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Astrophysics - Maths Equation


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So I've been trying to solve this question, which feels really simple, but there's a lot of room allocated for it so something feels wrong. 

Q: Stars must have an apparent magnitude equal or greater than +6.0 to be seen by the naked eye, whereas the Hubble Space Telescope can view stars as faint as +30.0. How much fainter are the faintest objects visible to the Hubble Space Telescope relative to the dimmest stars visible to the naked eye?

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While im here if someone could check this question as well 

Q: If the Sun and the full moon have apparent magnitudes of -26.71 and -12.24 respectively, then how many times brighter is the sun compared to the full moon?

it's not just -26.71--12.24 right,,, something just definitely feels too simple.

Any help is really appreciated <3 
Thank you for your time 

Frey~

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That is the approx difference in brightness between the brightest and faintest stars we can see. One of the Greeks classified  the brightest stars as first magnitude and the faintest he could see as sixth. When we formalised the system we used those approx figures - despite some stars being brighter than first mag. 100x brighter = 5 mags. As it is a logarithmic scale, 1 mag is the fifth root of 100 times brighter - about 2 5x.

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Another way, and the actual maths formula, of looking at the formula is 10 ^ (0.4x magnitude difference) which equates to approximately 2.512 ^ magnitude difference.  (^ = to the power of)

Hence 1 magnitude difference is 2.512 times brighter
2 magnitudes is 6.31 times brighter 
3 magnitudes is 15.85 times brighter
4 magnitudes is 39.82 times brighter
5 magnitudes is 100 times brighter
14.47 magnitudes is 614,163.67 times brighter.

The reason for why such a system is relatively arbitrary.  It is correct that the Greeks came up with the system however there is a natural reason for them choosing this system....It simply relates to how our eyes/brain works. 
Our brains/eyes naturally convert flux into a logarithmic scale.  In effect the brain does it's own non-linear stretch on the data to enhance dimmer objects and suppress brighter objects.  If you imagine what  a camera is like taking exposures during daytime then you can understand what would happen if we just used a linear scale.  At night we'd see nothing, during the day we'd be permanently dazzled.

As such the Greeks just interpreted what they saw and what we can still see today. A magnitude 4th star will look to us about twice as bright as 2nd magnitude star even though the actual flux is over 6 times less.

 

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