The Limit of Magnetic Helicity Estimation by a Footpoint Tracking Method during a Flux Emergence

Theoretically, the magnetic helicity transport flux through the solar surface into the upper atmosphere can be estimated indefinitely precisely by magnetic field footpoint tracking if the observational resolution is infinitely fine, even with magnetic flux emergence or submergence. In reality, the temporal and spatial resolutions of observations are limited. When magnetic flux emerging or submerging, the footpoint velocity goes to infinity and the normal magnetic field vanishes at the polarity inversion line. A finite observational resolution thus generates a blackout area in helicity flux estimation near the polarity inversion line. It is questioned how much magnetic helicity is underestimated with a footpoint tracking method due to the absence of information in the blackout area. We adopt the analytical models of Gold-Hoyle and Lundquist force-free flux ropes and let them emerging from below the solar surface. The observation and the helicity integration can start at different emerging stages of the flux rope, i.e., the photospheric plane initially cuts the flux rope at different levels. We calculate the magnetic helicity of the flux rope below the photospheric level, which is eventually to emerge, except the helicity hidden in the region to be swept by the blackout area with different widths. Our calculation suggests that the error in the integrated helicity flux estimate is about half of the real value or even larger when small scale magnetic structures emerge into the solar atmosphere.