Category:Essential Physics

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It appears to me that they who in proof of any assertion rely simply on the weight of authority, without adducing any argument in support of it, act very absurdly.
Galileo's father , quoted in Richtmyer & Kennard , Introduction to Modern Physics , 1942 .

"Numerous important discoveries have been made 'by investigating the next decimal place'. ... And ever since Kepler proved that the orbits of the planets are ellipses, relations expressible in quantitative form have carried greater weight than those which could be stated only qualitatively."
F.K. Richtmyer , preface to 1st ed , 1928 , Introduction to Modern Physics , 1942 .


Planetary temperature is essentially an issue of applied physics . The method of physics is to quantitatively understand geometrical arrangements so simple that they can be experimentally verified . We consider here first , the constraints imposed on a planet's temperature by the constraints on a ball enclosing the planet and its atmosphere . That is , we consider how to calculate the equilibrium temperature of a radiantly heated colored ball . It is worth noting that the mean temperature of a planet is a much simpler problem than the dynamics of its atmosphere just as determining the mean temperature of a volume of gas determined by the Gas Laws is much simpler than determining the details of its internal eddies . The necessary physics is more than a century old .

The most essential results are that the temperature of a planet attributable to the sun is linear with the temperature of the sun and inversely proportional to the square root of its distance from the sun .

Simply calculating the energy density in our orbit by adding up the energy ( almost totally from the Sun ) impinging on a point in it explains 97% , all but 10 degrees Celsius , of our temperature . This is the equilibrium temperature of a uniform gray ( flat spectrum ) ball in our orbit ; how light or dark gray does not enter the equation . This is frequently , incorrectly , asserted to be the temperature only of a black ( totally absorptive ) body .

The greenhouse effect is defined to be the difference from this equilibrium due to the difference in correlations between the spectra ( color ) of a ball and its surrounding radiant sources and sinks .


Wikipedia is a good source for much of the relevant basic physics up to the point at which they are diverted into the "warmist" insistence on a step function hypothesis for the spectrum of a naked earth . See , for instance , http://en.wikipedia.org/wiki/Talk:Black_body#Temperature_relation_between_a_planet_and_its_star_-_This_is_wrong
Another excellent resource is C.R. Nave's http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html .
On this site , each quantitative relationship is expressed in executable computer language . See #notation for specifics .

A number of significant questions will be suggested as exercises or points needing elaboration .

Contents

Radiant Heat and Energy

Radiation and Spectra

"resized from http://mauweb.net/3.0/images/random_img/gifs01/dark_side_of_the_moon.gif"
Around 1666-72 , Issac Newton showed that white light was the sum of the colors of a rainbow . We now know that the electromagnetic spectrum ranges far beyond the sliver which vibrate the molecules in the sensitive cells of our retinas . Here , we will often use color to mean the entire spectrum , not just those wavelengths we see , and white and gray to mean a flat spectrum with the same energy across the entire spectrum involved in a particular calculation .



Below are the sort of spectra involved in calculating the lumped earth/atmosphere temperature .
A couple points should be noted about these graphs . First , the Wavelength axis is logarithmic . Second , the Spectral Intensity curves have been scaled to height . As will be calculated below , the 5525k curve is so enormously larger than the ~ 300k curves that it is higher than them even at their peak wavelengths .

Our first task below is to show how to calculate the temperatures implied by such radiance and absorption spectra .

Ritchie's Experiment and Kirchhoff ( Stewart ) law

Considering first opaque bodies , a surface may either absorb radiant energy converting it to thermal energy , or reflect it .

We will call the proportion absorbed , the grayvalue, gv , ranging from 1 for a perfect black body to 0 for a perfectly white , reflective , body .

By 1860 , the crucial , but poorly understood by today's "climate scientists" , fact that a good absorber is a good emitter & vice versa had been codified by the great Gustav Kirchhoff and similarly by the lesser known Balfour Stewart .

The observational foundation of Kirchhoff's law was an elegant experiment a generation earlier by William Ritchie .
I've seen a better presentation of it than the image below but have not been able to find it again . This image is extracted from Demonstration Experiments in Physics by Richard Manliffe Sutton , 1938 . I'd appreciate a better source if someone finds one .

I find it notable that this law can be demonstrated with just the difference between hot and room temperature water .

I've added the neutral labels A and B because the implications of this experiment are far more general than just the extreme absorptive and reflective surfaces described . The brilliance of Ritchie's set up is that he effectively commutes absorptivity ( the transduction of radiant to material thermal energy ) and emissivity ( the transduction of thermal to radiant energy ) with respect to temperature .

