CL - Distances (remote)[91] (1)

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California State University, Bakersfield *

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1609

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Astronomy

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Apr 30, 2024

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docx

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Computer Lab – Distances (Virtual Lab Remote Edition) Introduction In this lab you will re-create experiments done by past astronomers to determine distances within our solar system. The same ideas are still used today to determine distances on Earth as well as in outer space. Aristarchus Experiment The Greek scientist Aristarchus (310–230 BC) realized that if we see the Moon half lit, the Sun, Moon, and Earth must make a right-triangle. See the figure below. The angle at the Moon’s vertex must be exactly 90° if we see the Moon exactly half lit. But the angle a next to Earth will be less than 90°, how much less depends on how far away the Sun and Moon are from Earth. The Sun is much further away from us than the Moon and the angle a is very close to 90°. Aristarchus did an experiment to measure a , he physically measured the angle between the Sun and Moon in our sky when the Moon appears half lit. Aristarchus measured a value of 87° from which he (properly) calculated that the Sun must be 20 times further away from us than the Moon. Because the Moon and Sun appear equally large in our skies, Aristarchus correctly reasoned that the Sun’s actual size must be 20 times larger than the Moon. Because he discovered the Sun was so large, Aristarchus proceeded to create a heliocentric model of the universe. The model was eventually revived by Copernicus and validated by Kepler and Galileo almost 2000 years after Aristarchus lived. Distances – 1 Moon

The Sun is actually 375 times further away (and wider) than the Moon. The experiment done by Aristarchus is challenging and difficult even with modern equipment. For the time, Aristarchus’s results were excellent and were a milestone in the development of the scientific method. Even using the Voyager 4 program, it is not easy to get really good results for this experiment, we will do a different experiment that uses the same idea. Copernicus Experiment We will do an experiment, first done by Copernicus, to measure the distance between Venus and the Sun. Launch the Voyager 4 program (csub.apporto.com). Select the Tools/Planet Report… menu and select Maximum Elongations from the pop-up menu (which is usually hidden behind the Time Panel). Find the date of the next maximum elongation for Venus and write the date and angle on the following line. Hmmm. The Venus values follow the Mercury values in that table but viewing them is tricky, changing to Full Screen mode ( ) worked for me. Here are the Venus values so you don’t have to try to find them yourself. You can just note which line gives the values for your “next” Venus maximum elongation. Planet Direction Angle Date Venus eastern 47.0° Oct 29, 2021 Venus western 46.6° Mar 20, 2022 Venus eastern 45.4° Jun 4, 2023 Venus western 46.4° Oct 23, 2023 Venus eastern 47.2° Jan 10, 2025 Venus western 45.9° Jun 1, 2025 The elongation angle is the angular separation between the Sun and Venus in the sky. Copernicus could measure this with simple. To get the maximum elongation, Copernicus just had to measure elongations day after day and watch the values for a maximum. While still in the Planet Report window, select Angular Separations from the pop-up menu, the white line shows how the elongation of Venus varies with time. you picked one of the dates when the elongation angle reached a maximum. Close the Planet Report window. Select the Chart/Set Time… menu, click on the Universal Time tab, and enter your date from above. Select the Distances – 2

Center/Planets/Venus menu. Click the Physical tab in Venus' Info Panel, the "Illumination" should be around 50%. Why does Venus appear about half-lit (50% illumination) when at maximum elongation? That is explained by the geometry shown in the figure below, if Venus was anywhere else in its orbit the angle e would be smaller. Copernicus lived before the invention of the telescope, he could have predicted that Venus would appear half-lit but had no way to check it. The distance x between the Sun and Venus is what we are trying to determine. The distance between the Earth and Sun is always close to 1 AU. The Venus-Sun line and Venus-Earth line are perpendicular because only that way would we on Earth see Venus at maximum elongation, hence the 90° angle. The angle e is the elongation, the angle between Venus and the Sun as seen from Earth, the angle from earlier. The mathematical relationship between the distances and angles for this triangle involve trigonometry, we need to use the trigonometric “sine” function. In particular, sine of angle e = Distances – 3

