Monday, February 8, 2021

3035 - STARS - How many stars are in the sky?

 -  3035  -  STARS  - How many stars are in the sky?   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.


---------------  3035  -  STARS  -  How many stars are in the sky?  

-  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.  

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-  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. 

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-   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.  

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-  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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-    “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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-------------------------  One parsec  =  3.26 lightyears

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-  One parsec  =  206,265 astronomical units.  And AU is the Sun-Earth distance of 93 million miles.

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-  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

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-  Arcturus -------------------  -.06  --------------------------  170

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-  Vega  ----------------------  0.04   --------------------------  60

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-  Capella  -------------------  0.85  --------------------------  77

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-  Rigel  ----------------------  0.14  -----------------------  70,000

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-  Procyon  -------------------  0.37  -------------------------  7.4

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-  Betelgeuse  ----------------  0.41  -----------------------  38,000

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-  Spica  ----------------------  0.91  ------------------------  23,000

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-  Deneb  ---------------------  1.26 -----------------------  170,000

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February 7, 2021             STARS  -  How many stars?                       3035                                                                                                                                                           

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