GEK1506
2005 - 2006
Group 16
http://www.shape9.nl/parallax/menu.html
Parallax
GEK1506 Group 16
CHIEN FU HOO
KON TZE SIANG
NGUYEN THI KHANH DUNG
TAY GUANG YU
TEO CHEA CHOON
WILLIAM YAP JUN-LIM
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Table of Content
1
Introduction
2
Historical Background
i.
ii.
Ancient period
Modern astronomy
3
Calculating distances using Parallax
4
Lunar Parallax
i.
ii.
iii.
iv.
v.
5
4
5
8
9
11
12
14
15
Solar Parallax
i.
ii.
iii.
iv.
v.
vi.
6
History of Lunar Parallax: Hipparchus
The facts behind lunar parallax
Observation of lunar parallax
Calculation of moon distance
Lunar Parallax Demonstration Project
3
Introduction
The Transit of Venus
The Transit of Mercury
Using Mars
Using Asteroid
Other Methods
19
20
26
27
28
30
Stellar Parallax
i.
ii.
iii.
iv.
v.
vi.
vii.
viii.
Definition
The History of Stellar Measurement
Pioneers of Stellar Parallax Measurement
Importance of Stellar Parallax
Calculation of the Distance
Types of Instruments Used
Modern Developments in Stellar Parallax Measurement
Limitations of Stellar Parallax Measurement:
31
32
33
34
34
36
38
39
7
Application
40
8
Bibliography
48
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1
Introduction
Parallax is a common phenomenon which is shown by the change of the angular position of
an object as observe by observers, due to the motion or position of observers. However, not
much people noticed it as the effect of parallax does not have a great impact to our daily life, but
it does exist around us, even on our body itself! By just applying a very easy experiment: Try to
lift a finger around 5cm in front of your eyes, then close your left eye, and then switch the closed
eye to open, and vise versa, you will find that as you only use the left eye, your finger switch to a
bit right, and when you change the relation, it will switch to the left. It is because when you use
the left eye, your observation position actually is on the left, thus the finger looks like it is at the
different side; and when you use your right eye, it is an inverse effect!
It may look like a very funny experiment, but when it occurs in a great distance of
observation, such as astronomy methods, the effects of parallax can be gigantic and affects the
measurements. It may be a disaster without knowing it, but to astronomers, rather than calling it
a curse, they may call it as the gift of the heaven! With the knowledge of parallax, we were
granted the ability to measure distance accurately by analyzing the parallax observation, we can
measure the distance to many objects, specifically celestial objects .The result is highly valued
and so this makes parallax one of the most important aspects in astronomy. If you still remember,
the little experiment and apply this idea on, it actually explain why our biological body consists
of two eyes but we may thought one is enough to see!
Since parallax is so close to our body, don’t you think it is important to study it and
appreciate the ability we have within us?
In this report, we will focus on astronomical usage of parallax. We will start off by the
definition of the parallax, the general application, and giving some examples to have a better idea
on this phenomenon. Later, we will divide into three main phenomena of parallax in astronomy
observation, there were Lunar Parallax, Solar Parallax and Stellar Parallax, which order are
ascending with the distance from us, the earth.
We hope this report would give the reader a better understanding of the idea of parallax by
presenting a clear picture of the topic. We hope to achieve this by having many diagrams so that
the reader is able to visualize and also to provide information that is understandable to the
layman.
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2
Historical Background
The parallax is related directly to the solar system. In this part, we try to discuss about how
astronomers come out with the proper solar system so that they can measure the parallax and
approach the accurate value.
i.
Ancient period:
In the ancient time, the awareness of people about the universe was very simple. Except the
terrestrial objects, people only knew or noticed three celestial objects: the Sun, the Moon and the
stars. Later on, the wonder about the sky and all the things outside our Earth made people to
observe carefully and realized another type of celestial objects: planet, which is very like the
stars but not exactly the stars and moves around the stars. Comet was considered as the ending
point of ancient celestial objects.
The stars seem to be fixed in the sky and the rest of celestial objects move against the
background of fixed stars within a narrow band in the sky (the Zodiac) and drift towards the east
(except the comets). Their speeds are different. The Moon is the fastest, completing a circuit
around the sky in a month; the Sun takes a year and the circuits’ time of the planets change from
each to each.
The very first model used to predict the movement of the planets was a large sphere with
the Earth at the center; other fixed stars were attached on this large sphere. This sphere rotates
once every 24 hours; the axis passes very close to the North Star. Every star revolves around the
Earth in this sphere. The Sun, the Moon and other planets also move through the sky but have
their own small sphere. This model is called a geocentric model, or earth centered.
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ii.
Modern astronomy:
The basic layout of the solar system was made by Copernicus in the early 1500s with
Ptolemy's analysis as a guide. He had the Sun in the center of his system and other planets
around it. The Earth is just only one planet orbiting around the Sun. Such a model is a
heliocentric model, or sun centered.
