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Venus transit-a1-earth-sun-distance-final
1.
EDUCATIONAL ACTIVITY 1.
Calculating the Earth-Sun distance from images of the transit of Venus.
Mr. Miguel Ángel Pío Jiménez. Astronomer Instituto de Astrofísica de Canarias, Tenerife.
Dr. Miquel Serra-Ricart. Astronomer Instituto de Astrofísica de Canarias, Tenerife.
Mr. Juan Carlos Casado. Astrophotographer tierrayestrellas.com, Barcelona.
Dr. Lorraine Hanlon. Astronomer University College Dublin, Irland.
Dr. Luciano Nicastro. Astronomer Istituto Nazionale di Astrofisica, IASF Bologna.
1 – Objectives of the activity.
Through this activity we will learn to calculate the Earth-Sun distance (Astronomical
Unit) from digital images using the method of the solar parallax during the transit of Venus.
The objectives are to:
- Apply a methodology for the calculation of a physical parameter (Earth-Sun
distance).
- Apply knowledge of mathematics (algebra and trigonometry) and basic physics
(kinematics) to derive this result.
- Understand and apply basic techniques of image analysis (angular scale, distance
measurement, etc.).
- Work cooperatively as a team, valuing individual contributions and expressing
democratic attitudes.
2 – Instrumentation.
The activity will be based on digital images obtained during the transit of Venus in June
2012 (see sky-live.tv). Please refer to the Glossary delivered with this document for a quick
reference to terms used, abbreviations and physical units.
3 – Phenomenon.
3.1. Occultation and Transits.
An occultation is the result of an alignment of one celestial body by another celestial
body as seen from Earth. A transit is a partial blackout phenomenon in which closer body des
not completely obscure the more distance body and the passage or transit of the closer one
projected onto the surface of the background one is observed (Figure 2).
From our planet we can only see the transits for the inner planets, Mercury and Venus,
over the solar disk. Mercury moves in a plane that is 7 degrees tilted with respect to Earth’s
orbit, so that most of the time Mercury goes “above” or “below” the solar disk, without causing
transits. Mercury tends to transit on average 13 times per century in intervals of 3, 7, 10 and 13
years. The last transit of Mercury occurred on November 8, 2006.
Venus
transit
1
2. 3.2. The Transit of Venus.
Venus, being closer to the Sun than Earth, also produces transists that are observable to us.
Venus's orbital plane is inclined by 3.4° to Earth’s. Otherwise, there would be only a transit of
Venus every 584 days (the time it takes Venus to return to the same position with respect to the
Sun as seen from Earth).
Each year Earth passes through the line of nodes (see Figure 1) of Venus’s orbit around
June 6-7 and December 9-10. If these dates coincide with inferior conjunction, i.e., when Venus
lies between the Sun and the Earth, then there will be a transit.
Figure 1: Illustration of the line of nodes of Venus' orbit, when it intersects Earth's orbit.
Transits of Venus are an extraordinarily rare phenomenon, since on average there are
only two per century. These two transits are separated by 8 years and the interval between pairs
of transits alternates between 105,5 and 121,5 years. Sometimes, as it happened in 1388, one of
the transits of the pair may not occur because it does not coincide with the passage by the node.
The last pair of transits of Venus occurred on December 9, 1874 and December 6, 1882.
The last transit visible from Europe took place on June 8, 2004 (Figure 2) and the next
one will be on June 6, 2012.
Venus
transit
2
5.
cleric in Liverpool (England), who had studied Astronomy and Mathematics, was able to follow
the transit of the planet when it had already begun.
18th Century. In the early eighteenth century, the English astronomer Edmund Halley
proposed to take advantage of the transits of Venus to determine with great precision, the solar
parallax, which would calibrate the size of the known solar system. The solar parallax is the
angle subtended by the Sun from the Earth's equatorial radius (Figure 5), and using this angle,
we can obtain the Earth-Sun distance, as we will see some paragraphs below.
Figure 5: Schematic representation to show the solar parallax, or angle ρ. This angle
is very small indeed, but for the sake of clarity it is shown enlarged.
Taking advantage of the transit of Venus, which was to occur in 1761, astronomers
around the world, commissioned by their governments, were prepared for the observation. In
total, the transit was observed from about 70 sites distributed around the globe, constituting the
first major international scientific enterprise. However, the results did not live up to
expectations. Bad weather in many places of observation, the difficulty of determining the
precise geographical location of the place where the observation was made and the black drop
effect invalidated the implementation of the Halley method.
There were 150 official observers and many other amateurs who observed the 1769
transit. Among the observers was the famous Captain James Cook, who performed the first of
his trips.
19th Century. The transits of 1874 and 1882 were also observed by hundreds of
astronomers sent by the scientific academies of many countries. The Bulletin of the
Astronomical Society of London records that 3440 photographs of the different aspects of the
phenomenon were obtained.
In the transit of 1882, Spain participated for the first time officially, sending two groups
of observers, one in Cuba and another one to Puerto Rico.
In any case, the black drop phenomenon affected the observations again, so that the solar
parallax was determined from a value of 8.790 to 8.880 seconds of arc, corresponding to a Sun-
Earth distance of between 148.1 and 149.7 million km, what was the best estimate since that day.
Transit of 2004. The parallax method is now obsolete, and current measurements made
with space probes and radar techniques tell us that the solar parallax has a value of 8.79415
seconds of arc or 149,597,892 km. During the 2004 transit observations and photographs were
Venus
transit
5
6.
made worldwide, creating an international educational network to determine the astronomical
unit as a global experiment and a noteworthy event.
