Download Penn State Astronomy 11 Laboratory

Penn State
Astronomy 11
Laboratory
Fall 2002 / Spring 2003
The Pennsylvania State University
Current editor: Michele Stark
Authors:
Many past and present ASTRO 11 TAs and instructors
have contributed to this work, including...
David Andersen
Roger Bartlett
Jason Best
Lee Carkner
David Chuss
Donald Driscoll
John Feldmeier
Mena Ferraro
Rajib Ganguly
Jason Harlow
Ian Hoffman
Anna Jangren
Karen Lewis
Suzanne Linder
Phillip Martell
Michael Sipior
Michele Stark
Dan Weedman
Michael Weinstein Darren Williams
c 2002 by the Department of Astronomy & Astrophysics, The Pennsylvania State
Copyright University
c 2002 by Hayden-McNeil Publishing, Inc. on illustrations provided
Copyright All rights reserved.
Permission in writing must be obtained from the publisher before any part of this work
may be reproduced or transmitted in any form or by any means, electronic or mechanical,
including photocopying and recording, or by any information storage or retrieval system.
Printed in the United States of America.
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ISBN 0-7380-0688-2
Hayden-McNeil Publishing, Inc.
47461 Clipper Street
Plymouth, MI 48170
www.hmpublishing.com
Stark 0688-2 F02
ii
Astronomy 11 Syllabus
A. Along with this laboratory packet, you need to purchase a planisphere from the bookstore
which will help you to locate stars and constellations this semester. You will also need a calculator
capable of scientific notation, and a small flashlight with some type of red filter on it (i.e., covered
with red cellophane, a red balloon, red nail polish, etc.).
B. Students will be required to prepare a written report for each lab, based upon answering
specific exercises and questions in the notebook. It is recommended that as much of this write-up
as possible be done during the lab meeting time, while instructors are available to answer questions.
C. Students must be prepared to go onto the roof of Davey Lab for observing with the telescopes
on any evening. Every lab will be expected to attempt observing until all of the objects in the
observing notebook are seen and described! This is an ongoing assignment that is not optional; it
is required. If the night is clear, it will usually be cold. Always bring adequate warm clothes for
going outside during the lab.
D. Laboratory activities this semester will include the following. Additions, subtractions, and
corrections to this list are likely. The labs may be done out of order.
1. The Semester Observing Project (page 1) — Observations of the Moon, planets, stars and
galaxies will be collected at the end of the semester.
2. The Changing Sky (page 7) — Use of the computer program Skyglobe to find different seasonal
constellations, and to demonstrate nightly and annual changes in the sky.
3. The Scale of Things: How Big Is It? (page 15) — A summary of the scale of things, from
the solar system to the universe, using scale models and analogies.
4. Angles, Navigation, and Data Analysis (page 21) — Celestial navigation and data analysis,
using observations in the planetarium.
5. Planetary Orbits and Kepler’s Laws (page 29) — An examination of Kepler’s three laws which
describe the motions of the planets using the program Orbit Maker.
6. Parallax and the Distances to the Stars (page 39) — Parallax concepts and their use for
measuring distances to the stars.
7. Spectroscopy of Stars and Galaxies (page 45) — How we use spectroscopy to tell us the
chemical content, velocity, and physical properties of distant stars and galaxies.
8. The Inverse-Square Law of Light (page 51) — Use of light meters to learn about the inversesquare relation for light.
9. Understanding the Stars (page 57) — The luminosity-temperature diagram, using data on
the closest and brightest stars, and what it shows about the nature of the stars.
10. The Lives of the Stars (page 63) — Stellar evolution and evolutionary states using astronomical data and luminosity-temperature diagrams.
11. The Structure of the Milky Way Galaxy (page 69) — Structure of our Milky Way Galaxy
through the study of globular clusters and young groups of stars.
12. The Local Group and the Hubble Deep Field (page 77) — Galaxy locations, distance measurements, and a look into the past using images from the Hubble Space Telescope.
13. Distances to Galaxies (page 83) — Visits to various web pages illustrate how to estimate the
large distances to other galaxies.
14. Distant Galaxies and the Expanding Universe (page 87) — Measurements of distant galaxies
and a derivation that the universe is expanding. Also, we determine the age of the universe.
15. The Search for Extraterrestrial Intelligence (page 93) — Using the Drake Equation to estimate
the number of other civilizations in our galaxy.
16. The Moon and Its Phases (page 97) — the phases of the Moon and the relationship between
the Sun, Earth, and Moon.
iii
The Greek Alphabet
α
β
γ
δ
ε
ζ
A
B
Γ
∆
E
Z
Alpha
Beta
Gamma
Delta
Epsilon
Zeta
η
θ
ι
κ
λ
µ
H
Θ
I
K
Λ
M
Eta
Theta
Iota
Kappa
Lambda
Mu
ν
ξ
o
π
ρ
σ
N
Ξ
O
Π
P
Σ
Nu
Xi
Omicron
Pi
Rho
Sigma
τ
υ
φ
χ
ψ
ω
T
Υ
Φ
X
Ψ
Ω
Tau
Upsilon
Phi
Chi
Psi
Omega
The Pleiades
Open Star Cluster
(see Lab 10, page 63)
M15
Globular Star Cluster
(see Lab 10, page 63)
iv
PENN STATE
ASTRONOMY LABORATORY
#1
SEMESTER OBSERVING PROJECT
I. Objective
Over the course of the semester, you will have the opportunity to observe many things, thereby
imitating astronomers throughout history. Although contemporary astronomy is largely an indoor
science, visual and telescopic observations are fundamental. Everything we think we know about
the universe is either supported by observations or hinges on the support of future observations.
Most of the observations you will make this semester will be similar to those that have been made
by countless numbers of astronomers throughout history. However, they will be unique to you, and
give you the opportunity to discover the nature of the universe for yourself.
Students often ask, “What is a good observation?” or “What should I draw?” Frustrated Astro
11 instructors typically respond, “Draw what you see.” The lesson is clear. An observation will
depend on an observer’s eyesight or other equipment used to make the observation, and one’s ability
to sketch in the dark. Even with this element of uncertainty, however, everyone’s sketch of a Full
Moon should be circular, the Andromeda Galaxy should not look like Jupiter, and the North Star
should always be drawn roughly North. You should strive to record what you see as accurately as
possible, but only in as much detail needed to clearly distinguish your object from another. You
need not draw every star in Orion to obtain the general shape, for example.
One note: You will see in Lab #2 that the position and visibility of the stars depends
greatly on when you are doing your observing. In order to be acceptable, every
drawing MUST have the following information:
1. The date,
2. The time,
3. The direction you are facing and the directions to your left and right
4. A label of what you have drawn.
Without this information, you have only made a drawing, not an Observation that
contains useful astronomical data!
II. Exercises
The semester observing project will consist of three activities.
1. naked eye observations of stars, constellations and planets
2. telescopic observations using the telescopes on the roof of Davey Lab
3. naked eye observations of the Moon and its phases
Each activity is described in more detail on the following pages, along with checklists to help you
keep track of what you have and haven’t done. Good Luck!
1
All observations will be collected at semester’s end. You only need to turn in one
observation of each object, but you are encouraged to observe objects more than once until you feel
that your sketch is a good one. Observations will be graded on completeness, accuracy, and clarity.
You will have a planisphere, which can be purchased from the bookstore, to help you find stars and
constellations. Most observations can be done during class time, weather permitting, but many
must be done outside of class time. For out-of-class observations, you are encouraged to observe on
your own or with a friend in a relatively dark area. Take advantage of any opportunity to observe
since it is cloudy over 70% of the time in State College. Enjoy your semester in Astro 11!
Activity 1: Naked Eye Observations of Stars, Constellations and Planets, Checklist
Fall Semester
Ursa Major (Big Dipper)
Ursa Minor (Little Dipper)
with Polaris
Auriga (the Charioteer)
with Capella
Cepheus (the King)
Cassiopeia (the Queen)
Cygnus (the Swan)
with Deneb
Boötes (the Herdsman)
with Arcturus
Lyra (the Lyre)
with Vega
Pegasus (the Winged Horse)
with the Great Square
Perseus (the Warrior)
Taurus (the Bull)
with Aldebaran
Scorpius (the Scorpion)
with Antares
Sagittarius (the Archer)
Summer Triangle
with Deneb, Vega, and Altair
Aquila (the Eagle)
with Altair
Virgo (the Maiden)
with Spica
Hercules (the Strongman)
Spring Semester
Ursa Major (Big Dipper)
Ursa Minor (Little Dipper)
with Polaris
Auriga (the Charioteer)
with Capella
Cepheus (the King)
Cassiopeia (the Queen)
Cygnus (the Swan)
with Deneb
Boötes (the Herdsman)
with Arcturus
Lyra (the Lyre)
with Vega
Pegasus (the Winged Horse)
with the Great Square
Perseus (the Warrior)
Taurus (the Bull)
with Aldebaran
Orion (the Hunter)
with Betelgeuse and Rigel
The Pleiades (the Seven Sisters)
Leo (the Lion)
with Regulus
Canis Major (the Big Dog)
with Sirius (the brightest star in the night sky)
Canis Minor (the Little Dog)
with Procyon
Gemini (the Twins)
with Castor and Pollux
2
Space has been left at the bottom as your instructor may suggest other objects, such as planets
or comets, which are only visible at special times. Also note that the shapes of these constellations
are subject to interpretation. The constellation of Ursa Major has been identified both as a large
animal and a kitchen utensil. Which is it? Again, the best rule of thumb is to see for yourself.
If you see a teapot instead of an archer when observing Sagittarius then remember it as a teapot.
When lying on its side, Orion may look to you like a giant bow tie rather than a hunter; feel free
to use your imagination.
Your sketches for Activity 1 will be semicircular sketches of the sky as shown in the sample
sketch on the following page. The semi-circle is used to depict the half of the sky that you are
facing. The horizon is depicted by the flat portion of the semi-circle. Objects seen directly overhead
should be drawn at the top of the frame. The corners of the semi-circle are reserved for objects
seen on the horizons directly to your left and right. Since the sky you observe depends on date,
time, direction, and observing locale, you must record this information near all sketches. You may
include as many stars, constellations, and planets in one sketch as you wish without making the
sketch incomprehensibly congested. All objects MUST be clearly labeled.
Sample Sketch for Activity 1
Title:
Naked Eye Sketch
Summer Triangle & Moon Phase
Observing Site: Old Main Lawn
Date and Time: Mon, Sept 24, 2001, 8:30pm Sky Condition:
very clear!!
Overhead
Vega
Deneb
Moon
Altair
The Summer Triangle
E
S
W
Your Left
In Front
Your Right
Description:
Drawing made from lawn of Old Main. Sky was clear,
but bright, only a few stars other than those in the summer triangle
were visible. Moon was half full, with right side (west side) illuminated.
3
Activity 2: Telescopic Observations of Planets, Galaxies, Stars and Clusters; Checklist
Fall Semester
Mizar and Alcor in Ursa Major (double star)
Albireo in Cygnus (double star)
M13 in Hercules (a globular cluster)
The Andromeda Galaxy (M31)
Spring Semester
Mizar and Alcor in Ursa Major (double star)
The Orion Nebula
The Pleiades (an open star cluster)
M15 in Pegasus (a globular cluster)
Again, space has been left intentionally blank for planets or other objects which may be visible at
special times. A sample observation for Activity 2 is given below. Telescopic observations capture
a very small portion of the sky, and thus should not be semicircular. Draw your observations within
a circle that represents the edge of your field of view. The same rules for time, date, etc. apply
here.
Sample Sketch for Activity 2
Telescope Sketch
Title:
Saturn & Titan
Observing Site: Davey Lab Roof
Date and Time: Tues, Nov 27, 2001, 9:00pm Sky Condition:
clear
Description:
Looking through 8−inch telescope on the
roof of Davey Lab. Saturn was
yellowish−grey. The rings were bright
and visible. Titan looked just like
Titan
a white dot, much like a star.
Saturn
4
Observing Template for Lab 1
(Please photocopy or reproduce manually as necessary)
Naked Eye Sketch
Title:
Observing Site:
Date and Time:
Sky Condition:
Overhead
Your Left
In Front
Description:
Telescope Sketch
Title:
Observing Site:
Date and Time:
Sky Condition:
Description:
5
Your Right
Activity 3: 10 Naked Eye Observations of the Phase of the Earth’s Moon
For this activity you will sketch the shape, appearance, and position in the sky of the Moon on
different dates. These observations may be done outside of class time; just go outside whenever it’s
clear and look around for the Moon.
As the Moon orbits the Earth once every month, we see the side of it that is illuminated by the
sun from different angles. That is what causes the Moon to appear to go through phases. Each
phase corresponds to a different angle between us (Earth), the Moon, and the Sun. For example,
Full is when the Earth sits between the Moon and the Sun; at that time, the illuminated half of the
Moon is facing us. If the Moon, Earth, and Sun make a right angle, then we see only half of the
side of the Moon illuminated by the Sun, and this is called first or third quarter phase (depending
on which side of the Moon is lit).
Each observation of the Moon must include:
1. date and time of the observation
2. description of observation
3. sky conditions
4. compass directions
5. sketch of the moon’s location in the sky — use a naked-eye constellation observation template
for this (see example on page 3)
6. sketch of the “shape” of the moon
Your instructor will provide you with the details of which phases to observe and/or when you
should make these observations.
