Calculating the Earth-Sun Distance From Images of the Transit of Venus

8/10/2019 Calculating the Earth-Sun Distance From Images of the Transit of Venus

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EDUCATIONAL ACTIVITY 1.

Calculating the Earth-Sun distance from images of the transit of Venus.Mr. Miguel ngel Po Jimnez. AstronomerInstituto de Astrofsica de Canarias, Tenerife.Dr.Miquel Serra-Ricart.AstronomerInstituto de Astrofsica 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 partialblackout 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 Earths

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.


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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 Earths. 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 Venuss 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.

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.

Figure 1: Illustration of the line of nodes of Venus' orbit, when it intersects Earth's orbit.


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From a visual point of view, the phenomenon of the transit of Venus is similar to

Mercurys: Venus is visible as a black circle moving slowly over the brilliant solar disk. The

transit of Venus lasts a maximum of 8 hours. During the transit, Venus has a very small

apparent diameter. However, it is clearly visible with the naked eye, properly protected, to

observe the Sun safely. A phenomenon called the Black Drop effect can also be observed at

the edges of the solar disk.

The Black Drop effect. Just after the internal contact between the disks of the Sun andVenus, the disk of the planet seems to remain attached to the rim of the solar disk for a couple

of seconds, becoming deformed and assuming the shape of a black drop. This phenomenon is

repeated right before the last internal contact (Figure 3). The black drop effect prevents theaccurate measurement of the time of contact between the disk of the planet and the disk of the

Sun1. This was the main cause of the inaccuracy in the observations used to calculate the

distance between Sun and Earth. This effect was first attributed to Venus atmosphere. However

using images of the transit of Mercury by the TRACE satellite (Transition Region and Coronal

Explorer, NASA, USA) it was found2 that the main causes of the black drop effect are image

blurring (due to atmospheric seeing and telescope diffraction) and solar limb darkening. This

implies that the development of the black drop effect as seen by an observer on Earth depends

mainly on the atmospheric conditions and the quality of instrument used (e.g. size and optics of

telescope).

1See a method to increase precision in contacts timing: http://www.transitofvenus.nl/blackdrop.html

2See the scientific paper: http://nicmosis.as.arizona.edu:8000/POSTERS/TOM1999.jpg

Figure 2: Venuss Transit on June 8, 2004 showing the path of Venus across the solar diskat intervals of 45 minutes. Credits: Juan Carlos Casado starryearth.com.


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The "Venus Aureole" effect. During the transits of Venus, a bright arc, about 0.1seconds of arc thick, has often been reported. It is seen around the circumference of Venus disk,

which is partially outside the solar limb. It was the Russian scientist Mikhail Lomonosov who

first described this effect when he observed the transit of Venus in 1761. Just after interior

contact at egress, this aureole effect starts with the appearance of a bright spot of light near one

of the poles of Venus. Generally, this spot gradually grows into a thin arc as Venus moves

further off the Sun (see Figure 4). At ingress, the phases occur in reverse order. The brightness

of the aureole is close to that of the solar photosphere, making it visible through a solar filter. It

can only be seen under good observing conditions, using an excellent telescope.

The Aureole effect is caused by refraction of the Sun light in the dense upper atmosphere

of Venus. Venus atmospheric conditions will determine the appearance of the aureole. If the

refraction index of the atmosphere is small, the aureole already breaks into bright spots when

the disk of Venus is just off the solar disk. But if the refraction index of the atmosphere is high,

the aureole will extend all around the planets limb as a complete arc (see Figure 4).

3.3. Previous transits.

17thCentury. The first recorded transit of Venus was on December 4, 1639. Horrocks, a

Figure 3: Evolution of the black drop phenomenon during the ingress of Venus on the solar disk. Credits: JuanCarlos Casado starryearth.com.

Figure 4: Venus Aureole effect detected in the 2004 Venus transit

using the 1m Swedish Solar Telescope located on Roque de LosMuchachos Observatory (La Palma , Instituto de Astrofsica de

Canarias). Credit: D. Kiselman, et al. (Inst. for Solar Physics), RoyalSwedish Academy of Sciences.


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cleric in Liverpool (England), who had studied Astronomy and Mathematics, was able to follow

the transit of the planet when it had already begun.

18thCentury. In the early eighteenth century, the English astronomer Edmund Halleyproposed 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.

