Basic Astronomy Labs

Astronomy Laboratory Exercise 31 The Magnitude Scale On a dark, clear night far from city lights, the unaided human eye can see on the order of five thousand stars. Some stars are bright, others are barely visible, and still others fall somewhere in between. A telescope reveals hundreds of thousands of stars that are too dim for the unaided eye to see. Most stars appear white to the unaided eye, whose cells for detecting color require more light. But the telescope reveals that stars come in a wide palette of colors. This lab explores the modern magnitude scale as a means of describing the brightness, the distance, and the color of a star. The earliest recorded brightness scale was developed by Hipparchus, a natural philosopher of the second century BCE. He ranked stars into six magnitudes according to brightness. The brightest stars were first magnitude, the second brightest stars were second magnitude, and so on until the dimmest stars he could see, which were sixth magnitude. Modern measurements show that the difference between first and sixth magnitude represents a brightness ratio of 100. That is, a first magnitude star is about 100 times brighter than a sixth magnitude star. Thus, each magnitude is 100115 (or about 2. 512) times brighter than the next larger, integral magnitude. Hipparchus' scale only allows integral magnitudes and does not allow for stars outside this range. With the invention of the telescope, it became obvious that a scale was needed to describe dimmer stars. Also, the scale should be able to describe brighter objects, such as some planets, the Moon, and the Sun. The modern magnitude scale allows larger magnitudes (for objects dimmer than sixth magnitude), zero and negative magnitudes (for objects brighter than first magnitude), and fractional magnitudes (for minor distinctions in brightness). As examples, the Sun appears at a magnitude of -26.7 in Earth's sky and the full Moon is 480 thousand times dimmer at magnitude -12.5. The ratio of the brightness of one object (B1) to the brightness of a second object (B2) is given by :~ =(woo' )(m,-m.), (1) 2 where m1 and m2 are the magnitudes of the first and second objects, respectively. Example 1 : How many times brighter is Sirius at magnitude -1.4 7 than the dimmest star visible to the unaided human eye (at about magnitude +6.0)? B ( )(+6 o-c -I 47)) Sirius = 1000.2 · · = 970. Bdim Thus, Sirius is almost a thousand times brighter than the dimmest star visible to the unaided human eye. 31- 1 The magnitude of an object (m2) can be determined by comparing its brightness with another object whose magnitude (m1) is known. Let BdB2 represent the brightness ratio of the known object to the unknown object. Then, = m 2 m 1 + 2.5·1og 10 (B~J . (2) Example 2: An observer with a small telescope sees a comet that is 120 times dimmer than a star with magnitude + 1. 5. What is the magnitude of the newly seen comet? The ratio of brightness of the brighter object to the dimmer one is 120 times. Thus, the magnitudJ of the comet is m 2 = +1.5 + 2.5·log10 (120) = +1.5+ 2.5· 2.079 = +6.7. Light-gathering power (LGP) is a measure of the amount of light collected by an optical instrument and is proportional to the area of the light-collecting surface. For most telescopes, this is proportional to the square of the diameter of the objective aperture (D). Mathematically, this is expressed as 2 LGP ex: D (3) Example 3: How many times more light is collected by a pair of 1OX70. binoculars than by fully dark-adapted human eyes? The diameter of the binoculars' objective aperture is 70. mm. Assume that the diameter of the fully-dilated pupil is 6.0 mm. Then, LGPbin = (DbinJ 2 = ( 70.-mm) 2 = 140 LGPeye Deye 6.0·mm With 10X70. binoculars, one can see objects that are 140 times dimmer than the dimmest objects visible to the human eye. Another way to look at this is that the dimmest objects seen with the human eye are 140 times brighter than the dimmest objects seen through 10X70. binoculars. Example 4: What is the dimmest magnitude visible to the dark-adapted, human eye (pupil diameter, 6.0 mm) through a 34 em diameter telescope? The telescope collects more light than the unaided eye. The LGP ratio of the telescope to the eye is LGP.,1 = e40· mm r= 3200 LGPeye 6.0· mm Thus, the dimmest object visible to the unaided eye is 3200 times brighter than the dimmest object visible through the telescope. The magnitude of the dimmest object visible through the telescope is m 2 = +6.0 + 2.5 ·log10 (3200) = + 15. A 34-cm diameter telescope can be used to view objects with magnitudes brighter (less) than + 15. 