3.4. The Transit of Venus in 2012.
The transit of June 5th − 6th, 2012 will be fully visible from the north of the Nordic
countries, the Far East, eastern Russia, Mongolia, eastern China, Japan, Philippines, Papua New
Guinea, central and eastern Australia, New Zealand, West Pacific Ocean, Alaska, northern
Canada and nearly all of Greenland. From Spain it is only visible the end of the phenomenon at
sunrise in the eastern region of the Iberian Peninsula and Balearic Islands (Figure 6). After this
transit, we must wait until the years 2117 and 2125 to see the next two transits of Venus, this
FIGURE 1
time in December.
Global Visibility of the Transit of Venus of 2012 June 05/06
Region X*
Greatest
Transit
at Zenith
Tra
t
et
nse
Tra
uns
nsit
t Su
nsit
at S
Beg
sa
End
ins
IV I IV I
ins
End
Transit No Entire
sa
Transit
Beg
III II III II
at S
Transit
t Su
in Progress
nsit
Transit in Progress
nsit
unr
at Sunset Visible
Tra
nris
Visible at Sunrise
Tra
ise
e
(June 05) (June 06)
Region Y*
F. Espenak, NASAs GSFC eclipse.gsfc.nasa.gov/OH/transit12.html
* Region X - Beginning and end of Transit are visible, but the Sun sets for a short period around maximum transit.
* Region Y - Beginning and end of Transit are NOT visible, but the Sun rises for a short period around maximum transit.
Figure 6: World visibility of the 2012 transit of Venus. Credits: F. Espenak, (NASA / GSFC).
4 – Methodology
4.1. Methods to calculate the solar parallax during a transit of Venus.
There are three main methods to calculate the solar parallax by combining observations
from two separate locations during the transit of Venus.
A fundamental principle to consider is that the more distant in latitude the two observers
are, the more accurate the measurement will be (e.g., one observer in the northern hemisphere
and the other in the southern hemisphere). This is the method we will use, considering the
locations of our observations.
I. Halley's method.
Halley’s method in of observing and comparing the total duration of the transit. The exact
times of the internal or external contacts of Venus and the solar disk must be calculated. The
observations should be carried out from two places on Earth where the entire transit can be
observed. However problems can arise because of bad weather that can prevent the observation.
Venus
transit
6
7. Figure 7: A diagram of the meaning of “Interior Contact” and “Exterior Contact”.
II. Delisle’s method.
In this method, the time of occurrence of the same contact event between the disk of
Venus and the solar disk is measured by geographically distributed observes. External contacts
are often difficult to determine so the inner contacts are the best choice. The advantage over the
Halley method is that it relies on only one contact being visible.
III. Direct measurement of the parallax of Venus through images.
Unlike the previous two methods, which rely on timing, in this method, simultaneous
images from two different geographical locations must be taken. The observable that is
measured is the distance between the centers of Venus’s shadow over the Sun's disk as seen
from the two locations. A full description of this method is given in Appendix I.
Figure 8: Simultaneous observation of the transit of Venus in front of the disk of the Sun from two different
locations M1 and M2 at the same instant of time.
We assume the geometry of the situation as shown in Figure 8. Point O is the centre of
the Earth, C the centre of the Sun and V1 and V2 the observed centers of the projection of Venus
seen from M1 and M2, respectively. The angles D1 and D2 are the angular separations between
the centers of Venus and the Sun seen from M1 and M2, respectively, i.e., the angles of parallax
CM1V1 and CM2V2. Similarly, we define the angles πs and πv as the angular separations between
Venus
transit
7
8.
M1 and M2 views from the Sun and from Venus, respectively, i.e., the angles M1CM2 and
M1VM2. By definition we have,
! !
sin !! = ; sin !! =
!! !!"
where rT is the Sun-Earth distance distance, rVT is the Venus-Earth distance, and d is the
distance between M1 and M2 in a straight line. Appendix I describes how d may be determinated.
We can make the following assumptions:
• Since the distances between the objects are large, and the parallax is small, we
can approximate the sin of the parallax to the parallax itself, i.e., sin πi ≈ πi.
• The Earth, Sun and Venus are aligned, so that rVT = rT – rV (where rv is the Venus
Sun distance).
• The observation points M1 and M2 on Earth are along the same meridian, so that
M1, M2, C and V are in the same plane (coplanar).
• We also assume that these points are coplanar during all the transit; in fact this is
not true since the Earth rotates and the geometry of the systems change during
this rotation.
We define Δπ = πV – πS. Since we have:
! !
!! = and π! =
!! !! − !!
by substitution we can set,
π! · r !
π! =
!! − !!
Since Δπ = πV – πS, we can substitute in for πv to get:
!! !!
!" = !! − 1 = !!
!! − !! !! − !!
And so,
!! !
!! = !" − 1 =
!! !!
Rearranging, we get the Earth-Sun distance, rT, at the time of observation to be:
!
!! = !" (! Equation [1]
! /!! ! !)
where Δπ is an observable quantity (distance between the centres of the shadow of Venus on the
solar surface in units of radians), d is determined from the locations (see Appendix II), and the
ratio rT / rV of the Sun-Earth and Earth-Venus distances can be obtained (see Appendix III). If
we express d in kilometres, then the Earth-Sun distance will be also in kilometres.
The observable Δπ can be calculated in two ways described below, in sections 4.2 and 4.3
(in practice in sections 5.2.1 and 5.2.2).
Venus
transit
8
9. 4.2. Method 1. Method of “The Shadows”.
In this method, the transit in photographed from two different places at exactly the same
instant, with the same type of instrument. The two images are then superimposed, and the
angular separation, Δπ, between the centres of the shadow of Venus can be found. Details of the
procedure to be followed are found in paragraph 5.2.1.
