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As the title suggests, I'm curious to know, approximately how many photons are emitted in a single lightning strike?

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  • $\begingroup$ Related: doi.org/10.1029/JC082i031p04967. PS sorry that it leads to a paywall but there is no legal way of displaying the document publically. $\endgroup$ – user79161 Jun 25 at 21:19
  • $\begingroup$ Counting photons runs into trouble at the very low-frequency end of the spectrum. Any reasonable attempt to count the softest photons finds that there are essentially an infinite number of extremely low-energy quanta emitted in any electrodynamic process. $\endgroup$ – Buzz Jun 26 at 0:14
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According to Could We Harness Lightning as an Energy Source?:

An average bolt of lightning, striking from cloud to ground, contains roughly one billion ($1,000,000,000$) joules of energy.

According to Visible light:

Red photons of light carry about $1.8$ electron volts (eV) of energy, while each blue photon transmits about $3.1$ eV.

So let's take an average photon energy of $2.5 \text{ eV}$.
Assuming all the energy of the lightning is converted to visible light, we can calculate the number of photons.

$$ N = \frac{10^9 \text{ Joule}}{2.5 \text{ eV}} = \frac{10^9 \text{ Joule}}{2.5 \cdot 1.6 \cdot 10^{-19} \text{ Joule}} = 2.5 \cdot 10^{27}$$

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  • $\begingroup$ I have seen lightning strikes where this is not even an approximate figure. An approximate figure is what was asked for. And to get an average figure,you would have to accurately measure a hell of a lot of lightning strikes,then average them. I doubt whether this has been done. $\endgroup$ – Michael Walsby Jun 25 at 22:28
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    $\begingroup$ @MichaelWalsby I think you are taking things way too seriously here $\endgroup$ – Aaron Stevens Jun 25 at 22:58
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    $\begingroup$ All of the energy is definitely not converted to light. $\endgroup$ – G. Smith Jun 26 at 0:15
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    $\begingroup$ When Bertrand Russel said: "Mathematics may be defined as the subject in which we never know what we're talking about, nor whether what we are saying is true," he must have had your figures for lightning strikes in mind. John von Neumann, American-Hungarian mathematician, said: "In mathematics you don't understand things, you just get used to them." Your figures are a confidence trick,and will fool naïve young students into thinking they represent an approximate figure for the luminosity of an average lightning strike when they don't. Strikes vary wildly in length,luminosity & side branches. $\endgroup$ – Michael Walsby Jun 26 at 8:01
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    $\begingroup$ @MichaelWalsby most things in the natural world which vary wildly still do so around an average. I submit that the fact that no lightning strikes have yet annihilated the planet and that lightning by definition dissipates a positive quantity of energy means that the probability distribution of lightning strike energies has finite support and therefore a population mean strike energy does exist. A student who sees an average, described as an average, and believes its existence to mean that no variance exists must be very naïve and very young indeed. $\endgroup$ – Will Jun 26 at 13:47
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From How Big Is A Lightning Bolt? we see that a lightining bolt is “an inch wide and five miles long”, and at “50,000 °F”. So in useful units, approximately 3 cm diameter, 8 kilometer long, 28000 K hot.

If we consider that the heat is mostly due to black body radiation (for a perfect black body with an emissivity of $\epsilon = 1$), then the power will be given by the Stef-Boltzmann law:

$$P = A \epsilon \sigma T^4$$

The area, $A$ of the lightning bolt (a cylinder, of course) is given by

$$A= 2 \pi\times(3 \text{ cm})\times 8 \text{ km} \sim 1500 \text{ m}^2$$

And so,

$$5.2 \times 10^{13}\; \text { Watts of power.}$$

Lets say, it lasts 10 miliseconds, so its around $\sim 5 \times10^{11}$ J.

Now to calculate it the amount of photons properly, you would have to consider the spectrum of the black body radiation, and convert the energy density to number of photons using Planks law. I will just use the rule of thumb that "1 Watt of monochromatic visible light is approx $10^{18}$ photons per second".

And so, it would be around:

$$\sim 10^{29}\ \text{ photons.}$$

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  • $\begingroup$ Note that your estimate of the energy in a lightning strike is different from the partially-sourced estimate in the answer by Thomas Fritsch. The difference is nearly three orders of magnitude. $\endgroup$ – rob Jun 25 at 23:53
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    $\begingroup$ Rakov & Uman (in Lightning: Physics and Effects, Cambridge University Press, 2003) put the range of total energy in a lightning strike at between 1 and 10 GJ, with somewhere between 1% and 70% going to optical power. Your answer is bigger than this by a factor of ~100. Could you comment on this discrepancy? $\endgroup$ – Emilio Pisanty Jun 26 at 9:23
  • $\begingroup$ rob & Emilio: Both good comments, I'm currently at the lab... I will answer once I get home. $\endgroup$ – Gyromagnetic Jun 26 at 9:36
  • $\begingroup$ My guess for the discrepancy is that the 5*10¹³ Watts is the peak power, and it only lasts for a few microseconds. Another source (public.asu.edu/~gbadams/lightning/lightning.html) says that lightning takes 30 microseconds - though with their estimate for power of 1 terawatt, it seems like they are two orders of magnitude low on energy. $\endgroup$ – Nikhil Murali Jun 26 at 12:11
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This looks like it's a quantity that we don't have a particularly good grip on. Quoting from Rakov & Uman's Lightning: Physics and Effects (Cambridge University Press, 2003),

