2023
Certificate Authority Cup International Mathematical Contest Mo d el i ng
http://mcm.tzmcm.cn
Problem A (MCM)
Sunspot Forecasting
Sunspots are phenomena on t h e Sun’s photosphere that appear as temporary
spots that are darker than the surrounding areas. They are regions of reduced
surface temperature caused by concentrations of magnetic flux that inhibit con-
vection. Sunspots appear within active regions, usually in pai rs of opposite
magnetic polarity. Their number varies according to the approximately 11-year
solar cy c le .
Individual sunspots or groups of sunspots may last anywhere from a few
days to a few months, but eventually decay. Sunspots expand and contract as
they move across the surface of the Sun, with diameters ranging fr om 16 km (10
mi)[1] to 160,000 km (100,000 mi). Some larger sunsp ot s can be visible from
Earth without a telescop e [ 2]. They may travel at relative speeds, or proper
motions, of a few hundred meters per second when they first emerge.
Solar cycles last typically about eleven years, varying from just under 10
to just over 12 years. The point of highest sunspot activity during a cycle is
known as solar maximum, and the point of lowest activity as solar minimum.
This period is also observed in most other solar activity and is linked to a
variation in t h e solar magnetic field that changes polarity wit h this period.
Sunspot numbers also change over long periods. For example, during the pe-
riod known as the modern maximum from 1900 to 1958 the solar maxima trend
of sunspot count was upwards; for the following 60 years the trend was mostly
downwards[3]. Overall, t he Sun was last as active as t he modern maximum over
8,000 years ago[4].
Due to their correlation w it h other kinds of solar activity, sunspots can be
used to help predict space weather, the stat e of the ionosphere, and conditions
relevant to short-wave radio propagation or sat e ll i t e communications. Many
models based on time series analysis, spectral analysis, and neural network s
have been used to p r ed ic t sunspot activity, but often with poor results. This
may be related to th e f act that most prediction models are phenomenology at
the data level. Although we generally know the length of the solar activity cycle,
the cycle is not comp le t el y stable, the maximum intensity of the activity varies
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