4.3. Method 2. Method of "The Strings".
In this case we will consider the whole trajectory of the shadow created by Venus on the
surface of the Sun (see Figure 2), created by Venus calling the line connecting the centre
positions of the shadow of Venus, string M1 or string M2, depending on the observing point on
the Earth to which it refers.
Given that the Earth-Sun distance changes only slightly over the course of transit, (the
change is only 7,500 km compared to the average Earth-Sun distance of 150 million km) we can
assume that the two strings are parallel and now the observable to be measured is not the
distance between the shadows of Venus but the distance between the two strings that are formed
on the surface of the Sun during the transit (see Figure 9).
Figure 9: A representation of the two
“strings” on the surface of the Sun, A1A 2
and B1B2 as seen by observers at locations
M1 and M2 on Earth.
Using Pythagora’s theorem, we can write the following expressions:
! ! !! !! ! ! ! !! !! !
!! ! = !
+ !
!! ! =
!
+
!
So we can express A'B' as:
! ! ! !
! ! ! !
! !! !! ! !! !!
!! =! !−!! = + − +
2 2 2 2
So measuring the length of the strings A1A2, B1B2, along with the diameter of the Sun
(D), we can then obtain the parallax Δπ from,
Venus
transit
9
10. 1
Δπ = ! ! − !! !! ! − ! ! − !! !! !
2
5 – Calculations for the Transit of Venus, on 5/6 June 2012.
5.1. Putting ourselves in position.
In this section we address, specifically, the next transit of Venus, trying to get as close as
possible to the situation we will find when we meet in June at the computer, watching the transit
and trying, with the images that will be taken, calculate the Earth-Sun distance. We begin with a
brief description of the instrumentation to be used as well as the latitude and longitude of the
locations where we will take the pictures, and all other information necessary to succesfully
complete the calculations.
5.1.1. Locations of the observations and instrumental description.
We begin by describing the observatiing locations where the photos are going to be taken.
As described above, to simplify the calculations as much as possible, we selected two places on
Earth's surface with similar longitude, values which are:
Cairns (Australia): Latitude: −16º 55' 24.237" Longitude: 145º 46' 25.864"
Sapporo (Japan): Latitude: 43º 3' 43.545" Longitude: 141º 21' 15.755"
The images will take in real time with a VIXEN telescope (model VMC110L), which has
a focal ratio of f/9.4, meaning a focal length of 1035 mm for its aperture of 110 mm. This
configuration ensures an acceptable size for the Sun’s image. A solar filter will be used for the
observations. The images will recorded with a Canon 5D 21-Mpixel camera attached to the
telescope.
With this Telescope and this camera, the Sun’s image has a size in the plane of the
camera, and therefore also in the image of 1630 pixels.
Considering that the angular size of the Sun in the sky is about 31.5 minutes of arc, then
the scale, ε, of the sun in the image will be:
31.5 !"#$%&' !" !"# (′) · 60 !"#$%&! !" !"# (")
!"#$% (!) = = 1.16 "/!"#$%
1630 !"#$%&
The telescope and camera will be mounted on an “Astrotrack” mount, which is very
stable and easy to assemble and tracks the Sun’s movement across the sky.
Images will be recorded every 5 minutes throughout the duration of the event, the order
of 5 hours. After some simple processing, they will be put in real time on a ftp server, to permit
easy and free access thereto, to acquire and make the practice. Each of the images, when saved,
will contain the time (in UT) when the image was taken in the filename.
Venus
transit
10
11.
Figure 10: A photograph of the Instruments. Credits: M. A. Pío (IAC).
5.2. What we need to do in practice.
In Sections 5.2.1 and 5.2.2 we explain the practicalities of determining Δπ using the two
methods described earlier. If time permits, we suggest using both methods and comparing the
results obtained.
5.2.1. Method 1. Method of “The Shadows”.
We start with two images taken at the same instant of universal time (or as close as
possible), one at each location. We have to determine the distance between the shadows of
Venus.
To calculate the distance Δπ we should align the two images (rotation and translation as
both have the same scale) and take the measurement of the distance between the shadows of
Venus using an imaging software package. However, to simplify the process and remove the
need for the images to be aligned, we have made some mathematical transformations to
determine the distance using, (i) the Cartesian (x, y) coordinates of the shadow of Venus; (ii) a
spot on the solar surface and (iii) the centres of the Sun in each image. Figure 11 shows the
Venus
transit
11
12.
observations (using astronomical software) during the transit (time 0:45 UT of the day June 6,
2012) from the two observation locations, in Cairns (Australia) and Sapporo (Japan). Note that
the distance which separates the two shadows (Δπ) will be very small, being of the order of 10
pixels maximum.
Figure 11: A real image of the Sun made with the instrumentation that will be used for the transit, with a fictional
representation of the phenomenon at 00:45 UT of June 6, 2012.
Following the calculations in Appendix IV, the observable Δπ is determined from the
expression:
!
Δ! = Δ!! ! + Δ!! !
where the components Δπx and Δπy can be expressed as:
Δ!! = !! − !!! cos ! + !! − !!! sin ! − !! + !!!
Δ!! = − !! − !!! sin ! + !! − !!! cos ! − !! + !!"
where (x1, y1) and (x2, y2) are the coordinates of the shadow of Venus in the images from
Sapporo and Cairns, respectively, while (xc1,yc1), (xc2, yc2) are the coordinates of the centre of
the Sun from Sapporo and Cairns, respectively, all referred to the coordinate system S.
In our case, for the day of June 6, 2012 and observing from Cairns in Australia, and
Sapporo in Japan, the angle θ is (see calculations in Appendix IV):
θ = 108º 4' 17.92"
So:
!
