globe of the Earth and the light-box as before in the “Motions of the Sun” lab; but
now include the foam ball to simulate the Moon. Position the globe such that it is the summer
solstice. Hold the ball and move it around the Earth to simulate the lunar orbit.
II.1
Position the globe such that Boulder is at noon and the Moon is at position #1 in its orbit.
What phase is the Moon?
At what time did the Moon rise; and when will it set? (Hint: You can rotate the Earth to
see where Boulder will be when the Moon is on the horizon.)
II.2
Position Boulder back at noon but now move the Moon to position #2. What is the
Moon’s phase now?
Where in the sky will the Moon appear to be from Boulder?
Where in the sky does the Moon appear to be in the country of Kyrgyzstan (remember
this place?)?
II.3
Now put the Moon at position #3. What phase is it?
At what time will it rise? (Hint: Think about what position Boulder would have to be at
on the Earth to see it on the eastern horizon.)
At what time will the Moon set? (This should make sense, as the Moon is on the
opposite side of the Earth relative to the Sun.)
What phase of the Moon do people in the southern hemisphere see right now?
II.4
Does the time of moonrise or moonset depend on where your are on the Earth? e.g. does
the Moon rise at the same local time for Boulder and Kyrgyzstan?
II.5
Does the Moon rise earlier or later each night?
The Moon completes one lunar orbit in 27.3 days. In this time the moonrise (and
moonset) time will have changed by 24 hours - which means it will appear to be right
back to where it started. How much earlier or later does the Moon rise each night?
II.6
The Moon is not the only Solar System object which shows phases - the planets do as
well. You’ve learned that the Moon shows a crescent phase when it is between the Earth
and Sun. Which of the planets can show a crescent phase? Why is this?
II.7
Even though it is usually regarded as a nighttime object, you can see the Moon during the
day! Go outside and check to see if the Moon is visible. If so, hold up the foam ball at
arm’s length in the direction of the real Moon. The foam ball should have the same phase
as the Moon! Explain why this is so.
APS 1010 Astronomy Lab
II.8
66
Motions of the Moon
What phase is the Moon today?
What phase will it be during next week’s lab?
Part III. The Size of the Sun and Moon
Warning! Looking directly at the Sun can damage your eyes!
In order to measure the diameter of the Sun we use a pinhole camera - a cardboard tube with a
piece of foil at one end with a tiny hole; the other end of the tube is covered in graph paper. Being
careful not to look at the Sun directly, aim the tube up to the Sun with the graph paper facing you.
You should see an image of the Sun on the graph paper.
III.1
Measure the diameter of the Sun’s image, counting the marks on the graph paper. Each
square on the graph paper is 1 mm across. What is the diameter of the Sun's image?
Distance from Sun to Earth
The Sun
Distance from
Hole to image
Pinhole Camera
Image of
The Sun
Examining the geometry of the pin-hole camera shown in the figure above, you can see that if the
distance to the Sun is known, we can use the similar triangles relationship to solve for the diameter
of the Sun. That is, we have the following:
III.2
Diameter of the Image = Distance of Image from Hole
Diameter of Sun
Distance of Sun from Earth
Assume that the distance between the Sun and Earth is 150 million kilometers (about 93
million miles). The distance of the image from the hole (i.e. the length of the tube) is 1
meter. Determine the diameter of the Sun and compare this to the true value.
The Moon has roughly the same angular size as the Sun. We can use the similar triangles
relationship again to determine the size of the Moon:
Diameter of the Moon = Distance of Moon to Earth
Diameter of Sun
Distance of Sun to Earth
III.3
If the distance of the Moon to the Earth is about 390 times less than the distance between
the Sun and the Earth, how big is the Moon? Check to see if your number makes sense:
the Moon's diameter is a little less than the width of the United States.
APS 1010 Astronomy Lab
67
Motions of the Moon
Part IV. Solar and Lunar Eclipses
Nerds watching a Solar eclipse.
Since the Sun and Moon have roughly the same angular size, it is possible for the Moon to block
out the Sun as seen from Earth, causing a solar eclipse.
IV.1
Hold the styrofoam Moon at arm’s length and move it through its phases; there is only
one phase of the Moon when it is possible for it to block your view of the Sun, causing a
solar eclipse. Which phase is this?
Now simulate this arrangement with the styrofoam Moon, the Earth globe, and the lightbox. Does the Moon eclipse the entire Earth, or does only a portion of the Earth lie in
the Moon’s shadow?
Does this mean that all people on the sunlit side of the Earth can see a solar eclipse when
it happens, or only some people in certain locations?
The Moon's orbit is not quite circular. The Earth-Moon distance varies between 56 and 64 Earth
radii. If a solar eclipse occurs when the Moon is at the point in its orbit where the distance from
Earth is “just right”, then the Moon's apparent size exactly matches the Sun’s. This produces a
very brief total eclipse - perhaps lasting only a few seconds.
IV.2
If the Moon were at its closest point to the Earth during a solar eclipse, would it appear
bigger or smaller than the Sun as seen from Earth?
How would this effect the amount of time that the eclipse could be observed?
If an eclipse occurred when the Moon was at its farthest distance from the Earth, describe
what the eclipse (called an annular eclipse) would look like from the Earth.
Balinese Hindus call Rahu, the eclipse demon, Kala Rau. In this
scene Kala Rau is losing his head as he is discovered sipping
from the elixir of immortality. In anger, he circles the sky in
pursuit of the two informers- the Sun and Moon. When he
catches either of them, his severed immortal head swallows
them and an eclipse takes place. Then the ingested Sun or
Moon drops out of Kala Rau’s severed throat; and the eclipse is
over.
APS 1010 Astronomy Lab
68
Motions of the Moon
It’s also possible for the shadow of the Earth to block the sunlight reaching the Moon, causing a
lunar eclipse.
IV.3
Once again, move the Moon through its phases around your head and find the one phase
where the shadow of the Earth (your head) can fall on the Moon, causing a lunar eclipse.
What phase is this?
During a lunar eclipse, is it visible to everyone on the (uncloudy) nighttime side of the
Earth, or can only a portion of the people see it? (Hint - could anyone “standing on your
face” see the Moon in shadow?)
Many people think that a lunar eclipse should occur every time that the Moon is full. This
misconception is due to the fact that we usually show the Moon far closer to the Earth than it
actually is, making it appear that an eclipse is unavoidable. But as mentioned above, the Moon’s
actual distance is roughly 30 Earth-diameters away.
IV.4
Hold the styrofoam Moon at its true (properly-scaled) distance from the globe of the
Earth. How big of an area would the shadow of the Earth be at this distance?
Demonstrate how the Moon can easily pass over or under this shadow during its orbit, so
that an eclipse does not occur.
A lunar eclipse over Tashkent, in what is now Uzbek S.S.R., was
witnessed on 16 December 1880. With noise of cymbals, tambourines
and drums they attempted to drive away whatever it was that was
attacking the Moon.
APS 1010 Astronomy Lab
69
Gravity
GRAVITY AND THE MASS OF THE EARTH
“To see the world in a grain
of sand,
And a heaven in a wild flower;
Hold infinity in the palm of
your hand,
And eternity in an hour.”
-W. Blake
SYNOPSIS: By measuring the time taken for an object to fall through a distance, you will
determine the acceleration due to Earth's gravity, and will use that information to calculate the mass
of the Earth. Using this information, you will compare the Earth’s gravity with that on the Moon
and on Jupiter.
EQUIPMENT: Falling body apparatus, tape measure, calculator.
Introduction
Acceleration is the change in speed (also called velocity) that a moving body experiences in a
given time interval. For example, suppose that an object starts at rest and accelerates at a rate of 10
meters per second per second (written m/sec2 ). After the first second, it will be traveling at a speed
of 10 m/sec. After the next second, its speed will be 10 m/sec plus 10 m/sec, or 20 m/sec, etc.
So, for constant acceleration of an object starting at rest,
speed = (acceleration) (time).
(1)
Because an accelerating object’s speed is constantly increasing, the distance it covers each second
is accordingly greater. For an object starting at rest and undergoing constant acceleration,
1
distance = 2 (acceleration) (time) 2.
(2)
Thus, an object accelerating at 10 m/sec 2 will travel a distance of 5 meters in the first second of
time; two seconds after starting, the distance covered is 20 meters, implying that the object
travelled 15 meters during the second second.
Every particle in the universe exerts a gravitational force on every other particle. A force gives rise
to an acceleration. The combined effects of all of the particles in the Earth acting upon us produces
an acceleration which we constantly experience; we refer to this acceleration simply as gravity, and
we call the force on our bodies weight..
If we drop an object and time how long it takes to fall a certain distance, we can use equation (2) to
measure the acceleration due to gravity:
acceleration = 2
(distance)
(time)2
(3)
APS 1010 Astronomy Lab
70
Gravity
Part I. Measuring the Acceleration due to Gravity
The falling-body apparatus uses a spring-loaded clamp to hold a steel ball above a landing pad.
When we release the clamp, the ball drops and the timer starts. The timer stops when the ball
strikes the landing pad, and the elapsed time in seconds appears on the readout.
Positioning
Clamp
Green
Pull
Knob
Dropping
Mechanism
Timer
Table
Red
Clamping Knob
Ball
Landing Pad
Floor
Work in a team of three individuals with one dropping apparatus. Measure the acceleration due to
gravity using the small steel ball as described below. After one individual has completed his/her
measurements, the other team members should independently conduct their own trial
measurements (step I.2).
I.1
I.2
Position the dropping mechanism over the edge of the table, and the landing pad (inside
the catch basin) on the floor directly underneath it. Switch the timer on.
•
While holding the small steel ball in the clamp of the dropping mechanism, pull on the
green knob (against the spring tension) so that the ball is captured. Tighten the red
clamping knob to hold the ball in place.
• Measure the drop-distance to the nearest millimeter: Use a metric tape measure to
determine the distance from the bottom of the ball to the top of the landing pad (another
team member may assist by holding one end of the tape, but you must actually read the
distance). Record your distance measurement in meters (1 mm = 0.001 meter).
• Press the reset button on the timer to initialize the display.
• Release the ball by loosening the red knob. Record the elapsed time in seconds as
indicated by the timer.
I.3
Write down each individual's name and his/her values for distance and time in Table 1.
For each individual trial, calculate the gravitational acceleration experienced by the small
ball using equation (3), and record the results in the table.
APS 1010 Astronomy Lab
Table 1:
Experimenter
Names
71
Small Ball
Distance
(meters)
Gravity
Long Distance
Time
(seconds)
Acceleration
(m/sec 2 )
Average
Presumably, each of you obtained similar, but not necessarily identical, values for the drop
distance, the elapsed time, and the calculated acceleration for each experimental trial. If the
measurements were not identical, it does not mean that the distance, time, or acceleration actually
changed between trials, nor does it necessarily imply that one or more of you made a mistake. The
results simply point out that there is no such thing as a perfect measuring instrument or procedure,
and therefore there is no such thing as a perfect measurement!
Since every measurement (distance, time) contains an experimental error (meaning an
uncertainty, not necessarily a mistake), then any information derived from those measurements
(acceleration) also contains an uncertainty. We can never know exactly what the "true" value is;
we can only try to get better and better results by improving our measuring equipment or
technique, and by averaging the results of our measurements to statistically reduce any random
errors that may have occurred. Note: You cannot claim to know the value of the acceleration to
any greater accuracy than the measurement error. It is senseless and misleading to list its value to
place-values smaller than the uncertainty. For example, if your calculator comes up with a figure
of 9.3487851 m/sec2 , while your individual trials ranged from 9.342 to 9.361 m/sec2 , you should
list your result as 9.35 m/sec2 , with an uncertainty of about ±0.01 m/sec2 . It may not look as
impressive, but it's honest!
I.4
Suggest at least one way in which the measuring equipment or your measurement
technique might be improved to produce a smaller experimental error.
I.5
Calculate an improved measurement for the gravitational acceleration by averaging all
three of your team's individual values, and enter the results in Table 1.
Aristotle claimed that heavier objects fall faster than ligher objects (that is, heavy objects
experienced a greater gravitational acceleration). Nearly two thousand years later, Galileo was the
first to dispute this belief, claiming that all objects fall at the same rate (and asserted that air friction
was responsible for the seemingly slow drop of light objects). According to lore, Galileo put his
hypothesis to the test by dropping balls of different weights off the leaning tower of Pisa.
I.6
Test Galileo's hypothesis. Repeat procedure I.2 using the large steel ball instead of the
small one. Once again, each member should make an independent distance measurement
(slightly different from the previous, since the size of the ball has changed), time
measurement, and calculation of the acceleration. Record all of your teammates results in
Table 2, and find the average acceleration.
APS 1010 Astronomy Lab
Table 2:
Experimenter
Names
72
Large Ball
Distance
(meters)
Gravity
Long Distance
Time
(seconds)
Acceleration
(m/sec 2 )
Average
I.7
Within your uncertainty of the measurements, does your experimental data support
Galileo's contention that the acceleration due to gravity is the same for all objects?
Explain.
Since both balls fell nearly equal distances, your measurements haven’t tested whether gravitational
acceleration was a constant value throughout the drop, but merely that it was the same for both
objects (it could have changed identically for both balls during the time that they fell).
I.8
Determine whether the acceleration due to gravity is a constant that is independent of the
drop-distance or duration of the time-of-flight: Reposition the dropping apparatus and
landing pad to perform a shorter drop to the tabletop as shown in the diagram. Using the
large ball, perform independent measurements as before, and record and average your
team's results in Table 3.
Table 3:
Experimenter
Names
Large Ball
Distance
(meters)
Short Distance
Time
(seconds)
Acceleration
(m/sec 2 )
Average
I.9
Within your uncertainty of the measurements, do your results support the contention that
all objects experience the same, constant acceleration due to Earth's gravitational
attraction? Why or why not?
I.10
Average all nine of your separate measurements together (or equivalently, average the
three averages) to come up with a final, best estimate of its value.
APS 1010 Astronomy Lab
73
Gravity
Part II. The Mass of Earth
Isaac Newton showed that the acceleration g experienced by an object due to the gravitational
attraction of a second body is directly proportional to the mass m of the attracting body, and
inversely proportional to the square of the distance r between the bodies (one form of Newton's
universal law of gravitation):
m
g = G 2 .
r
(4)
Here G is the constant of proportionality, called the gravitational constant. (The value of G
was determined very carefully by measuring the gravitational force between two large blocks of
metal hung very close together from wires.) Essentially, G converts the mass and the distance to
an amount of gravitational acceleration. In meters-kilograms-second (mks) units,
G = 6.670 x 10-11
m3
kg· sec 2
(5)
Important: When equation (4) is used with the above value for G, the mass m must be in
kilograms and r in meters; otherwise, any numerical values you obtain will have nonsense units.
Identical
Gravitational
Acceleration
One
Earth
Mass
M
E
Earth
Mass = M E
One
Earth
Radius
R
E
Using calculus (which he invented to enable him to solve his mathematical problems), Newton also
showed that the mass of a spherically-symmetric object, such as the Earth, gravitationally attracts
other objects as if all of its mass were concentrated at its center. Thus, the acceleration produced
by Earth's gravity is the same as that produced by a tiny object having the same mass as the Earth
(ME), and which is located at the center of the Earth a distance of one Earth radius (RE) away.
The Earth's radius is RE = 6,400 km = 6.4 x 106 meters. Since you have already measured the
acceleration due to gravity, g, we can re-write equation (4) to solve for the mass of the Earth:
2
g RE
ME = G
.
(6)
II.1
Calculate the mass of the Earth using equation (6) and your team's average value for the
acceleration of gravity.
II.2
Compare your result with the accepted value for the mass of the Earth: ME = 5.976 x
1024 kg. Is your result accurate to 50%? 10%? 1%? Is your result too large or too
small?
II.3
If you were to repeat this experiment on a mountaintop, would the acceleration due to
Earth's gravity be less than, equal to, or greater than that which you have measured?
