# BAFact Math: The Sun is 12 *trillion* times brighter than the faintest star you can see

By Phil Plait | August 28, 2012 10:00 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 12 trillion times brighter than the faintest star you can see with your naked eye.

In yesterday’s BAFact, I showed how the Sun is about 400,000 times brighter than the full Moon – and I showed my math. That’s an amazing brightness difference, but while I was writing it I had to wonder: how much brighter is the Sun than the faintest star you can see?

The faintest stars visible to the naked eye have a magnitude of about 6. This depends on lots of stuff, like how dark the sky is, how good your eyesight is, and so on. Some people with excellent vision can see stars down to magnitude 7, and there are reports of a few extraordinary people who can see even fainter. But on a dark night, the average person can just barely see 6th magnitude stars.

Let’s use that number then. All we have to do is plug that into the equation I gave yesterday (and remembering that the Sun has a magnitude of -26.7):

Brightness ratio = 2.512(6 – (-26.7)) = 2.51232.7 = 12 trillion

Now, to be fair, that’s not really the brightness range your eyes can detect. You can’t look right at the Sun easily or comfortably; it’s simply too bright. So the range of brightness your eye can see is probably smaller.

We can put a lower limit on it easily enough using the Moon. The Moon is the second brightest object in the sky, and we know we can look at that easily enough, so let’s do that math (the Moon’s magnitude is -12.7 when it’s full):

Brightness ratio = 2.512(6 – (-12.7)) = 2.51218.7 = 30 million

Wow. So you can comfortably see objects over a brightness range of 30 million. That’s impressive! The eye is a pretty cool little machine.

As an aside, your eye isn’t linear; it’s logarithmic (in reality, it’s more complicated than this, and I’m simplifying, but close enough). In other words, a star giving off twice as much light doesn’t look twice as bright as another. The way your eye responds to light squeezes down the scale, making it easier to see fainter and brighter objects at the same time.

So how faint do objects get? Ah, that’ll be tomorrow’s BAFact. Stay tuned!

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CATEGORIZED UNDER: Astronomy, BAFacts
MORE ABOUT: magnitudes, Moon, star, Sun

1. Messier Tidy Upper

Impressive stat indeed! 8)

Mind you, most of the stars we can see are actually much brighter than our daytime star but are also much, much more distant! By hundreds of light years rather than a mere single Astronomical Unit. (Captain Obvious speaking here but still.)

So we’re talking apparent magnitude NOT intrinsic luminosity clearly.

When it comes to the latter, Eta Carinae one of the most luminous stars can apparently boast something like five billion times our Sun’s power which blows my mind trying to imagine it every time!

With the faintest star we can see by unaided eyes – albeit barely on a dark night – being Epsilon Indi with about 70% or so of our Sun’s radience surely? Or, wait, maybe would that be 61 Cygni – Friedrich Bessel’s faint flying star ( a pair of nearby orange dwarfs we now know) and the first to have its distance measured? One of those two anyhow – or am I mistaken here?

2. pontoppi

The eye is good, but not that good. The moon is spatially resolved, so you’ll have to compare the effective surface brightnesses. In other words, you should factor in the size of the stellar point relative to the size of the lunar disk. The moons diameter is about 0.5 degrees, while the human eyes resolution is about 0.5 arcminutes (+/-). The ratio between the effective surface areas of a star and the moon is therefore about 60^2 = 3600. So the dynamic range between the moon and the faint star, as viewed by a human with good eyesight, is around 10,000. In comparison, modern astronomical detectors have dynamic ranges of around 100,000, so given that it is possible to look at something brighter than the moon, it is probably fair to say that the human eye and an astronomical camera are similar in this aspect. Still pretty impressive!

3. Bob

The human eye is truly amazing. Just think how long a digital camera takes to capture the light from such a dim object. That will tell you how much more sensitive your retina is to light than some of the best sensors available. Just don’t try looking at both at the same time. Since the human eye can only deal with a contrast ratio of 500: – 1000:1, the Moon would be so bright it would make the magnitude 6 object invisible.

4. Pete Jackson

The Suns disk is about 700 square arcminutes in size. So, a tiny portion of the Sun one square arcminute in size would have an apparent magnitude of about -26.7 + 2.5*log 700 = -19.6 magnitude, or just below -20 magnitude in brightness.

So, how to see the Sun look like this? One way is to travel to Neptune where the Sun would appear so small as to almost like a star in size, but still, at -20 magnitude, be 250 million times brighter than the faintest star we can see. If you are familiar with intensities in decibels (db), this would be a brightness range of 84 db!