Δ! = Δ!! ! + Δ!! ! = 8.4 !"#$%&
Appendix I presents a very precise method to determine the value of the distance d
between the two observers on Earth. In our case, the value of d is:
Venus
transit
12
13. d = 6662.9 km
Recalling Equation [1], we also require the ratio of the Earth-Sun and the Venus-Sun
distance (rT / rV) at the time of the observation.
The term of Δπ in this method, must be in seconds of arc, so you have to make use of the
scale value to be determined at the moment. For the case of the image that we produce using the
astronomical software, the value of Δπ is 8.4 pixels, and whereas the diameter of the sun in
pixels is 715 pixels, which gives a scale of:
31.5 !"#$%&' !" !"# ′ · 60 !"#$%&! !" !"# (")
!"#$% (!) = = 2.643"/!"#$%
715 !"#$%&
Note that this is only in this case, using the sizes of the Figure 11, and in the moment of
the Transit, the scale that we need to use, is the one that we put in Section 5.1.1.
So we use rT / rV = 1,39759 at 0:45 UT on June 6, 2012 (a value obtained from
ephemerides to that date), and substituting into Equation [1].
6662.9 !"
!! = = !"#. ! · !"! !"
!"#$%#&'( ! !"#
8.4 !"#$%& · ε · (1.39759 − 1)
!"#$%& 648000 !"#$%#&'(
can determine the value of rT, the distance from Earth to the Sun. Rcall that the value of
Δπ must be expressed in radians, hence the term π / 648000, which makes the change of units
(from seconds of arc to radians).
5.2.2. Method 2. Calculation of the Earth-Sun distance using the string method.
This method is easier than the previous one, since we need only to determine the lengths
of lines or strings that create the path of the shadow on the surface the Sun. For this reason, we
will not have the problem we had in the previous method in which we had to have a very tightly
synchronised observations between the two places on Earth, when taking the images, to ensure
that both have been taken at the same instant. However, the string method can only be applied
when the transit is completed. On the other hand, an advantage is that if the weather turns bad
during the transit or there are technical problems which result in some missing images, we only
have to extrapolate to the rest of the trayectory.
We must bear in mind that the images of each spot must be aligned, throughout the
transit, since due to the rotation of the Earth essentially, the Sun's image will rotate during the
time of the transit, so that the trajectory of the shadow instead of being rectilinear, is curved.
Be aware that the distance between the two strings will be very small, so the length of
both strings might be very similar.
We will need, as explained above, the value of the solar diameter (D), and the length of
the lines M1 and M2, all measured in the same units.
Figure 12 shows a representation of what might be seen using this method and an
astronomical software.
Venus
transit
13
14. Figure 12: An image made with astronomical software, with a simulated representation of the strings.
The lengths of the lines M1 and M2, based on this image, are join A1 to A2 and B1 B2,
respectively (see Figure 9), and can be measured, both in mm or pixels, depending on whether
the measurement is made with a ruler after printing the image, or using software that allows us
to represent and manipulate images. Proprietary software, such as Photoshop or Corel Draw, or
even Windows Paint, or free software like Gimp can be used for this task. Basically, any
software that allow us to calculate the sizes of objects within an image can be used.
Note: If you want to measure the longitude of the strings in mm, you must be coherent
with the units. This means you need to recalculate the scale factor (see page 17) using the
diameter of the Sun in mm. The scale, ε, will then be [arcsec/mm].
For the sample image, the value of Sun's diameter D, in pixels, is 711, and the string M1
(B1B2) measures 565 pixels, and the string M2 (A1A2) measures 578 pixels. We now need to
calculate the ratio A'B', where A'B' is, according to Figure 9, the distance between the two
strings, a distance which is directly related to the value of Δπ. Thus, the expression that we use
is:
1
Δ! = !! ! ! = ! ! − (!! !! )! − ! ! − !! !! ! = 8.79 !"#$%&
2
Finally, substituting into Equation [1]:
6662.9 !"
!! = = !"#. ! · !"! !"
!"#$%# ! !"#
8,79 !"#$%& · ε · (1.397589 − 1)
!"#$%& 648000 !"#$%#
where, again, the value of rT is the distance Earth-Sun distance,
d is the distance between
observers, determined according to Appendix II, ε is the scale value described, and the ratio
rT/rV is the average value for the transit.
One factor to take into account and not previously discussed is that the value of the radius
vector connecting the Earth to the Sun, and its counterpart linking Venus with the Sun, both
vary with time because the orbits of both Earth and Venus are elliptical. Therefore, in Method 1,
which considers a fixed point in time (0:45 UT in the example), the ratio rT / rV has to be the
instantaneous value at that time, but in the case of Method 2, the value of the ratio rT / rV to be
Venus
transit
14
15.
used is the average value over whole transit. However, we can see that both values differ only
slightly, because in such a short period of time (just over 5 hours of the transit), the variation in
the Earth-Sun distance is negligible (see Appendix III) .
6 – Useful Internet Resources.
• Online predictions of the Transit of 2012:
http://www.transitofvenus.nl/details.html
• General Information an data over the transit:
http://www.transitofvenus.org
• Safe methods for solar observation:
http://www.transitofvenus.org/june2012/eye-safety
• Data and Predictions:
http://eclipse.gsfc.nasa.gov/OH/transit12.html
• The live transmission of the transit through the Internet:
http://www.sky-live.tv
• Scientific expeditions of Shelios group to observe astronomical phenomena:
http://www.shelios.com
• Description of the Kepler’s laws.
http://csep10.phys.utk.edu/astr161/lect/history/kepler.html
• Description of the calculation referred to the solar parallax with examples:
http://serviastro.am.ub.es/Twiki/bin/view/ServiAstro/CalculTerrasolapartirDeVenus
and
http://www.imcce.fr/vt2004/en/fiches/fiche_n05_08_eng.html
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transit
15
16. APPENDIX I. Calculations of Method 1 in depth.
The determination of the Earth-Sun distance is based on the parallax effect (as seen
above) whereby, from two different locations, Venus is projected onto different locations on the
solar disk. Therefore it must combine observations from different places on Earth. The farther
apart are the two places of observation the more relevant is this effect of perspective and, thus,
we will be able to obtain a more accurate distance measurement.
