# Tag: brightness

## BAFact Math: The Sun is mind-crushingly brighter than the faintest object ever seen. Seriously.

By Phil Plait | August 29, 2012 10:10 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: How much brighter is the Sun than the faintest object ever seen? About Avogadro’s number times brighter.

Yesterday and the day before I wrote about how much brighter the Sun is than the Moon, and how much brighter the Sun is than the faintest star you can see (note that here I mean apparent brightness, that is, how bright it is in the sky, not how luminous it actually is). I have one more thing to add here.

Years ago, I worked on a Hubble Space Telescope camera called STIS – the Space Telescope Imaging Spectrograph. At the time, it was the most sensitive camera ever flown in space, and I was constantly amazed at what we saw using it.

Hubble did a series of observations called the Deep Fields: it stared at one spot in the sky for days, letting light from incredibly faint objects build up so that they could be detected. For the Deep Field South, STIS was used to observe a particular kind of galaxy, a quasar called J2233-606. The total observation time was over 150,000 seconds – nearly two days!

I worked on these images, and was chatting with a friend about them. We were astonished at the number of objects we could see, distant galaxies so faint that they were unnamed, uncategorized, because no one had ever seen them before. Playing with the numbers, we figured that the faintest objects we could see in the observations had a magnitude of about 31.5. That’s incredibly faint.

How faint, exactly?

The faintest star you can see with just your eye has a magnitude of about 6. Using the magnitude equation I wrote about earlier, plugging those numbers in we get

Brightness ratio = 2.512(31.5 – 6)) = 2.51225.5 = 16 billion

Wow.

But we can do better than that. A lot better. After all, the Sun is the brightest object in the sky, of course, with a magnitude of -26.7. Just for grins, how much brighter is the Sun than the faintest objects ever seen?

Brightness ratio = 2.512(31.5 – (-26.7)) = 2.51258.2 = 2 x 1023

Um.

That’s 200,000,000,000,000,000,000,000. 200 sextillion. Holy yikes.

That number is crushing my mind. It’s ridiculous. A sextillion is simply too big a number to grasp. And 200 of them? C’mon!

But hey, wait a sec…

Does the number 2 x 1023 look familiar to you? It does to me: it’s the same order of magnitude (factor of 10) as Avogadro’s number! It’s the number of atoms of an element in a mole of the element, where a mole is the number of atoms in 12 grams of pure carbon-12. I know, it’s an odd unit, but it’s handy in chemistry, and a lot of (geeky) folks have heard of it.

Avogadro’s number is actually about 6 x 1023. So if we could detect stars or galaxies just a hair more than a magnitude fainter, the ratio of the brightness of the Sun to those objects would be Avogadro’s number. Huh.

I’m not sure that helps, but it’s fun in a spectacularly nerdtastic kind of way.

Science, baby. I love this stuff!

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CATEGORIZED UNDER: Astronomy, BAFacts, Cool stuff, Science

## 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

## BAFact math: How bright is the Sun from Pluto?

By Phil Plait | March 15, 2012 11:07 am

[On January 4, 2012, I started a new features: BAFacts, where I write an astronomy/space fact that is short enough to be tweeted. A lot of them reference older posts, but some of the facts need a little mathematical explanation. When that happens I’ll write a post like this one that does the math so you can see the numbers for yourself. Why? Because MATH!]

Today’s BAFact:

From Pluto, the Sun is fainter than it is from Earth, but still can be 450x brighter than the full Moon.

I remember reading a science fiction story many years ago which took place on Pluto. The author described the Sun as being so faint that it looked like just another bright star (too bad I don’t remember the name of the story anymore). I was thinking about that again recently, and wondered just how bright the Sun does look from Pluto. This turns out to be pretty easy to calculate!

First, you need to understand how an object like the Sun — really, any source of light — dims with distance. The Sun emits light in all directions, so as you get farther away from the Sun, that light gets spread out. Imagine a sphere perfectly encasing the Sun right at its surface. Each square centimeter has a certain amount of light passing through it. If I double the size of the sphere, there’s a lot more surface area to that sphere, but the total amount of light passing through it hasn’t changed. Therefore the amount of light passing through each square centimeter has dropped. Since I doubled the sphere’s diameter, I can figure out how much its dropped, too!

The formula for the surface area of a sphere is

Surface area = 4 × π × radius 2

If I double the size of the sphere, everything on the right side of the equation stays the same except for the radius, which is now twice as big. Therefore the area increases by 22 = 4. So the light passing through each square centimeter of the bigger sphere drops by a factor of four. Someone standing on that sphere would see the Sun being 1/4 as bright as if they were on the surface.

If I make the sphere ten times bigger, the area goes up by 10 × 10 = 100 times, and the brightness drops by 100. You get the picture.

So now we’re ready to figure out how bright the Sun is from Pluto!

The Earth orbits the Sun, on average, at a distance of about 150 million km. Pluto has a very elliptical orbit, but has an average distance of about 5.9 billion kilometers, or roughly 39 times the Earth’s distance from the Sun. Using the method above, the Sun must be 392 = about 1500 times fainter, or more grammatically correctly, 0.00065 times as bright. That’s pretty faint!

Or is it? Well, let’s compare that to how bright the full Moon looks from Earth. To us here at home, the Sun is about 400,000 times brighter than the full Moon, so even from distant, frigid Pluto, on average the Sun would look more than 250 times brighter than the full Moon does from Earth!

Pluto’s orbit is also highly elliptical, stretching from 4.4 billion km to just over 7.3 billion km from the Sun. Doing the math again, that means the Sun goes from being 0.0012 to 0.0004 as bright as it is from Earth: a range of roughly 150 to 450 times as bright as the Moon from Earth. That’s a change in brightness by a factor of three!

Still, given that you can read by the light of the full Moon, obviously the Sun from Pluto is still pretty dang intense. It would hardly look like just any other star: it would greatly outshine everything else in the sky. Painful to look at, most likely. So the short story I read was wrong, but at least we learned something. That’s a decent trade.

And let me leave you with a question: From Pluto, how big would the Sun look? Ah, that’s a BAFact for another day. Tomorrow, actually!

Image credit: Dan Durda, showing Pluto, its moon Charon, and the Sun.

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