31- 2 The magnitude scale was used originally to distinguish stars according to how bright they appear from Earth. This is now called apparent magnitude (m). Table 1 lists the apparent magnitudes of some familiar objects. T abl e 1 S orne apparen t magm t u des. apparent object or description magnitude Sun -26.7 Moon (full) -12.5 Venus (brightest) -4.6 Jupiter (brightest) -2.5 Sirius (a Canis Majoris) -1.5 Canopus (a Carinae) -0.7 Vega (a Lyrae) 0.0 Unaided-eye limit +6.0 Proxima Centauri +11.1 Limit for 1O x70 binoculars +11.3 Limit for 15-cm telescopes +13.0 Limit for 5-m telescope visual +20.0 photographic +23.5 electronic +26.0 Limit for Hubble Space Telescope +30.0 How bright an object seems from Earth depends on both the intrinsic brightness of the object and its distance from Earth. A star that appears dim may be a nearby, intrinsically dim star or a distant, intrinsically bright star. What is needed is a scale for describing the intrinsic brightness of an object. Once the intrinsic brightness of a star is known, its apparent magnitude for any distance can be computed. The apparent magnitude of a star at a distance of 10 parsecs is called the absolute magnitude (M) of the star, and is a standard way of expressing the intrinsic brightness of a star. The Sun, for example, has an apparent magnitude of -26.7, but its absolute magnitude is +4.85. If the Sun were 10 pc from Earth, it would be only 2. 9 times brighter than the dimmest star visible to the unaided eye. Table 2 lists the apparent and absolute magnitudes of some stars. I 31- 3 T a bl e 2 The apparent an d a b so 1ute magmtu. des o f somest ars an d thetr . co or tn. d"tees. Name apparent absolute Color Index magnitude magnitude (B- V) Aldebaran (a Tau) +0.86 -0. 2 +1.54 Antares (a Sco) +0.92 -4.5 +1.83 Bellatrix (y Ori) +1.64 -3.6 -0.22 Betelgeuse (a Ori) +0.41 -5.5 +0.03 Can opus (a Car) -0.72 -3.1 +0.15 Deneb (a Cyg) +1.26 -6.9 +0.09 Fomalhaut (a PsA) + 1.19 +2. 0 +0.09 Mimosa (~ Cru) +1. 28 -4.6 -0.23 Pollux (~ Gem) + 1.16 +0.8 +1.00 Regulus (a Leo) +1.35 -0.6 -0.11 Sirius (a CMa) -1 .46 +1.4 0.00 Spica (a Vir) +0.91 -3.6 -0.23 Sun (Sol) -26.7 +4.85 +0.65 Wolf359 +13 .5 . +16.7 +2.01 The distance modulus (m- M) is the difference between the apparent magnitude and the absolute magnitude of a star, and is a measure of the distance to the star. These quantities are related by m- M = -5 + 5 ·log(_!_) , (4) 1· pc where d is the distance to the star measured in parsecs (pc). If the apparent and absolute magnitudes are known, the distance to the star can then be determined from m- M+5 5 d = 10 · pc. (5) Example 5: Deneb (Alpha Cygni) has an apparent magnitude of+ 1.26 and lies at a distance of 430 parsecs. What is the absolute magnitude of Deneb? If Deneb were at the same distance as the Sun, how bright would it appear in Earth's sky? 430 M=m+5-5·log( · pc) =-6.9. 1· pc At a distance of 10 pc, Deneb has a magnitude of -6. 9. At this distance, the Sun has a magnitude of +4.85 . B ( )(+4 s5-(- 6 9) ) Deneb = 1000.2 . = 5.0 ·104. BSun Hence, Deneb would appear 50. thousand times brighter than the Sun. 31- 4 Example 6: Betelgeuse has an absolute magnitude of -5 .5 and an apparent magnitude of +0. 41. How far away is Betelgeuse in parsecs? +0.41 - ( - 5.5)+5 d = 10 · pc = 150 · pc . Apparent visual magnitude (V) of a star is its apparent magnitude at a wavelength of 550 run, where the human eye is most sensitive. Early photographic film, however, was more sensitive to the blue region of the spectrum, with its greatest sensitivity around 450 run. So, when photographs were taken of the sky, blue stars appeared brighter photographically than they did visually. Although, today, astronomers use photometers to obtain accurate measurements of a star's brightness at these wavelengths, the magnitude of a star based on its brightness at 450 nm is still called its apparent photographic magnitude (B). See Exercise 12, Astrophotography, and Exercise 13, Electronic Imaging. The color index of a star is the difference between its apparent photographic magnitude and its apparent visual magnitude.

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