The observations must be complemented by Kepler’s laws which describe the orbits of
the planets around the Sun. These laws were discovered by Johannes Kepler (1571−1630) using
many observations of planetary motion. The law of universal gravitation, formulated by Isaac
Newton (1642−1727), applied to the case of two moving bodies around a common centre of
mass, explains the three empirical Kepler’s laws.
From two different locations M1 and M2 (see Figure 13 and Figure 8) and at the same
time t, Venus is projected in two different positions V1 and V2 on the solar disk due to the
parallax.
Figure 13: Observation of the transit of Venus in front of the solar disk from two different locations M1 and M2 at the
same instant of time.
Point O is the centre of the Earth, C the centre of the Sun and V1 and V2 the observed
centres of the projection of Venus as seen from M1 and M2, respectively. The angles D1 and D2
are the angular separations between the centres of Venus and the Sun seen from M1 and M2,
respectively, i.e., the angles of parallax CM1V1 and CM2V2. Similarly, we can define the angles
as πV πS and angular separations between M1 and M2 seen from the Sun and from Venus,
respectively, i.e., the angles and M1VM2 M1CM2.
Since the four points M1, M2, C and V are not in the same plane (the most common case
will not have both sites M1 and M2 on the same meridian, or Earth-Venus-Sun perfectly aligned),
the geometry of the problem is a bit complicated. In Figure 8 (and also in Figure 13) you can see
how the distance between the two centres of Venus Δπ = πV – πS is (hardly) the only observable
quantity which allows to calculate the distance to the Sun.
The practical realization of the measure of Δπ from the two images separately can be
made by measuring the position of the centre of Venus in each of them respect to a reference
point on the solar disk (a solar spot, for example) and comparing the two measurements.
Measured quantities are taken in units of length, for example in millimetres, and should be
converted to an angle that you can get by knowing the apparent diameter of the Sun.
Venus
transit
16
17. Let (x1, y1) and (x2, y2) be the separations between the centre of the disk of Venus and the
spot of reference, in mm, in the horizontal and vertical directions for each image. The
separations in arcseconds are obtained by multiplying each of the quantities x1 and y1 by the
factor of scale (ε).
!"#$% !"!"#$%& !"#$%&%' (!"#$%#)
!"#$% (ℇ) =
!"# !"#$%&%' (!! !" !"#$%&)
The distance between the centres of Venus in the two images will be:
Δπ (arcsec) = [(x2 − x1)2 + (y2 − y1)2 ]1/2 · ε
Figure 14: Positions of the projection of Venus over
the Sun's disk.
Suppose rV and rT are the distances between the centre of the Sun and Venus and Earth,
respectively, at time t of observation. Since the projection of the distance d between M1 and M2
in the plane perpendicular to OC is small compared to the distances Earth-Sun and Earth-Venus,
we can approximate:
πS = d/rT
πV = d/(rT − rV)
and from here, we can obtain:
πV = πS rT/(rT − rV)
Δπ = πS (rT/(rT − rV) − 1) = πS rV/(rT − rV)
so,
πS = d/rT = Δπ (rT/rV − 1).
The latter formula expresses that if we know the angular distance, Δπ, between the two
centres V1 and V2, and the ratio rT / rV between the distances Earth-Sun and Venus-Sun, the
parallax πS can be obtained, and knowing the projected distance d between the two locations, we
can calculate the distance rT. (In all these expressions the values of πV, πS and Δπ are given in
Venus
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17
18.
radians. To convert to arcseconds and make them compatible with equations, one need only to
multiply by 648000 and divide by the number π).
Δπ is the observable quantity, d can be determined using the locations in the Earth (see
Appendix II) and, therefore, the only quantity needed to solve the problem is the ratio rT/rV, the
Earth-Sun and Venus-Sun distances.
Determination of the average distance.
On other hand, we can also determine the average distance Earth-Sun (RT) and the
corresponding average parallax πo, which are related through Earth's equatorial radius R by:
πo ≈ R/RT,
and to do that, it is necessary to make some additional considerations.
The average distance Earth-Sun, RT, can also be defined as the radius that would have the
orbit of the Earth if it were a circle with the center coincident with the center of the ellipse that
defines the actual orbit. In this case the RT value matches the value of the semi-major axis of the
orbit a (a=1,000014 RT). So we can express the value of the medium parallax as:
! ! ! !! ! ! ! !!
!! = = · = !! · ⟹ !! = · · !!
!! ! ! ! !! ! !! ! !
where rT/a is the ratio between the Earth-Sun instant distance and the semi-major axis of Earth’s
orbit.
Based on the above expression πo ≈ R / RT, we can then derive the value of RT, which is
the average of the value of the Earth-Sun distance.
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19. APPENDIX II. Determine the value of d.
If you express the projection d of the distance between M1 and M2 in the plane normal to
the direction Earth-Sun in units of Earth’s equatorial radius and the Earth-Sun distance in units
of the average distance, we have:
πS = [(d/R) / (rT /RT)] (R/RT) ≈ [(d/R) / (rT /RT)] πo.
The ratio rT / RT can be calculated from Kepler's first law as:
rT/RT = 1 − eT cos ET(t)
and therefore we only need to calculate d/R (see Figure 8).
Figure 15: Projection of the distance between M1 and M2 in the plane normal to the Earth-Sun
direction.