Since the Moon is observable during the day and night, try to make at least three day-observations
of the Moon. If the moon is only partially illuminated, make a careful note of which side, left or
right, is lit.
6
PENN STATE
ASTRONOMY LABORATORY
#2
THE CHANGING SKY
I. Objective
You may not be aware of it, but the appearance of the sky is constantly changing. Which stars
and constellations you can see at night depends greatly on both the time, date, and location from
which you do your observations. In this lab, we will try to understand why the sky is different at
different times.
II. Exercises: Rotation of the Earth
During the course of a night or a year, the stars appear to change their positions in the sky.
This is not due to the stars moving through space, but rather the fact that the Earth is moving and
we are moving with it. In this exercise, we will use the computer program Skyglobe 1 to investigate
these apparent motions. Be sure to notice these motions yourself during your observations of the
real sky as you complete your semester observing project! We see two distinct types of motions in
the sky caused by the two different motions of the Earth.
The first motion is the rotation of the Earth about its axis, which runs through the North and
South Poles. This rotation produces the “diurnal” motion of the sky, that is, those motions we see
throughout a day (or night). The most familiar example of this motion is day and night. The Sun
rises in the east when the Earth rotates us around so that the Sun is shining down on us. It sets
in the west every day because we have rotated with the Earth so that we are in the shadow of the
Earth on the other side. The stars in the sky exhibit the same kind of motion. Every night, stars
will also rise in the east above the horizon and then set in the west as they pass below the horizon
and “behind” the Earth.
a. Based on the information given above, answer the following: Suppose you see a
star in a certain part of the sky (say, low in the south-east). About how many hours
will pass before you will observe that same star in the same position in the sky?
** Now log on to your computer and activate the Skyglobe program as your instructor demonstrates.
The map you see before you is a map of the sky, as it appears now, from State College, as if you
were facing south.
** Let’s take a look at some of Skyglobe’s buttons and features, the buttons and bars surrounding
the map. You may want to spend a few minutes “messing around” with these features to get a feel
for the program.
1
Skyglobe software 1997, by Mark A. Haney, a Shareware product of KlassM Software.
7
Skyglobe features:
ADVISORY: Double-clicking on any button causes the action to repeat rapidly (“Turbo”), and
the program’s settings run wildly away from you. Do multiple clicking SLOWLY, except where
indicated. To cancel a Turbo, click on the button once.
On TOP
→ The date and time is indicated, time is 24-hr based. (13:00 means 1:00 PM.)
→ Z – Zoom in (left mouse button) or out (right mouse button) to view less or more of the sky
in your window.
→ M – Magnitude: Add or subtract dimmer stars (right/left mouse button).
On BOTTOM
→ The NESW directional bar: click here to change your view to other compass directions.
At RIGHT
→ The various icons here add or subtract certain features from the map. Try them all.
→ You can add or remove constellation pictures, planets, star names, the horizon, and other
map features with the left mouse button.
→ The right mouse button here adjusts the completeness of the info shown. For example, clicking
on the Vega icon with the left mouse button will turn star names on and off, but the right
mouse button will include the names of dimmer and dimmer stars.
At LEFT
→ These buttons allow you to change the time of your observation.
→ 1 , 5 , 10 : Advance time by 1, 5, or 10 minutes (left mouse button = forward in time, right
mouse button = backward in time).
→ H , D , W , M , Y : Advance time by one hour, day, week, month, year (left/right mouse
button = forward/backward in time).
→ E : Enter any time and date.
→ R : Reset to the current time and date. Once you have reset, click R again to disengage –
otherwise it will periodically reset you, when you don’t necessarily want it to! (Another, way
to reset everything is to restart the whole program.)
** Having familiarized yourself with the program, click (SLOWLY), with the left mouse button,
on the arrow at the bottom ( 5 ), until the horizon looks flat.
** On the top border of the window, you will see the letter Z (for zoom) and a number. Click
(SLOWLY) on the Z with the right mouse-button to bring the zoom number down to 1.5. If you
overshoot 1.5, use the left mouse-button to regain it.
One important star is the North Star or Polaris. It is (almost) directly above the North Pole so
that it appears fixed while all the other stars appear to rotate about it.
b. Go to a north view of the sky and click on the H button (at left) with the left
mouse button, slowly (this advances the time of observation by hourly increments).
Watch the stars, particularly the constellation Ursa Minor (the Little Dipper). Do
the stars rotate clockwise or counterclockwise around Polaris?
8
Set the date and time to today, 21:00 (9 PM).
c. Face each of the four compass directions in turn, by clicking on the N, E, S, and W
letters that appear at the bottom of the window. For each direction, list two bright
stars that appear near the CENTRAL portion of the view (eight stars in all). Mention
the names of the constellations that contain these eight stars.
Do the following for EACH of the eight stars that you listed in part c.
d. Set the date and time to today, 21:00. Locate the star by facing the correct
direction. Now, double-click on the 1 (at left) with the left mouse button, to move
forward in time. Follow the motion of the star (if the star moves off the side of the
view, you may have to face another direction to find it again). Stop the motion of the
sky at 3:00 tomorrow morning (six hours after you started). Where is the star now?
(North, south, east, or west? High or low in the sky? Has the star passed below the
horizon?)
** Now move forward in time more quickly by double-clicking on 10 (at left) with the left mouse
button. While the sky is wheeling around, click on the N, E, S, W letters (at bottom) to look at
this diurnal motion of the stars. (Also note the rising and setting of the Sun.)
e. Describe the overall motions of the stars as the Earth rotates, when facing north,
east, south, and west.
9
III. Exercises: Revolution of the Earth
The second motion is the revolution of the Earth about the Sun. This rotation produces the
“annual” motion of the sky, that is, those motions we see throughout the year.
Because of the Earth’s motion around the Sun in a year, the Sun APPEARS to move through
the constellations, around the sky, slowly, over the course of one year. The path it appears to take
is called the ecliptic. Therefore, the plane of the Earth’s orbit is called the ecliptic plane.
** Reset Skyglobe to today, at 21:00. Face the northern part of the sky. Now move forward in time
by 3 months, by slowly clicking three times on the M (at left) with the left mouse button. Notice
that each step forward shows the night sky at the same time of night (21:00), but one month later.
f. Quantitatively compare the motion you see in the north sky over three months
(always looking at 21:00), to the motion you saw over six hours in one night (part b).
Do the stars move in the same way? in the same direction? by the same amount?
Again set the date and time to today, 21:00, but this time face south.
g. Over the course of three months (always looking at 21:00), which way do the stars
appear to move — towards the east, or towards the west?
h. If you always look at 21:00, in how many months will the stars be in the exact
same positions as they were today at 21:00? Find out using Skyglobe.
10
To help visualize the annual motion of the stars, perform the following “thought-experiment” (parts
i and j).
i. Draw a circle to represent the orbit of the Earth and put the Sun at the center.
Draw four small circles to represent the Earth during the four different seasons. Label
the top point “Spring,” the left point “Summer,” the bottom point “Fall,” and the
right point “Winter.” Darken the side of the Earth which is not being illuminated
by the Sun. On each of the four Earths, draw a stick figure or mark representing
a person observing the sky at midnight, and at sunset. Label these marks clearly.
(NOTE: To draw this correctly, you need to know that the Earth rotates and revolves
in a counter-clockwise fashion, as seen in this diagram.)
j. At sunset on New Year’s Day, you observe a star at its highest point in the sky.
At midnight that night, you see it low toward the western horizon. Later in the year,
you see the same star at sunset on April Fool’s Day. Where is it in the sky? (Use
your drawing in the last part as a reference. Keep in mind that for a person standing
outside at sunset, the Sun is always in the western part of the sky! Also keep in mind
that stars are VERY far away — to draw the above diagram to scale, the star would
need to be drawn many miles away from the Sun and the Earth.)
11
k. Review your answers to parts f, g, and h. Do they make sense in the context of
the thought-experiment in parts i and j? (Please do not simply answer “yes” or “no”
— elaborate!)
IV. Exercises: Phases of the Moon
Because of the Earth’s motion around the Sun in a year, the Sun APPEARS to move through
the constellations, around the sky, slowly, over the course of one year. The path it appears to take
is called the ecliptic. Therefore, the plane of the Earth’s orbit is called the ecliptic plane.
The Moon orbits the Earth about once a month (“moon”th) in a plane close to, but not exactly
aligned with, the plane of the ecliptic. The “phase” of the Moon depends on how much of the
Moon’s illuminated side is angled toward us, and thus it depends on the orientation of the Moon,
Earth, and Sun.
Another thought-experiment:
l. Draw a circle representing the Moon’s orbit with the Earth at the center. Draw an
arrow indicating which direction the Sun is in, and darken the appropriate side of the
Earth. Now draw four circles representing four different points in the Moon’s orbit
as before. Darken the appropriate side of each Moon and label their phases: New
Moon, First Quarter, Full Moon, and Last Quarter. Mark on the Earth the position
of a person observing the evening sky (a little after sunset).
12
** Reset the time on Skyglobe to 8:00 AM, this morning.
** View the eastern sky. The Sun should appear there. We will now seek the date of the next New
Moon.
** Go forward by days (by clicking slowly on the D at left, with the left mouse button) until the
Moon appears very close to the Sun. (Again, use the right mouse button to backtrack in time, if
you need to.) That will be the date of New Moon.
** Observe the rising of the Sun & Moon by using the 10 button. Go back or forward as necessary
until the Moon is very close to the horizon.
m. What time does the New Moon rise? Why is the New Moon not visible in the
night sky?
** Now click once on W to advance a week (when the moon will be in the NEXT phase, First
Quarter), then click on H slowly a few times until you see the moon rise. Then use the 10 button
to get the exact time of moonrise. (Repeat as necessary to answer the next question.)
n. At what time does the First Quarter Moon rise? The Full Moon? The Third
Quarter Moon? Can you see the First Quarter Moon at sunrise? Can you see a Third
Quarter Moon an hour after sunset?
A solar eclipse occurs when the Moon passes between the Sun and the Earth, casting a shadow
on the face of the Earth. A lunar eclipse occurs when the Earth passes between the Moon and the
Sun, casting the Earth’s shadow on the Moon.
o. What phase is the Moon in during a solar eclipse? How about during a lunar
eclipse? Refer to the diagram in part l.
p. Why do you think we don’t have eclipses every time the Moon is in those phases?
(This is a subtle, yet important, point — you may want to check your answer with
your instructor.)
13
l. Summarize the facts and ideas presented in this lab, including any additional questions you may have.
14
PENN STATE
ASTRONOMY LABORATORY
#3
THE SCALE OF THINGS: HOW BIG IS IT?
I. Objective
When you look up at the planets and stars, you are seeing things that are very far away. The
universe is a very big place. How big is it? If we were able to “zoom out,” and look back in at the
Earth and at its place in the solar system, or look at the solar system in the galaxy, what would
it look like? In this lab we will calculate the relative sizes of the various structures in the universe
by making scale models. We will try to comprehend the hugeness of space.
II. Exercises
We begin our exploration of the universe with the Earth: a ball of rock, mostly covered by
water, that is 13,000 kilometers (km) in diameter. Gravity pulls things on the surface toward the
center, and we live up on the surface.
Your instructor will show you something — a volleyball, an orange, maybe something really
weird — that represents a scaled down model of the sun. In real life the sun is a huge ball of
hydrogen gas 1,400,000 km in diameter. (That’s 1.4 × 106 km, or a little less than a million miles).
a. Draw an Earth which is approximately the correct size relative to the model used
for the sun. Explain your calculations and the scale transformations that you use to
make this drawing.
15
The sun is 150,000,000 km away from the Earth (that’s 1.5 × 108 km, or 150 million km).
b. Using the same scale as for part a., determine the distance that should be between
your drawing of the Earth and the instructor’s model of the sun. Explain your calculations. Assign two volunteers to hold the Earth drawing and the model of the sun
the correct distance apart.
Now we’ll use a smaller scale to visualize the solar system (the sun and its nine planets).
Below is a scale drawing of the “inner” solar system: the sun, Mercury, Venus, Earth, and Mars.
Astronomers often use a distance called the Astronomical Unit, or AU, to represent the average
distance between the sun and the Earth (1 AU = 1.5 × 108 km). The “outer” planets are more
distant from the sun, and would be located off the page: Jupiter, Saturn, Uranus, Neptune, and
Pluto.
Sun
✹
Mercury
Venus
Earth
Mars
c. If this drawing represents the “actual” size of the inner solar system, how far away
would Jupiter (5.2 AU) be? How about Neptune (30.1 AU)? Name an object that’s
about the same size as the orbit of Neptune on this scale. Show all work, and be sure
to explain how you got your answers.
16
The star Vega is one of the brightest in the summer sky. Its distance is so great that light itself,
which travels 300,000 km every second, takes 27 years to travel from Vega to the Earth. Thus
the distance to Vega is 27 “light years” (a unit of DISTANCE). The star Deneb appears almost
as bright but is at least 1,600 light years away. Astronomical distances are enormous; light takes
8 minutes just to get from the sun to the Earth. Note therefore that 1 AU is equal to 8 “light
minutes” or 0.000016 light years.
d. If the entire universe was shrunk down to the same scale as in part c., what would
be the scaled distances to Vega and Deneb? That is, if you placed the sketch of the
inner Solar System on the ground and had to walk the correct distance to get to Vega
or Deneb, how many kilometers would you have to walk? Show all work and explain
all of your calculations.