Aristarchus of Samos, around 280 B.C, also had this idea of heliocentric model but it
wasn’t seriously considered until Copernicus brought it back.
Copernicus
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But Copernicus used the circular orbits, the simple and inaccurate model. His observations
then were improving by Tycho Brahe, who became the world's best pre-telescope observer,
measuring the positions of planets at least ten times more accurately than ever before.
Tycho Brahe
The next step is Johaness Kepler’s work. Using Tycho's position measurements,
developing his own three laws, he was able to work out quite an accurate plan of the solar
system. It didn't come easy. All calculations had to be done manually and it took years to explore
various possible models to see if they fit observations.
Johaness Kepler
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The first problem is to allow for the moving earth, from which all the observations are
made. It requires determining the earth's path around the sun. Unlike the old astronomers, Kepler
used Mars and the Sun as two fixed points to obtain two lines that intersected at earth's position
in space. The major assumption needed is that each planet follows a repeatable path through
space, so after a whole number of orbital periods it is back to the same place. So after one Mars’
period, it would be at the same place while the Earth would be somewhere else in its orbit.
Repeating this observation many times, we can obtain enough points on the Earth’s orbit; allow
us to determine its shape and motion.
After Johannes Kepler discovered his three laws, it’s now possible to build a proper solar
system and approach the more accurate measurement of parallax.
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3
Calculating distances using Parallax
The most general way of calculating distance using parallax, be it in astronomy or in our
daily lives, would be through the application of basic trigonometry. For instance, if you want to
calculate the distance XZ, you can do so by measuring the distance XY and angle A.
We can then calculate XZ using the trigonometric formula:
tan(A) = XY/XZ
Y
A
X
Z
Similarly, this form of calculation can be applied in the field of astronomy to measure
distance of celestial object. Imagine the celestial object to be measured is at point Z, and the sun
and earth to be at points X and Y respectively. The angle A can be measured using advanced
telescopic equipment or the satellite, Hipparcos. Otherwise the distance XY has been calculated
to be approximately 1AU (Astronomical Unit). Thus, the distance of the celestial object from the
earth can be found.
The figure below illustrates what we mean by using trigonometry to measure distance of
celestial objects, in this case a star.
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4
Lunar Parallax
i. History of Lunar Parallax: Hipparchus
As the moon is the nearest celestial body to earth, The lunar parallax determination start in
an earlier time compare to the stellar and sun parallax, one of the famous is Hipparchus(190BC120BC).The measurement is done during a total solar eclipse at 1 Syene and a partial eclipse at
2
Alexandria. At the same time that an observer at Syene saw the entire Sun blocked by the
Moon, one at Alexandria saw 1/5th of the Sun's disk, that is 1/5th of 30 arc minutes of the Sun's
disk was visible (The Sun's angular diameter is 30 arc minutes or 1/2 degree). The angular size
of the visible Sun seen at Alexandria therefore is 1/10th of a degree (0.1 degree) and this angle,
expressed in radians and applying the small angle approximation gives the ratio of the SyeneAlexandria distance to the Earth-Moon distance.
1
2
Syene: now as Aswan, Egypt.
Alexandria: the second largest city and the main port of Egypt.
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His measurement ,which assume that sun being at infinity, led him to a value for the
distance to the moon of between 59 (58’)and 67 earth 3 radii which is quite remarkable (the
modern value is 60.2 earth radii, 57’02.6”), The main reason for his range of values was that he
was unable to determine the parallax of the sun, but only managing to give an upper value.
Hipparchus was born in (1) 4 Nicaea in Bithynia, is a Greek astronomer and mathematician
science who made the fundamental contributions to the advanced of astronomy as a
mathematical science and to the foundations of trigonometry. Although (2) 5 he is commonly
ranked among the greatest scientists of antiquity), it is very little is known about his life.Only
one of his work has survived, namely “Commentary on Aratus and Eudoxus” and this is certainly
not one of his major works. Most of the information is by (3) 6 Ptolemy's (4) 7 Almagest. However,
even Ptolemy’s himself is an argument of historians.
Another achievement of Hipparchus is measured the precession of the Earth's rotation axis.
Today we know that the precession period is about 26,000 years. While the North Celestial Pole
today is near the star Polaris, in 3000 B.C., it was near the star 8 Thuban in the constellation
9
Draco, and in 14,000 A.D. , it will be found near the star 10 Vega in the constellation 11 Lyra.
3
3 radii: (noun)radius.
Nicaea :now Iznik, north-western Turkey
5
It was ranked by Strabo (about 64 BC- 24 AD) a Greek geographer and historian.
6
Claudius Ptolemy (85-165): One of the most influential ancient Greek astronomers and geographers also known
as a mathematicians.
7
Almagest: major works of Ptolemy, treatise in thirteen books, based on mathematics.