Making the vector product between the vectors M1M2 and OC, we obtain the value of sin
θ, because:
M1M2 × OC = |M1M2| rT sin θ.
In Figure 15 you can observe that:
d = |M1M2| cos (90 − θ) = |M1M2| sin θ
and therefore,
d = M1M2 × OC / rT.
Now, we need to calculate M1M2 × OC.
The OC vector can be expressed from the equatorial coordinates of the Sun (α,δ) at time t
of observation as:
x = rT cos δ cos α
y = rT cos δ sin α
z = rT sin δ.
The position of each observer can be expressed as (see Figure 16):
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20. x = R cos φ cos (λ+TG)
y = R cos φ sin (λ+TG)
z = R sin φ,
where φ and λ are the geographical coordinates (latitude and longitude) of the observer and TG
is the sideral time of each point of the Earth who has a longitude λ. In our case, on June 6th 2012,
the value is TG (0h UT) = 16h 59m 12.495s.
360!
!! = !! 0 + · ! → !! = !! (0) + 1.00273791 !
23! 56! 4.1 !
M1M2 vector coordinates can be found easily as:
X = x1 − x2
Y = y1 − y2
Z = z1 − z2
!! !! = !! + !! + !!
!
!! !! = !! + !! + !!
and the coordinates for the unitary vector c, which connect the center of Earth with the Solar
center would be:
x = cos δS cos αS
y = cos δS sin αS
z = sin δS
So going back to the expression of d, it can be expressed as:
! = !! !! sin ! = !! !! × ! = !" − !" ! + (!" − !")! + (!" − !")!
Figure 16: Positions of a star (e.g. the Sun) and an
observer on Earth in equatorial coordinates.
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21. APPENDIX III. Kepler’s Laws.
The subject of planetary motion is inseparable from a name: Johannes Kepler. Kepler's
obsession with geometry and the supposed harmony of the universe allowed, after several failed
attempts, to create the three laws that describe with great precision the movement of the planets
around the Sun. Starting from a cosmological Copernican position, which at that time was more
a philosophical belief than a scientific theory, and using the large amount of experimental data
obtained by Tycho Brahe, Kepler created this wonderful, though entirely empirical, set of laws.
The first law states that the planets describe elliptical orbits around the Sun, which
occupies one focus. With Kepler’s disappointment, the circle occupied a privileged place; this
after multiple attempts to reconcile the observations with circular orbits.
1. – First Law: "The orbit that describes each planet is an ellipse with the Sun at one
focus".
Figure 17: Description of the elements of an object's orbit around the Sun.
Elliptical paths have very small eccentricity, so that differ little from a circle. For
example, Earth's orbit eccentricity is e = 0.017, and given the Earth-Sun distance of about
150,000,000 km, the distance from the Sun (focus) to the center of the ellipse is ae = 2,500,000
km.
The second law refers to the areas swept by the imaginary line connecting each planet to
the Sun, called the radius vector. Kepler found that planets move faster when they are closer to
the Sun, but the radius vector covers equal areas in equal times. (If the planet takes the same
time going from A to B in the figure, from C to D, the shaded areas are equal).
2. – Second Law: "Every planet moves so that the radius vector (line joining the center
of the Sun to the planet) sweeps out equal areas in equal times".
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22. Figure 18 : Graphical representation of Kepler’s 2nd law.
The radius vector r, ie the distance between the planet and the Sun (S) is variable, it is
minimum at perihelion and maximum at aphelion. As the areal velocity (swept area in unit time)
is constant, the velocity of the planet in its orbit must be variable. Under this law, if areas CSD
ASB and are equal, the arc AB will be less than the CD, indicating that the planet moves slower
at aphelion. That is, its velocity is maximum at the minimum distance from the Sun and
minimum to maximum distance.
Finally, the third law relates the semi-major axis of the orbit, R, the planet's orbital period
P, as follows: R3/P2 = constant. According to this law, the duration of the orbital path of a planet
increases with distance from the Sun and we know that the "year" (defined as the time taken by
the planet to return to the same point in its orbit) of Mercury has 88 days (terrestrial), Venus 224,
the Earth 365 and it continues to increase as we move away from the Sun. These laws also allow
to obtain the relative distances of objects in the solar system, if we know their movements.
3. – Third Law: "The square of the periods of revolution of two planets is proportional to
the cubes of their mean distances from the Sun".
Figure 19 :Relationship between the periods
and radii of the orbits around the Sun of two
objects, which graphically describes Kepler’s
3rd law.
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23. If R1 and R2 are the mean distances of two planets to the Sun, such as Mars and Earth,
and P1 and P2 are the respective times of revolution around the Sun, according to this law it is:
! !
!! !!
! = !
!! !!
where time is given in years and distance in astronomical units (AU = 150,000,000 km).
Following the statement of the law made by Kepler, Newton proved that in the equation
the masses of the bodies should be present, and thus he obtained the following formula:
! !
!! (! + !! ) !!
! = !
!! (! + !! ) !!
where M is the mass of the Sun (body located in the center of the orbit), equal to 330,000 times
the mass of the Earth, and m1 and m2 are the masses of the considered bodies that move in
elliptical orbits around it. This expression allows to calculate the mass of a planet or satellite, if
the orbital period P and its average distance to the Sun are known.
Overall for the planets of the solar system only the mass of Jupiter and Saturn are not
negligible with respect to the Sun. Because of this, in most cases (M + m) is considered equal to
1 (solar mass) so that the expression becomes the one originally given by Kepler.
For the first time a single geometric curve, without additions or components, and a single
rate law is sufficient to predict planetary positions. Also for the first time, the predictions are as
accurate as the observations.
