- How many stars can you count on a clear night. I sure you would estimate several thousand. Astronomers have been fascinated with counting the stars for centuries. Over recent decades they have even developed a mathematical formula for calculating the number of stars you can see.
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- Of course how dim a star is depends on how far away it is. How many you see depends on how sharp your naked eyes are. Many backward gazers can see stars that are a Brightness Magnitude of “6”, M = 6. With a small telescope you can see to a Magnitude “10”, M = 10. The larger the number the dimmer the star. The Hubble Space Telescope can see to a Magnitude of “ 25, M = 25.
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- What are these Magnitude numbers? To learn we need to explain “ Apparent Brightness” and “Absolute Brightness” in stars. Remember brighter stars further away can “look” dimmer. Then we need to explain “ parsecs” for measuring astronomical distances. These will be saved for the footnotes. Let’s first go for the number of stars ,”N” , we can count given the Magnitude of Apparent Star Brightness, “M”:
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- Log10(N) = -0.0003M^3 + 0.0019M^2 + 0.484 *M -3.82
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- This is a third order polynomial equation that is valid for star Brightness Magnitudes ranging from 4 to 25. The polynomial is Log10 (y) = x^3 + x^2 + x + constant which is an exponential function. It gives an answer of the number of stars per square degree in the sky. The Full Moon is ¼ square degree.
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- Let’s assume you do not have the sharpest eyes but on a clear night how many stars could you count up to a Magnitude of M = 5?
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- Log10(N) = -0.0003(5)^3 + 0.0019(5)^2 + 0.484 *(5) -3.82
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-Log10(N) = -0.0003(125) + 0.0019(25) + 0.484 *(5) -3.82
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-Log10(N) = -0.0375 + 0.0475 + 2.42 -3.82
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-Log10(N) = -1.39
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- N = 10^-1.39 = 0.0407 stars per square degree.
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- There are 41,253 square degrees in the night sky.
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- Number of stars you can count brighter that Magnitude 5 = 0.0407 * 41,253 = 1,679 stars. If you had sharp eyes and could see all the stars up to a Magnitude 6 you could count 5,077 stars
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- How many stars if you use a backyard telescope that can see up to a Magnitude 10?
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- Log10(N) = -0.0003(10)^3 + 0.0019(10)^2 + 0.484 *(10) -3.82
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-Log10(N) = -0.0003(1000) + 0.0019(100) + 0.484 *(10) -3.82
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-Log10(N) = - 0.3 + 0.19 + 4.84 - 3.82 = 5.03 - 4.12 =
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-Log10(N) = 0.91
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- N = 10^0.91 = 8.13 stars per square degree.
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- 8.13 per square degree * 41,253 square degrees - 335,316 stars
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- You can see over 60 times as many stars with a small telescope. What about the Hubble Space telescope that can see up to a Magnitude , M = 25?
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- Log10(N) = -0.0003(25)^3 + 0.0019(25)^2 + 0.484 *(25) -3.82
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-Log10(N) = -0.0003(15,625) + 0.0019(625) + 12.1 -3.82
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-Log10(N) = - 4.69 + 1.19 + 12.1 - 3.82 = 13.29 - 8.51
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-Log10(N) = 4.78
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- N = 10^4.8 = 60,255 stars per square degree.
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- Hubble’s view is ¼ square degree, about the size of the Full Moon. * 60,255 stars per square degree = 15,054 stars. So the telescope needs to point in many directions to count more stars. If it had the time to cover the sky we see it would see 60,255 * 41,253 = 2,487,100,000 stars. 2.5 billion stars. And, that is just the night sky we see. Hubble can see the entire cosmic sphere
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(1) “Apparent Brightness of a star is how bright it is to an observer on Earth. The actual or “ Absolute” or “Intrinsic” is the brightness, or “ Luminosity” at the power source. How bright we see it depends on how far away it is.
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- Astronomers invented a Magnitude scale. Giving the bright star Vega a zero and giving fainter or dimmer stars a higher positive number. In the beginning the number assignments were eyeball subjective. Eventually a mathematical formula was developed for this scale. The formula developed assumed that an increase of 5 Magnitudes corresponded to a decrease in brightness by 100.
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------------------------------- 100^ 1/5 = 2.5
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- To compare the brightness of 2 stars, b1 and b2 having brightness Magnitudes of M1 and M2:
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------------------------ b1/b2 = 2.5 ^(M2-M1) = 100 ^ ((M2-M1) / 5)
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- To compare al the stars as to their “ absolute brightness” astronomers arbitrarily put them at the same distance away from us, a distance of 10 parsecs.
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---------------------- 1 parsec = 3.26 lightyears distance.
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----------------------10 parsecs = 191,757,000 million miles away.
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----------------- Apparent Brightness Magnitude, m = Absolute Magnitude, M + 5 10g ^ d/10. “d” is the distance in parsecs.
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- So if you know the apparent brightness and the absolute brightness you can calculate the distance for the star using:
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------------------------- distance = d = 10 ^ ((m-M+5)/5)
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(2) One parsec = 3.26 lightyears
One parsec = 206,265 astronomical units. And AU is the Sun-Earth distance of 93 million miles.
As the Earth orbit’s the Sun nearby stars appears to change their location in the sky relative to the most distant stars. This is called “ parallax”. The angle of the parallax shift is the parallax angle which is usually measured in arc seconds.
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--------------------- distance in parsecs = d = 1 Astronomical Unit / parallax angle in arc seconds.
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--------------------- Angle in arc seconds = 206,265 * Distance in AU / distance to the star.
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- Our closest star, Proxima Centauri, has a parallax angle of 0.76 arc seconds. Some 7,000 stars have had their distances measured this way to an accuracy of better than 5%.
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---------------------------Apparent Magnitude -------------- Absolute Magnitude
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- Sirius -------------------- -1.46 ------------------------- 26
- Arcturus ------------------- -.06 -------------------------- 170
- Vega ---------------------- 0.04 -------------------------- 60
- Capella ------------------- 0.85 -------------------------- 77
- Rigel ---------------------- 0.14 ----------------------- 70,000
- Procyon ------------------- 0.37 ------------------------- 7.4
- Betelgeuse ---------------- 0.41 ----------------------- 38,000
- Spica ---------------------- 0.91 ------------------------ 23,000
- Deneb --------------------- 1.26 ----------------------- 170,000
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