Are humans brighter than the Sun?

By Phil Plait | December 30, 2009 7:02 am

You’re the only star in heaven
You’re the only star that shines
You’re the only star in heaven
Now that only star is mine
– Frankie Goes to Hollywood

Snuggler’s lament

These days of northern winter seem endless. It’s been brutally cold here in Boulder, causing much snuggling at night, both between humans and with Canises Major and Minor. Snuggling is fun, of course, but also useful: body heat shared is body heat doubled.

After a while it can get too hot, and even Mrs. BA with her ice cold feet will move away to cool off a bit. When that happens, of course, my mind turns to matters scientific. Our bodies generate a lot of heat. And with the Sun making only a desultory appearance every day, I was thinking recently about the energy generated by the Sun, versus that emitted by humans. I remember reading once that if you compare the heat coming a single square centimeter from the Sun to the same area on a human being, you’d find we actually put out more energy! As a skeptic I’m used to analyzing such claims; as a scientist I have the mad math skillz to work out the numbers; and as a communicator, I have the soapbox upon which I can talk about the whole thing.

So let’s get to it. Are humans more energetic than the Sun?

Yar, thar be math below, and plenty of it. Be ye fairly warned, says I.

Glowing places

As it happens, the math isn’t that hard. Objects that are warm (really, anything warmer than absolute zero) have a characteristic way they emit energy, called black body radiation. Both humans and the Sun are pretty close to being such radiators, and it’s not too bad to just assume they’re blackbodies. There’s a simple equation to calculate the amount of energy emitted per second, called the luminosity:

Luminosity = area of object x σ x temperature4

where σ, the Stefan-Boltzmann constant, is just a number (if you wanna check my math, it’s 5.67 x 10-5 erg cm-2 s-1 K-4, and K is Kelvins, the unit of temperature). That equation makes sense: at a given temperature, a bigger object will emit more energy. And if two objects are the same size, the hotter one will give off more energy (in other words, and to be a bit more vernacular, it’ll be brighter).

The Sun is big, so even if it were colder than a human it would win that fight! So we want to compare apples to apples here, taking one square centimeter of each and seeing who wins. That means we want the luminosity divided by the area, giving us the energy emitted per square centimeter in one second. Rearranging, we get

Luminosity / area = σ x temperature4

Hey, wait a sec! This makes something clear right away: if you want to compare the energy emitted per square centimeter from any two objects, all that matters is their temperatures. The hotter one wins.

That means the story I read — that humans emit more per square cm than the Sun — is wrong. The Sun is a lot hotter than a human, so it emits vastly more energy than a person does! In fact, it’s the ratio of the temperatures raised to the 4th power. The Sun’s temperature is 5780 Kelvins, and a human is 310 Kelvins. Plugging and chugging shows that the Sun gives off a whopping 121,000 times as much energy per square centimeter as a person does!

Yegads. And the Sun is a whole lot bigger. If you’re not careful, you may get the impression the Sun gives off quite a bit of energy.

Pump up the volume

A cubic Sun.

Anyway, far be it from me to simply say the story is wrong and drop it. There’s more science here! Instead of using the area, what about the volume? In other words, assume both a human and the Sun have the same temperature throughout (I mean, every chunk of a human is 310K, and the Sun is 5780 K). Would a cubic centimeter of the Sun outshine a cubic centimeter of human?

This is a little bit tougher to calculate. We need the total luminosity of the Sun and its volume, and the same for a human. For the Sun that bit’s actually easy, since we just use that luminosity equation above (knowing the Sun’s area is 6.1 x 1022 square centimeters, which I leave up to you to calculate if you want). The Sun’s energy emitted per second is then about 4 x 1033 ergs/second, where an erg is just a unit of energy astronomers like to use. It’s a small unit, but 4,000,000,000,000,000,000,000,000,000,000,000 of them is still a lot.

What about a human? Well, we need the area of a human to plug into the equation to get the luminosity, and we’ll have to estimate that. Let’s use me as an example, and assume I’m a big rectangular solid, like a shoebox (or a monolith). I’m 177 cm tall, about 50 cm across, and about 15 cm deep. That gives me an area of very roughly 25,000 square centimeters. That’s an estimate, but good enough — it won’t matter if I’m off by a factor of two either way.