Also, please complete the following sentence: “To a make a scale model of our Solar
System and the stars Vega and Deneb, to the same scale as the diagram in part c.,
I would need to place the Sun in State College, and the stars Vega and Deneb in
(some town) and (some other town).”
All of the stars we can see in the sky are part of our Milky Way Galaxy, a huge collection of
stars of diameter about 100,000 light years. All the stars in the Milky Way, including our sun, are
held together by mutual gravitational attraction. The Milky Way would be disk shaped if we could
travel outside it and look back, and brighter stars would form a nice spiral pattern when viewed
from above. The sun is located about 2/3 of the way out from the center toward one side of the
disk. A sphere of radius 1,600 light years (the distance to Deneb) would enclose most of the stars
we can see with the naked eye in the night sky.
e. Choose a scale, and make a sketch of the Milky Way Galaxy on the back of this
sheet. (You may represent the Milky Way as a plain old circle — nothing fancy is
needed.) Put a dot where the sun should be, and draw a circle around the sun, whose
radius is the correct scaled distance from the sun to Deneb. This circle represents
our local neighborhood of stars that can be seen with the naked eye. Be sure to show
how you determined the diameter of your Milky Way sketch, and the radius of your
“circle around the sun.”
17
Here you may draw your sketch for part e.
18
The observable universe is teeming with galaxies. Now, we think the universe is about 15 billion
(15 × 109 ) years old (see Lab 14). (That’s about 3 times the age of the Earth and Solar System,
which we measure to be 4.6 billion years old.) Since light can only travel at the speed of light,
and the universe did not exist more than 15 billion years ago, the farthest we can ever hope to see
with the largest telescopes is 15 billion light years. In this sense, the radius of the sphere around
us which is the observable universe is 15 × 109 light years.
f. If the Milky Way Galaxy were as small as a piece of paper (20 cm in diameter), how
far away would the most distant galaxy observable be located? Show your calculations
clearly.
g. Put it all together by imagining two trips by rocket or some other sort of spacecraft.
One is a trip to the sun. The other is a trip to the most distant observable galaxy.
How many times longer is the second trip?
19
h. Summarize the facts and ideas presented, including any additional questions you
may have.
20
PENN STATE
ASTRONOMY LABORATORY
#4
ANGLES, NAVIGATION, AND DATA ANALYSIS
I. Objective
In this lab, you will learn how to measure angular differences in positions, how to use this skill
for celestial navigation, and what the scientific concept of a measurement means.
II. Exercises
Your instructor will tell you why the angle of Polaris above the horizon is the same as your
latitude (how far north or south of the Equator you are). Observations will be made in the
planetarium to recreate an attempt at celestial navigation. For the first set of observations, your
instructor will set the sky to the way it would appear tonight at 9 PM in State College.
a. Use the simple sextant provided to measure the altitude of Polaris, in degrees,
above the northern horizon. Your answer should correspond to the latitude of State
College, or about 41 degrees. Practice with the sextant until you get about this
answer. For maximum accuracy, make your observation from as close to the center of
the planetarium as possible. Write down your best measurement.
Your instructor will now set the sky as it would appear from a different latitude on Earth at
the same time tonight. Your challenge is to determine your latitude from celestial observations, as
accurately as you can. Imagine you are the navigator of a ship travelling from Europe to America,
and you need an accurate latitude measurement to land the ship successfully.
b. Make your observation of the altitude of Polaris at the new position. List your
value as precisely as you can estimate it, within fractions of a degree if possible.
c. Explain in detail what factors (at least 3) you feel most affect the precision of your
measurements. (Do not simply say “human error” or “instrument error,” describe
what specifically about humans and instruments can lead to errors.)
21
Everyone’s measurements will now be compared to each other. They will all be somewhat
different from each other, because all measurements are subject to error. Whenever an experiment
is done, the measurements will be either too high or too low, compared to the correct value.
Most of the time, it’s just as likely that someone will be too high as too low. So the mean, or
average, is a good estimate of the “correct” answer. The mean is calculated by adding up all of the
measurements, and dividing the sum by the total number of measurements.
Next, we’d like to estimate how accurate our answer is. One way to estimate the amount of error
in a group of measurements is to look at how they are spread out. Let’s look at a few examples,
taken by imaginary classes doing the same experiment. If the classes’ latitude estimates looked like
this:
Class A: 25,26,27,26,24,27,25,28
Class B: 15,34,18,41,32,37,20,26
you’d probably think that the mean latitude of class A is fairly accurate. Because many people
are getting similar answers, and the spread of measurements are small, you tend to believe their
results. On the other hand, class B’s measurements have a bigger spread, which usually means
that their numbers are less accurate. From their measurements, it’s hard to tell what the correct
latitude is.
A way to measure this spread in a set of data is the standard deviation. To find the standard
deviation of a set of data, follow these instructions:
1.
2.
3.
4.
5.
Find the mean of your data.
For every measurement, take it, subtract the mean from it, and square the result.
Add up all of these squared terms (called the sum of the squared residuals).
Divide by: [the number of measurements minus one].
Take the square root of that result. That is the standard deviation!
The standard deviation gives you an estimate of the error inherent in a measurement. This means
that we don’t believe our result to any better than a standard deviation. Anyone who gives you a
scientific estimate without any error is trying to sell you something. . .
(Another meaning of the standard deviation: approximately 68% of the measurements should be
within one standard deviation of the mean value — see part e.)
22
d. Use everyone’s measurements of the “unknown” latitude, and follow the list of
instructions given on page 22 to find the mean and the standard deviation of the data
set. Show all your calculations (use a separate sheet if necessary).
23
e. Determine if your measurement falls within one standard deviation of the mean
value. (To do this, subtract your measurement from the class average. Compare the
absolute value of the result of this calculation with the standard deviation calculated
above.)
f. Convert the standard deviation from degrees to kilometers (km), explaining how
this conversion is done (remembering that there are 360 degrees in the circumference
of anything and 40,000 km in the circumference of the Earth).
g. Find the difference between the class average and your own measurement, convert
this difference from degrees to kilometers (using the same conversion as in the previous
question), and see how far away your measurement of the latitude was from the mean,
in kilometers. Explain clearly the different steps in this calculation.
24
h. Illustrate on the map: 1. where your ship would come ashore if the class average
is the correct value, 2. the range of uncertainty of landfall (as given by the standard
deviation), and 3. where your own measurement would have indicated a landing.
25
Now, let’s put everything together. Let’s take an imaginary trip to Planet X. A diagram of Planet
X is shown below. Your spaceship lands and you want to find out where on the planet you are. So
you look up and notice that none of the stars rise or set. Instead, they travel in circles that are
parallel to the horizon. All the stars seem to rotate about a stationary point directly above your
head.
Planet X
N
S
i. What are your possible landing spots (name both)? Pick one of these spots and label
it on the diagram above as your landing site. Label it with an ‘L’. (In the following
questions, this point will be referred to as point L.)
You decide to explore Planet X a bit. So, you move 5,000 km in one direction from your landing
site (L). At this new site, you find that some stars rise and set while others do not. All of the stars
seem to rotate about a stationary point that is 45◦ above your horizon. The stars that are near
this point are the same stars that you saw at your landing site.
j. What is your new latitude? Label this new site on the diagram above with a ‘1’.
k. What is the planet’s circumference? (Hint: What fraction of the circumference
have you moved around?)
26
Let’s now imagine a different case. Let’s start again at your landing spot, L. Let’s say that you
still move 5,000 km in one direction from your landing spot. This time, however, you notice that
all of the stars rise straight up in the east and set straight down in the west. You also notice that
the stars appear to be rotating around two stationary points: both are on the horizon — one is
due north, and one is due south.
l. What is your latitude now? Label this new site on the diagram above with a ‘2’.
What is the planet’s circumference in this case?
Let’s do this again. Starting again at point L, you move 5,000 km as before. But, now you find
that some stars rise and set while others do not. All of the stars seem to rotate about a stationary
point that is 30◦ above your horizon. However, the stars that are near this point are not the same
stars that you saw at your landing site, L.
m. Label this site on the diagram above with a ‘3’. What is the planet’s circumference
in this case?
Let’s repeat this one more time. You move 5,000 km from point L as you did before. Now you
notice that none of the stars rise or set. All the stars seem to rotate about a stationary point
directly above your head. These stars, however, are totally different from the stars that you saw
at your landing site, L.
n. Label this site on the diagram above with a ‘4’. What is the planet’s circumference
in this case?
27
o. Summarize the facts and ideas presented, including any additional questions you
may have.
28
PENN STATE
ASTRONOMY LABORATORY
#5
PLANETARY ORBITS AND KEPLER’S LAWS
I. Kepler’s 1st Law: Planets orbit the sun in ellipses (with the sun located at one of the foci)
Here’s an ellipse:
x
focus
b
focus
a
x
• Semi-Major Axis (a): Half the length of the longest dimension
• Semi-Minor Axis (b): Half the length of the shortest dimension
• Eccentricity (e): A number between 0 and 1 that describes how “squashed” the ellipse is
(an ellipse with a small eccentricity, near 0, is very round, whereas an ellipse with a large
eccentricity, near 1, is very elongated)
Here are some ellipses for you:
1.
2.
3.
Questions (note: you do not need to measure anything, this is qualitative)
a. List these ellipses in order of INCREASING eccentricity (i.e., write: “1, 2, 3”, or
“3, 2, 1”, or . . . ).
b. List these ellipses in order of INCREASING semi-major axis (i.e., write: “1, 2, 3”,
or “3, 2, 1”, or . . . ).
29
II. Kepler’s 2nd Law: The line between a planet and the Sun sweeps out equal areas in equal times.
One Month
✸
One Month
Sun
Planet
Planet’s Orbit
This law is a bit strange as stated, however, its meaning should become clear soon. In a weird
way, it qualitatively describes the speed of a planet at different parts of its orbit.
Using Orbit Maker1 , set up the following orbit. The Sun (or some other star) is represented by
“star1,” and “star2” is a planet. Adjust the Scale setting so that you can see the whole orbit, and
set the Step value so that one orbit is completed in a reasonable amount of time.
Name
star1 (Sun)
star2 (planet)
Mass
1
0.0001
x
0
3
y
0
0
z
0
0
vx
0
0
vy
0
5
vz
0
0
c. As you watch the planet orbiting around the Sun, describe the shape of its orbit.
According to Kepler’s 1st Law, the Sun is located at one of the foci of the planet’s
elliptical orbit. What is located at the other one?
d. Does the planet travel at the same speed during the entire orbit? If not, describe
how its speed changes at different points in its orbit (i.e., describe where it is when
it’s going fastest, and where it is when it’s going slowest). Where does the planet
spend most of its time — close to the Sun, or far from the Sun?
1
Orbit Maker software 1996, by Charles Meegan, distributed by Zephyr Services.
30
Adjust the Step value so that it takes roughly 60 seconds for the planet to complete its orbit.
Reset the orbit.
e. Draw an ellipse that roughly corresponds to the ellipse on the screen. Include the
location of the Sun and the initial location of the planet (please label these points).
Start the motion; let the planet orbit for about 5 seconds then stop it. Mark its new
position on your ellipse. Keep incrementing its orbit by 5 second intervals and continue
to record its position each time, until the orbit is complete. Draw lines between each
location of the planet and the Sun (just like in the figure at the beginning of this
section).
f. Look at the area contained in each sector (between two lines). Are they comparable
(the same)? How does this fit in with Kepler’s 2nd Law? Based on your observations,
do you think Kepler’s 2nd Law is correct? How does Kepler’s 2nd Law of “equal areas
in equal times” relate to your observations in question d. about the changing speed of
the planet in its orbit?
31
III. Kepler’s 3rd Law: For any planet in the Solar System, P 2 = a3
where:
P = orbital period (how long it takes to finish one orbit) in YEARS (Earth-years, that is)
a = semi-major axis (the average distance between the Sun and the planet) in
ASTRONOMICAL UNITS (AU)
This law describes quantitatively how a planet orbits the Sun. It says that as the average
distance between the planet and the Sun INCREASES (or as a gets larger), the time it takes for
the planet to complete its orbit also INCREASES (P gets larger).
All right, let’s make the Solar System. Well, part of it, anyway....
Enter these settings (“star1” = Sun, “star2” = Venus, “star3” = Earth, “star4” = Mars, “star5”
= Jupiter):
Name
star1 (Sun)
star2 (Venus)
star3 (Earth)
star4 (Mars)
star5 (Jupiter)
Mass
1
0.0001
0.0001
0.0001
0.0001
x
0
0.723
1
1.52
5.21
y
0
0
0
0
0
z
0
0
0
0
0
vx
0
0
0
0
0
vy
0
7.39
6.28
5.1
2.75
vz
0
0
0
0
0
g. Reset the screen and notice where the Earth (“star3”) is on the right side along
with all the other planets. Start the planets orbiting, let the Earth make a complete
orbit, then stop the motion; notice where the other planets are in their orbits. Which
planets have completed at least one orbit? Where is Jupiter (“star5”) along in its
orbit? (Has it gone very far?) Which planets complete their orbits the fastest: those
closer to the Sun (inner planets), or those farther away (outer planets)?
h. How does the motion you see relate to Kepler’s 3rd Law (the equation P 2 = a3 )?