Explain (hint: which parameters would be different in equation 4)?
APS 1010 Astronomy Lab
II.4
74
Gravity
Can you think of any reason(s) why your measurement made in Boulder might have
given results that were higher or lower than the "official value"?
Part III. Gravity on Other Planets
Newton's law of gravity (Equation 4) applies to everything and everywhere in the universe, not
just on the Earth (that’s why they call it the “universal” law of gravitation!).
III.1
Calculate the gravity (ie. acceleration) felt by the Apollo astronauts as they walked on the
Moon. To do this you'll need the mass of the Moon, 7.35 x 1022 kg, and its radius,
1738 km. Compare this acceleration to that here on Earth (don’t forget to convert from
kilometers to meters!). Does this explain why the astronauts "bounced" as they walked?
III.2
Now, find the “surface” gravity for Jupiter. Jupiter's mass is 1.9 x 1027 kg, and its
radius is 71,398 km. Aside from being squashed into a pancake in mere seconds, what
other difficulties would we face if we tried to stand on Jupiter's “surface”?
Part IV. Escape Velocity
When vehicles are launched into space, the rocket must have a minimum speed at launch to escape
the Earth's gravity. We call this velocity the “escape velocity” (vesc). If the rocket is moving
slower than this velocity, it will fall back to the Earth. From the equations given above, we can
calculate the escape velocity from any object, given by the following equation:
vesc =
2GM
R
(7)
“G” is the gravitational constant given in equation (5), and M and R are the mass and radius of the
object you are trying to “escape.”
IV.1
Calculate the velocity needed for a spacecraft to escape from the Earth. (Note that the
escape velocity does not depend on the mass of the spacecraft itself!)
IV.2
Now calculate the escape velocity for the Moon. (This should explain why the Apollo
astronauts needed only a small vehicle to leave the Moon’s surface and return to lunar
orbit, while the Saturn-V rocket used to lift everything off of the face of the Earth was so
huge!)
IV.3
How fast would your spacecraft have to travel to escape from Jupiter?
Individual atoms and molecules of gas in a planet’s atmosphere also have to reach these same
escape velocities in order to “leak off” into space; otherwise, they stay trapped close to the planet’s
surface, just like us.
IV.4
Use the concept of escape velocity to explain why the Moon has virtually no atmosphere,
Earth has a moderate amount, and Jupiter has so much.
APS 1010 Astronomy Lab
75
Kepler's Laws
Kepler's Laws
SYNOPSIS: Johannes Kepler formulated three laws that described how the planets orbit around
the Sun. His work paved the way for Isaac Newton, who derived the underlying physical reasons
why the planets behaved as Kepler had described. In this exercise, you'll use computer
simulations of orbital motions to experiment with the various aspects of Kepler's three laws of
motion.
EQUIPMENT: Computer with internet connection to the Solar System Collaboratory.
Getting Started
Here's how you get your computer up and running:
(1)
In the Computer Lab, make sure your computer is on the PC side (if on the Macintosh
side, simultaneously press the FLOWER and RETURN keys to bring up the PC).
(2)
Click on the Internet folder (not IE!).
(3)
Launch Netscape by double-clicking its icon.
(4)
Do not use "maximized" windows - if you don't see the "desktop" in the background,
click on the double-window button at the upper right of the Netscape window.
(5)
Go to the website http://solarsystem.colorado.edu/cu-astr
(6)
Click on "Enter Website - Low Resolution (1024x768)".
(7)
Click on the Modules option.
(8)
Click on Kepler's Laws.
Note: We intentionally do not give you "cook-book" how-to instructions, but instead allow you to
explore around the various available windows to come up with the answers to the questions.
But note: use the "applets" on the MAIN window, not the EXTRA window. If you use the
EXTRA window, you'll find that the HELP, HINT, and MATH information will be referring to
the wrong page.
The window placement is designed to facilitate access with one click of the mouse.
maximizing or moving windows; otherwise, it will only make your life harder!
Avoid
APS 1010 Astronomy Lab
76
Kepler's Laws
Part I. Kepler's First Law
Kepler's First Law states that a planet moves on an ellipse around the Sun.
I.1
Where is the Sun with respect to that ellipse?
I.2
(a) Could a planet move on a circular orbit?
(b) If your answer is "yes", where would the Sun be with respect to that circle?
I.3
What is meant by the eccentricity of an ellipse? Give a description (words, not formulae).
I.4
What happens to the ellipse when the eccentricity becomes zero?
I.5
What happens to the ellipse when the eccentricity becomes one?
I.6
On planet Blob the average global temperature stays exactly constant throughout the
planet's year. What can you infer about the eccentricity of Blob's orbit?
I.7
On planet Blip the average global temperature varies dramatically over the planet's year.
What can you infer about the eccentricity of Blip's orbit?
I.8
For an ellipse of eccentricity e = 0.9, calculate the ratio of periapsis to apoapsis (you may
want to look up periapsis and apoapsis in the ``hints'' section). Use the tick-marks to
read distances directly off the screen (to the nearest half-tick).
I.9
What is the ratio of periapsis to apoapsis for e = 0.5?
I.10
For e = 0.1?
APS 1010 Astronomy Lab
77
Kepler's Laws
The following questions pertain to our own Solar System. Remember that the orbits of the
different planets are not drawn to scale. We have scaled the diagram to the major axis of each orbit.
I.11
Which of the Sun's planets has the largest eccentricity?
I.12
What is the ratio of perihelion to aphelion for this planet?
I.13
Which planet has the second largest eccentricity?
I.14
What is the ratio of perihelion to aphelion for that planet?
I.15
If Pluto's perihelion is 30 AU, what is its aphelion?
I.16
If Saturn's perihelion is 9.0 AU, what is its aphelion?
Part II. Kepler's Second Law
Kepler's Second Law states that, for each planet, the area swept out in space by a line connecting
that planet to the Sun is equal in equal intervals of time.
II.1
For eccentricity e = 0.7 count the number of tick-marks on the speedometer between the
speed at periapsis and the speed at apoapsis.
II.2
Do the same for e = 0.4.
II.3
And again for e = 0.1.
II.4
For eccentricity e = 0.7, measure the time the planet spends
(a) to the left of the vertical line (minor axis): __________ .
(b) to the right of the vertical line: __________ .
II.5
Do the same again for eccentricity e = 0.2.
(a) to the left: __________ .
(b) to the right: __________ .
II.6
Where does the planet spend most of its time, near periapsis or near apoapsis?
APS 1010 Astronomy Lab
78
Kepler's Laws
Part III. Kepler's Third Law
Kepler's Third Law states the relationship between the size of a planet's orbit (given by itse semimajor axis), and the time required for that planet to complete one orbit around the Sun (its period).
III.1
The period of Halley's comet is 76 years. From the graph, what is its semi-major axis?
III.2
By clicking on the UP and DOWN buttons, run through all the possible combinations of
integer exponents available (1/1, 1/2, …, 1/9; 2/1, 2/2, …, 2/9; 3/1, 3/2, … 3/9; etc).
Which combinations give you a good fit to the data?
III.3
III.4
Using decimal exponents, find the exponent of a (the semi-major axis) that produces the
best fit to the data for the period p raised to the following powers:
(a) p0.6
__________
(b) p5.4
__________
(c) p78
__________
Why do you think Kepler chose to phrase his third law as he did, in view of the fact that
there are many pairs of exponents that seem to fit the data equally well?
Part IV. "Dial-an-Orbit" Applet
Here's where you can "play god": create your own planet, and give it a shove to start it into orbit.
IV.1
Start the planet at X = -80, Y = 0.
(a) Find the initial velocity (both X- and Y-components) that will result in a circular
orbit (use the tick marks to judge whether the orbit is circular):
V x = __________
V y = __________ .
(b) Using the clock, find the period T of that orbit:
T = __________ .
IV.2
Now start the planet at X = -60, Y = 0.
(a) Find the velocity that will result in an elliptical orbit of semi-major axis = 80
(attention: remember the definition of the semi-major axis!).
(b) Use the clock to find the period for that orbit.
IV.3
Would you expect the period you measured in question IV.2 to be the same as the period
you measured in question IV.1? Why?
APS 1010 Astronomy Lab
79
Temperature of the Sun
MEASURING THE TEMPERATURE OF THE SUN
SYNOPSIS: In this lab you will measure the solar flux, the amount of energy per unit area per
unit time that reaches the Earth from the Sun. From this you will calculate the temperature of the
surface of the Sun.
EQUIPMENT: Insulated cup, water, ink, plastic wrap, thermometer, watch, meter stick, gram
scale, graph paper, calculator.
Review: units (p. 11), temperature (p. 13), energy & power (p. 13), trig (p.19)
Introduction
The quantity of energy delivered by sunlight, per unit of area and per unit of time, is called the
energy flux F (in units of joules per square meter per second). If we multiply the flux F by the
area A onto which the light falls, and also multiply by the duration of time ∆ t that the area is
exposed to the light, we obtain the total amount of energy that has been delivered by the Sun:
Energy delivered by Sun = F . A . ∆t
.
The temperature increase ∆ T resulting from the sunshine depends upon two properties of the
absorbing material: its heat capacity C, and its total mass M. The more mass there is to be
heated, the less the temperature will rise for a given amount of energy. The heat capacity states
how much energy is needed to raise the temperature of a kilogram of the material by one degree;
for example, the heat capacity of water is
Cwater = 4,186
joules
,
kg . oC
meaning that it takes 4,186 joules of energy to raise the temperature of one kilogram of water by
one degree Celsius.
The amount of energy needed to produce a temperature increase of ∆ T in a material of mass M and
heat capacity C is then
Energy absorbed from the Sun = C . M . ∆ T
.
If all of the energy delivered by sunlight is completely absorbed, the two quantities above are
equal:
Energy delivered by Sun = Energy absorbed from Sun
F . A . ∆t
=
C . M . ∆T
The change in temperature of the material ∆T is then:
∆T =
F A ∆t
C M
Note that the increase in temperature is directly proportional to the rate F at which energy arrives,
the size of the energy-absorbing area A, and the duration of exposure ∆ t to the energy; the
APS 1010 Astronomy Lab
80
Temperature of the Sun
temperature increase is inversely proportional to C, the ability of the material to store energy
internally, and to the amount of material M that must be heated.
Of course, we don't need to calculate how the temperature of an object exposed to sunlight will
increase; all we need is a thermometer. But we can rearrange the equation to find out how to
measure the really difficult quantity: F, the energy per unit area per unit time arriving from the Sun:
F =
C M ∆T
.
A
∆t
(1)
Part I. Gathering Data
In this experiment, you will place a container of water out in the sunshine, and use a thermometer
to measure the increase in the water temperature ∆ T that occurs over a time interval ∆ t. You
already know the heat capacity of the water, Cwater. A scale will be used to determine the water's
mass M, and a meter stick can be used to determine the surface area A of the water. These bits of
information can then be combined in equation (1) to yield the solar flux F, the number of joules of
energy falling each second onto each square meter of the illuminated surface of the Earth.
Prepare your experimental apparatus:
I.1
Measure the inside diameter D of the top of the insulated cup in meters, and calculate its
surface area A cup in square meters:
D2
Acup = π 4
.
(2)
The water you use should start out a few degrees cooler than the outside (ambient) air temperature,
since the temperature increase will be the most accurate when the air and water temperatures are
approximately the same.
I.2
Measure the outside air temperature with your thermometer and measure the temperature
of the water. If the water is too warm, chill it with ice cubes or use water pre-chilled by
your teaching assistant.
I.3
Weigh the insulated empty container and write down its mass (in kilograms). Fill the cup
roughly three-fourths full with water, and add about three drops of ink to the water to
help it absorb energy more efficiently.
I.4
Weigh the container again, and calculate the mass of the water Mwater in kilograms.
Cover the top of the container with plastic wrap to help reduce heat losses from evaporation, and
poke the thermometer through the top. Carry the works outside to a sunny, level area protected
from wind.
You are now ready to monitor how the incident solar energy affects the temperature of the water.
APS 1010 Astronomy Lab
81
Temperature of the Sun
Solar
Energy
Flux F
Zenith
Angle
θ
θ
Watch
tan (θ ) = S/L
Thermometer
L
Meter
Stick
Effective
Area A
Mass
M
Shadow
Cup
Area
S
I.5
As the sunlight warms the water, keep a running record of the temperature T of the water
versus elapsed time t. Make a temperature reading once every minute. Between
readings, it is good practice to gently stir the water with the tip of the thermometer to keep
the temperature uniform. Continue to take measurements until the water is a few degrees
warmer than the ambient temperature; you will need about one-half hour of continuously
clear skies.
I.6
You will also need to know the zenith angle θ of the Sun (how many degrees it is from
directly overhead). The angle can be determined by measuring the shadow cast by the
meter stick L: hold the meter stick vertically and mark where the tip of its shadow falls
on the ground; then use the stick itself to measure the length S of its shadow. The
tangent of the zenith angle will be
S
tan(θ) = L .
(3a)
Note: If you’ve already measured the solar altitude for the Season’s lab, the zenith angle
is simply:
Zenith Angle θ = 90° - (solar altitude)
(3b)
You're through gathering data. Carry everything back inside.
Part II. Calculating the Solar Energy Flux
If the Sun had been directly overhead (which is not possible from Boulder's latitude), the energyabsorbing area of the water would equal the area of the cup A cup. However, the oblique angle of
the sunlight reduced the effective area A by an amount equal to the cosine of the solar zenith angle:
A = Acup . cos(θ)
II.1
(4)
Calculate the zenith angle θ from equation (3a) or (3b), and calculate the effective surface
area A using equation (4).
APS 1010 Astronomy Lab
II.2
82
Temperature of the Sun
On the graph paper at the back of this exercise, graph your temperature measurements in
degrees Celsius against the elapsed time in seconds, as shown below. (Don’t forget to
convert minutes into seconds!)
x
x
x
x
x
x
∆T
x
x
Temperature
(degrees C)
x
x
Ambient
Temperature
x
x
x
x
x
∆t
Time (Seconds)
During the period when the water temperature was close to the ambient temperature, the graph
should approximate a straight line, indicating a uniform rate of temperature increase.
II.3
Draw a best-fit straight line through this portion of the graph. Determine the slope of this
line: find the change in temperature ∆ T that occurs over the time interval ∆ t.
II.4
Use your experimentally-determined values for the quantities on the right-hand side of
equation (1) to calculate the solar flux F. If you have properly kept track of the units in
each of your parameters, the solar flux will have units of joules per square meter per
second (joules/(m2 . s)).
The watt is a unit of power, or energy per unit of time. One watt is equal to 1 joule per second, so
that your measurement of the solar flux in "joules/(m2 . s)" can also be written as "watts/m2". The
meaning of your measurement of the solar flux is then easier to visualize: it is the power in watts
delivered by the Sun to each square meter of the Earth's surface.
The solar constant E is the name given to the solar energy flux arriving just outside the Earth's
atmosphere. A portion of that energy is absorbed by the atmosphere (after all, the air is heated by
sunlight just like the water in the cup!), so your measurement of F will be somewhat smaller than
the value of E. The amount of energy loss in the atmosphere depends upon the zenith angle θ of
the Sun and the amount of water vapor in the atmosphere.
APS 1010 Astronomy Lab
II.5
83
Temperature of the Sun
Use the chart below to estimate what fraction of the incident solar radiation actually
reached the Earth's surface during your experimental measurements. Estimate whether
the sky was extremely clear, average, or hazy; then use the corresponding curve and your
measured solar zenith angle to determine the atmospheric transmission X.
Atmospheric Transmission X
Clear
0.8
Average
0.7
Hazy
0.6
0
10
20
30
Zenith Angle
II.6
40
θ
50
60
(Degrees)
Calculate the solar constant E by dividing your measurement of the flux F below the
atmosphere by the atmospheric transmission X :
F
E = X
II.7
70
(5)
Compare your calculated value for the solar constant with the generally accepted value of
1,388 watts/m2 (that's equivalent to nearly fourteen 100-watt light bulbs shining onto an
area 3 feet by 3 feet!). If your value is significantly different, suggest possible sources of
error.