8
Thuban: Alpha(the main star of) Draconi,. Fourth magnitude.
9
Draco: constellation,named as “the dragon”.
10
Vega: Alpha Lyrae
11
Lyra : constellation, represents the harp of the great mythical musician Orpheus.
4
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ii. The facts behind lunar parallax
The position of moon is appeared shifting to another position when view from two
different positions if compare to the background such as stars. This is because the distance of the
moon is relative close to earth. On the other hand, the background stars are at a long distance
from the earth.
There are few methods of calculating the distance of moon. A primitive way to determine
the distance of moon from one location is by using a lunar eclipse. The full shadow of the Earth
on the Moon has an apparent radius of curvature equal to the difference between the apparent
radii of the Earth and the Sun as seen from the Moon. This radius can be seen to be equal to 0.75
degree, from which (with the solar apparent radius 0.25 degree) we get an Earth apparent radius
of 1 degree. This yields for the Earth-Moon distance 60 Earth radii or 384,000 km.
Another way of determine the lunar parallax is by observing the lunar parallax and this is
the method which is related to lunar parallax. This method can be done by taking two pictures at
the same times at two different positions, and compare to the background stars.
As we know, The Earth's diameter is approximately 12,756km. So this would be the
maximum distance which separate observer 1 and observer 2. However in reality, the observers
are separated by less than the possible maximum distance.
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iii. Observation of lunar parallax
Procedure
To observe lunar parallax, at least two pictures have to be taken at almost the same time.
For attaining a better result, the separation between two observers should be as far as possible.
To determine lunar parallax, we have to use simultaneous photography. After that, the
parallax of the moon is determined through digital analysis. In addition, the distance of the moon
can be computed by using parallax angle.
When take the picture of moon, we also photographed the nearby stars or in other word,
the star background in order to determine the apparent location of the moon in the specific time.
However the most important issue is the pictures which are taken from different location should
be done on the same time. Due to the different observer locations, the moon appeared to be in
two different positions. The position shift is called the parallax. The amount of shift measured as
an angle is called the parallax angle.
Although it is very easy to capture the pictures of moon, but the brightness of the moon
will make the background stars to be dimmed and hard to be noticed. To overcome this problem,
normally these pictures are taken during lunar eclipse. The moon at this moment is considerably
dimmed compares to ordinary time. Hence the star background will be more noticeable.
However taking the moon picture during moon eclipse is not an essential step, as long as the star
background is noticeable.
After taking the pictures of moon from two different locations, the pictures have to be
combined to make the moon's parallax visible. The moon's parallax displacement has become
more obvious if the pictures are taken from two far locations.
For example
Picture taken at location A
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Picture taken at location B
Combination of two pictures
This procedure can be done by using any image processing software or, respectively, by
using a photocopier and putting the pictures on top of each other so that star A and star B
coincide on both pictures.
This procedure looked easy to be done, however a few question raise. Is it possible to see
lunar parallax using amateur equipment? The resolution of camera has to be high enough to take
the picture which the star background can be noticed. Besides, finding the partners which capture
the moon at different locations is also an important issue for those who want to do the project on
moon parallax have to be considered.
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iv. Calculation of moon distance
To compute the lunar parallax angle, the observers from two locations have to measure
the angular displacement of a distant bright star to moon. Let the angles be θ1, θ2, the parallax
angle, θ3 = θ2 - θ1 in which θ1 < θ2.
To compute the moon distance by using lunar parallax, we can use the mathematical
calculation. Let us consider the following diagram and calculate the moon distance of by using
the following diagram.
First of all, we have to calculate the angle between two stations. In order to obtain the
angle, we have to know the distance between two stations which the pictures are taken.
Angle between two station, θ =
distance between two stations
r
C = 0.5 r sin θ
b = c / cos (θ/2)
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Hence, the distance of moon is
d=
b
, where p = parallax angle
tan p
However there is something that we have to take note. Since we are measuring the
distance to the observers, there is an additional distance which must be added in to reach the
center of the earth
iv. Lunar Parallax Demonstration Project
There were many people on the world to do the Lunar Parallax Demonstration Project.
Let us discuss on the Lunar Parallax Demonstration Project done by the people from different
places on the world. The pictures were all capturing at moon eclipse so that the star background
can be noticed.
1.
Local Europe
Picture taken at Brussels, Belgium
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Picture taken at Maldon, Essex
Combination of two pictures
Brussels, Belgium <--> Maldon, Essex
Approximately 300km
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In this project, two pictures were taken at the same time at Brussels, Belgium and
Maldon, Essex. The pictures were captured approximately 300km apart. As we can see, the
moon appears to shift for short distance. However this is not distinct because the locations are
not far enough. Let us compare to the pictures which is taken from further locations at below.
