Relationship between the Sunspot Number and Active Day Fraction: An Application for the Maunder Minimum

Abstract Long-term solar activity can be studied using several parameters. Some of the most used are based on the sunspot counting. The active day fraction (ADF) is the simplest index derived from this counting. It is reliable in periods of low solar activity such as the Maunder minimum (MM). In this work, we study the relationship between the ADF and the sunspot number. We have obtained that the optimal fit of that relationship is an exponential function whose exponent is a degree 3 polynomial including all data except those with ADF equal to 100%. Then, we use that fit to estimate the sunspot number during the MM from the ADF calculated from the most recent sunspot group number database. Our estimations of the annual sunspot numbers are below 15, except that for 1656, which is 40.8, whereas our estimations of the triennial sunspot numbers are below 10 from 1648 to 1714. We have found peaks of the solar cycle in the middle of the 1650s, 1670s, 1680s, and 1700s but no clear evidence of solar cycle in the 1660s and 1690s, likely due to the scarcity of the available data. Our results agree with previous works obtaining values significantly higher than those of the group sunspot number derived by Hoyt and Schatten in 1998 but still fully compatible with a grand minimum period.