Starspots are areas on the surfaces of stars that are significantly darker than the surrounding photosphere; these are magnetic phenomena. These spots are analogous to sunspots on the Sun and could potentially provide insight into the differential rotation of the stars.  Differential rotation is the effect of the material at the poles and at the equator rotating at different rates around the star.  The star to be analyzed in great detail, II Pegasi (HD 224085) is a member of an RS Canum Venaticorum (RS CVn) binary system.  II Pegasi is of spectral class K2IV (Berdyugina, Ilyin, & Tuominen 1998).  RS CVn stars are chromospherically active binary systems that are important for the understanding of the magnetic dynamo activity of cool stars.  This activity is manifested, for example, in starspots, which are useful in the understanding of the magnetic activity of the Sun.

Within the solar atmosphere a magnetic field is caused by the movement of charged particles, which are then trapped by the magnetic field lines.  When the plasma gets compressed, the density of the magnetic field lines will increase, which strengthens the local magnetic field.  With the compression can come displacement of the magnetic field lines caused by the rotation of the Sun.  Because the Sun experiences differential rotation, the magnetic field linesget dragged around the star in a way as described by the Babcock Model.  

The Babcock Model (illustrated to the left; source:  Carroll & Ostlie) of stellar rotation describes that the Sun rotates at different rates at the equator and at the poles; the magnetic field lines that are embedded in the solar material begin to become tightly wound around the Sun, with particularly strong concentrations near the equator. The field lines are embedded in the solar material by flux freezing.  

Flux freezing (illustrated to the right) is illustrated by considering a bar magnet and its resultant magnetic field.  A metal loop being pulled through the magnetic field illustrates flux freezing.  The metal loop is simplified to be a perfect conductor, which has no resistance.  By Ohm’s Law, ε= I × R, where ε, is the induced emf, I is the current through the loop, and R is the resistance, having a resistance of zero would imply that the current is infinite.  This is not physical, so the emf is required to be zero. Having an emf of zero implies that the change in magnetic flux through the loop to be zero by Faraday’s Law, ε= −dΦ_B/dt, where dΦ_B is the magnetic flux.  This effect requires that the magnetic flux through the loopis constant.  On the Sun, the plasma acts as the metal loop.  Because the solar plasma is a good electrical conductor, the magnetic field lines that are threaded through the plasma will be dragged along as the plasma moves around the photosphere, or surface, of the Sun.  

Differential rotation on the Sun pulls the field lines at a faster rate at the equator than at the poles.  Consider a horizontal magnetic flux tube below the solar surface.  A pressure equilibrium exists between a flux tube and the surrounding plasma.  Outside, the pressure is due only to the plasma pressure.  Inside, the pressure is due to both plasma and magnetic pressure, which arises due to v × B forces exerted on currents flowing in the plasma.  As there is less plasma pressure inside the flux tube while the temperature inside is comparable to that outside, the density of the plasma within the tube is less than in the surrounding photosphere.  The lower density causes the tube to be buoyant and bulge out of the surface of the Sun.  As a result, the field lines are now perpendicular to the Sun’s surface.  There is much tension in the field lines, which is caused by the v × B force transmitted to the rest of the plasma via collisions.  The field lines resist being bent in regions of high magnetic field, so the plasma which they are embedded in can only move up and down along the field lines.  Horizontal motion across the field lines cannot as easily occur.  The perpendicular orientation of the field lines prevents convection in the plasma where the bulging occurs.  The Sun’s outer layers transport energy through convection. Since sunspots are areas located in the photosphere, part of the outer convective layers, the efficient energy transport of the convective zone is suppressed.  As a direct result, these areas are significantly cooler than the surrounding photosphere as a whole.  With a lower temperature, the Stefan-Boltzmann equation indicates that there will be a lower surface flux of the spot by the relation F_surf = σ  Teff⁴, where F_surf is the flux through the stellar surface (in W m^-2),. is the  Stefan-Boltzmann constant (5.670400 × 10^-8 W m^-2 K^-4), and Teff is the effective temperature of the surface (in K).  Effective temperature is defined to be the temperature of a star’s surface.  From the Stefan-Boltzmann equation, the temperature of the spot and surrounding photosphere can be determined.  In the analysis of II Pegasi, a spot temperature of 3500 K and a photosphere temperature of 4600 K were used.

When observing the dark spots more closely, a definite structure is evident.  Sunspots consist of two distinct parts:  the umbra and the penumbra (see image to the left; source:  The Royal Swedish Academy of Sciences).  The umbra is the visibly darkest portion of the sunspot where convection is halted most drastically.  Surrounding the umbra is a structure called the penumbra, which has a filament-like structure.  The filaments extend radially between the umbra and the photosphere.  The magnetic field lines of the umbra are oriented perpendicular to the Sun’s surface.  Those of the penumbra are oriented parallel to the surface, radially outward from the umbra.  

The sunspots follow a cycle that regulates the polarity of the spots (see image to the right; source:  Carroll & Ostlie). Every eleven years, the sunspot polarity reverses in accordance with the Sun’s magnetic cycle of twenty-two years. This means that over the course of just over one decade, the northern and southern magnetic fields on the surface of the Sun will switch.  During this time, the polarity of the leader and follower spots will reverse.  Once formed, a sunspot will remain at the same latitude for its entire lifespan, the longest of which does may exceed a few months in duration.  Subsequent spots will form at lower latitudes than their predecessors, until the spots are formed very near the equator.  When the formation of spots nears the equator, the sunspot cycle has ended and the magnetic field of the Sun is about to reverse.  The sunspot cycle will begin again with spots forming about 40° from the equator with the leading spot having the opposite polarity of those in the old cycle.

Sunspots are often found in groups.  There are typically leader and follower spots.  The dominant spot is the leader, which has the polarity of the hemisphere that the spot exists in and moves ahead of the follower spot or spots in the direction of the Sun’s rotation.  Leader spots are typically larger in area and fewer in number than the follower spots (Carroll & Ostlie 2007).

An unexceptional star, the Sun is not the only spotted star in existence. Stars similar to the Sun have also been found to have magnetic fields through various observations. Younger stars are found to have stronger magnetic fields. The spots that are produced by these stronger fields are significantly larger than sunspots. The age of the star and the strength of the magnetic field are related. Younger stars have stronger magnetic fields because they have a tendency to rotate much more rapidly than the Sun. This faster rotation and stronger magnetic field should result in a stronger magnetic dynamo than that of the Sun. A rotation period on the order of a few hours is typical for these young stars, as opposed to the more than twenty-six day period for the Sun (Lawrence, Cadavid, & Ruzmaikin 2008). The magnetic fields surrounding the stars drag matter through interstellar space creating an intricate spiral, known as the Parker spiral (see left; source:  NASA). This causes drag on the star and allows for angular momentum to be transferred away from the star to stellar wind particles traveling outward along the rotating field lines. The loss of angular momentum in the star will cause a reduction of the rotation rate. The result of this is that older stars rotate much more slowly than their younger analogs (Carroll & Ostlie 2007).