Wednesday, September 23, 2009

Light Bulbs and the Solar Energy Production

As of this September, regulations in the European Union ban the the manufacture and import of 100 Watt incandescent light bulbs, as a measure to cut down energy consumption. While this has created a bit of a fuss and lead people to hoard traditional light bulbs, I actually do not remember the last time when I had used a 100 Watt light bulb. I probably won't miss it – unless for a very nice comparison for the energy production of the Sun.

The Earth is at distance r = 150 million km = 1.5 × 1011 m from the Sun. The incoming total electromagnetic energy flux from the Sun at the Earth per unit area, the so-called solar constant, is C = 1360 W/m² = 1.36 × 103 W/m². Assuming that the energy flux from the Sun is the same in all directions, this means that the energy output per second of the Sun, called luminosity by astronomers, is L = 4 π r² × C = 3.85 × 1026 W. This corresponds, by the way, to the mass equivalent of roughly 5 million metric tons per second: dm/dt = L/c² = 4.27 × 109 kg/s. The Sun has a radius of R = 7 × 108 m. If we naively assume that energy production is the same throughout the whole volume on the Sun, the power density of the solar energy production would amount to ε = L/(4 π/3 R³) = 0.268 W/m³ This is a remarkably tiny number! Of course, energy production in the Sun happens only in the central part, where temperature and density are high enough to sustain nuclear fusion reactions. This central part extends to roughly 10 percent of the solar radius, so that we can estimate the energy production in the core to about ε ≅ 300 W/m³ This is the energy output of three 100 Watt bulbs per cubic metre!

Actually, this back-of-the envelope estimate is not that bad at all. Energy production in the Sun by nuclear reactions is now very well understood, in particular since the "Solar Neutrino Puzzle" has been solved. This knowledge about the Sun's inner parts is encoded in what is called the "Standard Solar Models".

A lot of information and papers on solar models are available from the web site of the late John Bahcall, and from this long list of models, I picked the data set for the model BP2004, which gives all kinds of physical quantities as a function of radial distance from the centre of the Sun.

Energy production can be inferred from the luminosity as a function of radius – there is difference between these quantities when heat is absorbed or released, but this difference is negligible for the current steady state of the Sun's interior. This yields the following figure:

Energy production in the Sun's centre drops to zero beyond roughly one quarter of the solar radius. And in the inner core, it is nearly 300 Watt per cubic metre.

Of course, beyond the energy balance, it's quite unphysical to imagine the solar interior as a vacuum lit by light bulbs. Due to the gravitational pull, density, pressure and temperature are enormous, and beyond anything we can imagine from everyday experience. Here are radial profiles of density, pressure, and temperature of the Sun. Data are taken again from solar model BP2004. Note that the plots now have a logarithmic scale. For better comparison with everyday numbers, I have added the density of water, atmospheric pressure multiplied by a factor of 1 million, and the melting point of iron, multiplied by 100.

There is, of course, another difference between the light from the Sun and a 100 Watt light bulb – that's the spectrum of the light. An incandescent light bulb is a quite inefficient light source, as most of the energy is radiated in the infrared. The solar spectrum, instead, peaks in the visible range.

But, leaving aside the huge differences in density, temperature and ambient pressure, and the different spectra, here is a nice comparison:

My small kitchen has a volume of about 25 cubic metre. So, I should light it with 75 bulbs of 100 Watt each to "simulate" the solar interior. This would be very bright, and blow the fuses, but it is a quantity conveniently to imagine, compared to the huge numbers we usually deal with in astronomy!

Here is another way to arrive at the order of magnitude of "100 W light bulbs per cubic metre" for the solar energy production – thanks to Bee for insisting on this estimate:

The solar disk in the sky has a diameter of half a degree. The incandescent inner part of a 100 Watt light bulb, with a diameter of about 2.5 cm, appears under an angle of half a degree in a distance of about 3 metres. A spherical cluster of 100 Watt bulbs at a distance r appearing under the same angle and containing (r/ 3 m)² bulbs will produce roughly the same apparent luminosity as the single bulb at a distance of 3 metres. At the distance of the Sun, such a cluster should contain 0.25 × 10²² light bulbs. Actually, the luminosity of the Sun is about 1600 times higher than that - meaning that the Sun is about 1000 times brighter than than a 100 Watt light bulb in a distance of three metres. This seems quite reasonable indeed!


