[BAFacts are short, tweetable astronomy/space facts that I post every day. On some occasions, they wind up needing a bit of a mathematical explanation. The math is pretty easy, and it adds a lot of coolness, which I'm passing on to you! You're welcome.]

Todays BAFact: The Sun is 400,000 times brighter than the full Moon in the sky.

If youve ever looked at the full Moon through a telescope you know how painfully bright it can be. But you can do it if you squint, or use a mild filter to block some of the light.

On the other hand, if you try the same thing with the Sun (hint: dont) youll end up with a fried retina and an eyeball filled with boiling vitreous humor.

So duh, the Sun is much brighter than the Moon. But how much brighter?

Astronomers use a brightness system called magnitudes. Its actually been around for thousands of years, first contrived by the Greek astronomer Hipparchus. Its a little weird: first, its not linear. That is, an object twice as bright as another doesnt have twice the magnitude value. Instead, the system is logarithmic, with a base of 2.512. Blame Hipparchus for that: he figured the brightest stars were 100 times brighter than the dimmest stars, and used a five step system [Update: My mistake, apparently he didn't know about the factor of 100, that came later.]. The fifth root of 100 = 2.512 (or, if you prefer, 2.5125 = 2.512 x 2.512 x 2.512 x 2.512 x 2.512 = 100), so there you go. Ill give examples in a sec

Secondly, the other weird thing about the magnitude system is that its backwards. A brighter star will have a lower number. Its like an award; getting first place is better than third. So a bright star might be first magnitude, and a dimmer one third magnitude.

To figure out how much brighter one star actually is than another, subtract the brighter stars magnitude from the dimmer ones, and then take 2.512 to that power. As an example, the star Achernar has a magnitude of roughly 0.5. Hamal, the brightest star in the constellation of Aries, has a magnitude of 2.0. Therefore, Achernar is 2.512(2.0 0.5) = 2.5121.5 = 4 times brighter than Hamal. So you can say its four times brighter, or 1.5 magnitudes brighter. Same thing.

Its weird, but actually pretty handy for astronomers. And it doesnt stop at 0. A really bright object can have a negative magnitude, and the math still works. For example, Sirius, the brightest star in the night sky, has a magnitude of about -1.5 (making it 6 times as bright as Achernar check my math if you want). Which brings us to the topic at hand

The Moon is pretty bright, and when its full has a magnitude of about -12.7. Thats bright enough to read by! But the Sun is way, way brighter. Its magnitude is a whopping -26.7. How much brighter is that?

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BAFact Math: The Sun is 400,000 times brighter than the full Moon | Bad Astronomy