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

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Astronomy

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Apr 30, 2024

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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
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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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method we just used but it has the same basic idea – determine when Earth-Sun- Mars form a right-angle triangle and use measured or calculated angles and trigonometry to determine the length of the planet-to-Sun side of the triangle. Parallax Basics The distance to objects can be determined using the technique of parallax or triangulation. Try this, hold a pencil in your outstretched arm. Close one eye then the other. As you switch eyes, the pencil will appear to jump between two locations. This parallax shift occurs because you keep changing your viewing location (which eye you are viewing from). If the pencil was held closer to you (or if your eyes were further apart – don’t try that at home), the shift would be larger. Knowing the distance between the two viewing locations (the “baseline” = b ) and the angular shift in the apparent direction to the object ( A = angular shift), the distance to the object ( d ) can be calculated. The mathematical formula relating these quantities is d = [ d = (b * 360) / (2 π A) ] To get proper results from this formula, • A must be measured in degrees. • The angular shift should be less than about 30° • The baseline must be perpendicular to the direction to the object. It is this last condition that will cause us the most difficulty. If we shift our viewing location directly towards or away from the object we are trying to view, the angular shift will be small or zero. Only if we are changing our viewing location “sideways” will the above formula give correct distances. Distance to the Moon Select the File/Open Recent Settings/Startup menu. Select the Chart/Set Location… menu, set your location to the North Pole by setting the Latitude to 90° 00' N. Change the Elevation Distances – 7
to 0, don't change the Longitude or Time Zone (change the Name only if you want to). Click OK. Select the Center/Planets/Moon menu, record the R.A. and Dec. of the Moon (round off to the nearest minutes of arc). R.A. = 6h 30m Dec. = +27° 39’ Now set your location to the South Pole (just change the "N" in Latitude to "S") and return to get the new R.A. and Dec. for the Moon. We expect the R.A. value to be the same, but the Dec. value should change. Likely both appear the same, try clicking on the Moon to force an update of the Info Panel display. R.A. = 6h 30m Dec. = +29° 18’ Again, the Right Ascension should be the same (if not repeat the process). The two Declination values should be different, likely both positive or both negative. If one was positive and one negative, the math gets a little tricky; if you have that case, start the experiment over on a different date. If both are negative, ignore the minus signs and treat them both as positive. We want to find the difference in these declinations measured in degrees. Convert both declinations (Dec.) into decimals using the following method. d° m' = d° + ° For example, 3° 41' would be d=3 degrees and m=41 minutes (remember, you were told not to include any seconds of arc). This converts according to the formula to 3.68 degrees, specifically 3° 41' = 3° + 41/60 ° = 3° + .68° = 3.68° Now you convert your two declinations above into decimal values. Decimal Declinations: 27.65° and 29.3° Subtract the two declinations (larger minus smaller) to get the shift in the Moon's position. Decimal Angular Shift = 1.65° Distances – 8
The baseline we're using is the diameter of Earth (the distance from North Pole to South Pole), which is 12,742 km. This angular shift should be around 2°, if you something more than 2.5 or less than 1.5, you likely have made an error. Now for the parallax formula calculation. Plug in your angular shift into the denominator and do the calculation to get the distance to the Moon. [This is two fractions multiplied by each other, one with a product of terms in the denominator. To avoid a math error, you might want to multiply all the top together, multiply all the bottom together, then do the top product divided by the bottom product.] Distance = = 7720.606 km Ideally, you should get a value between 350,000 km and 410,000 km, the Earth- Moon distance varies but is always within this range. You can check the distance given in the Moon's Info Panel to see how close you