An approximate range for the electrostatic energy available for a lightning flash lowering a charge Q to ground can be evaluated by multiplying Q by the upper and lower limits for V, the magnitude of the potential difference between the lower boundary of the cloud charge source and ground. Assuming that Q = 20 C, thought to be typical for a cloud-to-ground flash, and using the range of V from 50 to 500 MV estimated earlier in this section, we find that each flash dissipates an energy of roughly 1 to 10 GJ (gigajoules). Note that a flash is typically composed of three to five strokes, and that the first stroke is usually a factor 2 to 3 larger (in terms of peak current and peak field) than a subsequent stroke, that is, any stroke other than the first. The above energy range inferred from electrostatic considerations is for all processes involved in a lightning discharge. Specifically, this energy estimate may well be dominated by the energy dissipated in the formation of numerous filamentary channels in the cloud that serve, in effect, to funnel cloud charges into the narrow channel to ground. Marshall and Stolzenburg (2001), from their balloon soundings of the electric field through thunderstorms and assumed minimum and maximum values of charge transfer, estimated the energy available for lightning to be in the range from 10 MJ to 10 GJ, the energy available for intracloud flashes (Chapter 9) being usually larger than that available for ground flashes. There is no consensus regarding the proportion in which the total return stroke energy is converted to thunder, hot air, light, and radio waves. According to Paxton et al. (1986), who used a gas dynamic model of the lightning return stroke (subsection 12.2.2), almost 70 percent of the total energy input to the channel is optically radiated from the channel. However, Few (1995), in his theory of thunder (subsection 11.3.2), assumes that essentially all the input energy is delivered to a shock wave 116 4. Downward negative lightning discharges to ground that subsequently is heard as thunder. As discussed in the first part of subsection 12.2.6, the total lightning energy input estimates of Paxton et al. (1986) and others, who employed gas dynamic models, differ from that of Few (1995) by two orders of magnitude or so.

Krider and Guo (1983) and Krider (1992) estimated that the radio-frequency power radiated by a subsequent return stroke at the time of the field peak, 3 to 5 GW, is about two orders of magnitude greater than the optical power radiated in the 0.4 to 1.1 µm range at the time of the field peak. The average zero-to-peak risetime of the subsequent stroke field waveforms was 2.8 µs. The total optical power, however, was found to dominate at later times, the peak optical power occurring about 60 µs after the electric field peak (because the risetime of the optical signal was determined by the geometrical growth of the return-stroke channel)

(emphasis added).

That said, it does look like the estimates of about 100 MJ to 10 GJ radiated as optical power capture the rough ballpark; assuming a photon energy of 2.5 eV gives a rough total of some $10^{25}$ to $10^{28}$ photons per lightning strike as a starting ballpark estimate.

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Now after the question has been clarified, that is, you are asking for the number of photons emitted by the lightning, and seeing other answers likewise as mine before, I will try to edit the answer (citing references).

Assume visible photons (like in other answers too, I assumed the range average to be 2 eV), there is 2 eV in a photon (your question does not state visible or non-visible, but it would be very important to distinguish).

Now as Emilio Pisanty's answer seems to be the most correct one, stating that:

  1. there is no consensus on what portion of the lightning's energy is converted into photons and other forms of energy dissipation

  2. it is very important to understand that you are not specifying whether you are talking about visible or non-visible photons

A lightning has between 5 and 10 billion Joules (between 21.2*10^27 and 62.4*10^27 eV).

https://en.wikipedia.org/wiki/Harvesting_lightning_energy

This makes between 10.6*10^27 and 31.2*10^27 photons.

Now I have a very interesting additional reference for the spectrum of the lightning's emitted photons:

http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1964ApJ...139..994W&data_type=PDF_HIGH&whole_paper=YES&type=PRINTER&filetype=.pdf

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    $\begingroup$ You're assuming, without any evidence or referenced sources, that the totality of the energy ends up as light, which is clearly incorrect. $\endgroup$ – Emilio Pisanty Jun 25 at 20:28
  • $\begingroup$ @EmilioPisanty correct, though, I needed some average wavelength, otherwise can't really give a calculation, I assumed visible wavelength. It could be done with other average wavelengths too. $\endgroup$ – Árpád Szendrei Jun 25 at 20:36
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    $\begingroup$ You're neglecting the (likely substantial) energy that ends up as acoustic energy as well as thermal energy in the atmosphere and the ground. Without a well-referenced claim that those are negligible (and indeed without a suitable reference for the ~GJ figure for the energy content) this answer is unusable. $\endgroup$ – Emilio Pisanty Jun 25 at 20:40
  • $\begingroup$ @EmilioPisanty am I wrong that he is asking about how many photons are inside the bolt when it is created? $\endgroup$ – Árpád Szendrei Jun 25 at 20:43
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    $\begingroup$ ... and just in case it wasn't obvious, saying "I know what follows is wrong, but I'm going to keep it anyway" is not a solution. We shouldn't have to be telling you any of this. $\endgroup$ – Emilio Pisanty Jun 25 at 21:05

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