2. USA-Europe
Picture taken at Divide, Colorado
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Picture taken at Maldon, Essex
Combination
Divide, Colorado <--> Maldon, Essex
Approximately 7200km
In this part, one picture was taken at Divide, Colorado, USA, and another picture was
taken at Maldon, Essex, Europe. Two locations which the pictures were taken are approximately
7200km apart. As we can see, the moon was appear to move further compare to the picture taken
at Brussels, Belgium and Maldon, Essex. This is because the locations which the pictures were
taken are further.
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5
Solar Parallax
i.
Introduction
What Is Solar Parallax?
In general, the solar parallax can be understood as the difference in position of the Sun as
seen from the Earth’s center and from a point of observer’s location. If the sun is at the zenith
(directly overhead), the parallax is 0. The solar parallax reaches the maximum value when the
sun is seen on the horizon and is named the Horizontal Parallax or simply Parallax.
Solar parallax plays a very important role in some astronomic fields because it’s relating to
the distance of the Sun from the Earth. Knowing the solar parallax and the mean Earth radius
allows astronomers to calculate the AU (astronomical unit), the first, small step on the long road
of establishing the size – and thus the age – of the visible Universe.
Why Is It Difficult To Measure The Solar Parallax?
As the parallax is usually determined by the shift of the objects against the distant
background, it becomes difficult to measure the parallax of the sun because the sun is too bright
to see the background stars.
It is not possible even during the solar eclipse because the moon covers the entire sun,
including the edge. When the edge reappears, the eclipse is over and it turns be too bright again.
Moreover, unlike the stellar parallax, we can’t use the radar to measure directly the solar
parallax because the Sun has no solid surface to reflect the radar efficiently.
Review
Despite of all these problems, the requirement of measuring the size of the solar system and
the distance between celestial objects always forces the astronomers to find out the solutions.
• The first attempt was made by Aristarchus. He tried to determine the distance
from the Earth to the Sun by estimating the angular separation between the Sun and the
quarter-phase Moon. But his result is not accurate.
• Since the 17th century, Venus and Mars became the candidates for parallax
measurement. After many attempts, astronomers finally could obtain, at least, the
acceptable value of the solar parallax.
• Then, Eros was chosen to being used in the measurement of British Astronomer
Royal Sir Harold Spencer Jones.
• Other methods such as measurement speeds and using radar in the modern
astronomy made it more flexible to gain the accurate solar parallax.
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ii.
The Transit of Venus
What is the transit of Venus?
The transit of Venus happens when Venus passes through between the Earth and the Sun.
When viewed from the Earth, we will see a silhouette on the Sun. This concept is similar to that
of a solar eclipse but in this case the planet Venus looks much smaller because of its further
away from the sun. So during the transit, Venus would look like a small dot moving across the
sun.
The transit of Venus is a rare occurrence that occurs every 243 year with pairs of transit 8
years apart. Actually by looking a the relative speed of the Earth and Venus we should expect the
transits to occur every 584 days but in reality transits are much rarer than that because Venus’s
orbit around the sun is inclined to 3.4° to the Earth's.
When did we start to notice the Transit of Venus?
Johannes Kepler (1571 – 1630) (right), a German astronomer and
mathematician predicted the first transit of Venus on 1631 but the transit
was not visible then. It was only on 24 November 1639 that Jeremiah
Horrocks made the first observation of the transit of Venus.
The quest to observe the transit was heightened during the transit
of 1761. During that time, Britain and France were engaged in the Seven
years war and so they were also competing to acquire the timing of the
Venus’s transit in order to calculate the astronomical unit because that
meant scientific prestige. The transit of Venus after that happened on 1769, 1874 and 1882.
During those transits, astronomers continued to measure more and more accurate value of the
astronomical unit.
The first transit of Venus in 21st century occurred on June 8, 2004 and the next one on June
6, 2012.
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The Significance of the transit of Venus
The transit of Venus was an important because it can be used to measure the astronomical
unit (AU) which is the mean distance between the Earth and Sun. Kepler’s law have allowed us
to map out the entire universe in term of AU. Thus, the value of AU was an important because it
allowed us to calculate the distance of the solar system. This is because before then the value of
AU was not known and everything was in terms of the relative distance. The AU can be
measured by using parallax method.
Kepler’s Third Law
Kepler’s Third Law states that “The square of the sidereal period of an orbiting planet is
directly proportional to the cube of the orbit's semimajor axis.” 12
a3
P2
P = object's sidereal period in years
a = object's semi major axis, in AU
P
By using Kepler’s Third Law we can derive the relative distances of the planets in the solar
system in term of astronomical units (AU).
Now, I would use Kepler’s Third Law to calculate the distance of Venus in terms of AU,
P (years)2 = R(A.U)3
Ö
P(years)=R(A.U)3/2
Ö
R(A.U)=P(years)2/3
Ö
R(A.U)=( 0.615)2/3
Ö
R(A.U) = 0.723186
Therefore, the relative distance of Venus to the Sun is approximately 0.72
How to calculate the AU using solar parallax?
