Download Exercise 11

EXERCISE Introduction We use the relationship between the angular size of a
known object and its distance almost automatically
when we are driving an automobile. A car which
"looks half as big" (subtends an angle half as large as
another automobile) is approximately twice as far
away because automobiles are roughly the same
size. We use the fact that we know the approximate
linear sizes of common objects such as cars, build­
ings, and bicycles to estimate their distances from
their apparent angular sizes. The mental conversion
of an observed angular size into an estimated dis­
tance is done easily by most persons for common ob­
jects that are relatively dose. However, we would
have difficulty in estimating the distance to the
Goodyear blimp. We don't see it very often and, if it
is high in the sky, there is no common object at the
same distance to compare it to. The eye can measure
only the ANGULAR separations or angular sizes of
objects; it takes the mind to convert to real physical
measurements. We want to make this process more
systematic and quantitative, and use a simple and
practical technique for measuring angles to measure
distances and/or linear sizes of various objects on the
ground.
One of the basic problems in astronomy is deter­
mining distances to celestial objects. By definition,
parallax is one-half of the angle formed at the celes­
tial body by two intersecting lines drawn from the
ends of a baseline. In the case of objects in the solar
system, the baseline is a diameter of the earth, and in
cases of objects outside the solar system the baseline
is the diameter of the earth's orbit. Technically these
are called geocentrici (or horizontal) parallax and
heliocentric (or annual) parallax, respectively (Fig­
ures 11-1 and 11-2).
If the moon is observed from the two end points
of a diameter of the earth it will be seen in two dif­
ferent positions with respect to the background stars.
The total amount of apparent displacement (2p) is
about two degrees, the exact value depending on the
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78
PART
II
MEASURING IN THE SOLAR SYSTEM
2p
MOON
FIG U R E 11-1. Geocentric or horizontal parallax.
EARTH
STAR
EARTH
SI X MONTHS LATER
FIGURE 11-2. Heliocentric or annual stellar parallax.
earth-moon distance. Since parallax is defined as
one-half of this displacement (p), the parallax of the
moon is about one degree.
A similar shift is seen for all stars during the
course of one year. Such small displacements are
involved that the measurement of stellar parallax is
most commonly done by photographing a field of
stars at intervals of six months, and then measuring
the relative positions of the stars with a micro­
scope.
Even then, the largest parallax we measure for the
nearest star
Centauri), is less than one second of
arc: the size of a quarter (about 1 inch) seen from a
distance of about 3.25 miles!
In order to deal with such large distances we have
defined a distance unit: if the parallax is one second
of are, the distance is said to be one parsec, (a coined
word consisting of the first three letters of the words
parallax and second). Using the s r e equation and
a baseline of 1 AU we find that
1 parsec
206,265 AU,
which is equal to 3.086 X 1013 km. The parsec is a
commonly used unit by astronomers since it allows
us to use the simple relationship
d
«x
where p
d
1
P
parallax in seconds of arc
distance in parsecs.
One parsec is also equal to 3.26 light years.
EXERCISE ELEVEN
ANGLES AND PARALLAX
79
~CPROCEDURE
MEASURING ANGLES
The first step is to make and calibrate your own scale for measuring angles. A transparent
plastic strip marked off in 5° intervals is a simple and convenient scale, as it can be carried
with you easily. The calibration procedure is described in Exercise 2 Observing with Simple
Tools.
PARALLAX
Measuring distance using the parallax method involves measuring the angle an object ap­
pears to move through compared to a distant background when viewed from two different
places. It involves applying the s = r 8 relationship with a different pair of "knowns": now we
can measure sand 8, and solve for the distance r. A common example of parallax is to hold a
finger out in front of you and look at it first with one eye closed, then the other (Figure 11-3).
The finger appears to move back and forth even though we know the apparent motion is re­
ally caused by looking through each eye in turn. If you measured the parallax angle 8 your
finger appears to move through, and the distance between your eyes s (often called the base­
line), you could calculate the distance r to your finger.
Since
s=r8
we can solve for the distance r
s
r = "8 (8
.
ill
d.
ra tans)
or
r
s
.
= 57"8 (8 ill degrees).
We will use this method to measure the distance of a friend or object down a long hallway
or outdoors using a distant object as a reference point.
1. Place an object or have a friend stand partway down the hall. Mark off a baseline of sev­
eral feet.
~
S~
EYES
BASE· [
LINE
BACKGROUND
FINGER
DISTANCE
FIGURE 11-3. A common example of parallax.