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 ofastronomers 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 madewith space probes and radar techniques tell us that the solar parallax has a value of 8.79415seconds of arc or 149,597,892 km. During the 2004 transit observations and photographs were

Figure 5: Schematic representation to show the solar parallax, or angle !. This angleis very small indeed, but for the sake of clarity it is shown enlarged.


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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 ! 6

th, 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

time in December.

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 observationsfrom 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.

Halleys 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 beobserved. However problems can arise because of bad weather that can prevent the observation.

Transit

EndsatSunset

Transit

Begin

sa

tSunset T

ransitE

ndsatS

unris

e

TransitB

eg

insatS

unris

e

F. Espenak, NASAs GSFC eclipse.gsfc.nasa.gov/OH/transit12.html

I III IIIII III

IV IV EntireTransitVisible

NoTransitVisible

Transitin Progressat Sunrise

Transitin Progressat Sunset

GreatestTransit

at Zenith

(June 05) (June 06)

Region X*

* Region X - Beginning and end of Transit are v isible, but the Sun sets for a short period around maximum transit.

Region Y*

* Region Y - Beginning and end of Transit are NOT v isible, 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).


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II. Delisles 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 Venuss shadow over the Sun's disk as seen

from the two locations. A full description of this method is given in Appendix I.

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 V1and V2the observed centers of the projection of Venus

seen from M1and M2, respectively. The angles D1and D2are the angular separations betweenthe centers of Venus and the Sun seen from M1and M2, respectively, i.e., the angles of parallax

CM1V1 and CM2V2. Similarly, we define the angles "sand "vas the angular separations between

Figure 7: A diagram of the meaning of Interior Contact and Exterior Contact.

Figure 8: Simultaneous observation of the transit of Venus in front of the disk of the Sun from two differentlocations M1and M2at the same instant of time.


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M1 and M2 views from the Sun and from Venus, respectively, i.e., the angles M1CM2 and

M1VM2. By definition we have,

!"# !! !!

!!

! !"# !! !!

!!"

where rTis the Sun-Earth distance distance, rVT is the Venus-Earth distance, and dis the

distance between M1and M2in 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 "i#"i.

The Earth, Sun and Venus are aligned, so that rVT= rT rV(where rvis the Venus

Sun distance).

The observation points M1and M2on 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 $"= "V "S. Since we have:

!! !

!

!!

!"# !! ! !

!! ! !!

by substitution we can set,

!! !

!!! !!

!! ! !!

Since $"= "V "S, we can substitute in for "vto get:

!" ! !!

!!

!! ! !!

! ! ! !!

!!

!! ! !!

And so,

!! ! !"!!

!

!

! ! !!

!

!

Rearranging, we get the Earth-Sun distance, rT, at the time of observation to be:

!! !!

!"!!!!!!!!!

Equation [1]

where $"is an observable quantity (distance between the centres of the shadow of Venus on the

solar surface in units of radians), dis determined from the locations (see Appendix II), and the

ratio rT/ rVof the Sun-Earth and Earth-Venus distances can be obtained (see Appendix III). If

we express din kilometres, then the Earth-Sun distance will be also in kilometres.

The observable $"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).


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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, $",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 M1or 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).

Using Pythagoras theorem, we can write the following expressions:

!!! !

!

!

!

! !!!!

!

!

!!! ! !

!

!

!!!!!

!

!

So we can express A'B' as:

!!!

!! !

!! ! !

!! !

!

!

!

!!!!!

!

!

!

!

!

!

!!!!!

!

!

So measuring the length of the strings A1A2, B1B2, along with the diameter of the Sun

(D), we can then obtain the parallax $!from,

Figure 9: A representation of the two

strings on the surface of the Sun, A1A2

and B1B2as seen by observers at locations

M1and M2on Earth.


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!! !!

!!! ! !!!!

!! !! ! !!!!

!

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: !16 55' 24.237" Longitude: 145 46' 25.864"

Sapporo (Japan): Latitude: 43 3' 43.545" Longitude: 141 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 Suns 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 Suns 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, ", of the sun in the image will be:

!"#$%!!! !!"!! !"#$%&'!"!"#!!! ! !" !"#$%&!!"!"#!!!

!"#$!"#$%&

! !!!"!!!"#$%

The telescope and camera will be mounted on an Astrotrack mount, which is very

stable and easy to assemble and tracks the Suns 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.