Part III. The Temperature at the Surface of the Sun
Your calculated value of the solar constant E measured the number of watts striking one square
meter of surface at a distance of one astronomical unit (1.5 x 10 11 m) from the Sun. If you
multiply the energy hitting one square meter of the Earth by the total number of square meters on a
sphere of radius 1 AU (reminder: the area of a sphere of radius R = 4π R2), you obtain the
luminosity, or total power output in watts emitted by the Sun!
III.1
Calculate the luminosity of the Sun in watts:
Luminosity = E x 4π(1.5 x 1011 m)2
.
(6)
Compare your measurement of the total power output from the Sun with the value given
in your textbook.
APS 1010 Astronomy Lab
84
Temperature of the Sun
Earth
Surface Area
of Sun
r = 1 AU
E = Solar Constant =
Flux at Earth
R
Sun
Surface Area
of Sphere
of Radius 1 AU
E sun =
Flux at Sun Surface
Now that you know how much total energy the Sun puts out, you can also determine how much
energy Esun is emitted by each square meter of the solar photosphere! Since the radius of the Sun
is about 7.0 x 108 m, we divide the total energy output by its surface area:
Esun =
Luminosity
4π(7.0 x 10 8 m) 2
.
(7)
III.2
Calculate the power per unit area Esun, (watts/m2) leaving the Sun's surface.
III.3
How many 100-watt lightbulbs would be required to equal the energy output from each
square meter of the solar photosphere?
The Sun radiates energy according to the Stefan-Boltzmann law: the power radiated per unit area
(Esun) is proportional to the fourth power of its absolute temperature T measured in degrees
Kelvin:
Esun = σ T4
where σ (called the Stefan-Boltzmann constant) is known from experiment to be 5.69 x 10-8
watts/(m2 K4).
III.4
Calculate the temperature of the Sun's photosphere in degrees Kelvin:
T = (Esun/σ)1/4
(8)
Does your answer make sense? Compare your measured value with the accepted average
solar photosphere temperature of 5800 K.
Temperature (degrees C)
Elapsed Time (seconds)
Solar Heating of a Cup of Water: Temperature versus Time
APS 1010 Astronomy Lab
85
Temperature of the Sun
Temperature (degrees C)
Elapsed Time (seconds)
Solar Heating of a Cup of Water: Temperature versus Time
APS 1010 Astronomy Lab
86
Temperature of the Sun
APS 1010 Astronomy Lab
87
Spectroscopy
SPECTROSCOPY
“I ask you to look both ways. For the road to a knowledge of the starts leads through the atom;
and important knowledge of the atom has been reached through the stars.” - A.S. Eddington
SYNOPSIS: In this lab you will explore different types of emission spectra and use a
spectroscope to identify an unknown element. You will also observe and analyze the spectrum of
the Sun using the observatory's heliostat to study the Sun's composition.
EQUIPMENT: Hand-held spectroscope; spectrum tube power supply and stand; helium, neon,
nitrogen, air, and "unknown" spectrum tubes; incandescent lamp; and the heliostat.
WARNING: There is high voltage on the spectrum tube. You can get a nasty shock if you touch
the ends of the tube while it is on. Also, the tubes get hot and you can burn your fingers trying
to change tubes. Let the tubes cool or use paper towels to handle them.
Most of what astronomers know about stars, galaxies, nebulae, and planetary atmospheres comes
from spectroscopy, the study of the colors of light emitted by such objects. Spectroscopy is
used to identify compositions, temperatures, velocities, pressures, and magnetic fields.
An atom consists of a nucleus and electrons which “orbit” around the nucleus. An atom emits
energy when an electron jumps from a high-energy orbit around the nucleus to a smaller lowenergy orbit. The energy appears as a photon of light having an energy exactly equal to the
difference in the energies of the two electron levels. A photon is a wave of electromagnetic
radiation whose wavelength (distance from one wave crest to the next) is inversely proportional to
its energy: high-energy photons have short wavelengths, low-energy photons have long
wavelengths. Since each element has a different electron structure, and therefore different electron
orbits, each element emits a unique set of spectral lines.
Electron
Small Electron Transition
Medium Electron Transition
Nucleus
Allowed
Electron
Orbits
Low Energy, Long Wavelength
Photon (Red)
Medium Energy, Medium Wavelength
Photon (Green)
Large Electron Transition
High Energy, Short Wavelength
Photon (Violet)
The light produced when electrons jump from a high-energy orbit to a smaller low-energy orbit is
known as an emission line. An absorption line is produced by the opposite phenomenon: a
passing photon of light is absorbed by an atom and provides the energy for an electron to jump
from a low-energy orbit to a high-energy orbit. The energy of the photon must be exactly equal to
the difference in the energies of the two levels; if the energy is too much (wavelength too short) or
too little (wavelength too long), the atom can't absorb it and the photon will pass by the atom
unaffected. Thus, the atom can selectively absorb light from a continuum source, but only at the
very same wavelengths (energies) as it is capable of emitting light. An absorption spectrum
appears as dark lines superimposed against a bright continuum of light, indicating that at certain
wavelengths the photons are being absorbed by overlying atoms and do not survive passage
through the rarefied gas.
APS 1010 Astronomy Lab
88
Spectroscopy
The human eye perceives different energies (wavelengths) of visible light as different colors. The
highest energy (shortest wavelength) photon detectable by the human eye has a wavelength of
about 4000 Angstroms (one Angstrom equals 10-10 meters) and is perceived as "violet". The
lowest energy (longest wavelength) photon the eye can detect has a wavelength of about 7000
Angstroms, and appears as "red".
A spectroscope is a device which allows you to view a spectrum. Light enters the spectroscope
through a slit and strikes a grating (or prism) which disperses the light into its component colors
(wavelengths, energies). Each color forms its own separate image of the opening; a slit is used to
produce narrow images, so that adjacent colors do not overlap each other.
Hand-Held
Spectroscope
Spectrum
Red
Green
Grating
Eye
Violet
Source
of
Light
Slit
A grating is a sheet of material with thousands of evenly spaced parallel openings. In general,
the crests and troughs of the light waves passing through adjacent openings destructively interfere
with each other: they cancel each other (crest on trough) so that no wave survives. At a unique
angle for each wavelength, however, the waves constructively interfere (crests add to crests,
troughs to troughs); at that angle we see a color image formed by that wavelength of light. Longer
wavelengths of light are "bent" (diffracted) the most, so a spectrum is formed with violet seen at
small angles, and red at large angles.
Part I. Continuum and Emission Line Spectra
I.1
Install the helium discharge tube in the power supply and turn it on. Describe the
apparent color of the glowing helium gas. Now view the gas using the hand-held
spectroscope, and note the distinctly separate colors (spectral lines) of light emitted by
the helium atom. Make a sketch of the spectrum and label the colors. Compare the
apparent overall color of the glowing gas with the collection of actual colors contained in
its spectrum.
I.2
Turn off the helium tube and replace with the neon gas tube. Judging from the
appearance of glowing neon, what wavelengths would you expect to dominate its
spectrum? Observe and sketch the spectrum using the spectroscope as before. Judging
from the number of visible energy-level transitions (lines) in the neon gas, which element
would you conclude has the more complex atomic structure: helium or neon?
A number of neon spectral lines are so close together that they are difficult to separate or resolve.
This blending of lines becomes even more dramatic in molecules of gas: the constituent atoms
interact with each other to increase the number of possible electron states, which allows more
energy transitions, which in turn produces wide molecular emission bands of color.
I.3
Install the nitrogen gas tube containing diatomic (two-atom) nitrogen (N2 ) molecules.
Observe and sketch the spectrum, and identify the broad emission bands caused by its
molecular state.
APS 1010 Astronomy Lab
89
Spectroscopy
A solid glowing object such as the tungsten filament of an incandescent lamp will not show a
characteristic atomic spectrum, since the atoms are not free to act independently of each other.
Instead, solid objects produce a continuum spectrum of light regardless of composition; that is,
all wavelengths of light are emitted rather than certain specific colors.
I.4
Look through the spectroscope at the incandescent lamp provided by your instructor.
Confirm that no discrete spectral lines are present. What color in the spectrum looks the
brightest? How does this color compare to the overall color of the lamp (i.e. the lamp’s
color without using the spectroscope)?
Fluorescent lamps operate by passing electric current through a gas in the tube, which glows with
its characteristic spectrum. A portion of that light is then absorbed by the solid material lining the
tube, causing the solid to glow or fluoresce in turn.
I.5
Point your spectroscope at the ceiling fluorescent lights, and sketch the fluorescent lamp
spectrum. Identify which components of the spectrum originate from the gas, and which
from the solid.
Part II. Identifying an Unknown Gas
II.1
Select one of the unmarked tubes of gas (it will be either hydrogen, mercury, or
krypton). Install your "mystery gas" in the holder and inspect the spectrum. Make a
sketch of the spectrum and label the colors.
II.2
Identify the composition of the gas in the tube by comparing your spectrum to the spectra
described in the table below. (The numbers to the left of each color are the wavelengths
of the spectral lines given in Angstroms).
Hydrogen
Mercury
Krypton
6563
Red
6073
Orange
6458
Red
4861
Blue-Green
5780
Yellow
5871
Yellow-Orange
4340
Violet
5461
Green
5570
Green
4101
Deep Violet (dim)
4916
Blue-Green (dim)
4502
Violet (dim)
4358
Violet
4459
Violet (dim)
4047
Violet (dim)
4368
Violet (dim)
4320
Violet (dim)
4274
Violet (dim)
II.3
Install the tube of gas marked “air” and inspect the spectrum. Compare it to the spectrum
for nitrogen. What does this tell you about the composition of our atmosphere? What
molecule(s) do you think are responsible for the spectral lines in air?
APS 1010 Astronomy Lab
90
Spectroscopy
Part III. The Solar Continuum
Now you'll apply what you learned about gases in the laboratory to gases in the Sun. First, let's
study the Sun's continuum spectrum. The gas in the interior of the Sun is so dense that the atoms
cannot act independently of each other and therefore they radiate a continuum spectrum just like a
solid. The solar continuum is a blackbody spectrum. Like all blackbody spectra, the shape of
the solar continuum depends only on its temperature.
The graphs below show the blackbody spectra seen for any object (not just stars!) with different
temperatures. The temperatures for the graph on the left are chosen to show the large variation in
radiated energy for minor variations in temperature. The graph on the right uses a logarithmic scale
to encompass a larger range, and shows the temperatures for “O” stars (50,000 K - the hottest
known stars), the Sun (5783 K), and a person (310 K).
Blackbody
Curves for Various
Temperatures
Blackbody
Curves
for
Various
Objects
50000 K
5783 K
310 K
Curves for Various Temperatures
Blackbody Blackbody
Curves
near6000Solar
Temperatures
7000 K
K
5000 K
2020
7000 K
6
5783 K
(Sun)
L
o
g
6000 K
10
10
F
visible spectrum
4
4e14
l
u
x
2
2e14
5000 K
Visible
Spectrum
00
2000
2e3
III.1
50000 K
(O-Star)
15
Relative Intensity
F
l
u
x
Log Relative Intensity
6e14
4000
4e3
6000
6e3
8000
8e3
Wavelength (Å)
Wavelength (Angstroms)
5
00
10000
1e4
2
100A
310 K
(Human)
3
4
5
1000A
1µ
10µ
Log Wavelength (Å)
Wavelength (log scale)
6
100µ
7
1 mm
Why do you suppose we humans have evolved to see light in the wavelength range that
we do (4000 to 7000 Angstroms?) If the Sun were hotter, would it be useful for us to
see at longer or shorter wavelengths? Why can’t we see each other “glowing in the
dark”?
An object’s temperature can be determined from the wavelength at which the peak energy occurs
using Wien's Law:
Temperature =
2.90 x 10 7 (degrees kelvin x Angstroms)
Peak Wavelength in Angstroms
By measuring the peak energy of the solar continuum, we can determine the temperature of the
surface of the Sun.
III.2
Where in the solar spectrum does the continuum emission appear (with your eye) the
brightest? Estimate the wavelength using the Fraunhofer Solar Spectrum Chart below the
spectrograph screen.
III.3
Using a photometer (lightmeter) provided by your instructor, measure where the
continuum is the brightest. Estimate the wavelength. How does this compare to your
estimation with your eye?
III.4
Using Wien's law, calculate the temperature of the Sun using your measured estimate for
the peak wavelength. How does this compare to the accepted value of 5783 K?
APS 1010 Astronomy Lab
91
Spectroscopy
Part IV. Solar Absorption Lines
In the Sun we see both a continuum spectrum and absorption lines from gases in the tenuous solar
atmosphere, which selectively absorb certain photons of light where the energy is “just right” to
excite an atom’s electron to a higher energy state.
Of course, an atom can’t absorb light if the atom isn’t there. Thus, we can identify the elements
that make up the Sun simply by comparing the absorption lines in the solar spectrum with emission
lines from gases here on Earth. Your lab instructor will dim the solar spectrum by narrowing the
entrance slit, and switch on comparison lamps containing hydrogen and neon; their emission
spectra appear above and below the solar spectrum. If an element is present in the Sun (and if the
temperature of the gas is "just right"), its absorption line will appear at the same wavelength as the
corresponding emission from the comparison source.
IV.1
Which comparison spectrum (top or bottom) is due to hydrogen, and which is neon?
What is your evidence?
IV.2
Do you see evidence that hydrogen gas is present in the Sun? What about neon?
Even though your eye can't see it, the solar spectrum extends beyond the visible red portion of the
spectrum to include infrared light, and beyond the violet to include ultraviolet light. The
spectrum of ultraviolet light can be made visible by fluorescence (remember the solid material in the
overhead tube lights?).
IV.3
Hold a piece of bleached white paper up to the viewing screen where ultraviolet light
should be present in the spectrum. Note that there are many additional absorption lines in
the solar spectrum at wavelengths that are not normally visible to us! Use the Solar
Spectrum Chart to identify the element responsible for the two extremely broad, dark
lines in the ultraviolet.
IV.4
Using the Fraunhofer chart, dentify each of the following lines in the solar spectrum - for
each, give its wavelength, color, and include a brief descriptive phrase (dark, broad,
narrow, fuzzy, etc.).
Line Name
and Element
Abundance
per million
B
Oxygen (O2 ) - molecular
C
Hydrogen alpha (H α)
D
Sodium (Na)
2
E
Iron (Fe)
40
F
Hydrogen beta (H β)
Color
Descriptive
Appearance
Reason
none
900,000
900,000
H Calcium (Ca II) ionized
2
K
2
Calcium (Ca II) - ionized
Wavelength
(angstroms)
It seems reasonable to assume that stronger (darker, wider) absorption lines would indicate that
there are more atoms of that element present in the Sun - but as shown in the “abundance” column
of the table, we would be misleading ourselves. Although an atom must be present in order to
show its “signature”, that doesn’t necessarily mean that the element is abundant - otherwise, we
would probably assume (incorrectly) that the Sun is mostly made out of calcium!
APS 1010 Astronomy Lab
92
Spectroscopy
The analysis of spectral features is an extremely complex task, and there are several important
factors to consider. First of all, even if many atoms of an element are present, the temperature may
be too cool to “excite” them, so that they won’t emit or absorb (remember, the gas in the discharge
tubes was completely invisible until you turned on the power). Or, if the temperature is too hot,
the atoms will be ionized (electrons stripped away) so that the electrons responsible for the
absorption won’t be part of the atom anymore. Or, it could be that a spectral feature that we see
didn’t really come from the Sun at all, but from the gasses in our own Earth’s atmosphere!
Through careful analysis, physicists have figured out the temperature ranges necessary for certain
elemental species to “show themselves” through absorption, as indicated below.
