# What's the greatest range of orders of magnitude?

There's a famous claim along the lines of "40 dp of PI are sufficient to calculate the circumference of the Observable Universe to the width of a hydrogen atom"

I don't know the accuracy and detail of claim, but it prompted me to be curious ...

I assume that the claim (if it were true and accurately remembered) if equivalent to stating: "There are 40 orders of (decimal) magnitude difference between the diameter of the universe and the diamater of a hydrogen atom".

But that's not the biggest possible difference between interestingly measurable things, because the diameter of a hydrogen atom isn't the smallest length ... we could go smaller (protons, electrons, quarks, planck length)

I don't know astrophysics well enough to know whether there's anything that's interesting to describe bigger than the Observable Universe.

But, it seems that whilst considering length you can arrive at "The greatest possible difference in orders of magnitude".

But there are other things that can be measured. Time for example.

So question: What metric has the greatest range of orders of magnitude that are interesting to talk about? and how big is that range?

• You might not know astrophysics but have you done some research yourself to answer your own question? Feb 6, 2017 at 11:37
• I would say the ratio of the diameter of the visible universe to that of the planck length, about $10^{71}$
– user126422
Feb 6, 2017 at 15:58
• @AlbertAspect: How about the volume of the visible universe divided by the volume of a cube with each side equal to the planck length. Would that be about 10^(71*3)? Feb 6, 2017 at 16:53
• @James I believe he is asking for length only. In any case what is "interesting to talk about" is subjective.
– user126422
Feb 6, 2017 at 16:56
• @Bop_Bee that's the magnitude of error in our otherwise best theories compared to reality. Yes, it's possibly the most orders of magnitude error in any theory ever not instantly consigned to the mental trash-can of bad ideas. Feb 16, 2017 at 15:38

Your question is pretty vague, but I will restrict it to mean: what is the physical property with the largest range of measured values. This is still probably subjective, but it's a little more manageable and fun to think about anyway.

Here's one possibility: range of measured half-lives of radioactive isotopes (see wiki list). The shortest measured half-life (that of hydrogen-7) is order $10^{-23}$ seconds, and the longest (that of tellurium-128) is order $10^{31}$ seconds, so they span an amazing 54 orders of magnitude in all.

This is kind of ridiculous. It is more than the ratio between the size of a proton and the size of the observable universe, which are separated by a mere 41 orders of magnitude (maybe this is what your quote is supposed to say?), and it is about the difference between the Planck length and a light-year (!). It's fun to think about what the experimental challenges must be making measurements on both ends of that spectrum. Both ends (particularly the long-time end) are bounded by experimental ability, so this is not too far from being a list of the range of times over which we can measure anything. Naturally, that means it is subject to change. For example, we've been looking for proton decay for a long time, but all we can say right now is that the lifetime must be more than order $10^{39}$ seconds. If we ever find it this range will shoot at least another hundred million times larger.

• The decay time is exponentially related to the size of the potential barrier which has to be "tunnelled" in order for decay to happen. You can of course squash any range by taking logarithms, but in this case the logarithm has very significant physical meaning. Feb 16, 2017 at 15:32
• @nigel222 Yes, I drew out this point explicitly in another recent question: physics.stackexchange.com/questions/312406/… . Similar remarks apply to conductivity, which in many insulators will scale something like $e^{-L}$ where L is the macroscopic size of the system, and can be thought of (in a tight-binding model) as electrons tunnelling from site to site. It's a safe bet that most physical quantities that compete with these will also have a similar exponential underlying dependence on some parameter. Feb 16, 2017 at 16:23
• and at the extreme limit there is the lifespan of a black hole, which will eventually evaporate by emitting Hawking radiation. Well, it will if the theory is right. It's un-observable, for at least two reasons I can think of. Feb 16, 2017 at 16:29

Resistivity has a quite impressive range--for example, the resistivity of teflon is about $10^{30}$ times higher than the resistivity of copper.

So I think "resistivity of different materials" might be the winner, or at least a contender, for most orders of magnitude ratio of quantities that can and often do come up naturally in everyday life.

• You're underselling yourself. You've omitted superconductors, which have resistance not measurably higher than zero in that an electric current continues "forever", or at least until the liquid helium is delivered too late! Feb 16, 2017 at 15:45
• For any quantity that can be either zero or nonzero, it spans infinity different orders of magnitude. That's kinda cheating. So I'll say that "nonzero resistivities" is my non-cheating answer. Feb 16, 2017 at 17:04

Measured bulk baryonic densities vary by around 45 orders of magnitude - from around $10^{18}$ kg/m$^3$ in neutron stars to $4\times 10^{-28}$ kg/m$^3$ for the universe as a whole.

Observed energy of a single particle is interesting, because energy (and particles) are fundamental.

At one extreme, the IceCube observatory has claimed detection of neutrinos with energies of 0.001eV. I'm not sure if an energy difference counts, but the Mossbauer effect means that the change of energy arising from doppler-shifting gamma-photons from a radioactive source moving at a few centimeters per second is detectable: that's an energy difference of under $10^{-5}$ eV.

At the other extreme there are "OMG" cosmic rays with energies in excess of $10^{20}$ eV. We can't be absolutely certain that these are single protons. A mechanism for generating such particles is hard to envisage (and it has to be located in our immediate galactic proximity!). It's possible that it's spitting out atomic nuclei rather than protons, in which case we should maybe deduct a couple of orders of magnitude for safety.

Anyway, that's at least 23 orders of magnitude, maybe a few more.

We can of course detect electromagnetic radiation with frequencies of a few Hz and maybe lower, and one must assume that this corresponds to photons of femto-eV. However, we couldn't detect a single such photon, only the effect of large correlated numbers thereof.

Temperature achieved in man-made experiments ranges from half a nanoKelvin to five TeraKelvin (quark-gluon plasmas), or across 22 orders of magnitude.

The temperature at the start of the universe was a lot higher, but because the universe was opaque in its early days, any attempt to "measure" that temperature must be a theoretical derivation from other observations. Possibly, of the order of the Planck temperature $1.4\times 10^{32} K$, above which it is unclear whether "temperature" has any meaning.

But there are various ways of defining temperature, and under some definitions it is possible to create systems in which "temperature" reaches infinity, goes negative, and starts approaching zero from the other direction! Negative thermodynamic temperatures are measures of population inversions, as found in every lasing medium. Yes, this is perhaps cheating, from the perspective of this question.