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Figure 10: A photograph of the Instruments. Credits: M. A. Po (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 $"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 $"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) aspot on the solar surface and (iii) the centres of the Sun in each image. Figure 11 shows the


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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 ($") will be very small, being of the order of 10pixels maximum.

Following the calculations in Appendix IV, the observable $" is determined from the

expression:

!! ! !!!

!! !!!

!!

where the components $"xand $"ycan be expressed as:

!!! ! !! ! !!! !"# ! ! !! ! !!! !"# ! ! !! ! !!!

!!! ! ! !! ! !!! !"# ! ! !! ! !!! !"# ! ! !! ! !!"

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 ofthe 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 #is (see calculations in Appendix IV):

#= 108 4' 17.92"

So:

!! ! !!!!! !!!

!!

! !!!!"#$%&

Appendix I presents a very precise method to determine the value of the distance dbetween the two observers on Earth. In our case, the value of dis:

Figure 11: A real image of the Sun made with the instrumentation that will be used for the transit, with a fictionalrepresentation of the phenomenon at 00:45 UT of June 6, 2012.


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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 $"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 $" is 8.4 pixels, and whereas the diameter of the sun in

pixels is 715 pixels, which gives a scale of:

!"#$%!!! !!"!! !"#$%&'!"!"# ! ! !" !"#$%&!!"!"#!!!

!"#!"#$%&

! !!!"#$!!"#$%

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].

!! !

!!!"!! !"

!!! !"#$%& ! !!"#$%#&'(

!"#$%& !

!

!"#$$$

!"#

!"#$%#&'( !!!!"#$" ! !!

! !"#!! ! !"!

!"

can determine the value of rT, the distance from Earth to the Sun. Rcall that the value of

$"must be expressed in radians, hence the term " / 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 M1and M2, all measured in the same units.

Figure 12 shows a representation of what might be seen using this method and an

astronomical software.


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The lengths of the lines M1and M2, based on this image, are join A1 to A2 and B1B2,

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, ", will then be [arcsec/mm].

For the sample image, the value of Sun's diameter D, in pixels, is 711, and the string M 1

(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 $". Thus, the expression that we use

is:

!! ! !!!

!!

!

!!! ! !!!!!!

!! !! ! !!!!

!! !!!"!"#$%&

Finally, substituting into Equation [1]:

!! !

!!!"!! !"

!!!" !"#$%& ! !!"#$%#

!"#$%& !

!

!"#$$$

!"#

!"#$%#!!!!"#$%" ! !!

! !"#!! ! !"!

!"

where, again, the value of rTis the distance Earth-Sun distance,dis the distance between

observers, determined according to Appendix II, % is the scale value described, and the ratio

rT/rVis 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 / rVhas to be the

instantaneous value at that time, but in the case of Method 2, the value of the ratio rT/ rVto be

Figure 12: An image made with astronomical software, with a simulated representation of the strings.


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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 Keplers 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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APPENDIX I. Calculations of Method 1 in depth.

The determination of the Earth-Sun distance is based on the parallax effect (as seenabove) 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 Keplers 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 Keplers laws.

From two different locations M1 and M

2 (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.

Point O is the centre of the Earth, C the centre of the Sun and V 1 and V2 the observed

centres of the projection of Venus as seen from M1and M2, respectively. The angles D1and D2

are the angular separations between the centres of Venus and the Sun seen from M 1and M2,

respectively, i.e., the angles of parallax CM1V1and CM2V2. Similarly, we can define the angles

as "V "S and angular separations between M1 and M2 seen from the Sun and from Venus,

respectively, i.e., the angles and M1VM2M1CM2.

Since the four points M1, M2, C and V are not in the same plane (the most common case

will not have both sites M1and M2on 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 $"= "V "Sis (hardly) the only observable

quantity which allows to calculate the distance to the Sun.

The practical realization of the measure of $" 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.

Figure 13: Observation of the transit of Venus in front of the solar disk from two different locations M1and M2at thesame instant of time.


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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 thefactor of scale (").

!"#$%!!! !!"#$%!"!"#$%&!"#$%&%'!!"#$%#!

!"#!"#$%&%'!!!!"!"#$%&!