Probability of Absorption
1.0
Ca II
0.8
Na
Fe
H
O
2
0.6
0.4
0.2
He
0.0
12000
IV.5
10000
8000
6000
4000
Temperature (kelvin)
2000
0
For each spectral line in the table on the previous page give a reason (A, B, C, or D
below) why you think it appears in the solar spectrum.
A. Although only a very small fraction of the atoms absorb light at any one time, this atom is
very abundant and therefore shows its signature.
B. This spectral line is actually several lines that are closely spaced together. The sum of the
lines makes it appear to be stronger than it really is. It’s not really abundant in the Sun.
C. This spectral line is strong because the Sun's temperature is “just right” for this element
to absorb light at this wavelength, even though there aren’t that may atoms there.
D. The spectral “line” is actually an absorption band from molecules in Earth's atmosphere
rather than in the solar atmosphere. The Sun is too hot for molecules to exist.
IV.6
Although 10% (100,000 atoms out of a million) of the Sun is composed of helium (He),
we don’t see any helium lines in the visible solar spectrum. Inspect the graph of
absorption probabilities, and explain why not.
IV.7
However, during total solar eclipses astronomers do see helium in the solar corona, the
extremely tenuous outermost region of the Sun’s atmosphere (in fact, the element was
named for the Greek word helios, meaning “the Sun”. What does this imply about the
temperature of the gas in the corona? (If the answer surprises you, you’re not alone - it
also surprised astronomers!)
APS 1010 Astronomy Lab
93
Telescope Optics
TELESCOPE OPTICS
SYNOPSIS: You’ll explore some image-formation properties of a lens, and then assemble and
observe through several different types of telescope designs.
EQUIPMENT: Optics bench rail with 3 holders; optics equipment stand (flashlight, mount O,
lenses L1 and L2, image screen I, eyepieces E1 and E2, mirror M, diagonal X); object box.
NOTE: Optical components are delicate and are easily scratched or damaged. Please handle the
components carefully, and avoid touching any optical surfaces.
Part I. The Camera
In optical terminology, an object is any source of light: it may be self-luminous (a lamp or a star),
or may simply be a source of reflected light (a tree or a planet). Light from an object is refracted
(bent) when passed through a lens and comes to a focus to form an image of the original object.
Lens
Image
Object
Here’s what is happening: From each point on the object, rays are emitted in all directions. The
rays that pass through the lens are refracted into a new direction - but in such a manner that they all
converge through one single point on the opposite side of the lens. The same is true for rays from
every other point on the object, although these rays enter the lens at a different angle, and so are
bent in a different direction, and again pass through a (different) unique point. The image is
composed of an infinite number of points where all of the rays from the different parts of the object
converge.
To see what this looks like in “real-life”, arrange the optical bench as shown below:
Object O
Lens L1
Image Screen I
Flashlight
d obj = 20 cm
Holder 1
(10 cm)
d image
Holder 2
(30 cm)
Holder 3
First, turn on the flashlight by rotating its handle, and take a look at its illuminated face. The
pattern you see serves as the physical object we’ll be using in our study. The black ring around
the pattern is exactly one centimeter (10 mm) in diameter.
APS 1010 Astronomy Lab
94
Telescope Optics
Clamp the flashlight into the large opening of mount “O” as shown. Now install the mount and
flashlight into holder #1 so that the flashlight points down the rail. Then slide holder #1 to the 10
cm mark on the rail, and clamp it in place.
Position holder #2 at the 30 cm mark on the rail, and clamp it in place. Install lens L1 in the holder
so that one of its glass surfaces faces the flashlight.
The separation between the lens and the object is called the object distance dobject. In this
particular arrangement, since the lens is located at the 30 cm mark on the rail while the object is at
10 cm, the difference between the two settings gives the object distance: 20 cm.
Finally, put the image screen I (with white card facing the lens) in holder #3 on the opposite side of
the lens from the flashlight.
On the white screen, you should see a bright circular blob which is the light from the object that is
passing through the lens. Slowly slide the screen holder #3 back and forth along the rail while
observing the pattern of light formed on the screen. At some one unique point the beam of light
coalesces from a fuzzy blob into a sharp image of the object. Position the screen at this location
where the image is in best focus.
Not surprisingly, the distance from the lens to the in-focus image is called the image distance,
dimage. You can determine this distance by taking the difference between the location of the image
screen (from the rail markings) and the location of the lens (30 cm). Note that distances are always
given in terms of how far things are from the lens.
I.1
Determine the image distance from the lens to the nearest tenth of a cm (one mm).
I.2
Use a ruler to measure the diameter of the outer black circle of the image (remember, the
true size of this circle on the object is 1 cm across).
The term magnification refers to how many times larger the image appears compared to the true
size of the object:
Image Size
Magnification (definition) = Object Size
I.3
.
(Equation 1)
What is the magnification produced by this optical arrangement?
Magnification can also be calculated from the ratio of image distance to the object distance:
Magnification = dimage / dobject.
(Equation 2)
I.4
Show that equation 2 gives (at least approximately) the same value for the magnification
that you measured using equation 1.
I.5
Now reposition the object (flashlight) holder so that it is at the 3 cm mark on the rail; the
new object distance is now (30 - 3) = 27 cm. Refocus the image. What is the new image
distance (lens to white screen)? The new magnification?
Now let’s see what will happen if we use a different lens. Replace lens L1 with the new one
marked L2; but leave the positions of of the object (holder #1) and the lens (holder #2) unchanged
so that the object distance is still 27 cm.
I.6
Refocus the image; what’s the new image distance? The new magnification?
By now you should be convinced that, for any given lens and distance of an object from it, there
will be one (and only one) location behind the lens where an image is formed. By changing either
APS 1010 Astronomy Lab
95
Telescope Optics
the lens or the distance to the object, or both, the location and magnification of the image will also
change.
The optical arrangement you’ve been playing with is the same as that used in a camera, which
consists of a lens with a piece of photographic film behind it which chemically records the pattern
of light falling onto it. To achieve proper focus, the lens-to film spacing is adjusted by rotating a
ring on the barrel of the lens which moves the lens forwards or backwards towards or away from
the film, thus changing the image distance.
I.7
Where would you place a piece of photographic film in your arrangement in order to take
an in-focus picture of the object?
In many cameras it is possible to swap lenses so that the same camera will yield larger images of
distant objects (a “telephoto” lens).
I.8
Which of the two lenses you’ve used produced a larger image? Which would more likely
be considered a “telephoto” lens in a camera?
I.9
What modifications or additions to the apparatus would you have to make to built
yourself a practical, functional working camera?
Part II. The Lens Equation
There is a mathematical relationship between the object and image distances for any given lens,
called the lens equation:
1
f
1
1
= dobject + dimage .
(Equation 3)
The value f in the formula is called the focal length of the particular lens being used - it is a
property of the lens itself, regardless of the location of the object or the image.
II.1
Calculate the focal length f of lens L1, using your measured image distance dimage and
object distance dobject (either from step I.1, or step I.5, or both).
II.2
What is the focal length of lens L2 (from step I.6)?
HELPFUL HINT: When using your calculator with the lens formula, it is simpler to deal with
reciprocals of numbers rather than with the numbers themselves. Look for the calculator
reciprocal key labelled 1/x . To compute
1 1
2 +4 ,
hit the keys
2 1/x
+
4
1/x
=
.
You should get the result 0.75 ( = 3/4 ). Now if 1/f equals this quantity, just hit 1/x again
to get
II.3
1
f = 0.75 = 1.33.
Which lens, L1 or L2, has the longer focal length? Considering what you found in I.8,
does a telephoto lens have a longer or shorter focal length than a “normal” lens?
APS 1010 Astronomy Lab
96
Telescope Optics
Part III. The Refracting Telescope
In astronomy, we look at objects that are extremely far away - for all practical purposes, we can
say that the object distance is infinitely large. This means that the value 1/dobject in the lens
equation is extremely small, and for all practical purposes can be said to equal zero. Thus, for the
special case where an object is very far away, the lens equation simplifies to:
1
f
= 0 +
1
dimage
or simply
dimage = f
In other words, when we look at an object very far away, the image that is formed comes to a
focus at a distance behind the lens equal to the value of the focal length of the lens Since
everything we look at in astronomy is very far away, images through a telescope are always
formed at the focal length of the lens.
Let’s now look at an object that is far enough away to treat it as being “at infinity.” Remove the
flashlight and mount from holder #1. Instead, use the large object box at the other end of the table.
It should be at least 3 meters (10 feet) from your optics bench. Focus the screen. Your
arrangement should look like the following:
Lens L2
Large
Object
Box
dimage ≅ f
d obj ≅ 300 cm
Holder
2
III.1
Focal Plane I
Holder
3
Measure the image distance from the markings on the optics rail. Confirm that this is
approximately the same as the focal length of lens L2 that you calculated in II.2.
If the screen were not present, the light rays would continue to pass through and beyond the
image. In fact, the rays would diverge from the image as if it were a real object, suggesting that
the image formed by one lens can be used as the object for a second lens.
III.2
Remove the white card from the screen to expose the central hole. Aim the optics bench
so that the image disappears through the opening.
III.3
Place your eye along the optical axis about 20 cm behind the opening in the screen.
You'll be able to see the image once again, "floating in space" in the middle of the
opening! Move your head slightly from side-to-side to confirm that the image appears to
remain rigidly fixed in space at the center of the hole. (By sliding the screen back and
forth along the rail, you can also observe that the opening passes around the image, while
the image itself remains stationary as if it were an actual object.)
Notice that the image is quite tiny compared to the size of the original object (a situation that is
always true when looking at distant objects). In order to see the image more clearly, you’ll need a
magnifying glass:
III.4
Remove screen I from its holder and replace it with the magnifying lens E1. Observe
through the magnifier as you slowly slide it back away from the image. At some point, a
greatly enlarged image will come into focus. Clamp the lens in place where the image
appears sharpest. Your arrangement will be as follows:
APS 1010 Astronomy Lab
97
Telescope Optics
Objective Lens
Eyepiece Lens
Light from
Object
at Infinity
Eye
f obj
f eye
L
You have assembled a refracting telescope, an apparatus which uses two lenses to observe
distant objects. The main telescope lens, called the objective lens, takes light from the object at
infinity and produces an image exactly at its focal length f obj behind the lens. Properly focused,
the magnifier lens (the eyepiece) does just the opposite: it takes the light from the image and
makes it appear to come from infinity. The telescope itself never forms a final image; it requires
another optical component (the lens in your eye) to bring the image to a focus on your retina.
To make the image appear to be located at infinity (and hence observable without eyestrain), the
eyepiece must be positioned behind the image at a distance exactly equal to its focal length, f eye.
Therefore, the total separation L between the two lenses must equal the sum of their focal lengths:
L = fobj + feye .
III.5
(Equation 4)
Measure the separation between the two lenses, and use your value for the objective lens
focal length to calculate the focal length of eyepiece E1.
Earlier, we used the term "magnification" to refer to the actual size of the image compared to the
actual size of the object. This can't be applied to telescopes, since no final image is formed.
Instead, we use the concept of angular magnification: the ratio of the angular size of an image
appearing in the eyepiece compared to the object's actual angular size.
The angular magnification M produced by a telescope can be shown to be equal to the ratio of the
focal length of the objective to the focal length of the eyepiece:
fobj
M = feye .
III.6
(Equation 5)
Calculate the magnification using the ratio of the focal lengths of the two lenses.
Equation 5 implies that if you used a telescope with a longer focal length objective lens (make f obj
bigger), the magnification would be greater. This, however, would require that the eyepiece be
located farther from the objective lens, so that the entire telescope would be longer, bulkier, and
more costly.
But the equation also implies that you can increase the magnification of your telescope simply by
using an eyepiece with a shorter focal length (make feye smaller).
III.7
Eyepiece E2 has a focal length of 1.8 cm. Use equation 5 to calculate the magnification
resulting from using eyepiece E2 with objective lens L2.
III.8
Replace eyepiece E1 with E2, and refocus. Did the image get bigger or smaller than
before? Is your observation consistent with your prediction in III.7?
A note to the telescope shopper: Some inexpensive telescopes are advertised on the basis of their
“high power” (“large magnification”), because many consumers think that this means a “good”
APS 1010 Astronomy Lab
98
Telescope Optics
telescope. You now know that a telescope can exhibit a large magnification simply by switching to
an eyepiece with a very short focal length - it has nothing to do with the quality of the most
important part of the telescope: the objective lens that gathers the light and forms the initial image.
Part IV. The Reflecting Telescope
Concave mirrored surfaces can be used in place of lenses to form reflecting telescopes rather
than refracting telescopes. All of the image-forming properties of lenses also apply to reflectors,
except that the image is formed in front of a mirror rather than behind. As we'll see, this poses
some problems! Reflecting telescopes can be organized in a variety of configurations, three of
which you will assemble below.
The prime focus arrangement is the simplest form of reflector, consisting of the image-forming
objective mirror and a flat surface located at the focal plane. A variation on this arrangement is
used in a Schmidt camera to achieve wide-field photography of the sky.
Focal Plane
Light from
Object
at Infinity
Concave
Mirror
M
Screen X
f obj
IV.1
Assemble the prime-focus reflector shown above:
°
First, return all components from the optical bench to the stand.
°
Then move holder #2 to the 80 cm position on the optics rail, and install the large mirror
M into it.
°
Carefully aim the mirror so that the light from the object is reflected straight back down
the bench rail.
°
Place the mirror/screen X into holder #1, with the white screen facing the mirror M.
°
Finally, move the screen along the rail until the image of the object box is focused onto
the screen.
IV.2
Is the image right-side-up, or inverted?
IV.3
Measure the focal length of the mirror M (the mirror-to-screen spacing).
Since the screen X obstructs light from the object and prevents it from illuminating the center of the
mirror, many people are surprised that the image doesn't have a "hole" in its middle.
IV.4
Hold the white card partially in front of a portion of the objective mirror in order to block
a portion of the beam, as shown on the next page. Demonstrate that each small portion of
the mirror forms a complete and identical image of the object at the focus. When you
block some of the light, does the image change in brightness?
APS 1010 Astronomy Lab
99
Telescope Optics
White Card
Light
from
Object
light is blocked
Focal
Plane
light reaches mirror
Mirror
The previous arrangement can't be used for eyepiece
viewing, since the image falls inside the telescope tube;
if you tried to see the image with an eyepiece your head
would also block all of the incoming light. (This
wouldn't be the case, of course, if the mirror were
extremely large: for example, the 200-inch diameter
telescope at Mount Palomar has a cage in which the
observer can actually sit inside the telescope at prime
focus, as shown here!)
Isaac Newton solved the obstruction problem for small
telescopes with his Newtonian reflector, which
uses a flat mirror oriented diagonally to redirect the
light to the side of the telescope, as shown below. The
image-forming mirror is called the primary mirror,
while the small additional mirror is called the
secondary or diagonal mirror.
Light from
Object
at Infinity
Diagonal
Mirror
Focal Plane
Primary
Mirror
Eyepiece
Eye
IV.5
Convert the telescope to a Newtonian arrangement. Reverse the mirror/screen X so that
the mirror side faces the primary mirror M and is oriented at a 45° angle to it. Slide the
diagonal approximately 5 cm closer towards primary. Hold the image screen I at the
focal plane (see diagram) to convince yourself that an image is still being formed.
IV.6
Now remove the image screen and look into the diagonal mirror; if your eye is properly
placed and the optics are properly arranged, you can directly observe the image. You’ll
also be able to see the outline of the primary mirror, and the shadow of the diagonal.
APS 1010 Astronomy Lab
100
Telescope Optics
IV.7
Now try observing the image through the (hand-held) eyepiece E1. When the image is in
focus, why do you think that the primary mirror and the diagonal are no longer visible?
IV.8
Calculate the magnification produced by this arrangement.