Plugging away, I get that my luminosity is then 1.3 x 1010 ergs/sec. That’s a lot smaller than the Sun. But then, I’m not all that hot*.

I am your density

OK, almost there! All we need to do now is divide those numbers by the respective volumes and compare them. The Sun’s volume is 1.4 x 1033 cubic centimeters. That means that each cubic centimeter gives off 4 x 1033 ergs/second / 1.4 x 1033 cc = 2.8 ergs/second/cc. So every second, each cc of the Sun emits 2.8 ergs. OK then. What about me?

How I know humans and water
have the same density.

My volume is easy to estimate: I know humans are the same density as water, which is 1 gram/cc. I also know my mass is about 75,000 grams, so my volume must be 75,000 cc! Easy peasy.

Finally, dividing my luminosity by my volume yields 170,000 ergs/sec/cc.

Hey, wait a sec! That means not only am I brighter than the Sun, I’m a lot brighter! About 60,000 times brighter!

Make a gas of U and ME

So in that sense, the legend is right. If you want to think of it this way, a cubic centimeter of human gives off a lot more energy than the same volume of the Sun does!

But hold on there. Is this really a fair statement?

Well, not really. First, there are a whole lot more cubic centimeters in the Sun (about 1028 times as many, or ten billion billion billion times as many), so when you divide by such a big number the energy per cc for the Sun drops drastically. So even if we say, sure, humans are more luminous per cubic centimeter, it’s best not to get cocky. The Sun can still vaporize us with lots of cubic centimeters left to spare.

Second, remember the assumption I made, that the Sun has the same temperature everywhere? That’s not even close to being true. In fact, it’s whoppingly untrue. The core of the Sun is 15 million Kelvins hot, so each cc there is blasting out vast amounts of energy: about 5 quadrillion times what a cc of human flesh does. But outside of that region the Sun is much cooler, and each cc doesn’t contribute nearly as much. Over the entire Sun, that dilutes the amount of energy per cc quite a bit. Averaging over the volume of the entire Sun is not a great way to think about it, and makes comparisons difficult, if not really meaningless.

Like this exercise has any profound, deeper truths to it in the first place. Actually, it’s just an excuse to have some fun and do some mental gymnastics. And, in the end, it really comes down to this: the Sun is bright, and we are not.

But, you knew that. Of course, some humans are hotter than others. But I’m not sure I can do the math for that.

Tip o’ the pound of flesh to BABloggee Brad Stacey.



* You can take my word for it, or ask Mrs. BA. She’ll be honest, but her feet lie!

Swimmer image from dionhinchcliffe’s Flickr photostream.

Comments (68)

Links to this Post

  1. The List – Feb. 3, 2012 « Radio Acting | February 3, 2012
  1. Rachel Walmsley

    And just in case anyone tries to get cocky even so, it’s also worth noting that something like a bacterium is much brighter per unit volume than we are. It’s a surface area to volume ratio thing.

  2. Mike

    Very entertaining, Phil!

  3. Ha, great post! I think I’d heard the myth once, but I was skeptical enough to dismiss it. Glad to see the math isn’t too crazy.

    Looks like the real lesson is that most people (myself included) can always stand to learn a little more about the sun ;)

    (Oh and besides just size, there’s the fact that we last less than a hundred years usually, and the sun should last, oh, 10 billion)

  4. Mark Hood

    When I first read it, I assumed it meant the energy from the sun falling on a square centimetre of exposed skin was less than the energy emitted by that same square centimetre of skin…

    I can’t figure out the maths on the fly, but I’d have to say that if you feel the sun’s heat on your skin, it’s surely because there’s more energy in than out?

