Derive an expression for the shift \(\Delta \lambda\) in the peak wavelength of the Planck function for a blackbody of temperature \(T,\) corresponding to a small shift in temperature, \(\Delta T\)

To understand the shift in the peak wavelength of the Planck function, we first need to grasp Wien's displacement law. This law states that the wavelength at which a blackbody spectrum reaches its maximum intensity is inversely proportional to the temperature of the blackbody. Mathematically, this is expressed as: \[ \lambda_{\text{max}} = \frac{b}{T} \] where \[ \lambda_{\text{max}} \] is the peak wavelength, \[ T \] is the temperature, and \[ b \] is Wien's displacement constant, approximately equal to 2.9 \times 10^{-3} \text{m K}. In essence, as the temperature of an object increases, the peak wavelength of the radiation it emits shifts to shorter wavelengths. For example:

• A hot blue star emits most of its light in the ultraviolet to blue range.
• A cooler red star emits more of its light at longer wavelengths, in the red range.

This fundamental relationship helps astronomers determine the surface temperatures of stars and other celestial bodies.

When we consider a small change in temperature, denoted as \Delta T, we are interested in understanding how this change affects the peak wavelength of the emitted radiation. Using Wien's displacement law, we rewrite the peak wavelength equation to reflect a small shift in temperature as follows: \[ \lambda_{\text{max}} + \Delta \lambda \approx\frac{b}{T + \Delta T} \] Here, \[ \Delta \lambda \] represents the small shift in the peak wavelength. To find the exact relationship, we need to use differentiation. By taking the derivative of \[ \lambda_{\text{max}} \] with respect to temperature, we can see how a small change in temperature causes a change in the peak wavelength. This concept is crucial for fine-tuning measurements and understanding blackbody radiation more accurately.

To derive the shift in the peak wavelength, we differentiate Wien's displacement law. The law gives us the relation: \[ \lambda_{\text{max}} = \frac{b}{T} \] If we assume a small change in temperature \Delta T, we aim to find the corresponding change \Delta\lambda in the peak wavelength. First, express \Delta\lambda as a small perturbation: \[ \lambda_{\text{max}}_1 = \frac{b}{T + \Delta T} \] Take the derivative of both sides with respect to temperature T: \[ \frac{d\lambda}{dT} = -\frac{b}{T^2} \] Since \[ \frac{d\lambda}{dT} \approx \frac{-\lambda_{\text{max}}}{T}, \] using the chain rule, we find: \[ \Delta \lambda \approx \frac{-\lambda_{\text{max}}}{T} \Delta T \] Therefore, the expression for the shift \[ \Delta \lambda \] in the peak wavelength due to a small temperature change is: \[ \Delta \lambda = -\frac{\lambda_{\text{max}}}{T} \Delta T \] This derivative is vital in understanding the sensitivities of blackbody radiation and is widely applied in astrophysics and thermal studies.