Large reflecting telescopes (including the 16-inch, 18-inch, and 24-inch diameter telescopes at
Sommers-Bausch Observatory) are usually of the Cassegrain design, in which the small
secondary redirects the light back towards the primary. A central hole in the primary mirror
permits the light to pass to the rear of the telescope, where the image is viewed with an eyepiece,
camera, or other instrumentation.
Eyepiece
Light from
Object
at Infinity
Secondary
Mirror
Eye
Primary
Mirror
IV.9
Arrange the telescope in the Cassegrain configuration: Move the secondary mirror X
closer to the primary mirror M until the separation is only about 35% of the primary's
focal length. Carefully reorient the secondary mirror to reflect light squarely back
through the hole in the primary, and visually verify the presence of the image from
behind the mirror.
IV.10
Install eyepiece E1 into holder #3, and focus the telescope (be patient - this can be a rather
tricky operation).
IV.11
How does the magnification here compare to that in the Newtonian arrangement?
IV.12
How does the overall length of the Cassegrain telescope design compare with that of the
Newtonian style? Explain your reasoning.
IV.13
Why doesn't the hole in the primary mirror cause any additional loss in the lightcollecting ability of the telescope?
APS 1010 Astronomy Lab
101
Planetary Temperatures
PLANETARY TEMPERATURES
“Mars is essentially in the same orbit. Mars is somewhat the same distance from the Sun, which is
very important. We have seen pictures where there are canals, we believe, and water. If there is
water, that means there is oxygen. If oxygen, that means we can breathe.” -D. Quayle
SYNOPSIS: In the next two lab sessions we will explore the factors that contribute to the
average temperature of a planet using the EARTH/VENUS/MARS module of the Solar System
Collaboratory.
EQUIPMENT: Computer with internet connection to the Solar System Collaboratory,
photometer with light shade, soil/rock samples, calculator.
Part I. Temperature of a Planet
The module provides the tools you need to attack one of the grand questions of planetary science.
In this case, the grand question of the EARTH/VENUS/MARS module can be phrased in two
different but related ways:
•
Why can Earth support abundant life but Venus and Mars cannot?
- or -
•
What determines the habitable region around a star?
Although both questions are truly grand (meaning that it would probably take a number of major
research projects to answer them in detail) they can be broken down into a number of component
parts, each one of which is quite accessible. The module concentrates on one of these parts,
planetary temperature. Why temperature? Of all the factors that affect the presence of abundant life
on a planet (chemistry, radiation, biology, etc.) the presence of liquid water is of paramount
importance, and liquid water can only exist in a narrow temperature range.
I.1
Why do you think liquid water is thought to be so critical for life? (Of course, nobody
really knows for sure.)
I.2
Over what approximate range of temperatures do you think you could find liquid water?
I.3
Over what approximate range of temperatures would you expect to find life?
We will now explore the factors that determine the temperature of a planet, and to find the range of
parameters that would result in an average planetary temperature that allows liquid water.
In the Computer Lab, make sure your computer is on the PC side. Click on the Internet folder,
and launch Netscape (be sure that the window is not maximized).
Now go to the website http://solarsystem.colorado.edu/cu-astr/ . Click on "Enter
Website - Low Resolution (1024x768)", click on the Modules option, and finally, select the
Earth/Venus/Mars module.
APS 1010 Astronomy Lab
102
Planetary Temperatures
There are five parts to the module. In this lab we will concentrate on the first one (PLANET
TEMPERATURE - click on the words and you should bring up an “applet” - short for “little
application”) and use it to explore the way the temperature of a planet depends on
1) its distance from the Sun.
2) its albedo.
We will explore the effect of the atmosphere on a planet's temperature in the next lab session.
Part II. Distance from the Sun
This is the simplest and most obvious way to determine a planet's temperature. Like hikers who
build a fire and huddle around it at night, the inner planets huddle around the central fire in the
solar system. The closer you get to the fire, the warmer you get. Notice that the “model” used to
calculate the temperature of a planet in this applet is displayed at the bottom of the applet. It should
be set on “fast-rotating, dark planet”. The “fast-rotating” part of this just ensures that the planet
rotates fast enough that day-vs.- night temperatures are not too terribly different. The “dark” just
means that the planet absorbs (rather than reflects) all the sunlight that hits it.
The PHYSICS PAGE link provides background information on the physical processes that go into
the model. SHOW ME THE MATH provides the mathematical formulation that is programmed
into the computer.
Select the "Fast rotating dark planet" model from the menu at the bottom of the applet. Let’s
explore how the distance of a planetary orbit from the Sun affects the planet’s temperature.
II.1
Move Planet X to the orbit of the Earth (1 AU). What is the temperature of Planet X?
II.2
We know that if we move the planet to a larger orbit, its temperature will go down. But
by how much? Make a prediction: what do you think will happen if you double the size
of the Earth’s orbit? Think about it before you move Planet X! Here are come
possibilities:
(a) If the temperature follows an inverse law, doubling the radius should cause the
temperature to drop by a factor of 2.
(b) If the temperature follows an inverse square law, doubling the radius should
cause the temperature to go down by a factor of 4.
(c) If the temperature follows an inverse square-root law, doubling the radius should
cause the temperature to drop by a factor of “square root of 2” which is about 1.414.
Your prediction: ______________________________________________________
II.3
Now go ahead and move Planet X to 2 AU (twice Earth’s orbital distance) - what is the
temperature?
Which of the above laws does temperature follow? (To check, look at SHOW ME THE
MATH.)
APS 1010 Astronomy Lab
103
Planetary Temperatures
II.4
Do you think the temperature of a planet depends on the size of the planet in this model?
Have a look at the PHYSICS PAGE and/or SHOW ME THE MATH, and explain why
(or why not).
II.5
How good is this model? Bring up the FACT SHEET from the quick navigation frame
(the gray frame on the left). How does the temperature calculated by the model compare
to the measured average temperature of Mars?
What about Earth?
Venus?
II.6
So - is this model any good? Does it do what it claims to do (calculate the average
temperature of a dark body)? How can you tell? HINT: Of all the bodies mentioned in
the FACT SHEET, which is best described by the dark planet model?
Is the temperature of that body calculated accurately by the model?
So, what do you conclude about how well the model works?
Part III. Adding Albedo to the Model
Albedo (represented by the symbol A) is the fraction of sunlight falling on a surface that is
reflected back into space. (The word albedo comes from the Latin word for "white" - albus.) The
albedo represents the average reflectivity over the entire visible surface; hence it differs slightly
from the reflectivity of different portions of it. For example, the surface of the Moon has an albedo
of 0.07 (on average, 7% of the incident sunlight is reflected, 93% is absorbed), even though there
are bright and dark regions that have reflectivities different from the average.
Now let's consider how the planet's albedo affects its temperature. If you haven't done this
already, you may want to get more background information on the physics behind this applet.
Click on the PHYSICS PAGE link in the quick navigation frame (the gray strip on the left) and
look up albedo.
Select the "Fast rotating dark planet with adjustable albedo" model from the menu at the bottom of
the applet.
III.1
What value of albedo does the previous model ("Fast rotating dark planet") assume?
III.2
What is the present day albedo of Mars? (Hint: use the Fact Sheet)
III.3
What happens to the temperature of Mars if you increase its present day albedo by 0.05?
The temperature goes from _________________ to ______________.
APS 1010 Astronomy Lab
III.4
104
Planetary Temperatures
What happens to the temperature of Mars if you increase its albedo from 0.999 to
0.9999?
The temperature goes from _________________ to ______________.
III.5
In general, if you increase the albedo, you ___________________ the temperature.
Explain why this is the case.
III.6
If you put in the correct albedo for Mars, does the model give the correct temperature?
How about for the Earth?
And for Venus?
III.7
So does this model do what it claims to do (calculate the average temperature of a planet
with a non-zero albedo)? How can you tell?
Part IV. Measuring Albedos of Samples
The albedo of a planet or moon provides valuable information about its composition and
temperature. You will measure the albedos of various soil and rock specimens; this information
will be used to provide clues as to the nature of the surfaces of the planets and moons in our Solar
System.
A photometer (light meter) with a sunshade can be used to measure the reflectivity of different
materials, as shown below. The measurements are made as follows:
•
•
•
Measure the brightness of white paper
illuminated by sunlight. The paper has a
reflectivity of approximately 100%,
corresponding to an albedo of 1.00, and
thus gives you a convenient standard
against which to compare the reflectivity of
other surfaces.
Be sure that nothing
(including you or the photometer) is casting
shadows onto the area under observation!
Detector
Photometer
Meter
Incident Sunlight
Light
Shade
Measure the brightness of the sample. Be
sure to hold the photometer close enough to
the sample so that its detector only "sees"
the sample, not the surrounding area.
Divide your sample reading by the value
you obtained for the white paper. The
resulting fraction is the reflectivity of the
material.
Reflected
Sunlight
Specimen
APS 1010 Astronomy Lab
IV.1
105
Planetary Temperatures
Measure the reflectivities of the various specimens provided by your lab instructor: rocks
(vesicular basalt, dark basalt, marble, limestone, breccia), charcoal, wet and dry sand,
crushed ice or snow (if available). Enter your data in the table below.
Also note the color of the specimen. These samples have been chosen because they are
likely to be found in other objects in the Solar System.
IV.2
In addition, measure the reflectivity of some materials commonly found on the Earth's
surface which you think might significantly contribute to the Earth's albedo: e.g., water,
snow, grass and other plant life, etc. Add these data in the table below as well.
Specimen Identification
Granite
Basalt
Soil/Loam
Charcoal
Limestone
Sand
Marble
Sandstone
Mars Soil (Simulated)
Sample
Reading
Paper
Reference
Reflectivity
(albedo)
(A)
(B)
(A) / (B)
Color
APS 1010 Astronomy Lab
106
Planetary Temperatures
Part V. Comparing Solar System Albedos
We can measure the albedo of a planet or moon by comparing its brightness to how much sunlight
it receives. We know how much sunlight it receives because we know its distance from the Sun
(remember the 1/ R2 rule!); and we can measure how much light is coming from the object with a
photometer on a telescope. If the object has an albedo of 100% it will reflect all the sunlight
incident upon it. Note: We are assuming that all of the light coming from the object is reflected
sunlight; i.e. the object is not producing its own light. Planetary scientists are interested in
knowing the albedo of a planet or moon because it provides information on the composition of the
object. By comparing the albedo of a planet or moon to the albedos of substances found here on
Earth we can learn what substances may or may not be on that object.
V.1
What assumption are we making when we compare the albedos of substances on the
Earth to other objects in space?
The table below contains information regarding the measured albedo of various planets and their
moons. Use the table and information in the textbook to answer the following questions.
Object
MERCURY
VENUS
EARTH
Moon
MARS
JUPITER
Io
Europa
Ganymede
Callisto
SATURN
Mimas
Iapetus
URANUS
NEPTUNE
PLUTO
V.2
Albedo
0.06
0.76
0.40
0.07
0.16
0.51
0.60
0.60
0.40
0.20
0.50
0.70
0.05, 0.50
0.66
0.62
0.50
Why do you think the albedos of Mercury and the Moon are so close?
Look at photos of these two; do the pictures confirm your hypothesis?
APS 1010 Astronomy Lab
V.3
107
Planetary Temperatures
What material that you measured is closest to the albedo of Mercury and the Moon?
Does it have the same color as well? Do you think the Moon's surface contains this
material?
V.4
The Moon has a very low albedo, meaning that it is very dark.
suppose the Moon appear to be so bright?
Why then do you
V.5
Compare the albedos of the terrestrial planets: why do Venus and the Earth have much
higher albedos than Mercury and Mars?
What, if any, of the substances you measured in IV.2 might cause the Earth to have a
higher albedo than Mars?
Why does Venus have a higher albedo than the Earth?
V.6
The albedos of Earth and Mars are averages over the planet's disk as well as averages
over time. Average the albedos you measured of substances on the Earth’s surface in
IV.2 and compare it to the Earth’s average albedo. Why might they be different?
How does the Earth's reflectivity vary over the surface?
And Mars’? What is the likely reflectivity of the polar regions?
reflectivities change with time?
How do their
On Earth we see basalt in several forms. When it is first created (i.e. when it comes out of the
Earth in the form of lava) it is very dark; however, over time it gets “weathered;” exposure to
oxygen oxidizes basalt and turns it reddish.
APS 1010 Astronomy Lab
V.7
108
Planetary Temperatures
What planet do you think might have oxidized basalt based upon its color?
What can you say about that planet’s atmosphere?
V.8
The Moon contains basalt in its lunar maria which is very old. Why hasn't it turned red?
V.9
Compare the Jovian planets to the terrestrial planets: on average, why do the Jovian
planets have a higher albedo?
From this and your answer for question V.5 what can you conclude is the most important
factor in determining a planet's albedo?
APS 1010 Astronomy Lab
109
Galilean Moons
ENCOUNTER WITH THE GALILEAN MOONS
“All these worlds are yours, except Europa.” - A.C. Clarke
SYNOPSIS: Jupiter's four large moons were first discovered by Galileo in 1610 and looked at
in close detail by the Voyager I and II spacecraft in 1979. Launched in October 1989, NASA's
Galileo spacecraft entered orbit around Jupiter on December 7, 1995. Its mission is to conduct
detailed studies of the giant planet, focusing on its largest moons and the Jovian magnetic
environment. The objective of this lab is for you to have a look at some of the pictures of these
distant worlds and to learn what such pictures tell us about their surfaces, interiors and past
histories.
EQUIPMENT: Galileo photographs of the Galilean moons, a ruler and a calculator. The images
come from the Galileo website http://www.jpl.nasa.gov/galileo/sepo/
Please do NOT mark on the photographs!
Part I. Properties of the Galilean Satellites
Radius
(km)
Moon
Io
Europa
Ganymede
Callisto
Mercury
1738
1821
1565
2634
2403
2439
Average
Density
(kg/m3 )
3340
3530
3030
1930
1790
5420
Albedo
Orbital
(reflectivity) Distance, a
(Rjupiter )
0.11
--0.61
5.9
0.64
9.4
0.42
15.0
0.20
26.4
0.12
---
Orbital
period, P
(days)
--1.77
3.55
7.15
16.69
---
Orbital
period
(P/PIo)
--1.00
2.01
4.04
9.43
---
The table above shows properties of the Galilean satellites as well as the Moon and Mercury for
comparison. Figure 1 in the folder shows family portrait of the four Galilean moons. Note that Io
and Europa are similar in size to the Moon while Ganymede and Callisto are comparable to
Mercury in size. But when we look at other properties, such as density and albedo, the similarities
do not hold up.
Each of the four Galilean moons are believed to have some iron at their center, surrounded by rock
with perhaps a layer of water/ice on top. The density of iron is about 8000 kg/m3 , the density of
rock is 2000 to 3000 kg/m3 , the density of water/ice is about 1000 kg/m3 .
I.1
Considering just the average densities in the table above, (a) which Galilean moon would
you expect to have the most ice (least rock/iron); and (b) which moon could be all rock
and iron, with no ice on the surface?
I.2
All of the Galilean moons are “phase locked” with Jupiter. That is to say, one side of
each moon always faces towards Jupiter, and one side always faces away. Can you
think of another moon in the solar system that is also “phase locked” with the planet that
it orbits? What is its name?
APS 1010 Astronomy Lab
110
Galilean Moons
Part II. Io: The Jovian Cauldron
Io, the Galilean moon closest to Jupiter, is the most volcanically active body in the Solar System.
What makes Io's volcanic activity so interesting is the mechanism that causes it. For instance, our
Moon's interior cooled and most volcanic activity stopped over 3 billion years ago. Since Io is
about the same size as the Moon, its interior must have also cooled. So, why hasn't Io's volcanic
activity stopped? The answer lies with Jupiter and the other Jovian moons. Their strong
gravitational pull on Io “stretches and squeezes” the interior of Io, thus melting the interior and
causing volcanic activity. The process (known as tidal heating) is the same as the gravitational
effect of the Sun and Moon on the Earth, causing the oceans' tides.