    Mark

  5. Ray

    Someone needs a real job. ;)

  6. Cheyenne

    Can the above equations be re-computed using Karen Nyberg’s luminosity figures? ;)

  7. TSFrost

    Why’d you blur out your face, Phil? (As this year’s Mr. December, I wanted to know. ;) )

  8. Alan in Upstate NY

    Dr. Eric Chaisson made a similar point about energy flow in his talk “Cosmic Evolution: The Rise of Complexity in Nature” on October 13 as the first of Dudley Observatory’s three Skywatch Lectures for 2009.

    A diagram of energy flow as a function of complexity can be found in this paper…
    http://www.tufts.edu/as/wright_center/eric/reprints/big_history.pdf

    Clear skies, Alan

  9. Sir Craig

    Please, Phil, tell me that swapping “then” for “than” in your title of this post was done for irony’s sake (humans being “brighter” and all that).

    If not, I shall cry along with Baby Jesus…

  10. Nemo

    That’s not a blur, it’s the Sun.

  11. Ryan

    A trifle of a kibble really … capitalizing kelvins? And of course you’re also using ergs rather than joules. An astronomical convention?

  12. Sir Craig– Sorry, but that was simply a typo. I fixed it though. :)

  13. TheElkMechanic

    I would have gone with “Snuggler’s Blues” for the first subtitle, but that’s just me.

  14. Hello,

    Is there some estimated plot of the temperature of the sun from the surface to the core ? Just curious.

  15. Dave

    Phil,

    Very interesting! Just a few comments. I’ll put them in separate posts.

    (1) You imply that the assumption that the Sun has the same temperature throughout is important, but it’s not, not really. I mean, if you’re actually thinking about the energy that just leaves a given cc of Sun vs. that leaves a given cc of person per second, then even with the Sun at constant temperature, since a 1 cm cube of Sun has the same surface area as the corresponding cube of person, the cc of Sun is brighter by the 4th power of the ratio of the temperatures. So, clearly you’re just thinking about the energy that is leaving the whole system, averaged over all the cubic centimeters in the system, as opposed to the energy that leaves each individual cc (per unit time).

    In this case, if you’re measuring the rate at which energy is leaving the system (which is the typical measure of brightness), then the interior temperature of the Sun doesn’t change things at all. All that matters is the surface temperature, which gives rise to the luminosity of 4*10^33 erg/s, which you already calculated. Whether the Sun is really hot or really cold in the center is irrelevant to the computation of the overall luminosity of the Sun.

  16. Gebraden Kip

    Honestly, why would anyone use such obscure units as ergs and dynes? It’s not like those are units that are used in common speech like the pound or the mile, or am I wrong? Then what’s wrong with using SI units such as joules and newtons?

  17. Michel

    A long, long time ago someone told me a human gives of about the same amount of heat as 100 watt bulb.

  18. Andrew Perrin

    Phil, in the first section, your equation for the luminosity should probably use the *net* luminosity, since the surroundings irradiate the human as well as vice versa. (i.e. if the surroundings are warmer than the human, the heat flux should be *toward* the human, and the human would appear blue on a thermal imaging camera, or whatever color indicates “heat sink”) The equation would be:

    Luminosity = sigma * area * (T^4 – T_surroundings^4)

    For the sun it won’t matter since space is effectively 4K and therefore too cold to matter, but for the human it matters a lot.

  19. Dave

    (2) If, instead of looking at brightness per volume, we think about brightness per mass (define L’=L/M), then what’s the answer? For the Sun, it’s actually really easy to compute: Since the mass of the Sun is 2*10^33 g,

    Lsun’ = (4*10^33 erg/s) / (2*10^33 g) = 2 erg/s/g .

    For a person, if the mass is, say, 75 kg, then the L’ is,

    Lphil’ = (1.3*10^10 erg/s) / (7.5*10^4 g) = 1.7*10^5 erg/s/g .

    So, Phil’s L’ is 85,000 times bigger than the Sun’s.

    In general, let’s compare the L’ of two objects of different sizes, densities, and temperatures:

    (L1′/L2′) = L1/L2 / (M1/M2)

    And L = sigma T^4 A, where A is the surface area. And surface area goes as R^2, where R is “the linear size” of the object (strictly true only if the object is spherical, but hey let’s not quibble). So L is proportional to T^4*R^2.

