3 Pho­ton-Sur­face In­ter­ac­tions

Fig­ure 3: Pho­ton-sur­face in­ter­ac­tion mech­a­nisms. They de­pend on pho­ton wave­length. Pho­tos © Leon van Dom­me­len.

At this stage, the pho­ton-sur­face in­ter­ac­tion can be un­der­stood to the re­quired de­tail. Fig­ure 3 above shows how an in­com­ing pho­ton of light, $\gamma$, in­ter­acts with a white and a non­white Mi­ata. The white Mi­ata will re­flect or emit pho­tons of all col­ors. How­ever, a red Mi­ata will mostly emit red pho­tons.

If a red Mi­ata ab­sorbs a blue pho­ton and re­leases its en­ergy as heat, the mo­men­tum of the pho­ton is gone. The red Mi­ata has ef­fec­tively stopped the pho­ton. Be­cause of New­ton’s third law, ac­tion equals mi­nus re­ac­tion, the red Mi­ata ex­pe­ri­ences an op­po­site force slow­ing it down. That would hold even if all the pho­ton en­ergy was emit­ted again as om­ni­di­rec­tional in­frared ra­di­a­tion.

On the other hand, as il­lus­trated in fig­ure 3, the white Mi­ata leaves the back­ward mov­ing mo­men­tum of the pho­ton largely in­tact, as­sum­ing a pre­dom­i­nantly spec­u­lar emis­sion. The ver­ti­cal mo­men­tum does change more sig­nif­i­cantly. But that merely presses the white Mi­ata more strongly onto the road, pro­vid­ing an ad­di­tional mea­sure of safety at its higher speeds.

(As the Planck-Ein­stein and the Broglie re­la­tions show, pho­ton en­ergy is pro­por­tional to mo­men­tum. It is not pro­por­tional to square mo­men­tum, as New­ton­ian physics would sug­gest. So if a red Mi­ata re-emit­ted all in­com­ing pho­ton en­ergy as ra­di­a­tion, and if that ra­di­a­tion was pre­dom­i­nantly spec­u­lar with re­spect to the in­com­ing pho­ton, then there would not be a dif­fer­ence be­tween the Mi­atas. Un­for­tu­nately, nei­ther con­di­tion is true.)

You might of course won­der whether the ad­van­tage of the white Mi­ata would not be off­set by pho­tons com­ing from other di­rec­tions. There is some­thing to that. It should be stressed that a Mi­ata, of any color, in an equi­lib­rium sit­u­a­tion with black­body ra­di­a­tion com­ing from all di­rec­tions, will not ex­pe­ri­ence a net force. Any other state­ment would ob­vi­ously vi­o­late the sec­ond law of ther­mo­dy­nam­ics. And this pa­per would never sug­gest it would not. Only the high­est stan­dards of sci­en­tific in­tegrity are ap­plied in this work.

How­ever, for a mov­ing Mi­ata, the pho­tons com­ing from the front are blue-shifted. That is de­scribed by the rel­a­tivis­tic Doppler shift (2). This in­creases their en­ergy, and as a re­sult, a mov­ing Mi­ata on an oth­er­wise equi­lib­rium earth ex­pe­ri­ences a pho­ton drag slow­ing it down. Pho­tons like the ex­am­ple in fig­ure 3 dom­i­nate. As a re­sult, the Mi­ata will ex­pe­ri­ence a pho­ton drag. But a white Mi­ata will ex­pe­ri­ence less drag be­cause it slows down the dom­i­nant pho­tons less.

It is also in­ter­est­ing to look at non­equi­lib­rium sit­u­a­tions. In par­tic­u­lar, on sunny days the pho­ton dis­tri­b­u­tion is far from a black­body one. Pho­tons come pre­dom­i­nantly from a con­cen­trated source: the sun. Now if the sun is in front of the Mi­ata, fig­ure 3 showed that a white Mi­ata ex­pe­ri­ences less pho­ton drag.

You might now of course con­jec­ture that if the sun is in the back, then this ad­van­tage could re­verse. That in that case, a red Mi­ata might have an ad­van­tage. Un­for­tu­nately, that is not true. Fig­ure 4 shows what hap­pens. In this case, there is a so­lar sail ef­fect. This ef­fect, well es­tab­lished for in­ter­stel­lar travel, ac­tu­ally pro­vides a propul­sive force for the Mi­ata. Now if a pho­ton is sim­ply ab­sorbed by a red Mi­ata, its mo­men­tum adds to that red Mi­ata. But if a pho­ton is re­flected by a white Mi­ata, dou­ble its mo­men­tum is added to that white Mi­ata. Of course, the real sit­u­a­tion is more com­plex. A white Mi­ata is not a per­fect re­flec­tor, and a red Mi­ata not a per­fect ab­sorber. Fig­ure 4 tries to cap­ture the av­er­age sit­u­a­tion. Still, the white Mi­ata ex­pe­ri­ences a much greater so­lar sail ef­fect.

Fig­ure 4: The so­lar sail ef­fect.

There is an­other im­por­tant ef­fect that Mi­ata own­ers of­ten ig­nore. The high-en­ergy ra­di­a­tion ab­sorbed by a red Mi­ata comes out pri­mar­ily as heat. Now heated air at the sur­face of the red Mi­ata wants to rise away from the Mi­ata be­cause it is lighter than the sur­round­ing air at the same pres­sure. Clearly, that will pro­mote sep­a­ra­tion, the pri­mary ef­fect lim­it­ing the speed of a Mi­ata. True, the ris­ing air will also pro­mote tur­bu­lence, which might de­lay sep­a­ra­tion a bit. How­ever, it is to be ex­pected that global buoy­ancy will dom­i­nate and sep­a­ra­tion will be pro­moted. This will greatly in­crease the aero­dy­namic drag of a red Mi­ata.

Fig­ure 5: Görtler vor­tex sys­tem in the bound­ary layer on a white Mi­ata at speed. In­ferred from the­o­ret­i­cal ob­ser­va­tions by Van Dom­me­len and Ya­pal­parvi.

On the other hand, cur­rent work by the au­thor and Ya­pal­parvi, (in progress), sug­gests that the lower sur­face tem­per­a­tures of a white Mi­ata will gen­er­ate a Görtler vor­tex sys­tem that may be very ef­fi­cient in de­lay­ing sep­a­ra­tion. Fig­ure 5 shows the ex­pected mo­tion of the sys­tem. Note that this rep­re­sents just a very small seg­ment of the thin bound­ary layer, look­ing up­stream.

With the ad­van­ta­geous pho­ton in­ter­ac­tion, in ad­di­tion to the en­hance­ment of the aero­dy­nam­ics by the Görtler sys­tem, the much higher speed of white Mi­atas is clearly fully ex­plained.