Does what you see match what the equation predicts? (What does the equation
predict? — HINT: The x parameter in Orbit Maker is related to the distance from the
Sun, measured in AU’s.)
32
IV. Orbital Mechanics
These next parts of the lab show how different physical parameters affect the shapes of planetary
orbits. To begin, rerun the planet from Section II again; here are its Orbit Maker settings:
Name
star1 (Sun)
star2 (planet)
Mass
1
0.0001
x
0
3
y
0
0
z
0
0
vx
0
0
vy
0
5
vz
0
0
Make sure that you can see the whole orbit, and that the planet takes a reasonable amount of
time to complete its orbit.
Watch the planet orbit for a moment — you will be comparing the period and shape of this
orbit to other ones.
i. First, let’s increase the mass of the Sun and see what affect that has on the orbit of
the planet. Change Orbit Maker so that it has the following settings (Be Sure To Use
The Same Scale And Step Values That You Had For The Previous Orbit!):
Name
star1 (Sun)
star2 (planet)
Mass
2
0.0001
x
0
3
y
0
0
z
0
0
vx
0
0
vy
0
5
vz
0
0
Study the orbits carefully, then fill in the following table to indicate which of the
two planets has the “largest” and “smallest” values for each of the orbital properties
listed.
Semi-major Axis
Orbital Property
Period
Initial Velocity
Eccentricity
planet with
“normal” Sun
planet with
“massive” Sun
j. Now let’s see how eccentricity affects the orbits.
settings. (NOTE: the Sun’s mass is back to normal.)
Name
star1 (Sun)
star2 (planet1)
star3 (planet2)
star4 (planet3)
Mass
1
0.0001
0.0001
0.0001
x
0
3
15
30
y
0
0
0
0
z
0
0
0
0
vx
0
0
0
0
Change Orbit Maker to these
vy
0
5
1.9860
1.1472
vz
0
0
0
0
This time play around with the Scale and Step and adjust accordingly. Make sure
that you can entirely see all three orbits.
33
Study the orbits carefully, then fill in the following table to indicate which of the
three planets has the “largest,” “middle,” and “smallest” values for each of the orbital
properties listed (or if more than one have the same properties, then indicate that it
is the “same as star ”).
Semi-major Axis
Orbital Property
Period
Initial Velocity
Eccentricity
star2
(planet1)
star3
(planet2)
star4
(planet3)
k. Now examine how different initial velocities affect the orbits. Change Orbit Maker
to these settings. (NOTE: the initial velocities, vy , are different, but the distances, x,
are the same.)
Name
star1 (Sun)
star2 (planet1)
star3 (planet2)
star4 (planet3)
Mass
1
0.0001
0.0001
0.0001
x
0
3
3
3
y
0
0
0
0
z
0
0
0
0
vx
0
0
0
0
vy
0
5
4
3
vz
0
0
0
0
Study the orbits carefully, then fill in the following table to indicate which of the
three planets has the “largest,” “middle,” and “smallest” values for each of the orbital
properties listed.
Semi-major Axis
Orbital Property
Period
Initial Velocity
Eccentricity
star2
(planet1)
star3
(planet2)
star4
(planet3)
l. Now, to determine how different starting positions affect the orbits, create these
settings. (NOTE: this time, the distances, x, are different and the velocities, v y , are
the same.)
Name
star1 (Sun)
star2 (planet1)
star3 (planet2)
star4 (planet3)
Mass
1
0.0001
0.0001
0.0001
x
0
3
5
7
y
0
0
0
0
z
0
0
0
0
vx
0
0
0
0
vy
0
3
3
3
vz
0
0
0
0
As with the previous problem, study the orbits carefully, then fill in the following table
to indicate which of the three planets has the “largest,” “middle,” and “smallest”
values for each of the orbital properties listed.
34
Semi-major Axis
Orbital Property
Period
Initial Velocity
Eccentricity
star2
(planet1)
star3
(planet2)
star4
(planet3)
m. Summarize your observations from Section IV by explaining which physical factors
(semi-major axis, period, initial velocity, eccentricity, and/or mass of the sun) affect
the size, shape, and period of the orbit, and in what manner they are affected (i.e., it
makes the property bigger/smaller, faster/slower, etc.).
35
V. Optional Section: Binary Star Systems, and Fiddling with the Solar System (not a good idea!!)
This section is qualitative questions dealing with some really weird star systems. Not all star
systems are “nice” like our own Solar System, some have very strange and unusual orbits (but they
can still be described by modifications of Kepler’s Laws!). The following are just a few examples.
n. Here is a binary star system (two stars orbiting each other), with one star a lot
more massive than the other. Create these settings in Orbit Maker:
Name
star1
star2
Mass
30.7
1.25
x
0
0.8
y
0
0
z
0
0
vx
0
0
vy
0
39.72
vz
0
0
Describe what the orbits in this star system look like.
o. Now enter these settings for another binary star system, which is made up of two
stars that are the same mass:
Name
star1
star2
Mass
1.25
1.25
x
0
1
y
0
0
z
0
0
vx
0
0
vy
0
5
vz
0
0
Describe what the shapes of the orbits in this system are like (are they still ellipses?
how do their semi-major axes compare?). Compare it to the system in question n.
36
p. There has been a lot of talk about planets discovered around other Sun-like stars
(extra-solar planets). These new “solar systems” are very different from our own.
Almost all of them have planets the mass of Jupiter (or larger) in orbits much closer
to their star than Jupiter is to our Sun. Additionally many of these planets have
very elliptical orbits. We are going to “magically” transport one of those extra-solar
planets to the Solar System and let it orbit around the Sun with the other planets.
Your job will be to see what effect it has on the orbits of the inner planets (Venus,
Earth, and Mars). Here are the settings for the inner Solar System planets again:
“star1” is the Sun, “star2” is Venus, “star3” is the Earth, and “star4” is Mars. This
time, “star5” will be the extra-solar planet.
Enter these settings into Orbit Maker:
Name
star1 (Sun)
star2 (Venus)
star3 (Earth)
star4 (Mars)
star5 (massive planet with
small, eccentric orbit)
Mass
1
0.0001
0.0001
0.0001
0.005
x
0
0.723
1
1.52
0.3
y
0
0
0
0
0
z
0
0
0
0
0
vx
0
0
0
0
0
vy
0
7.39
6.28
5.1
15
vz
0
0
0
0
0
Let Orbit Maker run, watch how the system changes. Record your observations.
(Note: increase the Step value so the planets orbit really fast, also try turning the
trails on and off periodically to get a better sense of how the orbits change.)
Here are some things to consider in your observations: What effect does this new
planet have on the inner Solar System? What is the ultimate fate of Venus? What
about Mars? How about the Earth? Does this new planet make the Solar System a
“nice place to live?” What do you think happened to any possible “Earths” that may
have been formed in that planet’s own “solar system?” Any other comments?
After the system has “settled down,” zoom in on the Sun (go to something like Scale
= 1 AU). What effect does this new planet (“star5”) have on the Sun? (Note: astronomers can observe this effect by carefully watching other stars, and are using it to
find more extra-solar planets — they can actually see this effect as varying Doppler
shifts in the spectra of the star.)
(Continue on next page if needed.)
37
VI. Summary (mandatory, not optional)
p. Summarize the facts and ideas presented in this lab, including any additional
questions you may have.
38
PENN STATE
ASTRONOMY LABORATORY
#6
PARALLAX AND THE DISTANCES TO STARS
I. Objective
The question of distance determination lies at the heart of modern astronomy. As direct measurement is hardly an option at the cosmic scale, astronomers have been forced to develop indirect
techniques for establishing a distance to a given object. Geometric parallax is the oldest such
method, and, despite its limitations, has proven remarkably useful in gathering distance information on stars in the solar neighborhood.
II. Fundamentals
A simple way in which to visualize parallax involves the use of your thumb (or some other
socially acceptable digit). Hold your thumb at arm’s length and, with one eye closed, line it up
with a distant reference point — a spot on the blackboard, say. Now, if you switch eyes, you’ll
notice that your thumb has shifted with respect to the reference point. The apparent motion of
your thumb is known as parallax.
Now, try repeating the above exercise, but with your thumb closer to your eyes than before.
What is the relationship between the distance from your eyes to your thumb, and the size of
the resulting parallax? Eyes and brain working together routinely use this simple relationship to
estimate distances (what we call “depth perception”). And astronomers use the very same method
to determine the distances to stars.
However, the distances to even the closest stars are very great indeed; the star closest to the
Earth (besides the Sun) lies more than 4 light years (250,000 AU) away! Observing this star as
you observed your thumb would obviously result in a very tiny parallax — one too small to notice!
To see what we should do to remedy this problem, try the following: Again with your thumb at
arm’s length, close one eye and line up your thumb with a distant reference point. Now, move
your head, first to the left, then to the right. Try this for both small and large motions of your
head, and observe the resulting parallax. What is the relationship between the distance your head
moves and the size of the resulting parallax? Astronomers call the distance your head moves the
“baseline” of the parallax measurement, and it should be clear that, for a given distance, the larger
the baseline, the larger the parallax. To measure the parallaxes of stars, astronomers require the
largest possible baseline. Without leaving Earth, this astronomical baseline is the diameter of the
Earth’s orbit (2 AU), as Figure 1 shows.
Think of observing the target star from Earth at opposite sides of its orbit (i.e., from positions
separated by six months on the calendar). The dashed lines (“lines-of-sight”) in Figure 1 show
us the directions in space we must look from Earth to see the star. From the astronomer’s point
of view, the target star’s position has shifted against the background (which consists of distant
stars); this apparent shift is the star’s parallax. We are thus repeating — on an astronomical scale
— the simple exercise of observing your thumb from each of your eyes. In this case, your two
eyes become the Earth on opposite sides of its orbit; the diameter of the Earth’s orbit (2 AU) is
the baseline. The parallax shift is translated into an angle by observing that the angle formed
by the two lines-of-sight at the target star (θ) grows smaller as the distance to the star increases;
astronomers call half this angle (θ/2 = θp ) the parallax angle. To relate this angle to the star’s
distance, we note that θ/2 is one angle in a right triangle, and apply some simple trigonometry.
39
B = 1 AU
Earth
θ p= θ/2
Distance to Star (D)
✹
Sun
★
Star
θ/2
Earth
Figure 1: The parallax angle is θ/2 = θp .
Those of you for whom ‘simple’ and ‘trigonometry’ are practically antonyms are referred to the
Brief Review of Trigonometry that follows.
A (Very) Brief Review of Trigonometry
(or, an introduction to the Wages of Sin, and getting a good Tan)
c
b
90o
Θ
a
Figure 2: The ubiquitous right triangle.
We begin with Figure 2, the infamous ‘right triangle,’ aptly named from the requirement that
one of the angles be exactly 90◦ . The sum of the other two angles is 90◦ as well (you will recall
that the sum of the three angles in a Euclidean, i.e., flat, triangle must be 180 ◦ ). Right. Now, we
label the lengths of the three sides as a, b, and c. We can then form ratios of the side lengths, and
name them for convenience. We define the following:
tan Θ =
b
a
sin Θ =
40
b
c
cos Θ =
a
c
As a practical matter, suppose we were given side b and angle Θ, and asked to find side a. To
do this, simply invert the equation for the tan(gent) function:
tan Θ =
b
a
=⇒
a=
b
tan Θ
Despite what you were taught in high school, that’s really all there is to trigonometry, aside
from a few messy details that are not required for our purposes. Your calculator will do the rest.
What you should remember is that sine, cosine, and tangent are simply definitions which stand for
a ratio between the lengths of two specific triangle sides. Keep this in mind at all times!
Now, back to astronomy. If you look at the right triangle formed by the Earth, Sun and Star in
Figure 1, and then at Figure 2, you’ll notice that the distance we want is just the length of triangle
side a. We already know that side b, which we will call the “baseline” (or just B), is the radius of
Earth’s orbit, which is 1 Astronomical Unit (AU) in length (by definition). The relevant angle, Θ,
is just θ/2 (which from now on we’ll call θp , the “parallax angle”). Plugging all of these values into
the tan(gent) function, we have:
tan Θ =
b
a
=⇒
tan θp =
B
D
=⇒
D=
B
tan θp
=⇒
D=
1 AU
tan θp
So we see that getting the distance to a star involves nothing more than “solving” a simple right
triangle!
Enough theory! Now, for some practice. . .
III. Exercises
a. Your instructor will provide an artificial “star,” along with two protractors or
theodolites, used to measure the parallax angle. The instructor will place the “star”
some distance away, with the measuring instruments sitting some distance apart. Measure the parallax angle of the fake star. Also, record here the length of the baseline
according to this setup.
b. Calculate the distance to the fake star, using the baseline, the parallax angle, and
the tan(gent) function. Show your work.
41
c. Make a scale sketch of the experiment. This should look like a right triangle with
the distance (D) and baseline (B) as the two sides that make the right angle, drawn
to scale. Label the location of the fake star and the parallax angle. (Double-check:
the parallax angle on your sketch should be the same as the measurement of part a.