    So,

    (L1′/L2′) = (T1/T2)^4 * (R1/R2)^2 / (M1/M2) ,

    and since M = D * V, where D is density, we can write

    (L1′/L2′) = (T1/T2)^4 * (R1/R2)^2 / {(D1/D2)*(V1/V2)}
    = (T1/T2)^4 * (R1/R2)^2 / {(D1/D2) * (R1/R2)^3} ,
    = (T1/T2)^4 / {(D1/D2) * (R1/R2)} .

    Comparing the Sun to a person: the Sun’s density is about the same as a person’s (1.4 g/cc for Sun vs. 1 g/cc for water) so we can write (approximately):

    (Lphil’/Lsun’) ~ (Tphil/Tsun)^4 / (Rphil/Rsun) = (Tphil/Tsun)^4 * (Rsun/Rphil) ,
    where the “~” symbol means “approximately. The Sun’s temperature is only a factor of 20 higher. 20^4 is a big number — 160,000 — so the first term gives a factor of (1/160,000). But this is swamped by the ratio of the size scales. If Rphil ~ 100 cm (really an overestimate given how we’re thinking about this, but still), Rsun/Rphil = (7*10^10cm/100cm) = 7*10^8, which is much bigger than 160,000.

    So, although the Sun is somewhat hotter, it is MUCH bigger and VERY optically thick.

    In fact, it is the extreme optical depth from the center of the Sun to the outside that leads to the fact that photons traveling from the center of the Sun to the outside can take thousands of years or more to reach the surface. This is why the super-hot, super-emissive core does not make the surface brighter than it does — the photons bang around for practically an eternity before finally emerging.

  20. So, I think what we’ve learned here is more important than busting/confirming some silly urban legend… we’ve basically proven that Yoda was right.

    “Luminous beings are we.”

    Seriously, sir, I loved the mental gymnastics on this one.

    Now, if you put the whole of humanity into one huge snuggly mass (not to be confused with a Snuggie) would THAT mass outshine the Sun? (The ratio of politicians and bloggers in that heap of humanity may produce enough hot air to give Sol a run for its money.)

  21. Dave

    @Mark (#4):
    You’re basically right. In general, if you leave an object out in the sunshine, its temperature will rise (or fall) until the energy in equals the energy out. This is the condition called “energy balance” or simply “equilibrium”. On a hot sunny day, the equilibrium temperature is a good deal hotter than body temperature, which is why an object like a car or like the asphalt gets hot enough to cook eggs.

    @Stefano (#14):
    This isn’t great, but here: http://ioannis.virtualcomposer2000.com/astronomy/Polytrope.html
    you can find plots of density vs. radius and of pressure vs. radius. Qualitatively, a plot of temperature vs. radius looks similar.

    #Gebraden (#16):
    Astronomers always use “cgs” units (centimeters, grams, seconds). As you probably know, the cgs units of energy and force are the erg and the dyne. It turns out that there are a few nice things about cgs units, including that Maxwell’s equations look really nice when written in cgs. Phil being an astronomer, ….

  22. Andy Latto

    You say “if you want to compare the energy emitted per square centimeter from any two objects, all that matters is their temperatures”. But you can change the surface area of an object by an arbitrary amount, just by making it “crinkly”. In the discussion above, you model the sun as a perfect sphere. But suppose the surface of the sun was covered with little bumps, like a golf ball. It would have the same volume, but perhaps double the surface area. At the same temperature, would it then emit twice as much energy? This seems very surprising.

  23. GreenTom

    Fun exercise, it’s always nice to see what the back-o-the-envelope can do. Can I give you a harder (and IMHO, more interesting) question? Is there any wavelength in which human civilization is brighter than the sun?

    Also, just a sanity check on your numbers:

    1.3×10^10 ergs = 0.31 kcal
    0.31 kcal * 86400 sec/day = 26784 kcal/day

    Which makes it seem like either you’re eating an awful lot or not conserving energy! Or is 26,784 calories/day just the amount you’d need to eat to matain body temperature while suspended in an absolute zero void? Funny thought, that we get 10x as much energy from blackbody radiation falling on our skin than we do from food.

