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Distance to Venus

Goals of the Lab

• To virtually observe phases of Venus, calculate its distance from Earth, and create a Sun-Venus-Earth diagram.

Required Equipment: SC001 Map, Calculator, Straight Edge, Drawing Compass

Background: Venus is the second planet from the sun. Named after the Roman goddess of love and beauty, it rightly deserves its name when seen from Earth. Shining at an maximum apparent magnitude close to -5, it is only outshone by the Moon and Sun. Glittering in the morning or evening sky (you may have heard it termed the "morning star" or "evening star"), Venus's brightness is due to its proximity to Earth and it being completely enshrouded in clouds that give it a high albedo – the fraction of sunlight it reflects.

Venus is often termed "Earth's sister" because its diameter is only 650 km smaller than Earth's. The similarities end there. Due to a runaway greenhouse effect, the average temperature at the surface is around 850 °F (460 °C or 730 K), hot enough to melt lead. The thick clouds and atmosphere are made of toxic chemicals such as carbon dioxide and sulfuric acid, a primary constituent of batteries. Pressures at the surface are equivalent to being under a kilometer of ocean on Earth.

Venus is also unique in that it exhibits retrograde rotation - it spins backward on its axis, resulting in the sun rising in the west and setting in the east (don't get retrograde rotation confused with retrograde motion). Its day (243 Earth days) is longer than its year (225 Earth days). Since Venus' orbit is inside of Earth's (an average orbital radius of 107 million kilometers compared to Earth's 149 million), we should see it go through phases just like the moon, a hypothesis Galileo came up with to test the heliocentric model of the solar system. When he looked through his telescope and watched Venus over the course of its orbit, he indeed saw that the planet exhibited phases.

There have been quite a few attempts to land probes on the Venusian surface to do scientific analysis. Most were to no avail due to malfunctions or to the heat and pressure of the atmosphere essentially crumpling and melting the probes. Magellan was the first planetary orbiter to map the entire surface with cloud-penetrating radar. Results showed the surface to be very young since very few craters were found.

Part I: Plotting Venus's Position on the Sky

1. Since this is a "virtual" observation lab, we have the ability to look forward and backwards in time at the position of Venus. For this lab, we are going to make observations of Venus on two dates: April 30, 2007 and July 31, 2007. The reasoning behind choosing the two dates is that you will notice several different characteristics that ultimately give insight into how our solar system is arranged!

2. In order to determine the distance to the planet, you will need to know some of the angles in the triangle formed by Venus, Earth, and the Sun. You will use Sky & Telescope's Interactive Observing Tools to determine the Right Ascension (RA) and Declination (Dec) of Venus and the Sun on these particular dates.

3. Click on SkyTonight.com's Almanac. In the window, set your latitude to 36° 23' N, your longitude to 86° 46' W, and your time zone to -6 hours. Click the button to display your customized almanac.

4. When the almanac is displayed, click on More info. Next, set COUNTRY to USA, CITY to NASHVILLE, TN, and your TIME to 23:00. As you work through the lab, you will set the DATE to the appropriate dates listed above. After entering in the correct information, click on CALC to recalculate the ephemeris. Record the RA and Dec of the Sun and Venus for the two dates. You can now close the almanac window.

5. Next, print out the SC001 Map and plot their positions. Try to be as accurate as possible. It will help if you use a straight-edge to extend the right ascension and declination coordinate markers. Be sure to label the Sun and Venus positions and also record their coordinates and dates on the map.

6. Now you will need to determine the angular separation (in degrees) between the Sun and Venus for both dates. This is called the Sun-Earth-Venus (v) angle (see Fig. 2). Do this by converting the right ascension coordinates of both objects to degrees (This is easy if you remember that 24 right ascension hours = 360°). Next determine the difference in the right ascension coordinates as well as the difference in declination coordinates. Finally, using the Pythagorean Theorem, use your coordinate differences to determine the angular separation of Venus and the Sun. As a check to make sure your calculated angular separation values are reasonable, look that the declination scale of your starmap and guesstimate the separations - your two answers should be close.

Part II: Observing Venus and its Phase

Note: When referring to the two different observation dates, don't forget to mention which date you are talking about.

1. Below are simulated images of Venus on the two dates in question. Note that both images are one square arcminute.

   
   

   Venus on 4-30-2007

   
   

   Venus on 7-31-2007

2. You should immediately notice that Venus exhibits a phase. You should also notice that there is an angular size difference between the two dates. What do you think would be causing these two observable characteristics? Would other planets, say Mercury or Saturn, exhibit these characteristics based on your hypothesis of their causes? If so, would they exhibit size and phase changes like Venus?

Part III: Estimating the Distance to Venus

1. Now we will determine yet another angle in the geometry of the Sun-Venus-Earth system. This time we are going to use the phase of Venus to determine the Sun-Venus-Earth (theta) angle. Another way of thinking of this angle is to imagine yourself standing on Venus. This angle would be the angular separation of the Sun and the Earth in the Venusian sky. To find this angle, look at the previous two images and try to determine the fraction of Venus's disk that appears to be illuminated. If you look closely, you will see that the entire disk has been lightened so that you can make out the full circle. The easiest way to measure the fraction of illumination is to measure the diameter of Venus, measure how much of the diameter is illuminated, and divide the illuminated portion by the diameter. Thus you will get a result between 0 and 1. Find this fraction for both dates.

   
   

   Figure 1

   From Fig. 1, if W is the percentage of Venus' diameter that is visible to us from Earth (in the figure, W is about .66 or 2/3 of the diameter) and R is the radius of the planet, then

   
   

2. To ultimately solve for the distance (using the law of sines), we have to solve for theta

   
   

3. Finally, we can determine the remaining angle d in Fig. 2 through the formula

   

   Fig. 2 shows an example of a possible geometry of the system. As you can see, we have now determined all of the angles in the triangle. Note: In Fig. 2, the "180-" is needed so that we can use the same "theta" to refer to the phase of Venus and the angular separation of Earth and the Sun. You can verify this all works by plugging in 0 and 180 degrees to simulate superior and inferior conjunction.

   
   

   Figure 2

4. We already know that the value for E, the Sun-Earth distance, is 1 A.U. (A.U. = astronomical unit). Since we have all of the angles, we can find all of the other distances (in AUs) in Fig. 2 by using the law of sines:

   

   Determine the distance to Venus from Earth (D) and Venus' average orbital radius (V) Now that you have those two values, create diagrams showing the geometry of the Sun-Venus-Earth system for both dates using the angles and distances you calculated. It is highly recommended to use your compass to create an accurate drawing. Be sure to include labels and scales in your diagrams!

   Question: Looking at the completed diagrams, do the positions of Venus you found agree with the phases that you observed? Explain.

   Question: Do your values for the distances make sense? Explain.