These empirical laws found their physical and mathematical support in Newton’s
universal gravitation theory, who established the physical principles that explain planetary
motions. The construction of this body of ideas, that begins with Copernicus and culminated in
Newton's mechanics, is a prime example of what is considered a scientific procedure, which can
be described very briefly as follows: there is a fact, take measurements and draw up a data table,
then try to find the laws that relate these data and, finally, carefully investigate before to decide
to support or explain the law. On the other hand, new or more precise measurements can show
that a law or theory is wrong or approximate so that a new one is required. Einstein’s law of
gravitation is an example.
Application in the present case.
The orbits of Earth and Venus around the Sun are slightly elliptical and thus the ratio of
the distances rT/rV is not constant over the time. To find this ratio at time t of observation is
necessary to refer to Kepler's first law, which says that the Sun is one of the foci of the ellipse
and, thus, the distance between the Sun and a planet rp (t) can be obtained as:
rp(t) = Rp (1 − ep cos Ep(t)),
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24. Figure 20: Section of ellipse showing eccentric (E) and true (θ) anomaly.
where Rp is the semi-major axis of the orbit, ep is the eccentricity and Ep(t) the eccentric
anomaly (angle measured from the centre of the ellipse, which is the angle between the
projection of the planet on the so called auxiliary circle, and the ellipse major axis, see Figure
16) at time t. So, according to this
rT/rV = [RT (1 − eT cos ET)] / [RV (1 − eV cos EV)]
Kepler's third law links the semi-major axes of the orbits with periods of revolution Pp:
(RT / RV)3 = (PT / PV)2,
so that:
rT/rV = (PT / PV)2/3 (1 − eT cos ET) / (1 − eV cos EV) [2]
So far we have determined πS and rT, which are the parallax and the Earth-Sun distance at
the instant t of observation.
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25. APPENDIX IV. Calculation of the rotation / translation of the
images.
As we discussed in paragraph 5.2.1, if we take an image of the Sun from a place on
Earth's surface at a given instant of time t, and at exactly the same time we take another picture
from another location far enough from the first, these two images will be rotated by an angle θ
which depends directly on the separation of these two positions on the Earth. In addition, if the
telescope pointings are not exactly the same, a translation will also be present between the two
images. We must remember that the images were taken with the same instrumentation and,
therefore they have the same scale.
Coordinate Systems of S and S'.
Suppose two coordinate systems. We call one S' and is located in the center of the Sun
and the other S which, for convenience, place in the lower left corner of the image (see Figures
21 and 22). A really important thing is that S is the same for both images.
Figure 21: Diagram of the graphical representation of used systems.
The transformation equations for translation between the two systems are:
x'i1 = xi1 − xc1 ; x'i2 = xi2 − xc2
y'i1 = yi1 − yc1 ; y'i2 = yi2 − yc2
where (xc1, yc1), (xc2, yc2) are the coordinates of the center of the Sun as observed in Sapporo and
Cairns, respectively, in the coordinate system S. (xi1, yi1), (xi2, yi2) are the coordinates of a point
measured on the two images, S system, and (x'i1, y'i1), (x'i2, y'i2) are the corresponding
coordinates using the reference system S' centered on the Sun.
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26. Measurement of the angle θ.
Once we have all the items referred to the coordinate system S' centered on the Sun, we
will calculate the value of the angle θ by calculating the difference between the angles formed
by the vector T'2 relative to the Spot on the image of Cairns, and the same Spot on the image of
Sapporo T'1, through the expression (scalar product):
T! ! · T! ! = T! ! · T! ! · cos θ !′!! · ! ! !! + !′!! · ! ! !!
⟹ cos ! =
T! ! · T! ! = !′!! · ! ! !! + !′!! · ! ! !! !
!′! + !′! ·
!
!′! + !′!
!! !! !! !!
Δπ calculating the coordinate system S
Finally we are going to calculate the distance value Δπ referred to the system S, whose
zero is located in the lower left corner of the image.
Figure 22 : Vector diagram for each point according to the systems S' (centre of the Sun) and S (bottom left image).
According to Figure 22, we can express the vector T1, T2, R1 and R2 in terms of vectors S'
centred on the Sun, and the vector c from the origin of coordinates of S with S', as:
!! = ! + !! ! !! = ! + !! !
On the other hand, the expression of Δπ we have above can also be expressed in terms of
the coordinates, so that the total value is:
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27. !
Δ! = Δ!! ! + Δ!! !
So:
!! ! !! !! ! !!
Δ!! = !! − !! = !! − !! + !! Δ!! = !! − !! = !! − !! + !!
where (x2", y2") are the coordinates of the shadow of Venus in Cairns in the reference system S'
rotated by the angle θ, and using the transformation equations for rotation and translation to
coordinates on S we will have:
!! !
Δ!! = !! − !! = !! − !!! cos ! + !! − !!! sin ! − !! + !!!
!! !
Δ!! = !! − !! = − !! − !!! sin ! + !! − !!! cos ! − !! + !!!
where (x1, y1) and (x2, y2) are the coordinates of the shadow of Venus in the images of Sapporo
and Cairns, respectively, while (xc1, yc1), (xc2, yc2) are the coordinates of the centre of the Sun in
Sapporo and Cairns, respectively, all referred to the coordinate system S.
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28. APPENDIX V. GLOSSARY.
Arcminute (minute of arc)
Is a unit of angular measurement equal to one sixtieth (1 ⁄ 60) of one degree or (π ⁄
10,800) radians. Since one degree is defined as one three hundred and sixtieth (1 ⁄ 360) of a
rotation, one minute of arc is 1 ⁄ 21,600 of a rotation.
