Update #1
I now also tried to see if I can simply solve the problem through the operator NSolve but the system could not be solved given the methods available in this operator.
Update #1
I now also tried to see if I can simply solve the problem through the operator NSolve but the system could not be solved given the methods available in this operator.
Given a certain numerical value for the energy density flux $F$ of a star, e.g. the solar constant $s_\odot = 1362~\frac{W}{m^2}$, and a certain wavelength range corresponding to the irradiated energy $(\lambda_{min},\lambda_{max})$, how can one find the corresponding black body temperature?
For this task, I aimed to find a solution in Mathematica which would allow me to fit the planck law function
\begin{equation} B_\lambda(T) = \frac{2\,h\,c^2}{\lambda^5}\,\frac{1}{\exp(\frac{h\,c}{\lambda\,k\,T})-1} \quad, \end{equation}
to the numerical output of the energy density $F$. Assuming the star to have an isotopically emitting surface (i.e. approximation as black body), the energy density flux for a given wavelength window can be computed via 1
\begin{equation} F = \pi \, \int^{\lambda_{max}}_{\lambda_{min}} B_\lambda(\color{red}{T})\,d\lambda \quad. \end{equation}
In Mathematica I have tried the following -- so far -- unfruitful approach (assuming the irradiation power of $F_{emphir} = 1340~\frac{W}{m2}$ was measured in the range of $(0.1~nm, 2300~nm)$:
First defining any variables
refF = 1340.(*W/m^2*); c = 299792458(*m/s*); h = 6.62607015*10^-34(*Js*); k = 1.380649*10^-23(*J/K*);Defining the planck law function $B(\lambda,\,T)$and the energy density flux $F(T)$
B[\[Lambda]_, T_] := (2*h*c^2)/\[Lambda]^5*1/( Exp[(h*c)/(\[Lambda]*k*T)] - 1); F[T_?NumericQ] := \[Pi]*NIntegrate[ B[\[Lambda], T], {\[Lambda], 0.1*10^-9, 2300.0*10^-9}];Lastly, trying to find the corresponding effective temperature $T$ to the integrated area under the black body function, given a certain range of wavelengths from which the energy is emitted.
FindFit[F, {a* F[T_], {0.99 < a < 1.01}}, a, T]
Unfortunately, the FindFit[data, expr, pars, vars] function does not work with a single numerical value as data and also needs parameters and values. In this particular case, the parameter I aim to tweak for the fit to be successful is also the variable. However, I did not find any other (for me obvious) helper operators in Mathematica which would solve a problem like this.
(I used a(n unnecessary) pre-factor $a$ to have both, variables and parameters in the FindFit operator.)
Hence, my question: How would one go about this in Mathematica?
P.S. I also tried to directly integrate the planck law, given the effective temperature of the Sun for a certain wavelength range to see what wavelength range does contribute the most energy of the total solar energy flux, i.e. the solar constant. This at first did not work due to a wrong value of the speed of light or due to the extensive usage of subscript to label variables, which has now been amended. The cross-check calculation below yields $\approx 1313~\frac{W}{m^2}$, which -- given the large range of wavelength covered should be responsible for the emission of the bulk energy of the Sun -- already corresponds to the solar constant very well.
Footnotes & References:
1 Rybicki, G. B. & Lightman, A. P.: Radiative Processes in Astrophysics. Wiley-VCH Verlag, 2004, p. 7ff.
Given a certain numerical value for the energy density flux $F$ of a star, e.g. the solar constant $s_\odot = 1362~\frac{W}{m^2}$, and a certain wavelength range corresponding to the irradiated energy $(\lambda_{min},\lambda_{max})$, how can one find the corresponding black body temperature?
For this task, I aimed to find a solution in Mathematica which would allow me to fit the planck law function
\begin{equation} B_\lambda(T) = \frac{2\,h\,c^2}{\lambda^5}\,\frac{1}{\exp(\frac{h\,c}{\lambda\,k\,T})-1} \quad, \end{equation}
to the numerical output of the energy density $F$. Assuming the star to have an isotopically emitting surface (i.e. approximation as black body), the energy density flux for a given wavelength window can be computed via
\begin{equation} F = \pi \, \int^{\lambda_{max}}_{\lambda_{min}} B_\lambda(\color{red}{T})\,d\lambda \quad. \end{equation}
In Mathematica I have tried the following -- so far -- unfruitful approach (assuming the irradiation power of $F_{emphir} = 1340~\frac{W}{m2}$ was measured in the range of $(0.1~nm, 2300~nm)$:
First defining any variables
refF = 1340.(*W/m^2*); c = 299792458(*m/s*); h = 6.62607015*10^-34(*Js*); k = 1.380649*10^-23(*J/K*);Defining the planck law function $B(\lambda,\,T)$and the energy density flux $F(T)$
B[\[Lambda]_, T_] := (2*h*c^2)/\[Lambda]^5*1/( Exp[(h*c)/(\[Lambda]*k*T)] - 1); F[T_?NumericQ] := \[Pi]*NIntegrate[ B[\[Lambda], T], {\[Lambda], 0.1*10^-9, 2300.0*10^-9}];Lastly, trying to find the corresponding effective temperature $T$ to the integrated area under the black body function, given a certain range of wavelengths from which the energy is emitted.