Use a protractor to verify this.)
d. If you had done a parallax measurement that was exactly the same as what we
did in class, except that the fake star was placed further away from you, would the
following quantities be larger, smaller, or the same?
Baseline:
Distance:
Parallax:
42
By making the appropriate choice of units, we may simplify the parallax equation. It turns out
that, for small parallax angles, if we express the parallax angle in arc-seconds (where 1 arc-second
= 1/3600◦ ), then the equation for distance (D) may be simplified to:
D=
1
θp
The distance calculated using this equation has units of parsecs (or pc, for short); a parsec
is simply the distance at which a star’s parallax would be 1 arc-second. For example, if θ p = 1
arc-second, then D = 1 parsec; if θp = 0.2 arc-second, then D = 5 pc; and so on. It turns out that
1 parsec = 3.26 light years. For example the distance to the nearest star beyond our Sun, about 4
light years, is equivalent to about 1.2 pc.
e. The Hipparcos satellite (which carries a small, but very accurate telescope) can
accurately measure parallax angles as tiny as 0.01 arc-second — but no smaller. To
what distance does this parallax correspond? Carefully explain why this distance
represents the maximum distance Hipparcos can measure. The galaxy M31 (the Great
Andromeda Spiral) is the most distant object visible to the naked eye; it lies at
a distance of 890,000 pc. Could this distance have been measured by Hipparcos?
Explain your answer.
f. Another option for improving parallax determination is to increase the baseline.
Consider moving the space telescope from question e. to the orbit of Neptune (whose
mean distance from the Sun is about 30.1 AU). Now what is the maximum range for
effective parallax determination? Can you foresee any minor inconveniences caused
by having such an orbit? (Hint: how long is a Neptunian year, anyway? Answer:
1 Neptunian-year = 165.5 Earth-years.)
43
g. Summarize the facts and ideas presented, including any additional questions you
may have.
44
PENN STATE
ASTRONOMY LABORATORY
#7
SPECTROSCOPY OF STARS AND GALAXIES
I. Objective
Spectroscopy is the science of looking at rainbows. By splitting starlight into its different
wavelengths, or colors, we can learn a great deal. By looking at the relative brightness of the colors
in a star’s spectrum, and by measuring the wavelengths and strengths of absorption and emission
lines seen in the spectrum, we can tell what the star is made of, how hot it is, and how fast it is
moving.
We will look at the thermal spectrum of a standard light source, and at the spectra of a few
elements, and compare them with some spectra of stars and galaxies taken by astronomers. In this
way, we can learn what is going on in atoms that are millions of light years away!
II. Exercises
All things emit electromagnetic radiation of some type (radio, infrared, visible light, ultraviolet,
x-rays, gamma rays), because all things have some non-zero temperature. The human body’s
thermal radiation is mostly infrared, for example, whereas an incandescent light bulb radiates
visible light (as well as infrared) due to the high temperature of the bulb’s filament.
You will be shown an incandescent light bulb at different temperatures and you will look at the
different spectra using a diffraction grating.
a. Is white a single color? Does it have a place in the spectrum like red and blue do?
b. Describe the spectrum of the “bright” bulb. Are all the colors equally bright?
c. Now describe the spectrum of the “dim” bulb. How is it different from the “bright”
spectrum? (Of course all of the colors are dimmer, but do the RELATIVE strengths
of the colors change?)
d. If you were shown two light sources, how would you be able to tell which was hotter
if you were unable to touch either one? What type of instrument would you need?
45
You will now be provided with a hand-held spectrograph, and shown an emission line source.
Play with the spectrograph until you can see the emission lines from the source. Your instructor
will demonstrate how to use the spectrograph properly. You must line the source up with the right
hand edge of the spectrograph, and then look through the hole and off to the left to see the rainbow
spectrum. Also, notice that there is a scale which marks nanometers, or nm (1 nm = 1 × 10 −9 m!!);
this is the wavelength of the light.
e. Observe the emission lines from the first source. Sketch the brightest lines that
you see below, and label their colors. Your instructor will tell you what the source
is; try to match the lines you saw with the lines in the spectra on page 49. Label the
brighter lines with their true wavelengths (as given on page 49), in nanometers (nm).
700
600
500
400
Your instructor will now provide you with a source which you must try to identify yourself!
f. Observe the emission lines from the unknown source. Again, sketch the brightest
lines that you see, and label their colors. Compare with the given spectra, and determine which element it is. Label the true wavelengths of the lines (as given on page 49)
which helped you identify the source.
700
600
500
46
400
Atoms of a particular element can emit or absorb light at specific wavelengths, or colors. What
is really happening is electrons are jumping up or down in their atomic orbital levels: a jump down
causes light to be emitted, a jump up causes light to be absorbed. In the sources you have seen,
each element is emitting light at its characteristic wavelengths. In stars, most of the light is emitted
at all wavelengths in a smooth spectrum called “blackbody.” The atoms in the upper layers of the
star can then absorb light at these same characteristic wavelengths.
g. Look at the imaginary spectrum of “a bright star in a distant galaxy,” at the bottom
of page 49. Compare the pattern of absorption lines in this spectrum with the emission
lines in the spectra of hydrogen, helium, mercury and neon. Based on the pattern of
lines, what would you say is the most prominent element in this star?
h. There are three absorption lines in the spectrum of “a bright star in a distant
galaxy.” Measure and record their wavelengths. In part g, you identified a prominent
element in this star. Record the three wavelengths in this element’s emission spectrum
that correspond to the three wavelengths in the star’s spectrum. Compare the two
sets of wavelengths.
i. Based on the wavelength shift that you discovered in part h, would you say that
the distant galaxy that contains this bright star is moving towards us or away from
us? Hint: think about the Doppler shift. Explain your reasoning fully.
47
On the final page of this lab (page 50), you will see the spectral sequence as first described
by Annie Jump Cannon in the early 1900’s. The sequence, in order of decreasing temperature is:
O,B,A,F,G,K,M. This sequence can have numbers attached to it to subdivide the classes further,
with higher numbers meaning cooler temperatures. So the sequence of subclasses goes as follows:
O5, O6, O7, O8, O9, B0, B1, B2, . . . etc. The coolest star is an M9.
j. Can you find lines in any of the stars that indicate the presence of hydrogen? Please
label all of these lines on the spectra. Which classes (O,B,A,F,G,K,M) most prominently
show hydrogen?
k. Summarize the facts and ideas presented, including any additional questions you
may have.
48
Wavelength in nanometers (10−9 m)
700
600
500
400
Hydrogen
656
486
434
Helium
707
668
588
502
447
Mercury
707
580 575 546
436
Neon
703
640
580 541 535
454
Spectrum of a bright star in a distant galaxy:
700
600
500
Wavelength in nanometers (10−9 m)
49
400
The Spectral Sequence
50
PENN STATE
ASTRONOMY LABORATORY
#8
THE INVERSE-SQUARE LAW OF LIGHT
I. Objective
In this lab, we will attempt to understand how the flux from a light source varies with distance
and how that property helps us to determine the distances to astronomical objects.
II. Exercises
In looking at any object that gives off light, one notices that there is a relationship between
how bright that object appears and how far away it is. Obviously this relationship is such that
the farther one is from the source of the light the dimmer the light appears. How bright a light
source appears to an observer is referred to in scientific terms as the flux of the object. Flux is
measured in units of (energy/time/area), and depends on both the distance between the source
and the observer, and how much energy the source is emitting in a given amount of time. This
latter property is an intrinsic property of the light-source, known as the luminosity. It is measured
in units of (energy/time). Again, it is possible to get a feel for the luminosity dependence of the
flux. If two sources are at the same distance but have different energy outputs, the one with the
greater luminosity will appear brighter (i.e., it will have a greater flux.)
Instinctively it makes sense that the flux should increase with luminosity and decrease with
distance, but in order to approach this question from a scientific standpoint, it is necessary to
formulate this concept quantitatively. This is done below in what is called the “inverse-square law”
of light.
L
F =
4πd2
Where L is the luminosity (energy/time), and F is the flux (energy/time/area). The name
“inverse-square” comes about because if the luminosity of the light source is constant (as can be
taken to be true for astronomical objects), the flux decreases as the squared distance increases.
The inverse-square law can be used to determine one of the properties F , L, or d, if the other two
are known.
51
In the first set of exercises, you will become familiar with the inverse-square law by measuring
various quantities, and deriving unknowns from these. The equipment should be set up as follows:
a single light source, and two observing stations. Observing Station 1 is at distance d 1 away from
the light source, Observing Station 2 is at distance d2 , and d2 > d1 .
✹
x
x
Source
Station 1
Station 2
For the measurements done with the flux meters, the following conversion factors apply:
1 lux (the unit read by the instrument) = 0.0015 W/m2 (watts per square meter)
1 W=107 erg/s
1 m = 100 cm
1 W/m2 = 1000 erg/s/cm2
a. Using the flux meters (their operation will be explained to you), measure the flux
from the light source at Observing Station 1 and Observing Station 2. These fluxes
will be known as f1 and f2 .
b. Using parallax (you should remember this from a previous lab, but your instructor
will review), measure the distance from the light source to Observing Station 1. This
will be known as d1 .
c. Using the fluxes f1 and f2 from part a, the distance d1 from part b, and the inversesquare law, calculate the distance between the light source and Observing Station 2.
This will be known as d2 .
52
You instructor will now give you the true value of d2 , which will be known as d2t .
d. Using this true value (d2t ), and your measured flux at Observing Station 2 (f2 ),
calculate the luminosity at Observing Station 2 (which we call L 2 ).
e. Using the value of the distance from the light source to Observing Station 1 determined by parallax, as well as the flux measured at Observing Station 1, calculate the
luminosity at Observing Station 1 (which we call L1 ).
f. It is entirely possible that your L1 will not equal L2 . However, the luminosity of the
light source is an inherent quality of the light that DOES NOT CHANGE. There are
two places where there could have been errors while you were doing this lab so far.
Describe in detail these possible sources of error.
53
We now move on to the second setup, which mimics an astronomical application. NOTE: although
notation is re-used, neither the distances nor the fluxes may be the same as in the previous problems,
so DO NOT MAKE THAT ASSUMPTION!!!
The Observing Station (which in reality would represent Earth) is a certain distance d 1 from
one light source (star) and a distance d2 from a second light source (star), with d2 > d1 . The stars
are of the same luminosity.
✹
x
✹
Source 1
Station
Source 2
g. Using the flux meter, measure the flux f1 from the nearer light source, and f2 from
the more distant source.
h. Using parallax again, measure the distance to the nearer star (this distance will be
known as d1 ).
i. Using the fluxes f1 and f2 from part g, the distance d1 from part h, and the inversesquare law, calculate the distance between the Observing Station and the more distant
star. This will be known as d2 .
54
In the following examples, you will synthesize the knowledge you have gained in the hands-on part
of the lab.
j. The star Alpha (α) is a distance of 100 pc (parsecs) from Earth, and has a luminosity
(Lα ) of 1033 erg/sec. Assuming that the star Beta (β) has the same luminosity (Lβ =
Lα ), at what distance must Beta be from Earth such that the flux from Beta is 10×
the flux from Alpha (in other words Fβ = 10 × Fα )?
The star Gamma (γ), observed from Earth, has a parallax angle of 0.01 arcseconds, while star
Delta (δ) has a parallax of 0.35 arcseconds. Recall that the distance to any star, measured in
parsecs, equals 1 over the star’s parallax angle expressed in arcseconds (that is, D = 1/θ p ).
k. Using their parallaxes, calculate the distances (in parsecs) from Earth to the stars
Gamma and Delta.
l. The fluxes measured from Gamma and Delta are equal. Which star has the greater
luminosity? How many times greater?
55
m. If the value of the luminosity of Gamma is 1029 erg/sec, then what is the luminosity
of Delta?
Congratulations! With a very simple set-up, you have used the nature of light to calculate
distances, in much the same way as astronomers calculate the distances of stars that are too far
away to measure the parallax of.
n. Summarize the facts and ideas presented, including any additional questions you
may have.
56
PENN STATE
ASTRONOMY LABORATORY
#9
UNDERSTANDING THE STARS:
THE H-R DIAGRAM
I. Objective
Stars have temperature and stars have luminosity. To compare these properties of stars, we’ll
create a diagram of luminosity and temperature (a “Hertzsprung-Russell” diagram, named after
the scientists who invented it).
II. Exercises
First is a list of some of the nearest stars in the sky. Temperatures are in Kelvins (K), and
Luminosities are given relative to the Sun (how much brighter or fainter the star would appear if
located right next to the Sun, the unit is called: “Solar Luminosities,” which is abbreviated as L Sun
— note that the Luminosity of the Sun in this unit is 1 Solar Luminosity, or 1 L Sun ).
Star
Sun
Barnard’s
Sirius A
Sirius B
Wolf 359
BD+36◦ 2147
Eri
Ross 154
α Cen A
Temp.
5,800
3,100
10,300
24,800
2,700
3,500
4,800
3,300
5,800
Luminosity
1.0
0.000 44
22.9
0.002 6
0.000 021
0.005 7
0.294
0.000 55
1.60
Star
α Cen B
Proxima Cen
L 726–8A
61 Cyg A
Ross 248
Ross 128
Procyon A
Procyon B
van Maanen’s
Temp.
