Measurement of the open magnetic flux in the inner heliosphere down to 0.13 AU

Context. Robustly interpreting sets of in situ spacecraft data of the heliospheric magnetic field (HMF) for the purpose of probing the total unsigned magnetic flux in the heliosphere is critical for constraining global coronal models as well as understanding the large scale structure of the heliosphere itself. The heliospheric flux (Phi(H)) is expected to be a spatially conserved quantity with a possible secular dependence on the solar cycle and equal to the measured radial component of the HMF weighted by the square of the measurement's heliographic distance (BRR2). It is also expected to constitute a direct measurement of the total unsigned magnetic flux escaping the corona (Phi(open)). Previous work indicates that measurements of Phi(H) exceed the value predicted by standard coronal models (the "open flux problem"). However, the value of the open flux derived from in situ measurements remains uncertain because it depends on the method employed to derive it. Past derivations also pointed towards an increase in Phi(H) with heliocentric distance, although this may also be related to its method of computation. Aims. In this work, we attempt to determine a more robust estimate of the heliospheric magnetic flux (Phi(H)) using data from the FIELDS instrument on board Parker Solar Probe (PSP), to analyse how susceptible it is to overestimation and a dependence on time and space, as well as considering how it compares to simple estimates of Phi(open) from potential field source surface (PFSS) models. Methods. We compared computations of the heliospheric magnetic flux using different methods of data processing on magnetic field data from PSP, STEREO A, and Wind. Measured radial trends in fluctuations and background magnetic structure were used to generate synthetic data to analyse their effect on the estimate of BRR2. The resulting best estimates were computed as a function of time and space and then compared to estimates from PFSS models. Results. Radially varying fluctuations of the HMF vector as well as large-scale variations in the inclination of the Parker spiral angle are shown to have a non-trivial effect on the 1D distributions of BRR2. This causes the standard statistical metrics of the mean and mode (the most probable values) to evolve with radius, independently of the central value about which the vector fluctuates. In particular, the mean systematically underestimates Phi(H) for R < 0.8 AU and increases close to 1 AU. We attempt to mitigate for this by using the "Parker spiral method" of projecting the vector onto the background Parker spiral direction (which requires vector fluctuations to be evenly distributed about a central value). Even with this method, we find evidence of a small enhancement in flux close to 1 AU. The fraction of field which is locally inverted in a given time interval grows with radial distance from the Sun which remains a possible physical reason for this excess but is essentially negligible at PSP's perihelia distances where the impact of fluctuations in general is also much reduced. The Parker spiral method (PSM) and most probable values converge close to the Sun. Our derived best estimate for the time interval studied is similar to 2.5(-0.)(6)(+0.3)( )nT AU(2). To the extent probed by PSP, no strong dependence on latitude or longitude is apparent, although at 1 AU, the spread of measured values appears to grow at the highest latitudes. The best estimate of the heliospheric flux is significantly larger than estimates from PFSS models studied here, which predict values from 1.2-1.8 nT AU(2), depending on the choice of magnetogram or source surface height. Conclusions. Of the methods for computing the heliospheric flux over a wide range of heliocentric distances using only magnetic field data considered in this work, the most robust choice is to use the PSM. The decay of fluctuations and weakening importance of local flux inversions at smaller heliocentric distances indicate that the measurement is most accurate close to the sun and that it is justified for us to consider that Phi(H) similar to Phi(open), for these measurements. The determined value is too high to be explained via PFSS models. Contemporary magnetohydrodynamic models with the same photospheric input are unlikely to close this gap. Therefore, the most likely solutions remain in improvements of coronal models, for example, through improved boundary conditions via the direct measurement of the photospheric field in the solar polar regions or through the inclusion of missing physical processes such as time-dependent or non-potential effects, which can produce a contribution to the open flux that is not rooted in obvious coronal holes.