  1. Thanks! I learn a lot from reading this blog ;-)

  2. This is an end of an era and I don't like it at all. As I 'm getting older I realize that I don't want these small changes and I need to keep things around me the way they are. These pretty light bulbs is part of my world and I'll pile up a lot of these babies. I don't care about the EU firmans.

    Moreover what will be now the symbol of "I have an idea" in Europe?:-)

  3. Frankly, I never liked these bulbs. But I don't like the new ones either. I would really like to see a better selection of artificial light sources that have a spectrum more similar to sunlight.

  4. And how you gonna tell apart day from night:-)

  5. By the traffic ;-)

    But I was more thinking of lighting up the Winters than the nights.

  6. I have some lights with sunlight spectrum btw. But they were hard to get and imho unreasonably expensive. They are quite good, but I could need some more of those...

  7. Interesting Stefan, but your last rough calculation is insufficient IMO. As you note earlier, the spectrum from the sun (ca 5700 K) is different from that of a tungsten bulb (ca 3000 K), and much of the bulb spectrum is wasted for our perception of luminosity. (However, some atmospheric absorption yellows sunlight, even at noon, else the sun would look pastel bluish-green as it does in space, so I hear. Peak for 5700 K is in the blue-green 508 nm, not "yellow" as the star lore goes. Calculate at

    So you can't compare "the look" of so many bulbs versus the sun, of the same wattage. The bulbs are less apparently bright than the sun given a wattage, but perhaps just as annoying since we can still feel the energy. (Good sunglasses also absorb IR for greater comfort and retinal protection.)

    BTW, maybe incandescents should be taxed rather than banned. Sometimes you want to turn on and off a lot, sometimes you want the heat or the fuller spectrum, etc.

    PS: when I tried to post under my Google account, it kept refusing me. "There are errors on this form" - I checked, I submitted correctly.

  8. BTW Bee, some of the lights with "sunlight spectrum" really reallocate their luminosity in various ways, and others (like "plant bulbs") just absorb more red and orange to make up for the excess emission from 3000 K filaments. The latter are rather wasteful of course, but can look better and keep the heat more localized. They still don't emit much violet or near UV, so not "health ray" in effect.

  9. Hi Neil,

    but your last rough calculation is insufficient IMO. As you note earlier, the spectrum from the sun (ca 5700 K) is different from that of a tungsten bulb (ca 3000 K), and much of the bulb spectrum is wasted for our perception of luminosity.

    That is correct. The main point of the last estimate was to check that the comparison with 100 Watt light bulbs is not completely off. Given the huge exponents we have to deal with for the Sun, this should not be taken for granted...

    Actually, when we really want to compare visual brightness, we should take into account the luminous efficacies of the Sun and incandescent lamps, which are different by roughly a factor of 5, and compare the corresponding luminous fluxes.

    Taking data from this Wikipedia page, we find that the luminous flux is about 1.4e3 Lumen for the 100 Watt bulb, and 3.1e28 Lumen for the Sun.

    Thus, the luminous flux of the Sun would have to be 3.5e24 Lumen to appear as bright as the light bulb at a distance of 3 metres - where actually, it is about 1000 times brighter.

    It seems the estimate is astonishingly robust ;-)

    Cheers, Stefan

  10. RE Bee comment
    I would really like to see a better selection of artificial light sources that have a spectrum more similar to sunlight
    -- good point,
    it's interesting that on the black body radiation aspect and color rendition index, incandescents are supposed to be rather like the sun,
    but of course at a relatively low temperature they are red shifted

    with xenon gas can supposedly be
    close to sunlight in both colour temperature and rendition...

    but no doubt they are being banned too ;-)

  11. Actually, Panta Rei, some incandescent bulbs are efficient enough to pass the ban, for example see But that link of yours is full of wise counsel IMHO, and they think (like me) that a tax on incandescents is perhaps the best realistic solution:

  12. Please no discussion about the ban of incandescent light bulbs here... BTW, not all types of incandescent lamps are concerned - the halogen filled ones are exempt, for example.

