05Lab_Magnitudes-and-Distances

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Astronomy

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Oct 30, 2023

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M ONTGOMERY C OLLEGE – R OCKVILLE A STRONOMY 101 ASTR101 Laboratory 5 - Magnitudes & Distances 1 Name: Aditya Rana Millions of stars are scattered across the sky. Astronomers want to study these stars as carefully as possible. This means measuring everything we can possibly measure about them. Unfortunately, it's not easy to measure anything that's a septillion miles away! Astronomers use measurements of position, brightness, and color to deduce many other properties of the stars. We've already had practice using two methods to measure a star's position – the Altitude/Azimuth method and the Right Ascension/Declination method. Now it's time to look at a star’s brightness . Everyone agrees that stars come in different brightnesses – some are bright and some are dim. The brightness of a star as it appears in the sky is given by a number called its apparent magnitude , where smaller numbers mean brighter stars. Magnitudes can even be negative for very bright stars. Let’s explore this idea of magnitudes. Start Stellarium . Turn off the Atmosphere and Fog . Click on four stars randomly. For each star, when you've clicked on it and selected it, its Information will appear, as usual, in the upper left-hand corner of the screen. The star's “ Magnitude ” will be listed in the information below the star's name. This is the star's apparent magnitude . Enter the names and apparent magnitudes of your stars below. Click on both bright and dim stars to get a range of magnitudes, and then fill out the table below. Star Name Apparent Magnitude Hamal 2 OCC 9072 6.6 Al Kaff al Jidhmah IV 4.25 HIP 17000 6.7 1 Last edit Spring 2022. 1

ASTR101 L ABORATORY 5 Remember, there are two kinds of magnitudes – apparent magnitude , which is how bright the star appears to our eyes here on Earth, and absolute magnitude , which is how bright the star would be if it were 10 parsecs (32.6 light years) away. Let's examine more closely luminosity and distance . Luminosity is a measure of how much total light a star gives off every second. A star that looks dim to our eyes could be dim because it has a low luminosity, or because it is far away. Which one is it? If we could measure the star's distance, then we could answer that question. PART A Our first task is to measure a star's distance. The most accurate way to do this is to use parallax . Parallax is the word we use to describe how something appears to shift its position relative to the background when we look at it from two different angles. In the case of stars, those two angles are provided by looking at the star on two nights that are six months apart – that way we are seeing the star from two points on opposite sides of the Earth's orbit around the Sun. The figure below helps explain it: Unfortunately, the stars are all so far away that the amount they shift in the sky over the course of a year is very small – we're talking about arcseconds of angle (remember, one arcsecond is 1/3,600 of a single degree)! Astronomers have defined a distance, called a parsec , which is the distance a star has to be from Earth to shift its position by one arcsecond in the sky over the course of six months . 2

ASTR101 L ABORATORY 5 A parsec turns out to be 3.26 light years . To measure a star's distance, then, just observe that star for a year, and measure its parallax (in arc-seconds) during that time, and then you can use the following formula to calculate its distance: d = 1 p ( Eq. 1 ) Where d is the star's distance (in parsecs), and p is the star's parallax, in arcseconds. What the equation shows us is that the more a star appears to shift its position (in other words, the greater its parallax is), the closer it is! Let's calculate the distances to some stars. In Stellarium , search for the stars in Table 1 on the next page using the Search Window (or by simply pressing F3 ), and when you've selected them, look in the information given in the upper left-hand corner of the screen. The star’s parallax is listed near the bottom, and the distance (in light-years) is listed above that .  For each star record its apparent magnitude in the column labeled “Apparent Magnitude”, its parallax in the column labeled “Parallax (Arcseconds)”, and its distance in the column labeled “Distance from Stellarium (light-years)”. You will fill in the first three columns of the table. Please read the following carefully:  Stellarium will give you the parallax in units of milliarcseconds (mas). YOU MUST DIVIDE THIS NUMBER by 1000 to get it in arcseconds.  Stellarium will give you an error bar for the parallax and the distance. You do not need to keep the number after the “±”.  For example, for Sirius, Stellarium provides a parallax of 379.210±1.580 mas. You should enter 0.37921 into the table. (The row for Sirius has been completed for you as an example.) Table 1 Star Name Apparent Magnitude Parallax (arcsecon ds) Distance from Stellarium (light years) Distance (parsecs) Distance (light- years) Sirius -1.45 0.37921 8.60 2.64 8.60 Aldebaran .85 .048 66.64 20.43 66.63 Betelgeuse .45 .00655 497.95 152.67 497.94 Proxima 11 .7685 4.24 1.299 4.23 Spica .95 0.013 249.74 76.570 249.73 3

