Category:Essential Physics

Bob — Thu, 17 May 2012 16:57:33 GMT

Calculating the Temperature of a radiantly heated Gray Ball:

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	=== Stefan-Boltzmann law ===		=== Stefan-Boltzmann law ===

-	The Stefan-Boltzmann law is a simple equation which states that the '''P'''ower ( energy per unit time ) radiated by a body is proportional to its '''T'''emperature raised to the 4th power multiplied by the '''s'''tefan-'''b'''oltzmann constant derived from fundamentals which converts ~~degrees Kelvin~~ to watts per square meter .	+	The Stefan-Boltzmann law is a simple equation which states that the '''P'''ower ( energy per unit time ) radiated by a body is proportional to its '''T'''emperature raised to the 4th power multiplied by the [http://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_constant '''s'''tefan-'''b'''oltzmann] constant , derived from fundamentals , which converts temperature to energy flow . The standard temperature scale is [http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html Kelvin] . Energy flow ( or <i>flux</i> ) is measured in watts ( [http://en.wikipedia.org/wiki/Joule joules] per second ) per square meter . Note that energy flow per second can be converted to spacial energy density by dividing by a [http://en.wikipedia.org/wiki/Light-second Light-second] .

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-	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 .	+	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 ) .
-		+
-	~~Also note that energy flows in watts~~ ( ~~[http://en.wikipedia.org/wiki/Joule joules] per second ) per square meter can be converted~~ to ~~spacial~~ energy ~~density by dividing by a [http://en.wikipedia.org/wiki/Light-second Light-second]~~ .	+

	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 ( <small>except that { sb * T ^ 4 } bit solved more than a century ago , itself</small> ) .		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 ( <small>except that { sb * T ^ 4 } bit solved more than a century ago , itself</small> ) .

-	In classical physics fashion , one can determine the ~~power~~ 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 .	+	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 . <small>As a Gaussian surface I'm sure tons of related implications can immediately be made for spheres in place of points .</small>
		+
		+	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 .
		+
		+	: <code> + -1 -1 0 _' ( , Q ) , ( :: ; -': ) @\: T2Psb Q : 275 + ! 27 </code>
		+	<TABLE border=1 > <TR> <TD> 275 </TD> <TD> 276 </TD> <TD> 277 </TD> <TD> 278 </TD> <TD> 279 </TD> <TD> 280 </TD> <TD> 281 </TD> <TD> 282 </TD> <TD> 283 </TD> <TD> 284 </TD> <TD> 285 </TD> <TD> 286 </TD> <TD> 287 </TD> <TD> 288 </TD> <TD> 289 </TD> <TD> 290 </TD> <TD> 291 </TD> <TD> 292 </TD> <TD> 293 </TD> <TD> 294 </TD> <TD> 295 </TD> <TD> 296 </TD> <TD> 297 </TD> <TD> 298 </TD> <TD> 299 </TD> <TD> 300 </TD> </TR>
		+	<TR> <TD> 324.3 </TD> <TD> 329.04 </TD> <TD> 333.84 </TD> <TD> 338.68 </TD> <TD> 343.58 </TD> <TD> 348.53 </TD> <TD> 353.54 </TD> <TD> 358.6 </TD> <TD> 363.71 </TD> <TD> 368.88 </TD> <TD> 374.1 </TD> <TD> 379.38 </TD> <TD> 384.72 </TD> <TD> 390.11 </TD> <TD> 395.55 </TD> <TD> 401.06 </TD> <TD> 406.62 </TD> <TD> 412.24 </TD> <TD> 417.91 </TD> <TD> 423.65 </TD> <TD> 429.44 </TD> <TD> 435.29 </TD> <TD> 441.2 </TD> <TD> 447.18 </TD> <TD> 453.21 </TD> <TD> 459.3 </TD> </TR>
		+	<TR> <TD> 4.74 </TD> <TD> 4.79 </TD> <TD> 4.85 </TD> <TD> 4.9 </TD> <TD> 4.95 </TD> <TD> 5.01 </TD> <TD> 5.06 </TD> <TD> 5.11 </TD> <TD> 5.17 </TD> <TD> 5.22 </TD> <TD> 5.28 </TD> <TD> 5.33 </TD> <TD> 5.39 </TD> <TD> 5.45 </TD> <TD> 5.5 </TD> <TD> 5.56 </TD> <TD> 5.62 </TD> <TD> 5.68 </TD> <TD> 5.73 </TD> <TD> 5.79 </TD> <TD> 5.85 </TD> <TD> 5.91 </TD> <TD> 5.97 </TD> <TD> 6.03 </TD> <TD> 6.09 </TD> <TD> 6.15 </TD> </TR> </TABLE>
		+
		+	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 ===		=== Orbital geometry ===
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	Here are the radius of the sun and our distance from it in meters . ''mean'' is the mean of our aphelion and perihelion distances .		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 5.234958e-6 5.596861e-6 5.411374e-6	+	: <code> : SunSpecs[ `radius `ap `peri `mean ] : 6.96e+008 5.234958e-6 5.596861e-6 5.411374e-6</code>
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	<td>   radius </td>		<td>   radius </td>
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	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 .		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 ~~ ~~ ~~; &nbsp~~; SunSfrPrt.mean , 1 - SunSfrPrt.mean )	+	<code> : SunTemp : 5778 ; CBR : 3 </code>
		+
		+	: <code> : TP : ( SunTemp , CBR ; SunSfrPrt.mean , 1 - SunSfrPrt.mean ) </code>
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	<br>		<br>
	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'' :		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 )	+
		+	: <code> : T0 : ( ( TR 0 ) ^ 4 ; T2Psb TR 0 ) </code>
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	<br>		<br>
	Now we multiply each column by the portions in TP :		Now we multiply each column by the portions in TP :
-	: : r : ( TP 1 ) */: T0	+
		+	: <code> : r : ( TP 1 ) */: T0 </code>
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	<br>		<br>
	Now we sum across the partitions of the sphere :		Now we sum across the partitions of the sphere :
-	: : r : +/' r	+
		+	: <code> : r : +/' r </code>
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	Now converting back to temperature we have :		Now converting back to temperature we have :
-	: ( ( r 0 ) ^ % 4 ; P2Tsb r 1 )	+
		+	: <code> : ( ( r 0 ) ^ % 4 ; P2Tsb r 1 ) </code>
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	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 .		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 .	+	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 .
-		+	<br> <br>
	To summarize the computation into one function on any table , TP , of a partition of temperatures over the sphere :		To summarize the computation into one function on any table , TP , of a partition of temperatures over the sphere :
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Category:Essential Physics