Some physics laws are quadratic (*quadratum* is Latin for "square"), and others are cubic, and some are QUARTIC. It depends on whether a significant input to the formula is squared, or cubed, or quarted i.e. raised to the fourth power. One famous example of a quartic law is Ludwig Boltzmann's (and Josef Stefan's) "fourth power radiation law"--to find how intensely a hot object GLOWS you quart the absolute temperature. Square the temperature twice in succession, or raise it to the fourth power, or quart it--whatever you like to say. Further down the page I include a short discussion of a quartic law in GRAVITY.
Another example is in the way light is scattered by a molecule of air, more exactly by its surrounding electrons. An electron's ability to scatter light is essentially shown by its "Thomson cross-section" area. But in the (Rayleigh) scattering cross-section for the air MOLECULES there are also quartic terms showing how it varies with the frequency ƒof the incoming light. THIS VARIATION IS THE FUNDAMENTAL REASON WHY THE SKY IS BLUE.
The quartic frequency-dependent terms are made of just two things--an intrinsic resonant frequency ƒoof the molecule itself and the incoming frequency ƒ. The idea is simple: the fourth power of the frequency divided by the squared difference of the squares: ƒ4/(ƒ2 - ƒo2)2. In the case of air. the resonance is up in the UV so the denominator is smaller with blue light than it is with lower frequency red light. So blue scatters more easily.
So when you walk outdoors and see a blue overhead you are witness to a quartic law--quartic frequency-dependence of scattering.
BUT YOU ALSO witness a quartic law simply in the brightness of the sun itself. Quartic is how temperature relates to the brightness of shining.
A surface that is twice as hot as another will glow sixteen-fold brighter, other things equal. And one three times as hot will glow eightyone-fold brighter. It doesn't matter here whether or not you remember Boltzmann's (and Stefan's) quartic radiation law--the important thing to notice is that some laws do involve squaring a number twice in succession or, in other words, quarting it.
AN INTERESTING VARIANT of this law deals with the concentration of energy per unit volume in a room (or any hollow space) at a given temperature. One way of expounding this version is as follows. There is a primordial absolute concentration of energy--Planck energy density: one Stem of energy per cubic stitch of volume--that characterized the moment time began. And there is the primordial temperature Primal on which is based the natural temperature scale. If the room temperature and the concentration of heatshine are both expressed in natural terms--as a fraction of Primal temperature and as a fraction of absolute concentration--then their relation to each other is especially simple.
Whatever fraction the temperature is, you quart it and multiply by a number which is roughly two thirds (actually pi2over fifteen) and that gives the concentration as a fraction of the natural standard.
One very comfortable temperature (Fahrenheit 68, Celsius 20) to take as an example is 2.07 ten-to-thirtieth of Primal. Quarting 2.07 gives something like 18 and multiplying by the two-thirdish number gives 12--so the concentration of radiant warmth in the room turns out to be 12 times a ten-to-hundredtwentieth of absolute--obviously a very tiny fraction of the natural energy density unit.
For me, the quartic GRAVITY law is one of the most exciting ones. It applies to two bodies in circular orbits around each other and it connects the attractive force between them to their combined speed (the sum of their two individual speeds) and their two shares of the total. If speed and force are expressed in natural terms (as fractions of Top speed and Main force) then the relation is pretty simple: quart the speed and multiply by the two shares and that gives the force--as a fraction of the natural standard.
To take an example, suppose the combined speed is a ten-to-sixth of Top (which is about what sound travels in much of the earth's atmosphere) and the two shares are 1/3 and 2/3. To get the force you just quart ten-to-sixth, which gives ten-to-twentyfourth. and multiply by 1/3 and 2/3--same as multiplying by 2/9. The attraction between the two bodies, then, is 2/9 ten-to-twentyfourth of Main force. If you prefer, of course, say "Planck" force instead of Main.
Since we know the Main force in tons it is easy to work out the sample attraction in tons if that is wished. What impresses me is that the force of attraction (if you know the orbit speeds) does not depend on the masses and the distance of separation. They can as well be two massive things far apart or two unmassive things close together--what matters is how fast they are going.
So the three examples of quartic laws mentioned here describe how strongly orbiting things pull, how brightly hot things shine, and how effectively air scatters blue light compared with red. The last one is what makes the sky overhead blue (because it shines with scattered light) and the light of a sunset (which comes through a long stretch of atmosphere without getting scattered along the way) red.
I would like more examples of quartic laws evident in everyday experience. Please let me know if you come across any.