The distance between the centres of Venus in the two images will be:

$"(arcsec) = [(x2!x1)2+ (y2!y1)

2]

1/2&"

Suppose rVand rTare the distances between the centre of the Sun and Venus and Earth,

respectively, at time t of observation. Since the projection of the distance dbetween M1and M2

in the plane perpendicular to OC is small compared to the distances Earth-Sun and Earth-Venus,

we can approximate:

"S= d/rT

"V= d/(rT!rV)

and from here, we can obtain:

"V= "SrT/(rT!rV)

$"= "S(rT/(rT!rV) !1) = "SrV/(rT!rV)

so,

"S= d/rT= $"(rT/rV!1).

The latter formula expresses that if we know the angular distance, $", between the two

centres V1 and V2, and the ratio rT / rV between the distances Earth-Sun and Venus-Sun, theparallax "Scan be obtained, and knowing the projected distance dbetween the two locations, we

can calculate the distance rT. (In all these expressions the values of "V, "Sand $"are given in

Figure 14: Positions of the projection of Venus overthe Sun's disk.


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radians. To convert to arcseconds and make them compatible with equations, one need only to

multiply by 648000 and divide by the number ").

$" 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 "o, which are related through Earth's equatorial radius R by:

"o#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 RTvalue 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 Earths

orbit.

Based on the above expression "o#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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APPENDIX II. Determine the value of d.

If you express the projection dof the distance between M1and M2in the plane normal tothe direction Earth-Sun in units of Earths equatorial radius and the Earth-Sun distance in units

of the average distance, we have:

"S= [(d/R) / (rT/RT)] (R/RT) #[(d/R) / (rT/RT)] "o.

The ratio rT/ RTcan be calculated from Kepler's first law as:

rT/RT= 1 !eTcos ET(t)

and therefore we only need to calculate d/R (see Figure 8).

Making the vector product between the vectors M1M2and OC, we obtain the value of sin

', because:

M1M2( OC = |M1M2| rTsin '.

In Figure 15 you can observe that:

d = |M1M2| cos (90 !') = |M1M2| sin '

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 (),*) at time t

of observation as:

x = rTcos *cos )

y = rTcos *sin )

z = rTsin *.

The position of each observer can be expressed as (see Figure 16):

Figure 15: Projection of the distance between M1 and M2 in the plane normal to the Earth-Sun

direction.


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x = R cos +cos (,+TG)

y = R cos +sin (,+TG)

z = R sin +,

where +and ,are the geographical coordinates (latitude and longitude) of the observer and T G

is the sideral time of each point of the Earth who has a longitude ,. In our case, on June 6th

2012,

the value is TG(0hUT) = 16

h59

m12.495

s.

!! ! !! ! !!"#

!

!"!!"!!!!!! ! ! !! ! !!!!! ! !!!!"#$#%&!

M1M2vector 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 *Scos )S

y = cos *Ssin )S

z = sin *S

So going back to the expression of d, it can be expressed as:

! ! !!!! !"# ! ! !!!!!! ! !" ! !"

!! !!" ! !"!! ! !!" ! !"!!

Figure 16: Positions of a star (e.g. the Sun) and anobserver on Earth in equatorial coordinates.


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APPENDIX III. Keplers Laws.

The subject of planetary motion is inseparable from a name: Johannes Kepler. Kepler'sobsession 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 Keplers 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 onefocus".

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 = 2,500,000km.

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 centerof the Sun to the planet) sweeps out equal areas in equal times".

Figure 17: Description of the elements of an object's orbit around the Sun.


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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 tothe cubes of their mean distances from the Sun".

Figure 18 : Graphical representation of Keplers 2nd law.

Figure 19 :Relationship between the periods

and radii of the orbits around the Sun of two

objects, which graphically describes Keplers3rdlaw.


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If R1and R2 are the mean distances of two planets to the Sun, such as Mars and Earth,

and P1and P2are 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 timesthe 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 Newtonsuniversal 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. Einsteins 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 !epcos Ep(t)),


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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 !eTcos ET)] / [RV(1 !eVcos 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 !eTcos ET) / (1 !eVcos EV) [2]

So far we have determined "Sand rT, which are the parallax and the Earth-Sun distance at

the instant t of observation.

Figure 20: Section of ellipse showing eccentric (E) and true (!) anomaly.


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APPENDIX IV. Calculation of the rotation / translation of theimages.