came. This result might not be very accurate because our baseline (from North Pole to South Pole) will usually not be perpendicular to the line from Earth to Moon. Fun with Parallax Again, the key idea of this lab is that when we shift our viewing location, the positions of nearby objects in the sky appear to shift relative to more distant objects (stars). From the amount of the object's shift and the distance we moved (our "baseline"), we can calculate the distance to that nearby object. Select the File/Open Settings… menu, open the "110 Settings" folder in the “Quick access” sidebar (or else click the “ ≪ ” next to “Windows” and select Local Disk (C:) followed by Program Files > Carina Software > Voyager 4 > 110 Settings). Open the Settings File called "Moon in the Pleiades". The Pleiades is a nearby cluster of stars, also called the Seven Sisters. Here you can see a photographic quality picture of the Pleiades behind the Moon. Click and drag the location indicator within the Location Panel, watch how changing your position on Earth causes the Moon to shift relative to the Pleiades. Distances – 9
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This is parallax. We just used the same idea to measure the distance to the Moon. If you look closely, you'll notice that changing the position on Earth also slightly alters the side of the Moon displayed. Tycho Brahe and Parallax Tycho Brahe, from his observatory Uraniborg built on the island Hven in Denmark, observed celestial objects and measured many parallaxes. Based on the amount of parallax exhibited by celestial objects, he could determine which were closer to us and which were further away. Tycho searched for stellar parallax, parallax of stars. Assuming some stars are closer than others (likely the brighter stars are the closer ones), the Earth’s motion would cause parallax of stars. Even using the largest available baseline, the diameter of the Earth’s orbit, he could detect no stellar parallax. Tycho knew the lack of parallax of stars meant either ● the stars were so far away that their parallax was too small to measure; or ● the Earth was motionless at the center of the universe. Copernicus believed in a heliocentric cosmology and took the first position; later Kepler also agreed with that point of view. Tycho felt his measurements had proven that the Earth did not move. With his advanced but non-telescopic equipment, Tycho could measure positions accurate to about 2’ (two arcminutes). Nearby stars have a parallax angle of about 0.5” (half an arcsecond). (Q1) How many times smaller is 0.5” compared to 2’? [Hint: 1’ = 60”] The value 0.5" is 240 times smaller than 2 The true distances to stars are far, far greater than that imagined by any ancient astronomers. Even with telescopes (invented a few years after Tycho’s death), it was Distances – 10
more than two centuries (Friedrich Bessel in 1838) before stellar parallax was observed. In 1572, a bright new star appeared in the night sky, Tycho studied this extensively. In the book he wrote about the new star, he called it a “nova”, nova being the Latin word for “new”. Novae are erupting or exploding stars. The 1572 nova was an exploding star, what we today call a supernova. Tycho found that the nova exhibited no parallax, just like stars. The conclusion was that this was some real new star, not some nearby object that was being mistaken for a star. At that time, stars had been thought to be permanent and unchanging; this discovery was a revelation. In 1577, a bright comet appeared in Earth’s sky. Tycho studied it and measured a parallax that placed it beyond the Moon but closer than the stars. This was another revelation; comets had been assumed to be objects high in the Earth’s atmosphere, not cavorting out among the planets. I’m assuming you still have the “Moon in the Pleiades” settings open; if not, re-open that settings file. Let’s observe the parallax of a comet. Click the button along the right edge of the screen to turn on the display of comets (second button from the top in the picture). Select the Chart/Set Time… menu and change the date to 2061 Nov 10. Then select the Center/Comets/1P/1982 U1 (Halley) menu. Un-Lock Halley’s comet using the button in its Info Panel. If you drag the location marker around in the Location Panel, you’ll see Halley’s comet shift positions slightly due to parallax. It is much less parallax than we saw for the Moon because Halley’s comet is much further away. How much further away? I don’t know, but we can figure it out. (Q2) (a) What is the Distance to Halley’s comet as shown in the Info Panel? Halley’s Distance = 0.82042 AU (b) Convert this distance into km using that 1 AU = 149,600,000 km Halley’s Distance = 122,128,392 km (c) The Moon’s distance in the previous animation was 360,555 km, how many times larger is Halley’s distance compared to that Moon distance? Halley / Moon distance ratio = 338.78 The parallax we saw for the Moon was larger than Halley’s parallax by that same ratio. Distances – 11