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ASTR101 L ABORATORY 5 Barnard’s Star 9.5 .547 5.96 1.827344 5.95 Deneb 1.25 0.0023 1411.93 432.89 1411.89 Vega 0.0 .130 25.04 7.67 25.01 Arcturus 0.15 .088 36.71 11.255 36.7 Altair 0.75 .194 16.73 5.12 16.69  Now calculate the distances to each of these stars in parsecs , using each star's parallax angle and Eqn. (1) d = 1 / p . Enter the results in column titled “Distance (Parsecs)” of Table 1 . Finally, let’s check how well the calculated distances match with the distances from Stellarium .  Convert each distance from parsecs to light years, using the fact that one parsec = 3.26 light-years. Multiply each distance in the “Distance (Parsecs)” column by 3.26 to get the distance in light-years, and then enter the result in the last column of Table 1 . Do your calculated distances for the stars (Column 6) agree with what Stellarium lists for each star's distance (Column 4)? yes Parallax is a powerful and relatively simple way to measure the distance to stars. Unfortunately, most stars are too far away for us to measure their parallax – they just move too little in the sky to be observed. For these stars we will need to find another way to measure their distance. PART B Now let's think about the difference between absolute and apparent magnitude. Stellarium lists the star’s absolute magnitude below the apparent magnitude. The apparent magnitude tells us how bright the star appears to our eyes here on Earth, but some stars appear bright simply because they're nearby. If the stars in Table 1 were all put at the same distance away, say 10 parsecs ( 32.6 light years ), we would then be able to tell which star was intrinsically brighter. No star would have an unfair advantage in being bright by virtue of the fact that it was close by! Put a star 10 parsecs away and then measure how bright it appears to be, and you have that star's absolute magnitude . 4

ASTR101 L ABORATORY 5 If the star is farther than 10 parsecs away, which will be a larger number? Remember, the larger the number the fainter the magnitude. (check one) x apparent magnitude absolute magnitude If the star is closer than 10 parsecs away, which will be a larger number? Remember, the larger the number the fainter the magnitude. (check one) apparent magnitude x absolute magnitude Look at the distances and apparent magnitudes of the stars in Table 1 . Which of these stars do you think would have the largest absolute magnitude? In other words, which star would be dimmest if it were 10 parsecs away? Why? Proxima, as it is already dim, it would become even dimmer. Look at the stars in Table 1 . Which of these stars do you think would have the smallest absolute magnitude? In other words, which star would be brightest if it were 10 parsecs away? Why? Deneb because its apparent magnitude is the highest. There is a connection between a star's apparent magnitude, its absolute magnitude, and its distance. If you know any two of these three quantities, you can figure out the third! There's a neat mathematical way to connect these three quantities. Let's say M is a star's absolute magnitude, m is the star's apparent magnitude, and D is its distance, in parsecs. Then the following is true: m − M = 5 log ( d ) − 5 ( Eq. 2 ) 5

ASTR101 L ABORATORY 5 If you've never seen the math term log or logarithm of a number is the power that you have to raise 10 to get that number. In other words, log (10) = 1 , l og (100) = 2 , log (1000) = 3 , etc. Most scientific calculators have a log button. If you don't have one, you can use google.com . Just type “log” and then the number into google.com and you’ll get the correct value. With a little algebra, we can rewrite the above equation as d = 10 m − M + 5 5 ( Eq. 3 ) Note that this is 10 RAISED TO THE POWER OF m − M + 5 5 , NOT 10 MULTIPLIED BY m − M + 5 5 !  For each of the stars in Table 1 , use Equation 2 to calculate the star's Absolute Magnitude, M . Enter the results in Table 2 below. Remember, you have each star's Distance (in parsecs) and Apparent magnitude in Table 1 . (Sirius has been completed for you.) For Sirius : m − M = 5 log ( d ) − 5 M = m − 5 log ( d ) + 5 =− 1.45 − 5 log ( 2.64 ) + 5 ¿ − 1.45 − ( 5 ∙ 0.421 ) + 5 = 1.442 Table 2 Star Name Absolute Magnitude (M) Sirius 1.442 Aldebaran -0.7 Betelgeuse -5.47 Proxima 15.43 Spica -3.47 Barnard’s Star 13.19 Deneb -6.93 6

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ASTR101 L ABORATORY 5 Vega 0.57 Arcturus -0.11 Altair 2.2 Which of these stars has the greatest absolute magnitude (i.e., was the dimmest)? Proxima Which of these stars has the least absolute magnitude (i.e., was the brightest)? Deneb Were the predictions you made (on previous page) for the dimmest star and the brightest star correct? Yes Suppose I found a star whose absolute magnitude was -4, and its apparent magnitude was 6. How far away is that star? (Use Equation 3 ) For full credit, use proper units . 1000 parsecs away Suppose I found a star whose absolute magnitude was 4.75, and its apparent magnitude was 3.50. How far away is that star? (Use Equation 3 ) For full credit, use proper units . 7

ASTR101 L ABORATORY 5 5.62 parsecs away. In your own words, describe the difference between apparent magnitude and absolute magnitude. Explain the relationship between apparent magnitude, absolute magnitude and distance. Apparent magnitude is the brightness of the star from the observers view, where as absolute magnitude is the brightness of a start from 10 parsecs away. 8