5,000
3,100
3,000
4,600
3,000
3,200
6,500
9,700
13,000
Luminosity
0.46
0.000 058
0.000 059
0.088
0.000 11
0.000 35
7.38
0.000 55
0.000 19
Temp.
14,500
9,400
30,000
22,100
17,600
4,700
3,500
4,900
4,000
Luminosity
150
15
1,500
1,150
550
220
420
88
240
Next is a list of some of the visually brightest stars in the sky.
Star
Arcturus
Achernar
Canopus
Vega
Spica
Betelgeuse
Altair
Aldebaran
β Ari
β CMi
Temp.
4,500
18,800
7,200
10,800
24,200
3,300
8,200
3,800
9,000
13,400
Luminosity
150
1,260
12,600
50
2,400
8,700
13
180
26
180
Star
α Leo A
β Leo
τ Sco
κ Cen
ρ Car
β Cet
α Cet
ζ Hya
β UMi
57
a. Graph a luminosity-temperature diagram using the graph provided on page 62,
that includes all of the stars listed on page 57. Label the graph and axes (including
the correct units), as described by your instructor. You should use different symbols
for the two groups of stars (nearest and visually brightest). You should also label each
star’s point with its name.
b. Study your diagram and describe any trends that you see relating temperature and
luminosity in general for stars. Find and describe the “main sequence.”
c. Suggest which stars on your diagram seem most unusual. Describe what characterizes these stars.
d. Look at how the two groups of stars (nearest and visually brightest) are distributed
on the H-R diagram. Carefully describe any differences you notice between these two
groups.
58
e. Study your diagram and decide which star is the most similar to our sun, explaining
why you draw your conclusion.
f. Study your diagram and decide which star could be seen to the greatest distance,
explaining why you draw your conclusion.
g. If a star were seen in the sky and measured to have a temperature of 7,000 K,
predict, by examination of your H-R diagram, the luminosity which you think this
star should have, explaining the assumptions underlying your prediction.
59
The results you have obtained for this small sampling of stars are, in fact, representative of the
great majority of stars found anywhere. Stars are grouped into four main categories, depending
on where they are found in this diagram. These categories are: “main sequence” stars, “giants,”
“supergiants,” and “white dwarfs.”
h. On your H-R Diagram, mark and label the locations occupied by: the main sequence, giants, supergiants, and white dwarfs.
The overall color of a star is based on its surface temperature. Cool stars, around 3,000 Kelvins,
are reddish. Stars with temperatures of approximately 6,000 K are yellowish. Hotter stars, say
10,000 K, are pretty much white, but the very hottest stars — more than 20,000 K — have a bluish
tint. So observing the color of starlight is one way that astronomers can roughly gauge a star’s
temperature.
i. FALL SEMESTER: Look at your Lab 1 sketch and notes on the double-star Albireo,
in Cygnus. From your observations, and some careful thinking, can you suggest which
of the four categories (main sequence, giants, etc.) the two stars fall into? You must
explain your reasoning! (Caution: the two stars do not necessarily fall in the same
category.)
SPRING SEMESTER: Look at your Lab 1 sketch and notes on the stars Betelgeuse
and Rigel in Orion. Suppose you knew from other observations (like parallax) that
both of these stars were at approximately the same distance from Earth. From your
observations, and some careful thinking, can you suggest which of the five categories
(main sequence, giants, etc.) the two stars fall into? You must explain your reasoning!
(Caution: the two stars do not necessarily fall in the same category.)
60
j. Summarize the facts and ideas presented, including any additional questions you
may have.
61
10000
1000
100
10
1
62
0.1
0.01
0.001
0.0001
0.00001
30000
25000
20000
15000
10000
5000
0
PENN STATE
ASTRONOMY LABORATORY
# 10
THE LIVES OF THE STARS
I. Objective
In the labs you have done so far, you have examined the physical properties and classifications of
stars. All of these properties and classifications were based on a single instant in the star’s lifetime.
However, stars evolve and change over their lives.
In this lab you will be examining the ages, life-cycles, and evolutionary states of stars. You
will use two different groups of stars: one an open star cluster called the Pleiades, and the other a
globular star cluster called M15. By examining a luminosity-temperature diagram for each of these
clusters, you will see which of their stars have already completed their lives as main sequence stars.
This will tell you how long the stars in these clusters have existed until now, so you can determine
the ages of the Pleiades and M15.
II. Exercises: Open Star Clusters, the Pleiades
To see the Pleiades in the fall, you need to look well after midnight. In the winter and spring,
this group is conspicuous, high in the sky. The seven brightest stars in the Pleiades have names,
which originate in Greek mythology:
#1 . . . Merope
#5 . . . Maia
#2 . . . Alcyone
#6 . . . Calaeno
#3 . . . Electra
#7 . . . Pleione
#4 . . . Taygeta
These seven stars are often nicknamed the Seven Sisters. With exceptional eyesight, you could see
the seven brightest stars without a telescope; most people can only see five. However, the Pleiades
actually contains several hundred stars.
The following table lists the luminosities and spectral types of 24 stars in the Pleiades cluster:
Star
A
B
C
D
E
F
G
H
J
K
L
M
Solar
Luminosity
1,940
1,150
1,940
1,340
929
738
160
138
105
52.4
35.6
25.6
Spectral
Type
B7
B6
B8
B6
B6
B8
B9
B9
B9
A1
A5
A3
Star
N
P
Q
R
S
T
U
V
W
X
Y
Z
63
Solar
Luminosity
19.0
14.2
9.20
6.98
4.78
3.43
3.05
2.11
1.43
1.18
0.87
0.67
Spectral
Type
A6
A5
F1
A9
F2
F3
F6
F6
F8
G2
G6
G6
a. Using the table of stars on page 63, plot an H-R diagram using the graph given on
page 68. Your instructor will remind you how to plot a luminosity-temperature diagram, and how the spectral types listed here relate to the temperature scale you have
already learned about in the H-R diagram lab. Label each point with the appropriate
letter. Also label the axes on the graph (including units).
b. The solid line already plotted on your Pleiades H-R Diagram represents the location
of the main sequence. Compare the luminosities and temperatures of the Pleiades stars
to those of the main sequence. Determine from your own results which of the Pleiades
stars no longer appear to be main sequence stars and list their letters, explaining why
you conclude they are not main sequence stars.
For what follows, you need this information on the lengths of time that different kinds of stars
can survive as main sequence stars:
Spectral
Type
B0
B3
B5
B8
A0
A2
Main Sequence
Lifetime
12 million years
34 million years
100 million years
290 million years
540 million years
860 million years
Spectral
Type
A5
F0
G0
G5
K0
Main Sequence
Lifetime
1.4 billion years
7 billion years
11 billion years
18 billion years
26 billion years
c. Study your diagram and decide where the main sequence terminates (“turns off ”).
Using the information above, estimate the age of the Pleiades cluster. Explain the
reasoning behind your estimate.
64
III. Exercises: Globular Star Clusters, M15
Globular clusters contain some of the oldest stars known. They can be comprised of several
thousand to a million stars. Like we did for the Pleiades, an H-R diagram can be used to determine
the approximate age of a globular cluster, by finding where the main sequence terminates (“turns
off”). Below is an H-R Diagram of M151 a globular cluster orbiting the center of our Milky Way
Galaxy.
1000
100
10
1
0.1
0.01
|
|
|
|
|
|
|
|
|
|
|
d. Do you notice anything different about this H-R Diagram, compared to the one
you made of the Pleiades? Be as complete as possible!
You already found the age of the Pleiades. The Pleiades is an open star cluster in the disk of
our galaxy. Now, we will find the approximate age of M15, which is a globular cluster in the halo
of our galaxy.
e. Using the information given, determine the age of the globular cluster M15. Explain
your answer!
1
The H-R Diagram of M15 is adapted from Durrell & Harris (1993).
65
Now, if we assume that the entire universe has a finite age, the universe has to be at least as
old as M15, and is probably older. Let’s assume that M15 formed about a billion years after the
universe formed.
f. From the information given, what is your estimate for the age of the universe?
Explain your answer. Check to see if this answer makes sense by comparing it to
the estimated age of the Earth, which is 4.6 billion years — would you expect the
universe to be older or younger than the Earth? Do the estimated ages agree with
your expectations?
IV. Exercises: Stellar Remnants
A type B7 star like Alcyone in the Pleiades will end its stellar life by ejecting most of its mass
into a planetary nebula. The remaining mass will collapse into a white dwarf. A white dwarf is a
member of a class of objects called compact objects that also includes neutron stars and black holes.
Compact objects are incredibly dense.
g. Given that the density of a white dwarf is 109 kg/m3 and that the density of a
neutron star is 1017 kg/m3 , determine the weights, in tons, of: 1. a cubic centimeter
(cm3 ) of white dwarf matter, and 2. a cubic centimeter (cm3 ) of neutron star matter.
(1 ton = 900 kg)
66
Stars that are brighter than type B0 end their lives in huge explosions known as supernovae. A
neutron star or a black hole is the remnant that is left after this event.
The Schwarzchild radius (R) of a black hole (the radius of its “event horizon”) is related to its
mass (M ) by the formula:
2GM
R=
c2
where G = 6.674 × 10−11 m3 /kg/s2 and c = 2.998 × 108 m/s.
h. Given that the mass of the Earth is 5.973 × 1024 kg, determine its Schwarzchild
radius if it somehow collapses into a black hole. Do the same for the Sun (the mass
of the Sun is 1.989 × 1030 kg).
i. Summarize the facts and ideas presented, including any additional questions you
may have.
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68
PENN STATE
ASTRONOMY LABORATORY
# 11
THE STRUCTURE OF THE MILKY WAY GALAXY
I. Objective
In this lab, you will learn that we live in the Milky Way Galaxy. Our solar system and all the
stars you can see with your own eyes are in this galaxy. You will learn where we are in the galaxy,
what it looks like, and how old it is.
II. Exercises
Our Milky Way Galaxy is made up of a disk (which you see in the sky as the band of the
Milky Way), a bulge (a large clump of stars surrounding the galactic center), and a halo (a larger,
spherical cloud of stars that surrounds the entire galaxy). The halo is much larger than the bulge.
Our Milky Way Galaxy is made up of mostly stars, gas, and dust. The dust blocks out light from
distant stars, and makes it hard to see a lot of the galaxy, especially the bulge and parts of the
disk.
Now, we know that we live in a galaxy that has a disk-like shape. But, where in the disk do we
live? Are we at the center, or off on one side? In the early 1920’s, an American astronomer named
Harlow Shapley studied this question. He found that there were clusters of stars, called globular
clusters, that were spread around the center of the galaxy evenly in a spherical distribution in
the halo. Harlow Shapley realized that depending on where you are in the galaxy, the pattern of
globular clusters will look different.
Let’s try to perform Harlow Shapley’s experiment and find the center of the galaxy from the
globular clusters. We must start, however, with a discussion of the coordinates involved. Skip the
next text section if you are already familiar with right ascension and declination.
SKY COORDINATES
The celestial sphere is divided up in a system like the “longitude-latitude” system on the Earth’s
surface. The latitude of an object in the sky is called the declination; an object (like Polaris) whose
position is over the Earth’s north pole has a declination of +90 degrees; an object over the south
pole is at −90 degrees. Objects on the celestial equator have a declination of zero.
Longitude on the sky is called right ascension. On Earth, the line defining zero degrees longitude
is fairly arbitrary — it’s the circle that goes from the north pole to the south pole which passes
through Greenwich, England. Right ascension (RA) on the sky also has an arbitrary zero point —
it’s a circle from the north celestial pole (Polaris) to the south celestial pole that passes through
one of the points where the ecliptic crosses the celestial equator (in the constellation of Pisces).
The only difference between longitude on earth and right ascension on the sky is that longitude is
usually measured in degrees (from 0 to 360 degrees), while right ascension is measured in HOURS
(from 0 to 24 hours).
The point to the above discussion is that the coordinate system astronomers have for objects in
the sky is similar to the coordinate system that map makers have on earth. Just as every location
on earth has a longitude and latitude, every object in the sky has a right ascension and declination.
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6
The diagram below represents a view of Earth from outside the Milky Way, looking down at the
Earth’s North Pole. Earth is at the center, the lines show the various directions of right ascension,
and each circle represents two kiloparsecs of distance from the earth. (On this scale, the Earth
would be a VERY tiny point at the center.) The locations and distances for a sample of globular
clusters are plotted.
c. With an uncertainty of ± 12 hour, what is the right ascension of the galactic center?
70
Below is a diagram which views the Earth from above its equator. Again, Earth is a tiny dot
at the center, and each circle represents two kiloparsecs in distance, but this time, directions are
degrees of declination. The same sample of globular clusters is plotted again with declination
against their distance in this diagram.
d. With an uncertainty of ±10 degrees, what is the declination of the galactic center?
e. From both diagrams, what is the approximate distance to the galactic center?