Kepler's laws of planetary motion - Wikipedia, the free encyclopedia. 1 October 2005.
<http://en.wikipedia.org/wiki/Kepler%27s_Third_Law#Kepler.27s_third_law>.
12
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When two separate observations made from two different points on the Earth, we will be
able to observe the Venus moving to different paths depending on where you are on Earth. In the
diagram, at point N you will see the Venus moving at a lower path compared to the observation
made from point S.
When we view the Sun from Earth, we see it as a circle and because of that the two paths
seen by N and S will have different length. The length of the path N is long than the path S. To
measure this different length we can use the time taken for the transit by using also the four
phases of the transit as indicators.
As seen from the diagram above, there are four contact of Venus which is the 1st, 2nd, 3rd
and 4th contact.
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To calculate the distance from the Sun to Earth, we have two observers at two different
points on the Earth (A and B)
The angle between the two paths made on Earth is denoted E.
Using Kepler’s Third Law as explained earlier we will know the relative distances of all the
planets in the Solar system. For this case, Venus’s distance from the Sun is 0.72 times the Earth’s
distance from the Sun.
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Next we denoted the angle between the two paths as seen from Venus as angle V.
By using ratio, we can see a relationship between angel E and angle V. Let the Earth to Sun
distance be 1 and the Venus to Sun distance be 0.72. By ratio, V/1 = E/0.72, thus V=E/0.72. *
(valid for small angles only)
Next, we also need to find out the distance between the two observers on Earth which is the
difference from point A to point B. Lets denoted this distance as dA-B.
Using trigonometry and the earlier values when can now calculate the distance from Venus
to Earth.
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Therefore,
tan (V/2) = (1/2 d A-B) / (d Earth-Venus )
Ö d Earth-Venus = (1/2 d A-B) / tan (V/2)
Ö For small angle, tan (1/2 A) = ½ tan A
Ö Thus, d Earth-Venus = d A-B / tan (V)
By finding the value of d Earth-Venus , we can then calculate the distance between the Earth to
the Sun by using Kepler’s third law. As said earlier, by using the law we can tell that the distance
between the Earth and Venus is 0.28 the distance between the Earth and the Sun.
d Earth-Venus = 0.28 x d Earth-Sun
Therefore, d Earth-Sun = d Earth-Venus /0.28
Limitations – the Black-Drop effect
1
2
One of the limitations to using the transit of Venus to calculate the AU is the black-drop
effect. The black-drop effect happens when Venus is near to the rim of the Sun, basically during
the 2nd and 3rd contact phases. As seen from the diagram above, when the Venus is seen like an
oil drop (2) instead of a crisp image (1). When this happens it prevents astronomers from timing
the transit accurately which in the end affects the value of the astronomical unit.
In the earlier times, the reason this happening is thought to have been because of Venus
having an atmosphere. However, the black-drop effect was later seen on the transit of Mercury
also and Mercury does not have an atmosphere. However it is now known to be an optical effect
due to the Earth's turbulent atmosphere. 13
Black drop effect - Wikipedia, the free encyclopedia. 26 September 2005.
<http://en.wikipedia.org/wiki/Black_drop_effect>.
13
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iii.
The Transit of Mercury
What is the transit of Mercury?
The transit of Mercury happens when Mercury passes between the Earth and the Sun.
When we look at the Sun from Earth, we can see a tiny black dot moving across it and that black
dot is Mercury. The transit of Mercury is a rare occurrence and its one of the only two possible
transits, the other being the transit of Venus as explained earlier.
How rarely does it occur?
The transit of Mercury happens more frequent compared to the transit of Venus. This
happens because of Mercury is closer to the Sun and orbits at the faster speed compared to
Venus. The transit of Mercury happens roughly 13 or 14 times in one century.
When does it occur?
The transit of Mercury can occur in the month of May or November. November transits
occur at intervals of 7, 13, or 33 years; May transits only occur at intervals of 13 or 33 years. The
last two transits were in 1999 and 2003; the next two will occur in 2006 and 2016.
Who is the first person to observe it?
The transit of Mercury was first observed on November 7, 1631 by Pierre Gassendi. He
was trying to observe the transit of Venus a month before that but because of inaccuracies in his
calculation he did not realise that the transit of Venus could not be seen in more of Europe in
which he is in.
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iv.
Using Mars
The main problem of using Venus’ transit is the poor images which do not allow accurate
measurement. Using Mars is one way to avoid this limiting factor.
As Mars comes almost as close to the Earth as Venus and it doesn’t transit the Sun but pass
by the line of sight to other distant stars, which provide sharp clear point; this method is even
simpler than the transit method. Everything we need is to find a bright star in the same field of
the telescope as Mars itself and view these two objects at the same time from different parts of
the Earth. Knowing the angle between those observations, we can determine the parallax easily.
