THE ORBIT OF VENUS
Roland Boninsegna
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.
I) 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.
II) 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.
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.
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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.
III) The laboratory.
A) Calculating the elongation angle.
Due to Earth rotation, on 2004 January 31, the Sun and Venus cross the meridian (a NorthSouth 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.
DATE
VIEW FROM THE EARTH
TRANSIT TIMES
(U.T.)
2004 January 31
1
Sun: 11h55
Venus: 14h27
Elongation:
E
Sun: 11h45
Venus: 14h40
Elongation:
E
2004 March 31
2
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DATE
VIEW FROM THE EARTH
TRANSIT TIMES (U.T.)
2004 June 8
3
Sun: 11h40
Venus: 11h40
Elongation: 0°
2004 August 17
4
Sun: 11h45
Venus: 08h45
Elongation:
W
Sun: 11h27
Venus: 09h06
Elongation:
W
2004 October 15
5
2005 March 29
6
Sun: 11h46
Venus: 11h46
Elongation: 1° S
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 ’).
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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. 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 1meter 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
(1) 2004 January 31
(2) 2004 March 31
(3) 2004 June 8
(4) 2004 August 17
(5) 2004 October 15
(6) 2005 March 29
ELONGATION
(°)
38 E
44 E
0
45 W
35 W
1S
PHASE
(%)
75
50
0
50
75
100
APPARENT
DISTANCE FROM
DIAMETER (”).
EARTH (A.U.)
15
1.126
24
0.695
58
0.289
24
0.706
15
1.137
10
1.724
Table2: Complete data for the six dates presented in Table 1.
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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.
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.
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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).
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)?
V
E
Figure 4: Which is longer: synodic or sidereal period?
By the way, what is the value of the Earth sidereal revolution period? Right: 365.25 days!
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You can calculate the value of the sidereal revolution period of Venus (VSP) using the value
of its synodic period (VsP), obtained previously (see chapter III 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.
IV) 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)
29 March 2004
3 November 2005
9 June 2007
14 January 2009
20 August 2010
Western Elongation (morning)
17 August 2004
25 March 2006
28 October 2007
5 June 2009
8 January 2011
Table 3: Future dates for Venus greatest elongations.
V) 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
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