Endoscope system, processor, and method for estimating temperature of endoscope

A processing apparatus comprising:

• a processor comprising hardware, the processor being configured to:
• acquire heating element temperature data from a temperature sensor configured to estimate temperature of a heating element provided in an insertion portion of an endoscope inserted into a subject;
• output initial surface temperature, which is insertion portion surface temperature at a first time point prior to a start of heating of the heating element, based on the heating element temperature data at the first time point,
• estimate changes in the insertion portion surface temperature over time after the first time point based on the heating element temperature data after the first time point using the initial surface temperature as an initial value of the insertion portion surface temperature;
• and
• output a change in the insertion portion surface temperature based on a mathematical equation containing a function that has a gain and a time constant or a modification of the function, using the heating element temperature data as an input value to the mathematical equation.