Figure 2 shows Io photographed by the Galileo spacecraft in front of Jupiter's cloudy atmosphere.
The image is centered on the side of Io that always faces away from Jupiter. The color in the
image was derived using the near-infrared, green and violet filters of the Galileo camera, and has
been enhanced to emphasize variations in color and brightness that characterize Io's volcanopocked face. The near-infrared filter makes Jupiter's atmosphere look blue. The white patches are
sulfur dioxide frost. The black and bright red materials correspond to the most recent volcanic
deposits, some less than a few years old.
II.1
The dark spots are volcanoes. Make a rough count of the number of volcanoes in the
image.
The active volcano Prometheus is seen near the center of the disk (arrow). To appreciate the size
of the volcanoes on Io, we have to determine the scale of the photograph:
II.2
Measure the diameter of Io on photo #2 (measure in millimeters). Io is actually 3,630 km
in diameter. Determine the scale factor S, of the image: How many (real) kilometers
correspond to 1 millimeter on the photo?
S (km / mm) = Diameter of Io in km / Diameter of image in mm
II.3
Measure the diameter of the volcano Prometheus as seen on photo #2, including the
bright ring around it. Use the scale factor above to determine its actual size. What is its
actual size in kilometers? Compare this to the sizes of the volcano Mauna Kea in Hawaii
(~80 km across, about the size of a metropolitan area) and Olympus Mons on Mars (the
largest volcano in the solar system at 700 km across, about the size of Colorado).
Look at Photo #3. This image of Io, also taken by Galileo, shows a new blue-colored volcanic
plume, named Pillan, extending into space (see the upper inset, the lower inset is of Prometheus.)
The blue color of the plume is consistent with the presence of sulfur dioxide gas erupting from the
volcano, scattering sunlight (similar to water molecules making the Earth's sky blue). The sulfur
dioxide freezes into “snow” in Io's extremely tenuous atmosphere and makes the white patches on
the surface. These eruptions are common on Io and were first discovered when Voyager passed Io
in 1979. Interestingly, they were not discovered by Voyager scientists, but rather by Linda
Morabito, a young member of the Navigation Team. Morabito discovered them when she had
difficulties matching the edge of Io's image with a circle, and then enhanced the images to
investigate the problem.
These volcanic eruptions on Io are far larger than any on Earth. Because Io's atmosphere is
exceedingly thin, there is no air resistance and the gas molecules that erupt from a volcano follow
paths like shells fired from a cannon. These paths are called ballistic trajectories (after ballista, an
engine used in ancient warfare for hurling stones).
The maximum height H an object on a ballistic trajectory can reach is given by:
2
H= v
2g
APS 1010 Astronomy Lab
111
Galilean Moons
where v is the initial velocity of material exiting the volcano and g is the acceleration of gravity
which is 9.80 m/sec2 on Earth, but only 1.81 m/sec2 , about a factor of 5 less, on Io.
II.4
The scale factor S for the upper inset of photo #3 is about 10 km per millimeter.
Measure the height of the blue plume and determine its actual physical height. Convert
this height to meters (1 kilometer = 1000 meters).
II.5
Using your value for the height (in meters) of the blue plume, determine the initial vent
eruption velocity of the plume (in meters / second). For comparison, 1000 m/s equals
about 2000 mph.
II.6
Volcanic eruptions on Earth cannot throw materials to such high altitudes because of
atmospheric resistance and stronger gravity. Ignoring atmospheric friction and
considering the factor of 5 greater gravity on Earth, what would be the height of the
Pillan plume be if it had erupted from a similar volcano on Earth? (That is, what would
H be on Earth if the initial v were the same, but with the Earth value of g?) For
comparison, Mount Everest is just over 8 km high, commercial jets fly at about 15 km,
and the Space Shuttle orbits the Earth at 150 - 500 km. Would such plumes be a hazard
to commercial jets or the Shuttle?
II.7
Tall plumes (and high velocities of ejected material) are due to the pressures that build up
because of internal sources of heat. Compare the height of the Pillan plume, scaled down
to the Earth's gravity, with the height of the geyser “Old Faithful” (around 70 meters about the height of the Gamow Tower) and the plume from the eruption of Mt. St.
Helens (1000 meters). What does this tell us about the pressures that build up inside Io
compared with inside volcanoes on Earth?
Scientists are noting many changes that have occurred on Io's surface since the Voyager flybys 17
years ago, and even a few changes from month to month between Galileo images. Figure 4 shows
two images taken in 1997 on April 4 (upper) and September 19 (lower). The large black area in
the lower picture is material that has been ejected by the Pillan plume.
II.8
The diameter of the new black patch of volcanic material is 400 kilometers (half the state
of Colorado). This corresponds to about 1% of the surface area of Io. If such eruptions
occur at random over the surface every 6 months, estimate how many years it would take
to resurface the moon.
II.9
Do you see any impact craters on any of the Io pictures? Explain.
Part III. Europa: Water World?
Jupiter's moon Europa has a density closer to that of rock than water, yet spectroscopic analysis
had shown that the surface of Europa is composed almost entirely of water ice.
The ice seen on the surface can be the top of only a thin layer of water/ice over a rocky interior.
Recent measurements of the effects of Europa's gravitational pull on the Galileo spacecraft allow
us to put an upper limit of about 170 km for the thickness of the water/ice shell.
III.1
What percentage of Europa's radius is this outer water /ice shell?
APS 1010 Astronomy Lab
112
Galilean Moons
In Figure 5, Galileo pictures of Europa show an icy crust that has been severely fractured. The
upper image covers part of the equatorial zone of Europa, an area of about 360 by 770 kilometers
(220 by 475 miles, or about the size of Nebraska), and the smallest visible feature is about 1.6
kilometers (1 mile) across.
In the lower image the color has been greatly exaggerated to enhance the visibility of certain
features. The fractures, called linea, and the mottled terrain appear brown/red, indicating the
presence of contaminants in the ice. The blue icy plains are subdivide into units with different
albedos at infrared wavelengths, probably because of differences in the grain size of the ice. The
area shown is about 1,260 km across (about 780 miles - the size of Texas).
III.2
From these two images, does it look like the contaminants come from below (e.g.
seeping through the cracks) or from above (e.g. meteorite dust deposited on the surface)?
III.3
Figure 6 shows global views of Europa and of the Moon - Europa is only 10% smaller
than the Moon. Which moon has the older surface?
When asteroids and comets collide with planets and moons, the explosions resulting from the
collisions are extremely violent. Matter is excavated from great depths and can be flung wide
distances from the resulting crater. Larger impacts create larger craters and expose materials from
greater depths. The appearance of the crater resulting from such an impact depends, among many
things, upon the material properties of the material in which crater is formed. In Figure 7 there are
four examples of impact features, two on the Moon and two on Europa.
III.4
At 5.2 AU from the Sun, and having an albedo of 0.64, the temperature on the surface of
Europa is nearly two hundred degrees Celsius below the freezing point of water. The
lunar surface is composed primarily of a volcanically-derived type of rock known as
basalt. Given the similar appearance of the two small craters, what can you conclude
about the material properties of water ice at Europan temperatures and those of basalt?
Also shown in Figure 7 is the aftermath of two much larger impacts. The Europan impact feature,
known as Tyre Macula, looks very different from the lunar crater, but it can still be identified as an
impact feature from the clusters of surrounding small secondary craters that were created during the
big (primary) impact.
III.5
From seismology experiments carried out during the Apollo missions, we know that the
Moon's crust is very thick and is composed of the same material throughout. This is
why, up to a certain size, most lunar craters look very similar. Since larger impacts
excavate material from greater depths and "see" more of the interior, what does the
dissimilar appearances of these large impact structures lead you to conclude about the
water/ice layer of Europa? Do you think it is frozen solid all the way through? What do
you think happened to the impactor that made Tyre Macula?
Figure 8 shows an image of "chaos terrain" which looks as if it were created when the surface
surrounding the ice blocks was slushy and the ice blocks, or "ice rafts" were free to float about.
The thickness of these ice rafts probably does not exceed their width. (Why not? Think of a pencil
floating in water. In which way would it orient itself?)
III.6
The image covers an area of 35 by 55 kilometers. Pick the smallest "ice raft" that still
looks as if it was broken off from neighboring ice blocks. Measure a typical width for
this ice raft. From this measurement, make a rough estimate of the thickness the ice crust
(in kilometers) at the time that the "chaos terrain" froze.
III.7
The next issue is how long ago did the "chaos terrain" freeze? Could there be just a thin
layer of ice over a liquid ocean today? What is the evidence in Figure 8 that the chaos
terrain froze over relatively recently? (Millions rather than billions of years ago.)
APS 1010 Astronomy Lab
113
Galilean Moons
If such an ocean does exist, it might be a possible place for life to exist. Life, as we know from
studies of life here on the Earth, can be very hardy and can adapt to survive in many regions. Life
has been found near volcanic vents at the bottom of the ocean where no sunlight can reach, the
frozen lakes of Antarctica and even inside rocks!
III.8
What difficulties do you think life in a hypothetical Europan ocean would have to
overcome in order to survive?
Part IV. Ganymede: The Giant Moon
Ganymede is the largest moon in the Jovian system and also the entire Solar System. In fact, it is
even larger than Pluto and Mercury! Furthermore, the Galileo spacecraft detected a magnetic field
near Ganymede that indicates that the moon has a liquid iron core (similar to the Earth and to
Mercury). However, the average density of Ganymede is close to that of water so that the outer
layers of Ganymede must be water/ice rather than rock.
IV.1
Figure 9 shows global pictures of Ganymede and of Moon. Superficially, they look
rather similar. For the Moon, which is older: the lighter or darker regions?
Figures 10 and 11 are close ups of a darker and lighter region on Ganymede respectively. When
we look closer, any resemblance to the Moon disappears.
IV.2
The dark regions on Ganymede are older than the newer, lighter regions. The dark
material is probably meteorite dust that has accumulated on the surface. Looking at the
close up of the dark region in Figure 10, how do the craters on the icy surface of
Ganymede change as they get older? (Compare them to the "fresh" lunar craters shown
in Figure 7.)
IV.3
What has happened to the crust in the blown-up image of the light region in Figure 11?
The high-resolution image from the light region on Ganymede covers a region about the same size
as San Francisco Bay - compared in Figure 12. On the left the resolution is that of the best images
taken by Voyager in 1979 and on the right the images are at the Galileo resolution.
IV.4
For the cases of (a) San Francisco Bay and (b) Ganymede, describe how the higher
resolution images change our interpretation of the surfaces of these two regions.
You probably remember Comet Shoemaker-Levy-9 making a spectacular display when it
bombarded Jupiter in 1994. A couple of years before, the comet had been broken up into about 21
fragments when it had passed close by Jupiter. When it came back to Jupiter in 1994, the comet
fragments plunged (and exploded) in Jupiter's atmosphere. It was realized at this time that comets
are probably frequently broken up by Jupiter's gravity. Sometimes these broken up comets hit the
moons rather than the planet.
IV.5
Figure 13 shows a chain of craters on Ganymede that may have been produced by a
broken up comet. How many pieces of comet impacted Ganymede?
APS 1010 Astronomy Lab
114
Galilean Moons
Part V. Callisto: A Battered World
Callisto, the fourth Galilean moon, has the dubious distinction of being the most cratered moon in
the solar system. Callisto is a little smaller than Ganymede (about the size of the planet Mercury)
and is apparently composed of a mixture of ice and rock similar to Ganymede.
Unlike the other Galilean moons, Callisto has endured virtually no tidal heating. Callisto's albedo
is about half that of Ganymede but Callisto is still more reflective than the Moon. The darker color
of Callisto suggests that its upper surface is “dirty ice” or water rich rock frozen at Callisto's cold
surface. The abundance of craters on its surface suggests that its surface is the oldest in the
Galilean system, possibly dating back to final accretion stages of planet formation 4 to 4.5 billion
years ago. And, unlike the other Galilean moons, Callisto's surface shows no signs of volcanism,
tectonics or other geologic activity, further supporting the hypothesis that Callisto's surface took its
present form long ago, and is hence very old.
Valhalla, the prominent “bullseye” type feature in the top left image in Figure 14, is believed to be a
large impact basin, similar to Mare Oriental on the Moon and the Caloris Basin on Mercury. The
ridges resemble the ripples made when a stone hits water, but here they probably are the result of a
large meteorite. The fractured ice surface was partially melted by the impact, but they re-solidified
before the “ripple” could subside.
V.1
The four images in Figure 14 show increasing detail as Galileo zoomed in to smaller and
smaller scales. Notice how the top two pictures show the surface of Callisto to be
saturated with craters so that if another impactor hit, it would probably cover up on old
crater. But when we go to smaller scales, does the surface remain saturated with craters?
V.2
The dark material is probably dust produced by the break up of micro-meteroites (tiny,
dust-sized meteorites). It is probably very similar to the dust on the surface of the Moon.
What is the approximate size of the smallest crater that you can distinguish as a crater in
the bottom left image? If all smaller craters are covered up with meteorite dust, roughly
how thick must the dust layer be?
Earlier we learned that Io’s volcanic activity (hence resurfacing activity) is a result of the
gravitational interaction with Jupiter and its moons. The tidal heating which causes Io’s volcanic
activity is due to Io’s orbital resonance with the other Galilean moons. A moon is in orbital
resonance if its orbital period is (or nearly is) a whole-number multiple of the orbital period of
another moon (e.g. if one moon has twice the period of another moon they are in orbital resonance
with each other).
V.5
From the table at the beginning of this exercise, which of the Galilean moons is not in
orbital resonance with any of the other Galilean moons? How does the surface age of
this non-resonant moon’s surface compare to the others?
V.6
Now you are equipped to make some very general statements about how the
characteristics of the four Galilean moons vary with distance from Jupiter. First, how
does the degree of tidal heating vary with distance from Jupiter? How might this lead to
the observed differences in the amount of water on these four moons? How might this
result in the observed ages of their surfaces?
APS 1010 Astronomy Lab
115
Observing Projects
OBSERVING
PROJECTS
The Moon
SBO 16” Telescope
APS 1010 Astronomy Lab
116
Observing Projects
Messier 31, nearest Spiral Galaxy - CCD image using 180mm Telephoto Lens ...
... mounted on the ...
... SBO 18” Telescope
APS 1010 Astronomy Lab
117
Constellation Identification - Fall
CONSTELLATION AND BRIGHT STAR
IDENTIFICATION
SYNOPSIS: In this self-paced lab, you will teach yourself to recognize and identify a number of
constellations, bright stars, planets, and other celestial objects in the current evening sky.
EQUIPMENT: A planisphere (rotating star wheel) or other star chart. A small pocket flashlight
may be useful to help you read the chart.
Part I: Preparation, Practice, and Procedures
Part II contains a list of 30 or more celestial objects which are visible to the naked eye this
semester. You are expected to learn to recognize these objects through independent study, and to
demonstrate your knowledge of the night sky by identifying them.
Depending upon the method chosen by the course instructor, you may have the opportunity to
identify these objects one-on-one with your lab instructor during one of your scheduled evening
observing sessions. Alternatively, there may be the opportunity to take an examination over these
objects in Fiske Planetarium near the end of the semester. In addition, there may be the
opportunity to do both, in which case the better of your two scores would be counted.
If you are given the option, and if you wait until the end of the semester to take the verbal quiz and
then are clouded out, you have no recourse but to take the Planetarium exam. Do not expect your
TA to schedule additional time for you. If you have not taken the oral test and are unable to attend
the special exam session at Fiske, you will not receive credit for this lab!!! "Poor planning on your
part does not constitute an emergency on our part."
You can learn the objects by any method you desire:
●
Independent stargazing by yourself or with a friend.
●
Attending the nighttime observing sessions at the Observatory, and receiving assistance
from the teaching assistant(s) or classmates.