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 '

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.

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.

Figure 21: Diagram of the graphical representation of used systems.


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Measurement of the angle !.

Once we have all the items referred to the coordinate system S' centered on the Sun, we

will calculate the value of the angle 'by calculating the difference between the angles formed

by the vector T'2relative to the Spot on the image of Cairns, and the same Spot on the image of

Sapporo T'1, through the expression (scalar product):

!!!! !!

!! !!

!! !!

!! !"#!

!!!! !!

!! !!!!! !

!

!! ! !!!!! !!

!!

! !"# ! !!!!!! !

!

!! ! !!!!! !!

!!

!!!!

!! !!

!!

!!

! !!!!

!! !!

!!

!!

"#calculating the coordinate system S

Finally we are going to calculate the distance value $" referred to the system S, whose

zero is located in the lower left corner of the image.

According to Figure 22, we can express the vector T 1, T2, R1and R2in 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 $"we have above can also be expressed in terms ofthe coordinates, so that the total value is:

Figure 22 : Vector diagram for each point according to the systems S' (centre of the Sun) and S (bottom left image).


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!! ! !!!

!! !!!

!!

So:

!!! ! !!!!! !!

!! !!

!!! !! ! !! !!! ! !!

!!! !!

!! !!

!!! !! ! !!

where (x2", y2") are the coordinates of the shadow of Venus in Cairns in the reference system S'

rotated by the angle ', and using the transformation equations for rotation and translation to

coordinates on S we will have:

!!! ! !!!!! !!

!! !! ! !!! !"# ! ! !! ! !!! !"# ! ! !! ! !!!

!!! ! !!!!

!

!!!

!

!

!!!

!!! !"# ! ! !!!

!!! !"# !!

!! ! !!!

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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APPENDIX V. GLOSSARY.

Arcminute (minute of arc)Is a unit of angular measurement equal to one sixtieth (1 60) of one degree or (" 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 / 60

thof an arc minute or 1 / 3600

thof a degree. It is 1

3,600 of a degree, or 1 1,296,000 of a circle, or (" 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.

BlurringIt 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 massThe 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.

DiffractionDiffraction is the ability of a wave to bend around corners. The diffraction of light

established its wave nature

EclipseIt is the obscuration of a celestial body caused by the interposition of another body

between this body and the source of illumination.

EclipticThe 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.

EphemerisAn ephemeris is a table listing the spatial coordinates of celestial bodies and spacecraft as

a function of time.

FilterA 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.

FluxThe flux is the measure of the amount of energy given off by an astronomical object over

a fixed amount of time and area.

Galilean moonsThe name given to Jupiter's four largest moons, Io, Europa, Callisto & Ganymede.


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GravityGravity 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 conjunctionA conjunction of an inferior planet that occurs when the planet is lined up directly

between the Earth and the Sun.

Inferior planetA planet that orbits between the Earth and the Sun. Mercury and Venus are the only two

inferior planets in our solar system.

LatitudeLatitude is the angular distance north or south from the equator of a celestial object,

including the Earth.

LimbThe outer edge of the apparent disk of a celestial body.

LongitudeLongitude, 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.

MeridianThe meridian is an imaginary north-south line in the sky that passes through the

observer's zenith.

MicronA micron or micrometre is one-millionth of a metre.

NodeOne 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.

NucleosynthesisNucleosynthesis is the production of new elements via nuclear reactions. Nucleosynthesis

takes place in stars. It also took place soon after the Big Bang.

OccultationOccultation 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.

OppositionA planet is in opposition when the Earth is exactly between that planet and the sun.

Orbit


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The term orbit denotes the path an object follows around a more massive object or

common center of mass.

ParallaxParallax 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.

ParsecA 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.

PeriastronThe point of closest approach of two stars, as in a binary star orbit. Opposite of apastron.

PerigeeThe perigee is the point in the orbit of the Moon or other satellite at which it is closest to

the Earth.

PerihelionThe 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.

PlanetA 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.

RefractionRefraction 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.

RevolutionRevolution is the movement of one object around another.

SeeingThe 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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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 PlanetA planet that exists outside the orbit of the Earth. All of the planets in our solar system

are superior except for Mercury and Venus.

TransitTransit 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 timeUniversal 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).

Calculating the Earth-Sun Distance From Images of the Transit of Venus

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