71
On page 73 is a plot comparing the distributions in the sky of two classes of objects: globular
clusters and groups of young stars. This plot represents the whole sky, plotted in a different
“latitude-longitude” coordinate system, where the origin of the plot is the direction to the center of
the Galaxy (which you just found the coordinates of in the RA and Dec. system). “Longitude” is
measured along the plane of the Galaxy, in degrees ranging from 0◦ (at the center) to 180◦ (opposite
from the center) to 360◦ (back to the center). “Latitude” is measured above and below the plane
of the Galaxy in degrees and ranges from: 0◦ on the plane of the galaxy to +90◦ directly above the
plane, and −90◦ below the plane.
f. Compare the distribution of globular clusters with that of the young groups of stars.
(Are the two groups spread evenly over the sky or are they clustered in specific parts
of the sky?)
g. Which group would you study to learn about the properties of the disk of our
Galaxy and why? Which group would you study to learn about the halo of our Galaxy
and why?
h. In lab 10 you found the ages of a “young” group of stars (the Pleiades) and a
globular cluster (M15). Based on the ages you found in lab 10, and your answer to
question g., what can you conclude about the relative ages of the halo and disk of our
Galaxy?
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Distribution of Globular Clusters and Young Star Groups on the Sky
+90 o
Young Star Groups
+80 o x
x
x Globular Clusters
+60 o
x
x
x
+40 o
x
x
x
73
180o
150o
120o
90o
60o
+20 o
x
x
30o
0o
x
x
x
330 o
x
300 o
x x
240 o
270 o
x
xo
−20
x
x
x
x
−40 o
−60 o
−80 o
x
−90 o
x
x
210 o
180o
Assuming that young groups of stars trace the structure of our Galaxy’s disk, then by examining
their distribution, we we can study the structure of the disk of our Galaxy. The following plot shows
the spatial distribution of young star groups around the Sun. At the center of the graph is the
Sun. Distance away from the Sun is plotted radially in kiloparsecs (kpc). Around the circle are
angles measured along the plane of the Galaxy away from the galactic center (0 ◦ represents the
direction to the center of the Galaxy, 180◦ represents the direction exactly opposite the center of
the Galaxy). Think of this plot as a “picture” looking down on a little piece of the disk of our
Galaxy from far above the Sun.
i. Describe the distribution of young star groups near the Sun. (Are the young groups
of stars evenly distributed around the Sun or are they clumped? Are there any trends
in their arrangement?)
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Our Galaxy is classified as a spiral galaxy. This means that the disk of the galaxy is in the shape
of a “spiral,” “vortex,” or “whirlpool.” So within the disk of our Galaxy there are spiral arms,
which are long streams of denser concentrations of stars, gas, and dust.
j. In the diagram of young star groups near the sun, on page 74, there are stars from
parts of three different spiral arms near the sun. Mark the locations of the three spiral
arms in that diagram.
k. The circle below represents the entire disk of our Galaxy viewed from above. The
actual diameter of this disk is about 30 kpc. To scale, add and label the following
structures to complete this scale model of the Galaxy: 1. the central spherical bulge
(diameter = 2 kpc), 2. the position of the Sun (you found this earlier), 3. a circle
representing the extent of the young star groups near the Sun from the figure on
page 74 (NOTE: all of the stars we can see in the sky are contained within this circle).
l. Create and label an edge-on view of this model in the right-hand margin, next to
the face-on view above. Include: the thickness of the disk and bulge (the disk is about
0.6 kpc thick and the bulge is about 0.8 kpc thick), and the location of the Sun.
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m. Why can’t we see stars on the opposite side of the Galaxy from the Sun in visible
light? (Remember that all the stars in our galaxy’s disk that we can see in the sky
are contained within the circle defined by the young star groups that you drew on the
model of the Galaxy.)
n. Summarize the facts and ideas presented in this lab. Also include any additional
questions that you may have.
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PENN STATE
ASTRONOMY LABORATORY
# 12
THE LOCAL GROUP AND THE HUBBLE DEEP FIELD
I. Objective
In this lab, you will learn more about galaxies — what they can look like, and some of their
basic properties.
II. Exercises
Here is a list of most of the galaxies in the Local Group. This group is made up of our own
galaxy, the Milky Way, and its closest neighbors.
(Note: 1 kpc = 1 kiloparsec = 1000 parsec)
Name
Milky Way
Sculptor
Large Magellanic Cloud (LMC)
Carina
Draco
Ursa Minor
Sextans I
Small Magellanic Cloud (SMC)
Fornax
Leo II
Leo I
NGC 6822
IC 5152
WLM
Andromeda I
Andromeda II
Andromeda III
M32
NGC 185
NGC 147
NGC 205
M31 (the Andromeda Galaxy)
IC 1613
M33
DDO 210
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Distance
(kpc)
–
60
60
90
90
90
90
90
150
185
185
520
610
610
675
675
675
675
675
675
675
675
765
765
920
Diameter
(kpc)
40
0.3
6.1
0.2
0.2
0.3
0.9
4.6
0.9
0.2
0.3
2.5
1.5
2.1
0.6
0.6
0.9
1.5
1.8
3.1
3.1
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3.7
14
1.2
a. Which five galaxies in the Local Group are the largest? In a few sentences, compare
the sizes of the other galaxies in the Local Group to the two largest ones.
b. Which three galaxies have the largest angular size (not including the Milky Way)?
These galaxies are the ones that look the largest in the sky. Explain how you get your
answer.
c. Make a scale drawing of the Milky Way and the Large Magellanic Cloud, showing
their relative sizes and the distance between them (the galaxies can be represented by
circles). Write down the scale you use, and your calculations to find the scaled-down
sizes and distance.
78
As you have already seen, galaxies can vary a lot in size. Now, we will look at how their shapes
are different. Two basic types of galaxies are the spiral galaxies and the elliptical galaxies. Spiral
galaxies have a disk-like structure, and a central bulge. Our own Milky Way is a spiral galaxy.
Elliptical galaxies appear round, looking a lot like a football or an egg. Your instructor will show
you examples of what they look like, and explain how the appearance of a spiral galaxy changes
depending on its orientation.
Examples of both types of galaxies can be found in the research-quality Palomar Sky Survey
Photographs of the part of the sky that includes the center of the Local Supercluster. This cluster
is in the direction of the constellation Virgo, so it is sometimes called the Virgo cluster. The Local
Supercluster is 15 million parsecs away from the Local Group, and it contains thousands of galaxies.
d. Use the Palomar Sky Survey Print to locate two elliptical galaxies with different
ellipticities, a spiral galaxy seen face-on, and a spiral galaxy seen edge-on. Make a
diagram to show where these galaxies are on the picture, and sketch their appearances
here.
e. Sketch from the print a pair of galaxies that seem to be colliding or disrupting each
other. Describe the features that lead you to believe this. Mark the position of the
galaxy pair on the diagram you made in the previous question.
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Now, we’ll work with the cover of your lab manual. The picture on the cover is called the
Hubble Deep Field, and it was taken by the Hubble Space Telescope in December 1995. (The weird
L-shape of the picture is due to the fact that the telescope has four cameras: the Planetary Camera
— which is the small square in the upper-right — and three larger Wide Field Cameras.) It is a
panorama of galaxies, some close, most very far away. Almost everything you see on the cover is
a galaxy. Some of them are neither spiral nor elliptical galaxies; they look like irregular “blobs.”
Many of the galaxies in the picture appear so small that their shapes can not be distinguished.
f. Look at the cover of your lab manual, and choose a small region with about thirty
galaxies in it. In the table, list the color and shape of the galaxies: elliptical (E), spiral
(S), irregular (I), or too small to classify (TS). Draw a diagram showing the location
on the cover of the region that you studied.
#
Shape
Color
#
1
16
2
17
3
18
4
19
5
20
6
21
7
22
8
23
9
24
10
25
11
26
12
27
13
28
14
29
15
30
80
Shape
Color
g. From the sample of 30 galaxies you just made, look for trends. What galaxy shape
is the most common? Are there more spiral galaxies or elliptical galaxies? Are the
colors of the galaxies related to the shapes? If you can find any other trends, note
them too.
We will now use the Hubble Deep Field to make a rough estimate of the total number of galaxies
in the universe. We could count up all the galaxies in the picture, but that would take a long time.
Instead we will only count galaxies in the Planetary Camera, the small box in the upper right part
of the picture.
h. Count, as well as you can, the number of galaxies in the Planetary Camera. Knowing that the Planetary Camera is 35 arcseconds by 35 arcseconds in size, and that
the entire sky has an area of 41,253 square degrees (degrees2 ), how many galaxies are
there in the universe? Explain your calculations. (1 degree = 3600 arcseconds.)
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i. Summarize the facts and ideas presented, including any additional questions you
may have.
82
PENN STATE
ASTRONOMY LABORATORY
# 13
CEPHEID DISTANCE SCALE
I. Objective
Because galaxies are the most distant objects that can be seen and studied within the universe,
determining distances to them is equivalent to determining the extent of the universe. Our overall
knowledge of the universe’s size and age depends, therefore, on knowing how far away the galaxies
are. The exercises in this lab will give you some practice in understanding the techniques used to
measure galaxy distances.
The detailed instructions for this lab can be found on the web at the following URL:
http://www.astro.psu.edu/headta/labs/lab13/
It can be accessed from any computer with internet access and a web browser. Write your answers
to the questions on this answer sheet and turn it in.
NOTE: the complete questions and directions for this lab are only on the web page!
They are abbreviated on this answer sheet for the sake of clarity — refer to the web
page for the complete questions and instructions.
II. Cepheid Data Sheet:
Cepheid
Name
Grid
Number
P
mV
MV
(Period in days)
(Apparent Magnitude)
(Absolute Magnitude)
m V − MV
(Distance Modulus)
III. Distance Determination:
• Sum of all mV − MV :
• (mV − MV )average = (Sum of all mV − MV ) /5:
? Distance to M100 =
(Don’t forget units!!!)
Note: 1pc=3.26ly; 1kpc (kilo parsec)=1000pc; 1Mpc (Mega parsec)=1000kpc=1,000,000pc
83
IV. Distances to Other Galaxies
1. Near drawing:
Far Drawing:
2. In a sentence or two compare the apparent sizes of the images in your two drawings.
How did the apparent size of some part of the image change as you moved away from
it? Write a statement describing the relationship between the apparent size of an
object and your distance from the object.
3. The apparent diameter of M100 is:
(don’t forget units!)
4. The apparent diameter of the Coma Spiral is:
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5. The distance to the Coma Cluster is:
Calculations:
V. Summary
Write at least one paragraph summarizing the major points covered in this lab. Be
sure mention what a Cepheid is, how you found them, and how you used them to
find the distance to M100. Then describe one other method used to estimated the
distances to galaxies that we can’t see Cepheids in and the basic ideas and assumptions
behind this method.
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86
PENN STATE
ASTRONOMY LABORATORY
# 14
DISTANT GALAXIES AND THE EXPANDING UNIVERSE
I. Objective
You will verify the observed fact that all galaxies are moving away from us, and you will use
the Doppler shift formula to calculate how fast they are receding. You will prove for yourself that
the nature of their motion means that the Universe is expanding, and you will calculate the age of
the Universe.
II. Exercises
Galaxies form giant groups called galaxy clusters. The cluster closest to our own Local Group
can be seen (with a telescope) in the direction of the constellation Virgo. We call this cluster the
Virgo cluster; it is approximately 50 million light years (15 million parsecs) away. Many even more
distant clusters have been found in other directions. They all contain some very large elliptical
galaxies, and many smaller galaxies.
On the next page are pictures showing what an elliptical galaxy would look like if it were located
in different galaxy clusters. The farther away the cluster, the smaller the galaxy looks — there is an
inverse relationship between apparent size and distance. Next to each galaxy, there is a spectrum
of a bright star in the galaxy. The dark lines are “Balmer” hydrogen absorption lines. These lines
are not always found at the same wavelength; they are “shifted.” However, the general pattern
that the hydrogen lines form in each spectrum always stays the same. That is how we can tell if
a certain line is a hydrogen line, even though it is not always found at the same wavelength. The
“shift” of the pattern is called the Doppler Effect, and it is caused by the motion (relative to us,
the observers) of the star that is emitting the light.
When the wavelengths of hydrogen lines are measured in a laboratory, using a stationary hydrogen lamp, each line is always found at the same wavelength. We call this wavelength the “rest
wavelength” and denote it by λrest . The rest wavelengths of the hydrogen lines (from right to left)
are:
Hα (H-alpha)
Hβ (H-beta)
Hγ (H-gamma)
Hδ (H-delta)
λrest
λrest
λrest
λrest
=
=
=
=
656
486
434
410
nm
nm
nm
nm
[1 nm = 10−9 meter]
a. Label the four “Balmer” hydrogen lines with their names (Hα, Hβ, etc.) in each
galaxy spectra given on page 88 (match the line pattern as it is labeled on the spectrum
of the Virgo cluster galaxy). Also mark the rest wavelength, λrest , of each of these lines
on the scale below the spectra.
87
Elliptical galaxy in. . .
Virgo cluster
Distance: 15 Mpc
Hδ
350
Hγ
450
Hβ
Hα
550
650
750
[nm]
550
650
750
[nm]
550
650
750
[nm]
550
650
750
[nm]
Speed:
Ursa Major cluster
Distance: 190 Mpc
350
450
Speed:
Corona Borealis cluster
Distance: 270 Mpc
350
450
Speed:
Boötes cluster
Distance: 490 Mpc
350
450
Speed:
88
The Doppler effect does not only affect light, but occurs with waves of all kinds. A familiar
example is the change in pitch of the sound from a car as it moves towards you, passes you and
moves away from you. As the car moves towards you, the sound waves that move past you are more
closely spaced than normal — their wavelength is shortened. As the car moves away, the sound
waves move past you with longer spacing than normal — their wavelength is increased. Since a
high-pitched sound has a short wavelength, and a low-pitched sound has a long wavelength, we can
actually hear the Doppler effect.
