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Wednesday, 28 June 2017 00:11

Roland Boninsegna

"EAAE Summerschools" Working Group

Centre de Dépaysement et de Plein Air de Fleurus-Sivry (Belgium)

Abstract

This laboratory exercise will give the opportunity to use simple observations in order to confirm the heliocentric theory, define the Venus orbit and distance along with its revolution period. The students must be able to draw and measure angles with a protractor and to use a pocket calculator for simple operations.

Introduction

In December 1610, Galileo, using the first refractor, noticed for the first time the phases of Venus. Following observations were the first observational proof that the Sun was at the centre of our planetary system and not the Earth. The heliocentric theory already developed by Copernicus and, long time ago, by Aristarchus; was anyway rejected by ecclesiastic authorities. Some images of Venus phases are presented on Figure 1.

Figure 1 - A composite sequence of images of Venus photographed with the 50 cm reflector at the Torbay Astronomical Society Observatory during its evening apparition between early May and late October 2002. The Sun could be situated at the middle of the left border. North is down and thus the image must be rotate by 180° to mimic a natural view on the sky.Figure 1 - A composite sequence of images of Venus photographed with the 50 cm reflector at the Torbay Astronomical Society Observatory during its evening apparition between early May and late October 2002. The Sun could be situated at the middle of the left border. North is down and thus the image must be rotate by 180° to mimic a natural view on the sky.
Figure 1 - A composite sequence of images of Venus photographed with the 50 cm reflector at the Torbay Astronomical Society Observatory during its evening apparition between early May and late October 2002. The Sun could be situated at the middle of the left border. North is down and thus the image must be rotate by 180° to mimic a natural view on the sky.

Venus in the sky

As Venus is closer to the Sun (like Mercury), we can observe it just only a little time after sunset ("evening star") or a little time before sunrise ("morning star"). That planet, when visible, is always very bright due to its proximity of the Earth and also because Venus is surrounded by a dense atmosphere, which reflects 65% of the input sunlight (albedo 0.65). For more physical details concerning Venus, please consult the references chapter.

If you use a simple instrument with a 40x magnification, one interesting detail will appear: there is both a sunlit part and a dark shadow part, which relative fractions are varying weeks after weeks. The planet shows "phases" as our Moon do. As seen from the Earth, when the planet (or the Moon) is completely illuminated (by the sunlight), it is "full": the phase is 100%. When it is half illuminated, its phase equals to 50%. Similar to the Moon, the phases of Venus (or Mercury) change all the time, depending on where the body is placed relatively to the Earth and Sun (see Figure 1). If we were living on Saturn, we could observe phase phenomena for all the "internal" planets (Mercury, Venus, Earth, Mars, Jupiter). In reality each planet is always half illuminated.

The moment when the angular distance between the Sun and Venus is maximal is called "greatest elongation": "greatest eastern elongation" when Venus is observed as an "evening star", "greatest western elongation" when Venus is observed as a "morning star". For example, what is the date of the greatest elongation using the data of Figure 1? Sometimes between 12 and 31 August 2002 (exactly on August 22). Be careful, it is a "greatest eastern elongation" at that time. So the image has North down and must be rotate in order to place Venus left of the Sun as an "evening star" to imitate the natural view.

The measure of the separation between the two objects could give us some information concerning the trajectory of Venus. Meanwhile, the direct observation of the planet (with a small telescope) and the value of the phase and apparent diameter will allow us to draw a 2 D model of the orbit.

However, measuring the angular separation between Venus and the Sun is difficult: you must use a filter to aim the instrument (sextant) to the Sun and have e keen sight to pick out Venus on the bright blue sky. On a black sky, when Venus is easily seen, the Sun is below the horizon making the measurement impossible.

In that laboratory, we shall use the transit times of the Sun and Venus predicted for six dates, along with the physical appearance of our sister planet as if it was observed with a telescope (see Table 1). You will also need a protractor, a great ruler (1 meter or more), a candle, table tennis balls or styropor (polystyrene) globes and a pocket calculator.