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PART
II
MEASURING IN THE SOLAR SYSTEM
(EARTH IN
MARCH) BASELINE
(EARTH'S
ORBIT)
TO
REFERENCE
POINT
{
(OISTANT
BACKGROUNO
STARS)
~--~----------------------------(EARTH IN
SEPTEMBER)
TO
REFERENCE
POINT
FIGURE 11-4. Simple parallax measurements.
2. Standing at one end of the baseline, measure the angle between your friend and some ref­
erence point as far down the hall as possible (angle a in Figure 11-4). If you can go out­
doors, a distant building or radio tower will make an excellent reference object. Repeat the
measurement from the other end of the baseline (angle b in Figure 11-4).
3. Add your two measured angles (a and b) to get the angle e and find the distance to your
friend using s = r e.
~
~
~
~
.......
:J
~
~
;J
;J
~
=­
;:­
EXERCISE ELEVEN
II
ANGLES AND PARALLAX
81
Discussion Questions 1. List some possible reasons for differences in the observed and measured distances to the
"model star" or friend.
2. How does the distance of the "distant" source (representing the background stars) affect
parallax measurements?
3. The star Deneb is said to be at a distance of 1,600 light years. What should its parallax be?
Is it probable that this figure for distance was first obtained from parallax measurements?
4. What would be the parallax of the nearest star Alpha Centauri if measurements were made
from Mars? Refer to your text to obtain the distance to Alpha Centauri.
5. Calculate the maximum distance away a star may lie and still have a parallax measurable
by Hipparcos.
6. Why is it advantageous to have a satellite make parallax measurements?
EXERCISE Two
OBSERVING WITH SIMPLE TOOLS
15
...... PLASTIC STRIP
- - -- - - -:.e--------
- --
} y
= 0.087
X (KNOWN
DISTANCE)
KNOWN DISTANCE FIGURE 2-2. Calibration of the plastic strip.
3. Count the number of stars seen through the tube and record the approximate altitude and
azimuth for each area.
4. Determine the total number of stars counted, and calculate an estimation of the total num­
ber of stars visible to the unaided eye on the celestial sphere. Show all calculations.
5. If possible, do star counts and estimate the total number of stars visible both in the city and
at a relatively dark country observing site. Discuss the effects of "light pollution" on as­
tronomical observations in general.
A SIMPLE ANGLE-MEASURING DEVICE
The ability to construct devices to precisely measure angles in the sky occurred before the in­
vention of the telescope. The Dane Tycho Brahe (1561-1626) was recognized for his keen
eyesight and for the construction of massive and accurate quadrants. His ability to make and
record accurate positional measurement of the planets using the naked eye provided the data
from which the German Johannes Kepler (1571-1630) constructed the first empirical astro­
nomicallaws.
Angular measurements are still useful in astronomy. For example, being able to measure the
angular separation between double stars permits us to calculate their true separation, given that
we know their distance. While the device suggested here is very crude, it does permit the
measurement of some angles to give you a feeling for the utility of angular measurement.
1. The first step is to make and calibrate your own scale for measuring angles. A transparent
plastic strip marked off in 5° intervals is a simple and convenient scale since it can be car­
ried with you easily. The calibration procedure is illustrated in Figure 2-2.
Stand a known distance from a wall (e.g., 5 m) and hold the plastic strip vertically at
arms length between the thumb and index finger of each hand. Line up your right thumb
and forefinger with an upper mark on the wall and then your left thumb and forefinger
with the lower mark. If the marks on the wall are 0.087 times the distance from the wall
apart. Hold the
apart (.435 m or 43.5 cm for a distance of 5 m), your thumbs will be
strip flat on a table and mark the positions of your thumbs on the strip with a permanent
magic marker. Ten- and fifteen-degree intervals can be marked by doubling and tripling
the 5° interval, and 1° intervals can be obtained by subdivisions of a 5° interval.
2. Using the plastic strip, measure the angular height of a pole, building, or other object at a
number of different distances. Plot your data on a graph and look for a general relation­
ship. Is it linear? If not, can you redefine the variable plotted on one axis and replot the
graph so you get a linear relationship?
Exercise 11: Angles and Parallax
This exercise gives you hands-on experience, which is very helpful in
understanding of the work of an astronomer (or a topographer). It is easier
to do this lab with a partner.
The goal of the exercise is to find a distance from your position to a distant
point (a light pole on the figure) using a faraway object in the background (a
skyscraper on the figure).
The method is based on measuring the angles between the background
object and the point of interest from two different locations of the observer
(you). The procedure is explained pretty clearly in the manual. The only
difficulty is to find an instrument that allows you to measure angles. In the
lab you are suggested to prepare an instrument of your own and you are
referred to Exercise #2, where the simple calibrating procedure is explained.
In your report do not forget to write ‘Conclusions’ section and answer all six
questions.
Have questions? Here is my email: [email protected].