The Mars method was used from 1672 onwards. In the fall of 1672, Mars approached the
Earth at a distance of 0.37 AU, about 1/3 the Sun’s distance, which means that its parallax was
about three times of the Sun.
Jean Richer, a French astronomer, measured Mars’
position from Cayenne, French Guyana while Cassini
measured it from Paris at the same time. The result
obtained, which was the first reasonable estimate of the
actual scale of the solar system, deduced a solar parallax
of about 9½" with an error of around 25% to 30%.
In 2003, it was especially close to the Earth and this
year became an excellent time to
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v.
Using Asteroid:
In the first steps of the development of parallax measurement, familiar eyeable celestial
objects such as the Moon, Venus and Mars are favorable. After that, in order to reduce the
parallax, instead of using the old objects above; astronomers nowadays have chosen asteroids for
the precise determination of solar parallax.
The following table gives a summary of the various determinations of the sollar parallax
since the middle of the 17th century.
Year
Objects / Author
1672
Mars / Cassini
1857 - 1863
1862
1864
1888 - 1889
Moon’s orbit / Hanson
Mars / Hall
Venus transit / Powalky
Iris, Victoria, and Sappho
Parallax
9½"
(Error: 25% – 30%)
8.95"
About 8.90"
8.83"
8.802"
http://near.jhuapl.edu/eros/history/parallax.html
The very first idea of using asteroids to measure the solar parallax was expressed by Johann
Gottfried Galle (1812-1910), a German astronomer, in 1872. He recognized that there are two
useful advantages of using asteroid instead of other planets:
• Asteroid, such as Eros, comes much closer to the Earth than Venus or
Mars so that its parallax is bigger and easier to measure.
• Asteroid is enough small for its point-like images to become easier to
measure as compared to the disk of Mars.
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Galle first applied his method to the asteroid Flora in 1873, but was not so successful.
Later, the method was employed with great success to other asteroids such as Iris, Victoria,
Sappho in 1888 and 1889.
The other determination of solar parallax using asteroids, which is considered as the great
effort of 1930’s astronomers, was completed by British Astronomer Royal Sir Harold Spencer
Jones, (1890 – 1960).
He started his calculation with 433 Eros, which passed much closer to Earth than Venus,
about 22 million kilometres to the Earth. The observation of 433 Eros during its close approach
in 1930 – 1931 led him to the more accurate result. The value of AU was fixed at 150 million
kilometres.
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Based on the same basic idea, the measurement the parallax by using asteroids contains
these following steps:
• observe the asteroid from widely separated locations on the Earth
• measure the shift in its apparent position with respect to background stars
vi.
Other Methods:
a) Measuring Speeds Rather Than Distance
A measurement of the speed of a planet also can help us determine the distance from this
planet to the Sun, which is the product of the speed and the period, divided by 2 pi. Due to this
concept, the value of AU can be calculated from the speed of the Earth.
Bradley discovered a systematic displacement of the apparent positions of stars caused
directly by the speed of the Earth in its orbit. The displacement angle, in radians, is
approximately the ratio of the speed of the Earth to the speed of the incoming light. The angle
was measured, with commendable accuracy, as being close to 20 arc seconds, which is 20/(60 x
60 x 57.1) radians, =0.000097 radians (i.e. just under 1 in 10,000).
Since, as we now know, the speed of the Earth is 30 km per second and the speed of light is
nearly 300,000 km per second, it is clear that this was a very good result.
b) Radar:
As the astronomic science develops, radar can be used to support astronomers to approach
the high accuracy in measurement. Moreover, radar can also measure the speed, which offer us
the second and independent way of deriving the AU from Eros.
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6
Stellar Parallax
i. Definition
A general definition of stellar parallax would be the change in angular position of a
particular star relative to the more distant background stars as the earth revolves around the sun.
The greatest parallax occurs when the earth is at the ends of its orbit.
The parallax effect decreases with the increasing distance of star from earth.
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ii. The History of Stellar Measurement:
In 1609, Italian scientist Galileo Galilei was the first to attempt stellar parallax
observation. He used a 1-inch diameter telescope to observe the stellar parallax but was
unsuccessful as the telescope was not powerful enough to detect the tiny parallax motions.
Galileo was followed by other scientists such as Hooke, Flamsteed, Picard, Cassini, Horrebow,
and Halley during the next two centuries. They too were unsuccessful because their telescopes
were simply too small and did not have magnification large enough to measure the tiny parallax
motions. It was not until 1838 that the first parallax was measured. And it so happened that it
was done by three scientists independently using three different instruments to measure the
parallax motion. They were Bessel with his heliometer, Struve with his filar micrometer, and
Henderson with his meridian circle. Bessel made the most accurate measurement with his
heliometer. Bessel used the double star method. He observed two stars that appeared in his view
to be one star as seen in Figure 1. Over a six month period he found that two stars became visible
separately, the closer star moving relative to the background stars as seen in Figure 2.