The laboratory

A) Calculating the elongation angle

Due to Earth rotation, on 2004 January 31, the Sun and Venus cross the meridian (a North-South line) at 11h55 and 14h27 U.T. respectively for every place situated at a 4.6° east longitude. That is true even for northern European regions, where the Sun is below the horizon.

Table 1 - Observational data for the laboratory. From left to right: Date and number, phase and diameter of Venus as seen from the Earth, Sun and Venus transit times in Universal Time for a 4.6° East longitude and value of the elongation.  All the images have the same scale (1x1.5 ').
Table 1 - Observational data for the laboratory. From left to right: Date and number, phase and diameter of Venus as seen from the Earth, Sun and Venus transit times in Universal Time for a 4.6° East longitude and value of the elongation. All the images have the same scale (1x1.5 ').

Of course, if you are living in Portugal or Bulgaria, the transit time will be different for both, but the time difference between the Sun and Venus transit will remains the same (2h32min). We shall use that time difference to calculate the elongation angle (see Figure 2).

Figure 2 - Due to Earth rotation, the Sun and Venus cross the meridian at different moments.
Figure 2 - Due to Earth rotation, the Sun and Venus cross the meridian at different moments.

The times difference between these two passages give the elongation angle between the two objects.

The Earth achieves one rotation (360 degrees) in 24 hours. What is the value of the rotation angle in one hour? What is the angle value for a 2h32min transit time difference?

2h32min = 2.53h

Thus the elongation angle for 2004 January 31 is: 15° x 2.53 = 38°

As Venus cross the meridian later than the Sun, it is observed as an "evening star" with an eastern elongation. Complete the item "elongation" in the first line of Table 1.

B) Positioning Venus according to its elongation angle and its phase

During the measurements, always be accurate! On the floor of a dark classroom, draw a 1-meter long line. Put a candle (Sun) on one end and a protractor at the other end (Earth). From that point, measure the elongation angle eastward (left) of the Sun, extend the line up to 1.5 or 2 meters. Move a tennis table (or styropor) ball along this line up to a point where the shadow mimics the real phase of Venus. Mark the point and write down the date (or the number). You will need to do ball position evaluation in a dark room or replace the candle by a bright light bulb. Do the same (points A and B) for the other five dates. Don't hesitate to compare also the relative apparent diameter of Venus. You will notice that, for two dates, it will be impossible to decide where to put the ball with accuracy. Don't worry; you will be able to do it later. You will find the complete results for these six dates on Table 2.

Date Elongation (º) Phase (%) Apparent Diameter ('') Distance from Earth (A.U.)
(1) 2004 January 31 38 E 75 12 1.126
(2) 2004 march 31 44 E 50 24 0.695
(3) 2004 June 8 0 0 58 0.289
(4) 2004 August 17 45 W 50 24 0.706
(5) 2004 October 15 35 W 75 15 0.137
(6) 2005 March 29 1 S 100 10 1.724

Table 2 - Complete data for the six dates presented in Table 1.

C) The orbit of Venus and the heliocentric theory

The drawing complete (see Figure 3), measure each Sun-Venus and Earth-Venus distances. What is the difference between the two sets? The earth-Venus distances are very different but the Sun-Venus distances have remained almost constant. The four positions of Venus along with the phases and the apparent diameter allow us to imagine the trajectory of Venus as a circle. As a circle? Measure the four distances between the Sun and Venus and compute a mean value. Using that mean distance as the radius, draw a circle from the Sun. What about your feeling? Is it really a circle? Try to imagine another solution. Now, try to find the place of Venus for the dates 3 and 6. Is it fitting your solution?

Figure 3 - Elongation angles and position of Venus using the data from Table 1.
Figure 3 - Elongation angles and position of Venus using the data from Table 1.

As you may notice, these observational arguments are simple and they convinced Galileo that Aristarchus and Copernicus were right after all. Even though Galileo was forced to deny all his results, his work was later reconsidered and the face of the world changed.