●
Attending Fiske Planetarium sessions.
●
All of the above.
Observing and study tips:
●
It is generally be to your advantage to take the nighttime verbal quiz if it is an option,
since you control the method and order of the objects to be identified. In addition, it is
easier to orient yourself and recognize objects under the real sky rather than the synthetic
sky, since that is the way that you learned to recognize the objects in the first place. It
may also be possible to later take the Fiske written quiz and use it to replace your oral
quiz score if you feel it will improve your grade.
APS 1010 Astronomy Lab
118
Constellation Identification - Fall
●
The “Celestial Coordinates” section at the beginning of this lab manual (page 27) may
help you visualize the night sky and the location of objects in it.
●
We recommend using the large, 10" diameter Miller planisphere available from Fiske
Planetarium: it is plastic coated for durability, is easiest to read, and includes sidereal
times. The smaller Miller planisphere is more difficult to use but is handier to carry.
Other planispheres are available from the bookstore or area astronomy stores.
●
To set the correct sky view on your planisphere, rotate the top disk until the current time
lines up with the current date at the edge of the wheel. Planispheres indicate local
"standard" time, not local "daylight savings" time. If daylight savings time is in effect,
subtract one hour from the time before you set the wheel. (For example, if you are
observing on June 15th at 11 p.m. Mountain Daylight Time, line up 10 p.m. with the
June 15th marker).
●
The planisphere shows the current appearance of the entire sky down to the horizon. It is
correctly oriented when held overhead so that you can read the chart, with North on the
chart pointing in the north direction. The center of the window corresponds to the zenith
(the point directly overhead). When you face a particular direction, orient the chart so
that the corresponding horizon appears at the bottom. As with all flat skymaps, there will
be some distortion in appearance, particularly near the horizons.
●
Be aware that faint stars are difficult to see on a hazy evening from Boulder, or if there is
a bright moon in the sky. On the other hand, a dark moonless night in the mountains can
show so many stars that it may be difficult to pick out the constellation patterns. In either
case, experience and practice are needed to help you become comfortable with the objects
on the celestial sphere.
●
Learn relationships between patterns in the sky. For example, on a bright night it may be
virtually impossible to see the faint stars in the constellation of Pisces; however, you can
still point it out as "that empty patch of sky below Andromeda and Pegasus". You can
envision Deneb, Vega, and Altair as vertices of "the Summer Triangle", and think of Lyra
the harp playing "Swan Lake" as Cygnus flies down into the murky pool of the Milky
Way.
●
If you merely "cram" to pass the quiz, you will be doing yourself a great disservice. The
stars will be around for the rest of your life; if you learn them now rather than just
memorize them, they will be yours forever.
Good luck, good seeing, and clear skies ...
APS 1010 Astronomy Lab
119
Constellation Identification - Fall
Part II. Fall Naked-Eye Observing List
Constellations
Bright Stars
Ursa Minor (little bear, little dipper)
Polaris
Lyra (lyre, harp)
Vega
Cygnus (swan, northern cross)
Deneb
Aquila (eagle)
Altair
Cepheus (king, doghouse)
Capricornus
Aquarius
Pegasus (horse, great square)
Andromeda (princess)
Pisces (fishes)
Aries (ram)
Cassiopeia (queen, 'W')
Perseus (hero, wishbone)
Taurus (bull)
Aldebaran
Auriga (charioteer, pentagon)
Capella
Orion (hunter)
Betelgeuse, Rigel
Gemini (twins)
Castor, Pollux
Other Celestial Objects or Regions
Mercury, Venus, Mars, Jupiter, Saturn,
Uranus, Neptune, Pluto, Asteroids, Comets
any planets visible in the sky
(check the Celestial Calendar at the
beginning of this manual)
Ecliptic or Zodiac
trace its path across sky
Celestial Equator
trace its path across the sky
Great Andromeda Galaxy (M31)
fuzzy patch in Andromeda
Pleiades (seven sisters)
star cluster in Taurus
Hyades (closest star cluster)
near Aldebaran
Great Nebula in Orion (M42)
center 'star' in Orion's sword
APS 1010 Astronomy Lab
120
The Boulder night sky at 10 pm
October 15, 1999
Constellation Identification - Fall
APS 1010 Astronomy Lab
121
Telescope Observing
TELESCOPE OBSERVING
SYNOPSIS: You will view and sketch a number of different astronomical objects through the
SBO telescopes. The requirements for credit for telescope observing may vary from instructor to
instructor. The following is given only as a guideline.
EQUIPMENT: Observatory telescopes, observing forms, and a pencil.
Be sure to dress warmly - the observing deck is not heated!
Part I. Observing Deep Sky Objects
The two main SBO observing telescopes (the 16-inch and 18-inch) are both operated by computer.
The user may tell the computer to point at, for example, object number 206, or s/he may specify
the coordinates at which s/he wants the telescope to point. Deep sky objects are easily selected
from the SBO Catalog of Objects found in the operations manual of each telescope. Additional
objects may also easily be located with the 18-inch telescope using TheSky planetarium program.
Your instructor may point the telescopes to at least one of each of the following different types of
deep-sky objects (provided that weather cooperates and appropriate objects are "up" at the time);
distinguishing characteristics to look for have been included.
● Double or multiple stars. Separation of the stars, relative brightness, orientation,
and color of each component.
●
Open clusters. Distribution, concentration, and relative brightness and color of the
stars.
●
Globular clusters. Shape, symmetry, and central condensation of stars.
●
Diffuse nebulae. Shape, intensity, color, possible association with stars or clusters.
●
Planetary nebulae. Shape (ring, circular, oblong, etc.), size, possible central star
visible.
●
Galaxies. Type (spiral, elliptical, irregular), components (nucleus, arms), shape and
size.
For each of the above objects that you observe:
I.1
In the spaces provided on the observing form, fill in the object's name, type, position in
the sky (R.A. and Dec.), etc. Make sure you note what constellation the object is in; this
information is almost essential when using the reference books!
I.2
Observe through the telescope and get a good mental image of the appearance of the
object. You may wish to try averted vision (looking out of the corner of your eye) to aid
you in seeing faint detail. Take your time; the longer you look, the more detail you will
be able to see.
APS 1010 Astronomy Lab
122
Telescope Observing
I.3
Using a pencil, carefully sketch the object from memory, using the circle on the
observing form to represent the view in the eyepiece. Be as detailed and accuracy as
possible,
indicating
color,
brightness, and relative size.
I.4
Include an "eyepiece impression" of
what you observed: a brief statement
of
your
impressions
and
interpretations; feel free to draw
upon comparisons (eg., "like a
smoke ring", "a little cotton ball",
etc.), and to express your own
enthusiam or disappointment in the
view.
I.5
If you wish, or if your instructor has
required it, research some additional
information on your objects. The
Observatory lab room has some
sources, as does the Math-Physics Library. Specific useful books are Burnham's
Celestial Handbook, the Messier Album, and textbooks. Read about the object, then
provide any additional information you think is particularly pertinent or interesting.
Part II. Planetary Observations
Most of the other eight planets (besides the Earth) are readily observed with the SBO telescopes.
The difficult ones are Pluto (tiny and faint) and Mercury (usually too close to the Sun). Provided
that they are available in the sky this semester (consult the “Celestial Calendar” section at the
beginning of this manual):
II.1
Observe, sketch, and research at least two of the solar system planets as in paragraphs I.1
through I.5 above. Pay particular attention to relative size, surface markings, phase, and
of course any special features such as moons, shadows, rings. You may wish to use
different magnifications (different eyepieces) to attempt to pick out more detail.
APS 1010 Astronomy Lab
123
NAME:
Object Name
Object Type
Constellation
R.A.
Dec.
Date
Time
Telescope size
Sky Condition
Description of Observation
Additional Information
Object Name
Object Type
Description of Observation
Constellation
R.A.
Dec.
Date
Time
Telescope size
Sky Condition
Additional Information
Telescope Observing
APS 1010 Astronomy Lab
124
NAME:
Object Name
Object Type
Constellation
R.A.
Dec.
Date
Time
Telescope size
Sky Condition
Description of Observation
Additional Information
Object Name
Object Type
Description of Observation
Constellation
R.A.
Dec.
Date
Time
Telescope size
Sky Condition
Additional Information
Telescope Observing
APS 1010 Astronomy Lab
125
NAME:
Object Name
Object Type
Constellation
R.A.
Dec.
Date
Time
Telescope size
Sky Condition
Description of Observation
Additional Information
Object Name
Object Type
Description of Observation
Constellation
R.A.
Dec.
Date
Time
Telescope size
Sky Condition
Additional Information
Telescope Observing
APS 1010 Astronomy Lab
126
NAME:
Object Name
Object Type
Constellation
R.A.
Dec.
Date
Time
Telescope size
Sky Condition
Description of Observation
Additional Information
Object Name
Object Type
Description of Observation
Constellation
R.A.
Dec.
Date
Time
Telescope size
Sky Condition
Additional Information
Telescope Observing
APS 1010 Astronomy Lab
127
NAME:
Object Name
Object Type
Constellation
R.A.
Dec.
Date
Time
Telescope size
Sky Condition
Description of Observation
Additional Information
Object Name
Object Type
Description of Observation
Constellation
R.A.
Dec.
Date
Time
Telescope size
Sky Condition
Additional Information
Telescope Observing
APS 1010 Astronomy Lab
128
NAME:
Object Name
Object Type
Constellation
R.A.
Dec.
Date
Time
Telescope size
Sky Condition
Description of Observation
Additional Information
Object Name
Object Type
Description of Observation
Constellation
R.A.
Dec.
Date
Time
Telescope size
Sky Condition
Additional Information
Telescope Observing
APS 1010 Astronomy Lab
129
Observing Lunar Features
OBSERVING LUNAR FEATURES
SYNOPSIS: You will investigate the moon through telescopes and binoculars, and identify and
sketch several of the lunar features.
EQUIPMENT: Sommers-Bausch Observatory telescopes and binoculars, lunar map, lunar
observing forms, and a pencil.
Be sure to dress warmly - the observing deck is not heated!
Part I. Lunar Features
Listed below are several types of lunar features. Read the description of feature types, and identify
at least one example of each using either a telescope or binoculars. Locate and label each feature on
one of the lunar outline charts below. (One of the charts is presented in "normal" view, which
resembles the appearance of the Moon as seen through binoculars. The other is a "telescope
view", which is a mirror image of the Moon as it may appear through the telescopes. Use either or
both charts at your convenience.)
You will encounter a number of additional objects not shown on the outline charts, such as small
craters. Feel free to add them as you encounter them.
Feature types marked with "T" are best seen through a telescope, while those marked with "B" can
be seen with binoculars.
I.1
Maria: These are relatively smooth areas which comprise the "man-in-the-Moon".
They were thought to be seas during the early days of the telescope, but are now known
to be lava flows. (B) Shade in these dark patches with your pencil.
I.2
Craters with central peaks: Many large craters have mountain peaks in their centers
which reach 5000 meters in height. These peaks are produced by a rebound shock wave
produced by the impact that formed the crater. (T)
I.3
Craters with terraced walls: Often the inner walls of a crater collapse as in a
landslide. This sometimes happens several times, giving the inner crater wall a stairstepped appearance. (T)
1.4
Overlapping craters: An impact crater may be partially obliterated by a subsequent
impact, giving clear evidence of which impact occurred at an earlier time in the Moon's
history. (T)
I.5
Craters with rays: Some younger craters have bright streaks of light material
radiating from their centers. The rays are debris splashed out by the impact which
formed the crater that have not been obliterated by subsequent geologic or impact history.
Rays are most prominent near the time of the full Moon. (B)
I.6
Walled plains: A few craters are very large with bottoms partially filled by lava. The
appearance is that of a large flat area surrounded by a low circular wall. (T)
APS 1010 Astronomy Lab
130
Observing Lunar Features
I.7
Rilles: Rilles are trenches in the lunar surface which can be straight or irregular. Some
of them look like dried river beds on the Earth. It is generally believed that they were not
formed by water erosion, but rather by flowing liquid lava. Straight rilles are probably
geological faults, formed when the lunar surface collapsed on one side of the rille. (T)
I.8
Mountains and mountain ranges: Due to the lack of erosion processes on the
Moon, mountain peaks can reach heights of 10,000 meters. (B or T)
Part II. The Terminator
The terminator is the sharp line dividing the sunlit and dark sides of the Moon's front face.
II.1
If the Moon is not full on the night of your observations, carefully sketch the terminator
on the chart. Include irregularities in the line which give visual clues to the different
heights in the lunar features: high mountains, crater edges, low plains. (B)
II.2
Inspect the appearance of craters near the terminator, and those which are far from it.
How does the angle of sunlight make the craters in the two regions appear different? In
which case is it easier to identify the depth and detail of the crater? (If the Moon is full,
look at craters near the edge of the Moon and contrast with those near the center.) (T)
II.3
If you were standing on the Moon at the terminator, describe what would be happening.
Part III. Lunar Details
III.1
Select two lunar features of particular interest to you, and use the attached lunar sketch
sheet to make a detailed pencil sketch of their telescope appearance. Be sure to indicate
their locations on an outline chart so that you may later identify the features. (T)
Part IV. Lunar Map Identification
III.1
Compare your finished lunar outline charts and observing sheets with a lunar map, and
determine the proper names for the features you have identified and sketched.
APS 1010 Astronomy Lab
131
Observing Lunar Features
N
E
W
S
Binocular View
APS 1010 Astronomy Lab
132
Observing Lunar Features
N
W
E
S
Telescope (Inverted) View
North may not be "up" in the eyepiece
APS 1010 Astronomy Lab
NAME:
Object Name
Object Type
Comments
Object Name
Object Type
Comments
133
Observing Lunar Features
APS 1010 Astronomy Lab
NAME:
Object Name
Object Type
Comments
Object Name
Object Type
Comments
134
Observing Lunar Features
APS 1010 Astronomy Lab
135
The Messier Catalog
THE MESSIER CATALOG (CCD IMAGING)
SYNOPSIS: You have the opportunity to use the SBO telescopes to do electronic imaging of
celestial objects from the Messier catalog using a CCD camera.
EQUIPMENT: Sommers-Bausch Observatory telescopes, CCD camera, and image processing
software, a planisphere (rotating star wheel) or other star chart.
NOTE: This is an advanced lab exercise which may be possible only if a sufficient number of
clear nights are available. Students who pursue this exercise should be willing and prepared to
spend additional time and effort in order to process their work.
Part I. The Messier Catalog
During the years from 1758 to 1782 Charles Messier, a French astronomer (b. 1730 d. 1817),
compiled a list of approximately 100 diffuse objects that were difficult to distinguish from comets
through the telescopes of the day. Discovering comets was the way to make a name for yourself in
astronomy in the 18th century, so Messier's aim was to catalog objects that were often mistaken
for comets such that they would not be bothered with again. Ironically these are perhaps the most
interesting objects in the night sky and are well worth looking at!
The Messier Catalog soon became well-known as
a collection of the most beautiful objects in the
night sky, including a supernova remnant,
diffuse and planetary nebulae, open and globular
star clusters, and spiral and elliptical galaxies. It
was one of the first major milestones in the
history of the discovery of deep sky objects, as
it was the first comprehensive and reliable list of
its kind.
The study of these objects by
astronomers has, and continues to, lead to
important discoveries such as the life cycles of
stars, the reality of galaxies as separate “island
universes” and the age of the universe. Many of
the beautiful pictures taken by the Hubble Space
Telescope are of Messier objects. Today's
versions of the catalog usually also include
additional of objects observed by Messier and
his collegial friend, Pierre Mechain, but were
not included in his original list.
