Download Parallax and Distance

Lab Title: Parallax and Astronomical Distances
Equipment:
Sextant
Meter sticks (or tape measures)
Calipers
Magnetic compasses
Introduction:
Since we cannot travel to most celestial objects in order to learn about
their characteristics, we must gather information about them remotely from
Earth by relying on the information contained in their transmitted and
reflected radiation. As we shall see throughout the course, understanding the
characteristics of these objects is a slow but steady process with each new
measurement relying on observations and measurements that have come
before. The first of these observations is simply the position of celestial
objects and the corresponding local time. The second is a measurement of
the distance to nearby objects. The problems of determining the distances
to far away objects will run throughout our studies in astronomy. This is of
great importance because we can always measure the amount of energy
received at the Earth from a star, but to figure out how much energy is
actually emitted by a star, we must somehow determine its distance too!
The Parallax Method
One reason ancient philosophers discounted the notion that the Earth
moved about the Sun was that the stars showed no shift in their angular
position as the Earth "allegedly" moved in orbit about the Sun. In reality,
their angular positions did (and still do) shift, but by amounts too small to be
detected by the naked eye. Plato, Socrates, Aristotle, Ptolemy, and their
contemporaries would have been startled to learn of the actual fantastic
distances to even the nearest stars!
Using photography and telescopes, however, the distance to nearby
stars can be determined, using the Earth's orbit as a baseline:
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Earth's orbit
about the Sun
1 A.U.
1 A.U.
June
To distant "fixed" star
θ
Distance to nearby star in pc
L
θ
❊
θ
2θ
Nearby star
To distant "fixed" star
Dec
In the diagram above, the angle θ is called the parallax angle. The angle 2θ
is called the parallax displacement, or parallax shift.
In actual practice, this measurement requires astronomers to take two
photographs of the same region of the sky six months apart. During that
time interval, nearby stars will shift their position relative to the more distant
stars (see the star chart and transparency overlay in Part II).
In this lab, however, we cannot wait six months, so we will measure
much smaller distances and use a correspondingly smaller baseline for our
measurements. In Part II and Part III, we will make use of astronomical data
to determine parallax angles and stellar distances.
All methods of determining parallax rely on the very simple
θ ),
geometrical relationship between an arc length (d), a subtended angle (θ
and a radial distance (L):
θ
L = d/θ
(1)
L
L
d
θ
θ
d
L
L
If d is much smaller than L (such as in the triangle on the right) then
the arc length is approximately a straight line. Equation (1) can then be used
to determine L if both d and θ are known.
Throughout the course, we will be using angular (arc)
measurements. In the conventional system, there are 360 ° in a full circle.
Each circular degree (°) contains 60 arc minutes (') and each arc minute
contains 60 arc seconds (''). With these definitions we see that:
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1° = 60' = 3600''
360° = 21,600' = 1,296,000''
It will also be necessary to use a unit of angular measure known as
radians. Radians are simply another unit of angular measure defined by the
ratio of the arc length of a portion of a circle to its radius. For example, the
πR. The ratio of
arc length (circumference) of a full circle of radius R is 2π
the circumference to the radius for a complete circle is 2πR/R = 2π. So,
π radians or 57.3 degrees in one radian. For
there are 360 degrees in 2π
example, to compute the number of radians equal to 2 degrees, 20 minutes of
arc, we do the following:
2o 20' = 2 + (20/60) = 2 + 0.33 = 2.33 degrees
2.33 ° × (1 radian/57.3 °) = 0.041 radians
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Part I. Distance Determination Using the Sextant
Procedure:
1.
Review the instructions on how to use the sextant and first practice
using it indoors before heading outside. Make sure you understand how to
read the angle in degrees (°) and minutes (') off the scale.
2.
In the classroom, stand on one side of the room, and sight on an object
on the opposite side. This object (draw one with chalk if necessary) will be
your "nearby star".
3.
Using the meter sticks or tape measures, mark off a total baseline of
2 meters (1 meter on either side of the line to the object). Make sure that
the baseline is perpendicular to the line-of-sight to the object. Tables and
countertops can serve as references.
4.
Stand on one end of the baseline (i.e., the left side), and sight straight
ahead (perpendicular to the baseline). The nearby star should not be in view.
The angular measure on the sextant should be set at zero.
Now, rotate the sight until the nearby star comes into view. The angle
that you must rotate it will be your "angle left". Record this angle in
degrees and minutes of arc in Data Table 1.
5.
Move to the other end of the baseline and repeat the procedure in
Step 4. This angle will be your "angle right" in Data Table 1.
6.
Repeat Step 2 through Step 5 for two other objects of different
distances. One should be closer and one considerably farther (i.e., 20-50
meters). For the more distant object, you will have to go outside. Enter all
of your results in Data Table 1.
7.
Using the meter sticks or tape measures, measure the actual distances
to each of your three objects.
8.
Complete Data Table 1 and compare your distances determined using
the parallax method to the distances you actually measured using the tape
measures. Make use of the conversion between degrees and radians
(1 radian = 57.3 degrees), and equation (1) to determine the experimental
distance (L).
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Date:
Instructor's Signature: _______________
Lab Partners:
Data Table 1. Distances
Object
Baseline
Length
Angle
Right
Angle
Left
(m)
(° and ')
(° and ')
Avg.
Angle
(θ R + θ L)/2
(°)
Parallax
Angle
Exptl.
Distance
(radians)
(m)
Actual
Percent
Meas. Difference*
Distance
(m)
#1
(Near)
#2
(Medium)
#3
(Far)
* Percent Difference = Experimental distance – Measured distance × 100%
Measured distance
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Part II. Distance to Nearby Stars
Procedure:
Included in this part of the lab are two diagrams (made six months
apart) that show five nearby stars against a field of very distant stars. A
scale is included on each diagram in the form of a small line. The length of
this line corresponds to a given number of arc seconds on the scale of this
photograph.