A much easier way to see the Sun look like a star is to view a total solar eclipse! Just before and just after totality, a tiny part of the Sun’s photosphere is visible as a brilliant star-like glare of light superimposed on the chromosphere and corona circling the blackness of the Moon’s disk.
This phenomenon is called the ‘diamond ring’ with the tiny portion of the Sun’s photosphere appearing as the diamond. It is safe to view this without a filter for about ten seconds after the total eclipse is over before the amount of sunlight once again becomes dangerous to view.
(You wouldn’t want to look for this just before totality begins because 1) it is difficult to know when totality is only ten seconds away and it is safe to remove your filter and 2) it would ruin your dark adaptation for the total eclipse).

5. DanM

A healthy human eye can detect a single photon at a time, at least some fraction of the time, although the brain does not necessarily respond until there are at least a handful of photons at a time. This experiment has been done:
http://math.ucr.edu/home/baez/physics/Quantum/see_a_photon.html
Now THAT’S cool. It would seem to set a rigorous lower bound on visual sensitivity. An upper bound might be harder to define…

6. Andrew

Thirty million -whats-? I’m not grasping the concept here. My two objections:

If you were to take 30,000,000 stars, beginning from the dimmest and each brighter than the next by one -what-, I doubt the human eye could tell the difference between, say, stars #14,500,301 and #14,500,302.

Besides, as the arbitrarily-chosen base of 2.512 approches 1, this result of 30 million -whats- would decrease exponentially (e.g. 1.05^18.7 = 2.49, not nearly as impressive as 30 million). If the base itself is an arbitrary figure… I fail to see the significance of the 30 million figure

If anyone can straighten me out, I would appreciate it.

7. Infinite123Lifer

“Thirty million -whats-?”

thirty million times the faintest star you can see with your naked eye . . . i reckon

“If you were to take 30,000,000 stars, beginning from the dimmest and each brighter than the next by one -what-, I doubt the human eye could tell the difference between, say, stars #14,500,301 and #14,500,302.”

I am no optometrist but I think what Phil was getting at is though you probably could not distinguish between 14,500,301 and 14,500,302 (which would be 14,500,301 times the faintest viewable star with the naked eye) we still have this incredible vision which allows us to see the faintest you can actually see 1 (the faintest star you can see with your naked eye) and tolerate over 30,000,000 times (the second brightest object in the night sky, the moon) that lower bound. Which means the eye can detect and tolerate 30,000,000 times what would be considered its faintest detection, which at the very least is . . . a real eye opener

I wonder what the difference between a photon and the faintest star we can see is?

8. Nigel Depledge

Pete Jackson (4) said:

If you are familiar with intensities in decibels (db), . . .

Ahem.

If you are familiar with decibels, you use the correct notation, which is dB.

Pedantic? Moi?

9. Nigel Depledge

Andrew (6) said:

Thirty million -whats-?

As a ratio, it is a dimensionless quantity.

It matters not whether you measure the stars’ brightnesses in candela, lumens, watts or radioactive potatoes, the ratio is the same and has no units.

10. VinceRN

The eye most certainly is a “pretty cool little machine”.

11. Always wondered, why 2.512? Was that just a number someone came up with to approximate the 1-6 range they used before? Is there some logical, mathematical reason it had to be 2.512? Did whoever selected that number just like it? Was someone trying to incorporate the date of Christmas?

12. Pete Jackson

@11 Vince 2.512 is the 5th root of 100. To be precise, it is

2.5118864315095801110850320677993…

That way, a difference of 5 magnitudes is precisely a factor of 100 in brightness.

13. VinceRN

@12 Pete – Thanks, makes more sense than the Christmas explanation.

14. Nigel Depledge

@ Pete Jackson (12) -
Call that precise? You gave up after 32 sig figs!

15. @1 MTU : Faintest star you can see?

Well from this K5 & K7 dwarfs ^1 Cyni A & B :

With temperatures of 4450 and 4120 Kelvin, they shine only at luminosities of 15 and 9 percent solar, their masses only 60 and 50 percent solar, radii just 65 and 60 percent solar.

Versus Epsilon Indi :

http://stars.astro.illinois.edu/sow/epsind.html

class K (K4.5) dwarf, Eps Ind is one of the intrinsically least-luminous stars visible to the unaided eye. … (snip) .. With a cool temperature of 4620 Kelvin, Eps Ind shines with a luminosity that is just 22 percent that of the Sun, its radius 75 percent solar. This low wattage is the result of a low mass, about 70 percent solar as determined from luminosity, temperature, and theory.

Which compares with young K2 orange dwarf Epsilon Eridani :

http://stars.astro.illinois.edu/sow/epseri.html

which has a luminosity of 34 percent solar and a mass 83% solar.

Not sure too many other orange dwarfs are visible in our night skies seeing as these are the closest examples outside of being companions tobrighter stars such as Alpha Centauri B (linked to my name here) which in any case is a brighter K1 dwarf half as bright as our Sun.

So I think 61 Cygni has that record!

16. Matt B.

Hm, this means the sun would be barely visible at a distance of 54.75 c-yr. (That’s [the square root of 12 trillion] AU.)

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