# Dimensional analysis to estimate order of magnitude of quantities

In the Coursera course From the Big Bang to Dark Energy on several occasions dimensional analysis was used to estimate the scale of quantities. This almost seems like a contradiction in terms to me, since you cannot strictly infer anything about dimensionless factors. Nevertheless it seems to work if you know some rules of thumb.

The general procedure seemed to be to take the speed of light, Planck's constant and Boltzmann's constant, and then use dimensional analysis to relate these to the known quantities to get a quantity of the desired dimension.

Examples included

• wavelength of cosmic microwave background from temperature
• kinetic energy of quarks from proton radius
• energy of W and Z bosons from length scale of weak force
• radius of hydrogen atom from electron mass via electron wavelength

In the last case the dimensionless fine structure constant has to be thrown in.

I understand that when a force is involved, you will have to include some proportionality constant that cannot be obtained from dimensional analysis alone (like the fine structure constant above), but maybe that often is all that is needed.

Is there just a handful of rules of thumb, using which more or less reliable estimates can be made?

Apart from that, I would also be very interested in general comments on this theme.

Edit

As pointed out by QMechanic, a related (and very interesting) thread in this forum is found here In dimensional analysis, why the dimensionless constant is usually of order 1?. There certainly are similarities between the questions, but I am more specifically asking how to use dimensional analysis to obtain such estimates, and didn't find concrete answers in that thread.

I find it fascinating that it can be done, but wouldn't dare to apply it myself. Rather than asking why the constant is usually of order 1, maybe my question is when the constant is of order 1, or more generally, how to estimate its value.

• Possible duplicate: physics.stackexchange.com/q/13441/2451 – Qmechanic Oct 5 '13 at 23:21
• If I scale my model solar system's radius to the AU (i.e., $\xi=r/AU$, then when my output says $\xi=12.3456$, I know it means it is 12.3456 AU away (between Saturn & Uranus) $\Rightarrow$ inference from dimensionless quantity. – Kyle Kanos Oct 6 '13 at 2:29
• @Kyle: that is not what I meant; in your example you don't infer a value by using dimensional analysis, you just get a value out that you had put in yourself. For example, for the wavelength of the cosmic microwave background, given that the temperature is 2.7 K, we get and estimate for the wavelength: $\lambda \sim {hc\over k_BT} \approx 5 mm$ – doetoe Oct 6 '13 at 6:21