An Brief Introduction to Order of Magnitude Calculations

Order of magnitude calculations are an extremely useful tool when you want to make rough approximations to arrive at answers very quickly. In astrophysics, many measurements are not very precise, so order of magnitude calculations can provide a crude answer that is actually fairly accurate, since the degree of precision is not very high in the first place. In other disciplines, order of magnitude calculations are usually much quicker than precise calculations and can give you a rough ballpark estimate of things.

I'm going to do a quick order of magnitude problem to highlight their incredible power. Here's the question:

"The eye must receive ~10 photons in order to send a signal to the brain that says, 'Yep, I see that.' If you are standing in an enormous, completely dark cave and just barely discern a light bulb at a distance of 1 kilometer, approximately what is the power output of the bulb (assume the bulb is emitting light isotropically)?"

The first thing I'm going to do is a draw a (bad) picture. When dealing with problems like these, it's important to be able to visualize the scenario, making the subsequent calculations much more intuitive.

Pictured above is the light bulb, you, and d, the distance between the two objects (1km). Notice the circle with the light bulb in the center - this will be come important for solving the problem.

Necessary for this problem is the understanding of the relationship between light intensity and distance. It's similar to Gauss' Law for Electricity, by the fact that intensity from a point source follows an inverse square-law relationship. In other words, if you "surround" a source of light with a shell, the intensity of light "captured" by the shell is the same, regardless of its shape or size. For a spherical shell, each patch of shell of equal area will receive the same intensity of light, due to the spherical symmetry of the system. Mathematically, we can express it like this:

Where I is the intensity of light, and dA is the differential piece of area of the enclosing shell. More intuitively speaking, if you imagine a light bulb emitting light, the "density" of light will decrease as you move away from the bulb, since the light will be spreading out further. However, the "total" amount of light will stay constant, so if you enclose the bulb in a surface, it will collect a constant amount of light no matter its shape or size.

If we assume the light bulb is emitting light at a constant rate, the above equations hold for power output also. Now, how do we solve the problem?

We're given that the eye must receive 10 photons to "detect" light. If we imagine the light bulb to be surrounded by an imaginary spherical shell radius d=1km, the eye will be a tiny patch of that surface. Because of the above equations, we know that the power output of the light bulb times the area of the shell is constant; thus, we are allowed to arbitrarily choose this 1km radius shell. Since a spherical shell is radially symmetric (if you rotate it in any direction, it will still look the same), each patch of the shell of equal area must receive the same amount of power. Thus, we can state the following:

This is simply a proportion: the area of your eye divided by the total area of the shell is equal to the light power received by your eye divided by the total power output of the light bulb. If we solve for this total power, we have solved the problem. Thus:

So, if we can find the values of the three quantities on the right side of the above equation, we're good. Let's do that:

The area of the shell is simple. The spherical shell has a radius of 1km, or 1000m, so the surface area is:

For now, I'm going to leave everything in terms of variables, and plug in these numbers are the end. For the area of the eye, I'm going to estimate it as 1 square centimer:

The power received by the eye is a bit tricker. Using the definition of power, we can relate it to energy and time like so:

The energy received by the eye is the energy of 10 photons, the minimum number of photons the eye must receive to detect light. The energy of a photon is equal to Planck's constant times the speed of light divided by its wavelength. Because this light is visible, we will estimate its wavelength to be around 500nm:

(where n = 10 )

The length of time per reception of energy can also be determined. Most movies have a refresh rate of about 30 frames per second, and video games can operate on 60 frames per second. I can personally tell the difference between these frame rates, so I'll assume the refresh rate of the eye to be about 60Hz. So, each "frame" that your eye receives in this cave will contain the energy of 10 photons, and there are 60 frames per second. Thus:

We're almost done now. Plugging everything back into the equation, we get:

Let's plug in numbers now:

What's our answer? About 0.00003 watts. Does that make sense? Well, it's a light bulb in a pitch-black cave that you can barely see, so perhaps. More important is how we arrived at this answer - using just some physics and a few estimations, we arrived at an answer very quickly.