The time it takes for a spherical object to cool from an initial temperature of

Question

The time it takes for a spherical object to cool from an initial temperature of T_init to a final temperature of T_fin, caused entirely by radiation, is provided by Kelvin's cooling equation: t = Nk/2e sigma A[1/T_fin^3 - 1/T_init^3] t is the cooling time in years. N is the number of atoms. k is Boltzmann's constant (138 times 10^-23 m^2kg/s^2 K; note that 1 Joule = 1 m^2kg/s^2). e is emissivity of the object. sigma is Stefan-Boltzmanns constant (5.6703 times 10^-8 watts/m^2K^4). A is the surface area. T_fin is the final temperature. T_init is the initial temperature. Assuming an infinitely hot initial temperature, this formula reduces to t = Nk/2e sigmaAT_fin^3 Using this second formula, write a C++ program to determine the time it took Earth to cool to its current surface temperature of 300 degree K from its initial infinitely hot state, assuming the cooling is caused only by radiation. Use the information that the area of the Earth's surface is 5.15 times 10^14 m^2, its emissivity is 1, the number of atoms contained in the Earth is 1.1 times 10^50, and the radius of the Earth is 6.4 times 10^6 meters. Additionally, use the relationship that a sphere's surface area is given by this formula: Surface area of a sphere = 4 pi r^2

Explanation / Answer

/*
C ++ program that finds the time taken by
earth from infinite temperature to temperature
of present 300k .and print the t to console
*/
//temp.cpp
#include<iostream>
#include<math.h>
#include<iomanip>
using namespace std;
int main()
{

   //Set number of atoms in earth
   double N=1.1*10E50;
   //Set Boltzmann's constant
   double k=1.38*10E-23;
   //Set emmission
   double e=1.0;
   //Set constant for Stefan-Boltzmann's constant  
   double SB=5.6703*10E-8;
  
   //surface area of earth
   double SA=5.15*10E14;
   //temperature in kelvin
   double finalTemp=300;

   //calculate time taken by earth to reach
   //from infinite temp to current temp of 300k(26 deg)
   double t=(N*k)/(2*e*SB*SA*pow(finalTemp,3.0));

   //print to console
   cout<<fixed<<setprecision(3)<<"Time taken by earth : "
       <<t<<endl;

   //paue program output on console
   system("pause");
   return 0;
}

sample output:

Time taken by earth : 962641866729.235

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