This is analogous to what happens to light from a moving source. If a star is moving towards
us, its light will have a shorter wavelength — the light is blue-shifted. If the star is moving away
from us, the wavelength of the light is longer — the light is red-shifted. It is easiest to detect the
change in wavelength of the light from the shift of the spectral lines. (The shift of the line is the
difference between the observed wavelength and the rest wavelength.)
b. Compare the rest wavelength and the observed wavelength of the hydrogen lines in
the previous question. Which wavelength is longer? Are the galaxies moving towards
us or away from us?
The shift of the line gets larger as the speed of the light source (relative to us) increases. There
is a formula that makes it possible to determine how fast a source is moving by measuring the
change in wavelength.
v
(λobs − λrest )
=
Doppler formula :
λrest
c
where:
λobs
λrest
v
c
is the wavelength we observe,
is the wavelength from an object which is at rest,
is the speed of the object relative to us,
is the speed that the wave travels at.
A light wave travels at the speed of light, which is 300,000 km/sec.
This is a general formula; it can be used for both sound and light, and for any other kind of
wave phenomenon. If the speed of the object is zero, the shift in wavelength will also be zero. For
an object that is moving at high speed, v will be large, and the shift in wavelength will also be
large.
c. Use the Doppler formula to determine the speeds of the galaxies. (Write your
answer below each spectrum.)
89
d. Compare the distances to the galaxies and the speeds with which the galaxies
are moving away from us, and describe their relationship. For each galaxy, plot the
recession velocity (y-axis) versus the given distance (x-axis). Draw one straight line
which best fits the four data points you have plotted. What is the slope of this “bestfit” line?
Hubble Diagram
You have just done the same calculations that the astronomer Edwin Hubble did in the late
1920’s. The relation you described in d. between the distances and speeds of galaxies is called
Hubble’s Law, and the slope of the line in question d. is known as the Hubble Constant.
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What does Hubble’s Law tell us about the Universe? At first it may seem as if we (in the Milky
Way) are in a “privileged position” in the Universe, since all other galaxies are moving away from
us. Are we at the center of the Universe?? We will perform a “thought experiment” to find the
answer.
Imagine that A, B, C, D, and E are galaxies. The arrows represent the speeds of the galaxies as
seen from A (longer arrow = higher speed). This diagram represents what Hubble’s Law states.
A
B
C
D
E
e. What would an observer sitting in galaxy C see when s/he looked at the other
galaxies? Draw arrows to represent the speeds that this observer would measure.
A
B
C
D
E
f. Change your perspective again and do the same for an observer sitting in E.
A
B
C
D
E
g. Look at the diagrams in questions e. and f. What relation will observers in galaxies
C and E find between speeds and distances of galaxies? Is the Hubble law the same
for observers in all galaxies?
What you have seen in this thought experiment is precisely the explanation of why the proportionality between galaxy distances and speeds leads to the deduction that the Universe is expanding.
All galaxies are getting farther and farther apart all the time!
91
It is also possible to make an estimation of how long the expansion has been going on; this is
the time which astronomers take as the “age of the Universe,” or the time since the Universe began
to expand.
Note: There are 3 × 1019 kilometers in a Megaparsec (Mpc), and there are 3 × 107 seconds in a
year.
h. For any galaxy except the one in the Virgo cluster, take the distance and speed
you have calculated. By imagining backwards in time, calculate how many years this
galaxy has been moving away from the Milky Way. That is how long the Universe has
been expanding; your answer is the age of the Universe! Explain how your answer is
achieved. Does this age match the one you found from the globular cluster in Lab 10?
If it doesn’t match, try to explain the difference!
i. Summarize the facts and ideas presented, including any additional questions you
may have.
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PENN STATE
ASTRONOMY LABORATORY
# 15
THE SEARCH FOR EXTRATERRESTRIAL INTELLIGENCE
I. Objective
Humans have long wondered about the possibility of other intelligent life-forms elsewhere in the
universe. From UFO sightings to science-fiction movies, we see various interpretations of what such
life might be like. Is there any way to scientifically determine if life exists on other planets?
II. Exercises
We are going to attempt to estimate the number of civilizations in our galaxy with which we
could presently be able to communicate with (i.e. send a message and receive a reply). In order
to do this, we will need to determine values for the various factors that describe the chances of
such a civilization developing. Frank Drake, an early pioneer in the search for extraterrestrial life,
compiled a list of these factors in 1961. These factors draw upon knowledge from astronomy to
zoology, and are compiled into a formula known as the Drake Equation. While this is not the only
way to estimate the number of civilizations, it is a generally accepted method. The Drake equation
is as follows:
where:
N = N∗ × f p × n e × f l × f i × f c × f L
N = Number of civilizations we can communicate with in our galaxy at this time. This is
the answer we are looking for.
N∗ = Number of stars in the galaxy. Astronomers estimate this number as 4 × 1011 stars.
fp = Fraction of stars with planets. Of all the stars in the galaxy, what fraction have planetary
systems around them? This should be a number between 0 and 1.
ne = Number of suitable planets per system. In each planetary system, how many planets
have the right conditions for life to evolve? This should be a number between 0 and “a few”.
fl = Fraction of suitable planets on which life evolves. Of all the planets that are suitable for
life, on what fraction does life evolve? This should be a number between 0 and 1.
fi = Fraction of planets with life on which intelligence develops. Of all the planets which
have life, on what fraction will that life develop intelligence? This should also be a number
between 0 and 1.
fc = Fraction of intelligent worlds that develop the ability to communicate. Of all the
worlds with intelligent life, what fraction will have the ability and/or the desire to attempt
communication with other planets via radio signals. Again, this is a number between 0 and
1.
fL = Fraction of lifetime of star for which communication is possible. This is given by the
expression: fL = (Average lifetime of civilization) ÷ (Average lifetime of star)
For stars like the sun the average lifetime is 10 billion years (1×1010 years). You must estimate
the average lifetime of a civilization, from the time it develops the ability to communicate to
its extinction.
The value of each factor depends quite extensively on the definition of a few key terms, which
have caused quite a bit of controversy over the past 40 years.
93
a. In a paragraph (minimum 1/3 page), discuss in detail the factor in the Drake
Equation that you think we have the best grasp on, and why. In that same paragraph,
discuss in detail the factor that you think we have the weakest grasp on, and why.
b. In a paragraph (minimum 1/3 page), discuss in detail your definition of “intelligent
life,” and how this definition would affect the number you would put into the equation
as fi .
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c. To some extent, we can calculate the average lifetime of a civilization using our own
civilization as an example. In a paragraph (minimum 1/3 page), discuss the factors
that can extend and that can decrease the lifetime of our civilization, and how many
years you think that our civilization will live based on these factors.
One reasonable(?) estimate for N (the number of civilizations with which we can communicate)
is 1000. This is based on. . .
i. the most accurate estimate of the astronomical and biological factors based on current knowledge, and
ii. a choice of 2000 years as the average lifetime of a civilization. Recall that this average lifetime
is the period of time for which the civilization can communicate with other civilizations in
the galaxy, and that we on Earth have only had that ability for less than 100 years.
d. Is N = 1000 a promising number for interstellar communication? Based on this,
should we make an effort to attempt communication?
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e. The volume of the Milky Way is 2 × 1013 cubic light years. How many civilizations
are there per cubic light year in our galaxy, based on the information in part d? Call
your answer “E”.
f. Find 1÷E 1/3 (1 divided by the cube root of E). This is the average distance between
civilizations.
g. How long would it take for a light signal to make a round trip to the nearest
civilization, assuming that it is at the distance found in part f ? Is this time too long
for interstellar communication, based on the information in part d? What does this
say about the feasibility of communicating with extraterrestrial life?
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PENN STATE
ASTRONOMY LABORATORY
# 16
THE MOON AND ITS PHASES
I. Objectives
The purpose of this lab is to get you to understand the phases of the moon and the relationships
between the Sun, Earth, and Moon.
II. Size and Scale
a. Measure the diameter of a model Moon, enter it in the table below. Then, calculate
the corresponding model scale size of the Sun-Earth-Moon system. The “scale” of a
model is the scale size (or length) of one object divided by the actual size (or length)
of that object. To calculate the scale size of anything in this model, just multiply the
actual size (or length) of the object by the scale.
Actual Size
Diameter of Moon
3.5 × 103 km
Diameter of Earth
1.3 × 104 km
Diameter of Sun
1.4 × 106 km
Radius of Moon’s Orbit
3.8 × 105 km
Radius of the Earth’s Orbit
1.5 × 108 km
Scale
Scale Size
=⇒
III. Night and Day
For the first section of this lab you will be pretending that your head is the Earth (the North
Pole will be the top of your head, and the South Pole will be under your chin), and a light bulb
will be the Sun.
b. Stand facing the light bulb (Sun). Draw and label a diagram illustrating this setup
as seen from above your head (above the North Pole of the Earth). Shade the “day”
and “night” sides of your head (Earth), and indicate the position of a person who lives
on the end of your nose.
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c. As seen from above the North Pole, the Earth rotates counter-clockwise. In the
diagram from question b, indicate the spots on your head where it is “noon,” “sunset,”
“midnight,” and “sunrise”.
d. What time of the day would it be for a person that lives on the end of your nose
when you are facing the light bulb?
e. Turn a quarter of a circle to the left (so the light bulb is off your right ear). Now
what time is it for someone who lives on the end of your nose? (Make a new diagram
like question b if necessary.)
f. Turn another quarter of a turn to the left (now the light bulb will be behind you).
Now what time is it for a person who lives on the end of your nose? (Make a new
diagram like question b if necessary.)
g. Make another quarter of a turn to the left (so the light bulb will be off your left
ear). Now what time is it for a person who lives on the end of your nose? (Make a
new diagram like question b if necessary.)
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IV. The Phases of the Moon
Now we are going to examine the phases of the moon. Place a model Moon on the end of your
pencil. You will be holding the Moon out at arm’s length and describing/sketching its appearance
at different points in its orbit. Here is a diagram, as seen from above your head (above the Earth’s
North Pole), of the Moon’s orbit around the Earth with the eight different positions we will be
examining indicated (note: it is not drawn to scale). In this diagram the Earth rotates counterclockwise, and the Moon orbits the Earth in a counter-clockwise direction.
7
8
Moon’s Orbit
6
Moon
✹
1
5
Sun
("Light bulb")
Earth
("Your head")
4
2
Sunlight
3
The labeled positions (phases) of the moon have names:
1. New Moon
2. Waxing Crescent (actually any point between 1 and 3)
3. First Quarter
4. Waxing Gibbous (actually any point between 3 and 5)
5. Full Moon
6. Waning Gibbous (actually any point between 5 and 7)
7. Third Quarter (also called Last Quarter)
8. Waning Crescent (actually any point between 7 and 1)
h. In the diagram above, shade the “day” and “night” sides of the Earth as seen
from space, label the “noon,” “midnight,” “sunrise,” and “sunset” points on the
Earth. Then shade the “day” and “night” sides of the Moon at each point in its
orbit as seen from space.
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i. Starting with the Moon at position 1, follow the Moon through one orbit around
the Earth. Shade these circles to match the appearance of the Moon as seen from the
Earth at the eight positions in its orbit indicated in the diagram on page 99.
1. New
Moon
2. Waxing
Crescent
3. First
4. Waxing
Quarter
Gibbous
5. Full
Moon
6. Waning 7. Third
Gibbous
Quarter
8. Waning
Crescent
j. Describe how the apparent shape of the moon changes during one orbit as seen
from the Earth. Does the physical size and shape of the Moon change during its orbit?
k. At what time of the day would the New Moon be due South in the sky (directly
overhead for a person on the end of your nose)? (Look at the diagram for question
h, or draw one if necessary.) What about the First Quarter Moon? The Full Moon?
The Third Quarter Moon?
l. Hold the Moon at the First Quarter Phase position, turn your head to determine
what time the Moon rises for a person who lives on the end of your nose (look at
where the Sun is to figure out the time, make a diagram if necessary). Repeat with
the moon at the Third Quarter Phase position.
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m. Could you see the First Quarter Moon at Sunset? Could you see the First Quarter
Moon just after noon? (i.e., will the Moon be above the horizon at those times?)
n. Could you see the Third Quarter Moon at Sunset? Could you see the Third Quarter
Moon just after Sunrise?
o. An eclipse can occur when the Sun, Earth, and Moon line up (a lunar eclipse is when
the Moon passes into the Earth’s shadow, a solar eclipse is when the Moon’s shadow
falls on the Earth). At what phase(s) can a lunar eclipse occur? At what phase(s) can
a solar eclipse occur?
p. Simulate each of these types of eclipses with your model Moon. Which type was
easier to make? Based on this, which type do you think would be the most common
to see and why?
q. Extra Credit: Why don’t we see solar and lunar eclipses every month?
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r. Sumarize the facts and ideas presented, including any additional questions you may
have.
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