FindFit[F, {a* F[T_], {0.99 < a < 1.01}}, a, T]
Unfortunately, the FindFit[data, expr, pars, vars] function does not work with a single numerical value as data and also needs parameters and values. In this particular case, the parameter I aim to tweak for the fit to be successful is also the variable. However, I did not find any other (for me obvious) helper operators in Mathematica which would solve a problem like this.
(I used a(n unnecessary) pre-factor $a$ to have both, variables and parameters in the FindFit operator.)
Hence, my question: How would one go about this in Mathematica?
P.S. I also tried to directly integrate the planck law, given the effective temperature of the Sun for a certain wavelength range to see what wavelength range does contribute the most energy of the total solar energy flux, i.e. the solar constant. This at first did not work due to a wrong value of the speed of light or due to the extensive usage of subscript to label variables, which has now been amended. The cross-check calculation below yields $\approx 1313~\frac{W}{m^2}$, which -- given the large range of wavelength covered should be responsible for the emission of the bulk energy of the Sun -- already corresponds to the solar constant very well.
Given a certain numerical value for the energy density flux $F$ of a star, e.g. the solar constant $s_\odot = 1362~\frac{W}{m^2}$, and a certain wavelength range corresponding to the irradiated energy $(\lambda_{min},\lambda_{max})$, how can one find the corresponding black body temperature?
For this task, I aimed to find a solution in Mathematica which would allow me to fit the planck law function
\begin{equation} B_\lambda(T) = \frac{2\,h\,c^2}{\lambda^5}\,\frac{1}{\exp(\frac{h\,c}{\lambda\,k\,T})-1} \quad, \end{equation}
to the numerical output of the energy density $F$. Assuming the star to have an isotopically emitting surface (i.e. approximation as black body), the energy density flux for a given wavelength window can be computed via 1
\begin{equation} F = \pi \, \int^{\lambda_{max}}_{\lambda_{min}} B_\lambda(\color{red}{T})\,d\lambda \quad. \end{equation}
In Mathematica I have tried the following -- so far -- unfruitful approach (assuming the irradiation power of $F_{emphir} = 1340~\frac{W}{m2}$ was measured in the range of $(0.1~nm, 2300~nm)$:
First defining any variables
refF = 1340.(*W/m^2*); c = 299792458(*m/s*); h = 6.62607015*10^-34(*Js*); k = 1.380649*10^-23(*J/K*);Defining the planck law function $B(\lambda,\,T)$and the energy density flux $F(T)$
B[\[Lambda]_, T_] := (2*h*c^2)/\[Lambda]^5*1/( Exp[(h*c)/(\[Lambda]*k*T)] - 1); F[T_?NumericQ] := \[Pi]*NIntegrate[ B[\[Lambda], T], {\[Lambda], 0.1*10^-9, 2300.0*10^-9}];Lastly, trying to find the corresponding effective temperature $T$ to the integrated area under the black body function, given a certain range of wavelengths from which the energy is emitted.
FindFit[F, {a* F[T_], {0.99 < a < 1.01}}, a, T]
Unfortunately, the FindFit[data, expr, pars, vars] function does not work with a single numerical value as data and also needs parameters and values. In this particular case, the parameter I aim to tweak for the fit to be successful is also the variable. However, I did not find any other (for me obvious) helper operators in Mathematica which would solve a problem like this.
(I used a(n unnecessary) pre-factor $a$ to have both, variables and parameters in the FindFit operator.)
Hence, my question: How would one go about this in Mathematica?
