# BAFact Math: The Sun is 400,000 times brighter than the full Moon

By Phil Plait | August 27, 2012 10:02 am

[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.]

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

If you’ve 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: don’t) you’ll 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. It’s actually been around for thousands of years, first contrived by the Greek astronomer Hipparchus. It’s a little weird: first, it’s not linear. That is, an object twice as bright as another doesn’t 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. I’ll give examples in a sec…

Secondly, the other weird thing about the magnitude system is that it’s backwards. A brighter star will have a lower number. It’s 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 star’s magnitude from the dimmer one’s, 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 it’s four times brighter, or 1.5 magnitudes brighter. Same thing.

It’s weird, but actually pretty handy for astronomers. And it doesn’t 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 it’s full has a magnitude of about -12.7. That’s bright enough to read by! But the Sun is way, way brighter. It’s magnitude is a whopping -26.7. How much brighter is that?

Well, it’s 2.5(-12.7 – (-26.7)) = 2.514 = 400,000.

In other words, the Sun is 400,000 times brighter than the full Moon!

This would explain why you can look at the Moon easily enough with just your eye, but trying that with the Sun is not – wait for it, wait for it – a bright idea.

Image credit: NASA/SDO

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CATEGORIZED UNDER: Astronomy, BAFacts, Cool stuff
MORE ABOUT: brightness, magnitudes, Moon, Sun

1. Yes, but . . . !

How’s that square up with the fact that the Moon’s albedo is about 10%, which–since the Sun and full Moon have almost exactly the same apparent size in our sky–implies that the Sun is only 10x brighter?

I have an answer, but still think it’s an instructive question.

2. Keith Bowden

I deny this is true. It’s just like all your political posts and HIRGO and evolution posts. I’ll prove it, I’ll just go stare at the sun for a bit as as yuou can aplaintly see my eyesigt isa aknlsdavlas asdflka aapownds;la…

3. Not a comment on the above but an addendum for hobbyists.
Solar observation via telescope is possible with filters…
For most telescopes, if you want a solar filter, you want one on the objective end, not at the eyepiece end. Otherwise, you risk melting part of the telescope, as well as your eyeball!
Paper eclipse glasses which do no magnification at all but simply make the sun viewable can be had for a couple dollars from outfits that sell red-green 3-D glasses.

4. Blargh

Typo in the fourth to last paragraph: “It’s magnitude” instead of “Its magnitude”.

@ Brian
The main answer is that the moon is not a mirror. The light hitting it scatters in all directions (diffuse reflection) instead of reflecting off it like a mirror (specular reflection).
(It also isn’t flat (and aimed at the Earth ) but spherical, which means that even if the moon had been a perfect mirror, we would still only receive a fraction of the sunlight falling on it)

5. Robert

This feels a bit like comparing apples to oranges – one could argue that the Sun is infinitely brighter than an object that has no inherent brightness of its own, but rather only reflects the light of the Sun. In the same way – it is pointless to compare the brightness of a mirror compared to a 40w light bulb. One shines, the other one reflects.

6. Bryan

@Blargh

Without doing the math (which honestly, would be well beyond me these days), I would guess that the moon probably looks brighter as it is, then if it were a perfectly spherical mirror.

7. Blargh, that’s part of it (which I know you know because you wrote “main answer”). I think it’s kind of an interesting, complicated question, which is why I raised it.

I find it helpful to imagine looking at both the Sun and full Moon from sufficient distance that they’re equidistant point sources; then Phil’s result seems self-evident in a way it doesn’t quite (at least to me) from our POV on Earth. Why those perspectives differ is the key, I think.

8. hde226868

Nice discussion of magnitudes.

Just a quick correction: Hipparcos (and Ptolemy) did not know that the faintest stars were 100x fainter than the brightest ones. In his catalogue he just classified the stars in six(!) classes which is about the number of brightness differences one can reliably estimate with the eye. See the discussion on p. 259 of http://www.scribd.com/doc/46305490/Almagest about the completeness of the catalogue, which shows that the catalogue is more or less ok down to mag 4 or so.

The mapping from visually determined magnitudes to intensity and the realization that the magnitude scale was a logarithmic scale came only in the 19th century. In our current form it is due to Pogson (1856), see http://articles.adsabs.harvard.edu/full/1856MNRAS..17…12P (he notes that several other astronomers in the United Kingdom and Germany had already realized that the magnitude scale is logarithmic, but the factor 2.512 is due to him).

9. don w

clarification- your “not as bright” illustration is not a full moon

10. hde226868 (8): Ah, thanks. I updated the text. Note that I said a five step system, which results in six classes.

11. Bob

I’ve read about the magnitude of various objects many times over the years, but I don’t remember anyone ever mentioning what the reference object is. Perhaps I should research this when I have some free time…

Also, Phil, you bounce back and forth between using “x times as bright as”, and “x times brighter than”, and I wish you would stick with the first, since the second is poor grammar, and not the same value.

12. Pete Jackson

So if a solar panel generates 200 watts per square meter when the Sun is overhead, it will generate only 0.5 milliwatts when the full moon is overhead.

13. mike burkhart

Of course. If you look at the Sun wthout eye protecton you wll go blnd, but I’ve never heard of anyone going blnd from lookng at the Moon. Also the Sun radiates its own light from its Phtosphire near its surface ( the funy thing is the interor of the Sun is Dark, but hotter then the surface) The Moon just reflects the Suns light thats why it shines at night and sometimes in the daytime.

14. Georg

Hipparchos
did not know about logarithms, he just classified
the stars subjectively. Because brightness sense
depends logarithically on real brightness, he used
a logarithmic scale without knowing it.
Georg

15. Bob

Isn’t saying that the interior of the Sun is dark akin to saying the same about a lump of molten iron? If I recall correctly, many, if not most, of the photons given off by the Sun started their journey deep in the Sun’s interior. Everything that is hotter than blazes emits photons, so it would seem to me that the interior of the Sun must be quite bright.

16. hde226868

Bob (reply 11): the reference magnitude is Vega, which has 0mag by definition in all bands.

This usually results in much smaller magnitude values in bands where Vega is not that bright (e.g., in the infrared). One can get rather impressive negative magnitudes when extrapolating the scale and this zero point to the radio or X-ray bands. I have found this quite useful for undergraduate homework questions, by the way (as they cannot be looked up in wikipedia since no sane astronomer would use magnitudes when working in these wavebands ).

17. Thomas Siefert

Sometimes I get the feeling that you start with the punch line or the pun and then work your way backwards, coming up with an interesting article in the process.

18. carbonUnit

Speaking of the Moon not being a mirror (#4) how did those Hubble Space Telescope observations of the transit of Venus which used the Moon as a “mirror” turn out? All the hits I can find are articles written before the fact. I’m interested in the results..

19. Bob

hde226868 (reply 16): Thank you for that information!

20. Andrew

The eye is not very good at these sorts of comparisons: the iris opens, and the eye has a non-linear response.

And compared with the apparent magnitude of a domestic 60W lightbulb 10 metres away…?

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