D) The distance of Venus

We know that the Earth-Sun distance (discovered from observations during Venus transits across the solar disk) is around 150 millions of km (15 107 km). Astronomers define that distance as " 1 Astronomical Unit" (1 A.U.). Using your measures, could you calculate a mean Sun-Venus real distance? The official value is 0.723 A.U. You can use also your Earth-Venus measurements to compute the real Earth-Venus distances for points 1, 2, 4, 5. You can compare your results with the detailed data from Table 2.

E) The apparent revolution period of Venus (synodic)

From the Table 1 data and your drawing, you could compute the apparent revolution period of Venus also called the synodic period. Here "apparent" means, "as seen from the moving Earth". Which data are you going to use? You cannot choose whatever data. If you use the time interval between points 2 and 4, you are only taking into account the part of the orbit where Venus is nearer the Earth and where the apparent movement is faster. If you use points 1 and 5, the choice is still not correct because you avoid the part of the orbit where Venus is far away from Earth and where the apparent movement is slower. The best is to choose points 3 and 6 where you take fast and slow motions together into account. The time interval between these two points represents a half synodic period. The official value of the synodic revolution period is 584 days. Which value have you obtained? Less than 1 % difference?

F) The true revolution period of Venus (sidereal)

Figure 4 - Which is longer: synodic or sidereal period?
Figure 4 - Which is longer: synodic or sidereal period?

Of course, the synodic revolution period is not the real revolution period of Venus. The true one would be the value obtained from a non-moving Earth or from a fixed position in space. It is called the sidereal revolution period. Which period is longer: the sidereal one from a fixed point of view or the synodic one from a moving Earth?

You can test the difference by organizing a game. Draw two circles: one with 0.7 m radius inside the second with 1 m radius. You can use the drawing already made on the floor. Trace a 1 m straight line (the departure line) from the centre (see Figure 4). Choose two students with different feet size: the student with the longest feet size will play the role of Venus, the other one will play the role of the Earth which moves slower than Venus. Verify that the students will depart from their right orbit on the same departure line. Each student will advance by one foot on every second (or on each signal) on his own orbit. How much time have we to wait until Venus reaches again the departure (that's the sidereal period)? How many time to wait until the two students line up on a same common radius (that's the synodic period)?

By the way, what is the value of the Earth sidereal revolution period?

.../...

Right: 365.25 days!

You can calculate the value of the sidereal revolution period of Venus (VSP) using the value of its synodic period (VsP), obtained previously (see above point E), and the value of the Earth sidereal revolution period (ESP) using this relation:

1/VSP = 1/VsP + 1/ESP

The official value is 224.7 days.

One curious aspect of Venus is that its rotation period (243 days) is longer than its revolution period. Moreover, the sense of that rotation is clockwise or retrograde unless the other planets.

This laboratory is based on a 2 dimensions model of the orbit of Venus. We supposed that the orbits of Venus and Earth are on the same plane. That is not true! The orbit of Venus is tilted by 3.4°. That is why we cannot see a Venus transit across the solar disk every 584 days. That is why in Table 2 the value of the elongation is not zero on 2005 March 29 but 1° S. So, the elongation angles calculated using the transit times, the distances measured from the drawing are not very accurate but precise enough to obtain good values for that laboratory.

Acknowledgments

I would like to thank Mogens Winther whose work "Seeing is believing" is the framework of this laboratory. See the references chapter below for his web site. Thanks also to Bill Gray and his software Guide 8, which allows me to prepare some images, informations and animations, concerning Venus. Want to observe Venus? Here below (Table 3), you will find the dates of future Venus greatest elongations.

Eastern Elongation (evening) Western Elongation (morning)

29 March 2004

3 November 2005

9 June 2009

14 January 2009

20 August 2010

17 August 2004

25 march 2006

28 October 2007

5 June 2009

8 January 2011

Table 3 - Future dates for Venus greatest elongations.

References

Gray B.J., 2002, Guide 8 software: http://www.projectpluto.com/

Laurel C. et al., Celestia software: http://www.shatters.net/celestia

The Torbay Astronomical Society: http://www.halien.com/TAS

View of the solar system (in different languages): http://www.solarviews.com/ss.html

Winther Mogens, Seeing is believing: http://www.amtsgym-sdbg.dk/as/venus/ven-dist.htm