Viewing of Neptune Select the File/Open Recent Settings/Startup menu. Select Equatorial from the pop-up menu at the bottom of the Sky Chart window. Select the Center/Planets/Neptune menu. Select the Window/Planet Panel menu, click the box next to Neptune in the "Path" column, this turns on the path for Neptune, a line will now show the path followed by Neptune, then close the Planet Panel. Close the Info Panel, set the Time Step to 2 days, zoom to 45°, and animate. Watch until Neptune moves off the screen, then stop the animation. What were you seeing? Neptune appeared to oscillate back and forth while drifting eastward relative to the stars. The eastward motion of Neptune is because of Neptune’s motion around the Sun. The back-and-forth motion is because we are viewing from the moving Earth. The time between retrograde loops is about one year, the time it takes Earth to make an orbit around the Sun. The loops are the result of parallax. Like switching eyes with the pencil but now it’s Earth moving us from one side of the Sun to the other. Alternate Formula When you know the baseline length and angular shift, you can calculate the distance using our earlier formula. Ever since Bessel’s successful measurement of stellar parallax, astronomers have been using parallax to measure the distances to stars. For astronomers, they were always using a baseline of 2 AU and measuring parallax angles of a fraction of an arcsecond. If using our earlier parallax formula, they would constantly be repeating calculations and converting units. By defining a new unit of distance, a far simpler formula was created. The new distance unit was called the parsec (pc), it was the distance at which an astronomical object would have a parallax angle of one arcsecond when viewed using a baseline of 2 AU (‘parsec’, short for PARallax of one arcSECond). In terms of other units, 1 pc = 3.262 light-years = 3.086 x 10 13 km. The simplified parallax formula is d = 1/p ; p is the parallax angle in arcseconds when viewed with a 2 AU baseline and d is the distance to the object in parsecs (pc). As long as you are using the proper units (pc and “), you just put the numbers into this formula. Let's get you some practice using this simpler formula. Distances – 12
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(Q3) If a star has a parallax angle of 0.1 arc-seconds, how far away is it in parsecs? 10 parsecs (Q4) If 1pc = 3.26LY, how far away was the star in question 3 in light-years? 32.6 light-years away (Q5) If a star is 20 parsecs distant, what parallax angle would it have in arc-seconds? 0.05 arc-seconds Venus Transits and the A.U. The experiments you’ve reproduced above have found the distance to various solar system bodies measured in Astronomical Units (AUs), as a multiple of the Earth-Sun distance. Astronomers could determine, for instance, that Venus orbits 0.719 AU from the Sun but were embarrassingly uncertain of what an AU equaled. Kepler himself had estimated the AU to be 24 million kilometers (it is actually 150 million kilometers). In 1716 Edmund Halley (of comet fame), a colleague of Isaac Newton, explained how a transit of Venus (Venus seen moving across the face of the Sun) could be used to measure the AU. Transits of Venus, as seen from Earth, are exceedingly rare. Transits come in pairs that are 8 years apart with the pairs separated by more than a century. Venus transits occurred or will occur in these years: 1631, 1639, 1761, 1769, 1874, 1882, 2004, 2012, 2117, 2125. Why 8 years apart? In that time Earth orbits the Sun eight times (eight years!) while Venus orbits 13 times. Why the 105-year gaps? Venus’s orbit is tilted 3.4° relative to Earth’s. Usually when Venus passes between the Earth and Sun it appears above or below the Sun and no transit is seen. To illustrate Halley’s method of using a Venus transit to measure the AU, we’ll have you do the experiment for the 2012 Venus transit. Select the File/Open Recent Settings/Startup menu. Select the Chart/Set Time… menu; set the Local Time to June 5, 2012 at 3:00:00 PM. Center on Venus, zoom to 5’, lock Venus, and close Venus’s Info Panel. You should see Venus at the edge of the Sun. Set the Time Step to 1 second and animate. When the edge of Venus first touches the edge of the Sun is called 1st Contact. Let the animation continue until 2nd Contact – the first moment when Venus is completely in front of the Sun (the Distances – 13