Figure 1
Figure 2
The first attempts to determine parallaxes using photography were done during the period
1887-1889 by Pritchard at Oxford. There were initial doubts in using photographs to do
astronomy, but photography turned out to be an excellent way to measure parallaxes. The
accuracy was much greater than using visual methods. Furthermore, a permanent record of the
measurement was made, so the image could be examined and re-measured many times.
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iii. Pioneers of Stellar Parallax Measurement
Galileo Galilei (Pisa, February 15, 1564 – Arcetri, January 8, 1642), was a Tuscan
astronomer, philosopher, and physicist who is closely associated with the scientific revolution.
His achievements include improvements to the telescope, a variety of astronomical observations,
the first law of motion, and effective support for Copernicanism. He has been referred to as the
"father of modern astronomy," as the "father of modern physics," and as "father of science."
Friedrich Wilhelm Bessel (July 22, 1784 – March 17, 1846) was a German
mathematician, astronomer, and systematiser of the Bessel functions (which, despite their name,
were discovered by Daniel Bernoulli). He was born in Minden, Westphalia and died of cancer in
Königsberg (now Kaliningrad, Russia). Bessel was a contemporary of Carl Gauss, also a
mathematician and astronomer.
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iv. Importance of Stellar Parallax
Why do scientist and astronomers want to determine stellar parallaxes? Is it because they
have nothing else better to do? No. By knowing the parallax of a star, the distance of the star
from the earth can be determined using simple geometry and the properties of triangles. Nearby
stars are the stepping stones to measuring the distances to everything else in the universe.
Luminosity of stars can also be determined when the distance of the stars are known. Stellar
parallax thus helps astronomers to understand the stellar properties and underpin the whole
distance network for galactic and extragalactic astronomy. By mapping out the space galaxy
accurately we can then successfully launch our many space missions.
v. Calculation of the Distance
First let us introduce the units to be used in the calculation.
The parsec (symbol pc) is a unit of length used in astronomy. It stands for “parallax of
one arc second”. The parsec is defined to be the distance from the Earth of a star that has a
parallax of 1 arc second. Alternatively, the parsec is the distance at which two objects, separated
by 1 astronomical unit, appear to be separated by an angle of 1 arcsecond.
Arc second = A 60th part of a minute of arc
Astronomical unit (AU) = A unit of length used for distances within the solar system;
equal to the mean distance between the Earth and the Sun (approximately 93 million miles or
150 million kilometers
1pc = 206,265 AU = 3.08568×1016 m = 30.8568 Pm (Petametres) = 3.2616 ly (lightyears)
Next we take a look at the formulas used to calculate the distance.
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Some Notes in Stellar Parallax Measurement:
a)
For determination of a celestial distance, the base line is taken as long as possible
in order to obtain the greatest precision of measurement. For stellar measurement the base line is
taken to be the axis of the earth’s orbit.
The measured shifts tell us the size of the parallactic angle. We know how far the
Earth is from the sun which is approximately 1AU (Astronomical unit). From there we can use
simple geometry and the properties of triangles to work out the distance of the star.
b)
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vi. Types of Instruments Used:
The 1.55-m Kaj Strand Astrometric Reflector
This is the largest optical telescope operated by the U.S. Navy. It was designed and built
under the direction of the Scientific Director of the U.S. Naval Observatory from 1963 - 1977,
Dr. Kaj Aa. Strand. It was designed to produce extremely accurate astrometric measurements in
small fields, and has been used to measure parallaxes and therefore distance for faint stars. Over
1000 of the world's most accurate stellar distances and proper motions have been measured with
this telescope since 1964.
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Heliometer
This instrument was first used by Bessel in 1838 to measure the first accurate stellar
parallax
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vii. Modern Developments in Stellar Parallax Measurement:
In 1989, the European Space Agency launched a satellite called Hipparcos with the
purpose of obtaining parallaxes and proper motion of nearby stars to a higher degree of accuracy.
Hipparcos stands for High Precision Parallax Collecting Satellite. Hipparcos is able to measure
parallax angles for stars up to about 1600 light-years away. Parallaxes as small as 0.002" are now
possible and more precise compared to the 0.01" parallax measured from earth using advanced
telescopes. The higher accuracy of parallax measurements and the ability to measure stars much
further away has enabled us to understand our galaxy better and helped astronomers to calibrate
the distance scale of the galaxy.
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Some other advancement in stellar parallax measurement that we can expect in the next 5
years includes:
The GAIA mission is planned by ESA for launch in 2010. It will obtain positions, distances
and velocities for about 1 billion (1 000 million) stars, that is about 1% of the stars in the Milky
Way with an accuracy of 10 microarcseconds. This will allow astronomers to construct a
dynamic 3D map of our galaxy.