( ( A B ) T0 ) = ( ( B A ) T0 )


Stefan-Boltzmann law

The Stefan-Boltzmann law is a simple equation which states that the Power ( energy per unit time ) radiated by a body is proportional to its Temperature raised to the 4th power multiplied by the stefan-boltzmann constant , derived from fundamentals , which converts temperature to energy flow . The standard temperature scale is Kelvin . Energy flow ( or flux ) is measured in watts ( joules per second ) per square meter . Note that energy flow per second can be converted to spacial energy density by dividing by a Light-second .

Stefan Boltzmann Law
Notation :  APL evolute K
Executable Definition Descriptive Definition
sb : 5.6704e-8 Stefan Boltzmann constant : ( watt per square meter ) per Kelvin ^4
T2Psb : {[ T ] sb * T ^ 4 } Power ( watt per square meter ) for Temperature(s) in Kelvin
P2Tsb : {[ P ] ( P % sb ) ^ % 4 } Temperature(s) in Kelvin for Power ( watt per square meter )

The energy flows per unit area can then be summed up over surfaces . We will also need the inverse function to convert back from power to temperature . Note that the sb constant falls out when we make a round trip from temperature to temperature ( or energy to energy ) .

Since computations in energy are linear , the net rate of energy transfer between two objects is the simple difference of each of their individual rates , ie , {[ T0 ; T1 ] -/ T2Psb' ( T0 ; T1 ) } . The net energy flow , and therefore heat flow , is from whichever is warmer to whichever is cooler . No mystery ( except that { sb * T ^ 4 } bit solved more than a century ago , itself ) .

In classical physics fashion , one can determine the energy density , and therefore temperature , at a point by simply integrating ( summing ) the energy flux over its surrounding sphere , {[ T ; s ] +/ Psb T @ s } , where s is a partition of the sphere . As a Gaussian surface I'm sure tons of related implications can immediately be made for spheres in place of points .

Here's a table of energy densities , expressed in watts per meter squared , for temperatures near that of a point in our orbit , calculated below , and our observed mean temperature .

+ -1 -1 0 _' ( , Q ) , ( :: ; -': ) @\: T2Psb Q : 275 + ! 27
275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300
324.3 329.04 333.84 338.68 343.58 348.53 353.54 358.6 363.71 368.88 374.1 379.38 384.72 390.11 395.55 401.06 406.62 412.24 417.91 423.65 429.44 435.29 441.2 447.18 453.21 459.3
4.74 4.79 4.85 4.9 4.95 5.01 5.06 5.11 5.17 5.22 5.28 5.33 5.39 5.45 5.5 5.56 5.62 5.68 5.73 5.79 5.85 5.91 5.97 6.03 6.09 6.15

The third row is the difference between the sequential values in the second row . The total change in our observed mean temperature since before the industrial revolution is from about 288.0 to 288.8 degrees , or about 4.35 W%M^2 . The total difference between our temperature and a gray ball in our orbit is about 52.45 W%M^2 . By contrast a change of 1 degree in the Sun's measured 5778 is about 43764.2 W%M^2 .

Orbital geometry

Here are the radius of the sun and our distance from it in meters . mean is the mean of our aphelion and perihelion distances .

 : SunSpecs[ `radius `ap `peri `mean ] : 6.96e+008 1.520977e+011 1.4709807e+011 1.4959789e+011
  radius   ap   peri   mean
  6.96e+008   1.520977e+011   1.4709807e+011   1.4959789e+011

Over geological periods , the eccentricity of our orbit changes as does the tilt of our axis of rotation relative to our orbit . These have measurable effects on our mean temperature ( and global climate ) . See , for instance , the work of Nicola Scafetta . Currently we are at perihelion about January 3 , about 2 weeks after the northern hemisphere's winter solstice .

The "temperature" of a point in space may be calculated by adding up the radiant energy flows impinging from all directions around it . Almost all our energy comes from the sun , so the first task is to calculate the portion of the total celestial sphere around us it subtends . The solid angle subtended by a disk is simply its area , pi * R ^ 2 , projected onto a sphere of radius 1 , that is dividing it by its distance , D , squared ( because it's an area ) . The unit sphere has a total area of 4 * pi . So the portion of the total sphere covered by a disk is simply its solid angle divided by 4 * pi .

Portion of sphere covered by disk of radius R at distance D
Notation:  APL evolute K
Executable Definition
Descriptive Definition
Da : {[ R ] pi * R ^ 2 }  Area of disk of radius  R
SA : {[ R ; D ] ( Da R ) % D ^ 2 } Solid Angle ( steradians ) of disk from radius and distance
PoS : {[ R ; D ] SA[ R ; D ] % 4 * pi } Portion of sphere covered by disk of radius at distance D

So , the portions of the celestial sphere covered by the sun at ap- and peri-helion and mean distances are :

 : SunSfrPrt[ `ap `peri `mean ] : PoS . SunSpecs @' ( `radius ; `ap `peri `mean )
  ap   peri   mean
  5.234958e-6   5.596861e-6   5.411374e-6

Our planet is heated by just five millionths of the total sky .