picture shows Venus at second contact), the trailing edge of Venus should be just touching the Sun’s edge. Record this time, both the local time and the Julian Day (JD) number (found at the bottom of the Time Panel). 2nd Contact Time 3:23 PM JD 2456084.43316 Continue the animation until 3rd Contact – when the leading edge of Venus touches the Sun as Venus just starts to leave the Sun’s area. This will occur many hours later (around 9:30 PM) so temporarily increase the Time Step. 3rd Contact Time 9:29 PM JD 2456084.68730 We want to calculate the time between 2 nd and 3 rd Contact in seconds, this is most easily done using the Julian Day numbers you recorded. Take the difference between the two (they should only differ in the decimal part) and multiply by 86,400 (the number of seconds in a day). Time from 2nd Contact to 3rd Contact = 0.25414* (86,400) = 21,975.696 sec Halley’s idea for measuring the AU works as follows. Given any guess for the AU (like 1 AU = 100,000,000 km) the expected time for the transit can be calculated using Kepler’s Laws. For this transit and assuming this AU value, the expected time computes to 14,666 s. The effect of the Earth-Sun distance is such that twice as far away would mean only half the transit time (the further-away Sun appears only half as large). So, the Earth-Sun distance (AU) can be calculated from your measured time ( T ) by completing this calculation: AU = = 0.44519 km Many astronomers travelled to many far-flung locations to collect data in 1761 and 1769 but it did not work out. The atmospheres of Earth and Venus made images Distances – 14
fuzzy and, worse yet, an optical phenomenon called the black-drop effect made it impossible to determine exact times. As the 1874 and 1882 transits approached, it was hoped a new technique – photography – would solve previous problems and allow a far more precise measurement. But the same problems persisted and transit measurements managed only a minimal improvement to the AU value. Our modern value of the AU is mainly based on the timing of radar reflections from planets and the transit method is no longer important. Cepheid Variables Select the File/Open Recent Settings/Startup menu. Select the Center/Find and Center… menu, type in “delta cephei”, and click Search then Center. Okay, I was going to have you watch the Magnitude (brightness) value in the Info Panel as you advance the days one-by-one. But the Voyager program does not display the star’s changing magnitude. Delta Cephei is a pulsating variable star; it swells bigger and smaller, changing its magnitude from 3.48 to 4.37 and back to 3.48 with a period of 5.366 days. This turns out to be common behavior for many stars later in their lives. Stars behaving in this manner are collectively called Cepheid variable stars. Some Cepheid variables have longer periods and others shorter periods. The astronomer Henrietta Swan Leavitt studied thousands of Cepheid variable stars located in the Magellanic Clouds (small galaxies that orbit our Milky Way galaxy). From this work, she determined that the periods of the Cepheid variables are tightly correlated with the peak luminosity of these stars (see the figure). (Q6) If a Cepheid variable has a period of 10 days (look at the bottom scale), how many times more luminous would that star be compared to our Sun (those are the values along the vertical axis)? 10 3 times more luminous than our Sun Distances – 15
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This discovery was a breakthrough for humans in determining distances throughout the universe. That’s because the luminosity of a star, when combined with its apparent brightness in our skies, allows determination of the distance to that star (using the “inverse-square law”). But for most stars, we can’t tell if they are luminous and far away or dim and closer (parallax? That’s only useful for our nearest neighbor stars.). The “Period-Luminosity” relation for Cepheid variable stars means we can know their true luminosity just by timing how long it takes for them to go bright to dim to bright again. The number one purpose for which the Hubble Space Telescope was built was to spot Cepheid variable stars in other galaxies so that we could determine the distances to those other galaxies. Answers to Questions: 1. 240 How many times one is bigger or smaller than the other is determined by calculating their ratio, here 2’ / 0.5” = 120” / 0.5” = 240 2. Answer hidden. 3. 10 parsecs 4. Answer hidden. 5. Answer hidden. 6. Answer hidden. Distances – 16