P.S. I also tried to directly integrate the planck law, given the effective temperature of the Sun for a certain wavelength range to see what wavelength range does contribute the most energy of the total solar energy flux, i.e. the solar constant. This at first did not work due to a wrong value of the speed of light or due to the extensive usage of subscript to label variables, which has now been amended. The cross-check calculation below yields $\approx 1313~\frac{W}{m^2}$, which -- given the large range of wavelength covered should be responsible for the emission of the bulk energy of the Sun -- already corresponds to the solar constant very well.
Footnotes & References:
1 Rybicki, G. B. & Lightman, A. P.: Radiative Processes in Astrophysics. Wiley-VCH Verlag, 2004, p. 7ff.
Given a certain numerical value for the energy density flux $F$ of a star, e.g. the solar constant $s_\odot = 1362~\frac{W}{m^2}$, and a certain wavelength range corresponding to the irradiated energy $(\lambda_{min},\lambda_{max})$, how can one find the corresponding black body temperature?
For this task, I aimed to find a solution in Mathematica which would allow me to fit the planck law function
\begin{equation} B_\lambda(T) = \frac{2\,h\,c^2}{\lambda^5}\,\frac{1}{\exp(\frac{h\,c}{\lambda\,k\,T})-1} \quad, \end{equation}
to the numerical output of the energy density $F$. Assuming the star to have an isotopically emitting surface (i.e. approximation as black body), the energy density flux for a given wavelength window can be computed via
\begin{equation} F = \pi \, \int^{\lambda_{max}}_{\lambda_{min}} B_\lambda(\color{red}{T})\,d\lambda \quad. \end{equation}
In Mathematica I have tried the followfollowing -- so far -- unfruitful approach (assuming the irradiation power of $F_{emphir} = 1340~\frac{W}{m2}$ was measured in the range of $(0.1~nm, 2300~nm)$:
First defining any variables
refF = 1340.(*W/m^2*); c = 299792458(*m/s*); h = 6.62607015*10^-34(*Js*); k = 1.380649*10^-23(*J/K*);Defining the planck law function $B(\lambda,\,T)$and the energy density flux $F(T)$
B[\[Lambda]_, T_] := (2*h*c^2)/\[Lambda]^5*1/( Exp[(h*c)/(\[Lambda]*k*T)] - 1); F[T_?NumericQ] := \[Pi]*NIntegrate[ B[\[Lambda], T], {\[Lambda], 0.1*10^-9, 2300.0*10^-9}];Lastly, trying to find the corresponding effective temperature $T$ to the integrated area under the black body function, given a certain range of wavelengths from which the energy is emitted.
FindFit[F, {a* F[T_], {0.99 < a < 1.01}}, a, T]
Unfortunately, the FindFit[data, expr, pars, vars] function does not work with a single numerical value as data and also needs parameters and values. In this particular case, the parameter I aim to tweak for the fit to be successful is also the variable. However, I did not find any other (for me obvious) helper operators in Mathematica which would solve a problem like this.
(I used a(n unnecessary) pre-factor $a$ to have both, variables and parameters in the FindFit operator.)
Hence, my question: How would one go about this in Mathematica?
P.S. I also tried to directly integrate the planck law, given the effective temperature of the Sun for a certain wavelength range to see what wavelength range does contribute the most energy of the total solar energy flux, i.e. the solar constant. This alsoat first did not work: due to a wrong value of the speed of light or due to the extensive usage of subscript to label variables, which has now been amended. The cross-check calculation below yields $\approx 1313~\frac{W}{m^2}$, which -- given the large range of wavelength covered should be responsible for the emission of the bulk energy of the Sun -- already corresponds to the solar constant very well.
Given a certain numerical value for the energy density flux $F$ of a star, e.g. the solar constant $s_\odot = 1362~\frac{W}{m^2}$, and a certain wavelength range corresponding to the irradiated energy $(\lambda_{min},\lambda_{max})$, how can one find the corresponding black body temperature?
