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!

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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!

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