Yes there is. Here is the formula. It can be off by as much as a stop because not all strobe zoom lenses are equal.
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where E is the watt-seconds stored in the capacitor, and EFL is the effective focal length of the lens. (For a full frame 35 mm camera.)
For engineering Geeks: This is the set of assumptions and how the above formula was calculated.
First off, the vast majority of the electrical energy goes into the xenon tube and is converted to electro-magnetic radiation. This tube approximates a black body radiator at about 5600-6500K. For our purposes, that is close enough to the solar spectrum to use as a starting point for comparison. We can guess that 80% of the stored energy in the strobe capacitor goes into the tube. The main reason that it is not much closer to 100% is that after discharge, there is still quite a bit of voltage (50-100V) left on the capacitor because below this voltage the tube just doesn't continue to "fire."
We know how many watts per square meter are required for an exposure because someone has been kind enough to note that sun on the snow or light sand is an exposure value of EV 16 on a sunny cloudless day. We don't want white sand to come out gray, so we are going to use an EV15. (If our flash hits a white wall, we want to expose the wall as white.) We also know that the sun illuminates the ground with about 1360 watts per square meter. f/1 at 1/32,000 second is an EV15 exposure at ISO100.
So we know that 1/32,000 seconds x 1360 watts/m2 or 0.0425 Joules/m2 is the correct illumination level for an average exposure at ISO100 at f/1. Joules is another way to say "Watt seconds." Please note that Watts/second or Watts per second is incorrect, just as kilowatts per hour is incorrect to measure your electrical usage of your home.
Let me again state, that we could go through the black body radiation curve to find out how much of the 0.0425 Joules/m2 is in the visible range, but we don't need to do that. We are comparing two nearly identical spectra - the xenon flash and "daylight."
The "energy" of a strobe is measured in watt seconds (NOT watts and not watts/second.) This energy is actually the energy stored in the capacitor. A lot of strobes charge the capacitor up to 350-400 volts. The capacitors run from 330 microFarads up to maybe 1500 microFarads. The formula for calculating the energy is
E = ½ CV2
where C is the capacitance in Farads and V is the voltage and E is the energy in Joules or Watt seconds.
For a small strobe, the voltage may be 350 volts and the capacitance 330 μF for an energy of about 20 Joules. A large handle-mount strobe might have a 1500 μF capacitor with a voltage of 400, for a stored energy of 120 Joules.
So the strobe fires, and nearly all of the energy is converted to electro-magnetic radiation, because almost all of the energy is dumped into the tube. We will look later at strobe circuits to understand why this is so.
Where does the energy go? Most strobes are designed to cover the photographic field "seen" by a modestly wide angle lens. The reflectors try to compromise between getting the most energy into this angle and covering it as uniformly as possible. The better companies will control the reflector carefully, while the "no brand" companies will simply buy "standard" reflectors and take what they get. "Standard" is usually good enough.
In the last couple of decades, strobe makers have tried to play a game of getting the highest guide number. The easiest way to achieve this is to put a lens on the strobe that is moveable to control the illumination angle. It turns out that you can really only control the angle in one dimension. In the long dimension of the tube, it would be too difficult to control the angular spread, so we will just assume that it spreads more than enough to fill a 35mm frame with a 35mm focal length lens.
Let's start by assuming that the reflector does a fair job, and manages to spread about half of the energy into a 35mm lens's field of view. (The rest goes outside the lens's field of view and is mostly not useful to the photograph, except for a small amount of secondary bounce.) Let's guess that 60% of this total energy is directed into the frame. At one meter, the illuminated scene for a 35mm lens is 1 m*36mm/35mm x 1 m*24mm/35mm. The 36mm by 24 mm is the sensor size for a full frame 35mm camera. If you multiply that out, you have to fill 0.705 sq meters (at 1 meter from the lens) with .0425 Joules per sq meter, or a total energy at 1 meter of 0.03 Joules. And if we get 60% of the energy into the frame with an 80% conversion to electromagnetic radiation, that means that 0.0625 Watt seconds gives us a Guide Number of 1 meter.
Now, the amount of area covered is proportional to the square of the distance from the camera/strobe. So the Guide Number is proportional to the square root of the number of watt seconds. So we now have a formula for an average reflector for a 35mm field of view:
GN in meters=sqrt(E)*4
where E is the energy stored in the capacitor, in Joules (watt-seconds).
So for our 0.0625 watt second strobe, we
should get a Guide Number of 1:
GN= sqrt(0.0625)*4= 0.25*4 = 1
So far so good. Now let's also assume that the manufacturer puts a Fresnel lens on the strobe to try to compensate for longer focal length lenses by not spreading the light so much. The first thing we notice is that the tube is small in one axis but large in the other (horizontal axis.) When we try to use a Fresnel lens, we can reduce the vertical spread angle, but not the horizontal. Therefore, we expect that ideally, the area over which the light is spread goes not with the inverse square of the focal length, but only with the inverse of the focal length.
We can express this situation in the following approximate formula:
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where E is the watt-seconds stored in the capacitor, and EFL is the effective focal length of the lens. (For a full frame 35 mm camera.)
I have graphed up this formula for some common values of E and EFL for ISO 100. In this graph, the equation given above has been converted to feet by dividing by 0.3048. I have plotted the lines for 105mm, 85mm, 50mm, 35mm, 24mm, and 18mm lenses. These are the popular focal lengths for full frame 35mm DSLRs. If you have a smaller focal plane array sensor, you can use the next larger focal length setting.
I have plotted some real commercially available strobe together with their guide number ratings. Most of the ratings follow the equation above to less than one-half stop. The worst match is the Canon 580EX Speedlite at 18mm focal length. Canon seems to have spread the light more than necessary, resulting in an unexpectedly low guide number, more than a stop below the expected value.
The flash units plotted include the Rolei, a vintage 1970 very small flash unit, the Prinz, a 1971 low cost strobe, the Metz M24, M36, M44, M50, and M58 line, the 1980s Focal (a strobe with a zoom lens sold by Kmart), the Canon 580EX and its clone the Yongnu YN565EX, and the Nissan Di866 and the Nissan 450GTE "potato masher."
To me, the zoom lens is of very little use, since 99% of the time I am going to bounce flash.
I have also plotted several commercially available photographic flash guns as symbols to see how they fare against the formula. Please note that this is a log-log scale.
The top dark blue line represents a very aggressively focused flash gun set for a 135 mm focal length lens. This is a modest telephoto lens. There are a few flashguns with this capability and some are shown as circles on the plot, with Guide Numbers from 45 to 55.
The gold curve is for 85 mm EFL lenses. The black curve is for 50 mm EFL lenses.
35 mm is the normal "design to" focal length, and most strobes unless otherwise specified are rated for a 35 mm focal length lens. Along this curve, you can see built-in flashes and flashes in disposable cameras are shown in the light yellow shaded area, having guide numbers from about 10 to 14.
The orange area represents low cost flash guns with guide numbers up to about 20 or so.
The pink area represents small flash guns, but possibly with a number of features such as adjustable field of view, and possibly IGBT (see below) circuitry.
The green area represents premium flash guns with guide numbers around 30 at 35mm and up to 60 at 125 mm focal length.
The light turquoise colored area is the area occupied by the "potato masher" flash gun. I happen to have a Nissin 4500GTE, so I plotted it on the graph.