  13. Hi Stefan,

    “Actually, this back-of-the envelope estimate is not that bad at all. “

    Thanks for the excellent post and as I’ve said before you just have to tell me where you buy your envelopes :-) Seriously though, I enjoy pieces like this one, where things which we can get our heads around, such as a light bulb, are compared in some common physical aspect with those much harder to grasp as our sun.

    Something that should be reminded is how the spectra we see in is a relative thing, since our optical range is more indicative of the peak output of our particular star, rather than their being anything unique about this part of the spectra. As for instance, the Edison bulb would be considered much more efficient for creatures having a red sun. So we could say we shouldn’t fault the technology as much as our own physical limitations. It therefore might be more practical and economical to develop eyewear, which would have more of the spectra become visible and interpretive. So instead of switching on any light at all, we instead simply dawn the appropriate glasses.

    Of course many things which challenge us today could be dealt with in such ways, like grid lock and pollution, mitigated by our over use of cars, largely to have us reach destinations that we don’t need to travel to; since often much of the work could be done at home, simply by taking greater advantage of current technology. Perhaps we should be more mindful of what a light bulb represents to many, rather than for what it does, which is our species having an innate ability for coming up with better ideas and solutions.



  14. Incandescents
    with xenon gas can supposedly be
    close to sunlight in both colour temperature and rendition...

    There were incandescents with
    Krypton gas fill in Germany,
    but they are sold no longer
    (it is about 20 years since I saw one)
    But nevertheless, the atomic wheight of the fill gas in
    incandescents rules whether
    one can heat the filament to
    some hundred Kelvin more or not.
    Xenon is much too expensive to
    be used in such a way.
    I guess You mix up Xenon high
    pressure arc lamps, and
    incandescents, dont You?
    The latter are in fact very close to daylight.

  15. Panta: As you doesn't seem to have been around here before, please read our comment policy. We strongly disapprove of self-advertisements, especially if they are off-topic. Any such link counts as spam, and since blogger doesn't allow for editing of comments, your whole comment will vanish into digital nirvana. If you want to talk about something else than the content of our post, please do it elsewhere. If you want to stay, play by the rules.


  16. Not quite Bee. His deleted comment is alive and well and lives in my Google Reader:-)

  17. Yes Bee,
    I can understand that light bulb ban comments are not much about the post you made, though others did make them.

    You might allow a correction,
    with an EU link,
    since Stefan said halogens will remain allowed in the comment he put to what I said:

    All replacement halogens are to be banned too (with the most popular frosted type banned straight way)
    as anyone can see on the EU Commission ban specification, scroll to bottom:

  18. ah, the link url was too long!


    changing topic,
    I have had the same problem as Neil B, in posting using Google account, perhaps something you might want to look at...

  19. Hi Giotis:

    Yeah, but the point isn't that we don't want anybody to read whatever opinion people put forward, but that we don't want to support their website with having a link here. You see, we too know how Google works. But Panta seems to be a smart guy and understood quickly :-)


  20. Thank you Bee
    maybe that makes me one of those smartyPantas ;-)

  21. As usual, thanks for your insightful post Stefan. However I must protest in that you categorized this as "Astrophysics, Physics and Useless Knowledge" is certainly not useless knowledge! The Solar "Constant" (it's not really constant, but nearly so) is very important for computing the solar radiation perturbations on geostationary satellites (and other spacecraft of course).

    BTW, I think that 75, 100 W light bulbs in your kitchen will not only blow a fuse but cook not only whatever food is in your kitchen (without turning on the stove) as well as the cook him/herself.


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