How Well Can a Footpoint Tracking Method Estimate the Magnetic Helicity Influx during Flux Emergence?

As shown by Démoulin and Berger (2003), the magnetic helicity flux through the solar surface into the solar atmosphere can be exactly calculated if we can trace the motion of footpoints with infinite temporal and spatial resolutions. When there is a magnetic flux transport across the solar surface, the horizontal velocity of footpoints becomes infinite at the polarity inversion line, although the surface integral yielding the helicity flux does not diverge. In practical application, a finite temporal and spatial resolution causes an underestimate of the magnetic helicity flux when a magnetic flux emerges from below the surface, because there is an observational blackout area near a polarity inversion line whether it is pre-existing or newly formed. In this paper, we consider emergence of simple magnetic flux ropes and calculate the supremum of the magnitude of the helicity influx that can be estimated from footpoint tracking. The results depend on the ratio of the resolvable length scale and the flux rope diameter. For a Gold-Hoyle flux rope, in which all field lines are uniformly twisted, the observationally estimated helicity influx would be about 90% of the real influx when the flux rope diameter is one hundred times the spatial resolution (for a large flux rope), and about 45% when it is ten times (for a small flux rope). For Lundquist flux ropes, the errors incurred by observational estimation are smaller than the case of the Gold-Hoyle flux rope, but could be as large as 30% of the real influx. Our calculation suggests that the error in the helicity influx estimate is at least half of the real influx or even larger when small scale magnetic structures (less than 10,000 km) emerge into the solar atmosphere.