MIT OpenCourseWarehttp://ocw.mit.edu 12.002 Physics and Chemistry of the Earth and Terrestrial Planets Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.31 The Error Function We will use the conduction equation in a few more classes to relate lithospheric thickness to time and temperature so we might as well right down one very useful time-dependent solution, called the error function and written erf(y). ! erf (y) =2"e#$ y 2d$ y 0y%This function has the property that erf(0)=0, erf(+infinity)=1. Consider now the function: ! T(z,t) = Ts+ (Tin" Ts)erf (z /2#t ) = Ts+2(Tin" Ts)$e"% y 2d% y 0z / 2#t&This function has the property that its value at z=0 is Ts at all times, its value at t=0 is Tin for all values of z>0, and it is a solution to the heat conduction equation. (We will leave it to the homework to show that it is a solution and to calculate the rate of conductive cooling of a planetary surface.) Later on we will use this expression to calculate the thermal behavior of the oceanic lithosphere on earth. Figure by MIT OpenCourseWare.10.80.60.40.200 1 2 3Error
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