For this task, I aimed to find a solution in Mathematica which would allow me to fit the planck law function
\begin{equation} B_\lambda(T) = \frac{2\,h\,c^2}{\lambda^5}\,\frac{1}{\exp(\frac{h\,c}{\lambda\,k\,T})-1} \quad, \end{equation}
to the numerical output of the energy density $F$. Assuming the star to have an isotopically emitting surface (i.e. approximation as black body), the energy density flux for a given wavelength window can be computed via
\begin{equation} F = \pi \, \int^{\lambda_{max}}_{\lambda_{min}} B_\lambda(\color{red}{T})\,d\lambda \quad. \end{equation}
In Mathematica I have tried the follow unfruitful approach (assuming the irradiation power of $F_{emphir} = 1340~\frac{W}{m2}$ was measured in the range of $(0.1~nm, 2300~nm)$:
First defining any variables
refF = 1340.(*W/m^2*); c = 299792458(*m/s*); h = 6.62607015*10^-34(*Js*); k = 1.380649*10^-23(*J/K*);Defining the planck law function $B(\lambda,\,T)$and the energy density flux $F(T)$
B[\[Lambda]_, T_] := (2*h*c^2)/\[Lambda]^5*1/( Exp[(h*c)/(\[Lambda]*k*T)] - 1); F[T_?NumericQ] := \[Pi]*NIntegrate[ B[\[Lambda], T], {\[Lambda], 0.1*10^-9, 2300.0*10^-9}];Lastly, trying to find the corresponding effective temperature $T$ to the integrated area under the black body function, given a certain range of wavelengths from which the energy is emitted.
FindFit[F, {a* F[T_], {0.99 < a < 1.01}}, a, T]
Unfortunately, the FindFit[data, expr, pars, vars] function does not work with a single numerical value as data and also needs parameters and values. In this particular case, the parameter I aim to tweak for the fit to be successful is also the variable. However, I did not find any other (for me obvious) helper operators in Mathematica which would solve a problem like this.
(I used a(n unnecessary) pre-factor $a$ to have both, variables and parameters in the FindFit operator.)
Hence, my question: How would one go about this in Mathematica?
P.S. I also tried to directly integrate the planck law, given the effective temperature of the Sun for a certain wavelength range to see what wavelength range does contribute the most energy of the total solar energy flux, i.e. the solar constant. This also did not work:
Given a certain numerical value for the energy density flux $F$ of a star, e.g. the solar constant $s_\odot = 1362~\frac{W}{m^2}$, and a certain wavelength range corresponding to the irradiated energy $(\lambda_{min},\lambda_{max})$, how can one find the corresponding black body temperature?
For this task, I aimed to find a solution in Mathematica which would allow me to fit the planck law function
\begin{equation} B_\lambda(T) = \frac{2\,h\,c^2}{\lambda^5}\,\frac{1}{\exp(\frac{h\,c}{\lambda\,k\,T})-1} \quad, \end{equation}
to the numerical output of the energy density $F$. Assuming the star to have an isotopically emitting surface (i.e. approximation as black body), the energy density flux for a given wavelength window can be computed via
\begin{equation} F = \pi \, \int^{\lambda_{max}}_{\lambda_{min}} B_\lambda(\color{red}{T})\,d\lambda \quad. \end{equation}
In Mathematica I have tried the following -- so far -- unfruitful approach (assuming the irradiation power of $F_{emphir} = 1340~\frac{W}{m2}$ was measured in the range of $(0.1~nm, 2300~nm)$:
First defining any variables
refF = 1340.(*W/m^2*); c = 299792458(*m/s*); h = 6.62607015*10^-34(*Js*); k = 1.380649*10^-23(*J/K*);Defining the planck law function $B(\lambda,\,T)$and the energy density flux $F(T)$
B[\[Lambda]_, T_] := (2*h*c^2)/\[Lambda]^5*1/( Exp[(h*c)/(\[Lambda]*k*T)] - 1); F[T_?NumericQ] := \[Pi]*NIntegrate[ B[\[Lambda], T], {\[Lambda], 0.1*10^-9, 2300.0*10^-9}];Lastly, trying to find the corresponding effective temperature $T$ to the integrated area under the black body function, given a certain range of wavelengths from which the energy is emitted.
FindFit[F, {a* F[T_], {0.99 < a < 1.01}}, a, T]
Unfortunately, the FindFit[data, expr, pars, vars] function does not work with a single numerical value as data and also needs parameters and values. In this particular case, the parameter I aim to tweak for the fit to be successful is also the variable. However, I did not find any other (for me obvious) helper operators in Mathematica which would solve a problem like this.
(I used a(n unnecessary) pre-factor $a$ to have both, variables and parameters in the FindFit operator.)
Hence, my question: How would one go about this in Mathematica?
P.S. I also tried to directly integrate the planck law, given the effective temperature of the Sun for a certain wavelength range to see what wavelength range does contribute the most energy of the total solar energy flux, i.e. the solar constant. This at first did not work due to a wrong value of the speed of light or due to the extensive usage of subscript to label variables, which has now been amended. The cross-check calculation below yields $\approx 1313~\frac{W}{m^2}$, which -- given the large range of wavelength covered should be responsible for the emission of the bulk energy of the Sun -- already corresponds to the solar constant very well.