Arcsecond (second of arc)
An angular measurement equal to 1 / 60th of an arc minute or 1 / 3600th of a degree. It is 1
⁄ 3,600 of a degree, or 1 ⁄ 1,296,000 of a circle, or (π ⁄ 648,000) radians.
Astronomical Unit (AU)
The astronomical unit is a unit of length used by astronomers, usually to describe
distances within planetary systems such as our Solar system. One AU is equal to 149,597,871
km, and corresponds to the average distance from the Earth to the Sun.
Blurring
It is said when an image is not clear and it seems not well focussed. This is due to
atmospheric seeing and telescope diffraction.
Center of mass
The center of mass or barycenter of a body is a point in space where, for the purpose of
various calculations, the entire mass of a body may be assumed to be concentrated.
Diffraction
Diffraction is the ability of a wave to bend around corners. The diffraction of light
established its wave nature
Eclipse
It is the obscuration of a celestial body caused by the interposition of another body
between this body and the source of illumination.
Ecliptic
The ecliptic is the path that the sun appears to follow across the celestial sphere over the
course of a year. Indeed, it is the plane defined by the Earth's orbit around the Sun.
Ephemeris
An ephemeris is a table listing the spatial coordinates of celestial bodies and spacecraft as
a function of time.
Filter
A filter is an optical device that blocks certain types of light and transmits others. In
astronomy filters are mostly used to study light from a source in one particular colour, i.e. in a
particular wavelength region, which can yield information on the chemistry of the object.
Flux
The flux is the measure of the amount of energy given off by an astronomical object over
a fixed amount of time and area.
Galilean moons
The name given to Jupiter's four largest moons, Io, Europa, Callisto & Ganymede.
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29. Gravity
Gravity is a mutual physical force of nature that causes two bodies to attract each other.
The more massive an object, the stronger the gravitational force.
Inferior conjunction
A conjunction of an inferior planet that occurs when the planet is lined up directly
between the Earth and the Sun.
Inferior planet
A planet that orbits between the Earth and the Sun. Mercury and Venus are the only two
inferior planets in our solar system.
Latitude
Latitude is the angular distance north or south from the equator of a celestial object,
including the Earth.
Limb
The outer edge of the apparent disk of a celestial body.
Longitude
Longitude, on Earth, is a geographic coordinate that specifies the east-west position of a
point on the Earth's surface. It is an angular measurement, usually expressed in degrees, minutes
and seconds. Specifically, it is the angle between a plane containing the Prime Meridian and a
plane containing the North Pole, South Pole and the location in question. If the direction of
longitude (east or west) is not specified, positive longitude values are east of the Prime Meridian,
and negative values are west of the Prime Meridian. The closest celestial counterpart to
terrestrial longitude is right ascension.
Meridian
The meridian is an imaginary north-south line in the sky that passes through the
observer's zenith.
Micron
A micron or micrometre is one-millionth of a metre.
Node
One of the two points on the celestial sphere associated to the intersection of the plane of
the orbit and a reference plane. The position of the node is one of the usual orbital elements.
Nucleosynthesis
Nucleosynthesis is the production of new elements via nuclear reactions. Nucleosynthesis
takes place in stars. It also took place soon after the Big Bang.
Occultation
Occultation is an event that occurs when one celestial body conceals or obscures another.
For example, a solar eclipse is an occultation of the Sun by the Moon.
Opposition
A planet is in opposition when the Earth is exactly between that planet and the sun.
Orbit
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30. The term orbit denotes the path an object follows around a more massive object or
common center of mass.
Parallax
Parallax is the apparent change in position of two objects viewed from different locations.
It is caused only by the motion of the Earth as it orbits the Sun.
Parsec
A parsec is a unit of distance commonly used in astronomy and cosmology, the parsec is
equal to about 3.262 light years, or 3.09 × 1016 metres. It is the distance at which a star would
have a parallax of 1 arcsecond.
Periastron
The point of closest approach of two stars, as in a binary star orbit. Opposite of apastron.
Perigee
The perigee is the point in the orbit of the Moon or other satellite at which it is closest to
the Earth.
Perihelion
The perihelion is the point in the orbit of a planet or other body where it is closest to the
Sun. The Earth is at perihelion (the Earth is closest to the Sun) in January.
Planet
A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be
rounded by its own gravity, is not massive enough to cause thermonuclear fusion and hence it
does not shine on its own.
Refraction
Refraction is the change in direction of a wave due to a change in its speed.
Resolution (spatial)
The spatial resolution is the ability of an instrument mounted on a telescope to
differentiate between two objects in the sky which are separated by a small angular distance.
The closer two objects can be while still allowing the instrument to see them as two distinct
objects, the higher the spatial resolution.
Resolution (spectral or frequency)
The spectral resolution is the ability of an instrument mounted on a telescope to
differentiate two light signals which differ in frequency by a small amount. The closer the two
signals are in frequency while still allowing the instrument to separate them as two distinct
components, the higher the spectral resolution.
Revolution
Revolution is the movement of one object around another.
Seeing
The term seeing in astronomy is used to describe the disturbing effect of turbulence in the
Earth's atmosphere on incoming starlight.
Superior conjunction
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31. A conjunction that occurs when a superior planet passes behind the Sun and is on the
opposite side of the Sun from the Earth.
Superior Planet
A planet that exists outside the orbit of the Earth. All of the planets in our solar system
are superior except for Mercury and Venus.
Transit
Transit is when a smaller astronomical object passes in front of a larger one. During this
time, the smaller object seems to be crossing the disk of the larger one. Transit is also the
passage of a celestial body across an observer's meridian.
Universal time
Universal time (abbreviated UT or UTC) is the same as Greenwich Mean Time
(abbreviated GMT) i.e., the mean solar time on the Prime Meridian at Greenwich, England
(longitude zero).
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