Objects in the Messier catalog are given a number preceded by the letter “M” (for Messier, of
course!) For example, the Andromeda Galaxy is the 31st object in the Messier catalog and is hence
also known as “M31.” The table below lists all the Messier objects in order. The object type,
angular size (in arcminutes) and common name are listed, as well as the constellation in which the
object resides. Interestingly, for some of the catalog entries it has been difficult to determine
which object Messier originally saw, as his notes were sloppy! Thus the identity of some objects
are still under debate. For example, there is some question as to the identity of M102. Some
believe it is an accidental re-labeling of M101 while others believe it to be the galaxy NGC 5866
(NGC, the “new general catalog,” is another catalog of non-stellar objects). M104 through M110
are objects that were believed to have been originally seen by Messier but for some reason not
cataloged. They were later added to the Messier catalog by Mechain and other astronomers.
APS 1010 Astronomy Lab
136
The Messier Catalog
Object
Object Type
Angular Size (’)
Distance (kly)
Constellation
Common Name
M1
M2
M3
M4
M5
M6
M7
M8
M9
M10
M11
M12
M13
M14
M15
M16
M17
M18
M19
M20
M21
M22
M23
M24
M25
M26
M27
M28
M29
M30
M31
M32
M33
M34
M35
M36
M37
M38
M39
M40
M41
M42
M43
M44
M45
M46
M47
M48
M49
M50
M51
M52
M53
M54
M55
Supernova Remnant
Globular Cluster
Globular Cluster
Globular Cluster
Globular Cluster
Open Cluster
Open Cluster
Diffuse Nebula
Globular Cluster
Globular Cluster
Open Cluster
Globular Cluster
Globular Cluster
Globular Cluster
Globular Cluster
Open Cluster
Diffuse Nebula
Open Cluster
Globular Cluster
Diffuse Nebula
Open Cluster
Globular Cluster
Open Cluster
Open Cluster
Open Cluster
Open Cluster
Planetary Nebula
Globular Cluster
Open Cluster
Globular Cluster
Spiral Galaxy
Elliptical Galaxy
Spiral Galaxy
Open Cluster
Open Cluster
Open Cluster
Open Cluster
Open Cluster
Open Cluster
Double Star
Open Cluster
Diffuse Nebula
Diffuse Nebula
Open Cluster
Open Cluster
Open Cluster
Open Cluster
Open Cluster
Elliptical Galaxy
Open Cluster
Spiral Galaxy
Open Cluster
Globular Cluster
Globular Cluster
Globular Cluster
6
13
16
26
17
15
80
60
9
15
14
15
17
12
12
7
11
9
13
28
13
24
27
5
40
15
8
11
7
11
178
8
73
35
28
12
24
21
32
1
38
85
20
95
110
27
30
54
9
16
11
13
13
9
19
6.3
36.2
30.6
6.8
22.8
2
1
6.5
26.4
13.4
6
17.6
22.2
27.4
32.6
7
5
6
27.1
2.2
4.25
10.1
4.5
10
2
5
1.25
17.9
7.2
24.8
2200
2200
2300
1.4
2.8
4.1
4.6
4.2
0.825
0.3
2.4
1.6
1.6
0.5
0.4
5.4
1.6
1.5
60,000
3
37,000
7
56.4
82.8
16.6
Taurus
Aquarius
Canes Venatici
Scorpius
Serpens Caput
Scorpius
Scorpius
Sagittarius
Ophiuchus
Ophiuchus
Scutum
Ophiuchus
Hercules
Ophiuchus
Pegasus
Serpens Cauda
Sagittarius
Sagittarius
Ophiuchus
Sagittarius
Sagittarius
Sagittarius
Sagittarius
Sagittarius
Sagittarius
Scutum
Vulpecula
Sagittarius
Cygnus
Capricornus
Andromeda
Andromeda
Triangulum
Perseus
Gemini
Auriga
Auriga
Auriga
Cygnus
Ursa Major
Canis Major
Orion
Orion
Cancer
Taurus
Puppis
Puppis
Hydra
Virgo
Monoceros
Canes Venatici
Cassiopeia
Coma Berenices
Sagittarius
Sagittarius
Crab Nebula
Butterfly Cluster
Ptolemy's Cluster
Lagoon Nebula
Wild Duck Cluster
Hercules Cluster
Eagle Nebula
Omega Nebula
Triffid Nebula
Dumbbell Nebula
Andromeda Galaxy
Winnecke 4
Orion Nebula
Praesepe
Pleiades
Whirlpool Galaxy
APS 1010 Astronomy Lab
137
The Messier Catalog
Object
Object Type
Angular Size (’)
Distance (kly)
Constellation
M56
M57
M58
M59
M60
M61
M62
M63
M64
M65
M66
M67
M68
M69
M70
M71
M72
M73
M74
M75
M76
M77
M78
M79
M80
M81
M82
M83
M84
M85
M86
M87
M88
M89
M90
M91
M92
M93
M94
M95
M96
M97
M98
M99
M100
M101
M102
M103
M104
M105
M106
M107
M108
M109
M110
Globular Cluster
Planetary Nebula
Spiral Galaxy
Elliptical Galaxy
Elliptical Galaxy
Spiral Galaxy
Globular Cluster
Spiral Galaxy
Spiral Galaxy
Spiral Galaxy
Spiral Galaxy
Open Cluster
Globular Cluster
Globular Cluster
Globular Cluster
Globular Cluster
Globular Cluster
Open Cluster
Spiral Galaxy
Globular Cluster
Planetary Nebula
Spiral Galaxy
Diffuse Nebula
Globular Cluster
Globular Cluster
Spiral Galaxy
Irregular Galaxy
Spiral Galaxy
Elliptical Galaxy
Spiral Galaxy
Elliptical Galaxy
Elliptical Galaxy
Spiral Galaxy
Elliptical Galaxy
Spiral Galaxy
Spiral Galaxy
Globular Cluster
Open Cluster
Spiral Galaxy
Spiral Galaxy
Spiral Galaxy
Planetary Nebula
Spiral Galaxy
Spiral Galaxy
Spiral Galaxy
Spiral Galaxy
Spiral Galaxy
Open Cluster
Spiral Galaxy
Spiral Galaxy
Spiral Galaxy
Globular Cluster
Spiral Galaxy
Spiral Galaxy
Elliptical Galaxy
7
2
6
5
7
6
14
10
9
8
8
30
12
7
8
7
6
3
10
6
3
7
8
9
9
21
9
11
5
7
8
7
7
4
10
6
11
22
7
5
6
4
10
6
7
22
5
6
9
2
19
10
8
7
17
31.6
4.1
60,000
60,000
60,000
60,000
21.5
37,000
12,000
35,000
35,000
2.25
32.3
25.4
28.0
11.7
52.8
?
35,000
57.7
3.4
60,000
1.6
39.8
27.4
11,000
11,000
10,000
60,000
60,000
60,000
60,000
60,000
60,000
60,000
60,000
26.1
4.5
14,500
38,000
38,000
2.6
60,000
60,000
60,000
24,000
40,000
8
50,000
38,000
25,000
19.6
45,000
55,000
2200
Lyra
Lyra
Virgo
Virgo
Virgo
Virgo
Ophiuchus
Canes Venatici
Coma Berenices
Leo
Leo
Cancer
Hydra
Sagittarius
Sagittarius
Sagitta
Aquarius
Aquarius
Pisces
Sagittarius
Perseus
Cetus
Orion
Lepus
Scorpius
Ursa Major
Ursa Major
Hydra
Virgo
Coma Berenices
Virgo
Virgo
Coma Berenices
Virgo
Virgo
Coma Berenices
Hercules
Puppis
Canes Venatici
Leo
Leo
Ursa Major
Coma Berenices
Coma Berenices
Coma Berenices
Ursa Major
Draco
Cassopeia
Virgo
Leo
Canes Venatici
Ophiuchus
Ursa Major
Ursa Major
Andromeda
Common Name
Ring Nebula
Sunflower Galaxy
Blackeye Galaxy
Bode's Galaxy
Virgo A
Owl Nebula
Pinwheel Galaxy
Sombrero Galaxy
APS 1010 Astronomy Lab
138
The Messier Catalog
I.1
Read through the Messier catalog. Which objects are the farthest away? Is this what
you'd expect for the types of objects that are included?
I.2
Which types of object are the most nearby? Why don't we (or, more specifically, why
didn’t Messier) see these types of objects at greater distances?
Part II. The CCD Camera
Rather than use conventional photography, you will use a special type of electronic camera, known
as a CCD. In the last 15 years, the CCD (“charge-coupled device”) camera has revolutionized the
way astronomers do astronomy. Rather than preparing photographic plates or film using
specialized sensitizing techniques and then exposing for many hours to “capture” the sky,
observatories now use these electronic detectors to reconstruct an image of the sky in the form of
computer bytes and data arrays.
The advantages of using a CCD camera instead of conventional photographic film are numerous:
•
Sensitivity - An astronomical CCD detector is 20 to 60 times more sensitive to light
than film - allowing exposures to be made in seconds or minutes rather than hours.
•
Linearity - Twice the amount of light produces twice the signal value in a CCD, while
the behavior of photographic film is much more complex; it is therefore much easier to
analyze the results in a quantitative manner.
•
Dynamic range - The CCD detector can accommodate a far larger range of intensity
than film; useful information on both bright and dim objects can be obtained
simultaneously in the same field and at the same time.
•
Threshold - While photography requires a certain minimum of light before anything at
all is recorded, CCDs respond to any level of light reaching the detector.
•
Digital processing - Photography is a form of "analog" data, exhibiting a continuous
range of gray-scale brightness that does not lend itself well to quantitative analysis.
Further, development of film takes time and noxious chemicals, so you are unable to see
your results (or worse, mistakes) until well after the fact. CCDs produce "digital" data:
the image is acquired directly by a computer and the information is in the form of an array
of digital values, ready for immediate analysis and processing.
However, photography still offers some advantages over electronic imaging:
•
Size - The largest CCD detectors currently available are about the same size as an
exposed square of 35mm film; most are much smaller. This means that film can
photograph much larger regions of the sky, producing "panoramas" while CCDs usually
concentrate on "close-up" shots.
•
Resolution - The size of an individual CCD picture element, or "pixel", determines the
smallest feature that can be resolved. Photographic grain sizes are much smaller than
current-technology pixel sizes, so that photographs contain more spatial detail.
You will use the CCD camera on the 8-inch telescope that is mounted piggyback on the 18-inch
telescope; this permits the small ST-6 detector to see a larger patch of the sky, thus it is useful for
deep-sky imaging. Even though the 8-inch telescope has less light-gathering capability than the
18-inch, the sensitivity of the CCD detector allows you to get a much brighter image than what you
can see with your eye through the 18-inch. The field of view of the CCD on the 8-inch is similar
APS 1010 Astronomy Lab
139
The Messier Catalog
to that of the 16 or 18-inch with a 55mm eyepiece. This allows you to find your object with the 16
or 18-inch before taking a picture with the CCD.
II.1
How many times more light does the 18-inch telescope collect compared to the 8-inch?
Of prime importance is the field of view of the CCD camera. How much of the sky can it see? To
determine this you must know two things: the plate scale and the size of the detector. The plate
scale is a measure of the angular extent each pixel sees, which depends on the size of the pixels and
the telescope to which the CCD camera is attached.
II.2
III.
For the ST-6 CCD camera on the 8-inch f/6.3 telescope the plate scale is 3.8 arcseconds
per pixel. The ST-6 camera is a 375 x 242 pixel array (meaning it has 375 pixels in each
East-West row and 242 pixels in each North-South column). What is the field of view of
the ST-6 camera when it is on the 8-inch?
Observations
The goal of this lab is to get pictures of at least three Messier objects. Depending on the size of the
class, your instructor will gather you into groups of 2 to 4 students. Once in a group, follow the
following steps for each object. Note: It is essential that you keep good records on each exposure
and include all pertinent background information regarding equipment, technique, and exposure
with each image or photograph. Don't make the same mistakes that Messier made! It’s very
frustrating to create a beautiful image but not know what it is or how it was acquired. Fill in all the
information in the table on the next page.
III.1
Before you can look at an object you must first make sure that it is currently visible in the
night sky. Use the planisphere on the South wall of the observing deck (or your own if
you have one) to determine what constellations are currently visible. Also make sure the
constellation is visible to the 16 and 18-inch telescopes. For example, constellations just
now on the eastern horizon will not be visible (although they may be in a few hours.)
Also, the roof makes it difficult to see objects to the North. Look at the Messier catalog
and find objects within the constellations that you think are visible. Want help finding
objects that look interesting? There are several books at the observatory that have
pictures of many of the Messier objects. The display wall near the entrance of SBO may
also help. And of course your instructor will have his or her favorites.
III.2
Will your object be too big to look at? Some of the Messier objects are very large and
they may not fit in the field of view of the CCD camera. The Messier catalog in this lab
lists the largest angular size for each object in arcminutes (1 arcminute = 60 arcseconds).
Make sure your object will fit in the field of view of the CCD camera. If it doesn't you
may want to choose another object.
III.3
Once you have an object you'd like to observe, ask the instructor or the assistant to move
the 16-inch telescope to your object. This will confirm that your object is indeed visible
to and accessible by the telescopes. If not, return to step III.1 and start over. Look at the
object through the main eyepiece and confirm that it is there. It may be very faint and
hard to see. Allow your eye a few minutes to let your eye adjust and use the “averted
vision” technique to help you see it. Hopefully this will later give you an appreciation for
the power of the CCD camera!
III.4
Now go to the 18-inch telescope and ask your instructor or assistant to move the
telescope to your object. Guide the telescope so that your object is near the center of the
field of view as seen through the 18-inch eyepiece. Ask your instructor to take a 30second exposure. The picture will shortly appear on the screen. Ask the instructor to
APS 1010 Astronomy Lab
140
The Messier Catalog
save the image to disk. Be sure to record the filename of your object. Otherwise you
will be unable to later determine which file is yours!
III.5
Repeat steps III.1 through III.4 for at least two more objects. If time permits you may do
more if you like.
Catalog
Number
IV.
Object Type
Object Name
(if any)
Constellation
Weather (e.g.
clear, hazy, etc.)
Image
Filename
Image Analysis
Your instructor will return to you printed copies of you images. These are yours to keep so don't
write on them! But before you take them home and hang them on your wall there's some science
to be done with them. A property of an object that is of fundamental interest is its actual physical
size.
IV.1
Determine the largest angular size of three objects you observed. Do this by measuring
the width of your image to determining the scale of the image in arcminutes/cm
(remember that you already know the field of view from II.2). Now measure your object
along is longest length, and use the image scale to determine its largest angular size.
How do these compare to the values given in the Messier catalog? Why might there be
some discrepancy?
IV.2
Use the small angle approximation relationship (as discussed in other labs as well as a
section in the introductory section of the manual) to determine the actual physical size of
your three objects. The distances to each object are also listed in the Messier catalog in
units of kilo-light years (kly). Give your answers in appropriate units (light years,
Astronomical Units, etc.) e.g., centimeters are not suitable for galaxies!
V.
Fame and Fortune
The Observatory conducts a contest for the “Best Astrophoto of the Month”, which may be either a
CCD image taken at Sommers-Bausch, or a standard photograph of a celestial object. If you think
your finished work is of sufficient quality to have it exhibited for others to see, consult the contest
rules on the observatory’s website (located at http://lyra.colorado.edu/sbo/gallery/ ) and make an
entry or two! Besides notoriety, you’ll also receive a (very) modest reward if you win!
APS 1010 Astronomy Lab
University of Colorado
COURSE INFORMATION
NAME:
ADDRESS:
TELEPHONE:
COURSE TITLE:
LECTURE
Section Number:
Class Time:
Location:
Instructor:
Office Location:
Office Hours:
Telephone:
Lecture Teaching Assistant:
Office Location:
Office Hours:
Telephone:
LABORATORY
Section Number:
Lab Meets (Day & Time):
Lab Location:
Lab Instructor/TA:
Office Location:
Office Hours:
Telephone:
NIGHT OBSERVING SESSIONS
Dates and Times:
APS 1010 Astronomy Lab
University of Colorado