NASA is planning the Space Interferometry Mission, SIM, due for launch in 2009. Using
the principle of interferometry it will measure parallaxes to 25 000 pc at a 10% accuracy level
and stellar positions to within 4 microarcseconds.
viii. Limitations of Stellar Parallax Measurement:
One of the limitations would be for stars that are too far away, the parallax would be too
small to be able to measure accurately. Determination of stellar parallax is only possible when
the stars are not all at the same distance. It is much easier to detect the periodic shifting of a star
relative to the background stars, compared to measuring a tiny shift of the whole pattern.
Another limitation would be for the traditional ground based observation using telescopes, where
they face the problem of observing through a turbulent atmosphere, which adds further to the
error in parallax measurement.
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7
Application
In Greek, the word Parallax means "to change."
Examples in daily experiences:
For example if we are in a moving car, we can see the mountain move backward relative to
us, which is beneath a seemingly stationary moon. The moon is at a far distance from the
observer such that the angular change in position relative to us is extremely small compared to
the mountain that is much closer to us.
In other words, if we viewed from the car perpendicularly, distant objects appeared to be
moved with almost same speed and parallel relative to the car. However the nearby objects have
much grater apparently change in the relative angular position. So, in general, the angular
direction to the nearer object, as it is on a shorter distance to the viewer, will change more than
the angular direction to the farther object.
Pictures bellow may make you to understand this phenomenon better:
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(Simple Test:
With a nearby object in front of you, gaze at infinity. Cover one eye with your hand. Then
move your hand to cover your other eye instead. The nearby object will seem to ‘jump’
horizontally. )
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Application of the concept of parallax in science and technology:
We are always cautioned in science classes to "avoid parallax.”
a) For instance, we should always take measurements with one's eye on a line directly
perpendicular to the ruler, so that the thickness of the ruler does not create error in
positioning for fine measurements.
b) A similar error can occur when reading the position of a pointer against a scale in an
instrument such as a galvanometer. To help the us to avoid this problem, the scale is
sometimes printed above a narrow strip of mirror, and the user positions his eye so that the
pointer obscures its own reflection. This guarantees that the user's line of sight is
perpendicular to the mirror and therefore to the scale.
This is also the effect that enables us, and certain animals, such as cats, to see depth:
c) This concept is used in simple stereo viewing devices, such as the Viewmaster(TM)
used to view stereoscopic scenery in the form of two images taken from adjacent
locations.
d) The Apollo astronauts on the Moon knew how to take such stereo pairs, clicking two
frames of the same object in locations shifted slightly horizontally with respect to each
other.
e) Another way to allow a crowd of people simultaneously to view a stereoscopic scene
is to provide them with anaglyphic glasses. One glass is red, the other green, and the
stereo scene is produced by the printing process in a corresponding fashion. It is
generally we use monochrome to watch the stereo scene as mentioned-- red for the left
image, green for the right, but colour anaglyphic scenes also have been produced
today.
Generally, the idea of parallax has been developed , extended and applied widely into
various of fields nowadays, it is one of the greatest human breakthroughs.
Let us have a look on it:
anaglyphic glasses
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3-D image.
Communication by using VR( virtual reality)
Used in entertainment.
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Virtual learning
43
Medical visualization
Vocabulary:
Stereoscopic: able to see objects with lengths, width and depth.
Anaglyphic: related to the process of producing pictures in contrasting colours that appear
three-dimensional when superimposed and viewed through spectacles with one red and one
green lens.
Monochrome: having or appearing to have only one colour.
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The application of Parallax concept in Laparoscopic surgery:
Laparoscopic surgery is a surgical technique where the surgeon implement the operation
through a small opening (so-called key-hole) in the abdomen wall. It has great advantages for the
case of small scars, patients with less dehydration, etc. Furthermore, it gives faster recovery after
the operation.
Prototype for laparoscopy
Implementation of Laparoscopic surgery.
This the basic set up for the laparoscopic observation. The main idea or principle of it is
parallax shift is created in order to collect information about the spatial structure by moving the
laparoscope around the interest point after entered the abdomen. Generally it gives highly
reliable depth estimation, perception of spatial structure of certain point in the body, which
observed by only taking monoscopic image.
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Some examples of Theodolites that used today :
The Theodolites used today has high precision of optical alignment and measurement .The
azimuth and elevation axes is measured on a vernier scale with the aid of built-in magnifiers.
Both axes have slow motion controls and the telescope can plunge and reverse to reduce bearing
errors. The azimuth axis has a second spirit level mounted on the telescope with a mirror to assist
in leveling. The telescope utilizes precision optics and has a brass lens cover. A magnetic
compass is mounted on the azimuth axis.
Large Theodolite
Dumpy Theodolite
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Mini Theodolite
Alidate Theodolite
46
Have a full view of Theodolite!
You can know more complete about the how Theodalite works through the link bellow:
http://www.tpub.com/content/topography/TM-11-6675-200-10/
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