1.
Measure this line in millimeters and divide by the given number
of arc seconds to find the scale of the photograph. Your result will be the
number of arc seconds per millimeter on the photograph.
"=
mm
So the scale of the "photograph" is:
1 mm =
arc seconds
2.
Use a clear plastic sheet to make a copy of the second diagram and
then overlay the two diagrams. Determine the linear displacement (in mm)
for each of the numbered stars and record them in Data Table 2.
3.
Using your scale determined in Step 1, convert your linear
displacements into arc seconds and enter these into Data Table 2 as well.
4.
From the Pre-lab Exercises and your text, recall that equation (1) can
also be written as
θ
L = 1/θ
(2)
where θ is parallax angle (half of the parallax shift) in arc seconds, and L is
the distance in parsecs. Use this result to complete Data Table 2.
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Date:
Instructor's Signature: _______________
Lab Partners:
Data Table 2. Stellar Parallaxes and Distances
Star
Parallax
Shift
(mm)
Parallax
Shift
(")
Parallax
Angle
(")
1
2
3
4
5
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Distance
Distance
(parsecs)
(light-years)
Part III. Distance to the Moon
Included in the lab write-up are Voyager II sky charts of the region near
the Moon. These "photographs" are taken at the same time, but from two
different locations on the surface of the Earth. By overlaying these two
photographs you can determine the distance to the Moon.
First, you will need to determine the baseline distance, which is the
straight-line distance between the two locations. To do this, note that the
difference in longitude between the two cities tells you what fraction of a circle it
is between them. For example, if the difference in longitude is 100 degrees
between two cities near the equator, then the two locations are separated by
100/360th of the Earth's circumference as measured along the curved surface of
the Earth. Rermember to correct for the fact that the baseline must be the
straight-line distance between the two observation points.
Second, you need to overlay the two photos on top of each other and then
measure the parallax displacement between the two different apparent positions
of the Moon relative to the distant stars. To do this, you need to match up the
stars (not the photo frames). To determine the scale of the photographs, note that
the angular diameter of the Moon is 29.5 arc minutes. By using a ruler, or
calipers, to measure the diameter of the Moon on the photographs, you can now
convert from a linear distance scale to an angle. Remember that the parallax
angle is one-half of the measured parallax shift.
Finally, you can calculate the distance to the Moon from equation (1).
Don't forget to convert your angular measure to radians before using equation
(1)! Your classmates, and your instructor will be able to help if you are having
problems at this point.
Calculated distance to the Moon using equation (1) =
km
Percent difference from the accepted value of 400,473 km =
%
Show all of your work in this area:
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Conclusions and Comments
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Name: __________________
Lab Partner: __________________
Pre-lab Exercises: Parallax and Astronomical Distances
1. Hold one finger up at arm's length. View it with one eye open and the
other closed. Now open the closed eye, and close your open eye. Repeat
this a view times. Your finger will appear to shift back and forth. Estimate
what angle (in degrees) it shifts. Recall that your fist subtends an angle of
about five degrees when held at arms length.
Bring your arm in to about half the distance and repeat the exercise.
Now how much is the parallax shift?
Repeat for objects across the room, or outside. What, do you
estimate, is the smallest angular shift you can detect?
θ ) is fairly small,
2. Refer to the sketch shown below. If the parallax angle (θ
astronomers, surveyors, and you can all use a simple formula to relate θ to
the distance (L) to the object, and to half the baseline (d):
θ
L = d/θ
(1)
In Exercise 1, for example, the baseline is the distance between your two
eyes. The distance d is simply half that distance. θ is half of the shift angle you
estimated, and L is the distance from the center of your two eyes to your finger.
Note that if d is measured in centimeters, then L must also be measured in the
same units of centimeters. The angle θ must be in radians.
d
Baseline = 2d
θ
θ
L
d
❊
Distant Object
(a) If the angle θ is measured to be 5.73 degrees, what is θ in radians?
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(b) If the distance (2d) between your eyes is 10 cm, how far away is the
distant object (i.e., your finger)?
3. Refer to the sketch at the beginning of the lab write-up, showing the
Earth's orbit about the Sun and the parallax shift of a nearby star observed
from the Earth at two different times of the year separated by six months.
In this case, the distance d is one Astronomical Unit. Also, the
parallax angles are very tiny. Rather than convert Astronomical Units to
kilometers or meters, and angles from arc seconds to radians, astronomers
modify equation (1) by simply defining a new unit of distance called a
parsec. Equation (1) then becomes:
θ
L = 1/θ
(2)
where L is the distance from the Earth to the star now measured in parsecs,
and θ is the parallax angle measured in arc seconds. Using the diameter of
the Earth's orbit around the Sun as a baseline, one parsec turns out to be
3.26 light-years.
(a) If the parallax angle is 0.1 arc seconds, how far away (in parsecs) is the star?
(b) If the parallax angle is 0.01 arc seconds, how far away (in light-years) is
the star?)
4. Tycho Brahe used a parallax method similar to this to accurately measure
the distance to the Moon as early as 1590. His sextants had an accuracy of
1 arc minute (this means that the smallest parallax shift that he could
measure was about 1/60th of a degree). Estimate the minimum length
baseline (2d) on the Earth that Tycho used to obtain an accuracy of 10% in
his measurement of the distance to the Moon. You will probably want to
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make use of equation (1) and the fact that the distance to the Moon is about
400,000 km. SHOW ALL OF YOUR WORK BELOW.
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