  24. Sure are a lot of naked people in that post.

    If you’re soliciting opinions (and I’m sure you aren’t)…when there are naked people in your posts, I’d prefer skepchicks to skepdudes. But hey, that’s just me. It’s not my blog. Knock yourself out.

    By the way, that sure is a big telescope. What is that, 12, 14 inches?

    I feel faint.

  25. Thom

    “So let’s get to it. Are humans more energetic than the Sun?”

    What about the difference between the energy the Earth receives from the Sun, and the energy humans create/use (excluding of course, the solar energy we use)?

  26. @ Psyberdave:

    By the way, that sure is a big telescope. What is that, 12, 14 inches?

    Everyone knows it’s not the aperture that matters, it’s what you do with it.

  27. Denny

    Phil,

    All very interesting, but I read the original statement differently than you – I would have taken the question to be “does the amount of energy from the sun as impacted per square centimeter at the surface of the earth equal more energy than than emitted by a square centimeter of a human body.” To me that would be the more interesting question since I don’t think the answer is at all obvious (especially if you take lattitude into account)whereas the only real unknown regarding the comparison of the sq-cm of the suns surface compared to a sq-cm of human is just how much hotter.

  28. Dave

    @Andy (#22):
    It’s true that each square cm of an object radiates with a flux proportional to its temperature to the fourth. But this doesn’t mean that the total luminosity of the object goes as (sigma T^4 A) where A is the total area.

    A sphere is everywhere convex, so every photon that leaves the surface never hits the sphere again. But if you put in pockmarks like a golf ball has, then inside each little “crater” the surface is concave. Some photons leaving the surface then hit the other side of the crater, and therefore do not contribute to the apparent brightness. So the total luminosity is somewhat less than sigma T^4 A, because not all of A contributes to the luminosity.

  29. Steve

    Hi Phil. I know you used the rough estimate of a 177 cm by 50 cm by 15 cm rectangular solid to calculate your area, not your volume, but I’m going to use those numbers anyway to determine that you weigh 293 pounds.

    You appear slimmer in photographs. I suspect the use of Photoshop.

  30. Dave

    @Denny (#26):
    This isn’t quite true, but: the temperature of the land surface of the Earth more or less reflects the instantaneous radiative balance temperature. That means that the land temperature is such that the energy out (sigma times area times T^4) equals the energy in (the sunshine). Sometimes, the land temperature is hotter than a human body, indicating that the sunshine is greater than a human body’s emission. Other times, the land temperature is cooler than a person.

  31. jtradke

    Well, I’m just happy to learn that SOMEbody uses ergs besides crossword puzzle designers.

  32. Patrick

    @23

    That looks to be an Orion Deep Space Explorer 12.5 inch scope.

  33. Joakim Rosqvist

    Why is it not ok to divide power output with surface area?

    Sun’s output (number s from wikipedia) = 3.846e26 W /6.0877e18 m^2 = 63e6 W / m^2

    My output (roughly) = 100 W / 1.7 m^2 = 59 W / m^2

    That’d make the sun a million times more powerful per surface area. Phil got 121000. What is wrong with my calculation?

  34. Aerimus

    You’re a lot brighter than the sun? That would explain why I nearly went blind after seeing the above pic…

  35. Denny

    @Dave(#29)
    I’m not sure what you’re claiming isn’t quite true. However, taking the time to do some really rough seat-o-the-pants calculations and taking Phil’s estimate for human luminosity I think one can claim that the sun is hotter but not by that much.

    The quick version: taking the area of the sphere at the earths orbit as 73×10^26 sq-cm and using 4×10^33 ergs/sec I come up with about 5.5×10^6 ergs/sec/sq-cm as the suns radiative output at the earths orbit. Taking Phil’s human luminosity and dividing by his surface estimate of 25000 sq-cm I get an energy output of a human as 5.5×10^5 ergs/sec/sq-cm. So the sun’s output as received here is about 10 times the radiative output of a human.

    I don’t do this kind of thing for a living so if I’ve screwed up the calculations or made some invalid assumptions feel free to let me know.

