Upper limit of the total magnetic flux in an active region according to the photometric-magnetic dynamical model

The photometric-magnetic dynamical model handles the evolution of an individual sunspot as an autonomous nonlinear, though integrable, dynamical system. One of its consequences is the prediction of an upper limit of the sunspot areas. This upper limit is analytically expressed by the model parameters, while its calculated value is verified by the observational data. In addition, an upper limit for the magnetic strength inside the sunspot is also predicted, and then, we obtain the following significant result: The upper limit of the total magnetic flux in an active region is found to be of about 7.23 × 1023 Mx, namely, phenomenologically equal to the magnetic flux concentrated in the totality of the granules of the quiet Sun, having a typical maximum magnetic strength of about 12G. Therefore, the magnetic flux concentrated in an active region cannot exceed the magnetic flux concentrated in the photosphere as a whole.