or x = (sine of e ) (1 AU) = sin( e ) AU So, x , the distance between the Sun and Venus in AU, is just the sine of the elongation angle when Venus appears half lit. Use your calculator (make sure it is set for degrees, not radians) to compute this value or type “sine of 45.8 degrees” into Google – but use your angle from the earlier table. Venus to Sun Distance = 0.7169 AU Look up the actual Venus-to-Sun distance to check your result. Do that by selecting the Tools/Planet Report menu, then select Heliocentric Positions from the pop-up menu, and then you'll find the desired value in the Distance column. Actual Venus-to-Sun Distance = 0.72699 AU How well did your result work out? Our biggest error was assuming the Earth- Sun distance was exactly 1.000 AU. The same procedure works for determining the distance between Mercury and the Sun. And a similar procedure can be used to find distances to the outer planets. Distance to Mars In the Planet Report window, select Angular Separations from the pop-up menu and look for the orange/red line of Mars. You are going to determine the date, as accurately as you can, when Mars will have an angular separation from the Sun of 90°. Note that the line for Mars is actually a string of squares, each represents one day. Also note that the grid lines on the graph do not match the beginning and ending of most months as seen at the bottom of the window. Try to determine a date when Mars had a 90° value (note: on the graph 90° is only half-way up). Date of 90° Separation: February 1st 2021 Distances – 4

Close the Planet Report window. Set the date to what you just determined. Select the Center/Planets/Mars menu. Click on the Physical tab in Mars' Info Panel, record the percent Illumination value of Mars on this day. Illumination = 88.6% If we were watching Mars from the Sun, we would see only the lit side, 100% illumination. Because we are away from the Sun, we see part of the dark side of Mars, less than 100% illumination. Based on the illumination seen, we can figure out the angle we are away from the Sun relative to Mars. That is, we can figure out the angle e in the figure below from the illumination value. You can use the percent Illumination value to determine the angle e in the figure by using the following table: % Illum 75 76 77 78 79 80 81 82 83 84 Angle e 60.0° 58.7° 57.3° 55.9° 54.5° 53.1° 51.7° 50.2° 48.7° 47.2° 85 86 87 88 89 90 91 92 93 94 95 45.6° 43.9° 42.3° 40.5° 38.7° 36.9° 34.9° 32.9° 30.7° 28.4° 25.8° Distances – 5

Or you can use the following exact formula to calculate e , the formula involves an arccosine, so don’t attempt the formula unless you are familiar with those functions. If the fractional illumination of Mars is f (75% would be f = .75), the formula for calculating the angle e is: e = cos -1 (2 f – 1) Record the angle e you determined by formula or table. If you're using the above table and your percentage was between two of the percentages, interpolate to get a better value for the angle e . For example, if your percentage was 94.5, you might use an angle around halfway between the 94 and 95 values, like e = 27.0, don't worry about being exactly correct. e = 49.45° From trigonometry we can calculate the Mars-Sun distance ( d ), the formula is as follows. You can have Google do the calculation by searching for “one divided by sine of 39.3 degrees” (use your e value instead of 39.3). d = = 1.326 AU Compare your calculated value to the value given for Mars on the Heliocentric table of the Planet Report. Mars to Sun Distance = 1.54964 AU Your result will likely be off, because we had to estimate the date of the 90° separation and because we didn’t include the precise Earth-Sun distance for the time of the experiment. Still, it is a very good method. This method for Mars is really the same as the method we used for Venus just turned around. In both cases we knew we had a right angle (90° angle) between the Sun and planets, either because we saw a planet at maximum elongation or because we measured a 90° elongation angle in the sky. Copernicus and Outer Planets Copernicus actually used a different method to determine the distance to the outer planets (he couldn't use the above method because he didn't have a telescope and so he couldn't know the percent illumination). It's a bit more complex than the Distances – 6

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