  36. amphiox

    So, how many humans would it take to make a sphere of humans with a total energy output equal to the sun? And would such ball be massive enough to collapse on its own gravity, killing all the humans in a painful, gruesome fashion, and thus ceasing to emit any more metabolic energy?

    Oh, and would this be the reason why the Matrix machines decided to use human batteries instead of fusion power? (Or just send up satellites above the scorched sky layer to collect solar power?)

  37. timur

    Dave #15 :

    I would think that the surface temperature 6000 Kelvin is calculated from fitting the black body curve to Sun’s spectrum. If not, I think this fitted temperature will give more accurate estimation on Sun’s energy output.

  38. The Other Ian

    Latin nit: The plural of canis is canes, not canises.

  39. Gavin Flower

    Hmm…

    A couple of things:

    (1) How about using standard MKS units, in this case Joules and metres rather than ergs and centimetres (at least you were not using Imperial units – feet and pounds etc.), for the Stefan-Boltzmann constant. MKS units are a lot easier to deal with.

    (2) What is the relative heat perceived by your hand from the Sun versus the heat from Mrs. BA’s back at night? Well to be fair to the Sun, we should compare the heat your hand feels from Mrs. BA’s to that felt when you expose your hand to the Sun in daylight at noon. These are the burning questions!!! Besides, snuggling up to the Sun is fatal, at a far greater distance than you’ve ever been to Mrs. BA.

    Happy Mid Winter Feast days for you in the Northern Hemisphere.

  40. @ Jokim: I suppose the error is in considering that human body could stay at 37°C/100F/310K with only 100W radiating against cosmic background. In fact, you need 100W to stay at 310K when surrounded by walls at about 293K. It’s the (T^4-Tsurroundig^4) that was mentioned in former comments.

    @ Phil: come on, just to save some pi and e you can’t keep on with… erg! ;-)

  41. Harry Tuttle

    Of course, some humans are hotter than others. But I’m not sure I can do the math for that.

    It’s easy if you have some ships lying around waiting to be launched. You’ll need about a thousand of them to measure a real hottie, but luckily most people are a mere few hundred millihelens of hotness. Many much less.

    “With a surfeit of beats, I’m unlikely to run out;
    plus, I’m so bright it’s like redundant to have the sun out.”

    - M.C. Frontalot

  42. rmu

    Somehow you seem to neglect energy output of the human body that is transported via convection instead of radiation. Usually that is a significant factor.

    Perhaps more interesting would be a comparison of how much energy the fusion process in the sun generates per cubic meter (IIRC a surprisingly low number, less than 500W per m³) with the amount of heat generated in the human body (between 100W and 1kW, depending on activity).

  43. Joe Meils

    Isn’t that bad for the optics when you do that?

  44. JackC

    What? No one has said “Assume a spherical Phil….”

    JC

  45. Torbjörn Larsson, OM

    For the sun it won’t matter since space is effectively 4K and therefore too cold to matter, but for the human it matters a lot.

    Yes. As Intel likes to tell you, IIRC current microprocessors output a lot more power/area than the Sun. The difference is of course conducted, and later convected, away. Same goes for us, which aren’t all that much colder than the microprocessors in relation to the Sun. (But in relation to the surrounding environment, sure.)

    I have never checked that, since convection seems troublesome to estimate. Maybe Phil needs to do it though, considering his claims.

  46. Not to derail this fascinating thread, but I have a basic sun question (while we’re on the topic). After reading Phil’s book (and others!), I’ve never seen it adequately explained why the sun doesn’t simply fuse all available hydrogen nuclei at once. In other words, there is a significant mass of hydrogen that is simply “hanging out” with all the other hydrogen that’s fusing all around it, and yet it may not do the same for another 5 billion years? I find it questionable that such available fuel can stick around for so long without fusing.

    I have this image in my mind of a room full of explosives and someone tossing a grenade into it, and nothing else exploding. I’m sure I’m missing some basic process here or have failed to acknowledge the role the sheer mass/size of the sun plays in such processes. Any takers?