Calculating the Temperature of a radiantly heated Gray Ball

Now that we have the function for the conversion of thermal energy to radiant flux and the portion of the celestial sphere covered by the sun we can we can calculate the total energy impinging upon a point in our orbit . ( OT : The reason the sun appears uniform across its disk is Lambert's cosine law . )

Using SunSfrPrt.mean calculated above , we can make a table of the observed sun and cosmic background temperatures and portions of the celestial sphere they cover .

 : SunTemp : 5778  ; CBR : 3

 : TP : ( SunTemp , CBR  ; SunSfrPrt.mean , 1 - SunSfrPrt.mean )
5778 5.411374e-006
3 0.9999946


To show that the Stefan-Boltzmann constant , or any other just drops out of the overall equation , here are both the naked temperatures raised to the 4th , and the actual watts per square meter given by T2Psb :

 : T0 : ( ( TR 0 ) ^ 4  ; T2Psb TR 0 )
1.114577e+15 6.320098e+7
81 4.593024e-6


Now we multiply each column by the portions in TP :

 : r : ( TP 1 ) */: T0
6.031394e+9 342.0042
80.99956 4.592999e-6


Now we sum across the partitions of the sphere :

 : r : +/' r
6.031394e+9 342.0042


It's clear the 3 kelvin cosmic adds up to nothing , altho it represents all but 5 millionths of the sky . However , if you entered a background of , eg , 100 kelvin you'd start seeing it having an effect .

Now converting back to temperature we have :

 : ( ( r 0 ) ^ % 4 ; P2Tsb r 1 )
278.6791 278.6791


This approximately 279 kelvin is commonly asserted to be the temperature of a black body in our orbit . Note that throwing in any grayvalue , gv , ( other than 0 ) will drop out of the computation just like sb . Thus this is the temperature for any uniformly gray ball however dark or light .

We are about 10 degrees warmer than this , about 288.8 up from 288 since before the industrial revolution . That's a total change of less than 0.3% . The entire observed change would be explained by a change of 16 degrees in the temperature of the sun .

To summarize the computation into one function on any table , TP , of a partition of temperatures over the sphere :

Temperature of a radiantly heated gray ball
Notation : APL evolute K
Executable Definition Descriptive Definition
PointTemp : {[ TP ] ( +/ ( TP 1 ) * ( TP 0 ) ^ 4 ) ^ % 4 } temperature of a ball ( point ) from a table , TP , of temperatures and portions of the surrounding sphere .


Quiz


Planck radiation function

Planckf : {[ f ; T ] ( ( 2 * h * f ^ 3 ) % c ^ 2 ) % ( _exp ( h * f ) % Boltz * T ) - 1 }       / in terms of frequency 
Planckl : {[ l ; T ] ( ( 2 * h * c ^ 2 ) % l ^ 5 ) % ( _exp ( h * c ) % l * Boltz * T ) - 1 }   / in terms of wave length

We will work in terms of wavelength .

Calculating the Temperature of a radiantly heated Colored Ball


Semi-transparent ball with inclusions

Beer–Lambert law


Vertical Structure of Atmosphere

Atmospheric Convection

Other sources of heat

While much smaller than the radiant energy from the sun , there are several other sources of energy to the earth's surface , some of which may be locally significant , which must be considered in a complete audit .

Geothermal

Man Made

Particularly around urban areas , the gigawatts of energy released from fossil and nuclear fuels , not just changes in spectrum and terrain can be significant .

Solar and Other Particulate

Some amount of energy impinges on the earth in the form of kinetic particles in form of the Solar Wind and meteors .

CO2 & Atmosphere

CO2 respiratory cycle



Invitation


The initial content of this page is based on the analysis at http://cosy.com/Science/TemperatureOfGrayBalls.htm . In the spirit of Donald T Campbell's Fish Scale Model of collective Omniscience now enabled by the wiki , this site needs the knowledges of more minds . However , the purpose of this site is to nail down the physics . Thus only data and explicit experimentally verified or verifiable quantitative , computable assertions , ie , equations and their explanations and proofs are admissible . Tables of vocabulary implementing such relationships in various programming languages so individuals can "play" with the concepts themselves have been initialized .

Please post comments and suggested clarifications or additions to the discussion page for "peer review" before posting to this page . --Bob 15:47, 15 July 2011 (UTC)


notation

The executable notation used here is Arthur Whitney's K' evolved from Ken Iverson's APL array programming language . K summary

For important relations , tables have been structured to easily add translations into more accessible languages which are most welcome .

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