  47. Petrolonfire

    A hot topic Phil! Would having a fever make a difference? ;-)

    On a serious note – if we’re talking internal structure and density aren’t people also denser in some parts than others … ? ;-)

    … I was meaning bones actually but … :-)

    I thought the reason we float or sink was related to bouyancy as much as density although thinkng about it I’m not sure these aren’t the same thing?

  48. Mark Doyle

    Phil, I’ve never left a comment here before, but I’ve been reading your blog forever, and I just wanted to say it’s always interesting, and always entertaining. You don’t treat your readers like idiots, and you aren’t afraid to throw a little complex math at us. Love it. Keep it up.

  49. «bønez_brigade»

    I like how you used a 15cm estimate for your depth, Phil.

  50. Pete

    Is that a Newtonian in your pocket or are you just glad to see us?

  51. coolstar

    @Denny I like your formulation of this problem better as it seems to be much more informative. The easy way to do your problem is: solar constant = approx 1400 watts/m^2 at the top of the atmosphere, “human luminosity” = approx. 100 watts. So, assuming a spherical BA with radius 0.5 m , we get a “BA flux” of about 32 watts/m^2 or about a factor of 40 difference. “assuming a spherical BA” of 0.5 m radius apparently overestimates total surface area by about 70% (according to standard sources, but that’s not awful for a back-of-the-envelope calculation). As I think someone mentioned, this is actually a much better approximation than the one Phil used, as your body isn’t radiating into cold, dark interstellar space at 2.7 K. (Phil’s calculation of human luminosity means you’d have to eat about 10,000 calories per day. Hey, maybe he does, in which case my original estimate of his area wasn’t so far off…..)
    If you use the corrected value with a better average human (male) surface area, you get a ratio of about 25 between the solar flux at earth and a human flux. To change this to a ratio at the surface of the sun, you just need to multiply by the square of the ratio of the earth’s orbit to that of the sun’s radius, which is about 200, so that squared is 40,000 and the ratio of surface fluxes is about one million.
    Moral of the story: don’t let Phil calculate your share of the dinner check.

    (Phil could have found his error the same way I did, by using the Stefan-Boltzman equation to get a human luminosity of about 500 watts, an obviously wrong result. Back-of-the-envelope calcs are fun, but it’s always nice to do a “sanity” check.)

  52. nomuse

    Hrm. And how much radiant heat can we get off a human being? Solar constant 1.4 kW/m^2. I’ve seen figures on how much weaker the sunlight is on the ground, in the northern hemisphere, in winter, but I’m not looking them up now. Call it .8 kW/m^2.

    Human body about 100 W. Err…even if you throw out isotropy and put 100% of that energy towards your receiver, even the winter sun is easily beating the human radiator. I’m a pale European myself…just for a laugh, assume my (naked) albedo of .45, and use Phil’s cross-section for reception area…..that’s .85 square meters, x .45, or 306 watts. Assuming my albedo is zero in infrared, it would take three humans in close proximity to beat the warmth of the winter sun (and that’s still assuming they radiate 100% in my direction).

  53. coolstar, I’m not sure why you feel the need to insult me so much, though I have warned you about it before. I again suggest you familiarize yourself with my rules, linked in the sidebar. Were you to attack another commenter they way you do me I’d start marking your comments as spam.

  54. Flying sardines

    Are humans brighter than the Sun?

    Well it depends what you mean by “brighter” .. As far as we know, the Sun is not bright at all – it can’t even talk let alone work out equations like the BA has here! ;-)

    Oh & just on :

    It’s been brutally cold here in Boulder, causing much snuggling at night, both between humans and with Canises Major and Minor.

    What about Dinger? Your cat doesn’t snuggle too? Mine does. :-)

  55. Spectroscope

    What about stars with other spectral types? Can we apply these equations to them too & work out how we’d compare against say one of the brightest stars in our Galaxy Eta Carinae with 5 million times the solar luminosity? Or one of the faintest MOA-2007-BLG-192L (see : http://blogs.discovermagazine.com/badastronomy/2008/06/02/welcome-our-tiny-family/ ) which is less than 1,000 times as bright?

  56. CNH

    Ergs … I vaguely remember them from my childhood – forty years ago …

    And, for a previous comment as to why not all the hydrogen in the sun is fusing – the density, or the pressure, in the outer parts of the Sun is not sufficient.

  57. William Roeder

    He talks about the center being millions of degrees and the total volume of the sun.
    This is wrong. The center does not radiate in space so it is irrelevant, likewise everything below the photosphere.
    Therefor the volume calculation should be surface area times one cm.

  58. River Sol

    I agree with RMU. Conducted heat should not be discounted out of hand. http://www.nanomedicine.com/NMIIA/15.3.8.htm calculates the conducted body heat at 5-9 W/m2-K or about 100W/m^2 for a 20K body/environment difference (this would obviously depend on the medium a body would be immersed in) . http://www.xomba.com/solar_constant_and_the_amount_of_energy_produced_by_the_sun gives the solar luminosity per area as 63 x 10^6W/m^2 (using Phils 129,000 radiant heat ratio, that gives the value of human luminosity per area as 488 W/m^2) Thus the total human heat output (assuming ~ 0 convective heat) as 588W/m^2, quite less than 63 x 10^6 W/m^2 at the surface of the sun…or even the radiant heat that reaches Earth of 1366 W/m^2 (per http://en.wikipedia.org/wiki/Solar_variation).

    Still that’s quite interesting that the Sun’s heat that reaches our skin here on Earth is on the order of 3 x that which our bodies can put out through radiant and conductive heat mechanisms.

  59. River Sol: Interesting, yes, but maybe not surprising. When I go outside on a sunny day, my skin warms up considerably, even on a cold day when I can radiate/conduct heat away rapidly.

  60. Gary Ansorge

    46. artbot

    Good question. Fusion only occurs in the suns core, which, by most estimates, is at most a few hundred meters in diameter. Outside the core region, fusion rates fall exponentially. By the time we get a few thousand km away from the core(and Sol is about 1.2 million km in diameter), there isn’t sufficient temp/density to initiate fusion, so most of the mass of the sun isn’t subject to fusion. Now, if we just added a few negative muons to the mix, old Sol would likely go boom,,,(negative muons reduce the temp/density required for fusion to occur, by a couple of orders of magnitude).

    GAry 7

  61. amphiox

    “You’ll need about a thousand of them to measure a real hottie, but luckily most people are a mere few hundred millihelens of hotness.”

    But don’t forget to factor in the highly variable Agamemnon-Menelaus coefficient that impacts the launchibility of the ships.

  62. Maybe I’m slicing hairs (and maybe I’ve had one cocktail too many – it’s almost 1 a.m. and I rarely drink), but I’m curious as to the effect of distance on this comparison – I can say from personal experience that during this time of year, even in Oklahoma, the person next to me seems a LOT warmer than the winter sun – how can we take angle and distance into account in these calculations?

    Dr. Plait, thank you for all the great Bad Astronomy you exposed for us in 2009, and here’s to a 2010 better than any of us could ever imagine.

  63. “I am your density”. Nice Back to the Future reference.

  64. MadScientist

    cm^2 is a surface though, not a volume. Even if we were to consider insolation vs. what humans radiate, the sun still wins (except after sunset). Perhaps a human on an exercise bike (or a real bike) can generate enough power to compare with the sun shining on an equivalent area, but a human doing typical daily work (except perhaps for the athletes for whom typical work is their exercise regime) certainly doesn’t generate anywhere near the amount of power per square meter.

  65. hhEb09'1

    The real misconception here is dividing by volume. The interior mass is not contributing to the exterior luminosity (when you snuggle you exchange heat with those around you, with no net loss), so dividing by the total volume is meaningless in this case.

    Since the volume increases by the cube of the radius, large objects would be at a strong disadvantage in such calculations. We might just as well divide by IQ. *Then* the sun would win. :)

  66. I am shocked — SHOCKED, I tell you! — that the URL for this page still says “then” instead of “than”.

  67. tonyo

    “MAD MATH SKILLZ” yo!

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